{"http:\/\/dx.doi.org\/10.14288\/1.0088805":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Applied Science, Faculty of","type":"literal","lang":"en"},{"value":"Materials Engineering, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Johnston, Andrew A.","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2009-06-03T13:49:32Z","type":"literal","lang":"en"},{"value":"1997","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Doctor of Philosophy - PhD","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"Manufacture of large composite structures presents a number of challenges, one of the most critical of\r\nwhich is prediction and control of process-induced deformation. Traditional empirical techniques for\r\ntooling and process cycle development are particularly unsuitable for large parts, especially when\r\ndevelopment costs and process variability are key issues. Thus, there is a critical need to supplement\r\ncurrent techniques with a science-based manufacturing approach.\r\nIn the present work, a two-dimensional finite element model for prediction of process-induced deformation\r\nhas been developed. Integration of this model with analyses for heat transfer and resin cure and resin flow\r\nallows analysis of all major identified deformation sources. A 'virtual autoclave' concept is employed in\r\nwhich autoclave control algorithms and autoclave response are simulated to predict structure boundary\r\nconditions during processing.\r\nCharacterization of a carbon fibre\/epoxy composite is performed and models developed to describe material\r\nbehaviour during processing. An examination of autoclave heat transfer is also performed and a model\r\ndeveloped for the observed effect of pressure on heat transfer rates. Using these data as inputs, the process\r\nmodel is demonstrated through application to three case studies of varying complexity. In each, model\r\npredictions are compared to experimental results and the predicted sensitivity of processing outcomes to\r\nprocess parameter variation is examined. A good match between model predictions and experimental\r\nresults was obtained in most cases.\r\nThe developed model is expected to perform two complementary roles. First, the ability to analyse\r\nstructures of practical size and complexity makes the model a potentially useful process-development tool\r\nfor the industrial composites processor. Also, the integration of analyses for all major deformation sources\r\nallows examination of parameter interaction, potentially driving fundamental research into deformation\r\nmechanisms and the development of improved material behavioural models.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/8675?expand=metadata","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/extent":[{"value":"21802810 bytes","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/elements\/1.1\/format":[{"value":"application\/pdf","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"AN INTEGRATED MODEL OF THE DEVELOPMENT OF PROCESS-INDUCED DEFORMATION IN AUTOCLAVE PROCESSING OF COMPOSITE STRUCTURES by ANDREW A. JOHNSTON B.ScEng., The University of New Brunswick, 1992 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Metals and Materials Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1997 \u00a9 Andrew A. Johnston, 1997 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Me-fetls and frteci'eria.k Sv^ineerinj The University, of British Columbia Vancouver, Canada Date .jJL** V ^17 DE-6 (2\/88) Abstract Manufacture of large composite structures presents a number o f challenges, one of the most critical o f which is prediction and control of process-induced deformation. Traditional empirical techniques for tooling and process cycle development are particularly unsuitable for large parts, especially when development costs and process variability are key issues. Thus, there is a critical need to supplement current techniques with a science-based manufacturing approach. In the present work, a two-dimensional finite element model for prediction of process-induced deformation has been developed. Integration of this model with analyses for heat transfer and resin cure and resin flow allows analysis of all major identified deformation sources. A 'virtual autoclave' concept is employed in which autoclave control algorithms and autoclave response are simulated to predict structure boundary conditions during processing. Characterization of a carbon fibre\/epoxy composite is performed and models developed to describe material behaviour during processing. A n examination of autoclave heat transfer is also performed and a model developed for the observed effect of pressure on heat transfer rates. Using these data as inputs, the process model is demonstrated through application to three case studies of varying complexity. In each, model predictions are compared to experimental results and the predicted sensitivity o f processing outcomes to process parameter variation is examined. A good match between model predictions and experimental results was obtained in most cases. The developed model is expected to perform two complementary roles. First, the ability to analyse structures of practical size and complexity makes the model a potentially useful process-development tool for the industrial composites processor. Also, the integration of analyses for all major deformation sources allows examination of parameter interaction, potentially driving fundamental research into deformation mechanisms and the development of improved material behavioural models. Sommaire La fabrication de larges structures en composites comporte de nombreux defis dont l'un des plus critiques est la determination et le controle de la deformation induite durant le procede de fabrication. Dans une optique de reduction des couts de fabrication et des variations de qualite, les methodes empiriques traditionnellement utilisees pour la determination du cycle de cure et la conception de l'outillage sont particulierement inefficaces dans le cas de pieces aux dimensions importantes. Ainsi, il est necessaire de complementer les methodes actuelles par une approche scientifique aux problemes de production. Dans ce travail, un modele d'elements finis en deux dimensions calculant les deformations induites durant la fabrication a ete developpe. Ce modele permet l'analyse des principaux phenomenes contribuant aux deformations en combinant les effets du transfert de chaleur, de la polymerisation de la resine et de l'ecoulement de la resine. Un concept de \"l'autoclave virtuel\", dans lequel les algorythmes de controle de l'autoclave et la reponse de l'autoclave sont simules, est inclus afin de predire plus precisement les conditions frontieres de la piece durant la fabrication. Un composite fibre de carbone-epoxy est caracterise et des modeles decrivant son comportement au cours de la polymerisation sont developpes. Les caracteristiques du transfert de chaleur de l'autoclave sont determinees et un modele reliant Feffet de la pression sur le coefficient de transfert de chaleur de l'autoclave est developpe. En utilisant ces donnees, la performance du modele d'elements finis est demontree a travers trois cas ayant differents degres de complexite. Pour chacun des cas, les predictions du modele sont comparees aux resultats experimentaux. Sommaire L'effet de la variation des entrees du modele sur la stabilite des parametres du procede est egalement etudie. En general, les resultats experimentaux et numeriques concordent bien. Le modele developpe repond a deux besoins complementaires. D'une part, il est un outil pratique pour le manufacturier de pieces en composites, car il permet d'analyser des structures aux formes complexes et de grandes dimensions. D'autre part, il permet d'analyser l'interaction des parametres, car il integre les principaux mecanismes qui induisent des deformations. Ceci permettra d'encourager la progression de la recherche fondamentale sur les mecanismes causant les deformations et de developper de meilleurs modeles decrivant le comportement du materiau. - i v -Table of Contents ABSTRACT ii SOMMAIRE Hi TABLE OF CONTENTS v LIST OF TABLES xi LIST OF FIGURES xiii NOMENCLATURE xxi ACKNOWLEDGEMENTS xxvi 1. INTRODUCTION 1 1.1 A U T O C L A V E PROCESSING O V E R V I E W 1 1.2 R E S E A R C H OBJECTIVES A N D THESIS OUTLINE 5 2. LITERATURE REVIEW AND SYNTHESIS OF MODEL REQUIREMENTS 9 2.1 M O D E L L I N G A P P R O A C H 9 2.2 T H E R M O C H E M I C A L M O D E L S 11 2.2.1 Heat Transfer Modelling 12 2.2.2 Resin Cure Kinetics Modelling 13 2.2.3 Boundary Conditions 17 2.3 RESIN F L O W M O D E L 19 2.4 STRESS A N D DEFORMATION M O D E L S 20 2.4.1 Types of Residual Stress 21 2.4.2 Sources of Residual Stress and Deformation 22 2.4.3 Resin Mechanical Behaviour During Processing 31 - V -Table of Contents 2.4.4 Mechanical Property Models 33 2.4.5 Stress A nalysis Methods 38 2.4.6 Boundary Conditions 40 2.5 SUMMARY AND DISCUSSION 41 3. MODELLING APPROACH 43 3.1 FINITE ELEMENT MODEL 46 3.1.1 Discretized System of Equations 46 3.1.2 Element Description 48 3.1.3 Finite Element Description 50 3.1.4 Material Properties 52 3.1.5 PlyLayup 53 3.2 THE VIRTUAL AUTOCLAVE 54 4. THERMOCHEMICAL MODULE 58 4.1 FUNDAMENTAL EQUATIONS AND FINITE ELEMENT SOLUTION 59 4.1.1 Temperature Calculation 60 \"4.1.2 Degree of Cure Calculation 63 4.2 CURE KINETICS MODELS 65 4.3 BOUNDARY CONDITIONS 68 4.4 MATERIAL PROPERTIES 70 4.5 ITERATIVE SOLUTION 71 4.5.1 Module Solution Algorithm 72 4.6 MODEL VERIFICATION 73 4.6.1 Comparison with Exact Temperature Solution 74 4.6.2 Comparison with Literature 76 4.7 SUMMARY AND DISCUSSION 78 5. STRESS MODULE 81 5.1 FUNDAMENTAL EQUATIONS AND FINITE ELEMENT SOLUTION 81 - V I -Table of Contents 5.2 B O U N D A R Y CONDITIONS 84 5.3 M A T E R I A L PROPERTIES 85 5.4 SOLUTION A P P R O A C H 87 5.4.1 Module Solution A Igorithm 90 5.5 TOOL R E M O V A L SIMULATION 90 5.5.1 Tool Removal Algorithm 92 5.6 M O D E L VERIFICATION 93 5.6.1 Patch Test 93 5.6.2 Thermal A nisotropy Springback 94 5.6.3 Bi-material Strip 99 5.7 S U M M A R Y A N D DISCUSSION 101 6. MATERIAL AND BOUNDARY CONDITION CHARACTERIZATION 103 6.1 M A T E R I A L S C H A R A C T E R I Z A T I O N 105 6.1.1 Density 105 6.1.2 Specific Heat Capacity 107 6.1.3 Thermal Conductivity 110 6.1.4 Coefficient of Thermal Expansion 116 6.1.5 Mechanical Properties 119 6.1.6 Resin Cure Shrinkage 130 6.1.7 Resin Cure Kinetics 141 6.1.8 Summary and Discussion 150 6.2 A U T O C L A V E CHARACTERIZATION 152 6.2.1 Heat Transfer Coefficient Measurement 153 6.2.2 Autoclave A 155 6.2.3 Autoclave B 158 6.2.4 Autoclave C 158 6.2.5 Theoretical Heat Transfer Coefficient Calculation 159 6.2.6 Discussion 161 -vii-Table of Contents 7. EXPERIMENTAL AND NUMERICAL CASE STUDIES 164 7.1 C A S E STUDY 1: HYBRID SOLID L A M I N A T E \/ H O N E Y C O M B STRUCTURE 164 7. \/ . 1 Experimental Measurements 165 7.1.2 Model Predictions 169 7.1.3 Sensitivity Analysis 176 7.2 C A S E STUDY 2: A N G L E L A M I N A T E S 184 7.2. \/ Experimental Measurements 184 7.2.2 Model Predictions 192 7.2.3 Sensitivity A nalysis 200 7.2.4 Case Study 2 Summary\/Discussion 209 7.3 C A S E STUDY 3: F U S E L A G E SUBSTRUCTURES 210 7.3.1 Experimental Measurements 213 7.3.2 Model Predictions 220 7.3.3 Sensitivity Analysis 229 7.3.4 Substructure Analysis Using COMPRO 233 7.3.5 Case study 3: Summary\/Discussion 235 7.4 S U M M A R Y A N D DISCUSSION 235 8. SUMMARY AND DISCUSSION 239 8.1 FUTURE WORK 240 8.1.1 Numerical Modelling 240 8.1.2 Experimental 242 REFERENCES 244 A. COMPRO STRUCTURE AND OPERATION 251 A . l COMPRO SOFTWARE BASICS 251 A.2 COMPRO INPUTS A N D OUTPUTS 252 A.2.1 Initialisation Files 252 -vm-Table of Contents A.2.2 Input Files 252 A.2.3 Material Database Files 254 A.2.4 Output Files 254 A.2.5 COMPRO File Naming Conventions 255 A.2.6 Viewing COMPRO Output Results 256 A.3 COMPONENT FINITE ELEMENT DESCRIPTION 256 A.3.1 Element Definition and Node Numbering 257 A.3.2 Region Definition 257 A.3.3 Boundary Definition 259 A. 3.4 Element Local Axes Orientation and Ply-Element Intersection Calculation 260 A.4 TIME STEP CALCULATION AND MODULE ACTIVATION 264 A.5 THE VIRTUAL AUTOCLAVE - IMPLEMENTATION 267 A.5.1 Autoclave Simulation 268 A.5.2 Autoclave Environmental Control Software Simulation 269 A. 5.3 Virtual Instruments 274 A.6 ERROR HANDLING 274 A.7 COMPRO OVERALL PROGRAM FLOW 275 A. 8 THE COMPRO EDITOR 277 B. COMPRO EQUATIONS 279 B . l CALCULATION OF MATERIAL PROPERTIES AND PROPERTIES MATRICES 279 B. 1.1 Elastic Constants and Matrices 280 B. 1.2 Ply Micromechanics Model 289 B.l.3 Density 291 B. 1.4 Specific Heat Capacity 292 B.l.5 Thermal Conductivity 293 B. 1.6 Special Material Models 296 B.l.7 Cure Kinetics 296 B. 1.8 Cure Shrinkage Models 298 -ix-Table of Contents B. 1.9 Resin Modulus Development Models 299 B. 1.10 Resin Poisson's Ratio Development Models 300 B.2 THERMOCHEMICAL MODULE 301 B.2.1 Fundamental Equations 301 B.2.2 Finite Element Discretization 302 B.2.3 Calculation of Resin Degree of Cure 306 B.2.4 Solution Technique 307 B.2.5 Program Solution Algorithm '. 309 B.3 STRESS AND DEFORMATION MODULE 310 B. 3.1 Problem Finite Element Discretization 310 B.3.2 Solution Procedure 313 B. 3.3 Stress Module Solution Flow 314 B.3.4 Element Stiffness Matrix and Internal Load Vector Calculation 315 B. 3.5 Element Mechanical Strains Calculation 317 B.3.6 Element Mechanical Load Calculation 319 B.3.7 Tool Removal Simulation 321 B. 4 FINITE ELEMENT APPROACH 323 B. 4.1 Elements and Shape Functions : 324 B. 4.2 Element Integration 326 C. MATERIAL PROPERTIES AND SENSITIVITY ANALYSES RESULTS 330 C l THERMOCHEMICAL AND STRESS MODULE VERIFICATION TEST DATA 330 C.2 MATERIAL PROPERTIES USED IN NUMERICAL CASE STUDIES (CHAPTER 7) 332 C.3 SENSITIVITY ANALYSIS RESULTS SUMMARY 336 -x-List of Tables Table 2.1: Generalised boundary condition coefficients 9 Table 2.2: Necessary and sufficient conditions for modelling of the development of residual stress and deformation in industrial autoclave processing 9 Table 5.1: Summary of COMPRO predictions for patch test cases. F5x and F 5 : are the summation of element nodal loads at node 5 (the internal node) 94 Table 5.2: Effect of mesh density on COMPRO springback predictions after a temperature change of AT = 180 \u00b0C. Case: CTE, = 0, CTE2 = 0, and CTE3 = 100 x 10\"6 \/\u00b0C 98 Table 5.3: COMPRO springback predictions for various cured AS4\/8552 laminates subjected to a temperature change of AT= 180 \u00b0C 99 Table 5.4: COMPRO and ANSYS predictions for bi-material strip loading cases. Shown displacements are for node 1 (see Figure 5.8 ) 101 Table 6.1: Specific heat capacity measurements on uncured AS4\/8552 108 Table 6.2: Parameters used in 8552 resin modulus development model (Equation 6.6) 127 Table 6.3: Apparent total cure shrinkage after isothermal hold temperature reached 134 Table 6.4: Effective cure shrinkage test results 135 Table 6.5: Parameters used in 8552 resin cure shrinkage model (Equation 6.8) 139 Table 6.6: Parameters used in 8552 cure kinetics model (Equation 6.13) 148 Table 6.7: Autoclaves examined in heat transfer characterization tests 154 Table 7.1: Parameters varied in the hybrid solid\/honeycomb structure sensitivity analysis. Note that '*' in the Run Name is replaced with 'H' and 'L' for the high and low cases respectively 177 Table 7.2: Angle laminate experimental test matrix 186 Table 7.3: Parameters examined in angle laminate sensitivity analysis. Note that '*' in the Run Name is replace with 'H' and 'L' for the high and low cases respectively 201 Table 7.4: Parameters examined in stiffened honeycomb substructure sensitivity analysis. Note that '*' in the Run Name is replace with 'H' and 'L' for the high and low cases respectively 230 Table A.l: COMPRO file naming conventions 255 Table A.2: Example controller parameters (used to define process cycle in Figure A.l 1) 272 Table B. 1: Cure kinetics models available in COMPRO 297 Table B.2: Parameters used in COMPRO resin cure kinetics models 297 Table B.3: Resin cure shrinkage models available in COMPRO 298 Table B.4: Parameters used in COMPRO resin cure shrinkage models 298 Table B.5: Resin modulus development models available in COMPRO 299 Table B.6: Parameters used in COMPRO resin modulus development models 299 Table B.7: Resin Poisson's ratio development models available in COMPRO 300 Table B.8: Parameters used in COMPRO resin Poisson's ratio development models 300 -xi-List of Tables Table C.I: Material properties used in comparison of thermochemical module and exact predictions (Section 4.6.1) 330 Table C.2: Thermophysical properties of Hercules AS4\/3501-6 used in comparison of thermochemical module predictions with those of Bogetti, 1989 (Section 4.6.2). Cure kinetics parameters were also used in comparison of thermochemical and spreadsheet degree of cure calculations (Section 4.6.1) 330 Table C.3: Mechanical properties used in stress module patch test (Section 5.6.1) 331 Table C.4: Mechanical properties used in stress module thermal anisotropy springback tests (Section 5.6.2) 331 Table C.5: Mechanical properties used in stress module bi-material strip tests (Section 5.6.3) 331 Table C.6: Thermophysical properties of 'rubberized caul'. Material employed in case study #1 (Section 7.1) 332 Table C.7: Thermophysical and mechanical properties of 5052 aluminum. Material employed in case study #2 (Section 7.2) 332 Table C.8: Thermophysical and mechanical properties of invar 36. Material employed in case study #1 (Section 7.1) 332 Table C.9: Thermophysical and mechanical properties of HEXCEL HRP-3\/16-8.0 (3\/16 inch cell, 81b\/ft3, 0\/90 glass\/phenolic honeycomb). Material employed in case studies #1 and #3 (Sections 7.1 and 7.3) 333 Table C.10: Thermophysical and mechanical properties of 'resin transfer moulded J-frame' Material employed in case study #3 (Section 7.3) 333 Table C.I 1: Properties of Hercules AS4\/8552 unidirectional prepreg employed in case studies #1, #2 and #3 (Section 7.1, 7.2, and 7.3) 333 Table C.12: Properties of'adhesive'. Material employed in case study #3 (Section 7.3) 335 Table C.I3: Summary of results of hybrid solid laminate\/honeycomb structure sensitivity analysis (Section 7.1.3) 336 Table C.14: Angle laminate experimental test matrix (Section 7.2.1) 337 Table C.15: Summary of results from angle laminate experimental sensitivity analysis (Section 7.2.1). Shown results are measured springback angle 338 Table C. 16: Summary of results of L-shaped laminate numerical sensitivity analysis (Section 7.2.3).. 338 Table C.17: Summary of results of fuselage substructure sensitivity analysis (nominal maximum skin deflection = 119.4 microns) 340 -xu-List of Figures Figure 1.1: Component and tooling prepared for autoclave processing 2 Figure 1.2: Typical autoclave process cycle 2 Figure 2.1: Flow chart illustrating integrated sub-model approach as applied to the problem of composites processing modelling. After Springer (1986) 10 Figure 2.2: Schematic illustration of cure progression due to step-wise polymerization: a) prepolymer and curing agent prior to cure, b) curing started: molecular size increasing, c) gelation achieved: continuous network, d) full cure. From Berglund and Kenny (1991) 13 Figure 2.3: TTT diagram of thermoset cure (from Berglund and Kenny, 1991) 16 Figure 2.4: Warpage of a initially flat laminate due to flow-induced uneven resin distribution 19 Figure 2.5: Illustration of a part cross-section showing resin-rich and resin-poor regions caused by uneven resin flow 20 Figure 2.6: a) Micromechanical stresses in fibre and resin. The average, cr, is the ply stress, b) Macromechanical ply stresses. The average, Nlh, is the laminate stress. From Tsai and Hahn (1980) 22 Figure 2.7: Illustration of thermal strain anisotropy-induced springback (dashed line is the part after temperature decrease). 0 is the 'included angle' and Ad is the 'springback angle' 24 Figure 2.8: Theoretical influence of resin cure shrinkage on dimensionless stress free temperature, F n t - TRoom) I (TCim - TRoom). From Bogetti and Gillespie (1992) 26 Figure 2.9: Internal stress development during the curing and cooling process for low glass transition temperature (Tg = 165 \u00b0C) (C) and high glass transition temperature (Tg = 308 \u00b0C) (\u2022) resins. X and E are the points of vitrification, B-C is the time of cure and C-D is the cooling process. From Ochiet al. (1991) 27 Figure 2.10: Development of residual stress due to cure (and temperature) gradients, a) Initial state, both ply 1 and ply 2 uncured and unstressed; b) Ply 1 hardens and undergoes cure shrinkage while ply 2 is still pliant, resulting in minimal stress; c) Ply 2 hardens and shrinks resulting in significant stress since ply 1 is already stiff. 28 Figure 2.11: Contribution of tooling normal loads to process-induced deformation: a) Part heated to maximum temperature, tries to increase internal angle. Since part modulus is low, minimal force is required to maintain shape; b) Part cooled to initial temperature, tries to decrease angle. Since modulus is now high, tooling loads are large; c) Thus, residual normal loads exist after reaching initial temperature; d) Part is removed from tool. Relief of tooling loads results in springback 31 Figure 2.12: Stages in resin property development for a resin with a glass transition temperature greater than the maximum curing temperature. 'A' indicates the point of gelation (a = age\/), 'B' indicates vitrification (Tg> 7). Note the relatively short length of the viscoelastic region 33 Figure 2.13: Models for development of composite transverse modulus, all based on room temperature measurements 35 Figure 3.1: Scales of interest in composites process modelling 44 Figure 3.2: Schematic of COMPRO structure and program flow 45 -xiii-List of Figures Figure 3.3: Plane bilinear isoparametric element in global and natural coordinate systems 49 Figure 3.4: Composite element shown in: a) isometric view, b) plane view 50 Figure 3.5 Curved region, showing element local coordinates, local axes orientation and the region reference boundary 51 Figure 3.6 Temperature and degree of cure used for material properties calculations. The thermochemical module uses property values at the start o f the step k, while the stress module uses average values during the step 53 Figure 4.1: Example derivative calculation using the backward finite difference approximation 63 Figure 4.2: Example heat flow measurements from a dynamic D S C test (10 \u00b0C\/minute) 66 Figure 4.3: Comparison o f measured and predicted resin degree o f cure during a 150 \u00b0C isothermal cure using a kinetics model not accounting for vitrification 67 Figure 4.4: 1 - D transient temperature verification test 74 Figure 4.5: Comparison o f thermochemical module transient temperature predictions with exact solution for a 1-D non-curing slab 75 Figure 4.6: Comparison of thermochemical module isothermal cure rate predictions with exact solution (At = Is) 76 Figure 4.7: Finite element representation of 2.54 cm thick AS4\/3501-6 laminate used in comparison of thermochemical module predictions with Bogetti (1989) 77 Figure 4.8: Comparison o f C O M P R O thermochemical module predictions with Bogetti (1989). Temperature and degree of cure at centreline of a 2.54 cm thick laminate using (m:) (^-= 100 77 Figure 4.9: Effect of employed maximum time step on C O M P R O temperature predictions. Temperature and degree o f cure at centreline of a 2.54 cm thick laminate using (h\/k)eff= 100 78 Figure 5.1: Integration points for calculation o f element stiffness and load matrices for non-composite materials and elements containing only a single composite ply. Shown in both global (left) and element natural coordinates (right) 88 Figure 5.2: Integration points for calculation of element stiffness and load matrices for elements containing multiple composite plies. Shown in both global (left) and element natural coordinates (right) 89 Figure 5.3: Schematic of the tool removal process: a) prior to tool removal, part and tool in equilibrium, part conformed to tool shape; b) tooling removed, residual tool\/part interface forces remain; c) add negative of interface loads to obtain stress free interface; d) predicted part shape after tool removal 92 Figure 5.4: Schematic o f patch test for four noded plane strain elements (from Cook et al., 1989). Illustrated are the boundary and nodal load conditions for the 'uniform o^' case 93 Figure 5.5: Geometry employed for the thermal anisotropy springback verification test. Note that the finite element mesh shown is for the nominal case only 95 Figure 5.6: Predicted shape of anisotropic angle after a temperature change of A T of 180 \u00b0C. Case: CTEX = 0, CTEi = 0, and CTE3 = 100 x 10\"5 \/ \u00b0C, 6 elements in radial direction, 8 in circumferential direction. A l l displacements exaggerated by a factor of 10 97 -xiv-List of Figures Figure 5.7: Predicted variation of anisotropic angle springback with temperature during heating and cooling. Case: CTEX = 0, CTE2 = 0, and CTE3 = 100 x IO\"6 \/\u00b0C, 6 elements in radial direction, 8 in circumferential direction 97 Figure 5.8: Geometry employed for the bi-material strip verification test. In all cases, 10 elements (in total) are employed in the radial direction and 100 in the circumferential direction. 100 Figure 6.1: Effect of temperature on specific heat capacity of fully cured and uncured AS4\/8552 109 Figure 6.2: Schematic of thermal conductivity test. For kcii measurements, Pyrex 7740 standards were used as reference specimens, with Pyroceram 9960 used for kc\\ \\ tests 113 Figure 6.3: Assembly of composite longitudinal conductivity (kcU) test specimen from a unidirectional laminate 113 Figure 6.4: Measured AS4\/8552 transverse thermal conductivity (k^) with cured and uncured matrix resin 114 Figure 6.5: Measured AS4\/8552 longitudinal thermal conductivity (kcll) with cured matrix resin 115 Figure 6.6: Schematic of rectangular torsion test. Note that the composite fibres are aligned with the twist axis 121 Figure 6.7: Rectangular torsion test measurements for 135 \u00b0C isotherm test (specimen 1) 122 Figure 6.8: Rectangular torsion test results summary: calculated resin elastic modulus as a function of degree of cure at 135 \u00b0C and 160 \u00b0C 124 Figure 6.9: Illustration of degree of cure as a 'shift factor' in resin elastic response 126 Figure 6.10: Comparison of model predictions for resin modulus development with (normalized) experimental results. Also shown is the prediction for a 180 C isothermal cure 128 Figure 6.11: Process cycle used in effective cure shrinkage test 132 Figure 6.12: Specimen used in effective cure shrinkage test 133 Figure 6.13: Isothermal T M A cure shrinkage test (7\/= 150 \u00b0C) 134 Figure 6.14: Effective cure shrinkage specimen after second process cycle 135 Figure 6.15: Measured transverse cure shrinkage of AS4\/8552 as a function of predicted resin degree of cure 136 Figure 6.16: Transverse cure shrinkage of AS4\/8552 with cure (vertically shifted) 137 Figure 6.17: Finite element representation used for effective cure shrinkage specimen model. Note that in the actual specimen, 20 elements were used in the thickness direction and 75 for the length direction 140 Figure 6.18: Predicted effective cure shrinkage strain versus time during processing and comparison with measured value at completion of processing 140 Figure 6.19: Isothermal DSC scan of neat 8552 resin at 150\u00b0C 144 Figure 6.20: Comparison of measured cure rates at 170 \u00b0C vs. predictions of modified model from Scholz et al. (1994) 146 Figure 6.21: Calculation of cure equation activation energy from isothermal cure rates at three degrees of cure 147 -xv-List of Figures Figure 6.22: Comparison of cure kinetics model isothermal degree of cure predictions with experimental measurements 149 Figure 6.23: Comparison of cure kinetics model dynamic cure rate predictions with experimental measurements 149 Figure 6.24: Test apparatus used in measurement of autoclave heat transfer coefficient 154 Figure 6.25: Schematic showing relative location o f tool plates in autoclave A heat transfer characterization test 155 Figure 6.26: Autoclave A heat transfer coefficient measurement test 156 Figure 6.27: Calculated heat transfer coefficient during autoclave A characterization test 157 Figure 6.28: Comparison between measured heat transfer coefficients and model assuming a linear variation with pressure 157 Figure 6.29: Autoclave B heat transfer coefficient test, showing calculated heat transfer coefficients.. 158 Figure 6.30: Autoclave C heat transfer characterization test. Note the very poor agreement between measured heat transfer coefficients and fit equations and the different behaviour in regions 1 and 2 159 Figure 6.31: Measured heat transfer coefficient from autoclave A characterization test compared with best fits using turbulent and linear models 161 Figure 7.1: Schematic o f the hybrid solid laminate\/honeycomb structure (not to scale). 165 Figure 7.2: Schematic o f thermocouple placement during the hybrid structure experimental build (not to scale) 166 Figure 7.3: Instrumented hybrid structure prior to bagging (photo courtesy of P. Hubert) 166 Figure 7.4: Process cycle used to manufacture hybrid structure showing specifications for lead\/lag control (see Section 3.2, A.5) . Note that pressurization rate control is not used 167 Figure 7.5: Measured autoclave air temperature and temperature at three points through the thickness of the solid section o f the hybrid structure. Note the temperature overshoot at point A 168 Figure 7.6: Measured autoclave air temperature and temperature in the top and bottom skins of the honeycomb section o f the hybrid structure 169 Figure 7.7: Finite element representation used for modelling hybrid solid\/honeycomb structure (nominal case). Indicated T C ' s are 'virtual thermocouples' used by C O M P R O for process control 170 Figure 7.8: Predicted temperature and resin degree o f cure for hybrid structure and comparison with experimental measurements (with matching o f autoclave temperature) 172 Figure 7.9: Predicted temperature and resin degree of cure for hybrid structure and comparison with experimental measurements (using lead\/lag controller simulation) 173 Figure 7.10: Predicted temperature and resin degree of cure through the thickness of the solid section o f the hybrid structure. For comparison, experimental measurements for this section are shown in Figure 7.5 174 Figure 7.11: Predicted temperature and resin degree of cure through the thickness o f the honeycomb section of the hybrid structure. For comparison, experimental measurements for this section are shown in Figure 7.6 175 Figure 7.12: Predicted cure and temperature gradients through the thickness of the honeycomb section of the hybrid structure at three times near the point of maximum exotherm 175 -xvi-List of Figures Figure 7.13: Predicted maximum degree of cure gradients in hybrid structure during processing 176 Figure 7.14: Illustration of the effect of tooling thickness on process model temperature predictions for the hybrid structure. Note that the longer process cycle for the thick tool is due to the simulation of lead\/lag control 179 Figure 7.15: Illustration of the effect of tooling thickness on process model predictions of resin degree of cure in the hybrid structure. Note that most of the shift in time between the two sets of curves is due to the simulation of lead\/lag control 180 Figure 7.16: Predicted sensitivity of process cycle time to variation in process variables (hybrid structure) 181 Figure 7.17: Predicted sensitivity of maximum part 'exotherm' temperature to variation in process variables (hybrid structure) 182 Figure 7.18: Predicted sensitivity of maximum degree of cure gradient to variation in process variables (hybrid structure) 183 Figure 7.19: Predicted sensitivity of 'flow time' to variation in process variables (hybrid structure).... 183 Figure 7.20: a) Angle laminate (nominal case), and b) laminate mounted on solid convex aluminum tool. Not to scale 185 Figure 7.21: Thermocouple locations in angle laminate experiment (nominal case). Not to scale 185 Figure 7.22: Process cycles used for angle laminate experiments (cycle #1 used in nominal case) 187 Figure 7.23: Measured temperatures during processing of angle laminate (process cycle #1). Where two T.C.'s (e.g., T.C.'s #1 &7) are indicated, the shown temperatures are the average 188 Figure 7.24: Measurement of springback angle of trimmed angle laminates. PI - P4 are measurement points used by the image analysis software 189 Figure 7.25: Measured angle laminate springback for all specimens 189 Figure 7.26: Example part profiles after processing: a) typical part profile, b) part showing corner thinning (no displacement exaggeration). From Hubert (1996) 191 Figure 7.27: Finite element representation of angle laminate (nominal case) 193 Figure 7.28: Predicted temperature and resin degree of cure during processing of the angle laminate (nominal case) 194 Figure 7.29: Predicted part temperature contour and temperature gradients through part thickness. Taken at time of maximum through-thickness gradient 195 Figure 7.30: Comparison between predicted and measured temperatures during processing of angle laminate (nominal case) 195 Figure 7.31: Comparison of experimental and predicted springback angles for angle laminates. Error bars on experiment represent +\/- l a variation from the mean. Also shown is the predicted springback from strain anisotropy alone 196 Figure 7.32: Predicted warpage of angle arms, nominal case (perfect bonding) 197 Figure 7.33: Effect of shear layer modulus on predicted maximum angle laminate arm warpage for three different layups 198 Figure 7.34: Effect of shear layer modulus on predicted angle laminate springback angle for three different layups 199 -xv i i -List of Figures Figure 7.35: Comparison of measured arm warpage with model predictions for the [907+45\/-45\/0J6 laminate using a shear layer shear modulus of = 6 MPa 199 Figure 7.36: Predicted flow-induced change in [90\/+45\/-45\/0J6 angle laminate thickness and new thickness profile in corner region. Thickness profile changes exaggerated by a factor of 10 200 Figure 7.37: Predicted sensitivity of angle laminate springback to variation in analysis parameters 203 Figure 7.38: Predicted sensitivity of angle laminate springback to variation in composite thermophysical properties 204 Figure 7.39: Predicted sensitivity of angle laminate springback to variation in boundary and initial conditions 205 Figure 7.40: Predicted sensitivity of springback angle to resin and composite mechanical properties... 206 Figure 7.41: Predicted sensitivity of springback angle to layup, thickness and tool material 206 Figure 7.42: Comparison of predicted and measured springback angles for layups studied in sensitivity analysis. Experimental error bars represent a +\/-1 covariation from the mean 207 Figure 7.43: Comparison of plots of predicted springback angle versus experimental measurements for three cases: COMPRO with perfect tool\/part bonding, COMPRO with calibrated shear layer and strain anisotropy equation 208 Figure 7.44: Comparison of measured arm warpage with model predictions for the [90]i2[0]n laminate. Note that warpage was not significantly affected by shear layer modulus for this case 209 Figure 7.45: Illustration of a large, complex fuselage structure and a typical substructure showing 3-D shell element 'global' discretization and 'local' (COMPRO) discretization 211 Figure 7.46: Representative substructures examined in case study 3: a) unstiffened skin laminate, b) stiffened skin laminate, c) unstiffened honeycomb structure, d) stiffened honeycomb structure.... 212 Figure 7.47: Pre-cured resin transfer moulded J-frame bonded to stiffened substructures 212 Figure 7.48: a) Schematic of placement of representative substructures on tooling including thermocouple locations, b) end view of aluminum tool, c) top view of aluminum tool showing detail of 'egg-crate' base 213 Figure 7.49: Representative substructures assembled and bagged prior to being subjected to autoclave processing. Photo courtesy of The Boeing Company 214 Figure 7.50: Measured temperatures at selected locations on tool and representative parts. Note that TC#12 is not on the same part as TC's #10 and #7 (see Figure 7.48) 215 Figure 7.51: Typical contour plot of warpage of stiffened honeycomb structure 216 Figure 7.52: Measured warpage of unstiffened skin structure specimens showing second-order fit to specimen displacement 217 Figure 7.53: Measured warpage of unstiffened honeycomb structure specimens showing second-order fit to specimen displacement. Note the very small scale of the deformation; in this case only a few times larger than accuracy of the measurement technique 218 Figure 7.54: Measured warpage of stiffened skin structure specimens showing second-order fit to specimen displacement 219 Figure 7.55: Measured warpage of stiffened honeycomb structure specimens showing second-order fit to specimen displacement 219 Figure 7.56: Finite element representation of stiffened honeycomb structure 220 -xviii-List of Figures Figure 7.57: Comparison of predicted and measured tool and part temperatures during processing of representative substructures 222 Figure 7.58: Comparison of predicted residual mechanical \u00a3 strain profiles (s^) in unstiffened skin panel sections using models with a) low shear layer modulus, b) high shear layer modulus 223 Figure 7.59: Shear layer calibration using unstiffened skin part 224 Figure 7.60: Warpage prediction for unstiffened skin with calibrated shear layer 224 Figure 7.61: Predicted resin degree of cure and resin modulus in top and bottom skins of unstiffened honeycomb structure 225 Figure 7.62: Model warpage prediction for unstiffened honeycomb panel 226 Figure 7.62: Model warpage prediction for unstiffened honeycomb panel 226 Figure 7.63: Predicted post-processing shape of J-stiffened skin part with modelling of adhesive noodle (displacements exaggerated by a factor of 10) 227 Figure 7.64: Model warpage prediction for J-stiffened skin structure showing best fit prediction and prediction with crude adhesive noodle model 228 Figure 7.65: Model warpage prediction for J-stiffened honeycomb panel (nominal case) 229 Figure 7.66: Predicted sensitivity of maximum deformation of stiffened honeycomb structure to variation in thermophysical properties 232 Figure 7.67: Predicted sensitivity of maximum deformation of stiffened honeycomb structure to variations in boundary and initial conditions 232 Figure 7.68: Predicted sensitivity of maximum deformation of stiffened honeycomb structure to variations in mechanical properties 233 Figure 7.69: Substructure analysis using COMPRO. The behaviour of the global model elements (left hand side) is matched to give an equivalent response to the output of the local model (right hand side) 234 Figure A. 1: COMPRO input file hierarchy 254 Figure A.2: Use ofLOOKUP.DAT as a pointer to material database files 254 Figure A.3: Counterclockwise node numbering sequence required by COMPRO 257 Figure A.4: Curved region, showing element local coordinates, local axes orientation and the reference boundary 259 Figure A.5: Two finite element meshes each with two defined regions: a) continuous (recommended), b) discontinuous (permissible but not recommended) 259 Figure A.6: Example boundary definitions, showing unordered (input) and ordered node lists 260 Figure A.7: a) Permissible and b) Impermissible boundary definitions 260 Figure A.8: a) Calculation of nodal distance from reference boundary, b) Calculation of element local axes orientation using orientations of reference boundary elements 263 Figure A.9: Plies in an element and their intersection with the element sides 263 Figure A. 10: Communication between environmental control hardware and the control system 270 Figure A.l 1: Process cycle defined by lead\/lag control parameters shown in Table A.2 272 -xix-List of Figures Figure A. 12: The COMPRO Editor Interface 278 Figure B.l: Calculation of element stiffness matrix and change in internal load vector during a stress step 316 Figure B.2: Calculation of the change in the element internal strain vector during a stress step 318 Figure B.3: Calculation of the a c t u a l change in the element strain vector during a stress step 318 Figure B.4: Schematic of the tool removal process: a) prior to tool removal, part and tool in equilibrium, part conformed to tool shape; b) tooling removed, residual tool\/part interface forces remain; c) add negative of interface loads to obtain stress free interface; d) predicted part shape after tool removal 322 Figure B.5: Plane bilinear isoparametric element in global (left) and natural (right) co-ordinate systems 324 Figure B.6: Composite element shown in: a) isometric view, b) plane view 325 Figure B.7: Integration point locations for element integration using Gaussian quadrature. Shown in both global (left) and element natural coordinates (right) 328 Figure B.8: Integration point locations for element integration using Gauss-Trapezoidal integration. Shown in both global (left) and element natural coordinates (right) 329 -xx-Nomenclature A Area A\/ Resin cure kinetics model constant ak Element conductivity smearing factor aEr Resin modulus development model constant A W Resin Poisson's ratio development model constant b Bond probability (used in calculation of relaxed shear modulus) B Resin cure kinetics model constant B Resin cure shrinkage model constant bk Element conductivity smearing factor by,. Resin Poisson's ratio development model constant [BT] Temperature shape function derivative matrix [BS] Displacement shape function derivative matrix CPc Composite specific heat capacity (scalar) Cpj Fibre specific heat capacity (scalar) CPr Resin specific heat capacity (scalar) CSCic Composite cure shrinkage coefficients in material principal directions CTEjC Composite coefficients of thermal expansion in material principal directions CTEjj Fibre coefficients of thermal expansion in fibre principal directions CTEr Resin coefficient of thermal expansion (isotropic) [Ctp] Ply plane strain stiffness matrix (3 x 3) in element local axes [C ] Ply stiffness matrix [Cp ] c Element thermal mass matrix Resin cure rate dt E' Storage modulus (elastic component of complex modulus) E\" Relaxation modulus (viscous component of complex modulus) Ea Elastic moduli Er Resin elastic modulus (isotropic) E^ Resin modulus development model constant Erx Resin modulus development model constant {F} Stress module global nodal load vector {f}e Stress module element nodal load vector {f[}c Contribution of element internal strains to stress module element nodal load vector -xxi-Nomenclature {FB} Global body force vector {FT} Thermochemical module global nodal load vector {fnje Element convective heat transfer load vector {FT H} Global convective heat transfer load vector {FTR} Tool removal global nodal load vector {f T q } e Element boundary heat flux load vector {FT Q} Global boundary heat flux load vector {fro},; Element internal heat generation load vector {FTQ} Global internal heat generation load vector {fCT}e Element nodal mechanical load vector G' Storage shear modulus (elastic component of complex shear modulus) G\" Relaxation shear modulus (viscous component of complex shear modulus) Shear moduli GSL Shear layer shear modulus h Heat transfer coefficient heff Effective heat transfer coefficient H I s 0 Total resin heat generation during an isothermal cure (finite time, see Equation 6.11) HT Total resin heat generation during an isothermal cure (infinite time, see Equation 2.8) HR Resin heat of reaction (total heat evolved from a = 0 to 1) HResiduai Resin residual heat of reaction after isothermal cure [h]e Element convective heat transfer matrix [H] Global convective heat transfer matrix [J] Jacobian Matrix k Effective plane strain bulk modulus K, Cure kinetics model parameter kiic Composite thermal conductivities in material principal directions kjij Fibre thermal conductivities in fibre principal material directions kr Resin thermal conductivity (isotropic) [k]e Stress module element stiffness matrix [K] Stress module global stiffness matrix [kr]e Thermochemical module element stiffness matrix [KT] Thermochemical module global stiffness matrix [KTR] Tool removal simulation global stiffness matrix [kK]e Element thermal conductivity matrix [KK] Global thermal conductivity matrix -xxii-N o m e n c l a t u r e {Ka} Cure rate coefficient vector \/ Resin cure kinetics model constant m Resin cure kinetics model constant n Resin cure kinetics model constant Nu Nusselt number [N] Element shape function matrix [NT] Temperature shape function matrix [N5] Displacement shape function matrix P Autoclave gas pressure qs Surface heat flux q Rate of convective heat transfer q.n Measured total DSC heat input 4baseiim DSC apparatus baseline heat flow Q Heat generation rate from the resin exothermic reaction R Galerkin residual for heat transfer equation R Gas constant Re Reynold's number [S] Ply compliance matrix [S ] Ply 3-D compliance matrix (6 x 6) in ply local axes [S'] Ply 3-D compliance matrix (6 x 6) in element local axes [S^] Ply plane strain compliance matrix (3 x 3) in element local axes \/ Time T Temperature T0 Initial temperature or reference temperature T* Difference between resin temperature and its instantaneous Tg TA Autoclave air temperature TC\\a Resin modulus development model constant TC\\h Resin modulus development model constant TC2 Resin modulus development model constant Tg Glass transition temperature Tga Resin modulus development model constant Tgh Resin modulus development model constant TQ Torque Ts Aluminum plate calorimeter average temperature To, Far-field or 'bulk' air temperature -xxni-Nomenclature {T}C, Element nodal temperature vector {T} Global nodal temperature vector [T^] Stress\/strain transformation matrix (ply local axes to element local axes) [Te] Stress\/strain transformation matrix (element local axes to global axes) [T^] Conductivity transformation matrix (ply local axes to element local axes) [TAO] Conductivity transformation matrix (element local axes to global axes) {u} Element displacement matrix (i.e. u, w within an element) U System strain energy V Volume Vj Fibre volume fraction Vr\u2122 Resin cure shrinkage model constant Vrs Resin volumetric shrinkage strains Vco 'Bulk' autoclave air velocity Wk Galerkin weight functions a Resin degree of cure cto Initial resin degree of cure aco Resin cure kinetics model constant ccc\\ Resin cure shrinkage model constant or resin modulus development model constant ac2 Resin cure shrinkage model constant or resin modulus development model constant OCT Resin cure kinetics model constant P Angle between element local and global coordinate systems AE; Activation energies for resin curing reaction. At Time step AtSA Calculated maximum allowable stress module time step AtrA Calculated maximum allowable thermochemical module time step A{a}mat Maximum allowable change in norm of degree of cure solution vector between iterations A{T},\u201eax Maximum allowable change in norm of temperature solution vector between iterations A9 Springback angle (positive springback defined as a reduction in included angle) {8}E Element nodal displacement vector {5} Global nodal displacement vector {8TR} Global tool removal nodal displacement vector {dg} Differential operator (displacement) {dj} Differential operator (temperature) \u00a3 3 \/ Total specimen linear cure shrinkage strain in the 3 - direction -xxiv-Nomenclature Se\/ Laminate effective cure shrinkage strain Si Composite cure shrinkage strains in material principal directions ef Composite thermal strains in material principal directions Sr Resin linear cure shrinkage strain (isotropic) {e}c Element strain vector {eo} Ply internal strain vector (3 x 1) in global axes {\u00a3o*} Ply 3-D internal strain vector (6 x 1) in ply local axes {so'} Ply 3-D internal strain vector (6 x 1) in element local axes {ztf} Ply plane strain internal strain vector (3 x 1) in element local axes {s<,}e Element mechanical strain vector {\u00ae}e Element surface traction vector {O} Global surface traction vector yr Resin modulus development model constant r Surface [K] Element thermal conductivity matrix in global axes [K*] Ply 3-D thermal conductivity matrix (3 x 3) in ply local axes [K'] Ply 3-D thermal conductivity matrix (3 x 3) in element local axes [K\u00a3] Element plane temperature gradient thermal conductivity matrix (2 x 2) in element local axes [Krp] Ply plane temperature gradient thermal conductivity matrix (2 x 2) in element local axes \/j Dynamic viscosity vf\u00b0 Resin Poisson's ratio development model constant Vy Poisson's ratios Qw Potential energy of work done by external loads to a body D_ Domain Yip System potential energy 9 Ply orientation angle 6 'Included' angle for a curved shape pc Composite density (scalar) Pf Fibre density (scalar) pr Resin density (scalar) co Frequency (rad\/s) - X X V -Acknowledgements The work presented in this thesis owes a significant debt to a number of individuals to whom I would like to express my most sincere gratitude. Thanks first of all to my co-supervisors Dr. Anoush Poursartip and Dr. Reza Vaz i r i for their technical and non-technical support and guidance throughout this work. Many thanks to Dr. Pascal Hubert, the co-developer of C O M P R O , for his continuing friendship, guidance, assistance and support over the years of our work together. Thanks to M r . Roger Bennett and M r . Serge Mila i re for their technical support and M r . Robert Courdji and Dr. Goran Fernlund for putting the process model through its paces and helping to guide the final stages of its development. Much of this work would not have been possible without the technical guidance and assistance o f several employees of The Boeing Company. I would especially like to thank Dr. K a r l Nelson for his assistance with a large part of the experimental component of this work, and M r . GaryDuschl for performing much of the materials characterization. Thanks also to Dr. Larry Ilcewicz for giving me the opportunity to see a bit of the industrial application side o f composites and to M r . Brian Coxon and M r . Don Stobbe o f Integrated Technology Inc. for introducing me to the art of composites manufacturing. Many thanks to the numerous fantastic members of the Composites Group over the years. Your friendship, enthusiasm, and encouragement made my experiences at U B C some of the most special of my life. Finally, I would like to acknowledge the Natural Sciences and Engineering Research Council o f Canada and The University of British Columbia for their generous financial support. - x x v i -1. Introduction High performance fibre reinforced polymer matrix composite materials have found widespread use in military applications and secondary aerospace structures due to their many advantages over traditional materials: they are light-weight, high stiffness, high strength and very durable. In commercial applications, however, these advantages are often overshadowed by a less positive characteristic of composite materials: their very high base cost. In order for composites to attain their full potential in such applications, significant reductions in manufacturing costs relative to competing materials must be achieved. One of the most promising paths to achieving required cost reductions is by exploiting the inherent suitability of composites to be manufactured directly into large, complex structures without employing extensive machining and fastening operations. A major challenge associated with this task is the development of robust manufacturing processes that can consistently produce high-quality structures meeting strict tolerance requirements. Modern manufacturing techniques and the current generation of material systems have made quality goals increasingly achievable; dimensional consistency remains difficult to attain. 1.1 Autoclave Processing Overview A number of potential techniques may be used to manufacture fibre reinforced plastic composite components. For high-performance applications, especially involving large complex structures, autoclave processing is usually the method of choice. Using this technique, thin layers of high modulus fibre impregnated with partially cured resin (prepreg) are cut and stacked to form a component of desired shape. In addition to uncured prepreg, typical laminated composite structures also include such materials as honeycomb core, pre-cured composite stiffeners and structural adhesives to bond together the various Chapter 1: Introduction parts. After assembly, the structure is covered with various layers of cloth (bleeder and breather1) and sealed inside a vacuum bag, as illustrated in Figure 1.1. The entire assembly, including any tooling used to maintain the structure shape, is then placed inside an autoclave and subjected to a temperature\/pressure cycle similar to that shown in Figure 1.2. Vacuum Bat? Breather Bleeder Vacuum plug Sealant Dam Tool Core-Insert \\ Laminate Figure 1.1: Component and tooling prepared for autoclave processing 1.2 \u00b0C\/min 107 \u00b0C - 1 hour 2 \u00b0C\/min 170 kPa Full vac. 177 \u00b0C- 2 hours 375 kPa Vent bag Time Figure 1.2: Typical autoclave process cycle. The objective of this 'process cycle' is to cure the resin and promote resin flow such that an optimum resin content and a void free part are obtained. At the same time, the dimensions of the produced Bleeders assist in achieving an optimal fibre volume fraction by absorbing excess resin. Breathers provide a path for removal of air and volatile gasses from the part during cure. -2-Chapter 1: Introduction structure must not vary beyond pre-set tolerance limits. Meeting these objectives has traditionally entailed trial-and-error modification of a baseline process cycle and design o f tooling and structure design (Purslow and Childs, 1986). This same iterative procedure might be repeated when modifications were made in materials, component dimensions, tooling or even the autoclave in which the part was processed. This technique is both expensive and time consuming, especially when applied to large complex structures. Using this technique, even after an 'optimum' combination o f process cycle, tooling and structure design is chosen, it is difficult to gauge the robustness of the developed process, let alone forecast potential problems which might arise due to random variability in process variables such as material properties, part layup or even autoclave loading. One way to help minimize process variation is through use o f 'intelligent process control' (Hol l and Rehfield, 1992; Kalra , et al., 1992; LeCla i r and Abrams, 1988). Using this approach, various types o f instruments such as thermocouples or dielectric sensors are typically embedded in the part and their measurements used for real-time autoclave process cycle control. These systems often employ expert systems, sometimes in conjunction with simple process models (Kenny, 1992; Kl ine and Altan, 1993, 1995; Pi l la i et al., 1993, 1997) to meet certain pre-defined objectives such as minimum process cycle time or minimum temperature gradients within a part. While potentially very useful, this approach is not a panacea for the composites processor. One deficiency of such systems is that they neither allow a priori prediction of processing outcomes (such as process-induced deformation) nor provide any insight into robust process design. A l so , the rules on which such systems are based are usually very material dependent as well as inflexible and difficult to modify (Pi l la i , etal . , 1993). Many of the shortcomings of the intelligent process control approach can be addressed through the use of autoclave process modelling. Using this approach, computational models are developed for the various phenomena involved in autoclave processing such as heat transfer, resin cure, resin flow, and the development of residual stress and deformation. These models are then used to create 'virtual parts' Chapter 1: Introduction which can be subjected to a simulated autoclave process cycle. Using these models, a very large number of potential processing scenarios, part and tooling designs can be examined for such things as uniformity of resin cure and structure warpage before a single real part is constructed. In combination with processing trials, this can greatly reduce the number of process and structural design iterations and consequently both cycle time and development costs. By exploiting the inherent suitability of such models for sensitivity analyses, 'sweet spots' can be identified in which the process is relatively insensitive to material and manufacturing variability. The major remaining variability drivers can then be targeted for tight monitoring and control. As mentioned, one of the most critical issues in the autoclave processing of composite structures is process-induced residual stress and deformation. Residual stresses in themselves can have a significant impact on the performance of composite structures, potentially reducing both ultimate strength and fatigue life (Hahn, 1984). Process-induced deformation has become an increasingly important issue as composite structures are employed in applications with strict tolerance requirements. Potential problems are especially acute for large, stiff structures where the loads required for force-fits are often prohibitive and a 'shim-to-fit' strategy is not economical. While process-induced stress and deformation is tosome extent unavoidable, foreknowledge of the amount of deformation can be used to compensate in structure and tooling design. Thus, the relevant issue with deformation is not necessarily reduction of the mean, but reduction of its variation. Previous analyses of composites processing have revealed a number of sources of process-induced stress and deformation including such effects as thermal and resin cure shrinkage strain anisotropy, uneven resin flow, cure gradients and tooling effects. Any analysis of process-induced stress and deformation which is to include all of these effects must also include examination of: \u2022 Heat transfer within the structure and tooling and at their boundaries \u2022 The kinetics of the matrix resin curing reaction and resin cure shrinkage Chapter 1: Introduction \u2022 Flow of the resin within the curing composite and at its boundaries \u2022 Tool thermal strains Over the past two decades steady progress has been made in the field of composites process modelling, and a number of analyses have been developed which examine one or more of the phenomena listed above, often in conjunction with an analysis of process-induced stress. Most currently available process models, however, were developed largely for the purpose of research. While the underlying science and mathematics are often well developed, these models typically focus on a few phenomena (e.g. resin flow) to the exclusion of others, use very simple boundary conditions and are generally capable of modelling only simple shapes. Thus, despite the significant advances in the science of process modelling, composites processors remain without an effective analytical tool for assisting in autoclave process development including tool and structural design. 1.2 Research Objectives and Thesis Outline The objective of the current research is to develop an integrated model for prediction of the development of process-induced deformation in industrial autoclave processing of composite structures. This is to be accomplished by building on the established foundation of composites process modelling, taking advantage of recent advances in available computing power, and extending current modelling capabilities by: \u2022 Developing methods for examining structures of practical complexity \u2022 Incorporating analyses of all identified sources of process-induced stress and deformation \u2022 Improving models for structure boundary conditions during autoclave processing If these objectives are achieved, the developed process model is expected to provide a number of potential benefits to composites processors. These include reduction of the cost and experimental effort required for process development, optimized processing cycles, reduced component variability, and improved Chapter 1: Introduction product quality. The research presented in this thesis is organized as following: Chapter 2 - Literature Review: A review is presented of the process modelling literature focusing on aspects most pertinent to this work: heat transfer and resin cure models and modelling of the development of residual stress and deformation. Chapter 3 - Modelling Approach: The structure and operation of the developed process model are outlined and the 'virtual autoclave' concept presented. Chapter 4 - Thermochemical Module: The development of a two-dimensional finite element model of heat transfer and resin cure kinetics is outlined. Included is a discussion of the model boundary conditions, modelling of material properties and model verification. Application of the thermochemical module is demonstrated in Chapter 7. Chapter 5 - Stress Module: The development of a two-dimensional finite element model for the development of residual strain and deformation in composite structures is presented. Included is a discussion of model boundary conditions, material properties and model verification. Also included is a discussion of the simulation of tool removal after the completion of processing. Model application is discussed in Chapter 7. Chapter 6 - Material and Boundary Conditions Characterization: Characterization of Hercules AS4\/8552, a second generation carbon fibre epoxy composite material, and development of material behavioural models are presented. Also discussed is the characterization of the heat transfer characteristics of three different autoclaves. Chapter 7 - Experimental and Numerical Case Studies: Three experimental and numerical case studies are presented to illustrate the application of the developed process model to different types of processing problems. Also presented for each case study is a sensitivity analysis illustrating the importance of Chapter 1: Introduction various input parameters to model predictions. Chapter 8 - Discussion and Conclusions: A discussion of the significance of the developed model and of the obtained insights into autoclave processing is provided. Recommendations for improvements and extensions to the current analysis and for further required experimental work are also presented. Appendix A - COMPRO Operation: Details of the operation of the COMPRO software are discussed including the input file structure, finite element description, error handling, and program algorithms. Appendix B - COMPRO Equations: A detailed discussion of the development of important model equations is provided. This includes such things as the derivation of the system of equations for the thermochemical and stress modules, transformation equations, and presentation of the equations used by various material models. Appendix C - Material Properties and Sensitivity Analyses Results: Presents, in tabular form, the material properties used in various verification tests and case studies as well as experimental and numerical sensitivity analyses results. The work described in this thesis represents a continuation of several years of autoclave process model development in the UBC Composites Group. The first model developed in the group was LAMCURE (Smith, 1992; Smith and Poursartip, 1993), a 1-D model including analyses of heat transfer and resin cure, resin flow, and fibre bed compaction. Work on the current model, COMPRO, was begun in 1993 in conjunction with another researcher, Pascal Hubert (Hubert, 1996). The contribution of the author to COMPRO consists of much of the overall model structure, the autoclave simulation (Section 3.2), as well as the major contribution of the thermochemical and stress modules (Chapters 4 and 5). All experimental work described in this thesis was performed in conjunction with Dr. Hubert. Characterization tests of AS4\/8552 (Chapter 6) were performed at facilities of The Boeing Company in Renton, WA, by Boeing employees at the behest of the author. All analysis of the obtained data, including material behavioural model development, was performed by the author. The experimental Chapter 1 : Introduction builds described in case studies #1 and #2 (Chapter 7) were carried out at the facilities of Integrated Technologies, Inc. in Bothell, WA. The parts described in case study #3 were manufactured at the facilities of The Boeing Company. These three sets of experimental builds were all carried out with the input and assistance of the author. 2. Literature Review and Synthesis of Model Requirements The inherent limitations and inefficiencies of the traditional trial-and-error approach to autoclave process development have long been understood, as have the potential benefits of a more scientific approach to this problem. Thus, modelling of various aspects of autoclave processing has a long history, spanning more than 2 decades. An early example is an analysis of residual stress generation by Hahn and Pagano (1975). As most early models, this analysis was simplified in several respects, employing a linear elastic constitutive model, assuming a stress-free state prior to the final 'cool-down' stage of processing, and using a LPT (laminated plate theory) - based stress-calculation technique. Since this model was introduced, autoclave process models have become increasingly sophisticated, incorporating analyses of such phenomena as internal heat transfer and resin cure kinetics, resin flow, and void generation, and employing complex mechanical property models and finite element techniques. This chapter reviews major developments in the modelling of autoclave processing, with an emphasis on the subjects of heat transfer and resin cure and the development of residual stress and deformation. A synthesis of previous work is also generated, thereby establishing the foundation for the current model and outlining both the necessary conditions and the sufficient conditions for any industrial autoclave processing model. For example, a discussion of the major sources of residual deformation is presented which includes not only examples of those sources which have been previously examined, but also those which are generally recognized but for which formal analyses have not been published. A complementary literature review and synthesis on aspects of resin flow modelling has been developed by Hubert (1996). 2.1 Modelling Approach An important consideration in any complex model incorporating analyses of multiple phenomena is model 'structure', that is, how the various components of the model are combined to form a coherent whole. The structure generally used in the most comprehensive composites processing models is the 'integrated sub-model' approach, first applied to this problem by Loos and Springer (1983). Using this Chapter 2: Literature Review and Synthesis of Model Requirements approach, a very complex problem is tackled by dividing it into a series of simpler problems, each of which may be examined more or less independently. As illustrated inFigure 2.1, a model for composites processing may thus be divided into sub-models for such phenomena as heat transfer and resin cure (a 'thermochemical model'), flow of the matrix resin (a 'flow model'), residual stress and deformation (a 'stress model'), etc.,. A complete process cycle lasting perhaps several hours may be divided into a large number of small 'time steps' during each of which one or more of the program sub-models is solved to update predictions for the modelled parameters. Process Variables: Temperature and Pressure 1 Component Geometry Boundary Conditions I Thermochemical Model I Heat Transfer Cure Kinetics Temperature Viscosity Degree of Cure Flow Model Resin Pressure Fibre Volume Fraction Resin Mass Loss Stress Model Ply Stresses Ply Strains Component Strains Material Properties Material Properties Material Properties F i g u r e 2.1: F l o w c h a r t i l l u s t r a t i n g i n t e g r a t e d s u b - m o d e l a p p r o a c h as a p p l i e d to the p r o b l e m o f c o m p o s i t e s p r o c e s s i n g m o d e l l i n g . A f t e r S p r i n g e r (1986). In addition to Springer and co-workers, the integrated sub-model approach has been more recently used by Bogetti and Gillespie (1991, 1992), and White and Hahn (1992a, 1992b) in two of the most comprehensive autoclave process models described in the literature. Another important characteristic of an analytical model is its dimensionality. Most previous processing models have employed one-dimensional analyses (e.g., Loos and Springer, 1983; White and Hahn, 1990; -10-Chapter 2: Literature Review and Synthesis of Model Requirements Mijovic and Wang, 1988). Such models are unfortunately limited to very simple component shapes. At the other end of the scale of complexity are three-dimensional analyses such as those of Chen and Ramkumar (1988). These models can be applied to structures of any shape, but have had to employ very large-scale elements if they are to be computationally viable and are thus unlikely to capture important small-scale effects. Fortunately, the geometry of most composite structures is such that gradients in at least one direction are small and they can thus be adequately modelled by 2-D analyses. Such analyses have been performed in the past by Bogetti (1989) and Bogetti and Gillespie (1991, 1992). Another important characteristic of a model is the method used to solve the relevant equations over the domain of interest. The equation solution method has a significant impact on the complexity of the problem that can be modelled. Exact solutions (Levitsky and Schaffer, 1975; Mabson and Neall, 1988), for example, can only be used for simple combinations of component geometry, material properties and boundary conditions. Incremental LPT-based stress analyses (Bogetti and Gillespie, 1992; Loos and Springer, 1983) can be employed for fairly complex combinations of part layup and material behaviour, but can only be used for very simple shapes and boundary conditions. For more complex problems, finite difference (Bogetti and Gillespie, 1992) or finite element (Bogetti, 1989; Chen and Ramkumar, 1988) techniques are generally employed. These methods are quite involved, but allow modelling of complex combinations of geometry, material properties and boundary conditions. Division of a problem into separate sub-models for different phenomena allows ^combination of solution methods and even problem geometric descriptions to be used in the same model. Bogetti (1989), for example, employed both 1-D and 2-D finite difference heat transfer analyses in combination with both LPT and finite element stress models. 2.2 Thermochemical Models Component internal heat transfer and cure of the matrix resin have been the most extensively modelled processing phenomena, either by themselves (Mijovic and Wang, 1988), or as part of analyses which also Chapter 2: Literature Review and Synthesis of Model Requirements examined other phenomena such as resin flow and residual stress development (Loos and Springer, 1983; Bogetti and Gillespie, 1992). One reason for the large amount of interest in such 'thermochemical' models is their relative simplicity as compared to models for other processing phenomena. Another important factor is that component temperature and resin degree of cure are two of the most important composite material 'state' variables. These parameters are thus both important in themselves and are vital to modelling of other autoclave processing phenomena such as resin flow and the development of residual stress and deformation. Thermochemical models generally consist of a combination of 'sub-models' for heat transfer and resin reaction kinetics. In most analyses, these sub-models are treated as 'uncoupled' during individual time-steps of the transient solution (Bogetti, 1989). Thus, cure reaction and heat generation rates are assumed constant during the time step. This approximation is quite good if time steps are small and allows a more flexible, simpler and less time-consuming computational approach than a more rigorous 'coupled' analysis. 2.2.1 Heat Transfer Modelling The majority of heat transfer models for composites processing consider heat flow in the through-thickness direction only (a one-dimensional model) or make the even simpler assumption of a uniform laminate temperature (White and Hahn, 1992a, 1992b). More sophisticated models examining heat transfer in two and three dimensions have also been developed (Bogetti and Gillespie, 1991; and Chen and Ramkumar, 1988). The governing equation of the heat transfer sub-model is the transient Fourier anisotropic heat conduction equation, with a heat generation term from the exothermic resin cure reaction: - T). N o t e t he r e l a t i v e l y s h o r t l e n g t h o f the v i s c oe l a s t i c r e g i o n . 2.4.4 Mechanical Property Models A fundamental characteristic of any stress analysis is the constitutive model used to describe material mechanical behaviour. While the general mechanical behaviour of composite materials during processing is well understood, a number of quite different constitutive models have been used in the analyses of process-induced stress performed to date. A discussion of the models used and their advantages and drawbacks is provided in this section. 2.4.4.1 Elastic Mechanical Property Models Although the mechanical behaviour of the matrix resin (and thus that of the composite in resin-dominated directions) is known to be viscoelastic throughout much of processing, the majority of the composite stress analyses in the literature actually employ elastic constitutive models. This is done for two major -33-Chapter 2: Literature Review and Synthesis of Model Requirements reasons. First, viscoelastic stress analyses are much more complex than elastic ones, both in terms of mathematical modelling and materials characterization. Second, despite resin viscoelastic behaviour, elastic analyses can in many cases quite accurately predict process-induced stress. This will be true, for example, if the residual stresses generated prior to cool-down have mostly relaxed during the high temperature hold period. In this case, a purely elastic stress analysis considering only the cool-down process may be quite accurate. Such analyses have been performed by a number of researchers including Hahn and Pagano (1975), Griffin Jr. (1983), and Stagano and Wang (1984). Examination of stress generation during the cool-down process only will not, however, be adequate if stresses generated prior to this point in the process are significant. Calculation of these stresses requires examination of stress development throughout the complete process cycle. However, by using an elastic model for resin behaviour during the entire cycle, such as Bogetti and Gillespie (1992), the potential exists to significantly overestimate residual stresses unless some allowance for stress relaxation is made. Elastic Property Development Models The low importance traditionally assigned to stress generation prior to cool-down has resulted in only very limited examination of composite mechanical properties early in the processing cycle. The only in-depth investigations of elastic mechanical properties over a wide range of cure states have been performed by Kim and Hahn (1989) and White and Hahn (1990, 1992b). Lee and Springer (1988) also examined composite properties at various degrees of cure, but only for a> 0.6. As expected, composite transverse modulus and strength were found to be highly dependent on resin degree of cure while fibre-direction (longitudinal) properties were much less affected. Ply longitudinal properties were, however, much more highly dependent on resin properties than predicted by standard micromechanics models. This is believed to be a result of poor fibre\/matrix bonding, and thus poor load transfer, at low degrees of cure (White and Hahn, 1990, 1992b). Examples of the different models that have been used for composite elastic property development during -34-Chapter 2: Literature Review and Synthesis of Model Requirements cure are shown in Figure 2.13. All of these models are based on room temperature measurements performed using standard ASTM elastic tensile tests (except that of Bogetti and Gillespie; these authors did not measure modulus). This raises some doubt as to the applicability of this data to the current problem since material behaviour may be very different at the processing temperatures at which stresses are actually generated. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Resin degree of cure F i g u r e 2.13: M o d e l s f o r d e v e l o p m e n t o f c o m p o s i t e t r a n s v e r s e m o d u l u s , a l l ba sed o n r o o m t e m p e r a t u r e m e a s u r e m e n t s . 2.4.4.2 Viscoelastic Mechanical Property Models Viscoelastic constitutive models have been employed previously for prediction of process-induced stresses by Weitsman (1979) who examined microscopic stresses during cool-down, and White and Hahn (1992a, 1992b) who analyzed macroscopic stress development throughout processing. Both investigations revealed significant internal stress relaxation which would not have been accounted for in a purely elastic stress analysis. While allowing for potentially improved modelling of mechanical behaviour, such viscoelastic stress analyses are also much more complex and computationally expensive -35-Chapter 2: Literature Review and Synthesis of Model Requirements than equivalent elastic models. Viscoelastic Models of Property Development Characterization of the viscoelastic response of a composite material is a complex problem, and is doubly difficult for a curing composite. The most serious difficulties arise from the need to measure material behaviour over a wide range of temperatures and degrees of cure and the fact that curing of the resin changes its material response even as it is being tested. The amount and type of data collected, and the characterization method used depends on the viscoelastic model chosen. A commonly used procedure, outlined in Flaggs and Crossman (1981), is to use experimental data to develop a master relaxation curve of modulus (or compliance) versus time (or frequency) at reference values of, in the current case, temperature and resin degree of cure. Relaxation behaviour at other values of T and a can then be determined using temperature and resin degree of cure 'shift factors'. At a given degree of cure, the temperature portion of this shift factor can be calculated using such relations as the Williams, Landel and Ferry (WLF) equation (Ferry, 1980): \\ o g a S^LzIsL (2.12) C2+(T-TS) where aT is the temperature shift factor, Ts is the reference temperature and C\\ and C? are material-dependent constants. Calculation of the combined temperature\/degree of cure shift factor would be somewhat more complex. A viscoelastic model developed specifically for curing resins is that of Dillman and Seferis (1989). These workers propose the following expression for describing DMA (dynamic mechanical analysis) data for a reacting system: f =J'-iJ\" = J+ (Jf~J\"] (2.13) ' [X+iimYY where J\u201e is the 'unrelaxed' compliance (i.e. the compliance at time = 0), Jr is the 'relaxed' compliance -36-Chapter 2: Literature Review and Synthesis of Model Requirements (the compliance at time = oo), a> is the experiment frequency, r is the mean relaxation time of the system, and a and 6 are parameters accounting for relaxation time distribution. Another model outlined in the literature is that of White and Hahn (1992b). In this model, composite longitudinal modulus and the Poisson's ratio, v12, were assumed to be time independent, with the following power law relation used to describe transverse compliance with time: S22(a,t) = S22i(a) + D(a) aTa(T,a) (2.14) where S22j(a) is the compliance at time = 0, aJa(T,a) is the temperature\/degree of cure shift factor, and D(a) and q(a) are equation constants. Equation 2.14 provides a less accurate description of viscoelastic behaviour than the other models outlined, but requires significantly reduced material characterization and computational effort. Other Mechanical Property Models As discussed, there are challenges to applying either purely elastic or purely viscoelastic analyses to the problem of modelling process-induced stress and deformation development. Purely elastic analyses, while relatively simple, are unlikely to be accurate in all cases, whereas rigorous viscoelastic models are very complex and computationally expensive. An alternative approach to using either of these models is to employ a 'hybrid' constitutive model which is neither truly elastic nor viscoelastic, but combines features of both. Such a model is described by Chapman et al. (1990). These investigators employed the concept of a viscous-elastic transition for calculation of stress development in a thermoplastic component during cool-down. Although the examined material exhibited clearly viscoelastic behaviour during much of the cool-down process, the rate of increase in relaxation times with temperature reduction were so great that a transition temperature could be defined above which all induced stresses could be considered to relax -37-Chapter 2: Literature Review and Synthesis of Model Requirements instantly and below which the material exhibited purely elastic behaviour. 2.4.5 Stress Analysis Methods The two types of analyses most useful for prediction o f process-induced stress and deformation in composite materials are finite element (F.E.) techniques and laminate plate theory (LPT)-based analyses. Closed-form solutions are also available in some simple cases. LPT-based techniques can provide fairly accurate estimates o f both overall laminate behaviour and individual ply stresses (away from component edges) in cases where laminate cross-section is uniform and part thickness is small compared to in-plane dimensions. For components of greater complexity, finite element techniques are generally required, especially near component boundaries and where in-plane stress gradients are significant. Nearly all residual stress analyses described in the literature are based on one of these two methods with the majority o f integrated processing models using an LPT-based approach. 2.4.5.1 LPT-Based Analyses Stress analyses based on laminated plate theory are relatively simple and computationally inexpensive. For this reason, a large number of LPT-based models have been used in the past, both in analyses which considered cool-down stresses only (e.g., Hahn and Pagano, 1975), as wel l as those that examined the more complex case o f stress development throughout the process cycle (Bogetti and Gillespie, 1992). In both types o f problems, a technique referred to as 'incremental L P T analysis' is generally employed. Using this method, the process cycle is divided into a number o f temperature and\/or time steps and standard techniques are used to calculate the change in ply stresses and strains during each step. If an elastic constitutive model is used, total ply stresses and strains after a given step may be determined by summing those calculated at all previous steps, e.g.,: -38-Chapter 2: Literature Review and Synthesis of Model Requirements (of)\u201e = X(Aa*), (2.15) 1=1 where k is the ply number, n is the calculation step, and \/ is the material direction. A detailed description of an elastic incremental LPT analysis is provided by Bogetti and Gillespie (1992). If viscoelastic constitutive models are employed, stress calculation is more complex. In this case, relaxation of previously-generated stresses must be accounted for. There are a number of different ways to account for this stress relaxation. In the only viscoelastic analysis of process-induced stress described in the literature, White and Hahn (1992a, 1992b) employed what they term a 'quasi-elastic' method. Using this approach, an equation similar to the following is employed6: ( ^ ) . = i : R ( ' ' ) f x ( ^ - A ^ ] ( 2 i 6 ) where k, n, and \/' are as defined in Equation 2.15, Q*(0 is the ply viscoelastic stiffness matrix, (AfJ)* are the incremental unconstrained ply strains during step \/ and (Ae.)* are the incremental constrained ply strains after stress relaxation. The terms %(t) and %(z) are 'reduced' times which account for the variation in material stress relaxation characteristics with changes in temperature and resin degree of cure, defined as: (2.17) where t is the current time and r is the time at which a strain was generated. The term aTa[T,a] is a temperature\/degree of cure 'shift factor' as described previously. This procedure is significantly more The equation actually used by White and Hahn is somewhat different from that shown here since it was developed for a specific laminate lay-up. -39-Chapter 2: Literature Review and Synthesis of Model Requirements complex and computationally expensive than a comparable elastic analysis since stress calculation at a given step requires re-calculation of the contributions of the stresses generated at all previous steps. An alternate approach for viscoelastic LPT-based analyses is the 'hereditary integral' method, applied to stress analysis of composite materials by Flaggs and Crossman (1981). This approach has the advantage that stress relaxation calculations can be performed over the current time step only and do not require consideration of stresses generated over a number of previous time steps. However, this method is even more complex conceptually and mathematically than the model of White and Hahn. Also, the accuracy of this approach in cases where stress relaxation is very large, such as in the current problem, is unclear. 2.4.5.2 Finite Element Analyses Finite element analysis has been widely applied to composite components, both in structural analysis and in the examination of process-induced stresses and deformation. A number of specialized techniques have been specifically developed to deal with the special problems of composite stress analysis and many commercial finite element codes such as ANSYS and ABAQUS include special elements for layered composite materials. The main difficulties in applying F.E. techniques to the modelling of process-induced stress are the complexity of the analysis and the large amount of computational effort potentially required. Thus, while a number of investigators have used finite element analysis to predict cool-down stresses (Griffin Jr., 1983; Stagano and Wang, 1984), only a single reference to an analysis of stress development throughout processing has been found (Bogetti, 1989). In this analysis, Bogetti used a 'quasi 3-D' elastic finite element model to examine stress development in thick-section unidirectional laminates. This model was limited to rectangular shapes and required very large amounts of computational effort (13 hours on a VAX mainframe) to perform about 50 stress calculations for a small problem. No reference to a viscoelastic finite element analysis of process-induced residual stress has been found in the literature. -40-Chapter 2: Literature Review and Synthesis of Model Requirements 2.4.6 Boundary Conditions Use of representative boundary conditions is critical to ensuring accurate stress and deformation predictions. Most previous analyses have made the assumption that there are no external constraints (other than required to prevent free body motion) or boundary forces acting on the part of interest and that it is free to deform as desired. The finite element analysis performed by Bogetti (1989) allowed for both 'fixed' and 'sliding' (i.e. no normal displacement) constraints. The latter of these could conceivably be used to simulate the constraint of a stiff tool, but there was no indication that this was done. 2.5 Summary and Discussion The potential of process modelling when applied to the problem of autoclave process development has become increasingly apparent in recent years, as evidenced by the growing pace of research in this area. Thus, significant fundamental research has been performed in a number of areas of autoclave process modelling, especially heat transfer and cure kinetics, and resin flow (see Hubert, 1996). To the present, however, most work has been focused on small, simply-shaped laminates. Relatively little work has been directed toward 'applied issues'; that is in the application of process modelling knowledge to industrial autoclave processing of structures of practical size and complexity. Modelling of the development of residual stress and deformation has lagged behind analyses of other processing phenomena such as heat transfer and resin cure, and resin flow. Thus, no current model can either be applied to complex structures or can analyse all major known sources of residual stress and deformation. However, based on the material presented in this chapter, it seems reasonable to postulate the following 'necessary' and 'sufficient' conditions for an autoclave process model which is capable of meeting these objectives, as outlined in Table 2.2. -41-Chapter 2: Literature Review and Synthesis of Model Requirements Table 2.2: Necessary and sufficient conditions for modelling of the development of residual stress and deformation in industrial autoclave processing. Necessary Condition Sufficient Condition Must examine development of stress and deformation throughout the manufacturing process. Perform calculations throughout the autoclave process, including the tool removal and layup\/bagging process. Must incorporate analysis of anisotropic strain effects. Integration with models for heat transfer and resin cure, employing transversely isotropic (or orthotropic) material behaviour models. Must incorporate analysis of cure and temperature gradients. Integration with models for heat transfer and resin cure. Must examine the effects of resin flow and fibre bed compaction. Integration with models for resin flow and fibre bed compaction (e.g., Hubert, 1996). Must examine the effects of process tooling. Incorporation of tooling directly into the model; integration of stress model with models for heat transfer and resin cure; consideration of tool\/part interaction; and simulation of tool removal process. Must examine the effects of resin cure shrinkage. Incorporation of models for resin cure shrinkage strains. Must be capable of modelling structures of significant size and geometric complexity. Two-dimensional intermediate-scale model incorporating multiple composite and non-composite materials. Must have an accurate model for composite and resin mechanical behaviour during processing. A cure-hardening\/instantaneously linear elastic model for mechanical response (e.g. Bogetti, 1992). In some cases, a viscoelastic material model may be required. Must have an accurate representation of boundary conditions during processing. Simulate autoclave response, control systems and autoclave internal heat transfer. Must be capable of performing analyses using practical levels of computational effort. Simulation of autoclave processing for structures of intermediate size and complexity using a personal computer or workstation in less than 24 hours. Simulations for sensitivity analyses in less than 30 minutes. -42-3. Modelling Approach This chapter outlines the approach used in the current process model, COMPRO, including a description of the overall model structure, the finite element representation employed and a discussion of the 'virtual autoclave' simulation used to predict structural boundary conditions during processing. As illustrated in Figure 3.1, the processing of composite structures can be examined over a wide range of scales of size and structural complexity. Most previous autoclave process models have focused on the lower end of this scale, ('fundamentals') examining processing behaviour at the fibre and matrix level or the level of individual plies. Analyses at this level can be used to predict matrix cracking and delamination and small-scale warpage, but geometric effects and large-scale deformation cannot be considered. At the opposite end of the scale, the processing of large complex structures can be analysed employing plate and shell elements in a 'global discretization' of the entire structure. At this level, however, it is not currently feasible to employ a sufficiently fine discretization to capture small-scale interactions that are critical to the development of residual deformation. The current model focuses on a scale mid-way between these extremes, employing a 'local discretization' to permit modelling of structures of intermediate size and complexity. At this level, it remains possible to analyse both small-scale interactions and structures of practicable size and complexity. In all cases, a 2-D cross-section of the structure of interest is analysed, as illustrated in Figure 3.1. Since at least one dimension of composite structures is generally quite large, this 2-D representation is usually sufficient. To model large-scale structures, a substructuring technique outlined in Section 7.3.4 is proposed. Using this method, a complete structure is divided into a number of'representative' substructures, each of which is examined individually using the current model, then model predictions combined in a larger-scale analysis. -43-Chapter 3: Modelling Approach FLM)AMENTALS PRESENT MODEL APPLICATION Micromechanics Local discretization Structure of interest Ply level Global discretization IO\"6 IO 4 io-3 IO\"2 10 (m) Figure 3.1: Scales of interest in composites process modelling. The overall structure of the process model is similar to the 'integrated sub-model' approach, first applied to composites process modelling by Loos and Springer (1983). Using this approach, a complex coupled problem such as the current one is divided into a series of simpler problems that are tackled independently in a series of 'sub-models', or 'modules' (to emphasize the modularity of the approach). Coupling between modules is maintained by solving each in sequence as the solution marches forward in time. In this way the predictions of one module can be used by another1. In COMPRO, modules were created for not only the analysis of the structure of interest, but also for simulation of the autoclave response and for other essential program tasks. As shown in Figure 3.2, six different modules are currently employed including: 1. Autoclave Simulation Module: predicts the variation of autoclave internal temperature and pressure and the vacuum bag pressure with time. Also predicts other parameters associated with structure boundary conditions such as local heat transfer coefficient (see Section 3.2). 2. Thermochemical Module: predicts internal temperature in the structure and tooling and the resin degree of cure in composite structural components (see Chapter 4). -44-Chapter 3 : Modelling Approach 3. Flow Module: uses a percolation model to predict resin flow and fibre bed compaction in composite components (see Hubert, 1995). 4. Stress Module: models the development of residual strain and deformation in the structure and any process tooling (see Chapter 5). 5. Input Module: controls acquisition of all input parameters required for a program run. This may include translation of the problem finite element description from an external preprocessor to a form used internal by COMPRO. For input module details, see Appendix A. 6. Output Module: controls output of all parameters to external devices, for example, to the screen and hard drive. For output module details, see Appendix A. Figure 3.2: Schematic of COMPRO structure and program flow. Each module is called as needed by the controlling routine as the solution marches forward in time. The internal operation of each program module is completely independent, with module integration occurring Note that using this technique, highly coupled phenomena cannot be solved in separate sub-models unless very small steps in time ('time steps') are used. -45-Chapter 3: Modelling Approach via the updates that each module makes to a 'central database' containing a description of the system state. To minimize computational effort, the overall program dynamically calculates its own time step based on the 'suggestions' of the various modules. When called in turn, each module decides whether or not it 'needs' to run. If not, it simply skips its computations for the current step. A more detailed discussion of the specifics of program operation is provided in Appendix A. An important focus of the current model is accurate prediction of part boundary conditions during processing. To facilitate simulation of the actual conditions encountered in the autoclave, the model makes use of a 'virtual autoclave' concept. As will be discussed in Section 3.2, using this approach, the structure of interest is treated as a component in a virtual autoclave which employs analogues for most important autoclave system components including environmental systems, instrumentation and control algorithms. This permits much improved prediction of the actual autoclave time-temperature history, including the potential to model internal variation in temperature and heat transfer rates. 3.1 Finite Element Model All three of the current modules examining the behaviour of the structure of interest (the thermochemical, flow and stress modules) employ finite element methods to solve the relevant governing equations. This section briefly outlines some aspects of the finite element method as applied to the current problem, focusing on the thermochemical and stress modules that were developed as part of the current research. A detailed description of the theory of the finite element method is not provided here, but it is discussed in depth in numerous publications such as Cook et al. (1989). 3.1.1 Discretized System of Equations Using the finite element method, the governing equations describing a phenomenon are 'discretized' into a system of algebraic equations that can be solved numerically. This is done by subdividing the problem domain into a number of simply-shaped elements, connected at nodes at which the parameters of interest (e.g., temperature, pressure, or displacement) are actually calculated. The values of these parameters at -46-Chapter 3: Modelling Approach any point in the element are interpolated from nodal values using simple polynomial interpolation functions. Thus, using this approach, the exact value of a parameter within the element is approximated by: 0 = \u00a3\/v-.ci>. (3.1) where * is the approximate solution, TV, are the element interpolation or 'shape' functions andO, are the parameter values at the element nodes. Using matrix notation, the above equation becomes: d = [N]{ 1 1 Figure 3.3: Plane bilinear isoparametric element in global and natural coordinate systems. To minimize the number of elements required to discretize a given problem, multiple plies o f any orientation may be contained within each composite material element as shown in Figure 3.4. It is not required that a whole number of plies be contained within an element (e.g., a ply can contain 2.6 plies), but the ply plane must be parallel to the element local coordinate x 'axis (see Figure 3.4). Another important characteristic of the finite elements used in this model is that they all employ the assumption that derivatives of the calculated nodal parameters are zero in the direction normal to modelled x-z plane. For the thermochemical and stress modules we have: dy dy dy (3.7) where T is the element temperature and u and w are the element displacements in x and z directions respectively. -49-Chapter 3: Modelling Approach Ply i Ply orientation \u2022 1 z a b Figure 3.4: Composite element shown in: a) isometric view, b) plane view. The shape function matrix [N] for a particular problem will depend on the number of degrees of freedom at each node. For example, the temperature shape function matrix, used in the thermochemical module, is defined as: [NT] = [N, N 2 N 3 N4] (3.8) where N b N 2 , N 3 , N 4 are elements of the shape function matrix in Equation 3 . 6 . The displacement shape function matrix, used by the stress module, is defined as: N, 0 N 2 0 N 3 0 N 4 0 0 N, 0 N 2 0 INL 0 N. (3.9) 3.1.3 Finite Element Description As mentioned, in the finite element method the domain of the problem of interest is sub-divided into a series of finite elements connected at their nodes as illustrated in Figure 3 .5 for a simple curved shape. The finite element description of a problem, however, consists of more than just a list of elements and the spatial locations of their nodes. The other information required depends on the model being used. The -50-Chapter 3: Modelling Approach components of the finite element description for the current model are: \u2022 Definition of node locations in x and z coordinates. \u2022 Definition of element connectivity: This is an ordered list of the nodes attached to each element. These nodes should be listed in counterclockwise order. \u2022 Definition of model regions: A 'region' is defined as a group of elements made of a single material. If a region is comprised of a composite material, this element grouping must be continuous (see Appendix A). \u2022 Definition of the model boundaries: In the current model, this consists of a list of the nodes which define each boundary. \u2022 Definition of the reference boundaries: Reference boundaries must be defined for each region containing a composite material or any other anisotropic material. These boundaries are used to determine the element local axes orientation with respect to the global axes (angle\/? in Figure 3.5). Also, for composite material regions, these boundaries are used to calculate which plies are contained in each element. Figure 3.5 Curved region, showing element local coordinates, local axes orientation and the region A more detailed discussion of the finite element description used by the current model is provided in Appendix A. z reference boundary -51-Chapter 3: Modelling Approach 3.1.4 Material Properties Each time a module is run, the relevant material properties for each element are recalculated from state variable values (e.g., temperature, degree of cure and volume fraction) at the element centroid. As outlined in Chapter 6, a number of models are used to determine resin, ply and composite properties. Orthotropic material behaviour is assumed in all cases2. Material principal directions are assumed to be coincident with the element local axes except for composite plies which are oriented at an angle 6> to the element x'-z' plane (see Figure 3.4). Material properties in the global axes are calculated using a series of transformations as outlined in Appendix B. In the calculations of the thermochemical and stress modules, material properties are assumed to be constant throughout the module time step. In all cases, the state variable values used to calculate these properties are the 'best' available at the start of the step. Since the thermochemical module runs first, the properties it uses (i.e., density, specific heat, conductivity) are calculated from the material state at the start of the module time step since the state at the end of the step is unknown. However, by the time the stress module is run, some information about the material state at the end of the stepw available since the thermochemical and flow modules have already done their calculations. Therefore, the properties used by the stress module are calculated from average state variables during the module time step. This concept is illustrated in Figure 3.6. Orthotropic composite properties may be either specified directly or calculated from micromechanics models which use as input fibre volume fraction, isotropic resin properties and transversely isotropic fibre properties. -52-C h a p t e r 3: M o d e l l i n g A p p r o a c h S-i u o lk-\\ lk Time Figure 3.6 Temperature and degree of cure used for material properties calculations. The thermochemical module uses property values at the start of the step A, while the stress module uses average values during the step. 3.1.5 Ply Layup For elements containing composite material, another piece of information required by the model is the list of plies in each element and their orientations with respect to the elementx'-z' plane. As illustrated in Figure 3.4, ply orientation is defined according to the right-hand rule with the angle measured from the positive x' axes. Ply layup in a region is defined with respect to the reference boundary for that region (i.e. ply 1 starts at the reference boundary). The plies contained within each element are determined by the distance of the nodes of that element from this reference boundary (see Appendix B). Using this approach, the 'thickness' of the composite region need not be constant as long as the maximum region thickness equals the sum of the ply thicknesses in the layup. Thus, ply drops can be accommodated, but they are always assumed to occur from the end of the layup sequence (i.e. the last plies defined are dropped first). A few limitations of the current model related to layup should be mentioned. At present, only uni--53-Chapter 3: Modelling Approach directional plies can be modelled. It is suggested that for now woven plies be modelled as sets of unidirectional plies which give nearly equivalent properties; for example, a woven ply may be replaced with four unidirectional plies with a [0\/90\/90\/0] layup, each with a thickness % of the woven ply. Another limitation is that all plies in a region must be of the same initial thickness and be composed of the same material. If desired, however, there is no prohibition against 'stacking' multiple composite regions (effectively different laminates). 3.2 The Virtual Autoclave The use of 'representative' boundary conditions is central to the relevance of any finite element analysis. In previous process models, autoclave temperature, autoclave pressure and vacuum bag pressure were assumed to be uniform and to vary with time according to a pre-defined process cycle. Heat transfer rates were generally assumed to be constant and uniform throughout an autoclave and process tooling effects were usually not considered. In a real process, air temperature and heat transfer rates often vary significantly within an autoclave (Ghariban et al. 1992; Roberts, 1987) and are both a function of autoclave 'loading' (Roberts, 1987). Also, process cycles are rarely actually implemented on a strict time basis; rather feedback control systems are usually employed to compensate for process variation. While using such a control system helps ensure component quality, large variations in the process cycle can be encountered simply by processing a component in a different autoclave or even on a different tool (Hubert et al., 1995). To assist in prediction of the boundary conditions seen by the structure during processing, we introduce the concept of a 'virtual autoclave'. Using this approach, the entire autoclave system is modelled including the structure of interest, which becomes a component of the virtual autoclave system, albeit the most important and intensely modelled one. In principle, the virtual autoclave simulation should be quite simple; unlike the part, once a cycle has begun an autoclave is virtually a 'closed' system, with easily described characteristics, inputs and outputs. The autoclave itself, for example, has a known geometry -54-Chapter 3: Modelling Approach and thermal mass and the rate of heat loss through its walls can easily be determined. The characteristics of its heating, cooling and pressurization systems and basics of its internal air flow can similarly be measured. Knowing the characteristics of each of these systems, the conditions to which the part will be subjected throughout processing can, theoretically, be completely described. Practically, of course, the situation is much more complex since most system characteristics arenot easy to determine, nor do they even necessarily remain constant (e.g., autoclave loading). Even if it were possible to obtain a perfect system description, the computational effort required to obtain heat transfer coefficients alone would be tremendous, as demonstrated by the simplified analysis of Telikicherla et al. (1994a, 1994b). However, by simply using the framework of such a 'virtual autoclave' and employing even very simple models for major system components, a significant advance over traditional approaches to boundary condition modelling can be obtained. An example of a system which is easily modelled and can significantly improve boundary condition predictions is the autoclave environmental control system. As mentioned, most industrial autoclaves employ feedback control to implement the process cycle, often using a 'lead\/lag' control algorithm. Using such a control system, thermocouples are strategically placed throughout the structure being processed, within the autoclave and on the process tooling. Measured temperatures are used by the control system to control air temperature, autoclave pressure and vacuum pressure. Such systems can partially compensate for variations in autoclave heat transfer and tooling thermal mass by adjusting the process cycle to allow sufficient time for laminate compaction and full cure to be attained throughout a structure. As demonstrated in Chapter 7, even a relatively crude model of such a control system can generate much better process cycle predictions than a traditional approach. The components of the current virtual autoclave simulation and the models used for each are as follows: 1. Structure of Interest - The most intensely modelled component with analyses for heat transfer and resin cure, internal resin flow and development of residual stress and deformation. All of these -55-Chapter 3: Modelling Approach phenomena are modelled using 2-D plane finite element analysis. 2. Process Tooling - All process tooling used to manufacture the part (substrate tooling, fixtures, vacuum bag, bleeder\/breather, etc.) can in theory be included directly in the same finite element model as the structure of interest. In practice, for very large tools, a simplified tool geometry is employed, using 'equivalent' material properties. 3. Virtual Instruments - Virtual thermocouples can be placed at any finite element node within the structure of interest or the process tooling. Autoclave air temperature and pressure and vacuum bag pressures are automatically measured. No other types of virtual instruments such as dielectric sensors are currently employed nor is simulation of measurement error included. 4. Autoclave Hardware Simulation - The current simulation of the autoclave and its environmental control hardware (i.e. heating, cooling and pressurization devices) is primitive at present. The autoclave air temperature is assumed to respond instantly to controller 'requests' and maximum temperature change rate is controlled only by limits set by the user for each run. The situation is similar with both autoclave gas temperature and vacuum bag temperature. Such parameters as autoclave heat loss thermal mass, and internal air flow are not currently considered. Autoclave temperatures, pressures and heat transfer rates are thus assumed to be uniform. 5. Autoclave Environmental Control Systems - A basic lead\/lag control system is simulated as described in Appendix A. This system reads the described virtual instruments to determine autoclave temperature and pressure and vacuum pressure setpoints. While not comprehensive, the employed algorithm should allow fairly accurate simulation of most lead\/lag systems. Even in its current very simplified form, the virtual autoclave simulation permits much improved prediction of part boundary conditions than previously available. Improved accuracy could be obtained with the introduction of slightly more complex models of autoclave and environmental control hardware response. This modification alone would allow the current process model to be used in a number of new -56-Chapter 3: Modelling Approach applications including: \u2022 As a test bed for simulation of intelligent process control concepts. A module for the algorithm for such a control system could be easily 'plugged into' the current virtual autoclave simulation module and its effectiveness gauged by processing various virtual parts. Algorithm response to variations in material properties and random error in virtual instrument measurements could also be examined. \u2022 To test new tooling concepts such as heated tools to assess potential reduction in process time and component warpage minimization. Both hardware and control systems could be tested prior to creating a prototype. A more complete discussion of the details of the current implementation of the non-finite element components of the virtual autoclave is provided in Section A.5. -57-4. Thermochemical Module The thermochemical module is responsible for calculation of temperature in the structure of interest and the modelled tooling as well as the degree of resin chemical advancement (degree of cure) in composite components. Accurate prediction of these parameters is potentially useful to the composites processor in a number of ways. First of all, one of the main objectives of processing thermoset composite materials is to achieve 'full' and uniform cure of the matrix resin so that a structure can attain its maximum stiffness, static and fatigue strength and resistance to moisture and chemical degradation1. Achieving maximum degree of cure in minimum time would seem to indicate that a high temperature cure cycle be used. However, this approach has other, potentially negative, implications for the cure process. For example, the rapid heat evolution of the resin's exothermic reaction at high temperatures can lead to reaction 'runaway' as heat is generated more quickly than it can be removed, potentially resulting in resin thermal degradation. It has also been found that the large spatial and temporal gradients in resin degree of cure and temperature induced by rapid cure can be an important source of process-induced stress (Levitsky and Shaffer, 1975; Bogetti, 1989), especially in thick-section composites. Thermochemical module predictions are also important to the simulation of other processing phenomena such as resin flow and the generation of residual stress and deformation. One reason this is so is that temperature and degree of cure are two of the most important 'state' variables used to predict composite material properties during processing2. Thus, prediction of everything from resin viscosity to thermal expansion and resin cure shrinkage strains are dependent on thermochemical module predictions. The thermochemical module consists of a combination of analyses for heat transfer and resin reaction kinetics. Every effort has been made to incorporate the most useful aspects of past models while 'Full' cure does not necessarily mean a degree of cure of 1.0. Many resin systems do not approach a degree of cure of 1.0 at normal curing temperatures. At present, the only other state variable used to calculate material properties is fibre volume fraction (Vj), although other state variables such as pressure, moisture content or strain could conceivably be used. -58-Chapter 4: Thermochemical Module extending their capabilities to allow examination of realistic manufacturing problems. Just as the model of Bogetti and Gillespie (1991), this model can be applied to general two-dimensional cross-sections. It has the added capability of incorporating multiple composite and non-composite materials as well as process tooling. Other important features include the potential for improved boundary condition modelling using the autoclave simulation outlined in Chapter 3 and consideration of material property variation during processing. Also, by integrating this analysis into a model which considers resin flow, the effect of fibre volume fraction variation during processing can also be considered. This chapter outlines the development and implementation of the current thermochemical module including fundamental equations, boundary conditions, and material properties. 4.1 Fundamental Equations and Finite Element Solution This section introduces the fundamental equations of the thermochemical model and discusses the development of the discretized system of equations required to solve the problem numerically. A more detailed discussion of the discretization procedure as well as the specifics of the solution algorithm employed by the program thermochemical module are provided in Appendix B. The governing equation of the thermochemical module is the unsteady-state 2-D anisotropic heat conduction equation with an internal heat generation term from the resin's exothermic curing reaction, as follows: d , \u201e ^ d f, dT dt dx dx d dz dr \" dz d dx dT\\ \" I f*i u 1 U 1 V Ul U Ul U . Ul \u2022 , dz d dz dr dx where p is the composite density, Cv is the composite specific heat and kv are the composite anisotropic thermal conductivities. At each time step of the numerical solution, these properties are calculated from local resin and fibre properties and the fibre volume fraction, Vf(see Section 4.4). The resin heat generation term Q in Equation 4.1 is calculated from: -59-Chapter 4: Thermochemical Module 2 = ^ ( 1 - F > A (4.2) where pr is the resin density, a is the resin 'degree of cure', and HR is the resin heat of reaction, defined as the total amount of heat evolved during a 'complete' resin reaction. The degree of cure, a, is a measure of the degree of completion of the curing reaction and is defined as the ratio of the cumulative heat evolution of the reaction to the total heat of reaction, i.e.,: (4.3) where dq\/dt represents the specific heat generation rate. Degree of cure and temperature are assumed in the current model to be sufficient to completely describe the resin state. Thus, at a given temperature a one-to-one relationship is assumed to exist between resin cure rate (and all other resin properties) and its degree of cure. A number of different models are available from the literature for calculation of cure rate, the most appropriate for a particular case depending on the details of the resin curing reaction. In the current analysis, any of six different kinetics models may be employed (see Section B.1.7). 4.1.1 Temperature Calculation The governing equations of the heat transfer portion of the problem are solved employing the finite element approximation. This is a standard method for solving similar heat transfer problems, although most previous composites processing models have employed finite difference techniques (Bogetti and Gillespie, 1991; Loos and Springer, 1983). A very brief description of the finite element method and its application in the current processing model is provided in Section 3.1. Using this technique, the actual temperature at any point in the element, T, is approximated by: -60-Chapter 4: Thermochemical Module .\/=' where T is the approximate solution, NTj are the element interpolation or 'shape' functions for the temperature solution and 7} are the temperatures at the element nodes. The discretized equations for the thermochemical module are obtained using the Galerkin weighted residual method where the residual, R, is defined as: R=T-f (4.4) In this case, this expression can be replaced by (using a more general expression than shown in Equation 4.1): R = ^ (pCPT)-V(kVT)-Q (4.5) Using the weighted residual methods, the weighted average residual of the error over the domain of interest is set to zero, i.e., jWkRdCl = 0, k = \\,2,....n (4.6) n where n is the number of degrees of freedom in the model, Wk are the Galerkin weight functions and Q represents the volume of the domain. Substituting the residual expression in Equation 4.5 into Equation 4.6 and integrating by parts (Section B.2.2) we obtain: \\wkpC,j dQ - \\wkkVT dT + \\kVTVWk dQ - \\\\VkQ dQ = 0 (4.7) n r n n The second term in Equation 4.7 accounts for boundary heat flows, where T represents the surface of the domain. -61-Chapter 4: Thermochemical Module Rewriting Equation 4.7 in matrix form and replacing the weight functions Wk by the element (temperature) shape functions [NT] we obtain: J [NT ]1 pCP [NT ] ft} dQ - J [NT ] T M [BT ] {T} dT (4-8) +J[BT]' M[BT]{T} dn - j [N T ] 7 QdQ = 0 n o. where {T} is the nodal temperature vector, {T} is the derivative of the temperature vector with respect to time, [K] is the anisotropic thermal conductivity matrix, [BT] is the derivative of the element (temperature) shape function matrix, and Q is the volumetric heat flux from the curing reaction. Performing the indicated integrations and separating the terms of the surface heat flux term into its components (see Section B.2.2), Equation 4.8 can be rewritten as: ([K J + [H]){T} + [CP ]{t} = {FTq } + {FTQ } + {FTh } (4.9) where: [KK] is global conductivity matrix, [H] is the global convective heat transfer matrix, [CP] is the global thermal capacity matrix and {FTq}'{FTo} a n a \" { FTh} a r e 'load' terms. The definitions of each of these matrices are provided in Section B.2.2 Time Integration Equation 4.9 contains first-order derivatives and must be integrated in time. The technique used here is the implicit (backward) Euler method. This method is unconditionally stable3 and allows much larger time steps to be taken than the simpler but potentially unstable forward Euler approach. Using the backward Euler approach we can write: ffi^ifW'-W*-) (4.10) This does not imply unconditionally accurate, so care still must be taken to ensure convergence is achieved. -62-Chapter 4: Thermochemical Module where k is the number of the 'time step' (see Figure 4.1). Time Figure 4.1: Example derivative calculation using the backward finite difference approximation. Substituting into Equation 4.10, we obtain: ([K J + [H]){T}A + [CP ]\u00b1({T}k - {T}A_,) = {FTQ } + {FTQ} + {FTH } (4.11) Simplifying, we arrive at the equation for the temperature in the domain at the end of time step k: {T}k=[KTYki{FT}k (4.12) where [KT]t is the global temperature equation stiffness matrix defined by: [KA=^[CP] + [K,] + [H] (4-13) and {Fx}* is the global temperature load vector calculated as: FT}* -{FTQ} + {FTQ} + {FTH} + [^CP]{T}A_, (4.14) 4.1.2 Degree of Cure Calculation As can be seen from the heat transfer equation (Equation 4.1) and the cure kinetics equations outlined in Chapter 2, the solutions for temperature and the degree of cure are 'coupled'; that is, each depends on the -63-Chapter 4: Thermochemical Module other. Ideally, therefore, a coupled solution technique would be employed in which both temperature and degree of cure would be solved in a single calculation. For the current model, however, this route was not chosen since it was necessary to define discretized temperatures and degrees of cure at different spatial locations within the domain. This is because temperature spatial derivatives are continuous throughout the domain but degree of cure is discontinuous at material and domain boundaries. Therefore, in this analysis degree of cure is defined as an element state variable rather than a nodal variable. As outlined in Table B.l, element degree of cure may be calculated using any of six different kinetics models. The fundamental equations for each are similar in form as is the solution technique used. For illustrative purposes, the solution method is demonstrated for the simplest of these equations, that for kinetics model 1: \u2014 = Ae-*E\/xr(l-a.y (4.15) dt ' Species migration via either resin flow or diffusion is not considered. Since the temperature and degree of cure solutions are decoupled within an iteration in this analysis, degree of cure at any point can be solved for explicitly, thus eliminating the need for a matrix solution. However, to maintain a consistent notation with the rest of the analysis, we replace Equation 4.15 with a matrix representation as follows: {a} = {Ka}r({l}-{a})\" (4.16) where the discrete values of degree of cure, {a}, are defined at the element centroids, {d} is the cure rate vector, and {Ka} is the vector of cure rate coefficients such that: K a i = A i e - ^ ' R T ' (4.17) where I) is the temperature at the node centroid. The vector {1} is a vector where all values are 1. -64-Chapter 4: Thermochemical Module Time Integration Equation 4.16 contains first-order derivatives and must be integrated in time. Once again a backward-Euler technique is used, i.e., - \u00ab . ) (4-18) where k is again the number of the 'time step'. Substituting the above into Equation 4.16 we obtain: {a}, =A,{Ka}r({l}-{a}J' + {a}4_, (4.19) The presence of the non-integer exponential n in this equation makes solving this equation in terms of {a}k very difficult for the general case (and this is the simplest of the cure rate equations). Thus, instead of solving directly for {cc}*, an iterative solution technique is employed in which: {a}; = A4Ka}r({l}-{a};-')\" + {<_, (4.20) where \/' is the iteration number. Note that in the first step in this iterative procedure, {ct}^ 1 is replaced with {a},_,. 4.2 Cure kinetics models The basics of the cure process for thermosetting resins and some models employed in previous analyses for cure rate prediction are outlined in Section 2.2.2. Five of the six cure equations which may be used in the current model are taken from the literature as outlined in Table B.l. An important limitation of most kinetics models discussed in the literature is their inability to account for glass transition and the accompanying shift from a reaction kinetics-controlled to a diffusion-controlled process. As a result, model predictions might accurately fit experimental heat flow measurements and still perform poorly under actual processing conditions. To see how this can be so, consider that kinetics models are usually developed using heat flow measurements from dynamic (i.e., temperature-varying) -65-Chapter 4: Thermochemical Module differential scanning calorimetry (DSC) tests. These tests typically involve heat flow rate measurements during 'scans' from room temperature to about 300 \u00b0C at rates ranging from 1 \u00b0C\/min. to 10 \u00b0C\/min. as shown in Figure 4.2. Since the resin temperature continuously increases as it cures, vitrification usually does not occur during a test (unless the resin Tg is very high or scan rates very low). Thus, the cure rate will be reaction kinetics-dominated at all times during the test and can be predicted quite well by models ignoring diffusion. During the autoclave process, however, cure temperatures rarely exceed 200 \u00b0C and the likelihood of encountering vitrification is much higher. As shown inFigure 4.3, models which do not account for glass transition will greatly overpredict cure rates after it occurs and thus predict a much higher final degree of cure than actually obtained. 50 100 150 200 250 300 Temperature {\u00b0C) Figure 4.2: Example heat flow measurements from a dynamic DSC test (10 \u00b0C\/minute). -66-Chapter 4: Thermochemical Module 0 50 100 150 200 250 Time (minutes) F i g u r e 4.3: C o m p a r i s o n o f m e a s u r e d a n d p r e d i c t e d r e s i n degree o f c u r e d u r i n g a 150 \u00b0 C i s o t h e r m a l c u r e u s i n g a k i n e t i c s m o d e l not a c c o u n t i n g f o r v i t r i f i c a t i o n . The magnitude of the noted cure rate over-prediction depends on both the resin and cure cycle; it may be small in some cases and unimportant if the absolute value of the degree of cure is not an important simulation result. However, if the resin degree of cure is important to other model predictions, such as in the current model, it is vital that this parameter be accurately predicted at all times during cure. As discussed by Cole et al. (1991), the method typically used to account for glass transition is to determine the resin instantaneous Tg using DiBennedeto's equation (Equation 2.9), then express the diffusion-controlled cure rate equation in terms of T - Tg using a WLF-type expression (Equation 2.12). This method, however, can lead to extremely complex relations requiring many constants. A simpler approach suggested by Cole et al. is to account for the gradual shift from a kinetics-dominated to a diffusion-dominated reaction near the point of vitrification by modifying the reaction kinetics-controlled rate equation using: -67-Chapter 4: Thermochemical Module where \\ ^ - \\ is the effective cure rate, j-^\u20141 is the reaction kinetics-controlled cure rate and \/Ta,!) is a I dt J. I dt L A 'diffusion factor'. The diffusion factor is a function of how far the resin is from a 'critical' degree of cure (or temperature) marking the boundary of the diffusion-controlled regime. The expression for the diffusion factor proposed by Cole et al. is: f(a,T) = F r - | w (4.22) where C is a diffusion constant, ccco is the critical resin degree of cure at T = 0 K and etc? is a constant accounting for the increase in critical resin degree of cure with temperature4. Equation 4.23 defines an S-shaped curve, approaching unity at T \u00bb Tg and increasing sharply as vitrification is approached. Diffusion factors of this type can easily be added to any of the kinetics equations employed with COMPRO to account for the glass transition effect. To date, only kinetics model #2 (Table B.l) has been modified, resulting in a sixth cure model equation as follows: ^ - K a : { ] - a T n (4.23) Application of the above equation is shown in Section 6.1.7. Much better prediction of cure behaviour after glass transition is demonstrated here than could be achieved using standard models. 4.3 Boundary Conditions Accurate modelling of structural boundary conditions during processing is a major focus of the current model. By employing even a simple virtual autoclave simulation as described in Section 3.2, far better prediction of the actual autoclave process cycle (i.e. the air temperature and pressure and vacuum bag pressure with time) can be obtained than employing traditional approaches. -68-Chapter 4: Thermochemical Module In the current analysis, three basic types of heat transfer conditions may be represented: \u2022 Convective heat transfer (q = h^fT^-T), where hejT is the effective heat transfer coefficient and T\u201e and T are the air and boundary temperatures, respectively). Convective heat transfer is usually the dominant heat transfer mechanism in an autoclave at temperatures normally encountered in thermoset processing. By employing an effective heat transfer coefficient it is possible to account for the thermal resistance of low thermal mass surface layers such as breathers and bleeders as well as part\/air resistance. \u2022 Adiabatic boundary (q = 0, or dTldn = 0, where n is the surface normal vector). This condition is typically applied when part symmetry is exploited. \u2022 Prescribed temperature boundary (T = T^). The boundary temperature is set to the autoclave air temperature. This represents the case of very low resistance to heat transfer between the boundary and the autoclave air. The last two types of boundary conditions are limiting cases of the convective heat transfer boundary condition. The adiabatic condition can be obtained by setting hcff to zero and the prescribed temperature condition by setting to infinity. For cases where heff is far from either extreme, it is necessary that its actual value somehow be determined. As will be discussed in Section 6.2, in the majority of practical cases this is far from trivial. The major complicating factor is that the part\/air heat transfer coefficient is not only a function of the part geometry, but also of local gas velocity. This velocity is often quite variable within an autoclave and may also vary significantly between autoclaves. Also, as demonstrated in experiments conducted as part of this work, heat transfer rates are strongly dependent on autoclave gas pressure (see Section 6.2). One drawback of this model is the implicit assumption of a linear dependence of glass transition temperature on degree of cure. In practice, it would be preferable if Tg were first determined externally using an appropriate model, then used in this equation. -69-Chapter 4: Thermochemical Module Ideally the heat transfer coefficient would be calculated directly as part of the virtual autoclave simulation. Unfortunately, the complexity of autoclave internal air flow makes this impractical at this time. Instead, local heat transfer coefficients are experimentally determined and c^alculated using: hff = {\\lh + tlk)~] (4.24) where t and k are respectively the conductivity and thickness of any surface resistive layer5. The 'actual' heat transfer coefficient, h, is currently calculated as a linear function of autoclave pressure, i.e., h = a + b*P (4.25) A limitation of this approach is that heat coefficient is dependent on autoclave loading as well as local part or tooling geometry. Thus, the Abused is probably correct only on an average sense at best. This problem is discussed in more depth in Section 6.9. 4.4 Material Properties In addition to model boundary conditions, another set of model inputs whose importance should not be overlooked are the thermophysical properties of the various materials included in the model. The properties required by the thermochemical module for all types of materials include: \u2022 Specific heat, CP \u2022 Mass density, p \u2022 Thermal conductivities in the material principal directions, kh k2, k3 Most previous processing models have made the assumption that these properties are constant throughout processing or that any variation would not significantly affect model predictions. However, as shown by Mijovic and Wijaya (1989) and Twardowski (1993), these properties can sometimes vary enough during processing to cause significant errors in model predictions. Thus, in the current process model, simple Components with significant thermal mass should be included directly in the finite element model since a resistance boundary condition cannot fully account for thermal capacitance. -70-Chapter 4: Thermochemical Module models for variation in material behaviour during processing were implemented. These material properties are calculated at the beginning of each thermochemical module time step and are assumed to remain constant throughout the step. For non-composite materials all thermophysical properties are assumed to be linear functions of temperature, e.g., CP=CP0+aCp(T-T0) (4.26) For composite materials, the calculation is more complex since the state of such a material is described not only by its temperature, but also by the fibre volume fraction and resin degree of cure. To provide the user with some flexibility, most composite properties may be specified either as linear functions of temperature and resin degree of cure only, e.g., CP = CP0 + aCp(T- T0)+bCp(a-aa) (4.27) or, alternatively, the effect of Vf can be included if fibre and resin properties are both available. For composite materials resin cure information is also required. To calculate the resin cure rate (and the degree of cure), specification of the appropriate cure model equation and associated equation constants is required. To determine the heat generation rate, the total resin heat of reaction and the resin density are also required. 4.5 Iterative Solution The coupling of the temperature and degree of cure equations (Equations 4.12 and 4.20) and the use of an implicit time integration technique necessitates the use of an iterative solution procedure. In the current model, a simple iterative scheme is employed in which temperature and degree of cure are alternately solved until a converged solution is obtained. The technique employed is as follows: In iteration \/', the resin degree of cure at the end of time step k is estimated using temperature and degree of cure values from iteration z-1, i.e., -71-Chapter 4: Thermochemical Module {a}i=A\/({K a};) 7({l}-{a}r)\" +{a| k-\\ (4.28) The temperature at the end of the time step is then determined for that iteration using: m;=([KT]i)\" { F T I ; (4.29) This iterative procedure is repeated until the 'norms' of the change in the temperature and degree of cure vectors during the iteration are smaller than user-defined limits, i.e., At this point the solution is deemed to have converged. As will be discussed in Section A.4, the rate of convergence, i.e., the number of iterations required to achieve a converged solution, is then used to determine the maximum permissible time step (the faster the convergence, the bigger the step) and the solution moves to the next time step. 4.5.1 Module Solution Algorithm Just as all other program modules, the thermochemical module is called at every time step of the problem solution. Unlike the others, however, this routine also runs every step (the others simply RETURN if they are not 'ready' to do a calculation). This is done because of the need for temperature and degree of cure information in all other modules. The solution algorithm used by the thermochemical module is as follows5: 1) Update element thermal properties (CPc, p\u201e [xe], see Appendix B) Note that in the following, the subscript for the time step number (e.g. k, k-1) has been dropped from all but the final equation for convenience. W i - l a } : 1 < A { a } (4.30) and (4.31) -72-Chapter 4: Thermochemical Module 2) \/\/\"program in an error state then i) RETURN 3) Loop over all model elements 4) Calculate [kj c and [CP]e 5) Add contributions of [kj,, and [Cp]c to [kT]t, and [CP]e to {fT}c 6) Assemble [kT]t, and {fT}e into global matrices [KT] and {FT} 7) Apply essential boundary conditions 8) Add contribution from external load vectors J F X q } 7 and {FTh}to total thermal load vector {FT} 9) Factor the thermal stiffness matrix [KT] 10) Calculate {d}\u00b0 and {a}0 (Equation 4.20) 11) Begin iteration loop 12) Loop over all model elements 13) Calculate change in internal heat generation vector AJfTQ} from last iteration (for the first iteration, this is the full vector) 14) Add element A{fTQ}' to global thermal load vector, {FT}' = {FT}'~' + A{FTQ}' 15) Solve for {T}; =[KT]-'{FT}' 16) Calculate {d}^ and {a}^ 17) Check convergence (Equation 4.30 and Equation 4.31) 18) If not CONVERGED GOTO 11 19) Calculate allowable thermochemical module time step based on convergence rate 20) RETURN 4.6 Model Verification This section outlines three simple case studies examined to verify thermochemical module predictions through comparison with exact solutions and previous analyses. Further verification of the kinetics portion of the module is provided in Section 6.1.7 and comparison of module predictions with experimental results is demonstrated in three test cases discussed in Chapter 7. Currently, applied heat fluxes cannot be specified so this vector is always zero. -73-Chapter 4: Thermochemical Module 4.6.1 Comparison with Exact Temperature Solution In the first verification test, C O M P R O transient temperature predictions were compared with exact solutions for a simple 1-D case. In this test, from Bogetti (1989), a non-curing slab initially at a uniform temperature of 0 \u00b0C is subjected at time t = 0 + to temperatures of 0 \u00b0C and 10 \u00b0C on its top and bottom surfaces respectively. The exact transient temperature solution for any point in the thickness o f the slab is given by: T(z,t) = 2T(z = 0)^sm(Bnx) (4.32) where B = \u2122 (4.33) and a (4.34) A s shown in Figure 4.4, for this test, a slab of thickness 0.1 m was examined using 10 elements through the thickness. The material properties used are shown in this figure and are provided in Appendix C . r=o\u00b0c T5 #-T3 k T2 \u00a5 \/ = 0.1 m Properties: k = 5 W\/mK p= 1000 kg\/m 3 CP= 1000 J\/kgK -> x T= 10\u00b0C Figure 4.4: 1-D transient temperature verification test. -74-Chapter 4: Thermochemical Module A comparison of the temperatures predicted by COMPRO with the exact solution at 5 points through the part thickness is shown in Figure 4.5. As shown in this figure, excellent agreement is obtained. 8 T Exact Time (minutes) Figure 4.5: Comparison of thermochemical module transient temperature predictions with exact solution for a 1-D non-curing slab. A second test of the thermochemical module was performed by comparing COMPRO resin degree of cure predictions at several isothermal temperatures with explicit MS EXCEL\u2122 spreadsheet solutions. For this test, COMPRO cure kinetics model #1 (Table B.2) was used by both COMPRO and EXCEL, with kinetics constants for AS4\/3501-6 taken from Lee et al., 1982 (see Table C.2). As shown in Figure 4.6, excellent agreement was obtained between spreadsheet calculations and COMPRO kinetics model predictions using a constant time step of At = 1 second. -75-Chapter 4: Thermochemical Module 0 20 40 60 80 100 120 Time (minutes) Figure 4.6: Comparison of thermochemical module isothermal cure rate predictions with exact solution (At = Is). 4.6.2 Comparison with Literature The final verification test performed was a comparison of COMPRO 1-D temperature and cure rate predictions with those of Bogetti (1989). In this test, the processing of a 2.54 cm thick laminate of AS4\/3501-6 is simulated. The finite element representation used for this case is shown inFigure 4.7. As shown in this figure, a total of ten elements were employed through the part thickness and a convection heat transfer boundary condition was applied to the top and bottom part surfaces. The material properties used for this test are outlined in Appendix C. Figure 4.8 shows the simulated autoclave air temperature cycle for this test and a comparison between COMPRO predictions and those of Bogetti. As can be seen, excellent agreement between the two models was obtained. The small discrepancy between degree of cure predictions early in the process cycle could be caused by errors in transcribing the data from the reference or a difference in the initial degree of cure -76-Chapter 4: Thermochemical Module used in the two simulations (Bogetti does not specify the initial degree of cure used, the COMPRO simulation used OQ = 0.01). AS4\/3501-6 1.27 cm Adiabatic I I Convection )^ (h\/k33\\Jf= 100 x Figure 4.7: Finite element representation of 2.54 cm thick AS4\/3501-6 laminate used in comparison of thermochemical module predictions with Bogetti (1989). 250 150 200 250 Time (minutes) 300 350 400 Figure 4.8: Comparison of COMPRO thermochemical module predictions with Bogetti (1989). Temperature and degree of cure at centreline of a 2.54 cm thick laminate using (Ji\/k)eff= 100. -77-Chapter 4: Thermochemical Module The COMPRO predictions shown in Figure 4.8 were obtained using the same constant time step of A? = Is used by Bogetti. Explicit numerical techniques, such as that of Bogetti, often require time steps of this order or smaller for numerical stability, especially for problems with large heat source terms. To examine the effect of time step on COMPRO predictions, simulations were run using a number of different specified maximum time steps, with the model allowed to dynamically modify step size in critical solution regimes. As shown in Figure 4.9, using maximum time steps as high as 100 seconds resulted in only small changes in model temperature predictions (degree of cure predictions were similarly effected). This demonstrates that the employed solution can not only remain stable, but also give very accurate predictions using far fewer time steps than explicit methods. 250 0 50 100 150 200 250 300 Process time (minutes) Figure 4.9: Effect of employed maximum time step on COMPRO temperature predictions. Temperature and degree of cure at centreline of a 2.54 cm thick laminate using (h\/k)eff= 100. 4.7 Summary and Discussion The current thermochemical model can be applied to the analysis of general two-dimensional cross-sections with multiple composite and non-composite materials including tooling. A total of six different -78-Chapter 4: Thermochemical Module types of cure kinetics equations are available for calculation of the cure rate of various types of matrix resins. An iterative backward-Euler solution technique employing a skyline matrix solver and adaptive time-stepping results in a stable, accurate and efficient solution. The use of a virtual autoclave simulation allows much better prediction of model boundary conditions than possible using traditional approaches. Still, the user should keep a number of limitations in mind. Employing a 2-D model limits the model to cases where out-of-plane temperature gradients are negligible. As discussed previously, this is not seen as a serious limitation since the majority of composite structures are relatively large in at least one dimension. One possible issue of concern is heat transfer coefficient variation within an autoclave. If this is significant then the 2-D assumption may be violated. Another simplification employed is that heat transfer within the structure is assumed to be purely conductive. This is probably a very good approximation except perhaps within honeycomb structures when internal free convection and radiation may play a role. Species transfer via either resin flow (convection) or diffusion is neglected. This is again a good approximation in most cases. The employed cure kinetics models should not be applied at temperatures well beyond those used to obtain the equation constants. Thus, models developed from high-temperature measurements should not be used to predict room temperature cure rates over long time periods. Despite the advances offered by the virtual autoclave simulation, boundary condition modelling remains imperfect. Even if average temperature and heat transfer coefficients are known, these values may vary significantly throughout the autoclave. The current model still has no method for predicting this variation let alone accounting for it in an analysis. The best approach at present is probably to assume uniform air temperature and determine local heat transfer coefficients using methods such as outlined in Section 6.2 and predict processing outcomes based on maximum and minimum values. -79-Chapter 4 : Thermochemical Module A final important limitation is the lack of a radiation heat transfer model. This is not likely to be an issue at the temperatures normally encountered in autoclave processing of thermosets (unless convective heat transfer is very poor), but may become important at higher temperatures such as encountered in thermoplastic composite processing. -80-5. Stress Module The stress module is responsible for prediction of process-induced strain and deformation throughout the structure of interest and the process tooling. Through integration of this module with analyses of component temperature and resin degree of cure (the 'thermochemical module' - see Chapter 4) and resin flow and fibre bed compaction (the 'flow module' - see Hubert, 1996), this module can examine all five major sources of process-induced stress and deformation identified from the literature (see Section 2.4.2). Four-noded plane strain isoparametric elements are employed, allowing modelling of complex 2-D cross-sections composed of multiple composite and non-composite materials including process tooling. Instantaneously linear elastic constitutive models assuming orthotropic material behaviour are employed for all types of materials. For non-composite materials, mechanical properties at any time during processing are calculated as linear functions of temperature. For composites the isotropic matrix resin is modelled as a 'cure-hardening\/instantaneously linear elastic' (CHILE) material. Calculated elastic constants are used in micromechanics models to determine transversely isotropic ply properties. Numerous techniques are employed to minimize required computational effort including use of an adaptive 'time-stepping' algorithm, using an efficient 'skyline' matrix solver, and allowing incorporation of multiple composite plies within each element. Coupled with recent advances in available computing power, these methods allow examination of structures of intermediate size and complexity using only a mid-range personal computer. The current chapter outlines the development of the stress module including fundamental equations, boundary conditions, and material properties. Also included is a discussion of the major assumptions employed, the solution algorithm used and application and limitations of the current module. 5.1 Fundamental Equations and Finite Element Solution This section introduces the fundamental equations of the stress module and outlines the development of the discretized system of equations by employing the finite element approximation. Provided detail is -81-Chapter 5: Stress Module limited since finite element techniques for linear elastic structural analyses such as employed for this problem are well known and are described in depth in numerous texts (e.g. Cook et al., 1989). A more detailed discussion of the discretization procedure and specifics of the solution algorithm are provided in Appendix B. As described in Section 4.1, a Galerkin weighted residual technique was employed to develop the discretized system of equations for the thermochemical module. For structural analysis, 'variational' or Rayleigh-Ritz techniques are typically employed. Using this approach requires development of a functional, which is an integral expression that, upon minimization, results in the governing equations and boundary conditions for the problem at hand. For structural mechanics problems, the most widely used functional is the expression for the potential energy of the system: where U is the strain energy of the system, and Q_,\u201e represents the work done by body forces and surface tractions during deformation. The complete potential energy expression for a given domain can be expressed as: up = u+n (5.1) (5.2) j{u}T{FB}dn-j{u}7{\u00ae}aT n r For the current plane problem, the various terms in the above equation are: {8} = {sx \u00a3z], the strain field [C] = {so} , {o-0} the material stiffness matrix (plane strain). initial strains and initial stresses {0} = (u} = {F B } = {u w}T, the displacement field {Fx Fz}, body forces {<&x <\u00a3>z], surface tractions {5} = nodal degrees of freedom (d.o.f.) of the structure -82-Chapter 5: Stress Module Using this approach, displacements at any point in an element are interpolated from those at the element nodes, {5}, using: u} = [N6]{5} where [N5] is the matrix of displacement interpolation or shape functions defined in Section B.4.1. (5.3) Employing a small displacement assumption, element strains can be calculated from the nodal displacements using the equation: e} = [B6]{5} where [B5] is the matrix of displacement shape derivatives defined as: [B 8] = [as][N8] The matrix [d 5] in Equation 5.5 is the 'differential operator', which for plane problems, is: (5.4) (5.5) [96] = d_ dx 0 o \u00a3 dz d_ d_ dz dx Substitution of the above expressions for {e} and {u} into Equation 5.2 yields: 1 nele nele n,=YZ{s}.rM.{8}.-Z{s}:[a z e=l e=l (5.6) (5.7) where nele is the total number of elements in the complete discretized structure and [k]e and {f}e are the element stiffness matrix and load vectors respectively, defined as: M\u00ab = j[Bs]7'[C][B6]JQ (5.8) {f}, = J[B5]7[C]{80}^Q-J[B6]7{a0}^Q+ J[N 6f[FB] JQ+J[N6f[0] ^ (5.9) n e n\u201e n, r e -83-Chapter 5: Stress Module Combining the load and stiffness matrices for all elements the potential energy expression for the complete structure is obtained: n,=i[8]r[K]{8}-{8}r{F} (5.10) where [K] and {F} are the structure (or 'global') stiffness and load matrices. The system of algebraic equations for the discretized problem is obtained from the potential energy expression in Equation 5.10 by employing the principle of stationary potential energy. Making n p stationary with respect to small changes in nodal displacements, Sh the equilibrium configuration of the system is defined by the n equations: ^ = 0 fori = 1,2, ,n (5.11) dS. where n is the number of degrees of freedom in the model. Performing this differentiation, we obtain the classical finite element equation {F} = [K]{5} (5.12) The set of nodal displacements for the complete structure can then be determined by solving: {8} = [K]-{F} (5.13) It should be noted that in practice Equation 5.13 is not solved through matrix inversion but rather via techniques such as Gaussian elimination. In all COMPRO solution modules, a skyline solution routine is employed to minimize both stiffness matrix storage requirements and solution time. 5.2 Boundary Conditions In its current form, the COMPRO stress module only allows specification of very simple boundary conditions. As elsewhere in the model, stress module boundary conditions are applied on a 'boundary' -84-Chapter 5: Stress Module basis rather than at individual nodes (although, of course, a boundary may consist of a single node: see Appendix A). Currently, the user may specify the following boundary conditions: \u2022 Fixed (prescribed zero nodal displacements). All nodal displacements are suppressed. This condition is normally applied to a single node to prevent free body motion of the structure. \u2022 Sliding (prescribed zero normal displacement). Boundary nodes are prevented from moving in a direction normal to the boundary. It should be noted that at least 2 nodes are required to reliably define a normal. \u2022 Free (no constraints, no nodal loads). This is the default boundary condition for all external nodes. It should be noted that different mechanical boundary conditions must be specified for the main solution and the 'tool removal' process (see Section 5.5). Also, since no external nodal loads may be applied in the current model, the global load vector is composed entirely of contributions from the element thermal and cure shrinkage strains. 5.3 Material Properties Three types of material properties are required by the calculations of the stress module: elastic constants, thermal expansion coefficients, and, for composite materials, cure shrinkage coefficients. Orthotropic material behaviour is assumed in all cases. All mechanical properties are re-calculated at the start of each stress module time step and assumed to remain constant during the time step. As discussed in Chapter 3, for these calculations, a uniform 'effective' element temperature and resin degree of cure are employed. For mechanical properties these 'effective' values are based on the average element temperature and degree of cure during the time step. All materials are assumed to be 'instantaneously linear elastic'. This means that, although the modulus may not be constant during processing, at any a given instant (i.e., during any time step), they exhibit -85-Chapter 5: Stress Module linear elastic behaviour. Thus, although there is no guarantee that cr = Es, it is always true that do~ = Eds, where E is the 'instantaneous modulus'. For non-composite materials, thermal expansion coefficients, CTE\\, CTE2, and CTE3 are calculated as linear functions of temperature, e.g., CTE^CTEm + aCTE](T-T0) (5.14) where CTE\\^) is the thermal expansion coefficient at T0 and aCTE\\ is the variation of CTE\\ with temperature. A similar linear variation with temperature is assumed for all nine orthotropic elastic constants (Eu, E22, \u00a3 3 3 , G]2, Gi3, G2h vn, vn, v23), e.g., Eu=Eum + aEU(T-T0) (5.15) For composite materials, thermal expansion coefficients may be determined in one of two ways. If 'lumped' properties are chosen, orthotropic ply thermal expansion coefficients can be calculated as linear functions of temperature and resin degree of cure, e.g., CTE, = CTEm + acn:i (T-T0) + bCTFI (a-a0) (5.16) If preferred, the user can specify ply and resin thermal expansion behaviour and transversely isotropic ply thermal expansion coefficients will be calculated using a continuous fibre micromechanics model1 (Bogetti and Gillespie, 1992) as described in Section B. 1.2.2. The isotropic matrix resin in composite materials is modelled as a so-called 'cure-hardening\/ instantaneously linear elastic' (CHILE) material. This designation indicates that the modulus of the instantaneously linear elastic resin increases monotonically with the progression of cure. Two models for prediction of resin modulus development are employed here, one from Bogetti and Gillespie (1992) and another developed as part of this work, as outlined in Section 6.1.5. The equations employed by these Currently, only micromechanics models for uni-directional plies are available in COMPRO. -86-Chapter 5: Stress Module models are outlined in Table B.5. The two available models for calculation of resin Poisson's ratio are outlined in Section B.1.10. Transversely isotropic fibre elastic constants are calculated assuming a linear variation with temperature only. Using as inputs fibre and resin elastic properties and instantaneous fibre volume fraction V\/, transversely isotropic ply elastic constants are determined using the Bogetti and Gillespie micromechanics model. 5.4 Solution Approach The solution technique used by the stress module can be described as an 'incremental instantaneously linear elastic plane strain analysis'. Using this approach, independent elastic analyses of solution change are performed during each stress module time step2, and the results 'summed' to determine the total solution at the end of the step. Thus, for example, the total nodal displacement at the end of module time step k is determined from the summation of calculated displacements at all previous steps, i.e., {S} = i>{5},. (5.17) The change in nodal displacements, A{5}, during a time step is determined using: A{5} = [K]-'A{F} (5.18) where A{F} is the change in the global force vector during the step and [K] is the global stiffness matrix, calculated from material elastic constants which are assumed to be constant during the step. In the current analysis, no external boundary loads can be applied to the structure. Thus, the change in the element nodal force vector for each step, A{f}<,, is composed entirely of contributions from element thermal and cure shrinkage strains, i.e., Recall that the stress module may not actually run during every model time step. - 8 7 -Chapter 5: Stress Module A{f}e = j[Bj[C]A{s0}dQ (5.19) where A{e0} is the change in element thermal and cure shrinkage strains during the step. For elements containing non-composite materials or composite elements with only one ply (or a portion of one ply) calculation of [k]e and A{f}c is quite straightforward. For these elements, material properties are uniform throughout (but not necessarily isotropic) and a standard Gaussian integration technique is employed using four integration points per element. 4 Figure 5.1: Integration points for calculation of element stiffness and load matrices for non-composite materials and elements containing only a single composite ply. Shown in both global (left) and element natural coordinates (right). For elements containing more than one composite ply, the calculation is slightly more complex. Since material elastic properties are discontinuous at ply boundaries, of which there may be any number in an element, a Gauss-Trapezoidal integration technique is employed. As shown in Figure 5.2, a total of four integration points (2 x 2) are employed for each composite ply, two in the standard Gauss locations in the element \u00a3 axis and one at each ply interface in the n direction. A more detailed discussion of the calculation of [k]c and A{f}c. for both composite and non-composite materials is provided in Appendix B. -88-Chapter 5: Stress Module Integration points for Figure 5.2: Integration points for calculation of element stiffness and load matrices for elements containing multiple composite plies. Shown in both global (left) and element natural coordinates (right). After solving for the change in element nodal displacements, A{8} in each step, the change in element mechanical strains, A{Ect}<,, and element mechanical loads, A{FCT}t, are determined as outlined in Sections B.3.5 and B.3.6. The element mechanical strains are defined as the difference between the total element strains and the element thermal and cure shrinkage strains. Similarly, the element mechanical loads are the difference between the total element nodal loads and the element nodal loads induced by internal thermal and cure shrinkage strains. Element stresses are not currently output by COMPRO since these values will be different for each ply in an element and indicating 'representative' stress values could be very misleading. At the completion of the process cycle, the stress module performs two additional calculations to simulate manufacturing processes which occur after the completion of the cure cycle. The first is a 'cool-down' step, in which the component temperature is uniformly reset to its initial value (usually room temperature), removing any elastic thermal strains remaining in the structure. The second step is a simulation of the 'tool removal' process whereby the modelled part and tooling are separated. This procedure is discussed in more depth in Section 5.5. Throughout the solution, the stress module dynamically adjusts the time step to optimize computational effort. As detailed in Section A.4, maximum time step size is calculated on the basis of the rate of -89-Chapter 5: Stress Module solution change (e.g. temperature or degree of cure) and the rate of change in mechanical properties. Using this approach, stress calculations are performed more frequently when the solution is changing quickly (e.g. the part is curing rapidly) and less frequently during periods when it is changing slowly (e.g. long temperature holds). 5.4.1 Module Solution Algorithm Similar to the thermochemical and flow modules, the stress module is called by the M A I N program (see Section A.7) at every time step of the problem solution. When called, the module checks whether a stress solution should be performed during the current step and the maximum size o f the next overall time step (see Section A.4) . The solution algorithm used by the module is as follows: 1) Calculate allowable stress module time step, AtSA (Section A.4) 2) Check i f stress module is to be R U N ; i f not, R E T U R N 3) Update mechanical properties (E,h E22, \u2022 \u2022 \u2022 vl2, CTEh CSCh Section B . l ) 4) Reset [K] and {F} 5) Loop over all model elements 6) Calculate element stiffness matrix [k] e and change in nodal load vector, A{f} B (Section B.3.4) 7) 8) 9) 10) Calculate change in element thermal and cure-shrinkage strains, A{eo} e (Section B . 3 . 5 ) Apply essential boundary constraints (i.e. fixed and sliding nodes) Assemble [k] e and A{f}\u201e into global matrices [K] and A{F} 11) 12) Solve for change in nodal displacements, A{8} = [K]\"'A{F} Calculate the change in element mechanical strains, A { e a } e (Section B . 3 . 5 ) Calculate the change in element mechanical loads {Fa}e (Section B . 3 . 6 ) 13) Calculate total nodal displacement {8} = ]T A{8}. (do not update node locations) 14) R E T U R N -90-Chapter 5: Stress Module 5.5 Tool removal simulation At the end of a process cycle, the manufactured structure will typically be removed from the autoclave, still affixed to any process tooling, and allowed to cool to room temperature. At this point, depending on the case, large residual stresses might still exist between part and tooling, sometimes making separation quite difficult. Consequently, the stress state in the processed structure, and thus the residual deformation, will be quite different before and after tool removal. The situation in the simulated process cycle is quite similar: at the end of the process cycle simulation, the tooling and processed structure remain attached with residual stresses remaining between them. Thus, to predict the 'final' post-processed component shape and residual strain state, a simulation of the tool removal process is undertaken. The tool removal simulation is performed in a single elastic step and the calculated change in displacements added to the pre-tool-removal displacements to give the final component shape. The procedure used is as follows: 1. Create a finite element description of the tool removal problem. The new F.E. mesh is identical to that used in the main stress module simulation, except that the elements identified by the user as 'tooling' are not included. A new set of displacement boundary conditions is also required since the part will see different boundary conditions during tool removal than during the process cycle. 2. Assemble the tool removal global stiffness matrix, [ K T R ] , and set the tool removal global force vector, {F T R } , to zero. 3. Add the tooling element nodal forces, {fa}e, at all nodes along the tool\/part interface to the tool removal global force vector. Since these forces will be the negative of the forces which the tool applied to the part (from equilibrium), the effect of adding these forces is to force a stress-free interface. 4. Solve for the part displacements during tool removal ({5 T R } = [K T R ]\" ' {F T R }) -91-Chapter 5: Stress Module 5. A d d the calculated displacements to those calculated prior to the tool removal simulation to determine the post-tool-removal part shape. This procedure is illustrated schematically in Figure 5.3. a) Remove tooling elements 1 \u00ab -1 -T < -Residual tool force b) \\T - y e o f t^i * Residual Y tool force _<_I Add negative of Tool-Part forces c) d) Figure 5.3: Schematic of the tool removal process: a) prior to tool removal, part and tool in equilibrium, part conformed to tool shape; b) tooling removed, residual tool\/part interface forces remain; c) add negative of interface loads to obtain stress free interface; d) predicted part shape after tool removal 5.5.1 Tool Removal Algorithm The algorithm used in the tool removal simulation is as follows: 1) Generate the tool removal ID array and skyline vector 2) Initialize the global tool removal stiffness and nodal force matrices, [KT R] and {FTR} 3) Loop over all non-tooling model elements 4) Calculate element stiffness matrix [k]e (Section B.3.4) 5) Apply essential boundary constraints (i.e. fixed and sliding nodes) 6) Assemble [k]e into global matrix [KT R] -92-Chapter 5: Stress Module 7) Add all tooling element nodal loads to {FTR} 8) Solve for the tool removal nodal displacements, {8TR} = [KTR]\"'{FTR} 9) Add tool removal displacements to total pre-tool removal displacements 5.6 Model Verification In this section, verification of the current element formulation and its implementation in COMPRO is outlined using three simple test cases. Demonstration of module predictions is also provided in case studies examined in Chapters 6 and 7. 5.6.1 Patch Test The first case examined was a simple patch test, illustrated in Figure 5.4. This test, from Cook et al. (1989) employs a rectangular structure composed of four, four-noded elements sharing a common internal node, loaded such that a uniform stress state is induced. Just enough nodal constraints are employed to prevent free-body motion. For the case of plane strain elements, three different loading cases are required, one for each ax, az, and ixz. Figure 5.4: Schematic of patch test for four noded plane strain elements (from Cook et al., 1989). Illustrated are the boundary and nodal load conditions for the 'uniform cr* case. In all three load cases examined, a uniform stress of 1 MPa was applied to an isotropic material with\u00a3 = 1 GPa and v= 0.3. As shown in Table 5.1, the COMPRO predictions are in nearly perfect agreement with the exact solutions for element strain and the total nodal force at the internal node (node 5). This test -93-Chapter 5: Stress Module thus confirms element compatibility and ensures that the numerical solution will converge to the exact solution for the isotropic case. Table 5.1: Summary of COMPRO predictions for patch test cases. F5x and FSz are the summation of element nodal loads at node 5 (the internal node). COMPRO Predictions (Exact results) Load Case sz Yxz Fsx(N) F5z(N) ax = 1 MPa -3.90 x 10\"4 -3.90 x 10\"4 = 0 = 0 = 0 (-3.90 x1a\") (-3.90x10-\") (0) (0) (0) CJz = 1 MPa -3.90 x 10\"4 -3.90 x 10\"4 = 0 = 0 = 0 (-3.90x10\"') (-3.90x1 OT1) (0) (0) (0) rx: = 1 MPa = 0 = 0 2.60 x 10\"4 = 0 = 0 (0) (0) (2.60 x Iff4) (0) (0) Note: = 0 indicates a value of < 10\"s 5.6.2 Thermal Anisotropy Springback As discussed in Section 2.4.2, an important source of residual deformation in curved composite sections is thermal strain anisotropy. The second verification test demonstrates 'springback' in a simple 90\u00b0 angle shape made of a material with anisotropic thermal expansion coefficients. A number of different case studies are performed in order to: \u2022 Compare model predictions with exact solutions \u2022 Examine the effect of mesh density on springback \u2022 Examine the effect of composite laminate layup on springback \u2022 Demonstrate the validity of the employed integration approach for elements with multiple plies The part geometry and nodal constraint conditions employed in all examined cases is shown inFigure 5.5 along with the finite element mesh for the 'nominal' case. In all cases, the part inner surface is used as the region 'reference boundary'. Thus, the material '1' direction is parallel to this boundary and the -94-Chapter 5: Stress Module material '3' direction is perpendicular to it (the '2' direction is perpendicular to the plane of interest). In all cases, a total temperature difference of 180 \u00b0C is applied. 1 mm 5 mm Reference boundary \u2022 R, = 1 mm 0\u00b0 AT= 180 \u00b0 C 3 3 Figure 5.5: Geometry employed for the thermal anisotropy springback verification test. Note that the finite element mesh shown is for the nominal case only. The first case examined is a material with thermal expansion coefficients of CTE\\ = 0, CTE2 = 0, and CTE?, = 100 x 10\"6 \/\u00b0C. As outlined in Section 2.4.2, the exact springback angle for this case can be calculated using: AG=&\\ (CTE.-CTEAAT 1 + CTE AT (5.20) where 0 is the 'included' angle, which is 90\u00b0 in this case. For aAJof 180 \u00b0C, this results in a calculated springback angle of-1.591\u00b0 (note that the convention used here is that positive springback corresponds to a decrease in included angle). -95-Chapter 5: Stress Module In the case chosen as 'nominal', a finite element mesh of 6 x 8 elements (radial x circumferential) was employed. As illustrated in Figure 5.6, the total springback angle predicted by COMPRO for this case is -1.616\u00b0, representing a difference of approximately 1.55% from the exact solution. Figure 5.6: Predicted shape of anisotropic angle after a temperature change of A7of 180 \u00b0C. Case: CTEX = 0, CTE2 = 0, and CTE3 = 100 x 106 \/\u00b0C, 6 elements in radial direction, 8 in circumferential direction. All displacements exaggerated by a factor of 10. The predicted variation of springback angle with temperature as the part is heated from 20 \u00b0C to 200 \u00b0C, then cooled, is shown in Figure 5.7. The paths during heating and cooling are shown to be identical, and the predicted residual springback was approximately zero (<10~14 degrees) as expected. -96-Chapter 5: Stress Module \u2014 i 1 1 1 1 1 1 1 1 1 1 0 20 40 60 80 100 120 140 160 180 200 Temperature (\u00b0C) Figure 5.7: Predicted variation of anisotropic angle springback with temperature during heating and cooling. Case: CTEX = 0, CTE2 = 0, and CTE3 = 100 x 10 6 \/\u00b0C, 6 elements in radial direction, 8 in circumferential direction. The effect of mesh density on predicted springback angle for this case was then examined by varying the number of corner elements in the radial and circumferential directions. As shown in Table 5.2, at least 5 elements in the radial direction are required to obtain good springback predictions for this case. Another interesting observation is that the predicted springback angle is largely independent of the number of circumferential elements used. This is not expected to be true in cases where other deformation mechanisms such as tool\/part interaction are important. -97-Chapter 5: Stress Module Table 5.2: Effect of mesh density on COMPRO springback predictions after a temperature change of AT= 180 \u00b0C. Case: CTEX = 0, CTE2 = 0, and CTE3 = 100 x 106 \/\u00b0C. Number of circumferential elements Number of radial elements Predicted springback angle (degrees) Prediction error (%) 0* 6 -1.031 35.2 1 3 -0.757 52.4 2 2 -1.427 10.4 2 4 -1.428 10.3 2 6 -1.429 10.2 3 6 -1.545 2.92 5 6 -1.596 0.27 8 6 -1.616 1.55 20 6 -1.618 1.70 20 10 -1.618 1.70 In this case the corner was modelled as a sharp 'L' shape. The next set of analyses examines springback of the same geometry, composed this time of a number of cured AS4\/8552 laminates. These analyses demonstrate the effect of composite layup and number of plies per element on predicted springback angle. In this case, constant ply thermal expansion coefficients of CTE] = 0.6 x IO\"6 \/\u00b0C, CTE2 = 28.6 x IO\"6 \/\u00b0C, and CTE3 = 28.6 x IO\"6 \/\u00b0C were used, (other composite mechanical properties are indicated in Table C.4 in Appendix C). As shown in Table 5.3, as expected from theory, the [90]n laminates exhibit no springback whatsoever due to their (assumed) isotropy in the plane of interest. The greater predicted springback of the [0]n laminates as compared to the quasi-isotropic laminates ([0\/45\/90\/-45]s and [0\/45\/90\/-45\/-45\/90\/45\/0]s) is initially surprising since the difference between circumferential and radial thermal strains is normally greater for the latter layup due to Poisson effects (Stephan et al., 1996; Fahmy and Ragai-Ellozy, 1974). The opposite is found in this case due to the use of a plane strain formulation which suppresses both the -98-Chapter 5: Stress Module large out-of-plane thermal strains of the [0]n laminates and the much smaller strains of the quasi-isotropic laminates in this direction3. Table 5.3: C O M P R O springback predictions for various cured AS4\/8552 laminates subjected to a temperature change of AT= 180 \u00b0C. Laminate lay up Number of plies per element Predicted springback angle (degrees) [0] 0.167 -0.6578 [0]io 1.67 -0.6580 [90] 0.167 0.0000 [90] 1 0 1.67 0.0000 [0\/45\/90\/-45] s 0.5 -0.5850 [0\/45\/90\/-45] s 1.00 -0.5855 [0\/45\/90\/-45] s 2.00 -0.5852 [0\/45\/90\/-45\/-45\/90\/45\/0] s 4.00 -0.5857 The number of plies per element is shown to have very little effect on predictions. This is especially significant considering that a Gaussian integration technique is used when there is only one ply in an element and a Gauss-Trapezoidal integration technique is employed when an element contains multiple plies. 5.6.3 Bi-material Strip In the final set of verification tests, model predictions in the case of bonded materials are examined through the use of a curved 'bi-material' strip subjected to thermal and point loads. The geometry examined and the finite element representation used in all tests is illustrated inFigure 5.8. As shown in the figure, two semi-circular regions are defined for this case, with region reference boundaries specified on the inner and outer radii of the structure. In each test, the predicted deflection at node 1 of the F.E. It should be noted that this is not, in fact, an error in the formulation, but is rather a necessary consequence of using a plane strain assumption. -99-Chapter 5: Stress Module mesh (see Figure 5.8) is examined when the structure is subjected to either a point loadFx of 10000 N at this node or a temperature change of AT= 100 \u00b0C. A total of 10 radial elements and 100 circumferential elements are employed for all cases. z Figure 5.8: Geometry employed for the bi-material strip verification test. In all cases, 10 elements (in total) are employed in the radial direction and 100 in the circumferential direction. In the first set of tests, COMPRO predictions are compared with predictions of the commercial finite element code ANSYS\u2122 for a bi-material strip of aluminum (region 1) and invar (region 2). As shown in Table 5.4, excellent agreement is obtained between the predictions of the two finite element codes for both loading cases. COMPRO predictions were also examined for the case of a bi-material strip composed of aluminum (region 1) and two unidirectional [0]n laminates (region 2) of cured AS4\/8552. Comparisons of deflection predictions for the case of a [0] laminate and a [0]2o laminate in Table 5.4 again show excellent agreement, providing further evidence for the validity of the multiple-ply integration technique. -100-Chapter 5: Stress Module Table 5.4: COMPRO and ANSYS predictions for bi-material strip loading cases. Shown displacements are for node 1 (see Figure 5.8). COMPRO ANSYS Region 1 Material Region 2 Material F ( N ) Ar(\u00b0c) 8X (mm) 8Z (mm) 8X (mm) 8Z (mm) Aluminum Invar 0 100 9.581 15.26 9.582 15.26 Aluminum Invar 10000 0 18.99 25.09 18.98 25.08 COMPRO [0] COMPRO [0]20 Region 1 Material Region 2 Material F ( N ) A r ( \u00b0 c ) Sx (mm) b\\ (mm) 8X (mm) 8Z (mm) Aluminum AS4\/8552 0 100 10.68 17.04 10.67 17.03 Aluminum AS4\/8552 10000 0 22.17 29.13 22.15 29.11 5.7 Summary and Discussion The developed stress module can be applied to the prediction of process-induced strain and deformation throughout autoclave processing in complex 2-D cross-sections with multiple composite and non-composite materials. An incremental instantaneously linear elastic plane strain formulation is employed in which orthotropic behaviour is assumed. Via the integration of this module with other analyses for heat transfer and resin cure, and resin flow, all five major stress development mechanisms outlined in the literature can be examined. Several strategies have been employed to reduce computational effort and optimize solution efficiency including: adjusting the frequency of stress module calculations to account for the rate of solution change, employing a skyline matrix solver, and incorporating of multiple plies in each composite material element to reduce the number of elements required to model a given problem. These optimization strategies, combined with the tremendous increase in available desk-top computing power in recent years, permit examination of process-induced deformation in structures of practical size and complexity using even mid-range personal computers. However, as with any model, some important limitations of the current analysis should be kept in mind. -101-Chapter 5: Stress Module \u2022 Use of a 2-D plane strain analysis limits the model to applications where the out-of-plane dimension is both large and relatively uniform in the region of the examined plane. While this is true for most regions of most composite structures, this limitation should always be kept in mind. \u2022 Also, the relatively small size of the elements required to adequately examine most phenomena means that very large structures cannot be examined at one time. For such structures, a sub-structuring method outlined in Chapter 7 is proposed. \u2022 COMPRO does not contain failure models nor does it compute stresses, although both of these options may be implemented if required. \u2022 Plies are always assumed to correspond to the element natural co-ordinate (cj -1|) axes and plies in a region are always assumed to be 'dropped' from the top of the ply stack. The latter of these limitations is probably not difficult to address but the former would require a modified element integration strategy. \u2022 There is some doubt as to the validity of standard micromechanics models, such as the one used here, at low degree of cure due to poor load transfer when the matrix is very weak (White and Hahn, 1990, 1992b). Also, the current model includes only models for unidirectional plies. \u2022 The use of an incremental instantaneously elastic analysis prevents direct modelling of stress relaxation. Employing appropriate elastic constants (see Section 6.1.5) can account for most of this stress relaxation in many cases, but not in all. An example of a situation where model predictions are not to be trusted is when part temperature is increased from below instantaneous Tg to significantly above. \u2022 The properties required by the stress model for composite materials are numerous, not generally available, and somewhat difficult to obtain and analyze. As will be discussed in Chapters 6 and 7, however, there are numerous possible ways of reducing materials characterization efforts. -102-6. Material and Boundary Condition Characterization No matter how rigorous and inclusive, the usefulness of a process model as an engineering tool is limited without accurate representations of material behaviour and boundary conditions. One major criticism of process modelling is that obtaining these model inputs is often impractical due to both the volume of tests required and the difficulty of performing many of these tests. There are a number of reasons why material characterization for autoclave process models need be so extensive: \u2022 The large number and diversity of important phenomena in processing. To simulate these phenomena, a correspondingly large number and variety of properties are required. For the thermochemical and stress modules alone, the properties required include: density (p), specific heat capacity (Cj>), thermal conductivity (k), coefficient of thermal expansion (CTE), mechanical properties (E, v), resin cure kinetics and resin cure shrinkage behaviour. \u2022 The range of states over which properties must be known. For non-composite materials, characterization over the whole range of processing temperatures is required (e.g., 20 \u00b0C to 180 \u00b0C). For composite materials, characterization is also needed over a range of resin cure states (a = 0 to 1) and over a range of fibre Vj. The viscoelastic nature of resin mechanical response during much of the process cycle means that behaviour is also time dependent. \u2022 Composite material anisotropy. Laminated composite materials are usually anisotropic. Prepreg plies are typically (approximately) transversely isotropic or orthotropic. Even in the simpler of these cases, two constants are required to describe material conductivity, thermal expansion and cure shrinkage strains and five constants are needed to describe mechanical response. \u2022 The large number of types of materials in a model. Many composite structures are composed of a large number of different materials such as different types of prepreg, adhesives stiffeners, and sandwich cores. Properties for all of these materials as well as tooling and bagging may be required. -103-Chapter 6 : Material and Boundary Condition Characterization Characterization of component and tooling boundary conditions is also complex. Although some important issues are addressed in the current model via the virtual autoclave (e.g., simulation of air temperature\/pressure with time), characterization of autoclave heat transfer rates is far from trivial. As will be discussed in Section 6.2, non-uniform autoclave air flow can result in significant variations in temperature and effective heat transfer coefficients within an autoclave. Fortunately, there are a number of ways of reducing the required characterization effort for a given problem: \u2022 Make extensive use of published properties data. This is most applicable to non-composite materials such as substrate tooling which is often composed of a well-characterized material such as aluminum or invar. Even for most composite materials, basic properties information (e.g., cured density, specific heat capacity) is usually available from material producers or technical literature sources. Using literature data for similar well-characterized composites such as Hercules AS4\/3501-6 can also reduce required characterization effort. \u2022 Employ micromechanics models. Examining the properties of a composite's constituent fibre and resin separately, then using various models to predict overall composite behaviour, has several advantages. For one, since many composites use the same or similar fibres, often only the properties of resin need be re-examined in a new material. Also, this approach can eliminate the need to measure composite properties over a range of fibre Vj. It should be remembered, however, that standard models for laminate mechanical behaviour are suspect when the matrix is very weak (White and Hahn, 1990, 1992b). \u2022 Conduct sensitivity analyses. As will be outlined in Chapter 7, sensitivity analyses can be used to identify inputs which warrant detailed characterization and those which have little effect on process model predictions. -104-Chapter 6: Material and Boundary Condition Characterization This chapter outlines material and boundary condition characterization performed as part of the current work. Materials characterization is focused exclusively on properties related to the thermochemical and stress modules. Resin flow-related properties are discussed in Hubert (1996). 6.1 Materials Characterization The materials characterization work discussed here focuses exclusively on Hercules AS4\/8552, a uni-directional carbon-fibre epoxy prepreg material used in all three case studies discussed in Chapter 7. Properties examined here are: density, specific heat capacity, conductivity, thermal expansion, resin cure kinetics, resin cure shrinkage, and the development of resin modulus. A number of important assumptions underlie the experimental work and are fundamental to the current process model as a whole. It is assumed, for example, that all resin properties can be described completely in terms of its instantaneous temperature and degree of cure. Thus, the effect of such things as cure path dependence, moisture and volatile content are currently assumed negligible. It is also assumed that the micromechanics models used to relate fibre and resin properties to composite properties are valid for the entire range of cure state. Prepreg ply properties are also assumed to be transversely isotropic (2-3 plane is isotropic). 6.1.1 Density The material density (p) is a scalar quantity used in the calculations of the thermochemical module. As outlined in Appendix B, density is calculated in COMPRO as a linear function of temperature for non-composite materials and as a linear function of temperature and degree of cure for composite materials and resins. Given the fibre and resin densities (p\/, pr), the composite density for a given fibre volume fraction (V\/) is calculated using the rule of mixtures as: P c = VfPf + {\\-Vf)p, (6.1) -105-Chapter 6: Material and Boundary Condition Characterization 6.1.1.1 Previously Available Information Density is relatively easy to measure and experimental values are available for most composite and non-composite materials over a range of temperatures. Data as a function of degree of cure has been measured previously for MY720 CIBA-GEIGY by Mijovic and Wang (1988), but is not generally available for most systems. Resin density as a function of cure time has been measured by numerous other researchers (Lam and Piggott, 1989; Armstrong et al., 1986) but no correlation with degree of cure is generally made. For the studied material, AS4\/8552, the following information was available from Hercules Aerospace Company product data sheets (847-6 and H050-377): pc = 1.58x103 kg\/m3 @ r=20\u00b0C, a= 1.0 Pf = 1.79 xlO3 kg\/m3@r=20 \u00b0C Pr= 1.30 xlO3 kg\/m3 @T= 20 \u00b0C, a= 1.0 6.1.1.2 Tests Performed No density measurements were performed as part of this work since density over a range of temperature and degree of cure can be extracted from the performed thermal expansion and resin cure shrinkage tests. Examples of techniques employed for measurement of resin density are outlined in Mijovic and Wang (1988), Armstrong et al. (1986) and Russell (1993). Mijovic and Wang interrupted the cure process at several points and measured resin density employing a density gradient column in accordance with ASTM D1505. Armstrong at al. and Russell continuously measured resin density during the cure process using a volumetric dilatometer. -106-Chapter 6: Material and Boundary Condition Characterization 6.1.2 Specific Heat Capacity The material specific heat capacity1 (CP) is a scalar quantity used in the calculations of the thermochemical module. As outlined in Appendix B, specific heat capacity in COMPRO is calculated as a linear function of temperature for non-composite materials and as a linear function of temperature and degree of cure for composite materials and resins. If fibre and resin specific heat capacities (CP\/, Cj>r) and densities are known, the value for the composite material for a given Vjis calculated using the equation: CPc =(vfPfCPf + (\\ - Vf)prCPl)\/(vfPf+(\\ - Vf)pr) (6.2) 6.1.2.1 Previously Available Information Specific heat capacity data is available for most common materials, often at temperatures other than room temperature. Composite CP as a function of resin degree of cure has been measured by Mijovic and Wang (1988) but is not generally available for most composite materials and resins. For AS4\/8552, it was known from Hercules Aerospace Company Product Data Sheet 847-6 that the fibre specific heat capacity could be described by: Cpf = 750 J\/kgK + 2.05 J\/kgK\u00b0C * T(\u00b0C) This information was used in combination with data from previous tests on fully cured prepreg (Boeing, 1993a) to calculate fully-cured resin specific heat capacity as a function of temperature: CPr= 931 J\/kgK + 3.74 J\/kgK\u00b0C * 7T\u00b0C), a= 1.0 6.1.2.2 Tests Performed Previous measurements of resin and composite specific heat capacity by Mijovic and Wang (1988) show that this property can change considerably (more than 30%) during processing. Due to the magnitude of Often referred to more simply as 'specific heat'. -107-Chapter 6 : Material and Boundary Condition Characterization this change, it was decided to examine the effect of temperature on uncured AS4\/8552 specific heat capacity to add to the data on fully cured material. A commonly used technique for measurement of specific heat capacity (Mijovic and Wang; Campbell and Burleigh, 1982), and the method employed here, is to use differential scanning calorimetry (DSC) to measure the amount of energy required to increase specimen temperature by AT. As shown in Table 6.1, measurements were made on three different specimens over five temperature ranges from -40 \u00b0C to 125 \u00b0C using a Perkin-Elmer Tas 7 DSC. No data was obtained for higher temperatures as the increasing rate of heat given off by resin cure would interfere with heat flow measurements. 6.1.2.3 Results Test results for the three specimens are shown in Table 6.1 and Figure 6.1. Table 6.1: Specific heat capacity measurements on uncured AS4\/8552. Temperature Range (\u00b0C) Specimen #1 (J\/kgK) Specimen #2 (J\/kgK) Specimen #1 (J\/kgK) Average (J\/kgK) -40 to -20 1046 812 791 883 -20 to 20 1573 1301 1322 1399 20 to 60 1594 1477 1423 1498 60 to 100 1523 1351 1418 1431 100 to 125 1536 1439 1439 1471 -108-Chapter 6: Material and Boundary Condition Characterization 1600 1500 1400 -900 1 I 800 - .. 700 -| 1 1 1 ! 1 1 1 1 1 1 -25 -5 15 35 55 75 95 115 135 155 175 Average temperature fC) Figure 6.1: Effect of temperature on specific heat capacity of fully cured and uncured AS4\/8552. As shown in Figure 6.1, the specific heat capacity of the uncured composite appears to increase sharply with temperature at first, then remain nearly constant. The initial increase is likely due to vitrification of the resin as it passes through its glass transition temperature somewhere near 0 \u00b0C (this depends on the initial degree of cure). Thus, during the first measurement, the resin is below Tg and during the last 3 it is above Tg (the second measurement may partly straddle the transition). The data for the fully cured composite shows no such sudden transition since even the maximum testing temperature is well below the cured resin Tg. 6.1.2.4 Analysis Above Tg, the composite specific heat capacity is assumed constant, with a value of 1465 kJ\/kgK. Factoring out the fibre specific heat capacity, resin specific heat capacity as a function of temperature ata = 0.0 was determined to be: CPr= 2790 J\/kgK - 3.80 J\/kgK\u00b0C * T(\u00b0C) for T> Tg -109-Chapter 6: Material and Boundary Condition Characterization Below Tg, only a single data point was available (Ci>c = 880kJ\/kgK), so the effect of temperature in this regime could not be determined. Assuming the relative change in resin CV with temperature is the same below Tg regardless of degree of cure, CFr can be calculated as: CPr= 1300 J\/kgK + 4.29 J\/kgK\u00b0C * T(\u00b0C) - 369 * a for T< Tg 6.1.2.5 Discussion These results show that the change in composite specific heat capacity during processing is quite large. However, it should not automatically be assumed that the effect of this change on model predictions will be similarly large. In fact, as will be demonstrated in Chapter 7, since component thermal mass is normally small compared to that of the tool, specific heat changes may not be nearly as important as other process parameters. The observed effect of resin vitrification on composite specific heat capacity (a reduction of nearly 30%) should not be unexpected given the change in material state at this point. That this behaviour was not noted by either Mijovic and Wang (1988) or Campbell and Burleigh (1982) is probably attributable to their tests being done below Ts. This was partly to avoid influencing the DSC measurements by curing the resin. In the actual process, however, the resin temperature is usually above Tg for much of the process cycle so that it will cure. Unfortunately, an insufficient amount of testing was done here to establish either the magnitude or trend of specific heat capacity with temperature aboveTg nor the effect of degree of cure on this value. Clearly, the current linear equations for specific heat capacity used by COMPRO are insufficient to account for glass transition effects. After further verification of the observed trends is performed, a more complex model accounting for vitrification should be developed. 6.1.3 Thermal Conductivity The material thermal conductivity (ky) is a vector quantity, used in the calculations of the thermochemical module. Orthotropic conductivity is assumed for all materials so that values of^n, \u00a322, and \u00a333 need be -110-Chapter 6: Material and Boundary Condition Characterization specified. In COMPRO, these conductivities are calculated as linear functions of temperature for non-composite materials, and linear functions of temperature and degree of cure for composite materials and resins. Since it is a vector quantity, it is especially useful to analyse composite conductivity by examining fibre and resin properties separately. As outlined in Appendix B, the longitudinal conductivity of unidirectional prepreg can be predicted from fibre and resin properties using a simple 'rule-of-mixtures'. A number of different models are available for predicting transverse conductivity (\u00a322, o n e \u00b0f m e most popular being that of Springer and Tsai (1967), a corrected version of which (Twardowski et al., 1993) is shown in Equation B.70. 6.1.3.1 Previously Available Information Thermal conductivity of a number of resins is available in the literature. Mijovic and Wang (1988), found that thermal conductivities of epoxy resins reported in the literature range from about 0.17 W\/mK to 0.29 W\/mK. A high temperature dependence was reported in all cases. Less conductivity data is available for carbon fibres and values reported in the literature vary over a wide range due to the high dependence of this property on fibre structure. Butler et al. (1973) reported longitudinal thermal conductivity values (kn) of 102 W\/mK for HMS (High Modulus) fibres and 22 W\/mK for HTS (High Tensile Strength) fibres. Values cited for transverse carbon fibre conductivity show a similar range, varying from 2.39 W\/mK (Hasselman et al., 1993) to 26 W\/mK (Loos and Springer, 1983), with most values reported for HTS fibres tending toward the lower end of the range. Using data from previous transverse conductivity measurements on fully cured AS4\/8552 (Boeing, 1993b), and the tests described in this section, it was determined that fibre \/C22 in this case was near the lower end of the literature range. Choosing a value of 2.50 W\/mK (at 20 \u00b0C) from Behrens (1968), fully cured resin conductivity was estimated to be: kr= 0.199 W\/mK + 3.7xl0\"4 W\/mK\u00b0C * 7\/(\u00b0C) -111-Chapter 6: Material and Boundary Condition Characterization 6.1.3.2 Tests Performed While useful as a starting point, available experimental data provided no information regarding fibre longitudinal thermal conductivity, fibre transverse conductivity dependence on temperature, or resin conductivity dependence on degree of cure. As shown by Bogetti and Gillespie (1991) and Mijovic and Wijaya (1989), internal temperatures during processing can be sensitive to composite conductivities. These values are especially important for thick laminates and complex shapes. As part of this study, two sets of conductivity tests were carried out: 1. Composite through-thickness conductivity (\/cc33) was measured on both fully cured and uncured specimens. The cured resin measurements were compared with results obtained from previous tests to verify technique and material consistency. The analysis of uncured resin conductivity would provide information regarding the dependence of this value on degree of cure. 2. Composite conductivity in the longitudinal direction (kcU) was also measured for fully cured specimens. From this test, fibre longitudinal conductivity (\/cy,,) and its dependence on the temperature would be estimated. All specimens were tested in accordance to ASTM E 1225 using 5 cm diameter specimens. As shown in Figure 6.2, in this test the specimen is sandwiched between two 'reference' specimens of known conductivity and a temperature difference induced across the assembly (e.g., T\\ > T4). The specimen conductivity was calculated by comparing the temperature drop across it with that across the standards. For the cured specimen tests, average specimen temperatures ranged from about 50 \u00b0C to 200 \u00b0C. A lower temperature range was used for uncured specimens -50 \u00b0C to 5 \u00b0C to prevent the resin from chemically advancing during the test. -112-C h a p t e r 6: M a t e r i a l a n d B o u n d a r y C o n d i t i o n C h a r a c t e r i z a t i o n 5.1 cm Reference Specimen Reference Figure 6.2: Schematic of thermal conductivity test. For kcii measurements, Pyrex 7740 standards were used as reference specimens, with Pyroceram 9960 used for kcU tests. For through-thickness measurements, specimen manufacture was relatively simple, involving cutting and stacking circular sections until an acceptable thickness was obtained. Manufacture of the longitudinal conductivity specimens was more difficult. As shown in Figure 6.3, to make these specimens, a cured panel was sliced into 1.9 cm inch strips which were rotated 90\u00b0 and bonded with an epoxy adhesive. Cut laminate into strips Rotate 90 degrees Bond strips Figure 6.3: Assembly of composite longitudinal conductivity (kcn) test specimen from a unidirectional laminate. 6.1.3.3 Results The measured conductivities are shown in Figure 6.4 and Figure 6.5. Each data point represents the average values for the two specimens. From the figures, several interesting observations can be made. First, curing of the resin results in a small but consistent increase in composite conductivity. At the test temperatures used, this observation is not likely attributable to the effects of resin vitrification. As expected, composite thermal conductivity in the longitudinal direction was found to be almost an order of magnitude higher than in the transverse direction. This is because ku is dominated by the fibre -113-Chapter 6: Material and Boundary Condition Characterization conductivity while &33 is largely controlled by the much lower conductivity of resin. In all cases, the relationship between measured thermal conductivity and temperature was found to be very nearly linear. 0.80 0.75 0.70 2 0.65 E ~ 0.60 '> I 0.55 \u2022a s J 0.50 | | 0.45 0.40 0.35 0.30 -50 Cured Resin Uncured Resin kc33 = 0.432 + 5.66xl0\"4* T 50 100 Average specimen temperature (\u00b0C) 150 200 Figure 6.4: Measured AS4\/8552 transverse thermal conductivity (kc33) with cured and uncured matrix resin. -114-Chapter 6: Material and Boundary Condition Characterization 7.0 - r 6.5 I 20 40 60 80 100 120 140 160 180 200 Average specimen temperature (\u00b0C) Figure 6.5: Measured AS4\/8552 longitudinal thermal conductivity (kcU) with cured matrix resin. 6.1.3.4 Analysis From the experimental measurements, expressions were developed for the resin and fibre conductivities using either the Springer-Tsai conductivity equation (transverse direction) or a rule-of-mixtures (longitudinal direction) equation. The assumed V\/for all calculations was the measured 'nominal' value of 0.573. For all calculations it was assumed that the relative fibre conductivity change with temperature was the same in the longitudinal and transverse directions. The expressions developed for resin and fibre conductivities are as follows: kr= 0.148 W\/mK + 3.43x10\"4 W\/mK\u00b0C * T(\u00b0C) + 6.07xl0\"2 W\/mK * a kfl3 = 2.40 W\/mK + 5.07x10\"3 W\/mK\u00b0C * T(\u00b0C) * \/ i , = 7.69 W\/mK + 1.56x10\"2 W\/mK\u00b0C * T(\u00b0C) -115-Chapter 6: Material and Boundary Condition Characterization 6.1.3.5 Discussion A large change in composite thermal conductivity was observed over the range of temperature and resin degree of cure encountered during processing. Under these conditions, AS4\/8552 transverse conductivity (\/cc33) was found to increase by about 68% and longitudinal conductivity (7ccl) by about 32%. Changes of similar magnitude were also observed by Mijovic and Wang (1988) and suggest the importance of modelling the effect of temperature and degree of cure on conductivity during processing. It is suggested that any future tests use neat resin instead of prepreg. This would have the advantage of reducing errors due to uncertainty in and making specimen preparation easier. 6.1.4 Coefficient of Thermal Expansion The material coefficient of thermal expansion (CTE,) is a vector quantity, used in the calculations of the stress module. Orthotropic thermal expansion is assumed for all materials (in the material principal directions) so that values of CTE\\, CTE2, and CTE3 need be specified (for isotropic materials, CTE\\ = CTE2 = CTE?). As outlined in Appendix B, these thermal expansion coefficients are calculated as linear functions of temperature for non-composite materials, and linear functions of temperature and degree of cure for composite materials and resins. Since it is a directional quantity, it is especially useful to analyse compositeCTE's by examining fibre and resin thermal expansions separately. A number of models have been developed for calculation of the thermal expansion coefficients of unidirectional plies from resin and fibre CTE's, Vj and resin and fibre mechanical properties. In the current analysis, a micromechanics model taken from Bogetti and Gillespie (1992) is used. 6.1.4.1 Previously Available Information Thermal expansion data for AS4\/8552 was obtained from two sources prior to beginning this work. Bogetti and Gillespie (1992) reported 'graphite' fibre CTE's of -9.0xl0\"7 \/\u00b0C in the longitudinal direction (CTE]) and 7.2x10\"6\/\u00b0C in the transverse direction (CTE3). Previous thermal expansion measurements on -116-Chapter 6 : Material and Boundary Condition Characterization unidirectional AS4\/8552 prepreg tape (Boeing, 1994) found C7\u00a33 to be approximately 28.6x10\"6\/\u00b0C and CTE] to be about 0.6x10\"6\/\u00b0C. These measurements also indicated a temperature effect on CTE, but the magnitude of this effect was not determined. Using known fibre and resin mechanical properties and the transverse CTE values from above, it was calculated that fully cured resin CTE is about 42xlO\"6\/\u00b0C (average for -73 \u00b0C to 93 \u00b0C). Also, from the composite CTE], the fibre CTE] was found to be slightly higher than reported in Bogetti and Gillespie at about 3xl0\"8\/\u00b0C (both numbers are very close to zero). 6.1.4.2 Tests Performed To obtain a better description of the change in fibre and resin C T E ' s with temperature, and resin C T E with degree of cure, a series of tests of partially cured composite and neat resin specimens were planned. Originally, these tests were to be done in the same manner as the previous analysis of AS4\/8552 prepreg C T E ' s . These previous tests had been carried out on long, thin specimens using a Netzsch Model 402 dilatometer in accordance with ASTM E 228. However, during the initial stages of the current testing, it was discovered that specimens which were not fully cured did not possess adequate strength and tended to buckle during the test under the applied end load. This made accurate thermal expansion measurements impossible. Other potential methods for measurement of thermal expansion coefficients include TMA (thermo-mechanical analysis) or DMA (dynamic mechanical analysis). However, measurement of fully uncured prepreg CTE] using these techniques is quite difficult due to specimen preparation problems. Also, the small specimen size that could be used would make transverse measurements less accurate than using a dilatometer. Since some good measurements of CTE were already available, these problems and resource constraints led to abandonment of further CTE measurements. -117-Chapter 6: Material and Boundary Condition Characterization 6.1.4.3 Discussion Thermal expansion strains are one of the most important sources of process-induced deformation in many cases. Thermal expansion coefficients should therefore be known to a high level of accuracy. While a single coefficient of thermal expansion (i.e. no temperature-dependence) is usually quoted for most materials, this may not be appropriate. As shown by Hyer et. al (1983), and Springer (1984), thermal expansion strains of the T300\/5208 graphite-epoxy system are highly non-linear, with CTE3 increasing significantly with temperature. As discussed in a review paper by Maguire and Kulacki (1985), similar behaviour has also been observed for other carbon-fibre epoxy systems. The observed thermal strain non-linearity means that CTE's should be known not only at room temperature and full cure but over the entire relevant temperature range and over a range of degrees of cure. Currently, while fully cured AS4\/8552 thermal strain behaviour near room temperature is well characterized, nothing is known about this behaviour in other cure states or at elevated temperature (e.g., near the maximum curing temperature). Also, no information about thermal strains aboveTs is available, although it is expected that CTE's would be much higher, at least in the transverse direction2. Thus if any further characterization of AS4\/8552 is to be done, it is suggested that some effort be expended on CTE measurements, especially on fully cured or nearly fully cured material at high temperature to determine its temperature dependence. Any future tests on partially cured material should be done using DMA or TMA to avoid specimen buckling problems. In addition to prepreg, some analysis of neat resin CTE may also be useful as errors due to fibre volume fraction variations would be minimized. At the same time, however, resin modulus above Tg will be very low, so the absence of good data here is probably not critical. -118-Chapter 6: Material and Boundary Condition Characterization 6.1.5 Mechanical Properties Instantaneously linear elastic mechanical properties (Ea, Gy, Vy) are used in the calculations of the stress module. Orthotropic material behaviour is assumed for all materials such that values for E\\E22, E3i,Gu, G\\3, G21, V\\2, v13 and v2i are required by the stress module calculations. As outlined in Appendix B, these mechanical properties are calculated as linear functions of temperature for non-composite materials and fibres. As discussed in Chapter 5, the resin is modelled as a 'cure-hardening instantaneously linear elastic material' (CHILE), with an isotropic resin modulus and Poisson's ratio calculated using models outlined in Sections B.1.9 and B.1.10. Instantaneous composite material elastic constants are determined using a micromechanics model from Bogetti and Gillespie (1992), described in Section B.1.2. As outlined in the literature review in Chapter 2, the model used for prediction of the development of composite mechanical properties during processing is one of the most critical components of any process-induced deformation model. Since the mechanical properties of most fibres are relatively constant throughout processing, efforts to model composite mechanical response necessarily focus on resin behaviour. 6.1.5.1 Previously Available Information As discussed in the literature review, most previous measurements of the development of composite properties during cure were elastic tests performed at room temperature (e.g., Kim and Hahn, 1989; White and Hahn, 1990, 1992b). No references to continuous measurements of composite elastic response during the cure process were found. Only limited mechanical properties data for AS4\/8552 in particular was available prior to these tests. Fortunately, this information included at least approximate data for all required AS4 fibre mechanical properties as these are not easy to measure. The following 'graphite fibre' mechanical properties were obtained from Bogetti and Gillespie (1992): G13 = 27.6 GPa, G23 = 6.89 GPa, vu = 0.20. A value for AS\/4 fibre \u00a311 of 221 GPa was obtained from Hercules Aerospace Company product data sheet No. 847--119-Chapter 6: Material and Boundary Condition Characterization 6. Although these properties were presumably based room temperature measurements, they were assumed to be constant for all processing temperatures. Some data for the 8552 resin was available from Hercules Aerospace Company Product Data Sheet H050-377. This document gives resin Eu as 4.67 GPa and a value of resin Poisson's ratio of 0.37 could be inferred from data for similar resins. Both numbers are for fully cured material at room temperature. 6.1.5.2 Tests Performed Since fibre properties are already known and are relatively constant during processing, the best approach to the characterization of composite mechanical properties evolution would be to examine resin behaviour by itself. Composite behaviour for any temperature, resin degree of cure and V} could then be estimated from standard micromechanics models. A potential problem with this approach is that standard micromechanics models may underestimate the contribution of resin properties to those of the composite when the matrix is very weak (White and Hahn, 1990, 1992b). Also, the large change in resin mechanical behaviour over the complete cure range (i.e. a = 0.0 - 1.0) would be very difficult to measure using the available apparatus. For these reasons, all mechanical properties characterization tests were performed on composite prepreg only. The tests performed as part of this analysis were as follows: 1) The development of prepreg shear modulus was measured through the use of a 'rectangular torsion test'. This test involves subjecting a rectangular specimen to a sinusoidally varying torsional strain and measuring the required torsional load. The test in this case was performed using a Rheometrics RMS-800 rheometer on an 8-ply unidirectional specimen (fibres aligned with the twist axis) with approximate dimensions of 45 mm x 12.5 mm. The applied strain was varied from 1% to 10% with a frequency of 10 rad\/s. A schematic of the test set-up is shown in Figure 6.6. -120-Chapter 6: Material and Boundary Condition Characterization T, Q(onlput) input 8 plies 45 mm 0\u00b0 Figure 6.6: Schematic of rectangular torsion test. Note that the composite fibres are aligned with the twist axis. Measurements were taken of storage and relaxation shear moduli (G'and G\") as the specimen cured isothermally at 135 \u00b0C and 160 \u00b0C. Two specimens were tested for each data point. All specimens were debulked prior to test. Although not a standard test for composite materials, this method was employed because it allowed measurement of development of instantaneous composite modulus over a full range of resin mechanical behaviour from viscous to viscoelastic to elastic. Also, since the directly-measured property is resin-dominated ply G 2 3 ' and G 2 3 \" a large change in measured response during processing is ensured. 2) A second set of tests was also performed to obtain composite in-plane properties as a function of temperature and resin degree of cure. In this test, a Perkin-Elmer DMA-7 was used to apply a sinusoidally varying tensile load to 2 ply rectangular (10 mm x 3 mm) specimens and measuring the induced strains. From this information, composite storage and relaxation moduli (E' and\u00a3\") in the ply longitudinal and transverse directions were to be calculated over a range of temperatures and frequencies. All specimens used in this test were slightly pre-cured to prevent them from being damaged early in the test when the matrix would be very weak. However, the specified pre-cure was found to result in an -121-Chapter 6: Material and Boundary Condition Characterization insufficiently strong resin matrix and severe specimen damage during testing was encountered. Thus, the results of this set of tests were discarded. 6.1.5.3 Results A plot of measured storage and loss shear moduli (G'and G\") versus calculated resin degree of cure is shown in Figure 6.7 for one of the 135 \u00b0C isothermal tests. As can be seen, the specimen is very compliant for much of the test, then hardens quickly to a large percentage of the final modulus before slowly approaching a limiting value. This same type of behaviour was observed at both test temperatures, with very repeatable results obtained from different specimens. Some damage of the specimen during test was apparent in most cases, but was always relatively minor. 4.5E+09 0.60 100 150 Time (minutes) 200 250 Figure 6.7: Rectangular torsion test measurements for 135 \u00b0C isotherm test (specimen 1). 6.1.5.4 Analysis As discussed in Chapter 2, previous researchers have typically assumed that the instantaneous resin elastic modulus is a function only of degree of cure. Thus, it was expected that a model similar in form to -122-Chapter 6: Material and Boundary Condition Characterization those of Bogetti and Gillespie (1992) or White and Hahn (1992b) would be adequate to predict modulus development during processing. To obtain specimen elastic shear modulus, G(a>), from storage and loss modulus measurements, the following expression is often employed (Ferry, 1980): G(co) = G'(co) - 0.4G\"(0.4co) + 0.014G\"(1 Oco) (6.3) However, as discussed in Adolf and Martin (1996), for most thermosets the approximation G(aS) = G{co) will suffice. Thus, in this case, the measured value of storage modulus, G', was assumed to be equal to composite G 2 3 3 . Calculation of resin modulus was then obtained from this value using the micromechanics model outlined in Appendix B. In this calculation, the composite AS4 fibres were assumed to have the properties discussed in Section 6.1.5.1 and resin Poisson's ratio, vr, was determined employing a constant bulk modulus assumption, i.e.,: v = \u2014 4 r-\u00b1 (6.4) where E\u2122 is the 'unrelaxed' resin modulus at full cure and vmr is the 'unrelaxed' resin Poisson's ratio. After accounting for apparent specimen damage by vertically 'shifting' the modulus curves, the calculated resin elastic moduli were plotted against the predicted degree of cure as shown in Figure 6.8. This is not strictly correct due to the rectangular cross-section of the test specimen. -123-Chapter 6: Material and Boundary Condition Characterization 5.0E+09 ^ 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 Degree of cure Figure 6.8: Rectangular torsion test results summary: calculated resin elastic modulus as a function of degree of cure at 135 \u00b0C and 160 \u00b0C. It can be seen that quite good agreement was obtained between the two specimens measured at each temperature, especially with respect to the shape of the modulus development curve. The difference in the measured modulus magnitude in the two 135 \u00b0C runs is thought to be at least partly attributable to specimen damage and possibly slightly different resin cure rates. The shape of the modulus development curves at both temperatures are very similar. In both cases, the modulus is initially very low and remains so until a certain degree of cure is reached. At this point, the modulus increases approximately linearly with degree of cure. Since cure rate slows dramatically above glass transition and the specimens were not cured for a sufficiently long time, it is not clear what happens after this point. However, other tests indicate that a limiting value of modulus is reached after which it remains constant with time. While the general trend in modulus development is similar for all specimens, it is apparent from Figure 6.8 that the curves at the two different temperatures are somewhat 'shifted' from each other. Thus, contrary to initial expectations, resin modulus is not only a function of its degree of cure, but also of its temperature. -124-Chapter 6: Material and Boundary Condition Characterization This result should not be surprising recalling the inherently viscoelastic nature of the resin mechanical response. Consider a fully cured resin with a temperature initially above its glass transition temperature. Gradually decreasing the resin temperature to below Tg would result in the same modulus trend observed in Figure 6.8. In the isothermal tests performed here, while the resin temperature is held constant, the resin instantaneous Tg increases with time, with the same result. Interestingly, the 'unrelaxed' resin modulus (i.e. EratT< Tg) does not appear to be greatly affected by the degree of cure at (the limiting value of Er is about the same at both temperatures). This conclusion is supported by work by Dil Iman and Seferis (1989) and Adolf and Martin (1996). As discussed in Adolf and Martin (1996), the relaxed resin shear modulus is equal to zero prior to gelation and thereafter increases with resin advancement according to the relation: G , = G\", relax relax (b-b ,,. gelation bgelation J (6.5) Greim is the relaxed modulus and where b is a 'bond probability' whose relationship to degree of cure is a function of reaction type (e.g., b = a or ot ). However, as also noted by Adolf and Martin (1996) the overall significance of the stresses that arise when the resin is at its relaxed modulus is generally small. Thus, the variation in this modulus can be ignored in most cases. This implies that it is reasonable to model modulus development as a function of degree of cureplus a temperature 'shift factor'. Alternatively, taking advantage of the relationship between resin degree of cure and the instantaneous glass transition temperature, it is also possible to express modulus development at a function of temperature using a degree of cure 'shift factor' as illustrated in Figure 6.9. -125-Chapter 6: Material and Boundary Condition Characterization l.OE+00 _ Temperature Figure 6.9: Illustration of degree of cure as a 'shift factor' in resin elastic response. The latter of these equivalent representations was chosen in this case since it has a more-straightforward physical interpretation. To collapse the modulus development curves, a parameterT* is introduced which is equal to the difference between the instantaneous glass transition temperature, Tg, and the current temperature (i.e., T = Tg - T). A linear variation of modulus with T was assumed between 'critical' values T*cx and TC1. For simplicity, a linear relationship between Tg and degree of cure was assumed such that Tg = + aTg * a. The resulting equation resin modulus development is: K = K r T'a (6 6 ) E R = E \\ [ \\ + ai,(T-T0)] where: T' =(T\u00b0 +aTg*a)-T T* \u2014 T\" I T* * T J C I JC\\a ^ 1C\\b 1 -126-Chapter 6: Material and Boundary Condition Characterization In this equation, E\u00b0 is the 'relaxed' resin modulus (Er at T \u00bb Tg) and L\\ is the 'unrelaxed' modulus (Er at T \u00ab Tg). Allowance is also made in Equation 6.6 for any effect of temperature on E\u2122 and on the limiting values of T, but these were not used in this case. Based on data supplied by Hercules, the dependence of glass transition on degree of cure was estimated to be Tg = 268 + 220 * a (K). The 'unrelaxed' modulus was chosen to be E\u2122 = 4.67 GPa, also from Hercules data, with the 'relaxed' modulus arbitrarily chosen to be a relatively small\/i^ \/lOOO = 4.67 MPa. The 'critical' values for T* were chosen using an optimization routine, with a best fit obtained with T*CXa = -45.7 \u00b0C, T*C1 = -12.0 \u00b0C. A summary of the values used for the parameters in Equation 6.6 is provided in Table 6.2 following. Table 6.2: Parameters used in 8552 resin modulus development model (Equation 6.6). Parameter Value Comments Unrelaxed resin modulus \u00a3\/\u00b0 = 4.67 x 109 Pa Resin modulus at t = 0 (or T\u00ab Tg) and T= 20 \u00b0C Relaxed resin modulus \u00a3,\u00b0 = 4.67x 106 Pa Resin modulus at t = oo (or T\u00bb Tg) and 7= 20 \u00b0C. Arbitrarily chosen. This value is actually a high function of degree of cure. Lower critical value for T* at OK. Tc,a = -45.7 K Prior to Ta, Er = E\u00b0 Variation of lower critical value for T with increase in temperature. Tc,b = 0.0 Accounts for the variation in slope of d\u00a3^d7'* curve (between critical points) seen in tests on other materials. Upper critical value for T 7b = -12K After 7b, Er = E^ Glass transition temperature at a= 0. Tg\u00b0 = 268 K Linear dependence of Tg on degree of cure assumed. Variation of glass transition temperature with a = 0. aTg = 220 K Zero point temperature for calculation of secondary variation of Er with temperature. T0 = 20 \u00b0C Variation of thermoset moduli with temperature above Tg is sometimes observed. Relative secondary variation of Er with temperature aEr = 0 -127-Chapter 6: Material and Boundary Condition Characterization A comparison between the predictions of the developed model and the calculated resin modulus values from the rectangular torsion experiments is shown in Figure 6.10. Note that in this figure, the experimental results have been normalized so that the unrelaxed modulus matches with the model prediction. 5.0E+09 4.5E+09 4.0E+09 _ 3.5E+09 Q . 1 3.0E+09 4-3 \u20223 E 2.5E+09 \u20225 2.0E4-09 \"K cC 1.5E+09 1.0E+09 5.0E+08 0.0E+00 Model o Experiment 0.20 0.30 0.40 0.50 0.60 Degree of cure 0.70 0.80 0.90 Figure 6.10: Comparison of model predictions for resin modulus development with (normalized) experimental results. Also shown is the prediction for a 180 C isothermal cure. 6.1.5.5 Discussion The results of the DMA tests were very disappointing since these were to be the main sources of information regarding resin property development during processing. However, much useful data was still obtained from the rectangular torsion tests. These tests showed that previous resin development models which are based on the assumption that modulus is a function of degree of cure only (e.g., Bogetti and Gillespie (1992), White and Hahn (1992b)) fail to include important temperature effects. By employing the relationship between glass transition -128-Chapter 6: Material and Boundary Condition Characterization temperature and resin degree of cure, a temperature-based resin modulus model was developed. This model provided an excellent fit with experimental results. However, a number of concerns about the developed model should also be raised at this time. One problem is the very low number of data points (only 2) on which the model is based. Given the relatively weak theoretical basis for the model, this raises concerns about predictive accuracy outside the measured temperature range (135 \u00b0C - 160 \u00b0C) and under non-isothermal conditions. Another point of concern is that the effect of test frequency was not examined. Since the model is used to predict aneffective elastic modulus for an inherently viscoelastic material, an 'appropriate' test frequency should be chosen. Choosing the most appropriate test frequency is problematic since it will vary both within a process cycle and between different cycles. In any case, the frequency used in this test (10 rad\/s) is probably too high, a better value for a typical autoclave cycle might be somewhere in the range of 10\"1 to 10\"4 rad\/s. Despite the mentioned problems, the experimental results are very exciting in that they may indicate a method for a simplified analysis of resin viscoelastic response using degree of cure 'shift factors' in a method analogous to time\/temperature superposition. While this analysis is not likely valid before gelation (Dillman and Seferis, 1989), it would potentially allow resin viscoelastic response over a range of temperatures and degrees of cure to be predicted from standard characterization tests on fully cured material and a few isothermal tests on curing material. Given the importance of resin mechanical property development to the prediction of residual stress generation, it is apparent that further study on mechanical property development of 8552 and other resins should be performed. It is suggested that both rectangular torsion and in-plane DMA tests (at higher starting degree of cure than used here) be used. The rectangular torsion test was shown to be a very good method for obtaining shear modulus measurements over a wide range of resin degrees of cure from a = 0 to full cure. The in-plane DMA tests would be of special interest for evaluating the accuracy of standard micromechanics models at low degrees of cure, especially in fibre-dominated directions. -129-Chapter 6: Material and Boundary Condition Characterization Two important changes are suggested for future mechanical property measurements. First, it is suggested that the test frequency be lowered to 10\"' to 10\"4 rad\/s so that more representative data may be obtained. Using a technique such as FTMA (Fourier Transform Mechanical Analysis; Ganeriwala and Rotz (1987), Malkin et al. (1992)) would allow maximum information to be obtained from this test by scanning the whole range of frequencies at once. Some testing on neat resin is also suggested to eliminate fibre effects and attempt to assess the accuracy of employed micromechanics models. 6.1.6 Resin Cure Shrinkage As discussed in Chapter 2, during the cure process thermosetting resins undergo a significant increase in density and a corresponding reduction in volume commonly referred to as cure shrinkage. At the beginning of each time step, ply 'cure shrinkage coefficients' in the material principal directions, CSC, are calculated from fibre and resin properties and Vj using the same micromechanical model used to determine thermal expansion coefficients (see Section B.I.2.). In the stress module, the change in ply cure shrinkage strains are calculated from the values of CSC, and the change in resin linear cure shrinkage strain, e.g., Asscl = CSC, * Assr. It should be noted that the micromechanics model used assumes transversely isotropic behaviour such that CSCi - CSC3. These resin cure shrinkage strains are calculated from cure shrinkage models developed from experimental measurements. The two models currently available in COMPRO are shown in Section B.1.8. 6.1.6.1 Previously Available Information Measurements of resin or composite shrinkage during cure have been performed by a number of researchers (e.g., Lam and Piggott, 1989; Armstrong et al., 1986; Mijovic and Wang, 1988; Russell, 1993; White and Hahn, 1992b). Of these Mijovic and Wang, and White and Hahn developed a correlation between measured shrinkage strains and resin degree of cure. These researchers found very different -130-Chapter 6: Material and Boundary Condition Characterization relationships between shrinkage strain and cure extent, although both found that most strains were developed quite early in the cure process. No cure shrinkage tests had previously been performed on AS4\/8552. 6.1.6.2 Tests Performed Two different sets of tests were performed as part of the current cure shrinkage analysis. In the first, ply cure shrinkage strains were calculated from continuous measurements of the change in the dimensions of a test specimen during cure. In the second set of tests, effective cure shrinkage strains were examined at the completion of a test. Several methods may be used for cure shrinkage strain measurement. White and Hahn (1992b), for example, attached strain gages directly to the surface of a unidirectional laminate and measured ply strains during cure. This method was not used due to concerns about the integrity of the gage\/specimen bond early in the cure process when the resin is in an essentially viscous state. A preferred method, also used previously, is to directly measure neat resin volumetric cure shrinkage using a volumetric dilatometer as described by Russell (1993). Unfortunately, such a device was unavailable so other alternatives had to be pursued. In this case, shrinkage strains were measured using Thermo-mechanical Analysis (TMA). For this test, a Perkin-Elmer TMA-7 was used to measure specimen strain by monitoring the displacement of a small probe pressing lightly on the specimen surface (in this case P < 1 kPa). All specimens were cured isothermally to eliminate thermal strain effects. Transverse strain measurements were performed at three different temperatures: 130 \u00b0C, 150 \u00b0C, and 170 \u00b0C. Longitudinal measurements were also performed, but at 170 \u00b0C only. The 'effective cure shrinkage' experiments were performed using a procedure similar to that outlined in Daniel et al. (1989). These researchers first cured a unidirectional laminate using a standard processing cycle. A second identical laminate was then laid-up on the cured laminate and the assembly subjected to -131-Chapter 6: Material and Boundary Condition Characterization a second processing cycle. Having the same lay-up, both parts of the assembly underwent (nearly) identical amounts of thermal deformation throughout the second process cycle. However, since the second laminate experienced cure shrinkage during this process cycle, and the already-cured first laminate did not, a warped final shape resulted. The magnitude of this warpage will indicates the 'effect' of the cure shrinkage. This test was performed here as a 'sanity check' to ensure that the effective cure shrinkage predicted by COMPRO was reasonably close to that measured. To determine the effective cure shrinkage strains in the ply longitudinal and transverse directions, both [0]n and [90]n specimens were examined. As shown in Figure 6.12, long thin specimens (305 mm x 38 mm) were employed to magnify the relative deformation in the direction being examined. Both the bottom (cured in first cycle) and top (cured in second cycle) laminates were three plies thick. The process cycle used for both cycles is shown in Figure 6.11. In all cases a layer of Teflon was placed between the part and the substrate tool to minimize tooling constraint effects. CL u Air temperature \\ 07\u00b0C - Ihr l.ll\u00b0C\/min. y 177\u00b0C-2hr Autoclave pressure l.ll\u00b0C\/min. \u2022 170kPa Full vacuum J 375 kPa Vacuum pressure \u2014 \u2022 101 kPa Cycle time Figure 6.11: Process cycle used in effective cure shrinkage test. -132-Chapter 6: Material and Boundary Condition Characterization Figure 6.12: Specimen used in effective cure shrinkage test. 6.1.6.3 Results TMA Tests As shown in Figure 6.13, specimen thickness dropped rapidly during the heat-up period, then decreased much more slowly after the isothermal hold temperature was reached, for a total thickness change of slightly less than 6% in the case shown. This same trend was observed at all three isotherms. Repeatability initially appeared very poor as the total change in measured specimen thickness varied greatly from specimen to specimen. However, change in thickness after the isotherm was reached, was found to be very consistent for both specimens at all three isotherms. The total composite linear cure shrinkage strains for each specimen after reaching the isothermal temperature are shown in Table 6.3: -133-Chapter 6: Material and Boundary Condition Characterization 0 -I i 1 1 1 i 1 1 1 1 1- -6.0 0 10 20 30 40 50 60 70 80 90 100 Time (minutes) Figure 6.13: Isothermal TMA cure shrinkage test (T= 150 \u00b0C). Table 6.3: Apparent total cure shrinkage after isothermal hold temperature reached. Temperature Specimen #7 Specimen #2 Transverse 130 \u00b0C 1.83% 1.89% 150 \u00b0C 1.22% 1.19% 170 \u00b0C 1.89% 1.85% Longitudinal 170 \u00b0C 0.014% *0% The 130\u00b0C and 170\u00b0C transverse measurements initially appear to give quite similar results, with the 150\u00b0C specimens exhibiting somewhat different behaviour. As shown in the Analysis section, a comparison of total shrinkage strain is deceptive in this case; the trend vs. degree of cure is much more revealing. As expected, ply shrinkage in the fibre direction is much lower than in the transverse direction, in this case, indistinguishable from noise. -134-Chapter 6: Material and Boundary Condition Characterization Effective Cure Shrinkage Specimens After completion of the second cure cycle, the shape of the effective cure shrinkage specimens was similar to that shown in Figure 6.14. The effective cure shrinkage strain in each specimen was determined from the warpage displacement, 5, using Equation 6.7 (from Daniel et al., 1989): Figure 6.14: Effective cure shrinkage specimen after second process cycle. For the parts used in this test, the chord length, L, was 307 mm in each case and the total specimen thickness, t, was 1.11 mm. The measured deflections and calculated strains are shown following in Table 6.4. Table 6.4: Effective cure shrinkage test results. Specimen Maximum Deflection (mm) Effective Shrinkage Strain [90]n Specimens (Transverse) Specimen #1 17.85 0.112% Specimen #2 18.15 0.114% Specimen #3 18.40 0.116% [0]n Specimens (Longitudinal) Specimen #1 *0.0 \u00ab 0 . 0 Specimen #2 \u00ab 0 . 0 *0.0 Specimen #3 \u00ab 0 . 0 *0.0 As shown in Table 6.4, the deflection of the [0] specimens was so small (<0.1 mm) that it was not measurable with any level of accuracy. In contrast, deflection of the [90] specimens, was quite large and -135-Chapter 6: Material and Boundary Condition Characterization found to be very consistent. Note that the measured effective shrinkage strains are more than an order of magnitude lower than the total shrinkage strains shown in Table 6.3. 6.1.6.4 Analysis TMA Tests To compare the results from the three different isothermal TMA tests, specimen strain was plotted versus resin degree of cure. Cure shrinkage was measured from the start of the isothermal hold since it is not possible to differentiate shrinkage and thermal strains prior to this point. Resin degree of cure was calculated using the cure kinetics model developed as part of this work as described in Section 6.1.7. The change in measured strains as a function of resin degree of cure is shown in Figure 6.15. 2.0 1.8 1.6 ~ 1.4 1 1.2 S. \u00ab 1 0 c \u00a3 0.8 in o \u201e o o \u2022 \u2022\u00ab \u2022 <>\u2022\u2022 o \u00b0 . 0 ^ 0 o as a 130 C - Specimen 1 A 130 C - Specimen 2 o 150 C - Specimen 1 . 150 C - Specimen 2 0 170 C - Specimen 1 , 170 C - Specimen 2 0.1 0.2 0.3 0.4 0.5 Resin degree of cure 0.6 0.7 0.8 Figure 6.15: Measured transverse cure shrinkage of AS4\/8552 as a function of predicted resin degree of cure. As can be seen in the preceding figure, the results at 170 \u00b0C and 150 \u00b0C match quite well up to about a = 0.3, after which they begin to diverge. The results at 130 \u00b0C are very different from the other two, showing much more apparent shrinkage at very low degrees of cure. Since the resin remains in a viscous -136-Chapter 6: Material and Boundary Condition Characterization state for an extended period at this temperature, it is likely that the high apparent initial shrinkage is the result of specimen and\/or probe settling. Accounting for this settling by vertically 'shifting' the plots as shown Figure 6.16 results in a much better fit between shrinkage strains at all three temperatures. 2.0 1.8 1.6 I I1\" \u2022i 1.2 8f 1 0 0.8 U 0.6 0.4 I 0.2 0.0 \u2022 \u2022 \u2022 \u2022 \u2022 \u2022 8o0io8o ' > * \u2022 o \u00b0 \u00b0 o .if 1 o A 130 C - Specimen 1 A 130 C - Specimen 2 0 150 C - Specimen 1 a 150 C - Specimen 2 0 170 C - Specimen 1 \u00bb 170 C - Specimen 2 0 0.2 0.3 0.4 0.5 Resin degree of cure 0.6 0.7 0.8 Figure 6.16: Transverse cure shrinkage of AS4\/8552 with cure (vertically shifted). Figure 6.16 suggests that the relationship between cure shrinkage strain and degree of cure is not the same at all temperatures, with less shrinkage occurring at lower temperatures. It is possible that with the approach of vitrification at each temperature, the resin macromolecular structure will become sufficiently rigid as to prevent immediate accommodation of the molecular chain length reduction that accompanies cure. This hypothesis, supported by the work of Pang and Gillham (1989), would mean that, although the equilibrium level of shrinkage would be the same at each temperature, the apparent strains would be different. This implies that the significant observed shrinkage strains that sometimes accompany a high-temperature resin post-cure (Stone et al., 1994) might be largely the manifestation of 'frozen' non-equilibrium shrinkage strains generated earlier in the process. -137-Chapter 6: Material and Boundary Condition Characterization If the non-equilibrium nature of cure shrinkage is important, simple models such as those of Bogetti and Gillespie (1992) or White and Hahn (1992b) cannot be used to predict shrinkage strains for all cases. However, for the process cycles employed for this material, the highest hold temperature will probably be maintained for a sufficiently long period so that the resin can re-equilibrate. Shrinkage strains can be then be treated as if they all occurred at this temperature. Taking this to be the case, a shrinkage equation similar to that of Bogetti and Gillespie (1992) is proposed, different only in that a quadratic term is added to this earlier model as follows: Vf =0.0 a < a Vf = A*as+ (VrSm -A)* a] aciaC2 ( 6 ' 8 ) a - ar, as = iJ\u2014 From the measured transverse ply shrinkage, resin volumetric cure shrinkage strains were calculated using: \\ 3 + 1 -1 (6-9) where: V* is the volumetric resin shrinkage, e\\c is the measured transverse shrinkage strain and vr is the resin Poisson's ratio. This equation was developed using the assumption that shrinkage strains in the fibre direction were completely constrained and that strains in the 2 and 3 directions were equal. In the current calculation, resin Poisson's ratio was assumed to be constant at vr = 0.37. Equilibrium considerations, discussed above, mean that the 170 \u00b0C test measurements give the best approximation to the 'actual' resin shrinkage behaviour during processing. Thus, only these measurements were analysed in this case. Using the 170 \u00b0C test measurements only, the total resin volumetric shrinkage was calculated to be an unexpectedly-high 9.9%. This reason for this large amount of shrinkage is unknown, but could be related -138-Chapter 6: Material and Boundary Condition Characterization to such things as specimen settling prior to gelation. The best fit to the experimental results using Equation 6.8 was obtained using the parameters listed in Table 6.5. Table 6.5: Parameters used in 8552 resin cure shrinkage model (Equation 6.8). Parameter Value Comments Lower critical value for start of resin cure shrinkage aa = 0.055 Prior to this point, no shrinkage strain is assumed to occur. Upper critical for end of resin cure shrinkage aC2= 0.651 After this point, no additional cure shrinkage strain is assumed to occur. Total resin volumetric cure shrinkage VrSm = 0.099 Much of the shrinkage occurs prior to gelation when it will result in no significant contribution to stress. Factor accounting for the non-linearity of shrinkage strain with degree of cure A =0.173 If set equal to VrSco , a linear variation of strain with cure will be modelled. Effective Cure Shrinkage Tests Two important uses are seen for the effective cure shrinkage tests. First, they can serve as a quick way to estimate the relative importance of cure shrinkage and thermal strains for a given material and cure cycle. In cases where thermal and shrinkage strains are the dominant stress generation mechanisms, this can allow quick estimates of residual warpage. This test could also give the process modeller an indication of the amount of effort to expend on cure shrinkage strain characterization. For the [90]n laminates in this case, the ratio of the effective cure shrinkage strains to the thermal strains during cool-down (i.e. from 177 \u00b0C to 20 \u00b0C ) is about 0.26 for the process cycle used. This indicates that these strains will contribute quite significantly to the final stress state. The cure shrinkage test results can also be used to verify the accuracy of the combined modulus development and cure shrinkage models by comparing experimental measurements to COMPRO effective cure shrinkage predictions. In this case, the effective cure shrinkage test specimen was modelled using the finite element representation shown in Figure 6.17. For this model, symmetry was employed and only a small portion of the rest of the specimen length was modelled. In the model, an uncured [90]3 laminate was placed on top of a [90]3 laminate that had been pre-cured using the process -139-Chapter 6: Material and Boundary Condition Characterization cycle in Figure 6.11. Both parts were then subjected to a second identical process cycle. No tooling was included in the model. 1 1 1 1 1 1 1 1 1 1 1 1 i L J I 1 L - U I C U i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 M\u00abM 1 4 ffi \u00a3 irccurea r - -s - -xiu t 11 n r 1.1 mm |-< 12.7 mm \u2022] Figure 6.17: Finite element representation used for effective cure shrinkage specimen model. Note that in the actual specimen, 20 elements were used in the thickness direction and 75 for the length direction. The effective cure shrinkage strains predicted by the model were calculated using Equation 6.7. As shown in Figure 6.18, the predicted final effective shrinkage strain using the resin cure shrinkage and modulus development models developed here is slightly low (0.095% compared to 0.114%). A good match between model predictions and experimental results was obtained by slightly increasing the shrinkage model's second critical degree of cure value, ac2, from 0.65 to 0.67. 0.90 Experiment _ 0.12 Cycle time (minutes) Figure 6.18: Predicted effective cure shrinkage strain versus time during processing and comparison with measured value at completion of processing. -140-Chapter 6: Material and Boundary Condition Characterization 6.1.6.5 Discussion The outlined tests have been used to determine the relationship between cure shrinkage and resin degree of cure and the relative contribution of this cure shrinkage to process-induced stress and deformation for this material. Previous studies (Mijovic and Wang, 1988; White and Hahn, 1992b) have shown that for some materials virtually all cure shrinkage occurs early in the cure process before the resin modulus becomes significant. In this case, however, enough shrinkage occurs sufficiently late in the process for the contribution of cure shrinkage to residual stress to be quite significant. Further tests examining non-equilibrium cure shrinkage strains (if indeed they exist) would be useful since these strains could be important to the effect of cure temperature and post-cure on effective cure shrinkage strains. An issue of concern with the current tests is the magnitude of the calculated resin volumetric cure shrinkage. The nearly 10% resin volume reduction found in this case is larger than expected from values cited in the literature, although experiments by White and Hahn (1992) using a BMI resin found even larger composite transverse shrinkage strains than measured here (2.9% vs. 1.9%). Also, model effective cure shrinkage strain predictions using this value of shrinkage strain was in quite good agreement with experimental measurements. 6.1.7 Resin Cure Kinetics The resin cure kinetics model is used by the thermochemical module to calculate the resin cure rate and the degree of cure. An accurate model for resin cure kinetics is critical not only for accurate calculation of resin degree of cure and component temperature, but for all other process model predictions as well. This is true since resin degree of cure (a) is one of two state variables (the other is temperature) which are assumed in the current model to be sufficient to completely describe the resin state. Thus, all resin material properties used in the model are assumed to be dependent on resin temperature and degree of cure only, with no history dependence. -141-Chapter 6: Material and Boundary Condition Characterization As outlined in Section B.1.7, six different models for cure kinetics are currently available in COMPRO, one of which (model 6) was developed as part of the current work. 6.1.7.1 Previously Available Information Previous data for 8552 resin cure kinetics was already available prior to beginning this work (Scholz et al., 1994). In this previous analysis, an autocatalytic resin kinetics equation from Lee et. al (1992) (included in COMPRO as cure model 2) was employed: \u2014 = Kam(l-a)H dt (6.10) K = Ae-AE,Rr where AE is the activation energy and^ 4, m, n are experimentally-determined constants. Model constants were determined using data from isothermal scans of 8552 resin and dynamic DSC data for AS4\/8552 prepreg. The authors found that the best fit to the measured data was obtained using: AE = 73.1 kJ\/gmole, A = 1.905x105 \/s, m = 0.79, and n = 2.16. A later analysis of the same data by the authors4 yielded updated values for activation energy and the pre-exponential constant of: AE = 66.9 kJ\/gmole, A = 5.333xl05\/s. The average measured resin heat of reaction in these analyses was about 531 kJ\/kg. Also available prior to this analysis was the raw data used to develop the above model and data from other 8552 resin and AS4\/8552 prepreg DSC scans performed by Hercules. 6.1.7.2 Tests Performed As discussed in Section 2.2.2, cure kinetics models of the type expressed in Equation 6.10 are based on the assumption that the kinetics of bond formation is the reaction rate controlling factor throughout the cure process. While this is usually true early in the cure process, at some point the resin will normally undergo vitrification and the rate-controlling mechanism will shift from bond-formation to species Personal communication with Karl Nelson of The Boeing Company. -142-Chapter 6: Material and Boundary Condition Characterization diffusion. In this case, by not accounting for vitrification, the model of Scholz et al. (1994) would probably still generate good predictions early in the cure process, but would tend to over-predict cure rates at high degree of cure (see Section 4.2). The importance of the cure prediction to calculation of resin properties led to the decision to conduct further 8552 kinetics tests. As part of the current experimental work, a total of eight additional DSC (differential scanning calorimetry) 'scans' were performed. Of these, 6 were isothermal tests, two at each of 130 \u00b0C, 150 \u00b0C, and 170 \u00b0C. In these tests, the specimens were heated rapidly to the desired temperature where they were maintained for a total of 2 hours, then rapidly cooled. Two dynamic DSC scans were also performed in which specimens were heated at rate of 10 \u00b0C\/minute from room temperature to about 290 \u00b0C at which point cure was essentially complete. All tests were performed on neat 8552 resin using a Perkin-Elmer DSC-7. Neat resin was used rather than AS4\/8552 prepreg since this eliminated the effect of specimen fibre volume fraction variations and simplified sample preparation. This assumes that the effect of fibres on resin reaction kinetics would be small as indicated by Mijovic (1986) and Mijovic and Wang (1988). 6.1.7.3 Results Raw data from the DSC experiments consisted of measurements of heat flow and total resin heat of reaction (HR) as calculated by the apparatus software. For isothermal tests, measurements of the residual heat of reaction after completion of the experiment were also performed. A plot of measured heat flow with time for an isothermal run is shown following inFigure 6.19 (a dynamic scan in shown in Figure 4.2 in Section 4.2). -143-Chapter 6: Material and Boundary Condition Characterization 22.4 22.2 22.0 21.8 | 21.6 \u00ab = 21.4 21.2 21.0 1 20.1 20 40 60 Time (minutes) 80 100 120 Figure 6.19: Isothermal DSC scan of neat 8552 resin at 150\u00b0C. Measurements of heat flow and residual heat of reaction were very consistent for both of the 150\u00b0C and 170 \u00b0C isothermal tests. The two 130 \u00b0C isothermal runs gave slightly different heat flow curves and quite different residual heat values. The two dynamic runs produced very similar heat flow curves and the measured 77f was nearly the same in both cases: 553 kJ\/kg and 559 kJ\/kg. These values compare quite well with both previous measurements and two later scans which produced heats of reaction of 550 kJ\/kg and 556 kJ\/kg. 6.1.7.4 Analysis As discussed in Barton (1973), the first step in the calculation of exothermic heat flow from DSC measurements is determination of the appropriate 'baseline' heat flow for the instrument. Heat flow at any time can then be calculated from the difference between the established baseline and the actual measured heat flow. For dynamic scans, the baseline is calculated by measuring the heat flow at the start of the reaction and at its completion and assuming a linear variation between the extremes. For an -144-Chapter 6: Material and Boundary Condition Characterization isothermal experiment, this baseline is constant and can be determined from the measured flow at the completion of the reaction. In the current case, however, the isothermal runs were not carried out long enough, so reaction never went to completion for any run. Thus, while the DSC software had no difficulty calculating baseline for the dynamic scans, for the isothermal tests, this had to be done 'manually'. This was done by estimating the total heat evolved by the resin during the test by subtracting the measured residual heat of reaction (after test completion) from the known total resin heat of reaction. Using this information, the value of the baseline heat flow, qbaseline, was determined iteratively using: n-\\ H,so = H R - H^suud = Stew\/* - qin)l msampkAt (6.1 1) ;\"=1 where n is the number of data points, qjn is the total heat flow, mspi.cimm is the specimen mass and At is the time interval between heat flow measurements. From the baseline heat flow and the total heat flow, the resin cure rate was then determined using: dtX (Ahasdim tfiii) ^ m'sample ,^ dt ~ HRl(\\-a0) where a0 is the starting resin degree of cure, assumed to be 0.05 in all cases. Resin degree of cure as a function of time was the determined by integrating the calculated cure rate. The next step in the analysis was to compare the calculated cure rate measurements with the modified model from Scholz et. al. (1994). While quite good agreement was found for the dynamic scansFigure 6.20 shows that the predicted cure rate was much too high for the isothermal tests. -145-Chapter 6: Material and Boundary Condition Characterization 6.0E-04 ^ Degree of cure Figure 6.20: Comparison of measured cure rates at 170 \u00b0C vs. predictions of modified model from Scholz et al. (1994). Re-examination of the isothermal DSC scans on which this cure model was (partially) based (Hercules, 1992) showed that these previous tests had measured much higher cure rates than measured here. The source of the discrepancies is not known, but it is possible that the 8552 resin used in the two sets of experiments are slightly different formulations. The large difference between the predictions of the Scholz model and the measured cure rates emphasized the importance of developing a new cure model for the 8552 resin. The model equation chosen (Equation 6.13) is simply the equation used by Scholz et al., modified to account for a shift from kinetics to diffusion control, as outlined in Section 4.2. da _ Kam(\\-a)\" dt ~ 1 + e c{\u00ab-(\u00abco+\u00abcTr)} (6.13) K = Ae(-Af:iRT) The significance of the various terms in Equation 6.13 are discussed in Table 6.6. -146-Chapter 6: Material and Boundary Condition Characterization The first step in calculation of the appropriate model constants is determination of the activation energy, AE. As shown in Figure 6.21, this parameter was calculated from the slope of the natural logarithm of the isothermal cure rate, ln(da\/dt), vs. MT at a number of resin degrees of cure. A Alpha = 0.20 \u00bb Alpha = 0.25 H Alpha = 0.30 2.25E-03 2.30E-03 2.35E-03 2.40E-03 1 \/T(1 \/K) 2.45E-03 2.50E-03 Figure 6.21: Calculation of cure equation activation energy from isothermal cure rates at three degrees of cure All other model constants were determined using a weighted least-squares analysis which included data from both isothermal and dynamic DSC measurements. The calculated equation constants are shown in Table 6.6. -147-Chapter 6: Material and Boundary Condition Characterization Table 6.6: Parameters used in 8552 cure kinetics model (Equation 6.13). Parameter Value Comments Activation energy AE = 66.5 kJ\/gmole Calculated from slope of line of ln(da\/di), vs. MT Pre-exponential cure rate coefficient A = 1.53xl05\/s Calculated from weighted least squares analysis First exponential constant w = 0.813 Calculated from weighted least squares analysis Second exponential constant \u00ab = 2.74 Calculated from weighted least squares analysis Diffusion constant C = 43.1 Accounts for rate of shift from kinetics to diffusion control Critical degree of cure at T = OK. aco = -1.684 Note that this parameter is invalid below 35 \u00b0C since degree of cure cannot be negative Constant accounting for increase in critical resin degree of cure with temperature a c r = 5.475xlO\"3\/K When a approaches OQ at a given temperature, cure rate slows dramatically as the reaction becomes diffusion-controlled. As shown in Figure 6.22, the model provides an excellent fit to the isothermal degree of cure at all three examined temperatures. Figure 6.23 shows that a good fit to the dynamic DSC cure rate measurements is also obtained up to about 200 \u00b0C. After this point, cure predictions become progressively worse with increasing temperature. Since maximum 8552 curing temperature is never likely to exceed 200 \u00b0C in normal circumstances (maximum cure temperature is 177 \u00b0C), this is not a point of concern. -148-Chapter 6: Material and Boundary Condition Characterization 0 20 40 60 80 100 120 Time (minutes) Figure 6.22: Comparison of cure kinetics model isothermal degree of cure predictions with experimental measurements. 3.0E-03 - r 130 150 170 190 210 230 250 270 290 Temperature (\u00b0C) Figure 6.23: Comparison of cure kinetics model dynamic cure rate predictions with experimental measurements. -149-Chapter 6: Material and Boundary Condition Characterization 6.1.7.5 Discussion The developed cure kinetics model provides an excellent fit to the isothermal cure rate measurements at all three measured temperatures. By employing an equation capable of accounting for the shift in reaction-controlling mechanism from kinetics to diffusion, this model can provide very good cure rate predictions even after resin vitrification. This is believed to be a critical requirement in models for stress development since most stress development takes place near and after this point. The reason for the poor fit of the new model to dynamic DSC scans above 200 \u00b0C is not certain, but could be attributable to secondary reactions which occur only very slowly below this temperature as evidenced by the shape of the measured heat flow curves. Of greater concern is the relatively low number of experiments on which this model is based. More isothermal test data would have been useful, especially including at least one run at or above the normal maximum processing temperature of 177\u00b0C. Also, the 150 \u00b0C and 130 \u00b0C runs should have been allowed to continue until the reaction was essentially complete so that more information about diffusion-controlled kinetics could have been obtained. In addition, a DSC scan simulating a standard non-isothermal cure cycle would have been useful for comparison purposes. Another point of concern is the significant difference between the isothermal cure rates measured as part of the current analysis and those measured previously by Hercules. It is known that Hercules was investigating modified versions of the 8552 resin to facilitate easier automatic tape lay-up and it is possible that the 8552 resin tested in these experiments is slightly different from that examined previously. 6.1.8 Summary and Discussion The preceding sections outline the characterization of Hercules AS4\/8552 composite prepreg and 8552 neat resin properties performed as part of the current work. All material properties required by COMPRO thermochemical and stress modules were examined at some level, with special emphasis given to those -150-Chapter 6: Material and Boundary Condition Characterization properties expected to have the greatest impact on residual stress and deformation predictions, i.e., resin mechanical behaviour, resin cure shrinkage, and resin cure kinetics. Also discussed was the development of models to describe the observed material behaviour, including some models unique to this work. In general, very good agreement was obtained between developed models and experimental results. The sheer volume of the characterization and analysis work outlined here and the associated expense would seem to support the contention that material behaviour complexity alone limits the range of applicability of autoclave process modelling, especially for prediction of stress and deformation. However this is quite misleading. While there have been previous process models which have examined process-induced stresses (e.g., Bogetti and Gillespie, 1992; White and Han, 1992a, 1992b), understanding of resin mechanical property development during processing and how this behaviour may be modelled is still very limited. Thus, much of the characterization effort outlined here was relatively novel and future characterization efforts need not be repeated to the depth examined here. Also, as outlined earlier, a number of methods may be used to reduce the required level of effort which were not employed here, such as simply making use of available information, using simplified or more efficient test methods (e.g., the 'effective' cure shrinkage test or FTMA analysis), or conducting sensitivity analyses to determine the important properties for a given case. Although much insight into resin behaviour during processing was obtained as a result of the tests outlined here, limited resources prevented in-depth experimental examination of a number of issues of importance. Areas in which it is suggested that additional efforts be focused include: \u2022 Further examination of resin modulus development using 'rectangular torsion' tests or a technique such as Dynamic Mechanical Thermal Analysis (DMTA, Wetton et al., 1989) to examine behaviour over a range of isothermal and non-isothermal conditions and various test frequencies. \u2022 Additional resin cure shrinkage tests, especially using volumetric dilatometry, to examine shrinkage behaviour over a range of temperatures and longer cure times. Examination of the effect of relaxation of non-equilibrium shrinkage strains on post-cure behaviour is also an issue of interest. -151-Chapter 6: Material and Boundary Condition Characterization \u2022 Additional resin and prepreg CTE measurements. Neither the non-linearity of composite thermal strains with temperature nor the thermal strain behaviour at T> Tg were examined here. \u2022 Measurement of resin Tg development with cure (Tg =j{a) using DSC or DMA). A single model for material Tg should then be introduced into COMPRO, replacing different methods now used to account for glass transition currently incorporated in material behaviour models such as cure model 6 and resin modulus development model 2. New material behavioural models can easily be incorporated into COMPRO as they are developed. 6.2 Autoclave Characterization As discussed in previous chapters, most early autoclave process modelling efforts focused on analysis of the composite part itself, expending less effort on consideration of boundary conditions. To extend this previous work and make process modelling more useful in a manufacturing environment, a more detailed examination of these boundary conditions is required. This has become increasingly apparent to the process modelling community as evidenced by the numerous recent modelling efforts (Ciriscioli et al., 1992; Telikicherla et al., 1994a, 1994b; Kline et al., 1995; Pillai et al., 1996) in which tooling thermal effects were considered. A more complex analysis of boundary conditions by directly modelling autoclave internal air flow and heat transfer has been proposed by Kline et al. (1995), and simple preliminary examinations of these phenomena have been performed by Telikicherla et al. (1994a, 1994b). In autoclave processing of thermosetting resin composite materials, the dominant source of heat transfer between the part and its tooling and the surrounding autoclave air is (generally) forced convection heat transfer5. The rate of convection heat transfer in an autoclave is described by the equation: q = hA(TA-Ts) (6.14) -152-Chapter 6: Material and Boundary Condition Characterization where A is the surface area, TA is the 'bulk' autoclave air temperature, Ts is the surface temperature, and h is the 'heat transfer coefficient'. Empirical correlations have been developed for calculation of heat transfer coefficients for numerous simple cases (e.g., laminar flow over a flat plate), but this parameter is not easy to predict for this case. As discussed in Ghariban et al. (1992), air flow patterns inside an autoclave can be complex and highly variable, even when it is empty. When an autoclave is filled with parts for processing, even more complex flow patterns are developed due to the presence of flow obstructions, often resulting in the generation of areas of stagnant air, a phenomena known as 'shadowing' (Roberts, 1987). These internal variations in air flow can result in autoclave air temperature variations of 6 - 8 \u00b0C (Roberts) and a factor of 3 variation in heat transfer coefficients (Ghariban et al.). Given this level of complexity and variability, it is easy to see why autoclave heat transfer rates are often not examined in depth in most composites process models. However, the importance of this factor on processing outcomes dictates that it be examined at some level if only to gage its nominal value and variation. In this section, a preliminary examination of convective heat transfer rates in three different autoclaves is presented. The objective of these experiments was to obtain model inputs and to investigate the variation of heat transfer within an autoclave and between different autoclaves. 6.2.1 Heat Transfer Coefficient Measurement Heat transfer coefficient measurements were performed over a range of temperatures and autoclave air pressures in the three different autoclaves described in Table 6.7. In these tests, a crude version of the calorimeter discussed in Ghariban et al. (1992) was employed. As shown in Figure 6.24, the test apparatus consisted of a thick (3.2 cm) square aluminum plate, 32 cm on a side. The plate temperature was measured at three locations from the plate edge to the centre, midway through its thickness. As discussed in Monaghan et al. (1991), at the higher temperatures involved in thermoplastics processing, radiation can also be a very important mode of heat transfer, perhaps more important than forced convection. -153-Chapter 6: Material and Boundary Condition Characterization Table 6.7: Autoclaves examined in heat transfer characterization tests. Autoclave Primary Usage Dimensions (internal) Autoclave A Production 1.8 m x 4.5 m Autoclave B Development 0.9 m x 1.8 m Autoclave C Development 1.5 m x 2.4 m Breather layers for Figure 6.24: Test apparatus used in measurement of autoclave heat transfer coefficient. The top and bottom surfaces of the plates were exposed to the autoclave air while the edges were heavily insulated, creating an essentially 1-D heat transfer condition (assuming no significant variation in heat transfer coefficient across the plate surface). Plate thermal mass was great enough so that a significant lag could be created between part and air temperatures, allowing accurate determination o f heat transfer coefficients. The very high plate thermal conductivity allowed use o f a ' lumped' thermal mass assumption (maximum Biot number during any test was about 0.04). The heat transfer characteristics of each autoclave were characterized by subjecting the calorimeters to temperature\/pressure cycles such as that shown in Figure 6.26. Heat transfer coefficients during the cycle were evaluated using the equation: h = a*- (6.15) ATA-TS) -154-Chapter 6: Material and Boundary Condition Characterization where: p is the plate density, Cp is its specific heat capacity, V is plate volume, A is the exposed plate area (top and bottom surfaces), TA is the measured autoclave air temperature and Ts is the plate temperature, taken as the average of the three thermocouple measurements. The temperature of the autoclave air was typically measured at a single point between the inner and outer autoclave walls as shown in Figure 6.25. 6.2.2 Autoclave A Autoclave A is a medium-sized production autoclave, approximately 1.8 m in diameter and 4.5 m long, owned by Integrated Technologies Inc. (intec) of Bothell WA. As shown in Figure 6.25 two plates were used in this test, plate 1 near the front of the autoclave and plate 2 placed near the back. Both were placed midway between the autoclave walls about 30 cm from the 'floor'. Figure 6.25: Schematic showing relative location of tool plates in autoclave A heat transfer characterization test. The autoclave temperature\/pressure cycle used for this test and the average measured plate 1 and plate 2 temperatures are illustrated in Figure 6.26. As shown, autoclave air temperature was increased at a rate of 1.11 \u00b0C\/min. to a maximum of 180 \u00b0C, then quickly lowered. During the temperature ramp, autoclave pressure was slowly raised to a maximum of 790 kPa (100 psig), then cycled to isolate the effects of both temperature and pressure heat transfer rates. -155-Chapter 6: Material and Boundary Condition Characterization 0.0E+00 0 20 40 60 80 100 120 Time (minutes) 180 200 Figure 6.26: Autoclave A heat transfer coefficient measurement test. Figure 6.27 shows the calculated heat transfer coefficients for plates 1 and 2 using Equation 6.15. This figure shows clearly the significant influence of autoclave pressure on heat transfer coefficients, found to vary by about a factor of 4. The effect of temperature on heat transfer rate, if any, is not clear. A least-squares analysis was used to determine the best fit of the equation: h = a + b*P (Ta^ + c * T (K) (6.16) to the heat transfer coefficients measured by each plate. This analysis indicated only a small and inconsistent temperature effect; therefore the temperature coefficient,c, was dropped from Equation 6.16. The best fit to the measurements using the simplified 'linear model' was obtained with: Plate 1: Plate 2: hi = 37.4 W\/m'K + 1.83xl0\"4 W\/m2K7Pa * P (Pa.) h2 = 28.9 W\/m2K + 1.54x10\"4 W\/m2K7Pa * P (Pag) As shown in Figure 6.28, good agreement was obtained between the linear pressure model and measured values. -156-Chapter 6: Material and Boundary Condition Characterization Time (minutes) Figure 6.27: Calculated heat transfer coefficient during autoclave A characterization test. 180 ^ 0 20 40 60 80 100 120 140 160 180 200 Time (minutes) Figure 6.28: Comparison between measured heat transfer coefficients and model assuming a linear variation with pressure. -157-Chapter 6: Material and Boundary Condition Characterization 6.2.3 Autoclave B Autoclave B is a small development autoclave, about 0.9 m in diameter and 1.8 m in length. Because of its small size, only a single plate was used for heat transfer measurements. Figure 6.29 shows the temperature\/pressure cycle used to characterize this autoclave as well as the measured average plate temperature and the calculated heat transfer coefficient. Again, a very good fit to the measured heat transfer coefficients throughout the test was obtained using a linear pressure model, in this case: h = 30.1 W\/m2K + 1.61xl0\"4 W\/m2K7Pa * P (Pag) I 4.0E+05 Autoclave temperature Heat transfer coefficient -f ^ -5E+05 Plate temperature Autoclave pressure 1 3.0E+05 2.5E+05 2.0E+05 1.5E+05 1.0E+05 5.0E+04 O.OE+00 10 20 30 40 50 60 70 Time (minutes) 90 100 Figure 6.29: Autoclave B heat transfer coefficient test, showing calculated heat transfer coefficients. 6.2.4 Autoclave C Autoclave C is a small-to-intermediate size development autoclave, approximately 1.5 m in diameter and 2.4 m in length. Two aluminum plates were used to examine heat transfer in this autoclave, with plate 1 placed near the autoclave heat exchanger and plate 2 placed nearer the door. As illustrated in Figure 6.30, the heat transfer coefficients measured in this case are much lower than those found in the other two autoclaves and show a great deal of scatter. The best fit to the measurements was obtained using: -158-Chapter 6: Material and Boundary Condition Characterization Plate 1: Plate 2: hi = 13.5 W\/m2K + 3.02xl0\"5 W\/m2K7Pa * P (Pag) h2 = 12.6 W\/m2K + 3.16xl0\"5 W\/m2K\/Pa * P (Pag) As shown in Figure 6.30, a very poor fit was obtained using a linear model, indicating a much less consistent pressure effect in this case. Dividing the cycle into a two regions, one when the pressure is increasing (region 1) and one when it is decreasing (region 2), it can be seen that very distinct and quite consistent behaviour is found in each. The source of this anomalous behaviour is unclear, but it could be related to a shift in sensitive internal air flow patterns induced by the shift from a net gas inflow to an outflow. 4.0E+05 3.5E+05 3.0E+05 2.5E+05 L 2.0E+05 1.5E+05 1 .OE+05 5.0E+04 0.0E+00 80 100 120 Time (minutes) 180 Figure 6.30: Autoclave C heat transfer characterization test. Note the very poor agreement between measured heat transfer coefficients and fit equations and the different behaviour in regions 1 and 2. 6.2.5 Theoretical Heat Transfer Coefficient Calculation The observed effect of pressure on autoclave heat transfer rates, despite its significance, has not been noted in the process modelling literature, other than in a publication based on the work outlined here -159-Chapter 6: Material and Boundary Condition Characterization (Hubert et al., 1995). This is likely due to the relative lack of traditional focus on process model boundary conditions since the effect is apparently not unknown in industry (Roberts, 1987). The basis of the observed effect of pressure on heat transfer coefficient, as mentioned in Roberts, is the attendant increase in air density. For an ideal gas, density, p, is directly proportional to pressure and inversely proportional to temperature, i.e., P*j (6-17) For a given geometry, the Reynolds number describing the flow of a fluid is directly proportional to fluid density and the 'bulk' fluid velocity, V^, and inversely proportional to the dynamic viscosity, p, as presented in Equation 6.18: Recx^ (6.18) M As discussed in Incropera and DeWitt (1990), for fully-developed turbulent flows, the Nusselt number (Nu), is approximately proportional to the Reynolds number to the power 4\/5, i.e., Nu oc Re4\/S oc - (6.19) k where k is the fluid conductivity. Combining Equations 6.17 to 6.19, we obtain: {$}\\*~ AocJ\u2014>\u2014 (6.20) For air (and nitrogen), the value of kip is roughly constant over the range of interest. Thus, assuming constant bulk air flow velocity during a test, Equation 6.20 reduces to: p^4\/5 (6.21) -160-Chapter 6: Material and Boundary Condition Characterization This reveals the source of both the apparent linear dependence of the measured heat transfer coefficient on pressure and, paradoxically, its apparent temperature independence. This is because, while autoclave pressure varied by as much as a factor of 8, the difference between maximum and minimum absolute temperature was relatively small about 50%. Thus, pressure effects were quite evident while the effect of temperature variation was swamped by measurement noise. A comparison of the measured h\\ from the autoclave A test with best fits from the simplified version of Equation 6.16 (the 'linear' model) and from Equation 6.21 (the 'turbulent' model) is shown in Figure 6.31. Both models are seen to give very similar results with the fit of the turbulent model being the better of the two. B Measured h, Turbulent model: h = a*{PI1]>' \u2014 \u2014 Linear model: h \u2014 a + b*P 20 40 60 80 100 120 140 160 180 200 Time (minutes) Figure 6.31: Measured heat transfer coefficient from autoclave A characterization test compared with best fits using turbulent and linear models. 6.2.6 Discussion Autoclave heat transfer characteristics are an important parameter in the processing of composite materials, and one that has not received much attention in the process modelling literature. This section outlines a simple test for measurement of autoclave heat transfer coefficient over a range of temperatures -161-Chapter 6: Material and Boundary Condition Characterization and pressures using a crude aluminum plate calorimeter. This type of test can be used by composites processors for assessing and possibly enhancing the uniformity of heat transfer rates within an autoclave by such things as improved part placement and 'baffling'. Using this calorimeter, heat transfer coefficients were measured in three different autoclaves over a range of temperatures and pressures. Three main conclusions can be drawn from these tests: 1. Autoclave heat transfer coefficients are significantly influenced by pressure and are also effected by temperature, although to a much lesser degree. In one test, varying autoclave temperature and pressure resulted in a change of heat transfer coefficient of over 400%. 2. Heat transfer characteristics can be very different for different autoclaves. For example, at 375 kPa (40 psig), the average heat transfer coefficient in the three autoclaves examined varied by about a factor of 4. 3. Even within a single autoclave, significant variations in heat transfer rates may exist. The last two of these points highlight an important characteristic of every manufacturing process: inherent process variability. A valid criticism often levelled against process modelling in general is that there is no way for such models to deal with in-process variability so that process cycles developed using modelling must necessarily be conservative. As discussed in Chapter 2, this has lead to the development of numerous 'heuristics'-based process control systems (e.g., Abrams, 1987; LeClair and Abrams, 1988; Ciriscioli and Springer, 1991) which use on-line process measurements combined with expert-system based knowledge for real-time process optimization. However, as discussed in Pillai et al. (1994), the knowledge on which these systems are based is extremely process and material system specific. These authors suggest an alternative in which process models are directly integrated into a system for real-time process control. This approach combines the strengths of both of process modelling (scientific-based understanding of processing phenomena), and on-line process control (adaptability to variations in material properties and external boundary conditions). -162-Chapter 6: Material and Boundary Condition Characterization Some authors (Kline et al., 1995) have proposed using computational fluid dynamics (CFD) models to predict autoclave flow patterns during processing, then using these predictions to determine part boundary conditions. Although such an analysis has been conducted for a simple case (Telikicherla et al., 1994a, 1994b), the practicality of this approach at present is questionable. For example, even the very simple laminar analysis of Telikicherla et al. required massive amounts of computer time (8 hours on a Cray YMP) to analyse a single case. More comprehensive models examining realistic, unstable turbulent flows would require even more computational effort. Even then, such an analysis would only be valid for a single case and would have to be rerun each time autoclave loading was changed. Thus, at present, the best approach is probably experimental temperature and heat transfer coefficient measurement similar to that outlined here. A number of notes regarding the outlined technique should be made: \u2022 In these tests a single measured autoclave temperature was used as the 'bulk' air temperature. Given the known variation in air temperatures within the autoclave, heat transfer coefficients should be calculated from air temperature measurements much closer to the calorimeter (but still well above the part surface). \u2022 Plates should always be supported just at their edges by low thermal mass and\/or low thermal conductivity supports with as little of their surface area covered as possible. \u2022 Such phenomena as 'shadowing' and calorimeter orientation should be examined and more measurements at different autoclave locations should be performed. Measurements of the effects of breather and bleeder layers on the 'effective' heat transfer coefficient would also be useful. \u2022 The presented analysis technique is not valid for cases where radiation effects are significant (e.g., at the high temperatures seen in thermoplastics processing). A discussion of a similar technique for measurement of heat transfer rates in such cases is provided in Monaghan et al. (1991). -163-Chapter 7: Experimental and Numerical Case Studies 7. Experimental and Numerical Case Studies In this chapter, model capabilities are demonstrated through application to three case studies examining different types of processing problems. In each case study, model predictions are compared to experimental results and a sensitivity analysis is performed to determine the predicted sensitivity of processing outcomes to variations in process parameters. The case studies performed are presented in order of increasing analysis complexity as follows: 1. Hybrid solid laminate\/honeycomb structure: A heat transfer and cure analysis is performed for a hybrid structure which transitions from a solid composite section to a honeycomb sandwich section (Figure 7.1). 2. L-shaped angle laminates: An analysis of residual deformation in a number of small L-shaped 'angle' laminates with various ply layups is performed. The focus of this study is the springback of 24-ply quasi-isotropic ([90\/+45\/-45\/0]24) laminates (Figure 7.20). An experimental sensitivity analysis of springback angle is also presented. 3. Representative fuselage substructures: Residual deformation in four intermediate-scale substructures (Figure 7.46) of a large complex fuselage structure is examined. A potential technique for applying COMPRO predictions to residual deformation in very large, complex structures is also presented. In all case studies presented, the base composite material is unidirectional Hercules AS4\/8552 prepreg, whose properties were characterized as part of the current work (see Chapter 6). Properties for AS4\/8552 and other materials used in the described analyses are presented in Appendix C. 7.1 Case Study 1: Hybrid solid laminate\/honeycomb structure The first case study performed is the hybrid solid laminate\/honeycomb structure illustrated inFigure 7.1. This structure was manufactured as part of a project to develop an optimized process cycle for structures -164-Chapter 7: Experimental and Numerical Case Studies of this type (Schulz et al., 1994). Normally, the end sections of this part (i.e., the solid and the honeycomb section) would be processed using very different cycles. For example, for a solid section of AS4\/8552 of this thickness, the manufacturer (Hercules, 1993) recommends using a very high autoclave pressure of 1.14 MPa (150 psig) to ensure full laminate compaction. For a honeycomb section, a much lower pressure of about 375 kPa (40 psig) would normally be used to prevent crushing of the cell walls (known as 'core crush'). 12 plies Figure 7.1: Schematic of the hybrid solid laminate\/honeycomb structure (not to scale). 7.1.1 Experimental Measurements To manufacture this structure, 'kits' of AS4\/8552 prepreg tape were assembled around a pre-cut glass\/phenolic honeycomb core (HEXCEL HRP-3\/16-8.0). A ply of structural adhesive was placed between the prepreg and the core to be later co-cured with the rest of the assembly. During assembly, thermocouples were placed strategically throughout the structure as illustrated in Figure 7.2. After the structure was assembled, wooden blocks were placed around its edges to stabilize the core and a thin reinforced rubber caul was placed on the top surface to help equalize the applied autoclave pressure. The assembly was then vacuum bagged and placed inside the autoclave for processing. The substrate tooling in this case was a very large invar tool with an egg-crate substructure similar to that illustrated in Figure 7.48. -165-Chapter 7: Experimental and Numerical Case Studies A photo of the assembly prior to bagging is shown in Figure 7.3. Note that in addition to thermocouples this photo also shows sensors used to measure resin pressure during processing. 30.5 45.7 61.0 76.2 91.4 106.7 121.9 152.4 cm 4.8 mm thick caul 2 \u2022 J \u2022 | J 0 3 . K . J 2 J _ & 1 4 \u00a9 \u2014 \u2014 ~ T ~ i 17*18 1 9 \u00a9 ^ 1 23\u00ae 1 25\u00ae 1 27 1 1 29 r ' \u00ae 9\u00a9 ff\u00a9 a a \\tia 1 20\u00ae \" 21.22 1 Q 1 1 28 1 1 30 X Autoclave door 45.7 -\u2022 cm 22.9 -\u2022 11.4 -\u2022 0 -\u2022 1 2-6 7-10 11-16 17,19-21 23,24 25,26 27,28 29,30 t \u00a9\u2022 - - - - \u00a9 \u00ab \u00a9 18,22 Y X \u00ae Thermocouple Figure 7.2: Schematic of thermocouple placement during the hybrid structure experimental build (not to scale). Figure 7.3: Instrumented hybrid structure prior to bagging (photo courtesy of P. Hubert). The autoclave process cycle used in this case is shown in Figure 7.4. This process cycle represents a compromise between the ideal process cycles for each of the two end sections. Thus, a low autoclave pressure of 375 kPa (40 psig) is employed to prevent core crush and an intermediate-temperature (149\u00b0C) -166-Chapter 7: Experimental and Numerical Case Studies 'hold' is introduced to remove some of the energy of the resin exothermic reaction before the final cure at 177 \u00b0C. Segment U o D 3 13 938 + 2.56 * r(+25%) 1005 -369 * a m 790 m S1CPR* S1CPF* Thermal conductivity (W\/mK) Resin Fibre k-n Honeycomb 0.148 + 3 .43X10 - 4 * T+ 6.07x10\"2 *a 2.40 + 5.07xl0\"3 * T 0.0774 + 3.4X10 - 4 * T 0.185 + 4.293x10\"4 * T+ 7.59xl0-2*a(+25%) 3.00 + 6.34x10-3 * T (+25%) 0.0968 + 4.25X10\"4* T (+25%) 0.155 + 6.07xl0-2*a'2' 1.80 + 3.80xl0\"3 * T (-25%) 0.0581 + 2.55x10\u00b0 * T (-25%) S1KR* S1KF* SIKH* Fibre Vf{-) 0.573 0.602 (+5%) 0.544 (+5%) S1VF* Prepreg Material AS4\/8552 AS4\/3501-6 SI MAT* Resin heat of reaction (J\/kg) 540x103 590xl03(+10%) 490x103 (-10%) S1HR* Cure kinetics model Cure model #6, parameters in Table 6.6 Cure model # 2 ,3>, E a = 69.9 kJ\/gmole A = 5.333xl05\/s m = 0.79, n = 2.16 SICK* IfillllillB Effective heat transfer coefficients (W\/m2 K) P is gage pressure Top Side: 15.9 + 5.94xl0\"5* P Bottom Side: 45.6 + 1.31x10-'*\/' Top Side: 19.9 + 7.44xl0'5 * P (+25%) Bottom Side: 57.0 + 1 .64X10- 4* P(+25%) Top Side: 15.9 Bottom Side: 45.6 (-0 S1HT* Initial degree of cure (-) <% = 0.05 ab = 0.01 ob = 0.15 SI ALP* Tool thickness (cm) 2.54 3.81 0.635 S1TTK* Tool material invar aluminum S1TMT* IIIIIIRJI^ Heating rates (\u00b0C \/min.) 1.11 0.555 2.22 S1RT* Process control Lead\/lag Temperature\/time SI PC* Notes: < J > Measured variation above Tg (see Section 6.1.2) <2> Room temperature value for nominal case ( 3 > From previous cure kinetics analysis by The Boeing Company (see Section 6.1.7) ( 4 > Taken from the nominal case at atmospheric pressure 7.1.3.2 Sensitivity analysis results As expected, many of the process parameters examined had very little effect on any model predictions while others proved very important, affecting all aspects of the process. An example of the latter is tool thickness. As shown in Figure 7.14, a very different process cycle is effectively seen by a part processed -178-Chapter 7: Experimental and Numerical Case Studies on a 'thin' tool as compared to one processed on a 'thick' tool. Two of the most apparent effects of changing tool thickness are changes in the process cycle time (due to the use of lead\/lag control) and the maximum exotherm. Both of these are a result of different heat capacity: the thicker of tool, the more thermal energy it will absorb, both from the autoclave air and the part during the exotherm. 0 100 200 300 400 500 600 700 Time (minutes) Figure 7.14: Illustration of the effect of tooling thickness on process model temperature predictions for the hybrid structure. Note that the longer process cycle for the thick tool is due to the simulation of lead\/lag control. Another consequence of changing tool thickness, and one with implications for stress development, is the effect this has on part temperature and cure gradients. As illustrated in Figure 7.15, the large temperature gradients in the part processed on the thick tool give rise to correspondingly large gradients in resin degree of cure throughout much of the process. However, the maximum cure gradients at any time in the process cycle are actually observed in the part made using the thin tool due to the large exotherm it experiences. -179-Chapter 7: Experimental and Numerical Case Studies 0.9 TC#4 TC#27 0 100 200 300 400 500 600 700 T i m e ( m i n u t e s ) Figure 7.15: Illustration of the effect of tooling thickness on process model predictions of resin degree of cure in the hybrid structure. Note that most of the shift in time between the two sets of curves is due to the simulation of lead\/lag control. To summarize the predicted effect of other parameter variations on the cure process, four important process model predictions are examined: total process cycle time, maximum gradient in resin degree of cure in the component, maximum exotherm, and 'flow time1'. Figure 7.16 summarizes the predicted sensitivity to all parameter variations of process cycle time, found to be 628 minutes in the nominal case. The examined solution analysis parameters and composite thermophysical properties are shown to have very little effect on this outcome. The most important factors are predicted to be tooling thickness, heat transfer coefficient, and (not surprisingly) process cycle specifications. Variations in heat transfer coefficient, for example, result in a process cycle times ranging from about 600 - 700 minutes. Failing to simulate lead\/lag control resulted in an under-prediction of cycle time of over 3 hours. Defined here as the minimum total amount of time any point has a resin viscosity of less than 100 Pa*s -180-Chapter 7: Experimental and Numerical Case Studies \"1 700 . 650 Figure 7.16: Predicted sensitivity of process cycle time to variation in process variables (hybrid structure). Maximum exotherm is found to be more sensitive to process parameter variations. As shown in Figure 7.17, tooling and the process cycle are again shown to be important variables, but a number of other parameters also have important effects. The most important of these other parameters is the initial resin degree of cure, predicted to result in a range of exotherms from 5.8 \u00b0C (low case) to 15.0 \u00b0C (high case). -181-Chapter 7: Experimental and Numerical Case Studies 16 14 -Figure 7.17: Predicted sensitivity of maximum part 'exotherm' temperature to variation in process variables (hybrid structure). As illustrated in Figure 7.18, the maximum cure gradients during processing are also most strongly influenced by the tool and process cycle. Initial resin degree of cure is again an important parameter as are the resin reaction kinetics and the type of material employed (i.e., AS4\/3501-6 or AS4\/8552). Figure 7.19 shows the effect of variation in process parameters on predicted 'flow time'. This variable is relatively insensitive to variations in most parameters, but is highly influenced by the starting resin degree of cure. This fact is well understood by composites processors who regularly discard excessively 'aged' resin. Other important factors in predicted flow time are the type of material used and reaction kinetics. The process cycle is also predicted to have an important influence on flow time, but the process tooling does not have a large effect. -182-Chapter 7: Experimental and Numerical Case Studies 0.14 g 0.12 u C \u00a3 \u2022= 0 1 \u00b0 0.08 u fi co ra g ra g g> 5 a) o ra g ra g g> g CT g rag cn oi g O ^ 'iS ^ \" S ^ S ^ ^ 'S. ^ S ^ S ^ 'S- ^ c E o t I i j | f 8 \u00ab & \" i \u2022 1 i 1 t 1 s 5 I | I I | | i | f f I | I ! I I i | 5 5 \u00ab E ^ I f I I 1 I c \"\u2022 1 | ^ i I I E * : j :| \u00ab i I I t I I 1 I I I I I 2 ! \u2014 \" 3 \u00ab S 1 ? 2 ? 5 f ? \u00a7 \u2022 \u00a3 o ra ra o ra o o > 0 o > O c Figure 7.18: Predicted sensitivity of maximum degree of cure gradient to variation in process variables (hybrid structure). 300 , 200 E ~ 150 g 50 \u00a3 E- s \u00a3 T to ra o ra o \u2022= Q. W \u00ab I ! S | S 1 \u00ab I E | \u00a3 ih- U I I _0 ra o CTQ ra0 ra o 0 5 O ra Q S? 0 1 6 ra ra o CTCTQ cn Q a> o ra & S W .\u00a3 & .\u00a3 ^ \u00a3 . S S * * ^ - S \" ^ ^ J= o c \u2022 CL \u00ab * \" ? 1 1 1 1 I 1 ! | I I I I 2 I I I 1 1 i I i I I I = 1 | 1 i I f \u00a3 = 1 1 I 1 | 1 I 1 f 1 i | 1 \u00bb i 2 s \u00ab \u00b0 \u00a3 i - | - : , : x x Figure 7.19: Predicted sensitivity of 'flow time' to variation in process variables (hybrid structure). -183-Chapter 7: Experimental and Numerical Case Studies 7.1.3.3 Discussion A number of conclusions can be drawn from this sensitivity study. First of all, the analysis parameters appear to have been well chosen and even lower mesh densities and higher time steps could probably have been used without impairing predictive accuracy. The cure process is predicted to be quite insensitive to variations in most thermophysical properties such as material density, specific heat and thermal conductivity. Initial fibre volume fraction and resin heat of reaction are more important parameters, but are less critical in the modelled case than resin cure kinetics and the initial resin degree of cure. In this case, the most important process parameters were all related to part boundary conditions rather than material properties. These include variations in tooling thickness and material, heat transfer coefficients and in the specified process cycle, especially employing lead\/lag control rather than time\/temperature control. 7.2 Case Study 2: Angle Laminates The second case study illustrates the application of COMPRO to prediction of process-induced deformation in a number of variations of the simple L-shaped 'angle' laminate illustrated in Figure 7.20a. 7.2.1 Experimental Measurements In these experiments, L-shaped angle laminates of AS4\/8552 were layed up ply-by-ply on a pair of solid convex aluminum tools such as illustrated in Figure 7.20b. Parts were processed using one of the three process cycles defined in Figure 7.22 (cycle #1 was used in the 'nominal' case). Typical thermocouple placement for these tests are shown in Figure 7.21. Note that in all tests most thermocouples were placed on the tool rather than in the parts to avoid influencing part deformation. -184-Chapter 7: Experimental and Numerical Case Studies (1 cm inside) Figure 7.21: Thermocouple locations in angle laminate experiment (nominal case). Not to scale. The experimental test matrix examined is presented in Table 7.2. Parameters examined in this study include ('nominal' condition shown first, then variations): \u2022 Laminate layup: [90\/-45\/+45\/0]n, [0]n, [90]n, [90]n[0]n \u2022 Laminate thickness: 24, 12, or 48 plies \u2022 Effect of process cycle: process cycle #1, process cycle #2 or process cycle #3 as defined in Figure 7.22, where: -185-Chapter 7: Experimental and Numerical Case Studies \u2022 Process cycle #1 has a slow two-step heating rate to achieve uniform temperature distribution and to minimize exotherm. Low pressure is applied in order to minimize resin flow. \u2022 Process cycle #2 has a faster one-step heating rate to induce higher temperature gradients and exotherm. The same autoclave pressure profile as process cycle #1 was applied. \u2022 Process cycle #3 has the same temperature profile as process cycle #2, but higher pressure is applied to obtain more resin flow. \u2022 Effect of bagging condition: no bleed or bleed (AS4\/8552 is a no-bleed system). Table 7.2: Angle laminate experimental test matrix. Runs I and 2 (cure cycle I and 2 respectively) Specimen Layup Bagging 1 [90] 1 2 No-bleed 2 [90]24 No-bleed 3 [0],2 No-bleed 4 [0]24 No-bleed 5 [90\/-45\/+45\/0]3 No-bleed 6 [90\/-45\/+45\/0]6 No-bleed 7 [90\/-45\/+45\/0]6 Bleed Runs.{ and 4 (cure cycle 3) Specimen Layup Bagging 1 [90]24 No-bleed 2 [0]24 No-bleed 3 [90\/-45\/+45\/0]6 No-bleed 4 [90]12[0]l2 No-bleed 5 [90]48 No-bleed 6 [Ok No-bleed 14* [90]24[0]24 No-bleed * Note that specimen numbers are not continuous since the results presented here represent a subset of a larger experiment. -186-Chapter 7: Experimental and Numerical Case Studies Process Cycle #1 Process Cycle #2 Process Cycle #3 u B Air temperature \\ 1.1 l\u00b0C\/min. 107\u00b0C - lh 177\u00b0C - 2hr Autoclave pressure 1.1 l\u00b0C\/min. 375 kPa r 170 kPa Full vacuum Vacuum pressure 101 kPa Cycle time U 177\u00b0C -2hr Air temperature 2.78\u00b0C\/min. Autoclave pressure 375kPa 170 kPa full vacuum Vacuum pressure 101 kPa \u2022 \u2014 Cycle time 177\u00b0C - 2hr Air temperature 790 kPa Autoclave pressure Vacuum pressure 101 kPa Cycle time Figure 7.22: Process cycles used for angle laminate experiments (cycle #1 used in nominal case). The temperatures measured by several thermocouples during the process cycle are shown in Figure 7.23. The bag-side part temperature is shown to lead the tool temperatures by several degrees during all temperature ramps. The relatively large difference between the measured temperatures at the tool surface and those inside the tool is quite surprising given the very high aluminum thermal conductivity. The source of this difference is believed to be an imperfect connection between the tool and the inside -187-Chapter 7: Experimental and Numerical Case Studies thermocouples. Another interesting observation is the absence of any temperature exotherm. This is due to the relatively high tool thermal mass and the low part thickness. 180 - r . T i m e (minutes) Figure 7.23: Measured temperatures during processing of angle laminate (process cycle #1). Where two T.C.'s (e.g., T.C.'s #1&7) are indicated, the shown temperatures are the average. After processing, the laminates were removed from the tool and their edges trimmed. As shown inFigure 7.24, a digital image was taken of the edge of the specimen tool-side surface and digital analysis software used to determine the total 'included angle', 9, from points on the extreme ends of each trimmed 'arm'. The springback angle, A9, was then calculated by subtracting the measured angle from 90 degrees (thus, positive springback indicates a reduction in included angle). Three springback measurements were taken for each specimen and the average value recorded. Measurement repeatability indicates an accuracy of about +\/- 0.2 degrees for this technique. A summary of the measured springback angles for all specimens is shown in Figure 7.25. -188-Chapter 7: Experimental and Numerical Case Studies Trim Original Shape Figure 7.24: Measurement of springback angle of trimmed angle laminates. PI - P4 are measurement points used by the image analysis software. 5.0 4.5 4.0 in 3 5 a> a> i\u2014 D) 1 30 2 2.5 ra j* o | 2.0 c a 1.5 1.0 0.5 0.0 12 piies Run 1 Run 2 24 plies rj[90]n \u2022 [\u00b0]n gj [90\/+45\/-45\/0]n O [90\/+45\/-45\/0]n Bleed \u2022 [90]n[0]n Run 1 Run 2 Run 3 48 plies Run 4 Run 3 Run 4 Figure 7.25: Measured angle laminate springback for all specimens. While there were not enough parts of each type to accurately determine variability, springback measurements from runs 3 and 4 (same process cycle) indicate an approximate part-to-part variation of -189-Chapter 7: Experimental and Numerical Case Studies about +\/- 0.2 - 0.3 degrees2. This amount of variability makes it difficult to draw definite conclusions from the data, but some trends are apparent. The observed influences of various examined parameters are as follows: Laminate layup \u2022 As expected, parts with an unsymmetric layup exhibited much more springback than those with a symmetric layup. \u2022 Parts with a [90]n layup tended to spring back significantly less than [90\/+45\/-45\/0]n and [0]n specimens. Also, [90\/+45\/-45\/0]n parts showed a slightly higher average springback than [0]n specimens (average 1.92\u00b0 vs. 1.77\u00b0), although there is too much scatter in the data to be certain whether this is a general trend. Laminate thickness \u2022 A fairly strong tendency towards decreasing springback angle with increasing part thickness was observed. Process cycle \u2022 There was no apparent difference in the parts processed using process cycles #1 and #2. Slightly higher springback was observed for [90]n specimens processed with higher pressure process cycle 3 (average 1.10\u00b0 for runs 1 and 2 vs. 1.33\u00b0 for runs 3 and 4). The opposite trend was observed for [90\/+45\/-45\/0]n (2.04\u00b0 vs. 1.51\u00b0) and [0]n parts (2.08\u00b0 vs. 1.75\u00b0). Bagging Condition \u2022 Higher springback was observed for the two 'bleed' specimens than for 'no-bleed' parts (an average of 2.45\u00b0 vs. 2.08\u00b0). Although it is hard to tell how much of this is specimen variability and how much is measurement error. -190-Chapter 7: Experimental and Numerical Case Studies Another interesting phenomenon observed in this test was a change in specimen thickness in the region of the corner due to resin and fibre bed movement. For most specimens the observed thickness change was quite small, on the order of 0.1 - 0.2 mm, with no apparent trend, either positive ('thickening') or negative ('thinning'). For the 'high-pressure' runs, however, large pressure gradients at the corner resulted in a local thickness reduction of about 10-15% in all [90]n specimens. Measurements of part thickness profile showed that resin was being forced from the corner into adjacent areas. Examples of observed edge profiles of a typical component (no significant corner thickness change) and one that underwent large amounts of corner thinning are shown in Figure 7.26. This phenomenon is discussed in depth in Hubert (1996). Corner thinning U a Figure 7.26: Example part profiles after processing: a) typical part profile, b) part showing corner thinning (no displacement exaggeration). From Hubert (1996). Before proceeding to the numerical analysis, it is instructive to compare the results of these tests with those obtained by other researchers. In similar measurements of springback in L-shaped carbon fibre\/epoxy angles, both Rennick and Radford (1996) and Patterson et al. (1991) observed the same trend in springback angle with part layup as seen here, i.e., Zlf%0]n\u00ab ^ <%>]n < ^<%>o\/+45\/-45\/o]n- As discussed in Section 5.6.2, this trend can be explained by the relative difference in radial and circumferential thermal and cure shrinkage strains for each layup. -191-b Chapter 7: Experimental and Numerical Case Studies Strain anisotropy, however, cannot entirely explain the observed specimen behaviour. For one thing, the springback angles predicted for this phenomenon on its own are too small. Using measured AS4\/8552 properties measured from Chapter 6, springback angles of about 0\u00b0, 0.5\u00b0 and 0.7\u00b0 are predicted for the [90]n, [0]n and [907+45\/-45\/0]n laminates respectively (Equation 2.11). While the predicted differences between the springback angles for different layups is roughly the same as observed here (approximately the same difference was also seen by both Rennick and Radford, 1996 and Patterson et al., 1991), the predicted magnitude is about 1\u00b0 too small in all cases. Another problem is that strain anisotropy should result in a springback angle that is independent of part thickness, yet a definite thickness effect was observed. Clearly, there are some other important sources of residual deformation in this case. 7.2.2 Model Predictions The finite element representation used for the angle laminate analysis is shown in Figure 7.27. The 24-ply quasi isotropic laminate ([90\/+45\/-45\/0]6) was chosen as the 'nominal' case for this study, although parts with [0]24 and [90]24 layups were also examined. The boundary conditions employed for this case are as follows: Heat transfer: A convective heat transfer boundary condition was simulated on all model boundaries. In this case, a uniform heat transfer coefficient was employed where: heff = 20.1 (W\/mK) + 9.3x10\"5 * P (W\/mK\/Pa) This value was obtained from calculations based on tool temperature measurements during processing. This was required since no heat transfer coefficient measurements for this type of geometry were available. Flow: The inner surface of the part was fixed to the tool and autoclave air pressure was applied to the top surface of the part. All boundaries were made impermeable to resin flow to simulate a 'no-flow' condition. -192-Chapter 7: Experimental and Numerical Case Studies Mechanical: During processing, single points on the tool were set to 'fixed' and 'sliding' as shown in Figure 7.27. A thin 'shear layer' (0.37 mm in thickness) with a modulus set equal to that of the aluminum tool was placed between the tool and the part, simulating perfect tool\/part bonding3. During the tool removal process, all nodes but one along the part 'line of symmetry' were set to 'sliding'; this single node was set to 'fixed' to prevent free-body motion. For all simulations in this case study, the lead\/lag controller simulation was employed to predict the autoclave air temperature and pressure during processing. For this simulation, virtual thermocouples were placed at the locations shown in Figure 7.28. Figure 7.27: Finite element representation of angle laminate (nominal case). The predicted temperature and resin degree of cure at selected points in the part and tooling are shown in Figure 7.28. A small through-thickness variation in part temperature and resin degree of cure was This assumption was employed due to observed high degree of bonding between the tool and part at the end of processing, making tool removal quite difficult. -193-Chapter 7: Experimental and Numerical Case Studies observed, but no exotherm was predicted. A contour plot of the predicted part temperature field at one point during the process cycle is shown in Figure 7.29. \\ 3 \\ m \\ x \\ a \\ * \\ a \\ * \\ i \\ , \\ 0.90 0.80 0.70 0.60 0.50 0-40 = Q 4- 0.30 Predicted air temp. Predicted T l Predicted T2 Predicted T9 _, \u201e \u201e\u201e \\ % f 0.20 Predicted alpha (ply 1) ^ Predicted alpha (ply 24) N i O l O 0.00 200 250 Time (minutes) 350 400 450 Figure 7.28: Predicted temperature and resin degree of cure during processing of the angle laminate (nominal case). As shown in Figure 7.29, a maximum through-thickness temperature gradient of about 7 \u00b0C is predicted. Corner temperature is slightly higher than that in the flat section due to increased surface area for convective heat transfer in that region. Figure 7.30 shows that there is good agreement between experimental tool and air temperature measurements. Predicted and measured part temperatures did not agree as well, perhaps due to differences in effective heat transfer coefficients on the various sides of the tool. -194-Chapter 7: Experimental and Numerical Case Studies 180 _ 0 50 100 150 200 250 300 350 400 450 500 Time (minutes) Figure 7.30: Comparison between predicted and measured temperatures during processing of angle laminate (nominal case). After the simulation of part removal from the tool, the springback angle was calculated. This was done using a technique analogous to that used for the real part, moving measurement points inward from part -195-Chapter 7: Experimental and Numerical Case Studies edges to account for trimming. The predicted post-processing shape for the nominal case ([907+45\/-45\/0]6) is that shown previously in Figure 7.24 with predicted displacements exaggerated by a factor of 5. The predicted springback angle for this part was found to be 2.33\u00b0. Laminates with [0]24 and [90]24 layups were predicted to exhibit springback angles of 1.79\u00b0 and 0.05\u00b0 respectively. While the predicted value for the [90]24 layup is much lower than measured, as illustrated in Figure 7.31, the predicted springback angles for both the [90\/+45\/-45\/0]6 and [0]24 laminates are in general agreement with experimental measurements. 2.5 -r 2.0 a 1.0 CL in 0.5 pj Experimental \u2022 Predicted \u2014 Strain anisotropy (Equation 2.11) 0.0 J I \u2014 l I | L _ I I , I I I , [90\/+45\/-45\/0]6 [0]24 [90]24 Figure 7.31: Comparison of experimental and predicted springback angles for angle laminates. Error bars on experiment represent +\/- l a variation from the mean. Also shown is the predicted springback from strain anisotropy alone. As also shown in Figure 7.31, the springback angle predicted for the strain anisotropy mechanism alone is much lower than COMPRO predictions. An important reason for this is revealed by closer examination of Figure 7.24. While the strain anisotropy mechanism would result in changes in corner geometry o n l y , this figure shows that the model predicts that the laminate 'arms' will also warp slightly. Measurements of springback angle at various locations along the arm show that the springback angle at the corner is only -196-Chapter 7: Experimental and Numerical Case Studies approximately 1.06\u00b0. The difference between this value and the measured value of 2.33\u00b0 is due to arm warpage. Although this might seem to constitute a very large amount of warpage, the maximum deviation of the arm edge from a straight line is predicted to be only about 120 (am as shown in Figure 7.32. Distance from corner (mm) Figure 7.32: Predicted warpage of angle arms, nominal case (perfect bonding). The most important source of the predicted arm warpage was found to be tool\/part interaction. As mentioned earlier, tool\/part perfect bonding was simulated in this case by setting the shear layer modulus equal to that of the aluminum tool. The implications of this assumption were examined by varying the shear layer modulus over a wide range and observing the predicted warpage. As shown in Figure 7.33, the employed shear layer modulus has a tremendous impact on warpage, causing a change in maximum warpage magnitude of the [90\/+45\/-45\/0]6 laminate of greater than a factor of 6. -197-Chapter 7: Experimental and Numerical Case Studies 140 T Aluminum [90\/+45\/-45\/0]6 ; 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10 1.0E+11 1.0E+12 Shear layer shear modulus (Pa) Figure 7.33: Effect of shear layer modulus on predicted maximum angle laminate arm warpage for three different layups. As shown in Figure 7.34, shear layer modulus has a similar impact on predicted springback angle, resulting in values ranging from 0.57\u00b0 to 2.33\u00b0 for the quasi-isotropic laminate. The significance of the predicted effect of arm warpage on springback angle prompted re-examination of the experimental specimens. As shown in Figure 7.35, this examination revealed an initially-overlooked warpage, very similar in shape to that predicted by the model. The measured warpage could be closely matched by the model by employing a shear layer modulus of G$L = 6 MPa. -198-Chapter 7: Experimental and Numerical Case Studies 2.5 2.0 S 1.5 I \u2022o OD 1.0 I 0.5 a. -0.5 \\ -\\ ^ \\ Aluminum [90\/+45\/-45\/0J; i [0]24 [90]2 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10 Shear layer shear modulus (Pa) 1.0E+11 1.0E+12 Figure 7.34: Effect of shear layer modulus on predicted angle laminate springback angle for three different layups. Figure 7.35: Comparison of measured arm warpage with model predictions for the [90\/+45\/-45\/0]6 laminate using a shear layer shear modulus of GSL = 6 MPa. -199-Chapter 7: Experimental and Numerical Case Studies Two other sources of process-induced deformation are also evident from the model predictions. One is a uniform reduction in part thickness of about 3.75% caused by resin cure-shrinkage. Resin flow is also predicted to affect part thickness profde. Figure 7.36, shows that a maximum thickness reduction (thinning) of about 0.1 mm is predicted at the corner of the [90\/+45\/-45\/0]6 laminate due to the local pressure gradients. Both the predicted reduction in corner thickness and the thickening of adjacent regions are seen in some experimental specimens. \u2014 i 1 1 1 1 1 20 40 60 80 100 120 Distance S from untrimmed left edge (mm) Figure 7.36: Predicted flow-induced change in [90\/+45\/-45\/0]6 angle laminate thickness and new thickness profde in corner region. Thickness profile changes exaggerated by a factor of 10. 7.2.3 Sensitivity Analysis The sensitivity analysis presented in the previous case study focused on the predicted effect of variability of process parameters on the temperature and resin cure process. In this sensitivity study, a similar examination is performed of the effect of these variables on angle laminate springback. -120 4 0 -200-Chapter 7: Experimental and Numerical Case Studies 7.2.3.1 Process parameters examined Many of the parameters examined in the previous sensitivity study are again included here, as shown in Table 7.3. However, with the shift in focus of the current analysis to process-induced deformation, more emphasis is placed on composite mechanical behaviour. The nominal case for this study is taken to be the 24-ply quasi-isotropic ([90\/+45\/-45\/0]6) laminate examined in the previous section. To facilitate comparisons with experimental results, all laminates examined in the experimental study are also modelled here. The 'high' and 'low' process cycle cases are also chosen to correspond to experimental process cycles. Table 7.3: Parameters examined in angle laminate sensitivity analysis. Note that '*' in the Run Name is replace with 'H' and ' L ' for the high and low cases respectively. Parameter Nominal Value High Low Run Name Maximum Overall Time Step (s) 30 100 10 S2DT* Maximum Degree of Cure Step (Stress) 0.05 0.10 0.025 S2DALP* Maximum Percentage Change in \u00a3rcsin (Stress) 10 20 5 S2DMOD* B^SBlIiiBllllllii|llilll!iilIlHliH^ HI^ B^i Thermophysical properties Resin Specific heat capacity (J\/kgK) 1300 + 4.29 * T -369 * a 2790-3.80 * T\"> 1005 -369 * cP> S2CPR* Resin Thermal conductivity (W\/mK) 0.148+ 3.43X10-4 * T + 6.07x10-2 * a 0.185 + 4.293X10-4 * T+ 7.59xl0\"2* a(+25%) 0.155 + 6.07xl0-2* a'2' S2KR* Fibre Vf(-) 0.573 0.602 (+5%) 0.544 (-5%) S2VF* Resin heat of reaction (J\/kg) 540x103 590x103 (+10%) 490x103 (-10%) S2HR* Cure kinetics model Cure model #6, parameters in Table 6.6 Cure model # 2 Ea = 66.9 kJ\/gmole, A = 5.333x10s\/s m = 0.79, n = 2.16 S2CK* -201-Chapter 7: Experimental and Numerical Case Studies \" , '\" \u2022!', .' Mechanical'properties \u2022. , \u2022 , >f , Resin modulus development model Model #2, Parameters in Table 6.2 Model #1 \"\"(see Table B.5) a c i = 0.608, a C 2 = 0.750 S2EMDL* Resin modulus development: Timing (modulus development model #2) Parameters in Table 6.2 Ten' = -56.7 K TC2a = -23.0 K m Tcia* = -34.7 K TC2l, =-\\.0K'3> S2MOD* Resin modulus development: Initial modulus (modulus development model #2) Parameters in Table 6.2 E\u00b0 = E~\/IO2 \u00a3\u00b0 = \u00a3\"\/10J S2E0* Resin cure shrinkage: amount (shrinkage model #1) Parameters in Table 6.5, (aa - 0.67 actually used) V,s\"= 0.124 (+25%) Ks\" = 0.074 (-25%) S2CSA* Resin cure shrinkage: timing (shrinkage model #1) Parameters in Table 6.5 orci = 0.05, aa = 0.77 o t i = 0.05, o t 2 = 0.57 S2CST* Composite C7\u00a33 (xlO\"6\/\u00b0C) 28.6 27.4 + 6xl0-2* TIJ> 25.7 (-10%) S2CTE* , , ' Part geometry and tool mc Thickness Quasi 24 plies: [90\/+45\/-45\/0]6 48 plies: [90\/+45\/-45\/0],2 12 plies: [90\/+45\/-45\/0]3 S2TQ* Thickness 0\u00b0 24 plies: [0]24 48 plies: [0]48 12 plies: [0],2 S2TZ* Thickness 90\u00b0 24 plies: [90]24 48 plies: [90]4\u00bb 12 plies: [90],2 S2TN* Symmetry [90\/+45\/-45\/0]6 [90]24[0]24 [90],2[0]12 S2SYM* Layup accuracy Perfect [90\/+45\/-45\/0]6 [90\/+45\/-45\/0]6; -4\u00b0 to +4\u00b0 random angle deviation [90\/+45\/-45\/0]6 ; -4\u00b0 to +4\u00b0 random angle deviation S2LAY* Tool material aluminum invar S2TMT* Hi Ull Ul \\ f.\"J 'Mull . \u2022ndl'ini\". Effective heat transfer coefficients (W\/m2 K) 20.1 +9.3X10\"5* P 25.1 + 1.16X10\"4* P 20.1 S2HT* P is gage pressure Initial degree of cure (-) ab = 0.05 Qb = 0.01 \u00abb = 0.15 S2ALP* Resin flow No bleed Bleed No flow S2FLW* Process cycle . \u2022 Process cycle Cure cycle #1 (Figure 7.22) Cure cycle #2 (Figure 7.22) Cure cycle #3 (Figure 7.22) S2CYC* Notes: ( l > Measured variation above Tg (see Section 6.1.2) < 2 > Room temperature value for nominal case < 3 > Shifts modulus development curve by a = +\/- 0.05 at a given temperature. Estimated error is based on measurements of modulus development for other materials tested at a range of frequencies w Based on crude estimates of the variation of specimen CTE3 with temperature from the original ply measurements (taken from a single chart since no numerical data was available). < s > From previous cure kinetics analysis by The Boeing Company (see Section 6.1.7) < 6 > Combined with old calibrated resin cure shrinkage model (see Section 6.1.6) -202-Chapter 7: Experimental and Numerical Case Studies 7.2.3.2 Analysis results Sensitivity analyses results are summarized graphically in Figure 7.37 - Figure 7.41, with numerical results provided in Appendix C. As shown in Figure 7.37, the examined analysis parameters did not have a large effect on springback predictions. This indicates that the chosen 'steps' in time, modulus and degree of cure were adequately small. Figure 7.38 shows that with the exception of cure kinetics, variations in resin and composite thermophysical properties did not play a large role in model predictions. The approximately 0.08\u00b0 change in predicted springback caused by resin kinetics variations is most likely a result of changes in the timing of resin hardening and cure shrinkage strains rather than cure and temperature gradients. 2.334 2.332 2.330 t 2.328 2.326 2.324 S 2.322 2.320 2.318 2.316 2.314 < TO a \u2022o o 2 ro ro ai D Figure 7.37: Predicted sensitivity of angle laminate springback to variation in analysis parameters. -203-Chapter 7: Experimental and Numerical Case Studies 2.34 T Figure 7.38: Predicted sensitivity of angle laminate springback to variation in composite thermophysical properties. Variations in all examined boundary and initial conditions were found to have at least a small effect on springback angle. As shown in Figure 7.39, one of the most important parameters was predicted to be the resin flow boundary condition. Simulating a resin to bleed condition at all external boundaries resulted in an increase in springback angle of about 0.10\u00b0. Also important is the resulting thickness reduction of up to 0.3 mm and a relative 'thickening' of the laminate in the region of the corner. Initial resin degree of cure, boundary heat transfer and process cycle were also found to have small effects on predicted springback angle. -204-Chapter 7: Experimental and Numerical Case Studies 2.45 2.40 2.15 in + I o o Figure 7.39: Predicted sensitivity of angle laminate springback to variation in boundary and initial conditions. As shown in Figure 7.40, the sensitivity of predicted springback to resin and composite mechanical property variations was quite high. The two most important parameters were the model employed for resin modulus development and the initial resin modulus,\u00a30- This indicates that both the 'path' taken by the resin as it hardens and the starting point are important to stress development. The most significant factors in predicted springback angle were found to be the laminate layup and thickness and the tool material. As shown in Figure 7.41, the springback angles of both [90\/+45\/-45\/0]3 and [0]i2 specimens were predicted to be much larger than their 24-ply counterparts. The 48-ply versions of these parts were found to spring back much less. Changing the tool material from aluminum to invar also resulted in a large decrease in springback angle, mostly due to much closer matching of the laminate strains by the invar tool. -205-Chapter 7: Experimental and Numerical Case Studies 3.5 3.0 5 ,g> \u2022si <' l\" \u00a5 c) 15.2 c II n Jl \/ ' ' \u2022 \" 1 , II 1 ' \\ II \" rr ^ _ - n II n J \" , . \/ it \\ n ' \" \\ \" 1 i 1 1 1 i' ' \" i' i II ' 1 zn 1 N ^ _ ' II S ll n \\ ,** \\ 1 1 1 I ' ' \\\" ' i 1 \" i ' '!' ' Figure 7.48: a) Schematic of placement of representative substructures on tooling including thermocouple locations, b) end view of aluminum tool, c) top view of aluminum tool showing detail of 'egg-crate' base. -213-Chapter 7: Experimental and Numerical Case Studies Figure 7.49: Representative substructures assembled and bagged prior to being subjected to autoclave processing. Photo courtesy of The Boeing Company. Measured air temperature during processing and selected tool and part temperature measurements are shown in Figure 7.50. This figure shows that, as in the hybrid solid\/honeycomb part in case study 1, the tool temperature (and therefore that of the lower skin) lags significantly behind the temperature at the top-side of the part. Thus, the top skin cured much more quickly than the bottom one with associated implications for the development, residual stress The use of a non-lead\/lag temperature controller in this case resulted in another potential processing problem. Because of the slow rate of tool heat-up, the bottom skin did not approach the first hold temperature until just before the start of the second temperature ramp. Thus, the time this skin spent at low viscosity was quite short and inadequate compaction of the lower skins may have resulted. Temperature measurements not shown in Figure 7.50 also indicated a problem with uneven air temperature and\/or heat transfer rates within the autoclave. This was indicated by a several degree variation in tool temperature at different locations and differences in part temperatures measured on opposite sides of some J-frames. -214-Chapter 7: Experimental and Numerical Case Studies 180 -r 20 -? Autoclave air TC ToolTC#l PartTC#7 - - - PartTC#10 \u2014 - PartTC#12 0 0 50 100 150 200 250 300 350 400 450 500 Time (minutes) Figure 7.50: Measured temperatures at selected locations on tool and representative parts. Note that TC#12 is not on the same part as TC's #10 and #7 (see Figure 7.48). After processing, the undersides of all components were 'profiled' (by Boeing personnel) employing a laser-based co-ordinate measuring apparatus known as a SMART 310. For all parts, the locations of between 70 and 125 points on the bottom surface were obtained. Calculation of warpage was performed by finding the best fit of the measured points to an ideal cylinder representing the shape of the tool surface using the equation: The difference between the calculated z, values and the ideal cylinder were taken to be the component warpage. Each of the different types of structure was found to exhibit a distinctive warpage profile that was consistent for all specimens. A typical contour plot of the warpage on the underside one of the parts, the stiffened honeycomb structure, is shown in Figure 7.51. This part exhibits a warpage pattern centred -215-Chapter 7: Experimental and Numerical Case Studies about the J-frame, with deflection increasing roughly linearly with distance from the J-frame centre. The maximum deflection in the case shown is about 0.15 mm. J-Frame Foot Outline Z-displacement (urn mils) Figure 7.51: Typical contour plot of warpage of stiffened honeycomb structure. All four of the structures were found to exhibit this type of essentially one-dimensional warpage. Thus, for this case, a clearer way of illustrating component warpage is to 'collapse' the deformation to 1-D and look at the part edge-on. This also allows the warpage for all three specimens of one type to be viewed at the same time. Figure 7.52 shows the z-deflection of the three unstiffened skin specimens. A consistent downward (i.e., toward to tool) warpage was observed for all three specimens of this type, with a maximum deflection of about 0.4 - 0.5 mm. -216-Chapter 7: Experimental and Numerical Case Studies 200 _ -150 -100 -50 0 50 100 150 Distance from part centre line (mm) Figure 7.52: Measured warpage of unstiffened skin structure specimens showing second-order fit to specimen displacement. The measured warpage profiles for the other three types of structure are illustrated in Figure 7.53 to Figure 7.55. As shown in these figures, the unstiffened skin laminates warp in the Y-Z plane while the other three structures warp predominately in the X-Z plane. Of all the structures, the warpage of the unstiffened honeycomb (Figure 7.53) is the lowest and the least consistent, with the maximum absolute deflection of the three specimens varying from about -0.05 mm to 0.1 mm. Measurement scatter is also very large for this structure, and even the maximum deflections are of the order of the accuracy of the measurement technique (about 0.02 mm). -217-Chapter 7: Experimental and Numerical Case Studies 150 130 110 90 70 50 30 10 -10 -30 -50 t Specimen 1 a Specimen 2 4 Specimen 3 Fit to specimen 1 Fit to specimen 2 men 3 I d -150 -100 -50 0 50 Distance from part centre line (mm) 100 150 Figure 7.53: Measured warpage of unstiffened honeycomb structure specimens showing second-order fit to specimen displacement. Note the very small scale of the deformation; in this case only a few times larger than accuracy of the measurement technique. In contrast, warpage patterns for both the stiffened skin and stiffened honeycomb structure specimens are very consistent as well as of relatively large magnitude. As illustrated in Figure 7.54 and Figure 7.55, these structures both exhibit very nearly symmetric warpage about the centre line of the J-frame 'foot' (i.e., its base). In both cases, the warpage varies with distance from this centre line in a nearly linear fashion with some slight, but inconsistent, curvature as the edge is approached. As might be expected, the deflection of the stiffened skin is the greater of the two at about 0.7 - 0.9 mm maximum deflection as compared to about 0.15 mm for the much more rigid stiffened honeycomb structure. -218-Chapter 7: Experimental and Numerical Case Studies 1000 \u2014j-Distance from part centre line (mm) Figure 7.54: Measured warpage of stiffened skin structure specimens showing second-order fit to specimen displacement. -150 -100 -50 0 50 100 150 Distance from part centre line (mm) Figure 7.55: Measured warpage of stiffened honeycomb structure specimens showing second-order fit to specimen displacement. Similar measurements were also taken to assess the shape of the underside of the J-frame feet. It was found that these feet, expected to be essentially flat, were actually 'sprung-in' about 2\u00b0 . -219-Chapter 7: Experimental and Numerical Case Studies 7.3.2 Model Predictions The first step in the modelling phase of the case study involved development of the finite element discretization of each type of structure. Discretization of the unstiffened structures was performed using the COMPRO Editor pre-processor with PATRAN used for the more complex stiffened structures. Part symmetry was exploited for the unstiffened structures, but the entire stiffened parts were examined. In all cases, the complex structure of the tool was replaced with a 0.64 cm solid section with 'equivalent' thermal and stiffness properties. An example of the finite element discretization employed is illustrated in Figure 7.56. As shown, the adhesive layer between the J-frame foot and the top skin is modelled, but not the honeycomb\/skin adhesive nor the adhesive 'noodle' beneath the J-frame foot. Convection Figure 7.56: Finite element representation of stiffened honeycomb structure. The boundary conditions employed for this case study are as follows: Heat transfer: A convection heat transfer condition was simulated on all boundaries. Since no previous heat transfer characterization had been performed for the autoclave used in the experiment, both tool and part heat transfer coefficients were calculated from temperature measurements during processing. The effective heat transfer coefficients employed for the nominal case were: -220-Chapter 7: Experimental and Numerical Case Studies (Top-side) heJf= 4.05 (W\/mK) + 2.41xl0\"5 * P (W\/mK\/Pa) (Bottom-side) \/%\u2022= 36.6 (W\/mK) + 2.00xl0\"4 * P (W\/mK\/Pa) Although the 'shadowing' effect of the frame results in different heat transfer rates on the left and right hand sides of the part, for simplicity a uniform coefficient was applied to all top surface boundaries. The relatively low heat transfer coefficient on this surface indicates that the heat transfer characteristics of the autoclave used for this build are quite poor compared to that employed in the two previous case studies. Due to the odd behaviour of the measured air temperature during the process cycle, the lead\/lag controller simulation was not used in this case. Instead, the measured air temperature was simply matched in the analysis. Resin flow: Resin flow was not simulated because of the very low amount expected in this case. Mechanical: For all simulations, the bottom surface of the tool was set to 'sliding' except for a single point, which was held fixed. During simulation of tool removal, the unstiffened structures were constrained at their respective lines of symmetry while stiffened structures were constrained at the top surface of the J-frames. Tool\/part mechanical interaction was again simulated through the use of an elastic shear layer. As will be explained, the stiffness of the shear layer was not set arbitrarily as in the previous case study, but was 'calibrated' as part of the analysis. Figure 7.57 shows a comparison between experimental and predicted tool and part temperatures during processing. Good agreement was obtained at all three examined locations. -221-Chapter 7: Experimental and Numerical Case Studies 180 -,-- i i \u2014 i 1 1 1 1 i 1 1 1 0 50 100 150 200 250 300 350 400 450 500 Time (minutes) Figure 7.57: Comparison of predicted and measured tool and part temperatures during processing of representative substructures. Dividing the current analysis into a series of four sub-problems allowed independent examination of warpage sources in simple structures where perhaps only one mechanism was dominant before tackling the most complex problem (the stiffened honeycomb structure) in which several competing deformation mechanisms were encountered. The first of the substructures examined is the unstiffened skin. For this simple structure, the dominant source of deformation was found to be the tooling constraint. As discussed in case study 2, the level of tool\/part interaction in a given case, modelled via the shear layer, is a significant unknown. This structure is ideal for examining tool\/part interaction since this is the dominant deformation mechanism and the low thickness of the part (12 plies) and its large span (30 cm total) result in easily measured warpage. The effect of shear layer on the predicted unstiffened skin warpage was examined by varying its modulus over a wide range. As shown in Figure 7.58, using a low shear layer modulus resulted in a nearly symmetric residual strain profile through the skin thickness and thus only a very low level of warpage. -222-Chapter 7: Experimental and Numerical Case Studies The significant constraint of a high shear layer modulus, however, resulted in generation of an unsymmetric residual strain profile, and thus much higher warpage. 8.81E-3 7.91E-3 7.01E-3 6.11E-3 5.20E-3 4.30E-3 3.40E-3 2.49E-3 1.59E-3 6.87E-4 a) 9.30E-3 8.34E-3 7.38E-3 6.42E-3 5.47E-3 4.51E-3 3.55E-3 2.59E-3 1.64E-3 6.79E-4 b) Figure 7.58: Comparison of predicted residual mechanical c strain profiles (c\") in unstiffened skin panel sections using models with a) low shear layer modulus, b) high shear layer modulus. As shown in Figure 7.59, the maximum predicted deflection was found to be very highly dependent on shear layer modulus. Comparing predicted warpage to that actually measured, a shear layer shear modulus, Gs\/., of 5xl04 Pa was ultimately chosen. Figure 7.60 shows that this choice results in a very good fit to experimental measurements in both warpage shape and magnitude. -223-Chapter 7: Experimental and Numerical Case Studies l.E+03 l.E+04 l.E+05 1 .E+06 1 .E+07 l.E+08 1 .E+09 l.E+IO l.E+11 Shear layer shear modulus (Pa) Figure 7.59: Shear layer calibration using unstiffened skin part. 200 _ 100 1 -150 -100 -50 0 50 100 150 Distance from part centre line (mm) Figure 7.60: Warpage prediction for unstiffened skin with calibrated shear layer. Two major sources of process-induced deformation are apparent for the unstiffened honeycomb structure. One is tool\/part interaction, the same as for the unstiffened skin. The other is the rather large lag between the temperatures of the top and bottom skins due to the large thermal mass of the tool and the low air\/tool -224-Chapter 7: Experimental and Numerical Case Studies heat transfer rates in this case. As illustrated in Figure 7.61, this temperature gradient results in corresponding gradients in the cure rate of the two skins and consequently in differences in the rate of resin modulus development. As discussed in Chapter 2, the combination of these hardening gradients with the rapid change in thermal and cure shrinkage strains at this point in the process results in the development of residual stress. 5.0E+09 T 4.5E+09 4.0E+09 _ 3.5E+09 I \u00ab CL. 3 3.0E+09 3 \u2022o I 2.5E+09 I \u2014\u2022 \u2014 E Resin top skin \u2014\u2022\u2014 E Resin bottom skin Alpha top skin Alpha bottom skin 200 250 300 Process time (minutes) 350 400 450 Figure 7.61: Predicted resin degree of cure and resin modulus in top and bottom skins of unstiffened honeycomb structure. Figure 7.62 shows a comparison between measured and predicted warpage for this structure. It is difficult to say how good the agreement is in this case because of the large amount of scatter in the experimental data and since two of the specimens warped predominantly upwards and one downwards. All that can truly be concluded is that the deformation is of the same order of magnitude as measured. Two sources of potential error in this model should be pointed out. One is the rather large uncertainty in the honeycomb moduli employed in the model. Also, the adhesive layer between the honeycomb and skins were not modelled. It is unclear how predictions would be affected by either of these factors. -225-Chapter 7: Experimental and Numerical Case Studies 150 _ . -150 -100 -50 0 50 100 150 Distance from panel centre line (mm) Figure 7.62: Model warpage prediction for unstiffened honeycomb panel. The next part examined was the stiffened skin structure. For this structure, it was clear from the observed warpage pattern that the major source of warpage was an interaction between the skin and the J-frame. However, initial attempts to model this part resulted in very poor results, indicating warpage in the opposite direction to that actually measured. Closer examination of the problem revealed that an important factor had been overlooked. As shown in Figure 7.47, the J-frame includes on its underside an adhesive noodle that had not been included in the original model. Re-doing the analysis, this time incorporating a geometrically crude representation of this noodle, resulted in the predicted post-processing shape shown in Figure 7.63. -226-Chapter 7: Experimental and Numerical Case Studies Figure 7.63: Predicted post-processing shape of J-stiffened skin part with modelling of adhesive noodle (displacements exaggerated by a factor of 10). This magnitude of the predicted deflection from this new model agreed much better with experimental results. However, the bulk of the warpage was predicted to be much too localized, reflecting the crudity of the used geometric representation of the noodle. Unfortunately, model limitations on the number of elements prevented a more detailed geometric description of this region of the foot from being used. For this reason, the effect of the noodle was instead incorporated into the analysis by calibrating theCTE of the entire J-frame foot such that good agreement was obtained with measured total part deflection. Figure 7.64 shows that neither this 'distributed strain' approach nor the crude noodle model accurately predict the shape of the warpage directly beneath the foot, but act as bounds to the actual shape. A more accurate model of the noodle structure is expected to produce better fit in this region. -227-Chapter 7: Experimental and Numerical Case Studies 1000 \u2014 -150 -100 -50 0 50 100 150 Distance from part centre line (mm) Figure 7.64: Model warpage prediction for J-stiffened skin structure showing best fit prediction and prediction with crude adhesive noodle model. The final part examined, representing a synthesis of the other three, is the stiffened honeycomb structure. All of the deformation mechanisms acting in the other structures play a role in determining the final shape of this structure. Fortunately, since analyses of these deformation sources have already been performed in the simpler parts, modelling of this structure is quite straight-forward. The predicted underside deformation for this case is compared to measurements in Figure 7.65. As shown in this figure, a somewhat lower maximum deflection is predicted than obtained from experiment (about 0.12 mm versus 0.15 mm). Away from the foot of the frame, the 'slope' of the predicted deformation is in quite good agreement to that measured, but warpage in the area directly beneath the foot is not well predicted. This points to the most likely source of the observed disagreement as being the distributed foot CTE representation of the adhesive noodle. However, as is discussed in the sensitivity analysis following, the predicted warpage in this case is quite sensitive to input parameters, and other factors may also be involved. -228-Chapter 7: Experimental and Numerical Case Studies 250 200 c o \u2022\u2014 u \u2022a S3 Specimen 1 Specimen 2 Specimen 3 Model prediction (no noodle) \u2022 -50 0 50 Distance from panel centre line (mm) Figure 7.65: Model warpage prediction for J-stiffened honeycomb panel (nominal case). 7.3.3 Sensitivity Analysis The focus of the representative structure sensitivity analysis is again the effect of variability in process parameters on process-induced deformation. In this analysis the maximum underside deflection of the stiffened honeycomb structure is examined. Comparison of the results of this analysis with those of the angle laminates in Section 7.2.3 will provide a comparison of process sensitivity of two very different types of structures. 7.3.3.1 Parameters examined and their variation The parameters examined in this sensitivity study are outlined inTable 7.4 including the 'high' and 'low' values employed in each case. Many of the parameters examined are the same as included in case studies 1 and 2 to provide a basis for comparison of relative sensitivities. A number of additional parameters appropriate to this case were also examined such as honeycomb modulus, CTE, and conductivity and J-frame foot CTE. -229-Chapter 7: Experimental and Numerical Case Studies Table 7.4: Parameters examined in stiffened honeycomb substructure sensitivity analysis. Note that '*' in the Run Name is replace with 'H' and 'L' for the high and low cases respectively. Parameter Nominal Value High Low Run Name A nalysis parameters Maximum Overall Time Step (s) 30 100 10 S3DT* Maximum Degree of Cure Step (Stress) 0.05 0.10 0.025 S3DALP* Maximum Percentage Change in \u00a3 r e s l\u201e (Stress) 10 20 5 S3DMOD* IK I 'n;i>'\u00bb\\MI .\/\/ \u2022\u2022'oi till.'. Resin Specific heat capacity (J\/kgK) 1300 + 4.29* r-369 * a 2790 -3.80 * Tm 1005 -369 * a'2' S3CPR* Resin Thermal conductivity (W\/mK) 0.148 + 3.43X10- 4 * r + 6.07x10-2 * a 0.185 + 4.293x10-\" * T+ 7.59xl0-2 * a(+25%) 0.155 + 6.07xl0'2 * a'2' S3KR* Honeycomb 0.0774+ 3.4X10- 4 * T 0.0968 + 4.25X10- 4 * T (+25%) 0.0581 +2.55x 10\"3 * T (-25%) S3KH* Fibre Vf(-) 0.573 0.602 (+5%) 0.544 (-5%) S3VF* Resin heat of reaction (J\/kg) 540x103 590xl03(+10%) 490 xlO3 (-10%) S3HR* Resin modulus development model Model #2, Parameters in Table 6.2 Model # 1 \"\"(see Table B.5) oc i = 0.608, Oc2= 0.750 S3EMDL* Resin modulus development: Timing (modulus development model #2) Parameters in Table 6.2 Tc,a = -56.7 K TC2a' = -23.0 K <3> Tc,a = -34.7 K r \u201e \u201e = - i . 0 K w S3MOD* Resin modulus development: Initial modulus (modulus development model #2) Parameters in Table 6.2 \u00a3\u00b0 = \u00a3710 2 \u00a3\u00b0 = \u00a3V104 S3E0* Resin cure shrinkage: amount (shrinkage model #1) Parameters in Table 6.5, (aa= 0.67 actually used) K \/ \" = 0.124 (+25%) V,s=\u00b0= 0.074 (-25%) S3CSA* Resin cure shrinkage: timing (shrinkage model # 1 ) Parameters in Table 6.5 ac, =0 .05, Ofc2 = 0.77 oc, = 0.05, Qc2 = 0.57 S3CST* \u2022 Composite CT\u00a33 (xlO^C) 28.6 27.4 + 6xl0\"2 * T(J> 25.7 (-10%) S3CTE* J-Frame foot CTE, (xlO^\/\u00b0C) 8.50 (#) 2.9 12.0 S3CTEJ* Honeycomb moduli (Pa) \u00a3,,=43.6x10', \u00a3 3 3 = 113xl06, G , 3 = 16.6xl06 \u00a3 , ,= 87.6x106, \u00a3 3 3 = 226x10\", G l 3 = 33.2xl06 \u00a3\u201e=21.9xl0 6 , \u00a3 3 3 = 56.5xl06,Gl3 = 8.3xl06 S3HE* Honeycomb CTE, (xlO^\/\u00b0C) 10 15 5 S3F1C* Layup accuracy Perfect -4\u00b0 to +4\u00b0 random angle deviation, both faces -4\u00b0 to +4\u00b0 random angle deviation, both faces S3LAY* \u2022\u2022'r- , \u2022 Boundary anil ihi'tia^'conditions Tool material Aluminum 'equivalent' invar \"equivalent' S3TMT* Effective heat transfer coefficients (W\/m2 K) P is gage pressure Top Side: 4.05+ 2.41x10\" S*P Bottom Side: 36.6 + 2 . 0 0 X 1 0 - 4 * P Top Side: 6.08 +3.62x10-5 * P (+50%) Bottom Side: 54.9 + 3.00xl0'4* P(+50%) Top Side: 4.05 Bottom Side: 36.6 (j) S3HT* Initial degree of cure (-) a, = 0.05 \u00ab ) = 0.15 Qb = 0.01 S3ALP* -230-Chapter 7: Experimental and Numerical Case Studies Notes: Measured variation above Tg (see Section 6.1.2) ( 2 ) Room temperature value for nominal case < 3 ) Shifts modulus development curve by a = +\/- 0.05 at a given temperature. Estimated error is based on measurements of modulus development for other materials tested at a range of frequencies < 4 ) Based on crude estimates of the variation of specimen CT\u00a33 with temperature from the original ply measurements (taken from a single chart since no numerical data was available). ( 5 ) From previous cure kinetics analysis by The Boeing Company (see Section 6.1.7) ( 6 ) # - Calibrated value. 7.3.3.2 Analysis results Sensitivity analysis predictions for a number of parameter categories are summarized graphically in Figure 7.66 - Figure 7.68. Numerical results for all runs are provided in Appendix C. For the most part, the relative importance of parameter variations is quite similar to that found previously in case study 2. Analysis parameters again had little effect on predicted deflection, nor did any composite thermophysical properties with the exception of resin cure kinetics (Figure 7.66). As shown in Figure 7.67, boundary and initial conditions (including tool material) were of intermediate importance, with all parameters except initial degree of cure having a significant influence on predicted warpage. Varying the mechanical properties of the various materials in the structure (Figure 7.68) again had the greatest effect on deformation predictions. For the range of property values examined, predicted deflections ranged from -0.06 mm to 0.23 mm. There were also a number of differences in the predicted sensitivity of the stiffened honeycomb deformation as compared to the angle laminate springback. For example, variations in heat transfer coefficient (+ and -) resulted in about 1.5% variation in springback angle in case study 2; in this case the range of predicted maximum deflection was much higher at about 18% (with the same relative change in heat transfer coefficients). -231-Chapter 7: Experimental and Numerical Case Studies E 3-130 -125 120 115 \u00ab 110 105 -| S 100 9 5 ^ 90 o I f Figure 7.66: Predicted sensitivity of maximum deformation of stiffened honeycomb structure to variation in thermophysical properties. 1 40 130 120 1 \u00bb 110 .\u00a7 IOO 90 3 o CC I Figure 7.67: Predicted sensitivity of maximum deformation of stiffened honeycomb structure to variations in boundary and initial conditions. The predicted sensitivity of structure warpage to mechanical property variations (Figure 7.68) was similarly magnified, with virtually all parameters examined causing at least a 10% shift in model Chapter 7: Experimental and Numerical Case Studies predictions. The most significant was the J-frame foot thermal expansion coefficient, variations in which resulted in changes in model predictions of over 100%. Another important difference from the previous analysis is that the effect of parameters related to tool\/part interaction was much lower in this case due to the much lower shear layer modulus employed. Thus, while still not insignificant, neither tool material nor initial resin modulus were predominant sources of variation in this case. Figure 7.68: Predicted sensitivity of maximum deformation of stiffened honeycomb structure to variations in mechanical properties. 7.3.4 Substructure Analysis Using C O M P R O The preceding sections demonstrated the feasibility of applying the current model to deformation predictions in substructures of a large, complex structure. Attention is now turned to the process for transferring deformation predictions from the 'local' COMPRO model to a large-scale 'global' finite element model. To do this, the following technique is proposed: \u2022 The element nodes of the undeformed global model must correspond to strategically chosen positions ('reference nodes') in the undeformed local model, as shown in Figure 7.69. -233-Chapter 7: Experimental and Numerical Case Studies \u2022 The process-induced u, w displacements of the 'reference nodes' are determined using the current model. \u2022 Strains and curvatures ({s}, {K}) are determined such that the global element nodes see these u, w displacements (Figure 7.69b). \u2022 In the global model, the calculated strains and curvatures are specified as element 'properties' such as thermal expansion (or other) coefficients. These strains and curvatures can then be applied by increasing the component temperature, for example, by AT- 1. Given the local reference nodes, strains and curvatures for each element in the global model can be determined using only the local finite element mesh, predicted nodal displacements, and the global element shape functions. Figure 7.69: Substructure analysis using COMPRO. The behaviour of the global model elements (left hand side) is matched to give an equivalent response to the output of the local model (right hand side). -234-Chapter 7: Experimental and Numerical Case Studies 7.3.5 Case study 3: Summary\/Discussion In this case study, a procedure for examination of process-induced deformation in 'representative' substructures of a large, complex structure was demonstrated and a proposed technique outlined for applying the analysis results to deformation predictions in large structures. Experimental measurements of deformation in the examined structures indicated a consistent pattern of warpage distinctive for each structure type. Subsequent analysis indicated the dominance of only one or two sources of deformation in the simpler structures (the unstiffened skin, the unstiffened honeycomb, and the stiffened skin). This allowed these deformation sources to be addressed more or less independently, permitting detailed examination of such issues as tool\/part interaction and the effect of the J-frame noodle. Model deformation predictions in most cases compared very well with measured warpage in both warpage pattern and magnitude. Predicted total warpage in the stiffened honeycomb structure was slightly lower than measured, probably due to simplified modelling of the J-frame foot region and inaccuracies in employed material properties. The performed sensitivity analysis indicated that the deformation of this structure is relatively sensitive to process parameter variations as well as inaccuracies in finite element representation used for the structure. 7.4 Summary and Discussion Application of the developed process model is demonstrated for three very different case studies, presented in order of increasing model complexity. In each case study, model predictions are compared with experimental measurements and a sensitivity analysis is carried out to examine the effect on model predictions of variations in process and modelling parameters. In the final case study, a technique was proposed for examination of process-induced deformation in large complex structures by employing a substructuring technique. Model predictions were generally in good agreement with experimental measurements in all three presented case studies, particularly with respect to observed warpage patterns in different types of parts. -235-Chapter 7: Experimental and Numerical Case Studies However, some areas requiring future work were also noted. In particular, the effect of tool\/part interaction, now modelled via an elastic shear layer, should be examined. While it is possible to 'calibrate' the employed shear layer modulus for a particular case (demonstrated in case study 3), a much better understanding of tool\/part interaction is required before its effects can be predicted a priori. In addition, there are some concerns about the use of the shear layer in highly curved sections (such as examined in case study 2) if tool\/part interaction is weak. In such cases, since bagging constraint is not currently modelled, an unphysical separation of the simulated part from the tool can easily occur. Also, in neither the described experiments nor in the model predictions were moisture effects considered. Although it is not believed that this would have any significant effect on the parts examined in case studies #1 and #3, it is conceivable that this could have affected the deformation of the angle laminates studied in case study #2. A number of conclusions can be drawn from the results of the performed sensitivity analyses. For the examined cases: \u2022 Normal variations in composite thermophysical properties such as resin and fibre conductivity, specific heat and mass density have very little effect on process model predictions. Small variations in initial fibre volume fraction also do not seem to be an important factor, although the effects of potentially larger in-process variation's'due to resin flow were not considered. \u2022 The resin cure kinetics model employed plays an important role in process model predictions, especially warpage. This is more a result of the effect of cure model predictions on resin hardening and cure shrinkage behaviour rather than its effect on temperature distribution (at least in the cases examined). \u2022 The virtual autoclave simulation can be a useful tool for a priori prediction of the part boundary conditions during processing. This approach has significant advantages over the traditional approach in which process models have either employed either an unrealistic preset time\/temperature process cycle or used measured air temperatures after the fact (e.g., Bogetti, 1989). Improved process cycle -236-Chapter 7: Experimental and Numerical Case Studies prediction could be obtained by developing more sophisticated process controllers and simulation of the autoclave itself. \u2022 Process tooling has very important effects on both part temperature and boundary mechanical constraints that should not be overlooked. Although some of these effects are well known in the composites industry (Borstell and Turner, 1987, Pagliuso, 1982), tooling effects, especially tool\/part mechanical interaction, are not generally considered in autoclave processing models. One probable reason is that academic research has often focused on warpage of small parts, as generally appropriate for examining fundamental material behaviour. As shown by the results of case studies 2 and 3, even though the consequences of this warpage may be very important (e.g., significant effect on springback), relatively large parts are required before it is easily identifiable. \u2022 Initial resin degree of cure is predicted to have important effects on flow behaviour and a smaller but potentially important effect on other processing outcomes such as residual deformation. Since resin degree of cure often varies over a fairly wide range during a material's productive lifetime, it is important that further investigation into this factor be undertaken. \u2022 Small variations in resin modulus development and resin cure shrinkage behaviour generally had important, but not overwhelmingly large effects on predicted residual deformation. The exception to this was initial resin modulus, which had a large effect on the springback of the angle laminates due to significant tool\/part interaction effects in this case. This indicates that there is some value in reexamining the arbitrary assignment of a non-zero 'initial' resin modulus as a certain fraction (e.g., l\/1000,h) of the full modulus. \u2022 Boundary heat transfer rates had a significant effect on predictions of both part temperature and residual deformation. In particular, failure to include the effect of autoclave pressure on heat transfer coefficient was shown to have important implications for predicted processing behaviour. -237-Chapter 7: Experimental and Numerical Case Studies \u2022 Different levels of variability in process outcomes should be expected from structures of different types. This is true for a number of reasons. For one thing, the dominant source of deformation in one structure might be much more variable than that in another. For example, deformation due to component thermal strain mismatch, for example, is probably much more consistent than tool\/part interaction. Another problem arises when two equally dominant mechanisms compete. If mechanism A causes warpage of + 100 and B causes warpage of -99, then a 5% increase in either results in a 5-fold difference in observed residual deformation. \u2022 Examination of very local behavior may be required in some instances in order to get accurate predictions. This is clearly demonstrated in case study 3 by the effects of the J-frame noodle on part warpage. -238-8. Summary and Discussion An integrated model for prediction of process-induced deformation during the processing of composite structures has been developed. A 2-D finite element model employing an incremental instantaneously linear elastic plane strain formulation is used to analyse all five major sources of process-induced deformation identified from the literature including: thermal strains, resin cure shrinkage strains, gradients in temperature and resin degree of cure, resin flow, and tooling mechanical constraints. The integration of the analyses for all of these sources allows the model to be employed as a research tool to identify and focus effort on major process drivers. The combination of an optimized solution strategy and advances in desktop computing capabilities allows rapid analysis of composite structures of intermediate size and complexity using very modest computational resources. The developed model can therefore also be a useful tool for the composite processor, assisting in tool and structure design, and in the development of robust, optimized autoclave process cycles. Other accomplishments of the current work include: \u2022 Process model application was demonstrated for three numerical and experimental case studies, spanning the range of current model capabilities. In one case study, a substructuring technique was demonstrated for predicting process-induced deformation in large structures using intermediate-scale COMPRO analyses. \u2022 Numerical sensitivity studies were performed to examine the effect of variation in process parameters on process model predictions. These analyses show predictions to be quite insensitive to variations in thermophysical properties while process tooling (both thermal mass and constraint), part layup and part mechanical properties all showed important effects. \u2022 A mechanism for tooling constraint - induced warpage has been proposed and successfully modelled using an elastic 'shear layer' approach. -239-C h a p t e r 8: S u m m a r y a n d D i s c u s s i o n \u2022 Characterization of most material properties required by the thermochemical and stress modules was performed, and equations developed to describe observed material behaviour. New models for prediction of resin cure rates and resin elastic modulus development were introduced. \u2022 An autoclave simulation including modelling of a lead\/lag controller algorithm was implemented and its usefulness for prediction of industrial autoclave processing demonstrated. This represents a significant improvement in part thermal boundary condition modelling as compared to previous methods. \u2022 The heat transfer characteristics of three autoclaves of varying size were examined using a simple aluminum plate calorimeter. The heat transfer coefficient was found to be very highly dependent on autoclave pressure, changing by more than a factor of four over the temperature and pressure range. The observed effect was explained by appealing to established empirical correlations for fully-developed turbulent flows. 8.1 Future Work During the course of the current work, a number of issues arose which indicate the need for future investigations. Some areas of further work that might yield the greatest return for the composites process modeller are presented, divided roughly into numerical modelling and experimental sections: 8.1.1 Numerical Modelling \u2022 Development of improved techniques for modelling tool\/part interaction. While current models for tool\/part interaction can provide significant insights into the development of process-induced warpage via this mechanism, the need for improved methods is apparent. A logical first step would be to investigate improvement of the current shear layer approach employing such concepts as a hardening resin shear layer rather than the constant modulus material currently used. -240-Chapter 8: Summary and Discussion \u2022 Development of improved models for material mechanical response. Recent advances in computation power that have allowed much shorter COMPRO solution times compared with previous analyses have also made feasible more complex models of material behaviour, including viscoelasticity. The increased effort involved in such an analysis should be weighed against any potential accuracy improvements that might be derived. For example, it is not clear that viscoelastic models would be significantly more accurate than an elastic model in the examined cases due to vitrification early in the process cycle. \u2022 Application to real-time autoclave process control systems. Computational speed advances also make plausible incorporation of the process model into process-model based autoclave control systems such as that discussed in Pillai et al. (1996). \u2022 Increasing the sophistication of the autoclave simulation. The current autoclave simulation is very basic and should be upgraded to include more sophisticated models of autoclave internal heat transfer, thermal response and control systems. It might also prove useful to incorporate simulations of smart control systems into the autoclave model to allow evaluation of these systems and other advanced processing concepts. \u2022 Incorporation of a module for void generation and transport. The presence of voids is an important quality issue in composites processing which is not addressed in the current model. However, given the modular structure of the current process model, development and integration of a module for void generation and transport could be relatively easily accomplished. \u2022 Implementation of fabric micromechanics models. This would facilitate improved modelling of fabric materials. These are currently modelled in COMPRO using such techniques as replacing each fabric ply with four uni-directional plies with a layup of [0\/90\/90\/0]. There is also some doubt as to the validity of standard micromechanics models for prediction of ply properties at low resin degree of cure. This should be investigated. -241-Chapter 8: Summary and Discussion \u2022 Increase flexibility in boundary conditions modelling. It would be useful in some cases to allow the user to specify some boundary conditions now unavailable such as 'prescribed heat flux' (thermochemical module) and point and distributed loads (stress module). \u2022 Implementation of stress calculation. While the focus of COMPRO is prediction of process-induced deformation and residual strains are already calculated, it is often useful to know stresses as well. 8.1.2 Experimental \u2022 Further investigation of tool\/part interaction. Tool\/part mechanical interaction has been shown to be a very important parameter in model deformation predictions and there is reason to believe that this is an important source of manufacturing variability (Ridgard, 1993). However, little is currently known about the mechanics of the tool\/part interaction and much experimental study is required. \u2022 Further model validation using very simple parts. Some of the insights obtained through the application of the model to the case studies can be used to develop experiments to isolate and examine in detail individual deformation sources such as flow effects and temperature and cure gradients. \u2022 Comparison of model predictions with literature studies. A number of case studies are available in the literature that offer potential for qualitative comparison with model predictions. Although material properties data are most often not available, it is expected that trends could still be predicted using approximate material models. \u2022 Further investigation of resin and composite mechanical property development during processing. The characterization tests described in this thesis were performed for the practical purpose of obtaining process model inputs and, while adequate for present needs, highlighted the inadequacies in current understanding of such behaviour as resin cure shrinkage and modulus -242-Chapter 8: Summary and Discussion development. More fundamental investigations are likely to indicate the need for improved models and simplified characterization tests. \u2022 Further investigation of autoclave heat transfer. Increased understanding of autoclave heat transfer characteristics including internal temperature variations and the effects of such effects as 'shadowing' would result in improved boundary conditions modelling. -243-References F.L. Abrams, \"An Expert System Process Controller for Advanced Composites\", The Automotive Challenge and Plastics Response: Automotive Plastics RETEC '87, Dearborn, MI, Nov. 2-4, 1987, pp. 117-119. D. Adolf and J.E. 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Crossman, \"Analysis of the viscoelastic response of composite laminates during hygrothermal exposure\", Journal of Composite Materials 15, 1981, pp. 21-40. S.R. Ganeriwala and CA. Rotz, \"Fourier Transform Mechanical Analysis for Determining the Nonlinear Viscoelastic Properties of Polymers\", Polymer Engineering and Science 27 (2), 1987, pp. 165-178. N. Ghariban, D.Y.S. Lou, and A. Haji-Singh, \"Effect of Honeycomb Flow Straighteners on Turbulence and Heat Transfer in Autoclave Model\", Heat Transfer Effects in Materials Processing, ASME HTD233, 1992, pp.45-52. L.E. Govaert, L. Kodde, A.A.J.M. Peijs, \"A micromechanical approach to viscoelasticity of unidirectional hybrid composites\", Proceedings ofICCM\/8, Honolulu, July 15-19, 1991, pp. 33-A-l to 10. O.H. Griffin Jr., \"Three-Dimensional Curing Stresses in Symmetric Cross-Ply Laminates with Temperature-Dependent Properties\", Journal of Composite Materials 17, Sept. 1983, pp. 449-463. S.K. Ha and G.S. Springer, \"Time dependent behavior of laminated composites at elevated temperatures\", Journal of Composite Materials 23, 1989, pp. 1159-1197. H.T. Hahn and N.J. Pagano, \"Curing Stresses in Composite Laminates\", Journal of Composite Materials 9, Jan. 1975, pp. 91-108. -245-References H.T. Hahn, \"Effects of Residual Stresses in Polymer Matrix Composites\", Journal of the Astronautical Sciences 32 (3), 1984, pp. 253-267. N.L. Hancox, \"The Thermal Conductivity of Composite Materials\", Proceedings of the First Conference on Computer Aided Design in Composite Materials Technology, CA. Brebbie, W.P. de Wilde, and W.P. Blain, Eds., Southampton, England, 1988, pp. 189-207. B. Harper, D. Peretz, and Y. Weitsman, \"Assessment of Chemical Cure-Shrinkage Stresses in Two Technical Resins\", AIAA Paper 83-799, 24th Structures, Structural Dynamics, and Materials Conference, 1983, pp. 29-35. B.D. Harper and Y. Weitsman, \"On the effects of environmental conditioning on residual stresses in composite laminates\", International Journal of Solids and Structures 21, 1985, pp. 907-926. Z. Hashin, \"Analysis of composite materials - A survey\", Journal of Applied Mechanics 50, 1983, pp. 481-505. D.H.P Hasselman, K.Y. Donaldson, and J.R. Thomas Jr., \"Effective Thermal Conductivity of Uniaxial Composite with Cylindrically Orthotropic Carbon Fibres and Interfacial Thermal Barrier\", Journal of Composite Materials 27 (6) , 1993, pp. 637-644. Hercules Aerospace Company Product Data Sheet 847-6. Hercules Aerospace Company Product Data Sheet H050-377. Hercules Aerospace Company, Interoffice Memo from E.W. Rubie, April 1, 1992. Hercules Aerospace Company, \"8552 Composite Systems\", January 1993. M.W. Holl and L.W. Rehfield, \"An Evaluation of the Current Status of Automated Process Control for Thermosetting Composites\", Composites Materials Testing and Design, ASTM STP 1120, Philadelphia, 1992, pp. 308-319. P. Hubert, A. Johnston, A. Poursartip, and R. Vaziri, \"Two-Dimensional Finite Element Processing Model for FRP Composite Components\", Proceedings of The 10th International Conference on Composite Materials (ICCM\/10), Whistler, B.C., Canada, August, 1995, pp. 111-149 - III-156. P. Hubert, \"Aspects of Flow and Compaction of Laminated Composite Shapes During Cure\", PhD. Thesis, The University of British Columbia, Vancouver, B.C., May, 1996. W.H. Hyer, \"Calculations of the Room-Temperature Shapes of Unsymmetric Laminates\", Journal of Composite Materials 15, July 1981, pp. 296-310. M.W. Hyer, CT. Herakovich, S.M. Milkovich, and J.S. Short, Jr., \"Temperature Dependence of Mechanical and Thermal Expansion Properties of T300\/5208 Graphite\/Epoxy\", Composites 14, 1983, pp. 276-280 . Incropera and DeWitt, Introduction to Heat Transfer, Second Edition, Wiley, New York, 1990. A. Johnston and P. Hubert, \"Effect of Processing Variables on Springback of Composite Angles\", A Report to the Boeing Company, Oct., 1994. L. Kalra, M.J. Perry, and L.J. Lee, \"Automation of Autoclave Cure of Graphite\/Epoxy Composites\", Journal of Composite Materials 26 (17), 1992, pp. 2567-2585. J.M. Kenny, A. Trivisano, and L.A. Berglund, \"Chemorheological and Dielectric Behavior of the Epoxy Matrix in a Carbon Fiber Prepreg,\" SAMPE Journal 27 (2), 1991, pp. 39-45. J.M. Kenny, \"Integration of Process Models with Control and Optimization of Polymer Composites Fabrication\", Proceedings of the Third Conference on Computer Aided Design in Composite Materials Technology, 1992, pp. 530-544. -246-References K.S. Kim and H.T. Hahn, \"Residual Stress Development during Processing of Graphite\/Epoxy Composites\", Composites Science and Technology 36, 1989, pp. 121-132. Y.R. Kim, S.P. McCarthy, and J.P. Fanucci, \"Compressibility and Relaxation of Fiber Reinforcements During Composite Processing\", ANTEC, 1990, pp. 1252-1256. R.A. Kline and M.C. Altan, \"Computer-Controlled Method for Composite Curing\", United States Patent 5,207,956, May 4, 1993. R.A. Kline and M.C. Altan, \"Curing of Composite Materials Using Extended Heat Transfer Models\", United States Patent 5,453,226, Sept. 26, 1995. Y. Kuen, I. Lin, and LH. Hwang, \"Thermo-viscoelastic analysis of composite materials\", Journal of Composite Materials 23, 1989, pp. 554-569 Y. Kuen, I. Lin, and S. Yi, \"Analysis of interlaminar stresses in viscoelastic composites\",International Journal of Solids and Structures 27, 1991, pp. 929-945 P.W.K Lam and M.R. Piggott, \"The Durability of Controlled Matrix Shrinkage Composites Part I : Mechanical Properties of Resin Matrices and Their Composites\", Journal of Materials Science 24, 1989, pp. 4068-4075. S.R. LeClair and F. Abrams, \"Qualitative Process Automation\", Proceedings of the IEEE Conference on Decision and Control Including the Symposium on Adaptive Processes, 1(3), pp. 558-563, 1988. S.N. Lee, M.T. Chiu, and H.S. Lin, \"Kinetic Model for the Curing Reaction of a Tetraglycidyl Diamino Diphenyl Methane\/Diamino Diphenyl Sulfone (TGDDM\/DDS) Epoxy Resin System\", Polymer Engineering and Science 32 (15), 1992, pp. 1037-1046. S.Y. Lee and G.S. Springer, \"Effects of Cure on The Mechanical Properties of Composites\", Journal of Composite Materials 24, Dec. 1990, pp. 1270-1299. S.Y. Lee and G.S. Springer, \"Filament Winding Cylinders: I. Process Model\", Journal of Composite Materials 22, Jan. 1988, pp. 15-29. W.I. Lee, A.C. Loos, and G.S. Springer, \"Heat of Reaction, Degree of Cure, and Viscosity of Hercules 3501-6 Resin\", Journal of Composite Materials 16, 1982, pp. 510-520. M. Levitsky and B.W. Shaffer, \"Residual Thermal Stresses in a Solid Sphere Cast From a Thermosetting Material\", Journal of Applied Mechanics, Sept. 1975, pp. 651-655. A.C. Loos and G.S. Springer, \"Curing of Epoxy Matrix Composites,\" Journal of Composite Materials 17 (2), 1983, pp. 135-169. G.E. Mabson and E.P. Neall, III, \"Analysis and Testing of Composite Aircraft Frames for Interlaminar Tension Failure\", Presented at a meeting of the American Helicopter Society, Bridgeport, Connecticut, March 15-16, 1988. D.M. Maguire and F.A. Kulacki, \"Thermophysical Properties of Composite Materials - A State of the Art Assessment\", Fifth International Conference on Composite Materials (ICCM-V), 1985, pp. 1711-1726. A.Y. Malkin, S.A. Bolgov, V.P. Begishev, and V.A. Mansurov, \"Evolution of Viscoelastic Properties of Polyurethane in the Course of Curing\", Rheologica Acta 31, 1992, pp. 345-350. J.-A.E Manson and J.C. Seferis, \"Process Simulated Laminate (PSL): A Methodology to Internal Stress Characterization in Advanced Composite Materials\", Journal of Composite Materials 26 (3), 1992, pp. 405-431. J. Mijovic, \"Cure Kinetics of Neat Versus Reinforced Epoxies\", Journal of Applied Polymer Science 31, 1986, pp. 1177-1187. -247-References J. Mijovic and H.T. Wang, \"Modeling of Processing of Composites Part II - Temperature Distribution During Cure,\" SAMPE Journal 24 (2), 1988, pp. 42-55. J. Mijovic and CH. Lee, \"A Comparison of Chemorheological Models for Thermoset Cure,\" Journal of Applied Polymer Science 38 (2), 1989, pp. 2155-2170. J. Mijovic and J. Wijaya, \"Effects of Graphite Fiber and Epoxy Matrix Physical Properties on the Temperature Profile Inside Their Composite During Cure,\" SAMPE Journal 25 (2), 1989, pp. 35-39. R. Mohan and D.F. Adams, \"Nonlinear creep-recovery response of a polymer matrix and its composites\", Experimental Mechanics 25, 1985, pp. 262-271 P.F. Monaghan, M.T. Brogan, and P.H. Oosthuizen, \"Heat transfer in an autoclave for processing thermoplastic composites\", Composites Manufacturing, 2 (3\/4), 1991, pp. 233-242. S. Montserrat, \"Vitrification and Further Structural Relaxation in the Isothermal Curing of an Epoxy Resin\", Journal of Applied Polymer Science 44, 1992, pp. 545-554. R.H. Nelson and D.S. Cairns, \"Prediction of Dimensional Changes in Composite Laminates During Cure\", 34th International SAMPE Symposium, May 8-11, 1989, pp. 2397-2410. M. Ochi, K. Yamashita, and M. Shimbo, \"The Mechanism for Occurrence of Internal Stress during Curing Epoxide Resins\", Journal of Applied Polymer Science 43, 1991, pp. 2013-2019. S. Pagliuso, \"Warpage, a Nightmare for Composite Parts Producers\", Progress in Science and Engineering of Composites, ICCM-4, T. Hayashi, K. Kawata, and S. Umekawa, Ed., Tokyo, 1982, pp. 1617-1623. K.P. Pang and J.K. Gillham, \"Anomalous Behavior of Cured Epoxy Resins: Density at Room Temperature versus Time and Temperature of Cure\", Journal of Applied Polymer Science 37, 1989, pp. 1969-1991. J.M. Patterson, G.S. Springer, and L.P. Kollar, \"Experimental Observations of the Spring-back phenomena\", Eighth International Conference on Composite Materials (ICCM-VIII), 1991, pp. 10-D-l -10-D-8. D. Peretz and Y. Weitsman, \"The nonlinear thermoviscoelastic characterizations of FM-73 adhesives\", Journal ofRheology 27, 1983, pp. 97-114. V. Pillai, A.N. Beris, and P. Dhurjati, \"Heuristics Guided Optimization of a Batch Autoclave Curing Process\", Computers and Chemical Engineering 20 (3), 1996, pp. 275-294. V.K. Pillai, A.N. Beris, and P.S. Dhurjati, \"Implementation of Model-Based Optimal Temperature Profiles for Autoclave Curing of Composites Using a Knowledge-Based System\", Industrial and Engineering Chemistry Research 33, 1994, pp. 2443-2452. V. Pillai, A.N. Beris, and P. Dhurjati, \"Intelligent Curing of Thick Section Composites Using a Knowledge-Based System\", Journal of Composite Materials 31 (1), 1997, pp. 22-51. D.W. Radford and R.J. Diefendorf, \"Shape Instabilities in Composites Resulting from Laminate Anisotropy\", Journal of Reinforced Plastics and Composites 12, January 1993, pp. 58-75. D.W. Radford, \"Cure Shrinkage Induced Warpage in Flat Uni-Axial Composites\", Journal of Composites Technology and Research 15 (4), 1993, pp. 290-296. T.S. Rennick and D.W. Radford, \"Components of Manufacturing Distortion in Carbon Fiber\/Epoxy Angle Brackets\", 28th International SAMPE Technical Conference, November 4-7, 1996, pp. 189-197. C. Ridgard, \"Accuracy and Distortion of Composite Parts and Tools: Causes and Solutions\", SME Technical Paper EM93-113, 1993. -248-References G.D. Roberts, D.C. Malarik and J.O. Robaidek, \"Viscoelastic properties of addition-cured polyimides used in high temperature polymer matrix composites\", Proceedings of ICCM\/8, Honolulu, July 15-19, 1991, pp. 12-H-l to 12-H-10. R.W. Roberts, \"Cure Quality Control\", Engineered Materials Handbook, Vol. 1: Composites, Dostal, C. A., Woods, M.S., and Ponke, A.W., Eds., ASM International, Metals Park, Ohio, 1987, pp. 745-760. J.D. Russell, \"Cure Shrinkage of Thermoset Composites\", SAMPE Quarterly, Jan. 1993, pp. 28-33. E.P. Scott, \"Determination of Kinetic Parameters Associated with the Curing of Thermoset Resins Using Dielectric and DSC Data\", Composites: Design, Manufacture, and Application, ICCM\/VIH, Honolulu, 1991, pp.10-0- 1-10. E.P. Scott and J.V. Beck, \"Estimation of Thermal Properties in Carbon\/Epoxy Composite Materials during Curing\", Journal of Composite Materials 26 (1), 1992, pp. 20-36. R. A. Schapery, \"Stress analysis of viscoelastic composite materials\", Journal of Composite Materials 1, 1967, pp. 228-267 G.W. Scherer and S.M. Rekhson, \"Viscoelastic-Elastic Composites: I, General Theory\", Journal of the American Society 65 (7) , pp. 352-360. J. Shi and R. Flannagan, \"A Simple Spring-Damper-Slider Model For Laminate Slippage\", Proceedings of The 10th International Conference on Composite Materials (ICCM - 10), A. Poursartip and K. Street, Eds., Whistler, B.C., Canada, August, 1995, pp. Ill-197 - III-204. D. B. Scholz, K.M. Nelson, W.B. Avery, B.W. Flynn, C.C.M Eastland, D.W. Stobbe, P. Hubert, A.A. Johnston, and A. Poursartip, \"Material and Processing Developments for Composite Fuselage Sandwich Structure\", Proceedings of the 5th NASA DOD Advanced Composites Technology (ACT) Conference, Seattle, WA, August, 1994 G.D. Smith and A. Poursartip, \"A Comparison of Two Resin Flow Models for Laminate Processing\", Journal of Composite Materials, 27(17), 1993, pp. 1695-1711. G.D. Smith, \"Modeling and Experimental Issues in the Processing of Composite Laminates\", MASc Thesis, University of British Columbia, Vancouver, B.C., August, 1992. G.S. Springer and S.W. Tsai, \"Thermal Conductivities of Unidirectional Materials\", Journal of Composite Materials 1 (2), 1967, pp. 166-173. G. S. Springer, \"Moisture and Temperature-Induced Degradation of Graphite Epoxy Composites\", Environmental Effects of Composite Materials, 2, pp. 6-19. Technomic (1984). R.J. Stagano and S.S. Wang, \"Process-Induced Residual Thermal Stresses in Advanced Fibre-Reinforced Composite Laminates\", Journal of Engineering for Industry 106, Feb., 1984, pp. 48-54. A. Stephan, E. Schwinge, J. Muller, and H. Ory, \"On the Springback Effect of CFRP Stringers: An Experimental, Analytical and Numerical Analysis\", 28th International SAMPE Technical Conference, November 4-7, 1996, pp. 245-254. M.A. Stone, I.F. Schwartz, and H.D. Chandler, \"Residual Stresses Arising From Partial In-Service Post-Cure of GRP Structures\", Composites 25, 1994, pp. 177-181. H. Stutz, K.-H. Illers, and J. Mertes, \"A Generalized Theory for the Glass Transition Temperature of Crosslinked and Uncrosslinked Polymers\", Journal of Polymer Science: Part B: Polymer Physics 28, 1990, pp.1483-1498. L. Sun, S.C. Mantell, and D.R. Cohen, \"Resin Flow and Compaction in Filament Wound Cylinders\", Proceedings of The 10th International Conference on Composite Materials (ICCM - 10), A. Poursartip and K. Street, Eds., Whistler, B.C., Canada, August, 1995, pp. 111-189 - III-196. -249-References M.K. Telikicherla, X. Li, M.C. Altan, and F.C. Lai, \"Numerical Study of Conjugate Heat Transfer in Autoclave Curing of Thermosetting Composites\", Proceedings of the 1994 International Mechanical Engineering Congress and Exposition, ASME HTD 289, 1994, pp. 213-221. M.K. Telikicherla, M.C. Altan, and F.C. Lai, \"Autoclave Curing of Thermosetting Composites: Process Modelling for the Cure Assembly\", International Communications in Heat and Mass Transfer 21 (6), 1994, pp.785-797. S.W. Tsai and H.T. Hahn, Introduction to Composite Materials, Technomic Publishing Co., 1980. S.W. Tsai, Composites Design, 4th ed., Think Composites, Dayton, 1988. T.E. Twardowski, S.E. Lin, and P.H. Geil, \"Curing in Thick Composite Laminates: Experiments and Simulation\", Journal of Composite Materials 27 (3), 1993, pp. 216-250. M.E. Turtle, H.F. Brinson, \"Prediction of the long-term compliance of general composite laminates\", Experimental Mechanics 26, 1986, pp. 89-102. T. Uesaka, I. Kodaka, S. Okushima, R. Fukuchi, \"History-dependent dimensional stability of paper\", Rheologica Acta 28, 1989, pp. 238-245. R.C. Wetherhold and J. Wang, \"Difficulties in the Theories for Predicting Transverse Thermal Conductivity of Continuous Fiber Composites\", Journal of Composite Materials 28 (15), 1994, pp. 1491-1498. R.E. Wetton, G.M. Foster, V.R. Smith, J.C. Richmond, and J.T. Neill, \"Dielectric Monitoring of Epoxy Cure - Detailed Analysis\", 33rdInternational SAMPE Symposium, 1988, pp. 1285-1294. Y. Weitsman, \"Residual Thermal Stresses due to Cool-Down of Epoxy-Resin Composites\", Journal of Applied Mechanics 46, 1979, pp. 563-567. S.R. White and H.T. Hahn, \"Mechanical Property and Residual Stress Development During Cure of a Graphite\/BMI Composite\", Polymer Engineering and Science 30 (22), 1990, pp. 1465-1473. S.R. White and H.T. Hahn, \"Process Modeling of Composite Materials: Residual Stress Development during Cure. Part I. Model Formulation\", Journal of Composite Materials 26 (16), 1992a, pp. 2402-2422. S.R. White and H.T. Hahn, \"Process Modeling of Composite Materials: Residual Stress Development during Cure. Part II. Experimental Validation\", Journal of Composite Materials 26 (16), 1992b, pp. 2423-2453. S.R. White and Z. Zhang, \"The Effect of Mandrel Material on the Processing-Induced Residual Stresses in Thick Filament Wound Composite Cylinders\", Journal of Reinforced Plastics and Composites\\2, June 1993, pp.698-711. -250-A. COMPRO Structure and Operation This Appendix discusses the overall structure and operation of the COMPRO code which is introduced in Chapter 3 of the main body of this thesis. While not intended as a users guide, this Appendix provides some details that the user should be aware of before using the code. Topics discussed in this Appendix include: \u2022 Program structure and flow \u2022 Input and output files, including viewing COMPRO predictions \u2022 The finite element problem description (local axes orientation, regions, boundaries) \u2022 The autoclave simulation \u2022 Dynamic time step calculation \u2022 Program internal error handling The operation of the thermochemical and stress modules are provided in both the main body of this thesis and in Appendix B. A.1 COMPRO Software Basics The computational engine of COMPRO consists of approximately 21000 lines of code in total (including extensive in-line comments) written in FORTRAN 771. This language was chosen because it is easy to read, widely understood, very portable, and quite well suited to this type of numerically-intensive analysis. Care was taken throughout coding development to create a modular code which could be modified and extended as new knowledge comes available without requiring major code rewrites. Another major factor in code structure development was the facilitation of a 'virtual autoclave' approach (Section 3.2) in which the operation of an autoclave was simulated to allow better approximation of component boundary conditions. -251-Appendix A: COMPRO Structure and Operation A.2 COMPRO inputs and outputs Once it has been launched, virtually all communication between COMPRO and the 'outside world' is made via text files2. This route chosen chiefly for reasons of simplicity and portability. In a)), COMPRO utilizes 4 different types of input and output fdes as follows: 1. Initialization Files - Provide initialisation information required by COMPRO and the COMPRO Editor. 2. Input Files - Provide COMPRO with all required model input information with the exception of material properties data. 3. Material Database Files - Provide COMPRO with material property specification information. 4. Output Files - Provide the user with model predictions and the status of the simulation. A.2.1 Initialisation Files The initialisation file COMPRO.INI provides COMPRO with start-up information required before the program can begin a batch run. This includes such things as the directory locations of all input, output, output and database files, program version number, etc. A similar initialisation file, COM_EDlT.INI, is used by the COMPRO Editor (see Section A.8) for a similar purpose. A.2.2 Input Files With the exception of 'material database' files (Section A.2.3) which provide material property specifications, 'input' files provide all data required by COMPRO to perform a batch run. This There are a few exceptions to this, where language extension were used for functions unavailable in the standard code. These, however, are confined to a pair of subroutines and are clearly noted. The exceptions to this are transient summary results and error messages which are output to the screen and user-interrupts. -252-Appendix A: COMPRO Structure and Operation information includes such things as the list of projects to be run, the input files to be included in each project, a finite element description of the problem, etc. A total of 8 different input files are used by COMPRO, including: \u2022 The batch file (COMPRO.BCH) - Contains a list of all projects to be run in the current 'batch'. \u2022 Project files (PROJECT FILE.PRJ) - Contain a list of the input files containing the project description as well as other project-specific information (e.g. output file prefix) \u2022 Finite element description files (FE MESH FILE.FAT) - Contain the finite element (F.E.) description of the problem (see Section A.3). \u2022 Boundary and initial conditions specification files (BOUNDARY FILE.BCY) - Contain specifications for all model boundary and initial conditions. \u2022 Program control data files (CONTROL FILE.CTL) - Contain all information used by COMPRO to control its internal operation including modules to be executed, time step control data, output time intervals, etc. \u2022 Process cycle definition files (PROCESS CYCLE.CYC) - Contain all information related to specification of the autoclave process cycle including controller specifications, virtual instruments, etc. \u2022 Layup definition files (LAYUP FILE.LAY) - Contain ply layup specification information for all composite material regions. \u2022 Material identification files (MATERIAL ID FILE.MAT) - Contain the identification names of the materials in each model region. As illustrated in Figure A.l, these input files are arranged into a 3-level hierarchy in which the batch file comprises the list of projects to be modelled, and theproject files in turn contain the list of 6 input files describing the problem. Since the projects are performed sequentially and the input files included in a project need not have the same prefix, the same input files can be used in any number of projects in a single batch run. This feature is especially useful when performing parametric studies in which only a single parameter (and thus a single input file) is changed at any one time. -253-Appendix A: COMPRO Structure and Operation Batch File Project File A Project File X F.E. Model File Program Control Data File Layup Definition File Boundary and Initial Conditions File Process Cycle Definition File Material Identification File Figure A.l: COMPRO input fde hierarchy. A.2.3 Material Database Files In the material identification file (MATERIAL ID FILEMAT), the user provides a list of the identification names of the materials in each region, for example 'AS4\/3501-6, low viscosity'. The complete set of properties specification data for each material is contained in a material database file referred to as a material properties specification file (MATERIAL DATA FILE.DAT). To allow for descriptive material names such as the one above, while conforming to DOS file name limitations, we employ a database reference file (LOOKUP.DAT). This file essentially acts as a database of 'pointers' between the descriptive material names in the material identification file and the short-named material properties specification files. This concept is illustrated in Figure A.2. LOOKUP.DAT Material A Material ID File Material A = A.DAT Material B = B.DAT Material Q = Q.DAT Material R = R.DAT Material X = X.DAT A.DAT X.DAT Material X Figure A.2: Use ofLOOKUP.DAT as a pointer to material database fdes. A.2.4 Output Files During a batch run, COMPRO creates two types of output text files: -254-Appendix A: COMPRO Structure and Operation \u2022 The batch run summary file (SUMMARY.BCH) containing such information as the list of projects in the batch run, project runs times and generated error and warning messages. \u2022 System snapshot output files (RUNNAME.###) - containing 'snapshots' of all major state variable values at every model node or element as well as autoclave state variables, project comments, etc. These output files are generated at intervals specified in the program control data file (CONTROL FILE.CTL). The naming convention for these files is ' RUNNAME.###' where RUNNAME* is the name given for the project run, specified in each project file and '###' is the number of the output file (i.e. 001, 002, ..). A maximum of 999 system snapshot output files can be created for a given run due to DOS limitations. It should be noted that these files can be quite large (often 100 - 200 kilobytes each) and the user should take care to avoid running out of disk space, especially when doing a batch run. A.2.5 COMPRO File Naming Conventions A summary of the naming convention for COMPRO files and the information contained in these files is provided in Table A.l. Table A.l: COMPRO file naming conventions File Name File Type Information Contained COMPRO.im Initialization The directory location of all program input, database, and output files as well as other program start-up information (e.g. version, etc.). BOUNDARY FILE. B C I Input Boundary and initial condition specification data. CONTROL.CTL Input Program control data. PROCESS CYCLE FILE.CYC Input Process cycle specification information. LA YUP FILE. L A Y Input Ply layup specification information for all composite material regions. FE MESH FILE. P A T Input Problem finite element description. PROJECT FILE. P R J Input List of the input files to be used for a given project, the run name and a project description. -255-Appendix A: COMPRO Structure and Operation MATERIAL ID FILE.MAT Input List of the reference names for the material in each region. COMPRO.BCH Input A list of all the projects in the current 'batch' run. MATERIAL DATA FILE.DAT Database Material properties specification data. LOOKUP.DAT Database A list of all available materials and pointers between material names and their associated file names. RUNNAME.Um Output All state variables at output interval ###. Up to 999 RUNNAME.### files can be created. SUMMARY. BCH Output Batch file summary information such as a list of projects in the batch run, run times, error messages, etc. Notes: \u2022 File names in italics are set by the user, those not in italics are constant. For example, the layup specification file LAYUP FILE.LAY can have any name with the extension .LAY (e.g. FILE1.LAY), but the batch file is always COMPRO.BCH. \u2022 RUNNAME is the name of the run as specified in the project file. Note that the run name cannot exceed 8 characters in length (this will change in future versions for Windows 95 and NT only) A.2.6 Viewing COMPRO Output Results Rather than developing a new post-processor for viewing COMPRO outputs, it was instead decided to write 'translators' to extract the information from the COMPRO output files and rewrite it is a format compatible with commercial post-processing packages. Currently translators exist to create input files for: \u2022 MS EXCEL\u2122 - used to view the variation of a parameter with time at a single point. \u2022 Tecplot\u2122 - used to view contour plots of parameter values over the entire domain and for viewing deformed shape. A.3 Component Finite Element Description As discussed in Chapter 3, all three major solution modules (thermochemical, flow and stress\/deformation) employ the finite element method to solve for the component state variables during processing. The component finite element description can be created using virtually any finite element -256-Appendix A: COMPRO Structure and Operation pre-processor which can generate an output text file3. Translation subroutines can then be written internally to COMPRO which will extract the needed information from the pre-processor file and convert it into a form used internally by the program. Currently, translation routines are available only for PATRAN and the COMPRO Editor pre-processor. A.3.1 Element Definition and Node Numbering Currently, COMPRO computations can only employ four-noded isoparametric quadrilateral elements as depicted in Figure A.3. As shown in this figure, a list of nodes for an element (i.e., the element connectivity) must be provided in a counter-clockwise fashion. It does not matter, however, which node is provided first as this will not be used to indicate the material principal axes direction. Figure A.3: Counterclockwise node numbering sequence required by COMPRO. All COMPRO solution modules employ a 'skyline' matrix solution technique to solve their respective nodal variable equations and are most efficient when certain node numbering sequences are used. Wherever possible, node renumbering should be performed for skyline minimisation, or if this not available, for minimisation of the stiffness matrix bandwidth. A.3.2 Region Definition In COMPRO, a region is defined as a collection of elements containing a single type of material. In addition to defining element material type, regions are used in combination with reference boundaries Some editing may be required using the COMPRO Editor. -257-Appendix A: COMPRO Structure and Operation (see Section A.3.4) to define ply layup and element local axes orientation. Regions are also used to define which elements represent process tooling, usually removed at the end of the processing cycle, and which are part of the composite structure (whether specifically defined as composite materials or not). Important considerations in region definition include: \u2022 All elements containing one type of material need not be combined in a single region; in fact, it is usually best that groups of elements that are physically separated from each other are defined as belonging to different regions. \u2022 A given element may be part of only one region. \u2022 The initial conditions of all element-based state variables (e.g. resin degree of cure) will be the same for all elements in a region. \u2022 If a region contains a composite material, it must be defined such that its maximum thickness (as defined by the perpendicular distance from the reference boundary) is equal to the sum of the thicknesses of the plies that it contains. There is no limit on the minimum thickness of a region. \u2022 The local axes orientation of all elements in a region are defined by the reference boundary for that region (if one is defined), not by the element nodal connectivity. \u2022 While regions need not be 'continuous' (see Figure A.5), this is not recommended, especially for anisotropic or composite material regions. \u2022 The list of elements in a region need not be listed in any particular order in the input file. One consequence of the current region definition is that it is not possible to define plies of more than one type of material in a single region. To define a layup with multiple types of materials, it is necessary to define separate regions for each material. -258-Appendix A: COMPRO Structure and Operation z Figure A.5: Two finite element meshes each with two defined regions: a) continuous (recommended), b) discontinuous (permissible but not recommended). A.3.3 Boundary Definition 'Boundaries', as defined in COMPRO, are used for three main purposes: to specify 'boundary conditions' (e.g. heat flux, vacuum pressure), for definition of ply layup, and for definition of the element local axes orientation. As shown in Figure A.6, COMPRO boundaries are always defined by node groups (never by element 'faces' as in some finite element codes). The 'rules' for defining these boundaries are quite straightforward as follows: \u2022 Any number of nodes from a single element may be included in a boundary (a boundary may go completely around an element). \u2022 The nodes in any one boundary must form a continuous line with no gaps or 'branches' (see Figure A.7). \u2022 The nodes in a boundary need not be listed in any particular order in COMPRO input files. -259-Appendix A: COMPRO Structure and Operation The same node can be part of multiple boundaries. Boundaries need not be at the outside edge of the model (i.e. they can be defined between elements in the model) Boundaries used to define non-essential boundary conditions (e.g. heat flux), reference boundaries, and 'sliding' displacement must contain more than one node. The same boundary can be used for multiple purposes by COMPRO (e.g. to define a layup reference boundary, a heat flux boundary and a pressure boundary) \"\u2014nr 12 ' 30 24 36 T ' 17 ' ' 11 ' 1 29 ' ' li ' ' 35? 16 IU 1 11 ' 35 Y 15 y ' 11' 1\\ ' o 14 1 1 -t ^ 8 1 r-t lb ' ' TT< 2U 1 TTT< 1 J 2 T {B}, = {1 2 3 4 5 6} T {B}2 = {1 7 13 19 25 31} T { B * } , = {1 23 4 5 6} T {B*}2 = {1 13 725 19 31} T {B}3 = {31 32 33 34 35 36} T {B*}3 = {31 32 34 33 35 36} T Unordered boundary node list Ordered boundary node list Figure A.6: Example boundary definitions, showing unordered (input) and ordered node lists. 3 1 I 26 ' 1 21 1 13 \" 7 \" 25 _ 31\" 12 24 36 27 z-< 26 ' ^\u00ab r*-< {B*}, = {13 1 2 3 4 5 6 18}T {B4}2 = {7 25 26 21}T {B*}3 = {31} {B}, = {7 13 19 25 26 27}T {B}2= {12 24 36} 7 Figure A.7: a) Permissible and b) Impermissible boundary definitions. A.3.4 Element Local Axes Orientation and Ply-Element Intersection Calculation Since COMPRO permits the definition of anisotropic (orthotropic) materials, it is necessary that the material principal axes orientation (x'-z') be defined with respect to the global axes (x-z). As shown in -260-Appendix A: COMPRO Structure and Operation Figure A.4, in 2-D analyses such as this one, this relative orientation can be defined using a single rotation angle, \/J. In the current analysis, the rotation angle for the elements in a given region are determined through the use of the reference boundary defined for that region. As illustrated in Figure A.8, the local axes orientation assigned to each element in a region is that of the reference boundary element (defined as a consecutive pair of boundary nodes) whose centre is the closest to the element centroid, i.e. that boundary element for which dR is a minimum where: and where xh x-,+i, z\u201e z,+\/ are the coordinates of the z'th boundary element and xc, zc are the element centroidal coordinates. It should be noted that if a reference boundary is not defined for a region, the element local axes is taken to correspond to the global axes (i.e. the rotation angle fi is zero). For regions containing composite materials, the reference boundary is also employed for another task: definition of the layup reference surface. From this defined surface, the list of plies contained in each element and the location of the intersections of these plies with the element edges can be determined. The first step in determining the plies in each element is to calculate the perpendicular distance from the reference boundary to each element node. This is done by first finding the reference boundary element whose centre is closest to the node, i.e. that boundary element for which the distanced as defined in Equation A. 1 (replacing xc, zc with nodal coordinates x,\u201e z\u201e) is a minimum. The perpendicular distance from the boundary to the node, dz', is then determined using: ( A . l ) dz' = m { X n - X i ) - ( Z \u201e - Z i (A.2) Z; m = x, -261-Appendix A: COMPRO Structure and Operation where xh xi+1, zh zi+1 are as defined in Equation A . l and x\u201e, z\u201e are the nodal coordinates. The ply number at the node is then determined by dividing '^ by the initial ply thickness (all plies in the region start out with the same thickness). Once the ply numbers at all nodes in a region have been determined, those for each element are calculated from the average of the lowest ply numbers at the element nodes (this defines the 'first' ply in the element) and the average of the highest ply numbers at the element nodes (this defines the 'last' ply in the element). It should be noted that it is assumed that the plies are parallel to the element \u00a3 axis (see Figure A.9). This assumption can easily be verified for a given element by checking that the two highest and two lowest nodal ply numbers are equal. Once the maximum and minimum ply numbers in each element are known, the intersections of the plies with the element sides in local element coordinates are easily determined. Since the ply planes are parallel to the element \u00a3 axis, they thus intersect the element sides at\u00a3 = -1 and \u00a3,= 1 and at the same n location at both sides (Figure A.9) of the element. Thus the ply-element intersections can be defined as a vector {r)}int with nint elements4 where rj, = -1.0 and nnh,i = 1.0. The algorithm for calculation of the other intersections is quite straightforward and is not provided here. 4 The value nint is usually, but not always equal to nply +1 where nply is the number of plies in the element. -262-Appendix A: COMPRO Structure and Operation Boundary elements Boundary element axes orientations a b Figure A.8: a) Calculation of nodal distance from reference boundary, b) Calculation of element local axes orientation using orientations of reference boundary elements 1 Global coordinates L o c a l coordinates Figure A.9: Plies in an element and their intersection with the element sides The algorithm for calculation of element local axes orientation and the ply-element intersection vector is as follows: 1) Define reference boundaries (user input in BOUNDARY FILE.BCl) 2) Loop over all boundaries 3) Reorder the list of boundary nodes so that they are listed in the same order as they appear on the boundary 4) Calculate the local axes orientation for each boundary element accounting for which direction, if any, is 'inward' 5) Loop over all model regions 6) If a reference boundary is defined for the region then 7) Loop over all boundary elements in the reference boundary a) Calculate the location of the boundary element centres -263-Appendix A: COMPRO Structure and Operation 8) Loop over all elements in the region 9) Determine the element local co-ordinate axes orientation (angle \/ J - See Figure A.4) a) Calculate the location of the centroid of each element b) Find the boundary element whose centre is closest to the element centroid c) Assume the local co-ordinate axes orientation of the boundary element in 9b) 10) t^he material in the region is a composite then 11) Determine which plies are contained within each element and the ply-element intersection vector, {njint a) Loop over all nodes in the region i) Find the boundary element whose centre is closest to the node ii) Calculate the local region thickness at the node (Equation A.2) iii) Calculate the ply number at the node, checking that the calculated number does not exceed the number of plies defined for the region b) Check that the maximum thickness calculated anywhere in the region is not significantly less than that expected from the defined number of plies and the ply thickness c) Loop over all elements in the region i) Determine the 'first' and 'last' plies in the element from the ply numbers at the nodes ii) Calculate the ply-element intersection vector, {T| }INT 12) RETURN Notes: \u2022 The is algorithm is currently written exactly as shown, but all concepts used are the same. \u2022 If no reference boundary is defined for a region, the element local coordinates axes are defined to be the same as the global axes \u2022 Ply numbers at the nodes need not be integers. A.4 Time Step Calculation and Module Activation COMPRO does not employ a single user-defined time step throughout an analysis, but rather employs an algorithm which dynamically determines the time step based on the solution state and the current rate of solution change. The same conditions are used to determine which of the solution modules is executed at each time step. This procedure is employed to minimize computational effort and while at the same time maximizing solution accuracy by focusing effort where it is most needed. Long time steps may be used when the solution is changing slowly, such as during long 'hold' periods with shorter steps used when states are -264-Appendix A: COMPRO Structure and Operation varying quickly such as during periods of rapid resin cure. Allowing solution modules to act independently allows us to account for both the different relevant time scales for the various phenomena and the material states when change occurs most rapidly. For example, the bulk of resin flow takes places in a relatively short time period when the resin viscosity is near its minimum. On the other hand, prior to gelation, very little stress is built up in composite components due to the inability of the resin to carry shear stress. The thermochemical module, must be run at virtually every step since temperature change and\/or resin cure occurs throughout nearly the entire cycle5. During a solution step, k, the overall solution time step for the next step, A ^ ' is determined by taking the minimum of the 'allowable' time steps computed by each of the solution modules as they are called and the overall maximum time step, i.e., Atk+l = Min(AtTA,AtFA,AtSA,Atmax) (A3) where AtTA, AtFA, AtSA, are the calculated allowable temperature, flow and stress module time steps and Atmax is the maximum allowable time step (set by the user mCONTROL FILE.CTL,). Each module uses an independent set of criteria for calculation of their allowable time steps, based either on the rate of convergence of their solutions or the rate of change of the relevant solution variables since that module was last executed. The stress module, for example, calculates AtSA based on the rate of change of resin degree of cure, temperature, and modulus. It also checks to see how much time has elapsed since it was last executed as compared to the maximum specified time between steps. 5 In addition, since temperature and degree of cure are the two most important state variables for determining composite behaviour, it is important that accurate values of each be available to the 'downstream' modules at each time step. -265-Appendix A: COMPRO Structure and Operation To determine if a module is to be run during a given solution step, each module checks the current state of the model as well as the change in time or other parameters since it was last executed3. For example, the flow module solution is performed only when the resin viscosity (somewhere in the model) is less than a user-defined limit. The stress module, in turn, will only run after gelation of the resin occurs somewhere in the model (provided there are any composite materials being modelled, if not, it can run any time). Following are provided algorithms for calculation of the overall solution time step and that for calculation of the allowable stress module time step, including determination of whether the stress module is to be run during the current step. Algorithm for calculation of the overall solution time step (for the next time step): 1) Define maximum and minimum allowable solution time steps (Atmi,\u201eAtmax) and criteria for calculation of allowable time steps for all modules (user input in CONTROL FILE.CTL,) 2) START PROJECT SOLUTION LOOP 3) Calculate maximum allowable thermochemical module time step (AtTA) a) AtTA = AtT *\/(# of solution iterations) 4) Calculate maximum allowable flow module time step (AtFA) 5) Calculate maximum allowable stress module time step (AtSA) 6) If (PROJECT run incomplete and no serious errors) Then a) Calculate next model time step, Atk+1 from Minimum of Atmax, AtTA, AtFA, At$A b) GOTO 2 7) GOTO next PROJECT Algorithm to calculate the allowable stress module time step (AtSA) for the next solution step and to determine if stress module is to be run during the current step: 1) C A L L stress module 2) Check if stress module is active a) If NOT (gelation achieved anywhere or if no composite materials in the model) Then It should be noted that the thermochemical module and the virtual autoclave module are actually executed every time step since their predictions are so important to 'downstream' calculations. -266-Appendix A: COMPRO Structure and Operation i) Set maximum allowable stress module time step to user-defined value (AtSA = AtSmax) ii) RETURN 3) Calculate the elapsed time since the stress module was last executed (Ats = TIME - STRESS TIME) 3) Calculate maximum allowable stress module time step a) Calculate maximum allowable stress module degree of cure time step (AtaA) i) Determine largest a change since last stress calculation (Aas) ii) Calculate At^ = AaSmJAas *Ats # b) Calculate maximum allowable stress module temperature time step (AtrA) i) Determine largest temperature change since last stress calculation (ATS) ii) Calculate AtaT= ATSmJATS*Ats c) Calculate maximum allowable stress module modulus time step (AtEA) i) Determine largest percentage change in modulus since last stress calculation (AEPS) ii) Calculate AtEr= AEPSmJAEPS*Ats d) Calculate time remaining before maximum stress module time step is reached (At,A = AtSmax -Ats) e) Calculate AtXA = Minimum of Ats,\u201eax, AtaA, AtTA, AtEA, AtlA 4) Check if stress module is to be run during the current solution step a) If (Aas > Aamax OR ATS > ATSmax OR AEPS > AEPSmax OR Ats > AtSmax) Then i) Set STRESS TIME = TIME ii) Execute stress module 5) RETURN Notes: \u2022 The stress module is said to be 'active' after gelation has occurred. This does not mean that it will be executed during the current time step, only that it can be run. \u2022 By 'executing' the stress module it is meant that a complete solution is determined. The stress module is called by the main routine every step, but it is only executed some of the time. A.5 The Virtual Autoclave - Implementation The design of the process model is based on the concept of a 'virtual autoclave'. Employing this approach, whereby the structure of interest is but one part of a virtual autoclave with analogues for each major autoclave system, much improved predictions of component boundary conditions during processing can be obtained over traditional techniques. As is currently envisaged, the virtual autoclave consists of the following components: \u2022 The structure of interest -267-Appendix A: COMPRO Structure and Operation \u2022 Al! process tooling used to manufacture the structure (substrate tooling, fixtures, vacuum bag, bleeder\/breather, etc.) \u2022 Virtual instruments including thermocouples, pressure and vacuum gages, dielectric sensors, etc. \u2022 The autoclave itself including environmental control hardware (i.e. heating, cooling and pressurization devices) \u2022 Autoclave environmental control hardware control systems Outside of the structure being modelled (and the consideration of process tooling), the components of the current virtual autoclave are nearly all simulated using very simplified models. The only exception to this is the simulation of the operation of the autoclave environmental control system hardware controllers which is relatively complete. This section discusses the models currently used for the various components of the virtual autoclave with the exception of the structure of interest and its tooling which are part of the finite element model which is described in detail elsewhere. A.5.1 Autoclave Simulation The autoclave simulation includes the simulation of the behaviour of the autoclave during processing including all environmental control hardware (i.e. heating, cooling, and pressurization devices), but not the environmental control hardware controllers. The current autoclave simulation is quite crude, with no consideration given to finite response times of the autoclave and its systems to controller requests; rather temperatures and pressures are set instantly to the controller set points. The only consideration given to autoclave limitations is the limited heating, cooling and pressurization rates specified by the user in CONTROL FILE.CTL). In the near future it is anticipated that the current virtual autoclave simulation will be expanded to include such things as simulation of autoclave heat transfer, including heater input, heat absorption by the autoclave gasses, walls and process tooling, and heat loss to the external environment. The response of the environmental control hardware to controller electrical signals will also be simulated, including -268-Appendix A: COMPRO Structure and Operation consideration of such things as limitations on output power, etc. This improvement would permit more accurate calculation of the overall autoclave response and thereby improve prediction of the boundary conditions seen by the modelled structure. This would also permit the model to be used to simulate novel control strategies and possible process improvements such as heated tools or radiative heating. A modular structure similar to that used elsewhere in the model is envisaged for this simulation, whereby the user w i l l 'create' an autoclave by combining autoclave 'shells' with individual environmental control hardware components. Each autoclave component w i l l be separately characterized and its properties stored in a database in a fashion similar to that now used for materials used in the finite element models. A similar arrangement would also be used for other components of the virtual autoclave such as controllers and virtual instruments (discussed in the following). A.5.2 Autoclave Environmental Control Software Simulation A n important component of the virtual autoclave is the simulation of the control systems for the autoclave environmental control hardware. In a typical industrial or research autoclave, a simple feedback control algorithm is used to generate a pre-defined set of temperature\/pressure conditions within the autoclave (i.e., the process cycle), perhaps employing some sort of 'intelligent' algorithm. In the current simulation, autoclave environmental conditions are determine using lead\/lag control algorithm is used, based on a system employed in practice in a known research autoclave7. While quite complete, the current controller simulation is necessarily simplified by the lack of a realistic autoclave simulation. A s shown in Figure A . 10, in a real system the control software and the environmental control hardware appear to each other as 'black boxes' with unknown characteristics. The only thing the controllers 'know' about the hardware is that for a given electrical input, a certain response, measured via instrumentation, is obtained. In the current simulation, the hardware simulation is completely by-Autoclave operated by intec (Integrated Technologies Inc. of Bothell, WA). -269-Appendix A: COMPRO Structure and Operation passed and temperature and pressure values directly set by the controller simulation. For the most part, this simplified approach should provide reasonably good predictions, but will not be able to account for such things as thermal inertia and is quite poor in predicting the latter part of the cool-down process. An outline of the controller algorithm currently in place is provided in the following section. Temperature (t) \u2014 \u2022 < V T , IT (t) Air Pressure (t) \u2014 \u2022 < V P , IP (t) Vac. Pressure (t) \u2014 \u2022 <* V v ,I v(t) Autoclave including Controller algorithm environmental control hardware Figure A.10: Communication between environmental control hardware and the control system A.5.2.1 Controller Algorithm The controller uses as its input a parametrized version of the process cycle. For the current algorithm, this parametrization is performed by subdividing the process cycle into a series of independent steps referred to as segments. Two types of segments are defined: ramp segments and hold segments as described following. Ramp Segments During a ramp segment, one or more of the three environmental state variables (autoclave air temperature, autoclave air pressure, and vacuum bag pressure) may be changed. For each variable to be changed, a number of control parameters are required: \u2022 The target is the value that the variable is to ultimately attain during that ramp, for example, 350 \u00b0F. -270-Appendix A: COMPRO Structure and Operation \u2022 The criterion is the value of a parameter that must be attained before moving to the next segment. This parameter is used for two purposes: 1) As a tolerance on target values which may be approached asymptotically and never actually reached 2) To account for the fact that temperatures or pressures at all points within the autoclave (or component) will not be uniform. If, for example, component areas which are slow to react to temperature change are not accounted for, these 'lagging' sections may not be held at high temperature for long enough to fully cure. It should be noted that only after the criteria have been reached for all variables being changed will the segment be deemed complete. \u2022 The ramp rate (desired rate of change) of each variable must also be specified. In the current system, the rate requested will be exactly the rate obtained (up to the limits specified by the user elsewhere). \u2022 The control thermocouple is the thermocouple (either the air temperature thermocouple, the lead thermocouple or the lag thermocouple) used to determine when the target temperature has been reached. For example, if the lead thermocouple is used as the control thermocouple, the air temperature will continue to be increased (or decreased) until the leading part temperature reaches the target value. \u2022 The watch thermocouple is the thermocouple (either the air temperature thermocouple, the lead thermocouple or the lag thermocouple) that is used to determine when the criterion temperature has been reached. It should be noted that no analog to the control and watch thermocouples are used for controlling autoclave gas pressure and vacuum bag pressure since only single virtual instrument measurements are made for each of these variables. Hold Segments During a hold segment, all environmental state variables are held constant for the specified hold time. The only exception to this are parameters which have not yet attained the target values from the previous segment. These will continued to be changed until the old target values are attained. -271-Appendix A: COMPRO Structure and Operation The only other input presently required by the controller simulation is a list of the virtual thermocouples (other than the air temperature thermocouple) to be 'monitored' to determine lead and lag temperatures. These are input as a list of nodes in the finite element model at which the thermocouples are assumed to be placed. Table A.2: Example controller parameters (used to define process cycle in Figure A . l l ) Segment Number Hold time (minutes) Target temp. (\u00b0C) Heat, rate (\u00b0C\/min) Criterion temp. (\u00b0C) Target press. (kPag) Criterion press. (kPaJ Press, rate (kPa\/min) Target vacuum (kPaJ 1 120 2 110 70 65 100 -100 2 90 3 177 1.5 172 410 400 100 0 4 180 5 20 70 6 0 5 Air Temperature Ai r Pressure Vacuum Pressure 5 . 0 E + 0 5 4 . 0 E + 0 5 3 . 0 E + 0 5 oi 2 . 0 E + 0 5 \u00a3 e> 1 . 0 E + 0 5 | Q. 0 . 0 E + 0 0 - 1 . 0 E + 0 5 - 2 . 0 E + 0 5 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 Time (minutes) Figure A . l l : Process cycle defined by lead\/lag control parameters shown in Table A.2. Notes: Parameters for the autoclave controller simulation are provided by the user in CYCLE FILE.CYC -272-Appendix A: COMPRO Structure and Operation \u2022 Due to the very high actual vacuum bag pressurisation\/depressurisation rates, changes in this parameter are assumed to be instantaneous and no rates or criteria values may be specified. \u2022 If rates are left unspecified (e.g., segment #5, temperature change rate), the change will be made as quickly as possible. \u2022 Currently, controller time steps are assumed to be the same as the main overall solution time step. With a more advanced virtual autoclave simulation, autoclave control time steps and solution time steps can be made independent. \u2022 Hold times are specified for hold segments, targets and criteria values set for ramp segments. Both types of inputs cannot be specified for the same segment. \u2022 If no target or criteria values are set for a variable during a segment, that variable is maintained constant or, i f it has not yet reached its last target value, ramped as quickly as possible toward that target value. Controller Simulation Algorithm The algorithm employed by the controller simulation is as follows: Loop over all projects in batch run 1) Obtain the process cycle specification (user input in PROCESS CYCLE.CYC) 2) Check process cycle specification for consistency and perform setup functions 3) START PROJECT SOLUTION LOOP 4) Calculate autoclave air temperature, TAc a) Read virtual thermocouples b) Determine lead and lag temperatures c) If (not (LAST TARGET REACHED)) Then i) TAC = TAC +TRale * At d) Else l) TAC = P'fargel e) ^(CURRENT CRITERION REACHED) Then i) TEMP RAMP DONE = TRUE 5) Calculate autoclave air pressure, PAC a) Read virtual pressure sensor b) If (not (LAST TARGET REACHED)) Then ^PAC^PAC+PRCC* At c) Else i) PAC = Plargel d) ^(CURRENT CRITERION REACHED) Then i) PRESS RAMP DONE = TRUE 6) Calculate vacuum bag pressure, PVBAG -273-Appendix A: COMPRO Structure and Operation a) SetPVHAG = PVrarge, b) -VACUUM RAMP DONE = TRUE 7) \/\/(HOLD SEGMENT) Then a) Increment elapsed segment time {AtSEG(EL) = AtSEG(EL) + At) 8) Update segment number if it is time to do so. a) If (RAMP SEGMENT and (TEMP RAMP DONE and PRESS RAMP DONE and VACUUM RAMP DONE)) or (HOLD SEGMENT and AtSEC(EL) > AtSEG) Then i) Increment segment number (nSEG = nsnc + ') 9) If process cycle incomplete (i.e. nSEG < nSEG(Toiaij) Then a) GOTO 4 10) GOTO next PROJECT A.5.3 Virtual Instruments An important component of the virtual autoclave is its virtual instrumentation. The readings of these virtual instruments are used both as inputs to the environmental control hardware controller and model predictions. In the current model, the virtual instruments used include the autoclave temperature thermocouple, a user-specified number of thermocouples in the part and tooling, the autoclave air pressure gage and the vacuum pressure gage. The measured values are presently assumed to be measured exactly and the autoclave temperature and pressure and vacuum pressure to represent values valid for the entire autoclave or bag. Future anticipated improvements include such things as development of additional instruments such as dielectric sensors and introduction of random 'error' in instrument measurements. Both of these would assist in improving the simulation of actual controller response, especially if examining novel control strategies. A.6 Error handling COMPRO performs extensive internal error checking, both to verify the consistency of the program inputs and to ensure that the solution is proceeding as expected. When an error is detected, an 'error handler' is called which sets an error level 'flag' and outputs an error message (including error number) to -274-Appendix A: COMPRO Structure and Operation both the screen and the batch run summary file (SUMMARY.BCH). The subsequent behaviour of the program following error detection depends on the error level generated. The error levels and program response are as follows: \u2022 Error level 1 - Warning: Program operation continues uninterrupted. \u2022 Error level 2 - Serious Error: Project run halted at the end of the step and the next project begun. \u2022 Error level 3 - Fatal Error: Batch run halted immediately. It should be noted that the existence of this error handler does not mean that unexpected errors of every level cannot occur. Although it would be possible (and desirable) to implement a higher level of error checking in COMPRO, it would never be possible to prevent 'unhandled' errors from occurring altogether. A.7 COMPRO Overall Program Flow The overall flow of the COMPRO program is as follows: 1) START BATCH RUN 2) Obtain PROGRAM startup information a) Read PROGRAM initialisation file, COMPRO.INI b) Read BATCH file listing projects to be run, COMPRO.BCH c) Obtain BATCHrun starting time 3) Loop over all projects in batch run a) Obtain PROJECT startup information i) Read list of files in project from PROJECT.PRJ ii) Obtain PROJECT run starting time 4) Obtain PROJECT input data (Section A.2.2) a) Set default values and initialize variables b) Read the program control information from CONTROL FILE.CTL, c) Read finite element discretization information from FE MESH FILE.PAT d) Read boundary and initial condition information from BOUNDARY FILE.BCI e) Read process cycle control information from PROCESS CYCLE.CYC f) Obtain material properties specification information (Section A.2.3) -275-Appendix A: COMPRO Structure and Operation Loop over all defined model regions i) Read the names of the materials in each region from MATERIAL ID FILE.MAT ii) Find the data files associated with each material name from LOOKUP.DAT iii) Read properties specification information for each material from DATA FILE.DAT g) Read layup specification information from LA YUP FILE.LAY 5) Perform PROJECT setup functions a) Determine local axes orientation for all elements (Section A.3.4) b) Calculate ply-element intersections (Section A.3.4) 6) Perform Thermochemical Module setup functions a) Calculate ply conductivity 'smearing' factors for each element (Section B.1.5) b) Generate the thermochemical module ID array and skyline vector 7) Perform Flow Module setup functions 8) Perform Stress Module setup functions a) Generate the stress module ID array and the skyline vector b) Determine slide line vector directions at nodes where 'sliding' displacements have been prescribed. 9) START PROJECT SOLUTION LOOP 10) Update boundary conditions a) Calculate autoclave air temperature and pressure and vacuum bag pressure at the end of the time step b) Calculate boundary condition parameter values (e.g. heat transfer coefficients) 11) Call Thermochemical Module 12) Call Flow Module 13) Call Stress Module 14) Call Output Module 15) If {PROJECT run incomplete and no serious errors) Then a) Calculate next solution step time step (Section A.4) and next solution step elapsed time b) GOTO 10 16) Call Stress Module to determine cool-down state 17) Call Output Module to report results at end of cool-down 18) Call Stress Module to determine post-tool removal state 19) Call Output Module to report post-tool removal results 20) Write PROJECT summary information to SUMMARY.BCH 21) If BATCH run not complete then GOTO 2 22) Write BATCH summary information to SUMMARY.BCH -276-Appendix A: COMPRO Structure and Operation END Notes: \u2022 File names in bold italics are set by the user, those not in italics are constant. For example, the layup specification file LAYUP FILE.LAY can have any name with the extension .LAY (e.g., FILE 1.LAY), but the batch file is always COMPRO.BCH. A.8 The COMPRO Editor To create all the input files required for a COMPRO run, all the user needs, in theory, is a text editor and a knowledge of all required the input line headings. In practice, the number of inputs required by the program makes this impractical for all but the simplest cases. To perform all COMPRO input and output handling functions a 'COMPRO Editor' was developed. This routine, written using Microsoft Visual Basic 3.0, employs a graphical user interface (Figure A. 12) and incorporates a number of utilities which simplify the use of the COMPRO code. These utilities include: \u2022 Separate graphical forms for generating and editing each type of COMPRO input file (see example in \u2022 Figure A. 12). \u2022 A utility for viewing the problem finite element description and editing defined boundaries. \u2022 A utility for generation of the finite element description for simple geometries (for complex geometries an external pre-processor is required). \u2022 A 'translation' utility which converts COMPRO output files to formats readable by commercial plotting packages. Currently, routines for translation to Tecplot\u2122 and MS EXCEL\u2122 are implemented. -277-Appendix A: COMPRO Structure and Operation Fife Edit View Run SJtilities Help O @ B acr I CTI C Y C ! MAT PffJ \"2: Units\/Comments Then-nocherntcsl Module | Program Control \"\"jViow Module Autoclave Parameters ]J[Stress_Moduie Execute Module? '\"\"] fSYes P No Remove Tooling? + S{2 5,'3 0 0 '34 0 0 S'56 (B.12) Designating the first compliance matrix in EquationB.12 as [ S 3 ] and the second as [S^ ], this equation becomes: Substituting for {'s shown following: 1) Calculate the change in ply thermal and cure shrinkage strains in the ply principal directions: i) Compute the change in the thermal strains during the step in direction \/': where CTE, are the thermal expansion coefficients and 7*~' and 7* are temperatures at the start and end of the time step, respectively. The temperatures are calculated at the centroid of the element rather than the ply resulting in uniform thermal strains in all plies in an element (in the ply principal directions). The user may impose orthotropic ply cure shrinkage by inputting a 'Cure shrinkage ratio' (defined as CSC3\/CSC2) not equal to 1.0 (the micromechanics equation calculates CSC2). Note that although a 'A' designation is not employed the strains defined throughout this section are strain changes rather than total strains unless otherwise indicated. (B.28) -286-Appendix B: Equations and Algorithms ii) Compute the change in the cure shrinkage strains in direction \/ (non-zero for composite materials only): < = C5,C,*(<*)-^<*-\u00b0) (B.29) where CSd are the cure shrinkage coefficients and er(k) and Esrik'[) are the total resin linear cure shrinkage strains at the start and end of the stress time step respectively. Resin linear cure shrinkage strains are determined from resin volumetric strains, Vrs, using the relation: g>r={\\ + v:f-\\ (B.30) The volumetric strains are themselves determined using one of two cure shrinkage models described in Section B.l.8. It should be noted that since resin degree of cure is uniform within an element, cure shrinkage strains in the ply principal directions are the same for each ply in an element. 2) Define the 3-D ply internal strain vector, {\u00a3\u2022*} such that: =K(,) *0(2) 0 *0(3) 0 0 } ( B - 3 1 ) where %^ is the internal strain in the direction \/' which is equal to the sum of thermal expansion and cure shrinkage strains, i.e.: E m = sTl+sfl (B.32) All shear strain terms in this vector are zero since only normal strains are induced in the ply principal directions by thermal and cure shrinkage strains. 3) Calculate the transformed 3-D ply internal strain vector in the element local axes using: to} = [Tfl]-'{*;} (B.33) -287-Appendix B: Equations and Algorithms where [TJ is the stress\/strain transformation matrix defined in EquationB.8 in which #is the rotation of the ply local axes with respect to the element local axes as illustrated in Figure B.6. 4) Determine the element-axes plane strain ply internal strain vector, {sf}'-In plane strain problems, all strains out of the plane of interest are suppressed. Thus, in this case, all strains withy' terms, i.e., sy. , yxy , y ,,, are set to 0. Employing this plane strain assumption reduces the number of active strains from 6 to 3. Thus, we may define a plane strain ply internal strain vector as: {\u00a3o } { f o V ) f o( , ' ) T'oiy.-')} (B.34) To solve for this vector: i) Determine the change in in-plane strains induced by forcing all out-of-plane strains to zero as required by the plane strain assumption. This is done by applying the negative of out-of-plane strains and using the fact that change in in-plane loads during this procedure will be identically zero). Thus: 0 \u2022 = \u2022 0 \u2022 = A r o ( \u201e - ) . 0 c c c c ^ 1 4 ^ 4 4 0 0 0 0 c ^ 5 5 As, As, O(x') 0(0 c c ^ 1 2 ^ 1 3 c c 0 0 0 0 Y o(xy) \\J o(yz') J (B.35) where Cy are the terms of the transformed 3-D stiffness matrix defined by [C] = [S']\"'. ii) Designating the first stiffness matrix in EquationB.35 as [Cj] and the second as [Cj], the change in in-plane strains caused by forcing the out-of-plane strains to zero is calculated as: { A ^ ^ t q r ' t c i ^ , } (B.36) where {^ 0(I>)} are the in-plane strains and |^ 0(^ )} a r e m e out-of-plane strains (first and second strain vectors in Equation B.35 respectively). iii) Sum the initial in-plane strains and the change in these strains induced by the out-of-plane constraints, to obtain the plane strain ply internal strain vector. -288-Appendix B: Equations and Algorithms {^ } = Rw} + {A*i(W} (B.37) 5) Calculate the ply internal strain vector in the global axis is then determined using: {*0} = [T,f{tf} (B.38) where [T\/J is the transformation matrix defined in EquationB.20 with \/J defined as in Figure B.6 (Section B.4.1). B.1.2 Ply Micromechanics Model Elastic constants and strains for unidirectional plies are determined from the mechanical properties and strains of their constituent resin and fibre using micromechanics equations from Bogetti and Gillespie (1992). Currently no micromechanics models for non-unidirectional plies are implemented. B. 1.2.1 Calculation of Elastic Constants Given as inputs the transversely isotropic mechanical properties of the fibres (E\\\\f,E^j, G13\/, V\\y, and v23\/), the properties of the isotropic resin (Er, vr) and the fibre volume fraction, Vj, ply mechanical properties are calculated as follows: In-Plane Moduli where: Eu=EufVf + Er(\\-Vf) + 4(vr-v2nf)kfkrGr(\\-Vf)Vf {kf+Gr)kR+{kf-kr)GrVf ' 2(1 + vr) ^22 ~ 1 (l\/4kT) + (\\\/4G23) + (vzu\/Eu) (B.39) (B.40) (B.41) Shear Moduli G 1 2 =GI3 =G, (Gu\/ + Gr) + (Guf-Gr)Vf (Guf+Gr)-(GUf-Gr)Vf (B.42) -289-Appendix B: Equations and Algorithms Q Gr[kr{GR + G 2 3 \/ ) + 2G 2 3 jGr + kr{G23 f-GF)Vf] 2 3 kr(G23f + Gr) + 2G23fGr-(kr+2Gr)(G23f-Gr)Vf where: G23f = E ' 3 f (B.44) 2(l + v 2 3 \/ ) Poisson's ratios: vU = vU = v\\3fVf + vr(l-Vf) + (vr-vX3f)(kr-kf)Gr(l-Vf)Vf (kf + Gr)kr+{kf-kr)Gr\\ (B.45) v23 = 2EnkT EXXE22 -4vX3krE22 2ExlkT (B.46) In the above, k is the so-called (by Bogetti and Gillespie) 'isotropic plane strain bulk modulus' defined by: 2 ( 1 - v - 2 v z ) (B.47) and kT is the effective plane strain bulk modulus of the composite defined by: kr = (kf + Gr)kr+{kf-kr)GrVf (kf + Gr)-(kj-kr)Vf (B.48) B. 1.2.2 Calculation of Thermal and Cure Shrinkage Strains Ply strains in the material principal directions arising from strains of the constituent resin and fibre are calculated from the fibre and resin mechanical properties and strains as follows: ^fEiijVf + erEr(\\-Vj) ExxfVf + Er{\\-Vf) (B.49) \u00a3 2 = \u00a3 3 = ( \u00a3 2 f + V, 3 fSx f)Vf+(\u00a3r + Vr\u00a3r){\\ - V}) \u00a3\\JE\\fVf + S R E r ^ - V f ) [vx3fVJ + vr{\\-Vf)]\\ ft E,Vf + Er(\\-Vf) (B.50) -290-Appendix B: Equations and Algorithms To calculate thermal expansion coefficients (CTE,), replace resin and fibre strains by their respective thermal expansion coefficients. To calculate cure shrinkage coefficients (CSC,), set resin strains to 1 and fibre strains to 0. Note that the user is actually permitted to set different thermal and cure shrinkage strains in the 2 and 3 directions (for each s3 = Ratio *s2) B.1.3 Density Material density (p) is used in the calculations of both the thermochemical and flow modules. For non-composite materials, density is calculated using the equation: Pc=Pm+afiC(T-T0) (B.51) The density of composite materials may be calculated using either 'lumped' or 'non-lumped' models. Using the 'lumped' model, composite density is calculated as: Pc = Pcm + a^T- r0) + ^ (a - a0) (B.52) Using the 'non-lumped' model, resin and fibre densities are first calculated using: Pr = Prm + apr(T-T0) + b^a - a0) (B.53) P f = P \/ m + a^(T-T0) (B.54) The composite (ply) density is then determined using: Pc = VfPf + (\\-Vf)Pr (B.55) Composite Element Density: Since density is a scalar quantity and the density of all plies are calculated from an assumed uniform element temperature, degree of cure and fibre volume fraction, the density of the element is the same as that of any one ply. -291-Appendix B: Equations and Algorithms B.1.4 Specific Heat Capacity Material specific heat capacity (Cp) is used by the calculations of the thermochemical module (see Equation B.86). For non-composite materials, specific heat capacity is calculated using the relation: CPc=CPc(0) + acpc(T-T0) (B.56) The specific heat capacity of composite materials may be calculated using with 'lumped' or 'non-lumped' models. Using the 'lumped' model, composite Cp is calculated as: CPc = CPcW + aCpc(T-T0) + bCpc(a-a0) (B.57) Using the 'non-lumped' model, resin and fibre specific heat capacities are first calculated using: CPr =CPrm + aCpr(T-T0) + bCpr(a-a0) (B.58) CP, =CP\/(0)+aCp\/(T-T0) (B.59) The composite (ply) specific heat capacity is then determined using one of the following equations: a) If resin and fibre densities are known (i.e. composite density was given as 'non-lumped'): j,c,P,Hi-v,ycr,Pr VlPj+i\\-Vt)p, b) If resin and fibre densities are unknown (i.e. composite density was given as 'lumped'): CPc = VfCP\/+(\\-Vf)CPr (B.61) Composite Element Specific Heat Capacity: Since specific heat capacity is a scalar quantity and the CFc of all plies are calculated from an assumed uniform element temperature, degree of cure and fibre volume fraction, the Cp of the element is the same as that of any one ply. -292-Appendix B: Equations and Algorithms B.1.5 Thermal Conductivity Material thermal conductivity (k) is employed by the calculations of the thermochemical module (see Equation B.86) . Conductivity is a vector quantity and is thus described by ann x n matrix, [K] in an n-dimensional problem. For non-composite materials, conductivity in the principal material directions is determined using the equations: knc^kncio) + aknc(T-TQ) (B.62) k 2 2 c = k 3 3 c = k 3 3 c { 0 ) + a k , 2 c ( T - T 0 ) (B.63) The conductivity of composite materials may be calculated using either 'lumped' or 'non-lumped' models. Using the 'lumped model', composite conductivity in the ply principal directions is calculated as: kuc =kncm + akllc(T-T0) + bkUc(a-aa) (B.64) k22c =k33c = k 3 3 c m + amc(T-T0) + bmc(a-a0) (B.65) Using the 'non-lumped' model, isotropic resin conductivity and transversely isotropic fibre conductivities are first calculated using: *, = Km + akr(T-T0) + bkr(a - a0) (B.66) k n \/ = k n \/ m + a k n f ( T - T o ) (B.67) k22f - k33f = ^33\/(0) +Ctk33f(T ~ TQ) (B.68) Longitudinal conductivity is determined using a simple rule of mixtures: Kc = Vfknf+(\\-Vf)kr (B.69) and the composite (ply) transverse conductivities are calculated using the following equation from Springer and Tsai, 1967 (as corrected by Twardowski, 1993): -293-Appendix B: Equations and Algorithms ( l - 2 ^ 7 \u00a5 ) -B n -4 J\\-B2Vf\/n i =tan - , j\\-B2Vf\/n \\ + B^Vfln (B.70) where: B = 2 yKif j (B.71) Element Conductivity: For non-composite materials, the material principle directions correspond to the element local axes. Thus, the element conductivity matrix becomes: kn 0 0 (B.72) For composite material elements, the ply conductivity matrix in the local element axis is first calculated from the ply conductivity matrix in the material principle direction,[K*] using: M = [T A ( ,] 7 [^][T A ,] (B.73) where: [**] = ku 0 0 K, 0 0 K (B.74) and [TA.] = cost? sinf? 0 -sint? cosf? 0 0 0 1 (B.75) Employing the assumption that the temperature gradient in the out-of-plane direction (i.e-yO is equal to zero4, the conductivity matrix can be reduced from a3x3toa2x2 matrix using an analysis similar to This is equivalent to the plane strain assumption. Alternatively, it could have been assumed that heat flux in the out-of-plane direction was zero. -294-Appendix B: Equations and Algorithms that used to calculate the plane strain stiffness matrix in Section B.l. 1.1. For this case, the ply plane temperature gradient conductivity matrix is calculated from Equation B.73 to be: 0 XT' where: !,=\u00a3,, cos 0 + k22sin 6 and 3 3 = \u00a3 3 (B.76) (B.77) The element conductivity matrix is then calculated from the ply conductivity using: 0 0 k\u201e where: and *\u00a3 = -J- I (7\/-7\/-i)(*iT). 2 \/=i ^ 7 = 33 )\/ = ^33 (B.78) (B.79) (B.80) In the above equations, \/ is the ply number, nply is the number of plies in the element and rji are the ply-element r\/ intersections. Since the ply orientations and the ply-element intersections, t],, remain constant throughout the simulation, it is possible to define conductivity 'smearing' factors, ak and bk such that: where: and: 1 \"p'y, \\ 2 ak = - X (7\/-7\/-i) c o s \u00b0i 2 i=\\ 1 \" Z 7 ^ \/ 2 (7\/-V\/-i)sin 0, 2 \/=i (B.81) (B.82) (B.83) These factors can be calculated once for each element, then used to determine conductivities throughout the process cycle. -295-Appendix B: Equations and Algorithms Finally for both composite and non-composite materials, the element conductivity matrix in the global axes is calculated using: B.1.6 Special Material Models A number of special models for material properties are required to predict the behaviour of the matrix resin during processing. To emphasize the special nature of these models and their importance to the component behaviour during processing, the current sub-section is devoted to their discussion. B.1.7 Cure Kinetics The current analysis includes a total of 6 different models of various types for describing the kinetics of resin cure, as outlined in Table B.l. An accurate model for resin cure behaviour is one of the most important requirements for obtaining accurate model predictions. This is so not only due to the heat generation of the exothermic reaction Section B.2.1), but more importantly since the resin degree of cure is one of the two state variables which are assumed to completely describe the resin state (the other is temperature). Thus, if the resin degree of cure is not known accurately, good predictions of such vital properties as resin viscosity and modulus are difficult to obtain. It should be noted that a number of these model equations are similar enough in form that it may be useful at some time to combine them into a lesser number of more general equations. The specific models outlined below may then be described as specific cases of the general expressions. As an example, models 2 and 3 can both be identified as special cases of model 4. M = [ W [ v e ] [ T A \u00bb ] (B.84) where: cos B sin B -sin\/J cos\/? (B.85) -296-Appendix B: Equations and Algorithms Table B.l: Cure kinetics models available in COMPRO. Model Equations Parameters Model 1 Lee, Loos and Springer (1982) dec \u2014 = (Kl + K2a)(\\-a)(B-a) aac dt V . -AEJRT Ki = Ate \u2022 Ah Aj, A3, AEH AEJ, AE3, B, aQ HR Model 2 Lee, Chiu, and Lin (1992); White and Hahn (1992) \u2014 =Kam{\\-a)\" dt K = AE-AE\/RT A, AE, m, n, HR Model 3 Kenny (1992); Scott (1991) \u2014 = K(l- a)\" dt K = AE-<*IRT A, AE, n, HR Model 4 Scott (1991) ^L = (Kl+K2a\"')0-a)\" dt Ki = A,e \u2022 A\\, Aj, AEH AE2, m, n, HR Model 5 Lee, Chiu, and Lin (1992) ^ = ^ , ( l - \u00ab ) ' +K2am(\\-a)\" \u201e , -AEJRT Kt = Ap A i, A2 , AEH AE2, I, m, n, HR Model 6 Johnston and Hubert (1995) da Kam(l - a)\" dt 1 + ec{\u00ab4\u00abco+\u00abCTn} is A -AEJRT A, AE, m, n, C, CCQO, CCCT, HR Table B.2: Parameters used in COMPRO resin cure kinetics models. Variable Description Units a Resin degree of cure. -T Resin temperature K or R HR Total resin heat of reaction (a = 0 to 1) J\/kg or BTU\/lbm M Pre-exponential factor. \/s AEi Activation energy. J\/mol or BTU\/mol I Equation superscript. -m Equation superscript. -n Equation superscript. -R Gas constant J\/(mol K) or BTU\/(mol R) -297-Appendix B: Equations and Algorithms c Diffusion constant -aco Diffusion constant -OCT Diffusion constant \/ K or \/R B.1.8 Cure Shrinkage Models In the current analysis two different models for calculation o f volumetric resin cure shrinkage may be used. The reference from which these models were obtained and the equations used by each are given in Table B.3. A description o f the parameters used in each of the models is provided in Table B.4. It should be noted that the volumetric resin shrinkage is given as a ratio o f the initial resin volume rather than as a percentage. Linear resin shrinkage is determined from volumetric resin shrinkage using Equation B.30. Table B.3: Resin cure shrinkage models available in COMPRO. Model Equations Parameters Model 1 Bogetti and Gillespie (1992) V? = 0.0 a < a a Vrs = A* a s + (vrSm- A)*aj a a aC2 a - a r , aH = kJ\u2014 \u00ab C 2 - \u00ab C 1 VrSco, act, OCT, A Model 2 White and Hahn (1992) Vrs = -^\u2014 *\\0Ba aac VrSc\u00b0, ac, B Table B.4: Parameters used in COMPRO resin cure shrinkage models. Variable Description Units V? Resin volumetric cure shrinkage -ySco Total volumetric resin shrinkage from a= 0 to 1. -The 'cure shrinkage' degree of cure. -act Degree of cure after which the resin shrinkage begins (model 1). -aC2 Degree of cure after which the resin shrinkage stops (model 1). -A Linear cure shrinkage coefficient -\u00abc Degree of cure after which the resin shrinkage begins (model 2). -B Cure shrinkage model superscript. --298-Appendix B: Equations and Algorithms B.1.9 Resin Modulus Development Models Two different models are available for calculation of resin modulus during cure as outlined in Table B.5 and Table B.6. The model from Bogetti and Gillespie (1992) has been modified to include terms for resin temperature dependence. The second model shown was developed as part of the current work. Table B.5: Resin modulus development models available in COMPRO. Model Equations Parameters Model 1 Bogetti and Gillespie, 1992 (modified) E \\ = E \u00b0 a < acl K = ^ - a m B i ) E \u00b0 + a m a i E T + Tr amoi 0 ~ \" m o d ){Er ~ E r ) \u00abCI < \u00ab < \u00ab C 2 E \\ = E ; a > a C 2 E.=K[\\ + a\u00a3,(T-T0)] where: a m o i = C 1 Er,Er, yn c t c \\ , ccc2, T0, a E r Model 2 (Developed as part of current work) Er = E\u00b0r f TC2 E r = E r [ l + aEr(.T-T0)] where: f = (Tga + Tgb *a)-T; fcx = T*aa + T\"ah*T IT ^ rr ^ rj~i rf~f E'r i r^r \u2022 ' ga, ' gb, Tela, Tcib, TQTJ T0, ClEr Table B.6: Parameters used in COMPRO resin modulus development models. Variable Description Units a The resin degree of cure. -Er\" The resin elastic modulus at very low degrees of cure. Pa or psi Er The resin elastic modulus at T0 and a = 1.0 Pa or psi The 'modulus development' degree of cure. -c t c i Degree of cure above which the resin modulus begins to increase -CCC2 Degree of cure at which the resin modulus reaches full value. -yu The resin hardening rate factor. -T Resin temperature. \u00b0C or \u00b0F f Difference between resin temperature and resin instantaneous Tg \u00b0C or \u00b0F -299-Appendix B: Equations and Algorithms T Glass transition temperature at a = 0 Kor \u00b0R Tgb Factor expressing the degree of cure dependence of resin glass transition temperature. K or \u00b0R Tela f above which resin modulus begins to increase at T= 0 K. Kor \u00b0R Tew Factor expressing the temperature dependence of the T* above which resin modulus begins to increase. -TC2 T* above which resin modulus has reached its full value. Kor \u00b0R To Resin modulus development reference temperature. \u00b0C or \u00b0F Factor expressing the temperature dependence of resin modulus. Pa\/\u00b0C or psi\/\u00b0F B.1.10 Resin Poisson's Ratio Development Models As outlined in Table B.7 and Table B.8, two different models may be used for calculation of resin Poisson's ratio. In the first, vr is calculated as a linear function of temperature and degree of cure. The second assumes a constant resin bulk modulus so that the Poisson's ratio is a function Er. Table B.7: Resin Poisson's ratio development models available in COMPRO. Model Equations Parameters Model 1 Linear vr = vr0 +avr(T-T0) + bvr(a-a0) To, o\u201e, Oo, bw Model 2 Constant bulk modulus 1 v\u00b0\u00b0 (others from resin modulus development model). Table B.8: Parameters used in COMPRO resin Poisson's ratio development models. Variable Description Units Resin Poisson's ratio -VrO Baseline resin Poisson's ratio. -T Resin temperature. \u00b0C or \u00b0F To Resin modulus baseline temperature. \u00b0C or \u00b0F The factor expressing the variation in resin modulus with temperature. \/\u00b0C or \/\u00b0F a Resin degree of cure. -a0 Resin modulus baseline degree of cure. --300-Appendix B: Equations and Algorithms byr The factor expressing the variation in resin modulus with degree of cure. Pa or psi Er Resin elastic modulus. Pa or psi Ex Resin elastic modulus at a = 1.0. Pa or psi V Resin Poisson's ratio at a = 1.0. B.2 Thermochemical Module As discussed in Chapter 4, the thermochemical module is used to calculate the temperature and resin degree of cure in the structure of interest during processing. These parameters are calculated using the finite element method, employing the same discretized mesh as used in the stress module solution. The following section outlines in detail the development of the discretized equations for the finite element solution and provides details of the solution algorithm employed. It should be noted that some of the following discussion has already been presented elsewhere in the thesis, but is repeated here for completeness. B.2.1 Fundamental Equations The governing equation of the thermochemical module is the unsteady-state 2-D anisotropic heat conduction equation with an internal heat generation term from the resin's exothermic curing reaction, as follows: (B.86) where pc is the composite density, CPc is the composite specific heat capacity and ky are the composite anisotropic thermal conductivities. These properties are calculated at each time step from current local resin and fibre properties and the fibre volume fraction, K)(see Section B.l) The resin heat generation term Q in Equation B.86 is calculated as: Q=*B-(\\-Vf)prHR (B.87) -301-A p p e n d i x B: E q u a t i o n s a n d A l g o r i t h m s where pr is the resin density, a is the resin degree of cure, and HR is the resin heat of reaction, defined as the total amount of heat evolved during a complete reaction. The degree of cure, a, is defined as ratio of the heat given off by the resin reaction at a point during cure to the total resin heat of reaction, i.e.,: A number of different models are available for calculation of cure rate, the most appropriate for a particular case depending on the type of resin. In the current analysis, any of six different kinetics models may be employed (see Section B.1.7). B.2.2 Finite Element Discretization The governing equation of the heat transfer problem (Equation B.86) is solved using the finite element method. This is a fairly standard technique for analysis of heat transfer problems although finite-difference methods have been typically employed in previous composites processing heat transfer analyses. As discussed in Section B.4, the Galerkin weighted residual technique is employed to derive a system of algebraic equations from the governing differential equation for the problem. The first step in this procedure is establishment of the residual equation for the problem as follows (starting with a more general expression of the heat transfer equation than given in Equation B.86). Assuming that temporal variation in density and specific heat capacity can be neglected, this can be simplified to: (B.88) where dq\/dt represents the specific heat generation rate. R = -{pCpT)-V{kVT)-Q (B.89) -302-Appendix B: Equations and Algorithms R = pCPt-V(kVT)-Q (B.90) Introducing the weight functions we obtain: JW\u201eRdQ = \\w\u201epCPT dQ- jWkV(kVT) dQ-\\\\VkQdQ = 0 (B.91) n a n n Using integration by parts for the second term on the RHS, we obtain: \\\\VkpCPT dQ- j V ^ J f c V r ) dQ + \\kVTVWk dQ-jrVkQdQ = 0 (B.92) n n a n Using the divergence theorem on the new second term: \\wkpCPT dn - jWkkVT dT + jkVTVWk dQ. -\\wkQdn = 0 (B.93) n r n n We now rewrite Equation B.93 in matrix form, replacing the weight functions Wk by the element (temperature) shape functions [NT] and introducing the temperature shape function derivative matrix [BT] where: [ B T H d T } [ N T ] (B.94) and {di} is the temperature differential operator defined for the plane case being examined as: (B.95) Thus, Equation B.93 becomes: J [ N T ] 7 pCP [N T ] {t} dQ - J [ N T ] 1 M [B T ] {T} dT r. (B.96) +J [B T ] r M [B T ]{T} dil- \\[TXTYQdQ = 0 n a The surface integral term (the second term in Equation B.96) accounts for surface heat fluxes and can be more conveniently expressed rewriting it in conventional terms, accounting for individual sources of this flux, i.e. -303-Appendix B: Equations and Algorithms qs=qh+h(Tx-T) (B.97) where qs is the total surface heat flux, qh is the externally-imposed surface heat flux (e.g., inductive heating), h is the convective heat transfer coefficient and Tx is the temperature of the boundary fluid. Substituting these terms into Equation B.96 and integrating over the volume Ve and the surface Se of each element, we obtain: ([kj, +[h]e){T}e +[Cp]c{f}e ={fTq}t +{fTQ}\u00a3 +{fTh}e (B.98) where the following matrices have been defined for convenience: [CP], = J[NT]r\/X7p[NT] dSl (B.99) K [kj e = j[BTfM[BT]JQ (B.100) v, {fTQ}= J[NT]r0dO (B.101) ve {fTq}e=\\[KTfqbdT (B.102) Se { U e = j\\XT]ThTadT (B.103) [h], = J[NT]77z[NT]rfT (B.104) To obtain the system of equations for the entire domain, the contributions of the above matrices for all elements are summed to obtain the equivalent global matrices, e.g., nele [KJ = \u00a3 [ k k ] e (B.105) where nele is the total number of elements in the domain. -304-Appendix B: Equations and Algorithms Substituting the above matrix definitions into Equation B.98 and summing the contributions of all element stiffness and load matrices, the following global relation is obtained : ([K J + [H]){T} + [CP ]{f} = {FT Q} + {F T Q } + {FT H} (B.l 06) B.2.2.1 Time Integration Equation B.l06 contains first-order derivatives and must be integrated in time. This integration may be performed through the use of either modal or direct integration techniques. In this case, the presence of a highly non-linear heat generation term makes direct integration methods the more attractive option. These non-linearities and the large time-steps required for accurate and efficient problem solution further prompted the choice of the implicit, or backward, Euler method. Using this technique, the integration is performed using a backward finite difference approximation: W ^ ^ d T L - l T V , ) (B.l 07) where k is the number of the current 'time step' and At is the time step length. Substituting the above into Equation B.l06, we obtain: ([KJ + [H]){T}, + [C p]^-({T}, - {TV,) = {FT Q} + {FTQ j + {FT H} (B.l08) Simplifying, we arrive at the equation for the temperature in the domain at the end of time step k: { T W K ^ J F , } , (B.109) where [KT]A is the global temperature equation stiffness matrix, defined by: [ K T L = ^ [ C P ] + [KK] + [H] (B. l 10) and {FT}* is the global temperature load vector calculated as: {FT} = {FTQ} + {F T Q} + {FT H} + -J-[CP]{T},_, (B. l 11) -305-Appendix B: Equations and Algorithms B.2.3 Calculation of Resin Degree of Cure As can be seen from the heat transfer equation (Equation B.86) and the various available cure kinetics equations (Table B.l), the solutions for temperature and the degree of cure are 'coupled', that is, one depends on the other. Ideally, therefore a solution technique would be employed in which both temperature and degree of cure at each step would be solved for in a single coupled calculation. In this case, however, that route was not chosen since it was found necessary to define discretized temperatures and degrees of cure at different spatial locations within the domain. The reason for this is that while temperature spatial derivatives are continuous throughout the domain, those for degree of cure are discontinuous at material and domain boundaries. Therefore, unlike temperature, resin degree of cure cannot be defined as a single value at a boundary node and is instead defined in this analysis as an element property. Any of the six different models outlined in Table B.l may be used to calculate element degree of cure in this analysis. The fundamental equations for each are similar in form as is the solution technique used. For illustrative purposes, the solution method is demonstrated with the simplest of these equations, that for Model 1 as follows: Species migration via either resin flow or diffusion is not accounted for in this analysis. This greatly simplifies problem solution since the value of degree of cure at one point does not affect that at another (except indirectly via its effect on the internal temperature). Since the temperature and degree of cure solutions are decoupled within an iteration here, the result is that degree of cure at any point can be solved for completely on its own, thus eliminating the need for a matrix solution. However, to maintain a consistent notation with the rest of the analysis, we replace Equation B.l 12 with a matrix representation as follows: da = Ae dt -&E\/RT 0-a)' (B.l 12) {a} = {Ka}7'({l}-{a})' tl (B.l 13) -306-Appendix B: Equations and Algorithms where the discrete values of degree of cure, {a}, are defined at the element centroids, {d} is the cure rate vector, and {Ka} is the vector of cure rate coefficients such that: Kai=Aie-*E'\"a'> (B.l 14) where 7, is the temperature at the element centroid. The vector {1} is a vector where all entries are 1. B.2.3.1 Time Integration Similar to the temperature equation, that for degree of cure contains first-order derivatives and must be integrated with time. Once again, for consistency, a backward-Euler technique is used, i.e., {a}*=^({<-{a}*-,) (B.H5) where again k is the number of the current 'time step' and At is the time step length. Substituting the above into Equation B.l 13, we obtain: {cc}t = A\/{K a} r({l}-{a} t)B+{a} i. 1 (B.l 16) The presence of the non-integer exponential n in this equation makes solving this equation in terms of {a}* very difficult for the general case (and this is the simplest of the cure rate equations). Thus, instead of solving directly for {a}k, an iterative solution technique is employed in which: {a};' ,A4Ka}''({l}-{a};)\"+{a},, (B.l 17) where i is the iteration number. B.2.4 Solution Technique As discussed, the equations for temperature and the resin degree of cure in composite materials are coupled, that is, the value of each parameter affects the other. This coupling effect is shown in the term in Equation B.86 for heat generation (dependent on resin cure rate) and in the cure rate factor in Equation B.l 12 (dependent on temperature). Had a forward Euler time integration technique had been employed, -307-Appendix B: Equations and Algorithms the coupling of these terms would be completely accounted for in the shown equations and no solution iteration would be required. However, the extreme non-linearity of the cure rate equation and the often large heat generation terms would require that the forward Euler solution employ extremely small time steps to remain stable. The backward Euler integration scheme employed ensures solution stability (although not solution accuracy), however large the time steps used. This technique, however, requires that the heat generation and temperature values in Equations B.86 and B.l 12 be those at the end of the time step. Since these are unknown at the start of the step, an iterative solution is required. The technique used is as follows: In each iteration, the resin degree of cure at the end of the time step is calculated using the last iteration values for temperature and degree of cure, i.e., {a}i = A\/({Ka};)7({l}-{a}r)\" + {<_, (B.l 18) where k is again the time step number and \/ is the iteration number. The temperature at the end of the time step is then determined using: {T};=([KT];)-'{FT}; (B.l 19) The iteration is performed until the norms of the solution change between iteration steps are sufficiently small, i.e., IIWl-Wr-.II^ Aia}^ (B.120) and l l l T l l - m r . ' . ^ A i T ^ (B.121) -308-Appendix B: Equations and Algorithms B.2.5 Program Solution Algorithm The thermochemical module is called by the MAIN program (see Section A.7) every time step of the solution. Unlike other modules, this routine runs every step due to the importance of the calculated temperature and degree of cure to 'downstream' module calculations. The solution algorithm used by the thermochemical module is as follows: 1) Update element thermal properties (C]>c, pc, [Kc], Sections B.l.3, B.1.4, B.l.5) 2) \/\/\"program in an error state then i) RETURN 3) Loop over all model elements 4) Calculate \\kK\\e and [CP]<, 5) Add contributions of [Kc],, and [CP] e to [kT]e and [Cp],, to {fi}e 6) Assemble [kT]e and [fT]B into global matrices [KT] and {FT} 7) Apply essential boundary conditions 8) Add contribution from external load vectors {FTq}5 and {FTh} to total thermal load vector {FT} 9) Factor the thermal stiffness matrix [KT] 10) Calculate {d}\u00b0 and {a}\u00b0 (Equation B. 118) 11) Begin iteration loop 12) Loop over all model elements 13) Calculate change in internal heat generation vector A{fTQ} from last iteration (for the first iteration, this is the full vector) 14) Add element A{fTQ}^ to global thermal load vector, {FT}' = {FT}\"' + A{FTQ}' 15) Solve for {T}^ = [KT]_,{FT}'' 16) Calculate {ct}^ and {a}^ 17) Check convergence (Equations B.120andB.121) 18) If not CONVERGED, GOTO 1 1 19) Calculate allowable thermochemical module time step based on convergence rate (see Section A.4) 20) RETURN Currently, applied heat fluxes cannot be specified so this vector is always zero. -309-Appendix B: Equations and Algorithms B.3 Stress and Deformation Module As discussed in Chapters 2 and 5 of the main body of the thesis, the stress and deformation module (referred to in short as 'the stress module') is used to predict the development of strain and deformation in the structure of interest throughout the process cycle. An 'incremental' elastic plane strain finite element solution is employed, using the same discretized mesh as in the thermochemical module solution. The following section outlines in detail the development of the discretized equations for the finite element solution and provides details of the solution algorithm employed. Particular emphasis is placed on discussion of non-standard procedures such as the algorithm for simulation of tool removal and calculation of residual strains for hardening materials. B.3.1 Problem Finite Element Discretization As discussed in Section B.4, the discretized system of algebraic equations for the stress and deformation module was developed using the Rayleigh-Ritz technique, employing the functional expression for the system potential energy, i.e.: where U is the strain energy of the system, and Qw represents the external potential energy of body forces and surface tractions as a body undergoes deformation. When all terms are considered, the complete potential energy expression becomes: (B.l 22) (B.123) J{u} r {F\u201e}dn-J{u} r {\u00ae}dT n r For the current plane problem, the various terms in the above equation are: {e} = {sx sz], the strain field [C] = the material stiffness matrix (plane strain). {e0} , {a0} = initial strains and initial stresses -310-Appendix B: Equations and Algorithms {u} = {u w}r, the displacement field {F B }= {FXF:}, body forces {O} = {<3>x ,}, surface tractions {8} s nodal degrees of freedom (d.o.f.) of the structure Displacements at any point within an element are interpolated from the element nodal displacements {8}, using: u} = [N8]{5} (B. l 24) where [N5] is the matrix of displacement interpolation or shape functions defined in Equation B . l 54. Employing the assumption of small strains, element strains are calculated from the displacements using: B} = [55]{U} (B.125) often expressed as: \u00a3} = [B5]{5} (B. l 26) where [B8] is the matrix of derivatives of the displacement shape functions, defined as: [B8] = [S 6][N 6] (B. l 27) and [98] is the differential operator, which for the plane displacement problem is defined as: [3J = d_ dx 0 d 0 d_ dz dx dz d (B.l 28) Substitution of the above expressions for {e} and {u} into Equation B.123 yields: n,=^i{5}:LkL{6}.-\u00a3{6}:Lf]. (B. l 29) where nele is the total number of elements in the complete discretized structure. [k]t. and {f}e are the element stiffness matrix and load vectors respectively, defined as: -311-Appendix B: Equations and Algorithms [k]e= j[B8f[C][B5]<\/Q (B.l 30) {f}e = J [ B 5 ] 7 ' [ C ] { E 0 } ^ Q - J [B8]r{a0}^ Q+ J [ N 5 ] R [ F B ] J [ N 5 ] R [ ( D ] r f T (B.131) n e nc, n s r\u201e Combining the load and stiffness matrices for all elements (and adding externally applied loads), we obtain the potential expression for the structure as a whole: np=l[5]7'[K]{5}-{5}R{F} (B.132) nele where: [K] = ]\u00a3[k] e (B.l 33) e = \\ nele and: F} = {P} + \u00a3 [ f ] e (B.134) To obtain the system of algebraic equations for our discretized problem from the potential energy expression in Equation B.132, we employ the principle of stationary potential energy which states (Cook etal., 1989): Among all admissible configurations of a conservative system, those that satisfy the equations of equilibrium make the potential energy stationary with respect to small admissible variations of displacement. Thus, making Tlp stationary with respect to small changes in nodal displacements 5-,, the equilibrium configuration of the system is defined by the n algebraic equations: ^ = 0 fori = 1,2, ,n (B.135) (where n is the number of degrees of freedom in the model). Performing this differentiation, we obtain: {F} = [K]{8} (B.136) -312-Appendix B: Equations and Algorithms with {F} and [K] defined in Equations B.l 34 and B.133 respectively. The set of nodal displacements can then be solved as: {5} = [K]-'{F} (B.137) It should be noted that in practice this equation is not solved through matrix inversion but rather via techniques such as Gaussian elimination. B.3.2 Solution Procedure The solution procedure used for the stress module calculations can be described as an incremental linear elastic plane strain analysis. At each stress module time step6, the change in nodal displacements, A{8}\u201e is determined and the total displacement calculated using: {S} = \u00a3A{5}. (B.138) 1=1 where k is the number of the stress module time step7. To determine A{8}(note that the subscriptis being dropped for convenience), we use: A{8} = [K]~'A{F} (B.l 39) The element stiffness matrices [k]e are determined using Equation B.l30 in which the required material stiffness matrices, [C], are based on estimated average material properties over the last time step. In the current analysis, the change in element nodal force vector, A{f}e, is composed entirely of contributions\" from internal element thermal and cure shrinkage strains with no contribution from external loads, i.e., A{f}e = A{f}e = j*[BJr[C]A{e0} dQ. (B.140) Recall that the stress module may not actually run every model time step. ~> i ~> ol;>-Appendix B: Equations and Algorithms For non-composite materials, calculation of [k]e and A{fj}e are relatively straightforward since element properties are constant throughout their volume. For composite materials, the calculation is more complex since an element may contain several plies with different orientations with respect to the element local axis. A more detailed discussion of the calculation of [k]eand A { f i } e for both composite and non-composite materials is provided in Section B.3.4. At the completion of the process cycle, two final calculations are performed to simulate manufacturing processes which occur after the completion of the nominal cure cycle. The first is a 'cool-down' step, in which the component temperature is uniformly reset to its starting temperature to remove any residual thermal strains. The second step is tool removal, discussed in more detail in Section B.3.7. The tool removal step is used to simulate what will occur when the part is removed from the process tooling. It is required since large stresses may remain at tool\/part interfaces at the end of processing. Relieving these stresses by removing the tool may have a significant impact on part shape. B.3.3 Stress Module Solution Flow This section provides an outline of the operation of the stress and deformation module. As all other program modules, the stress module is called by the MAIN program (see Section A.7) at every time step of the problem solution. Although called, this routine does not necessarily run each time step, but calculates the allowable time step and checks whether it is time to perform a stress calculation (see Section A.4). The solution algorithm used by the stress and deformation module is as follows: 1) Calculate allowable stress module time step, AtSA (Section A.4) Since a small displacement formulation is used, the total {8} is just stored for output and the nodal locations are not actually updated each step. -314-Appendix B: Equations and Algorithms 2) Check if stress module is to be RUN; if not, RETURN 3) Update mechanical properties (En, E22, ..\u2022 vn, CTE,, CSCh Sections B.l) 4) Reset [K] and {F} 5) Loop over all model elements 6) Calculate element stiffness matrix [k]e and change in nodal load vector, A{f}e (Section B.3.4) 7) Calculate change in element thermal and cure-shrinkage internal strains, Ajsje (Section B.3.5) 8) Apply essential boundary constraints (i.e. fixed and sliding nodes) 9) Assemble [k]e and A{f}c, into global matrices [K] and A{F} 10) Solve for change in nodal displacements, A{8} = [K]_1A{F} 11) Calculate the change in element mechanical strains, A{sCT}e (Section B.3.5) 12) Calculate the change in element mechanical loads {FCT}e (Section B.3.6) k 13) Calculate total nodal displacement {8} = ^ A{8}. (do not update node locations) \/=i 14) RETURN B.3.4 Element Stiffness Matrix and Internal Load Vector Calculation As outlined in Section B.3.2 and illustrated in Figure B.l, the element stiffness matrix [k]c. and the change in the internal load vector A{fj}<, are calculated using equations B.130 and B.140 respectively. For elements containing non-composite materials, both of these calculations are quite straightforward since the material stiffness matrix [C] (see Section B.l. 1.1) and the internal strain vectors {so} (see Section B.l. 1.3) are both constant throughout the element volume. This is also true for composite material elements which contain only a single ply or plies of a single orientation. In either cases, volume integrations indicated in Equations B.130 and B.140 are performed using standard Gaussian integration as described in Section B.4. For composite material elements containing multiple plies, the procedure is more complex since each ply may be at a different orientation and thus have both a different stiffness matrix and a different internal strain vector. For this case, [k]e and A{f}c are determined through the use of a Gauss-Simpson (Trapezoidal rule) integration technique. Using this method, described in more detail in Section B.4.2.2, -315-Appendix B: Equations and Algorithms the contributions of each ply in the element to the matrices are calculated individually and summed to find the total element matrices. The technique outlined in this section is equivalent to: nply (B.141) nply -mil (B.142) where nply is the number of plies in the element and [k]e, = J[B 5 ] r [C] \/ [B i ]dn (B.143) Aft} = J[BJ r[C],A{E 0},dQ (B.144) where [C]\/ is the ply material stiffness matrix, A{e0}\/ is the change in the ply internal strain vector, and Q\/ is the ply volume. 4f:2 [k] e= J[B 5] 7[C][B 5]^Q Af A{lj}e= J[B8f[C]{e0}rffi Figure B.l: Calculation of element stiffness matrix and change in internal load vector during a stress step. The algorithm used to determine [k]c and A{f}t, and the global matrices [K] and A{F} is as follows: 1) Calculate [S ] and A{s0 } in ply local axes (Section B.l. 1.1 and Section B.l. 1.3) -316-Appendix B: Equations and Algorithms 2) Loop over all plies in the element8 3) Transform [S*] and A{s0*} to element local axes to obtain [S']\/ and A{s0'}\/ Section B.l. 1.1 and Section B.l. 1.3) 4) Calculate plane strain matrices [CEp]\/and A{\u00a30Ep}\/(Section B.l. 1.1 and Section B.l. 1.3) 5) Transform [CEp]\/ and A{e0Ep}\/ to global axes to obtain [C]\/ and A{e0}\/ (Section B.l. 1.1 and Section B.l. 1.3) 6) Calculate [k]e;and A{f:}e( (Equation B.l43 and Equation B.l44) 7) Add contribution of [k]e; and A{ff} to [k]e and A{f }e B.3.5 Element Mechanical Strains Calculation In spite of its name, the stress and deformation module nowhere actually calculates a stress. This is because for layered composites such as those examined here, in-plane ply-level stresses will be discontinuous as we move through the part thickness. Thus, for an element with multiple plies, no single representative stress vector can be computed. Strain, however, will be continuous within an element and can be represented as a single vector for each element. Thus, in this analysis, strain is always used to represent the state of an element rather than stress. In this analysis, we define both an element strain {s}c and an element 'mechanical' strain {sCT}e. This mechanical strain is defined as the difference between the element strains induced by internal thermal and cure shrinkage loads (i.e. non-stress-causing strains) and the total element strain, i.e., M.l=i(*{*L-H*i}.l) (B.145) where k is the number of the current stress module time step and A{ej}e\/- is the change in the element internal strain vector at time step i. For non-composite materials, we define only one ply per element with material principal directions which correspond to the element local axes. -317-Appendix B: Equations and Algorithms This mechanical strain is employed for two purposes: as a kind of isotropic representation of local composite stress and as part of the calculation of element mechanical loads which are used later in the analysis to simulate tool removal (see Section B.3.7) The first step in the mechanical strain calculation is to determine the change in the internal strain vector from the element stiffness matrix, [k]c, and the change in the internal load vector,A{f;}e. In order to do so, we solve the equation: applying the minimum number of permissible constraints to the element to prevent free-body rotation as shown in Figure B.2. Figure B.2: Calculation of the change in the element internal strain vector during a stress step. The change in the internal strain vector A{Sj}e is then determined using Equation B.126. As shown in Figure B.3, after solution of the global displacements, the actual change in strains for each element, A{e}<, are determined (again using Equation B.126) and the new mechanical strain vector for each element calculated using Equation B.145 above. Figure B.3: Calculation of the actual change in the element strain vector during a stress step. The algorithm used to calculate element mechanical strain is as follows: A{8,},=[krA{i;}f (B.l 46) A { e , } e = [ B 5 ] A { 8 , } e -318-Appendix B: Equations and Algorithms 1) Calculate element stiffness matrix, [k]t,, and change in internal load vector A{fj}<, (Section B.3.4) 2) Apply element essential boundary conditions 3) Calculate change in element nodal displacements and element strains due to thermal strains,A{5j}c, and Alsjjj, respectively (Equation B.l46 and Equation B.126) 4) Solve for change in global nodal displacements during step (Equation B.l40) 5) Calculate actual change in element strains A{e}e during step (Equation B.126) 6) Calculate change in element mechanical strains during step (A{eCT}e = A{s}e- A{Sj}e) 7) Add contribution of change in element mechanical strain during step to total mechanical strain It should be noted that if the plies in an element are of more than one orientation, it is not possible to use the results of the current analysis to determine ply stresses and thereby define failure. The reasons for this are twofold. First, for such elements, the computed e l e m e n t mechanical strain cannot easily be directly related to the strains seen by the plies within it. This is true since the plies in the element undergo different amounts of thermal and cure shrinkage strain depending on their orientation. For example, while the internal thermal strains in an element containing plies of a [0\/90] orientation will certainly be different, no mechanical strains will be calculated for this element unless it is constrained by external forces (such as an adjacent element). Even if the mechanical strains of each ply were known, their stress states at the end of processing could not be determined. This is true because a material's properties change continuously throughout processing, meaning that its stress strain relationship at a given point in time is not defined by {a} = [C]{s} but rather by d{a} = [C]c\/{e}. This is an important distinction which means that the stress of a ply cannot be extracted from its strain. If we want to know the total stress in a ply at a given time, it must be determined at each solution step and the values at each step summed1. B.3.6 Element Mechanical Load Calculation In order to perform the tool removal simulation (Section B.3.7), it is necessary that the mechanical loads applied to the nodes of each element by its neighbours be known. Due to varying material mechanical This could be done, but would require additional computational effort and larger amounts of computer memory. It was not done in the current due to the focus on overall deformation rather than ply failure. -319-Appendix B: Equations and Algorithms properties during processing, this calculation must be performed at each time step and the total element mechanical loads be determined from the sum of the loads at each step, i.e., The change in element mechanical loads at each step cannot be determined directly since at equilibrium the sum of the loads at each global model node will sum to zero. The value of A{fa}e must therefore be determined from the difference between the total element strains in the step and the part of those strains that were induced by the internal strains of the plies in that element. Calculation of the change in these mechanical strains in the step, A{sCT}e, is described in Section B.3.5. The change in element mechanical loads, A{fCT}e, is calculated from the change in the element mechanical strains, A{ea}<,, using the same basic method outlined in Section B.3.4 for calculating the change in internal loads, A{fj}e, replacing A{eo}\/ with A{eCT}e. Thus, Equation B.144 becomes: {U. t = 5>{U, (B.l 47) (B.148) Note that while [C]\/ varies from ply to ply, A{ea}c is the same for all plies in the element. The algorithm used to determine element mechanical loads, {fa} , is as follows: 1) Calculate [S ] in ply local axes (Section B. 1.1.1) 2) Loop over all plies in the element 3) Transform [S*] to element local axes to obtain [S']\/ (Section B. 1.1.1) 4) Calculate material plane strain stiffness matrix [CEp]\/ (Section B. 1.1.1) 5) Transform [CEp]\/to global axes to obtain [C]\/ (Section B.l. 1.1) 6) Calculate A{fa}e (Equation B.148) 7) Add contribution of ply load, A{f0} , to element load, {fa}c -320-Appendix B: Equations and Algorithms B.3.7 Tool Removal Simulation At the end of the process cycle, a simulation of the removal of the tool from the component is performed to determine post-processing component shape. The tool removal simulation involves performing a single elastic stress calculation step, simulating tool removal by forcing tool\/part interface stresses to zero. This is done using the following steps: \u2022 Create a finite element description of the tool removal problem. The new F.E. mesh is identical to that used in the main stress module simulation, except that the elements identified by the user as 'tooling' are not included. A new set of displacement boundary conditions are also required since the part will see different boundary conditions during tool removal than during processing. \u2022 Assemble the tool removal global stiffness matrix, [KTR], and set the tool removal global force vector, {FTR}, to zero. \u2022 Sum the tooling element nodal forces, {fCT}e, at all nodes along the tool\/part interface and add these to the tool removal global force vector. Since these forces will be thenegative of the force with which the tool pushes on the part (from equilibrium), the effect of adding these forces is to force a stress-free interface. \u2022 Solve for the part displacements during tool removal ({8TR} = [K T R] 1 {FTR}) \u2022 Add the tool removal part displacements to those calculated prior to stress removal to determine the post-tool removal part shape. Note that the tooling element displacements thus calculated are meaningless since tool elements were not included in the tool removal simulation. This procedure is schematically illustrated in Figure B.4. -321-Appendix B: Equations and Algorithms a) Remove tooling elements . Residual I tool force b) f k \\ C\\ i T rrr -ve of Residual tool force Add negative of Tool-Part forces c) d) Figure B.4: Schematic of the tool removal process: a) prior to tool removal, part and tool in equilibrium, part conformed to tool shape; b) tooling removed, residual tool\/part interface forces remain; c) add negative of interface loads to obtain stress free interface; d) predicted part shape after tool removal. The algorithm used in the tool removal simulation is as follows: 1) Generate the tool removal ID array and skyline vector 2) Initialize [ K T R ] and {F T R} 3) Loop over all non-tooling model elements 4) Calculate [k]e (Section B.3.4) 5) Apply essential boundary constraints (i.e. fixed and sliding nodes) 6) Assemble [k]e into global matrices [KTR] 7) Add interface forces to {FTR} 8) Solve for the tool removal nodal displacements, {5TR} = [K T R ]\"' {F T R} 9) Add tool removal displacements to total pre-tool removal displacements -322-Appendix B: Equations and Algorithms B.4 Finite Element Approach All program modules examining the processing behaviour of the structure of interest (i.e. the thermochemical, flow and stress modules) employ finite element methods in their solution10. This section briefly outlines some aspects of the finite element method as applied to the current problem. A detailed description of the theory of the finite element method is not provided as it is discussed in depth in numerous publications such as Cook et al., 1989. Using the finite element method, the governing equations describing the problem of interest are 'discretized' into a system of algebraic equations which can be solved numerically. This is done by subdividing the problem domain into a number of simply shaped elements, connected at nodes at which the parameters of interest (e.g., temperature, pressure, or displacement) are actually calculated. The values of these parameters at any point in the element are interpolated from nodal values using simple polynomial interpolation functions. Thus, the exact value of a parameter within the element, O, is approximated by: n ***, (B.l 49) where is the approximate solution, TV, are the element interpolation or 'shape' functions andO, are the parameter values at the element nodes. Using matrix notation, the above equation becomes: O = [N]{0} (B.l 50) where [N] is the shape function matrix (see Section B.4.1) There are a number of potential techniques for developing the system of equations describing a problem, the most appropriate depending on the problem. As discussed in Section B.3.1, for structural analysis, variational or Rayleigh-Ritz techniques are typically employed. In cases where differential equations are 1 0 This may not be true of models of other components of the virtual autoclave. -323-Appendix B: Equations and Algorithms available for a problem and a variational formulation o f the problem does not exist (e.g., the differential equation o f the problem contains derivatives of odd order), 'weighted residual' techniques are often employed. The most popular of the weighted residual techniques is the Galerkin method used to develop the system o f equations for the temperature solution of the thermochemical module as outlined in Section B.2.2. B.4.1 Elements and Shape Functions A large number o f different types of elements with various shape functions are employed in finite element analyses. In the current model, the only type o f available element is the four-noded plane (2-D) solid bilinear isoparametric element, used in all finite element solutions. The shape functions for this element, are (Cook et al., 1989): [N] = 1\/4(1-^(1-77)\" 1\/4(1 + ^)0-\/7) 1\/4(1 + ^ )0 + 7) 1 \/ 4 ( 1 - \u00a3 ( 1 + 7) (B.151) where \u00a3 and TJ are the element natural coordinates defined in Figure B.5 . Thus, for this element, parameter values at any point are calculated from nodal values using linear interpolation. V 4 \" 1 1 4 3 1 1 1 (1 2 1 1 > 1 4 Figure B.5: Plane bilinear isoparametric element in global (left) and natural (right) co-ordinate systems. -324-Appendix B: Equations and Algorithms In order to minimize the number of elements required to discretize a domain, it is permissible to incorporate multiple plies of any orientation within each composite material element as shown inFigure B.6. A whole number of plies need not be maintained (e.g., a ply can contain 2.6 plies), but the ply plane must be parallel to the element local co-ordinate x 'axis. Ply orientation \u2022 1 z a b Figure B.6: Composite element shown in: a) isometric view, b) plane view. Another important characteristic of the finite elements used in COMPRO is that they all assume that derivatives of the calculated nodal parameters in the out-of-plane direction (y) are zero. Thus, for the temperature and stress modules we have: dy dy dy (B.l 52) The shape function matrix for a particular type of problem will depend on the number of degrees of freedom at each node. The temperature shape function matrix, is defined as: [NT] = [N, N 2 N 3 N 4 (B.153) while the displacement shape function matrix is defined as: -325-Appendix B: Equations and Algorithms IN*1 = N, 0 N 2 0 N 3 0 N 4 0 0 Nj 0 N 2 0 N 3 0 N 4 (B.l 54) The matrices [BT] and [B5] contain the basis function derivatives for displacement and temperature respectively. These matrices are defined as follows: [B T | = Nu N2,x N3 v \/ W \u201e , (B.162) .k=l 1=\\ m=l where, as before \u00a3 = \u00b11 \/ V3 \u2022 Here k is the number of plies, rjkm is the matrix of integration points in the element TJ direction, and | J\\k is the determinant of the ply (rather than the element) Jacobian matrix. -329-C. Material Properties and Sensitivity Analyses Results This appendix outlines the material properties employed in numerical analyses in Chapters 4, 5 and 7 in the main body of the thesis. Also provided are tabular results of the experimental and numerical sensitivity studies outlined in Chapter 7. Sources of material properties data referenced in this section include: 1. Engineered Materials Handbook, Volume 2, Engineered Plastics, ASM International, Metals Park, Ohio, Nov. 1988. 2. Metals Handbook, Tenth Edition, Volume 2, Properties and Selection: Nonferrous Alloys and Special Purpose Materials, ASM International, Metals Park, Ohio, Nov. 1990 3. HEXCEL Corporation, Dublin, CA, product data sheet TSB -120. C.I Thermochemical and Stress Module Verification Test Data Table C.I: Material properties used in comparison of thermochemical module and exact predictions (Section 4.6.1). Parameter Value(s) Employed Density (kg\/m3) p= 1000 Specific heat (J\/kgK) Cp= 1000 Thermal conductivity (W\/mK) *33 = 5 Table C.2: Thermophysical properties of Hercules AS4\/3501-6 used in comparison of thermochemical module predictions with those of Bogetti, 1989 (Section 4.6.2). Cure kinetics parameters were also used in comparison of thermochemical and spreadsheet degree of cure calculations (Section 4.6.1). Parameter Value(s) Employed Density (kg\/m3) p= 1.52xl03 kg\/m 3 Specific heat (J\/kgK) Cp = 942 J\/kgK Thermal conductivity (W\/mK) k33 = 0.4457 W\/mK Heat of Reaction (J\/kg) HR = 198.9 x lO 3 Cure Kinetics (cure model equation 1) At = 3.503xl0 7 Is \/f2 = -3.357xl0 7 \/s \/f3 = 3.267xl0 3\/s . -330-Appendix C: Material Properties and Sensitivity Analyses Results AE, = 8.07x104 J\/gmol A\u00a3 2 = 7.78x104 J\/gmol A\u00a3 3 = 5.66xl0 4 J\/gmol ac = 0.3 B = 0.47 Table C.3: Mechanical properties used in stress module patch test (Section 5.6.1). Parameter Value(s) Employed Isotropic elastic modulus (Pa) E= l x l O 9 Isotropic Poisson's ratio (-) v= 0.3 Table C.4: Mechanical properties used in stress module thermal anisotropy springback tests (Section 5.6.2). Parameter Value(s) Employed Thermally anisotropic Material Isotropic elastic modulus (Pa) E= l x l O 9 Isotropic Poisson's ratio (-) v=0.3 Thermal expansion coefficients (xl0\" 6\/\u00b0C) CTE, = 0, CTE2 = 0, CTE-i = 100 Cured unidirectional AS4\/8552 plies Elastic constants (stiffnesses in Pa) E\u201e= 1 2 2 . 2 x l 0 9 , \u00a3 2 2 = 9.879x109, \u00a3 3 3 = 9.879xl0 9, G,2 = 5.179xl0 9, G 1 3 = 5.179xl0 9, G 2 3 = 3.357xl0 9, v,2 = 0.2684, v,3 = 0.2684, v 2 3 = 0.4712 Thermal expansion coefficients (xlO\"6\/\u00b0C) CTE, = 0.6, CTE2 = 28.6, CT\u00a3 3 = 28.6 Table C.5: Mechanical properties used in stress module bi-material strip tests (Section 5.6.3). Parameter Value(s) Employed invar A l l See Table C.8 Aluminum A l l See Table C. 7 Cured unidirectional AS4\/8552 plies A l l See Table C.4 -331-Appendix C: Material Properties and Sensitivity Analyses Results C.2 Material Properties Used in Numerical Case Studies (Chapter 7) Table C.6: Thermophysical properties of 'rubberized caul'. Material employed in case study #1 (Section 7.1). Property Model Source Density (kg\/m3) p = 1400 Estimated from properties of 8552 epoxy matrix with carbon reinforcement {Vf- 0.25) Specific Heat Capacity (J\/kgK) Cp= 1060 + 3.2 * T Estimated from properties of 8552 epoxy matrix with carbon reinforcement {Vf= 0.25) Conductivity (W\/mK) k\\\\ = k-22 = ^33 = 0.68 Estimated from properties of ABS (1) matrix with carbon reinforcement {Vf= 0.25) Table C.7: Thermophysical and mechanical properties of 5052 aluminum. Material employed in case study #2 (Section 7.2). Property Model Source Density (kg\/m3) p=2660 (2) Specific Heat Capacity (J\/kgK) Cp=960 (2) Isotropic thermal conductivity (W\/mK) k = 120 Estimated from similar alloys (2) Isotropic thermal expansion coefficient (xlO\"6) CTE = 24.0 (2) Isotropic elastic constants \u00a3 = 7 1 . 0 GPa, v = 0.334 (2) ' ' Table C.8: Thermophysical and mechanical properties of invar 36. Material employed in case study #1 (Section 7.1). Property Model Source Density (kg\/m3) \/9=8140 (2) Specific Heat Capacity (J\/kgK) Cp=515 (2) Isotropic thermal conductivity (W\/mK) k= 10.7 (2) Isotropic thermal expansion coefficient (xlO\"6) CTE= 1.54 Average over 0 - 215 \u00b0C (2) Isotropic elastic constants E= 150 GPa, v = 0.280 (2) -332-Appendix C: Material Properties and Sensitivity Analyses Results Table C.9: Thermophysical and mechanical properties of H E X C E L HRP-3\/16-8.0 ( 3\/16 inch cell, 81b\/ff\\ 0\/90 glass\/phenolic honeycomb). Material employed in case studies #1 and #3 (Sections 7.1 and 7.3). Property Model Source Density (kg\/m3) p= 128 By definition (i.e. 81b\/ft3) Specific Heat (J\/kgK) Cp= 1260 Estimated from values for a glass\/polyester system (Bogetti, 1989) Thermal conductivities (W\/mK) k\u201e = k21 = \/c33 = 0.0706 + 3.4xl0\"4 Estimated from charts in (3) Thermal expansion coefficient (xlO-6) CTE, = CTE2 = CTE-i = 10.0 Estimated from values for quasi-isotropic epoxy-glass laminates Elastic constants \u00a3 , , = 0 . 4 3 GPa, \u00a3 3 3 = 1.13 GPa, G, 3 = 0.166 GPa v\\1 = 2^3 = 0.1 Estimated from values of'stabilized compression modulus' of 1.13 GPa and 'plate shear moduli' of 0.227 GPa (L) and 0.131 GPa (W) (3). Table C.10: Thermophysical and mechanical properties of'resin transfer moulded J-frame' Material employed in case study #3 (Section 7.3). Property Model Source Density (kg\/m3) p= 1510 Estimated from values for AS4\/8552 prepreg tape Specific Heat (J\/kgK) Cp= 906 + 2.77 * T Estimated from values for AS4\/8552 prepreg tape Thermal conductivities (W\/mK) ku = k22 = 2.54 + 4.5x10-4 * T \/c33 = 0.45 + 8.0xl0\"4 * T Estimated from values for AS4\/8552 prepreg tape Thermal expansion coefficient (xlO\"6) CTE, = CTE2 = 2.9, CTE3 = 42.0 Estimated from values for a quasi-isotropic layup of AS4\/8552 Elastic constants \u00a3 , , = 4 8 . 0 GPa, \u00a3 3 3 = 10.5 GPa, G, 3 = 4.10 GPa, v,3= v23 = 0.438 \u00a3, , from data provided by The Boeing Company, others estimated from values for a quasi-isotropic layup of AS4\/8552. Table C . l l : Properties of Hercules AS4\/8552 unidirectional prepreg employed in case studies #1, #2 and #3 (Section 7.1, 7.2, and 7.3). Property Model Source Fibre volume fraction (Vf) 0.573 See Chapter 6 Resin cure kinetics Cure model #6 (Appendix B) where: HR = 540xl03J\/kg, A = 1.528x105, A\u00a3 = 66.5x103 kJ\/gmole See Chapter 6 -333-Appendix C: Material Properties and Sensitivity Analyses Results tn \u2014 0.8129, n \u2014 2.736 acr = 5.475xl0\"3, otco = -1.684 Resin viscosity (Pa*s) Resin viscosity model #2: Hubert (1996) M = Aue Eft\/RT (A+Ba) where: A\u201e = 3.25xl010, EM = 7.654x104, ag = 0.47, ,4 = 3.8,-8 = 2.5 Resin degree of cure at gelation 0.469 Hubert (1996) Fibre bed permeability Fibre bed permeability model #4: Hubert(1996) \u00b1 4\u00a3 v *K v, ^ + i where: rf= 4.2x10, V'a = 0.68, k'x = 6,k'z = 0.2 Fibre bed compaction Fibre bed compaction model #2: Ex = Ex and \u00a3 1 3 = \u00a3 i 3 \u00a3 2(e)=( o;+1 - o;)\/(si+,-Si) * (e- e,)+ a, where \u00a3t< \u00a3< eM See Hubert (1996) for values Hubert (1996) Resin modulus development Resin modulus development model #2 (Table B.5) where: Er\u00b0 = 4.67 MPa, E\" = 4.67 GPa, TC!u = -45.7 K, Tclb = 0.0, TC2 = -\\2K, T\u00b0 = 268 K aTg = 220 K, T0 = 20 \u00b0C, aEr = 0 See Chapter 6 Resin Poisson's ratio development Resin Poisson's ratio development model #2 (Table B.7): where: v_= 0.37 See Chapter 6 Resin cure shrinkage development Cure shrinkage model #1 (Table B.3): where: See Chapter 6 VrSco= 0.099, act = 0.055, ac2 = 0.670, A 0.173 -334-Appendix C: Material Properties and Sensitivity Analyses Results Density (kg\/mJ) Pf = 1790,pr = 1300 See Chapter 6 Specific Heat Capacity (J\/kgK) C \/ , \/ = 904 + (7,- 75)* 2.05 \/\u00b0C Cpr = 1005 + (7-20) * 3.74\/\u00b0C See Chapter 6 Thermal conductivity (W\/mK) kj, = 1.69 + T* 1.56xl0\"2\/\u00b0C kf, = 2.4+ T* 5.07xl0\"3 \/\u00b0C Ar = 0.148 + T* 3.43xlO\"4\/\u00b0C + a* 6.07xl0\"2 See Chapter 6 Thermal expansion coefficient (xlO\"6) CTE, = 0.6, CTE2 = CTE3 = 28.6 See Chapter 6 Fibre elastic properties \u00a3 n \/ = 210 GPa, \u00a3 3 3 \/ = 17.24 GPa, G,3f= 27.6 GPa, v13 = 0.2, v23 = 0.25 See Chapter 6 Table C.12: Properties of 'adhesive'. Material employed in case study #3 (Section 7.3). Property Model Source Fibre volume fraction (Vj) 0.001 Low, non-zero value chosen to allow use of composite material models in COMPRO Resin cure kinetics See AS4\/8552 properties See Chapter 6 Resin viscosity (Pa*s) See AS4\/8552 properties Hubert (1996) Resin degree of cure at gelation See AS4\/8552 properties Hubert (1996) Fibre bed permeability See AS4\/8552 properties Hubert (1996) Fibre bed compaction See AS4\/8552 properties Hubert (1996) Resin modulus development Resin modulus development model #1 (Table B.5) where: \u00a3 r\u00b0 = 4.67 MPa, \u00a3 ,\" = 4.67 GPa, YR = 0.0, aa = 0.79 \u00abC2 = 0.81, r 0 = 20 \u00b0C, dEr = 0 Simulated hardening of the adhesive resin late in the process. Resin Poisson's ratio development See AS4\/8552 properties See Chapter 6 Resin cure shrinkage development See AS4\/8552 properties See Chapter 6 Density (kg\/m3) See AS4\/8552 properties See Chapter 6 Specific Heat (J\/kgK) See AS4\/8552 properties See Chapter 6 Thermal conductivity (W\/mK) See AS4\/8552 properties See Chapter 6 -335-Appendix C: Material Properties and Sensitivity Analyses Results Thermal expansion coefficient (xlO\"6) See AS4\/8552 properties See Chapter 6 Fibre elastic properties See AS4\/8552 properties See Chapter 6 C.3 Sensitivity Analysis Results Summary Table C.13: Summary of results of hybrid solid laminate\/honeycomb structure sensitivity analysis (Section 7.1.3). Run name Cycle time (minutes) Maximum exotherm (\u00b0C) Flow time (minutes) S1NOM f>28 9.86 0.10 215 XunierLt.il Solution S1DTH 638 9.53 0.10 323 S1DTL 627 9.94 0.10 218 S1NEH 628 9.90 0.10 215 S1NEL 628 9.76 0.10 215 Thernwphyskal P)ofcrtiei S1RHOH 632 9.46 0.10 210 S1RHOL 625 10.26 0.10 215 S1CPRH 646 8.88 0.11 195 S1CPRL 625 10.55 0.10 215 S1CPFH 634 9.21 0.10 205 S1CPFL 625 10.56 0.10 215 S1KRH 627 9.09 0.10 215 S1KRL 629 10.85 0.10 215 S1KFH 628 9.67 0.10 210 S1KFL 628 10.18 0.10 215 S1KHH 628 9.86 0.09 220 S1KHL 629 9.86 0.10 205 S1VFL 628 8.48 0.10 215 S1VFH 628 11.40 0.10 210 SI MATH 614 10.17 0.13 260 S1HRH 627 11.40 0.10 210 S1HRL 629 8.43 0.10 215 S1CKH 626 12.00 0.13 150 -336-Appendix C: Material Properties and Sensitivity Analyses Results Tntii Clhirm [eristics S1TTKH 709 6.73 0.15 205 S1TTKL 545 15.22 0.07 205 S1TMTH 571 12.70 0.07 210 liiiundury and Initial C <>nditiniT< S1HTH 596 10.26 0.09 220 S1HTL 702 8.50 0.13 195 S1ALPH 627 5.80 0.07 0 S1ALPL 631 15.01 0.13 265 S1RTH 585 10.49 0.11 215 S1RTL 727 8.10 0.08 200 S1PCH 472 12.78 0.13 165 Table C.14: Angle laminate experimental test matrix (Section 7.2.1). Runs 1 and 2 (cure cycle 1 and 2 respectively) Specimen Layup Bagging 1 [90], 2 No-bleed 2 [90] 2 4 No-bleed 3 [0],2 No-bleed 4 [0]24 No-bleed 5 [90\/-45\/+45\/0]3 No-bleed 6 [90\/-45\/+45\/0]6 No-bleed 7 [90\/-45\/+45\/0]6 Bleed Runs 3 and 4 (cure cycle 3) Specimen Layup Bagging 1 [90] 2 4 No-bleed 2 [0]24 No-bleed 3 [90\/-45\/+45\/0]6 No-bleed 4 [90] l 2[0] 1 2 No-bleed 5 [90] 4 8 No-bleed 6 [0] 4 8 No-bleed 14* [90]2 4[0]24 No-bleed -337-Appendix C: Material Properties and Sensitivity Analyses Results The specimen naming convention for the angle laminate experimental sensitivity study, the results of which are shown in Table C.I5, is as follows: Tool id. (A, B, C, D) Tool type (M:male, Ffemale) Run(l-n) \u2014 ^ \u00a3 j\u2014 Test matrix sample 1AM1 Table C.15: Summary of results from angle laminate experimental sensitivity analysis (Section 7.2.1). Shown results are measured springback angle. Run #1 - Process cycle 1 I AMI 1AM2 1AM3 1AM4 1AM5 1AM6 1BM7 1.6 1.2 1.6 1.9 3.1 1.5 2.1 Run #2 - Process cycle 2 2AM1 2AM2 2AM3 2AM4 2AM5 2AM6 2BM7 1.2 1.0 2.2 2.2 2.8 2.1 2.8 Run #3 - Process cycle 3 3 AMI 3AM2 3AM3 3AM4 3AM5 3AM6 3BM14 1.5 1.7 1.7 4.2 1.0 1.4 2.9 Run #4 - Process cycle 3 4CM1 4CM2 4CM3 4CM4 4CM5 4CM6 4DM14 1.1 1.4 1.8 4.9 0.9 0.9 2.0 Table C.16: Summary of results of L-shaped laminate numerical sensitivity analysis (Section 7.2.3). Run designation Springback angle (degrees) Max. arm warpage (microns) S-munal ( use S2NOM (Quasi) 2.332 -123.6 Xumcrical Solution S2DTH 2.331 -123.4 S2DTL 2.333 -123.6 S2DALPH 2.332 -123.5 S2DALPL 2.333 -123.5 S2DMODH 2.321 -122.9 S2DMODL 2.323 -122.9 -338-Appendix C: Material Properties and Sensitivity Analyses Results 7 hermophysical Properties S2CPRH 2.332 -123.6 S2CPRL 2.333 -123.6 S2KRH 2.334 -123.6 S2KRL 2.331 -123.5 S2VFH 2.339 -124.4 S2VFL 2.327 -122.7 S2HRH 2.334 -123.6 S2HRL 2.332 -123.5 S2CKH 2.247 -116.5 Meehtmieal Properties S2EMDL 2.053 -109.8 S2M0DH 2.371 -121 S2MODL 2.289 -122.9 S2E0H 3.295 -187.4 S2E0L 1.706 -84.7 S2CSAH 2.371 -124.4 S2CSAL 2.262 -121.1 S2CSTH 2.417 -120.9 S2CSTL 2.262 -121.7 S2CTEH 2.365 -119.8 S2CTEL 2.298 -124.7 Purl fzeonicln S2TQH (Quasi) 1.125 -41.2 S2TQL (Quasi) 5.262 -325.7 S2TZH(0\u00b0) 1.026 -31.4 S2TZL (0 \u00b0) 4.183 -241.6 S2TNH(90\u00b0) 0.086 -5.8 S2TNL (90\u00b0) 0.037 0.6 S2SYMH 3.729 -225.09 S2SYML 2.221 -116 S2LAYH 2.35 -124.8 S2LAYL 2.312 -121.9 -339-Appendix C: Material Properties and Sensitivity Analyses Results linundcin and Initial Conditions S2TMTH 0.51 6.6 S2HTH 2.333 -123.4 S2HTL 2.272 -118.3 S2ALPH 2.285 -121.1 S2ALPL 2.342 -124.2 S2FLWH 2.43 -131.1 S2FLWL 2.333 -123.6 Proccw Cycle S2CYCH 2.313 -122.1 S2CYCL 2.352 -124.8 Table C.17: Summary of results of fuselage substructure sensitivity analysis (nominal maximum skin deflection = 119.4 microns). ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ S3DTH S3DTL S3DALPH S3DALPL S3DMODH S3DMODL 119.5 120.2 119.5 119.6 119.2 119.6 ThcrmophysLal Properties S3CPRH S3CPRL S3KRH S3KRL S3KHH S3KHL S3VFH S3VFL 119.4 119.4 119.5 119.6 121.0 119.4 122.0 117.4 S3HRH S3HRL S3CKH 120.3 119.5 100.8 Resin mechanicalpro\/niiies S3EMODH S3MODH S3MODL S3E0H S3E0L S3CSAH S3CSAL S3CSTH 97.1 108.3 131.4 136.3 97.2 114.5 127.5 104.7 S3CSTL S3CTEH S3CTEL 136.3 99.9 132.0 Other mechanical properties and part geometry S3CTEJH S3CTEJL S3HEH S3HEL S3HCH S3HCL S3 LA YH S3 LAYL 230.3 -61.1 111.4 130.2 115.7 123.6 132.5 133.9 liminclary and Initial C 'onditiom S3TMTH S3SLH S3SLL S3HTH S3HTL S3ALPH S3ALPLL 130.9 -908.9 127.9 122.4 100.1 123.6 119.1 -340-","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#dateIssued":[{"value":"1997-05","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0088805","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Metals and Materials Engineering","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"For non-commercial purposes only, such as research, private study and education. 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