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UBC Theses and Dissertations

Modelling and experimental issues in the processing of composite laminates Smith, Gregory David 1992

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MODELLING AND EXPERIMENTAL ISSUES IN THE PROCESSING OF COMPOSITE LAMINATES by Gregory David Smith B.A.Sc. The University of British Columbia, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE 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 August 1992 © Gregory David Smith, 1992 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 writ ten permission. (Signature) Department of Metals & Materials Engineering The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract Two published resin flow models for the autoclave/vacuum degassing process (AC/VD) have been implemented in a user friendly computer code. One, the sequential compaction flow model (SCM), includes a heat transfer model and the other, the squeezed sponge resin flow model (SSM), has been extended to include the same heat transfer model. An expression has been derived that makes the SSM mimic the resin pressure and lamina thickness predictions of the SCM. A parametric study of the effect of the lamina stress-strain behavior on the predicted resin pressure and laminae thickness profiles as a function of time has been conducted. The lamina stress-strain behavior has been found to greatly influence the compaction behavior of the laminate. Laminae with hardening stress-strain behavior, which is characteristic of real laminae, have the fastest compaction times. The predictions of the flow models have been compared to experimental laminates. Three laminates, two [0]24 and one [0/902/0]12, were laid-up with small postage stamped sized pressure sensors placed at the upper and lower surfaces and at the 1/4, 1/2 and 3/4 points through the thickness of the laminates and cured by the AC/VD process. The cure cycle was then simulated by the SCM and SSM models and their predictions compared to the sensor response monitored during the cure cycle. The experimental resin pressure profiles for both laminates showed that the resin flow begins very early in the cure cycle, much earlier than predicted by the SCM and about the same as predicted by the SSM. The laminate mass and thickness at the end of the cure cycle have been compared to the model predictions for all three laminates. For the 48 ply laminate the laminae thicknesses have also been compared to the model predictions. In all cases the best agreement was for the SSM model. ii Table of Contents Abstract ii List of Figures xi List of Tables xx Nomenclature xxii Acknowledgement xxix Chapter 1: Introduction 1 1.1. Economic Motivation to Use Composites 1 1.2. Transition from Materials User to Materials Producer 1 1.3. Process Models 2 1.4. Closed Loop Control 3 1.5. Material System 3 1.6. Autoclave/Vacuum Degassing Process 3 1.7. Goals of this Work 4 1.8. Thesis Organization 4 1.9. Figures 6 Chapter 2: Literature Review 7 2.1. Heat Transfer Models 7 2.2. Resin Flow Models 9 2.2.1. Sequential Compaction Model of Loos and Springer 10 2.2.2. Squeezed Sponge Model of Gutowski/Dave 13 Chapter 3: Modelling Details 17 3.1. On the Use of an Object Oriented Programming Approach 18 3.2. Comments on Nomenclature 19 3.3. Common Material Input Parameters and Calculated Properties 19 iii 3.3.1. Fiber Properties 20 3.3.1.1. Input Parameters 20 3.3.1.2. Calculated Properties 20 3.3.2. Resin Properties 21 3.3.2.1. Input Parameters 21 3.3.2.2. Heat of Reaction, Degree of Cure, Cure Rate and Viscosity Calculations 21 3.3.3. Resin Collections in the Bleeder and Laminae 23 3.3.4. Calculation of Resin Collection Properties 24 3.3.5. Bleeder Properties 27 3.3.5.1. Input Parameters 27 3.3.5.2. Calculated Property 28 3.4. Lamina Properties 29 3.4.1. Input Parameters 29 3.4.2. Calculation of Lamina Properties 30 3.5. Laminate Input Parameters 32 3.6. Autoclave Input Parameters 32 3.7. Integration of Heat Transfer and Resin Flow Models 33 3.8. Heat Transfer and Sequential Compaction Resin Flow Models 34 3.8.1. Heat Transfer Model 36 3.8.1.1. Initial and Boundary Conditions for the Heat Transfer Model 36 3.8.1.2. Nodal Equations 37 3.8.1.2.1. Zeroth Node, Adjacent to the Laminate/Bleeder Interface 37 3.8.1.2.2. Interior Nodes 38 3.8.1.2.3. Last Node, Adjacent to the Toolplate 39 3.8.1.3. Method of Solution 40 3.8.2. Sequential Compaction Resin Flow Model 40 3.8.2.1. Additional Input Parameters 40 3.8.2.2. Calculation of Additional Lamina and Laminate Properties 41 iv 3.8.2.3. Perpendicular Resin Flow: Sequential Compaction Flow Model 42 3.8.2.4. Parallel Resin Flow: Sequential Compaction Model 51 3.8.2.5. Method of Solution 54 3.9. Squeezed Sponge Resin Flow Model 54 3.9.1. Lamina Properties for the Squeezed Sponge Resin Flow Model 55 3.9.2. Specific Permeability 56 3.9.3. Resin Velocity for the Squeezed Sponge Resin Flow Model 57 3.9.4. Initial & Boundary Conditions for the Squeezed Sponge Resin Flow Model.... 58 3.9.5. Nodal Equations 58 3.9.5.1. Laminate/Bleeder Interface Nodes 59 3.9.5.2. Interior Nodes 60 3.9.5.3. Toolplate Nodes 62 3.9.6. Method of Solution 63 3.10. Verification of Heat Transfer and Resin Flow Models 65 3.10.1. Heat Transfer Model 67 3.10.2. Sequential Compaction Resin Flow Model 68 3.10.3. Squeezed Sponge Resin Flow Model 71 3.11. Summary 74 3.12. Figures 75 Chapter 4: Reconciliation of the Sequential Compaction and the Squeezed Sponge Resin Flow Models 92 4.1. Pressure Distribution in the Sequential Compaction Flow Model 92 4.2. Force Balance in the Compacted Zone 94 4.3. Derivation of Void Ratio Vs Fiber Bed Pressure Relationship 94 4.4. Idealized Stress Vs Strain Behavior 98 4.5. Comparison of the Flow Models 98 4.6. Resin Velocities 102 4.7. Summary 103 v 4.8. Figures 104 Chapter 5: Parametric Study of Lamina Stress-Strain Behavior and Permeability 110 5.1. Fiber Bed Stress-Strain Relationships 110 5.2. Sensitivity of the Compaction Time to the Lamina Stress-Strain Curve 115 5.3. Effect of Permeability 117 5.4. Summary 119 5.5. Figures 120 Chapter 6: Experiments and Results 132 6.1. Sensor Construction and Response to Pressure 132 6.2. Estimating the Permeability of the Prepreg 136 6.3. Laminate Experiments 138 6.3.1. Laminate #1: A 24 Ply Unidirectional Laminate 139 6.3.2. Laminate #2: A 24 Ply Unidirectional Laminate 142 6.3.3. Laminate #3: A 48 Ply Cross-Ply Laminate 145 6.4. Summary 150 6.5. Figures 151 Chapter 7: Conclusion and Recommendations for Future Work 170 7.1. Conclusions 170 7.2. Recommendations for Future Work 171 References ..172 Appendix A: User's Guide For LamCure 177 A.l. Requirements for LamCure 177 A.2. Installing LamCure 177 A.2.1. Adding the LamCure Icon to the Program Manager 177 A.2.2. Using WIN.INI to control Memory Allocation 178 A.2.3. Updating the WIN.INI File 178 vi A.2.4. Associating *.INP Files with LAMCURE.EXE 178 A.2.5. Starting an Input File from the Program Manager 179 A.2.6. Starting an Input File from the File Manager 179 A.3. Running LamCure-the Menu Options 179 A.3.1. File 179 A.3.1.1.0pen 179 A.3.1.2. Close 179 A.3.2. Run!/Resume! 180 A.3.3. Stop! 180 A.3.4. RunUntil 180 A.3.5. Step! 180 A.3.6. Reset! 180 A.3.7. DDE 180 A.3.7.1. Display Server(s) 180 A.3.7.2. Terminate DDE Links 180 A.4. The Input File Structure 181 A.4.1. Symbols and Units for the Input Values 181 A.4.2. General Layout of the file: Headings and Sub-sections 181 A.4.3. Order is Important for Certain Headings 181 A.4.4. Representation of Boundary Conditions 182 A.4.5. Time-Item Objects 182 A.4.6. Schedules: Collections of Time-Item Objects that Change Over Time 182 A.4.7. Schedule Objects within the Model 183 A.4.7.1.[toolplateTemp] 183 A.4.7.2. [interfaceTemp] 183 A.4.7.3. [bleederTemp] 183 A.4.7.4. [vacuumPressure] 184 A.4.7.5. [externalPressure] 184 vii A.4.7.6. [timeS tep] 184 A.4.7.7.[voidRatioVsFiberPressure] 184 A.4.8. Controlling execution of the program 185 A.4.8.1.flowModel 185 A.4.8.2.sequentialCompactionModel 185 A.4.8.3.parallelFlow 186 A.4.8.4.perpendicularFlow 186 A.4.8.5. mixing 186 A.4.8.6.permeabilityModel 186 A.4.8.7.thermalModel 187 A.4.8.8.checkDiagonalDominace 187 A.4.9. Specifying Autoclave Geometry and Physical Properties 187 A.4.9.1. Specifying the bleeder 188 A.4.9.1.1. Case 1: Mochburg bleeder cloth 188 A.4.9.1.2. Case 2: Generic bleeders 188 A.4.9.2. Specifying the fiber 188 A.4.9.2.1.Casel: Hercules AS4 fibers 188 A.4.9.2.2.Case 2: Generic fibers 188 A.4.9.3. Specifying the resin 189 A.4.9.3.1.Case 1: Hercules 3501-6 resin 189 A.4.9.3.2.Case 2: Generic resin 189 A.4.9.4. Specifying the laminate 190 A.4.9.4.1.Case 1: identical lamina 191 A.4.9.4.2.Case2: unique laminae 192 A.5. Specifying Output 192 A.5.1. Specifying Autoclave, Bleeder, Laminate and Lamina Properties 193 A.6. Example Input Files 194 A.7. Example Output Files and a Comment on DDE Links 200 viii A.8. Nomenclature and Units of the Input Quantities 201 A.9. Output Properties 203 A.9.1. Autoclave Properties 203 A.9.2. Bleeder Properties 203 A.9.3. Laminate Properties 204 A.9.4. Lamina Properties 205 A.10. List of Errors 207 A.11. Figures 212 A. 12. Conversion Formulae 217 Appendix B: Calculation of Additional Properties 218 B.l. Material Properties 218 B.2. Calculation of Lamina Properties 218 B.3. Calculation of Laminate Properties 220 B.3.1. density 220 B.3.2. Properties that Return a Statistical Value: Minimum, Maximum and Average.220 B.3.3. Properties that Return a Sum of Values 221 B.3.4. Properties that Return a Set of Values 221 B.3.5. Properties that Return the Maximum Value 222 Appendix C: Derivation of Lamina Thermal Conductivity Expressions 223 C.l. Thermal Conductivity Parallel to the Fiber Direction 224 C.2. Thermal Conductivity Perpendicular to the Fiber Direction 224 C.2.1. Parallel Thermal Resistances 225 C.2.2. Series Thermal Resistances 225 C.2.3. Square Fibers in a Square Packing Array 226 C.2.4. Circular Fibers in a Square Packing Array 227 C.2.5. Circular Fibers in a Hexagonal Packing Array 229 C.3. Comparison of Thermal Conductivity Expressions 231 C.4. Figures 232 ix Appendix D: Sensor Experiments 236 D.l. Sensor Construction and Principle of Operation 236 D.2. Calibration Apparatus 237 D.3. What Do the Sensors Measure? 237 D.4. Autoclave Testing of Sensors 1 to 5 237 D.4.1. Change in Resistance at Ambient Pressure and Temperature 237 D.4.2. Typical Sensor Response 238 D.4.3. Reproducibility of Sensor Response 238 D.4.4. Effect of Temperature on Resistance 239 D.4.5. Reproducibility of Sensor Initial Resistance 239 D.4.6. First Normalization Procedure 239 D.5. Autoclave Testing of Sensors 6 to 10 241 D.6. New Material and Curite Software 241 D.7. Summary 241 D.8. Figures 242 Appendix E: Determination of Prepreg Resin Mass Fraction 249 x List of Figures Figure 1-1: The relationship between the process model, the controller and the laminate for a closed loop system 6 Figure 1-2: A typical lay-up for producing a composite panel for the AC/VD process 6 Figure 3-1: The unit cubes for the fiber and resin geometries used for calculation of lamina thermal conductivity, k21. (a) parallel heat flow (b) series heat flow (c) square fibers in a square packing array (d) circular fibers in a square packing array 75 Figure 3-2: Simulation flowsheet showing the integration of the heat transfer and resin flow models 76 Figure 3-3: The node network for the heat transfer model with the laminate/bleeder interface, TIF{t), and toolplate, TTP(t), boundary conditions 77 Figure 3-4: Geometry of the 0,H node for the heat transfer model 77 Figure 3-5: Geometry of the interior nodes for the heat transfer model 78 Figure 3-6: Geometry of the nth - 1 node for the heat transfer model 78 Figure 3-7: Heat transfer model flowsheet 78 Figure 3-8: Possible direction for resin flow in a unidirectional lamina 79 Figure 3-9: Schematic of the apparatus used to study compaction in reference [25] 79 Figure 3-10: Compaction sequence during resin flow for the first three layers 79 Figure 3-11: Geometry of a lamina for the sequential compaction resin flow model 80 Figure 3-12: Geometry of the first and second laminae at the moment when the first lamina is fully compacted 80 Figure 3-13: The effect of the incremental and continuous lamina compaction assumptions on the compacted thickness of the laminate, Xl , over time 81 Figure 3-14: The lamina geometry and fiber distribution for the incremental lamina compaction model. Note there are no fibers in the resin channel 81 Figure 3-15: The applied pressures on a lamina and the pressure distribution in the resin channel for the incremental lamina compaction assumption 82 Figure 3-16: The lamina geometry and fiber distribution for the continuous lamina compaction assumption. Note the presence of fiber in the resin channel 82 xi Figure 3-17: Flowsheet of the integration of perpendicular and parallel resin flow for the sequential compaction resin flow model 83 Figure 3-18: The node network for the squeezed sponge resin flow model. Nodes are placed at the interfaces between laminae and at the mid-point of the laminae 84 Figure 3-19: Geometry of the nodes adjacent to the laminate/bleeder interface for the squeezed sponge resin flow model 84 Figure 3-20: Geometry of the interior nodes for the squeezed sponge resin flow model 84 Figure 3-21: Geometry of the nodes adjacent to the toolplate for the squeezed sponge resin flow model 84 Figure 3-22: The flowsheet for the squeezed sponge resin flow model 85 Figure 3-23: Comparison of the temperature profiles through the thickness of the laminate at different times for the analytic solution and three values of the time step for the current simulation 86 Figure 3-24: Comparison of the number of compacted laminae for the current simulation and the original work on the sequential compaction model for a 16 ply laminate for perpendicular flow 86 Figure 3-25: Comparison of the number of compacted laminae for the current simulation and the original work on the sequential compaction model for a 32 ply laminate for perpendicular flow 87 Figure 3-26: Comparison of the number of compacted laminae for the current simulation and the original work on the sequential compaction model for a 64 ply laminate for perpendicular flow 87 Figure 3-27: Comparison of the mass loss of a 64 ply laminate as a function of time for the current simulation and the simulation and experiments of the original work 88 Figure 3-28: Comparison of the mass loss of a 32 ply laminate as a function of time for the current simulation and the simulation and experiments of the original work 88 Figure 3-29: Comparison of the mass loss of a 16 ply laminate as a function of time for the current simulation and the simulation and experiments of the original work 89 Figure 3-30: Comparison of the squeezed sponge resin flow model with the analytical solution for different Ar values 89 Figure 3-31: Comparison of the resin pressure at the toolplate as a function of time for the current simulation and the results from reference [4] 90 Figure 3-32: Comparison of the resin mass loss as a function of time for the current simulation and the results from reference [4]. Note that the waviness in the simulation results is due to errors in digitizing the original data 90 xn Figure 3-33: Comparison of the laminate resin mass fraction as a function of time for the current simulation and the results from reference [4]. Note that the waviness in the simulation results is due to errors in digitizing the original data 91 Figure 3-34: Comparison of the laminate thickness as a function of time for the current simulation and the results from reference [4]. Note that the waviness in the simulation results is due to errors in digitizing the original data 91 Figure 4-1: Schematic of a compacting laminate/bleeder system and the corresponding resin pressure distribution 104 Figure 4-2: Free-body diagram of the pressures acting on the laminate/bleeder system at a cut through the compacted zone 104 Figure 4-3: The assumed resin pressure profile for the sequential compaction model and the corresponding implied fiber bed pressure profile required to satisfy equilibrium 105 Figure 4-4: Schematic of the resin pressure profile within the laminate for a constant viscosity condition 105 Figure 4-5: Void ratio, e, as a function of fiber bed pressure, pf, for an 8 ply laminate 106 Figure 4-6: Fiber bed pressure, pf, as a function of lamina strain for an 8 ply laminate 106 Figure 4-7: Fiber bed pressure, pf, as a function of lamina strain for the relationship used by Dave et. al. [4]. Also shown are two cases of the curves derived from the sequential compaction model 107 Figure 4-8: Resin pressure at the lower interface of each lamina in an 8 ply laminate for the SCM and SSM-SC cases. Note that the resin pressure for the last lamina is, by definition, equal to the external pressure for the sequential compaction model 107 Figure 4-9: Resin pressure at the center of each lamina, pr, in an 8 ply laminate for the SSM-SC case 108 Figure 4-10: Lamina thickness, X, for each lamina in an 8 ply laminate for the SCM and the SSM-SC cases 108 Figure 4-11: Resin velocities for the SCM case and for each lamina of the SSM-SC case. The integers denote the lamina number 109 Figure 5-1: The different lamina stress-strain curves used to generate the simulation results 120 Figure 5-2: The laminae resin pressures as a function of normalized time for a plastic/rigid stress-strain curve 121 xiii Figure 5-3: The laminae thicknesses as a function of normalized time for a plastic/rigid stress-strain curve. tcomp = 950 s 121 Figure 5-4: The laminae resin pressures as a function of normalized time for a hardening stress-strain curve. tcomp = 36.5 s 122 Figure 5-5: The laminae thicknesses as a function of normalized time for a hardening stress-strain curve. tcomp = 36.5 s 122 Figure 5-6: The laminae resin pressures as a function of normalized time for a linear stress-strain curve. tcomp = 150 s 123 Figure 5-7: The laminae thicknesses as a function of normalized time for a linear stress-strain curve. tcomp = 150 s 123 Figure 5-8: The laminae resin pressures as a function of normalized time for a softening stress-strain curve. tcomp = 400 s 124 Figure 5-9: The laminae thicknesses as a function of normalized time for a softening stress-strain curve. tcomp = 400 s 124 Figure 5-10: The laminae resin pressures as a function of normalized time for a rigid/plastic stress-strain curve. tcomp = 1 s 125 Figure 5-11: The laminae thicknesses as a function of normalized time for a rigid/plastic stress-strain curve. tcomp = 1 s 125 Figure 5-12: The laminae resin pressures as a function of normalized time for stress-strain curve based on the work of Dave et al. [36] and Gutowski et al. [50]. tcomp = 70 s 126 Figure 5-13: The laminae thicknesses as a function of normalized time for stress-strain curve based on the work of Dave et al. [36] and Gutowski et al. [50]. t = 70 s 126 Figure 5-14: Comparison of the compaction times for the last lamina for the five stress-strain curves considered 127 Figure 5-15: The family of stress-strain curves used for the sensitivity analysis of the compaction time 127 Figure 5-16: The resin pressures as a function of time for the first and last laminae. Note that the line styles in this figure correspond to the line styles of the stress-strain curves in Figure 5-15 128 xiv Figure 5-17: The laminae thicknesses as a function of time for the first and last laminae. Note that the line styles in this figure correspond to the line styles of the stress-strain curves in Figure 5-15 128 Figure 5-18: The laminae resin pressures as a function of normalized time for a Kozeny constant value of 6. tcomp =45 s 129 Figure 5-19: The laminae thicknesses as a function of normalized time for a Kozeny constant value of 6. tcomp =45 s 129 Figure 5-20: The lamina resin pressure as a function of normalized time for the first and last laminae for five values of the Kozeny constant 130 Figure 5-21: The lamina thickness as a function of normalized time for the first and last laminae for five values of the Kozeny constant 130 Figure 5-22: Compaction time as a function of the Kozeny constant 131 Figure 6-1: Schematic of the Force Sensing Resistor. The channel that leads to the interior of the device is formed by the break in the adhesive layer 151 Figure 6-2: Schematic of the lay-up used to calibrate the sensors. The sensor lead passes through the tape seal 151 Figure 6-3: Sensor resistance as a function of time for sensors si to s5 at ambient temperature (approximately 20°C ) and 0 psig. The vacuum pressure was approximately 29" Hg. The majority of the decrease in resistance occurs in the initial 10 minutes 152 Figure 6-4: Sensor resistance as a function of external pressure for sensors si to s5. The temperature increased linearly from 22°C to 38°C in 82 minutes (corresponding to the maximum pressure). The temperature returned to 24°C upon returning to atmospheric pressure for a total run time of 156 minutes 152 Figure 6-5: The resistance as a function of pressure of sensor si for all the isothermal runs 153 Figure 6-6: Schematic of the interaction between sensor, resin and fiber. The sensor bears the load carried by the fibers, which is transmitted to the sensor at the points where they come into contact, and the hydrostatic pressure of the liquid resin which surrounds it 153 Figure 6-7: Fitting the permeability for the sequential resin flow model to the laminate mass data of an 15 cm square 8 ply laminate. Also shown are the laminate mass predictions for the squeezed sponge resin flow model for two values of the Kozeny constant 154 Figure 6-8: The positions of the sensors though the laminate thickness for a 24 ply laminate 154 xv Figure 6-9: The sensor positions in the plane of the laminate for laminates #1 and #2, 24 ply unidirectional laminates 155 Figure 6-10: The manufacturer's recommended cure cycle for AS4-3501-6 graphite/epoxy prepreg 155 Figure 6-11: The cure cycle used for the 24 and 48 ply laminates 156 Figure 6-12: Schematic of the lay-up assembly for laminates #1 and #2, 24 ply unidirectional laminates 156 Figure 6-13: The cure cycle for Laminate #1, a 24 ply unidirectional laminate. The symbols superimposed on the curves are the discretized boundary conditions used in the simulation 157 Figure 6-14: The measured laminate mass and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #1, a 24 ply unidirectional laminate 157 Figure 6-15: A cross-section of laminate #1 in the xy-plane near the center of the laminate. The position in the laminate is indicated in Figure 6-16 158 Figure 6-16: The position of the laminate thickness measurements for laminates #1 and #2 158 Figure 6-17: The measured laminate thickness and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #1, a 24 ply unidirectional laminate 159 Figure 6-18: The normalized inverse resistances as a function of time for Laminate #1, a 24 ply unidirectional laminate 159 Figure 6-19: The resin pressure predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #1, a 24 ply unidirectional laminate 160 Figure 6-20: The cure cycle for Laminate #2, a 24 ply unidirectional laminate. Note the pressure steps at the end of the cure cycle. The symbols superimposed on the curves are the discretized boundary conditions used in the simulation 160 Figure 6-21: The measured laminate mass and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #2, a 24 ply unidirectional laminate 161 Figure 6-22: The measured laminate thickness and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #2, a 24 ply unidirectional laminate 161 Figure 6-23: The normalized inverse resistance as a function of time for Laminate #2, a 24 ply unidirectional laminate 162 Figure 6-24: The resin pressure predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #2, a 24 ply unidirectional laminate 162 Figure 6-25: The sensor position in the plane of the laminate for Laminate #3, a 48 ply cross ply laminate 163 xvi Figure 6-26: A schematic of the lay-up assembly for laminate #3, a 48 ply cross ply laminate 163 Figure 6-27: The cure cycle for Laminate #3, a 48 ply cross-ply laminate. The temperature cycles at the toolplate and the laminate/bleeder interface are different. The symbols superimposed on the curves are the discretized boundary conditions used in the simulation 164 Figure 6-28: The measured laminate mass and the predictions for the SCM, SSM-6 and SSM-1100 flow models for Laminate #3, a 48 ply cross-ply laminate 164 Figure 6-29: The position of parts A, F and H in laminate #3 165 Figure 6-30: A cross-section of laminate #3 part A near the center of the laminate. The position in the laminate is indicated in Figure 6-29 165 Figure 6-31: The measured laminae thicknesses and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #3, part A 166 Figure 6-32: The measured laminae thicknesses and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #3, part F 166 Figure 6-33: The measured laminae thicknesses and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #3, part H 167 Figure 6-34: The normalized inverse resistance and the pressure applied to the laminate as a function of time for the sensors at the toolplate and the laminate/bleeder interface and the control sensor for Laminate #3 167 Figure 6-35: The normalized inverse resistance and the pressure applied to the laminate as a function of time for the sensors 12 plies above the toolplate and the control sensor for Laminate #3 168 Figure 6-36: The normalized inverse resistance and the pressure applied to the laminate as a function of time for the sensors 24 plies above the toolplate and the control sensor for Laminate #3 168 Figure 6-37: The normalized inverse resistance and the pressure applied to the laminate as a function of time for the sensors 36 plies above the toolplate and the control sensor for Laminate #3 169 Figure 6-38: The resin pressure predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #3, a 48 ply unidirectional laminate 169 Figure A-l: LamCure's main window 212 Figure A-2: The File Open dialog 213 Figure A-3: The File Close dialog 213 Figure A-4: The RunUntil dialog 213 xvn Figure A-5: The Reset! dialog 214 Figure A-6: The Terminate DDE Links dialog 214 Figure A-7: The temperature schedule represented by the time-item objects 214 Figure A-8: The toolplate, interface and bleeder temperature cycle 215 Figure A-9: The external and vacuum pressure cycle 215 Figure A-10: The void ratio as a function of fiber bed pressure 216 Figure A-l l : The Error dialog 216 Figure C-l: A schematic of a lamina showing the three principal directions 232 Figure C-2: The unit cubes for the fiber and resin geometries used for calculation of lamina thermal conductivity, k22. (a) parallel heat flow (b) series heat flow (c) square fibers in a square packing array (d) circular fibers in a square packing array (e) circular fibers in a hexagonal packing array 232 Figure C-3: The thermal model for square fibers in a square packing array 233 Figure C-4: The thermal model for circular fibers in a square packing array 233 Figure C-5: The geometry of the infinitesimal slice, dy, for the center section for circular fibers in a square packing array 233 Figure C-6: The thermal model for circular fibers in a hexagonal packing array 234 Figure C-7: The geometry of the infinitesimal slice, dy, for the center section for circular fibers in a hexagonal packing array 234 Figure C-8: A comparison of the normalized thermal conductivity as a function of the fiber volume fraction for the five derived expressions. The values for the resin and fiber thermal conductivities were taken from reference [114] 235 Figure D-l: Schematic of the Force Sensing Resistor. Note the channel formed by the break in the adhesive layer that leads to the interior of the device 242 Figure D-2: Schematic of the lay-up used to calibrate the sensors. Note the sensor lead passes through the tape seal 242 Figure D-3: Schematic of the interaction between sensor, resin and fiber. The sensor bears the load carried by the fibers, which is transmitted to the sensor at the points where they come into contact, and the hydrostatic pressure of the liquid resin which surrounds it 243 xviii Figure D-4: Sensor resistance as a function of time for sensors si to s5 at ambient temperature of approximately 20°C at 0 psig. The vacuum pressure was approximately 29" Hg. The majority of decrease in resistance occurs in the initial 10 minutes 243 Figure D-5: Sensor resistance as a function of external pressure for sensors si to s5. The temperature increased linearly from 22°C to 38°C (corresponding to the maximum pressure) in 82 minutes. The temperature returned to 24°C upon returning to atmospheric pressure for a total run time of 156 minutes 244 Figure D-6: The resistance of sensor 1 as a function of pressure for the first two runs 244 Figure D-7: The resistance as a function of pressure of sensor 1 for all the isothermal runs 245 Figure D-8: The resistance as a function of pressure of sensor 2 for all the isothermal runs 245 Figure D-9: Normalized inverse resistance as a function of pressure for all isothermal runs 246 Figure D-10: Resistance as a function of temperature for sensors 1 to 5 at 20.1 psig +/-0.4psia 246 Figure D-l 1: Resistance as a function of temperature for sensors 1 to 5 at 80.2 psig +/-0.4psia 247 Figure D-12: The resistance as a function of temperature for runs 1 and 2 for the second batch of sensors, s6 to slO at a constant pressure of 20 psig +/- 0.2 psia for both runs 247 Figure D-13: The resistance and percent change in resistance of the sensors 6 to 10 as a function of time. Note the temperature and pressure are constant at 125 C and 20 psig. The results were obtained from the same run as Figure D-12 248 xix List of Tables Table 3-1: Fiber Input Parameters 20 Table 3-2: Resin Input Parameters 21 Table 3-3: Bleeder Cloth Input Parameters 28 Table 3-4: Lamina Input Parameters 29 Table 3-5: Equations Used to Calculate the Thermal Conductivity Values 32 Table 3-6: Autoclave Input Parameters 33 Table 3-7: Additional Lamina Input Parameters for the Sequential Compaction Flow Model 41 Table 3-8 : Fiber Input Parameters 66 Table 3-9: Resin Input Parameters 66 Table 3-10: Bleeder Cloth Input Parameters 66 Table 3-11: Common Lamina Input Parameters 67 Table 3-12: Additional Lamina Input Parameters for the Sequential Compaction Flow Model 67 Table 3-13: Laminate and Autoclave Input Parameters 67 Table 4-1: Lamina Fiber Bed Pressure at Full Compaction for the Sequential Compaction Flow Model 97 Table 4-2: Fiber Input Parameters 99 Table 4-3: Resin Input Parameters 99 Table 4-4: Bleeder Cloth Input Parameters 99 Table 4-5: Common Lamina Input Parameters 100 Table 4-6: Additional Lamina Input Parameters for the Sequential Compaction Flow Model 100 Table 4-7: Laminate and Autoclave Input Parameters 100 Table 5-1: Time Required for Compaction of the Last Lamina as a Function of Laminate Compaction Time 115 Table 5-2: Kozeny Constant Values, Time to Compaction and Permeabilities for Each Simulation Run 118 xx Table A-l: Possible Headings for the Input File 181 Table A-2: The Default Settings for the [execution] Heading 185 Table A-3: Units of the Input Quantities 201 Table A-4: Autoclave Properties, Required Arguments, Units and their Valid Flow Model(s) 203 Table A-5: Bleeder Properties, Units and their Valid Flow Model(s) 203 Table A-6: Laminate Properties, Units, Required Arguments and their Valid Flow Model(s) 204 Table A-7: Lamina Resin Collection Properties, Units, Arguments and Valid Flow Model(s) 205 Table A-8: Lamina Fiber Properties, Units, Arguments and Valid Flow Model(s) 205 Table A-9: Lamina Properties, Units, Arguments and Valid Flow Model(s) 206 Table B-l: Laminate Properties that Require a Statistical Argument 221 Table B-2: Laminate Properties that are Sums of Lamina Properties 221 Table B-3: Laminate Properties that are Sets of Lamina Properties 222 Table B-4: Laminate Properties that Return a Maximum Value 222 Table C-l: Maximum Fiber Volume Fractions for Each Packing Arrangement 225 xxi Nomenclature OC degree of cure of the resin in a lamina a, degree of cure of the resin in a laminate ab degree of cure of resin in the bleeder ccinil initial degree of cure of the resin in a lamina X cure rate of resin in a lamina ( x ) . c u r e r a t e °f resin in the bleeder (x ) , cure rate of resin in the laminate AE{,i = 1,2,3 activation energies in the resin cure rate equation At time step as a function of time £ porosity of a lamina eh porosity of the bleeder X thickness of a lamina Xc compacted thickness of a lamina Xchan channel thickness of a lamina X j thickness of the /* resin mass in a lamina X, thickness of the laminate X, compacted thickness of the laminate 0 r origin(s) of resin(s) in a lamina or the bleeder ji viscosity of resin in a lamina jib viscosity of resin in the bleeder fi, viscosity of resin in the laminate (ix pre-exponential factor in the resin viscosity equation xxii p density of a lamina pf density of the fiber in lamina p, density of the laminate pr density of the resin(s) in a lamina or the bleeder av compressibility coefficient of a lamina A area of a lamina Ab area of the bleeder A, area of the laminate , i = 1,2,3 cure rate equation pre-exponential factor B constant in the resin cure rate equation or constant in thermal conductivity expressions C transition degree of cure in the resin cure rate equation Cp specific heat of a lamina CP specific heat of the fiber in a lamina Cp specific heat of the resin(s) in a lamina Df diameter of the fibers in a lamina CC compaction coefficient of a lamina e void ratio of a lamina er resin ratio of a lamina = f(/?y) void ratio as a function of applied stress for the laminae Fp packing geometry of the fibers in a lamina Fp packing geometry of the fibers in the laminate hb height of the resin(s) in the bleeder HARC heat accumulation rate coefficient HARCf heat accumulation rate coefficient of the fibers HARCr heat accumulation rate coefficient of the resin HGR rate of heat generation in a lamina HR heat of reaction of the resin(s) in a lamina / the i'h lamina of a laminate, / = 0 —» « - 1 j the j ' h resin mass in a lamina, j = l—>p kn thermal conductivity of the lamina in the x -direction k ,k22 thermal conductivity of the lamina in the z -direction kf thermal conductivity of fibers in the x -direction kf thermal conductivity of fibers in the z -direction kr thermal conductivity of the resin K constant in the resin viscosity equation Kn Kozeny constant in x -direction of the laminae K22 Kozeny constant in z -direction of the laminae L length of a lamina Lb length of the bleeder L, length of the laminate m mass of a lamina mc compacted mass of a lamina mf mass of the fibers in a lamina mr mass of the fibers in the laminate ml mass of the laminate mr mass of resin(s) in a lamina or the bleeder xxiv mr excess resin mass in a lamina mr excess resin mass in the laminate mr mass of the resin (s) in the laminate ri mv volume change coefficient of a lamina Mf fiber mass fraction of a lamina Mf compacted fiber mass fraction of a lamina Mfi fiber mass fraction of the laminate Mr resin mass fraction of a lamina Mr compacted resin mass fraction of a lamina MT initial resin mass fraction of a lamina Mr resin mass fraction of the laminate n number of laminae in the laminate nc compacted laminae in the laminate p the number of resin masses in a lamina pext external pressure pf fiber bed pressure in a lamina pr resin pressure in a lamina p'r intermediate resin pressure pvac vacuum pressure pIF pressure at the laminate/bleeder interface pL resin pressure at the lower interface of the i'h lamina P parallel flow coefficient of a lamina xxv qin heat flux into a lamina qoul heat flux out of a lamina q rate of heat generation in a lamina due to resin curing qacc rate of heat accumulation in a lamina Qoul volume flow rate of resin out of the laminate in the z -direction for the S C M flow model Qz vo lume flow rate out of a lamina parallel to the fiber direction rH hydraulic radius of a lamina RN normalized inverse resistance—first method RN normalized inverse resistance—second method s width or diameter of fibers in thermal conductivity expressions Sb permeability of the bleeder in z -direction Sp permeabili ty of a lamina in the *-direct ion Sp, SP permeabili ty of a lamina in thez -direction S, permeabili ty of the laminate in the z -direction SCM sequential compaction resin flow model SSM squeezed sponge resin flow model t t ime tcomp compaction time of a laminate T temperature of a lamina Tb temperature of the bleeder Tinit initial temperature of a lamina T glass transition temperature of resin xxvi TlF temperature at the laminate/bleeder interface 7] temperature of the laminate TTP temperature at the laminate/toolplate interface U activation energy in the resin viscosity equation v volume of a lamina vc compacted volume of a lamina vf vo lume of the fibers in a lamina vc compacted volume of the laminate vz vo lume of the laminate vr volume of the resin(s) in a lamina or the bleeder vr compacted resin vo lume of the laminate vr excess resin volume of a lamina excess Vr compacted resin volume fraction of a lamina Vf fiber vo lume fraction of a lamina Vf compacted fiber volume fraction of a lamina Vf fiber vo lume fraction of the laminate Vr resin vo lume fraction of a lamina VT resin volume fraction of the laminate Vr initial resin volume fraction of a lamina # 2 2 velocity of the resin in the i'h lamina for the S S M $in velocity of the resin(s) into the bleeder for the S C M $out velocity of the resin(s) through the compacted zone of the laminate for the SCM XXVll &x velocity of the resin(s) in the x-direction for the SCM VI viscosity integral of a lamina VI) viscosity integral of the laminate VIb viscosity integral of the bleeder W width of a lamina Wb width of the bleeder Wt width of the laminate x,y,z coordinate axes xxviii Acknowledgement This thesis is not the product of a single person but the result of the accumulated effort, talent and knowledge of the Composites Group. There are many people I want to acknowledge and thank for their help and friendship. To Dr. Anoush Poursartip for taking me on as a grad student and his guidance and support through to the conclusion of this lengthy project. To Alcan Aluminum Limited and the Cy and Emerald Keyes Foundation for the financial support during part of this work in the form of fellowships. To Katherine MacKenzie, Dennis Chinatambi, Golnar Riahi, Roger Bennett, Livio Gambone, Scott Ferguson, Reza Vaziri, Serge Milaire, Ross McLeod, Daniel Delfosse and Mark Bailey who all played a part in bringing this work to fruition. And lastly to my fiancee Vicki Kerschbaumer for her patience while I was completing this project. xxix Chapter 1: Introduction 1.1. Economic Motivation to Use Composites Aircraft designers often choose to use composite materials because they are stronger and lighter, have better corrosion resistance and longer fatigue life than their metallic counterparts. A lighter structure leads to a more profitable airplane due to increased fuel efficiency and payload capacity. Better corrosion resistance and longer fatigue life of a structure leads to lower maintenance costs. Furthermore, composite components offer additional potential savings. A significant cost in the construction of any aircraft is its assembly. Composites decrease the cost of manufacturing by integrating many small parts into a single piece, thus eliminating the assembly cost. 1.2. Transition from Materials User to Materials Producer Traditional manufacturers are not materials producers. They acquire materials of the required physical properties, modify their shape and assemble the parts. In contrast, composites manufacturers not only produce the material but they must use it intelligently to fully exploit its potential. Hence manufacturers have not only become a materials producers but part designers as well. Therefore they must know the effect each processing operation has on their materials in terms of the physical properties of the part. 1 Generally, the goal of any processing method is to produce an acceptable part. For a composite material a good part might be one that has the correct fiber and resin distribution throughout the part, the correct placement and orientation of each lamina, the correct dimensions and shape, low porosity, low void content and finally the correct degree of cure. In addition to these goals, it is economically desirable to cure a part in the shortest possible amount of time. The additional savings of part integration can only be achieved if a part of acceptable quality is produced. Discarding an unacceptable part is now a much more expensive proposition than before. To optimize the manufacturing route, many process models of the various methods have been proposed. 1.3. Process Models A process model is a combination of mechanistic and/or empirical models that accurately portray the state of the material during processing. In multiphase materials like composites there is often a combination of processes occurring rather than a single distinct process. Thus, before any manufacturing process can be described mathematically, an intimate knowledge of the sub-processes and their interactions is required. Once a process model has been validated against experiment, it can be used to investigate the effect of the input parameters on the quality of the final part. The advantage of this approach over a trial and error method is that it can cope with change; a model can suggest a process cycle for a new part that has a high probability of producing a satisfactory part. However, process models are only predictive. The manufacturer can only use the model to generate a process cycle that will produce a good quality part provided it is processed exactly as prescribed. The manufacturer cannot respond to unanticipated variations or discrepancies in the 2 process cycle. Ideally one would like to go one step further and modify the process cycle in real time to achieve a good quality part. One technique for achieving this is closed loop control. 1.4. Closed Loop Control The ultimate goal of intelligent processing of materials is to modify the process cycle in real time according to the state of the material, so as to produce a part of acceptable quality in the shortest possible amount of time. This approach requires some way of monitoring the critical parameters during the cure of the material, perhaps using embedded sensors. This information would be used in a feedback loop to make suitable changes to the processing environment. The relationship between the control system and the laminate is shown in Figure 1-1. 1.5. Material System In this work the material system is Hercules AS4/3501-6 unidirectional "prepreg" tape. Hercules 3501-6 resin is a thermosetting TGDDM/DDS epoxy resin and was chosen since most of the process models have been developed for thermoset resins, and there is considerable material data available on this system. The raw material is referred to as unidirectional "prepreg" because the fibers are pre-impregnated with resin and all run in the same direction. 1.6. Autoclave/Vacuum Degassing Process Although there are many process routes available, the process examined in this work is the autoclave vacuum degassing (AC/VD) process. It was chosen for two reasons: First, it was the original process for producing high performance structural parts in the aircraft industry and is still the most widely used process today. Second, there are a number of models available in the open literature for the AC/VD process. 3 In the AC/VD process the prepreg laminae are laid-up in the prescribed orientations on a toolplate to form a laminate. The toolplate is either covered with a sheet of release cloth or has a special coating applied to it that prevents the prepreg from sticking to it. The laminate is covered with release cloth, bleeder layers, a caul plate (optional), a breather layer and finally a plastic sheet that acts as a vacuum bag. The laminate may be surrounded by dams to prevent flow in the plane of the laminate. The assembly is shown in Figure 1-2. The whole assembly is then placed inside an autoclave. The temperature is raised and external and vacuum pressures are applied to cure the composite. The variations with time of the temperature, external pressure and vacuum pressure are collectively known as the cure cycle. The quality of the laminate produced by this process varies greatly with the selection of the cure cycle. 1.7. Goals of this Work This work was undertaken with two goals in mind. The first goal was to directly compare two current AC/VD process models in the literature and improve them where possible. The two models are the integrated model of Loos and Springer [1] and the consolidation model of Gutowski [2] and Dave et al. [3,4]. The theory and predictions of each of these models are examined and compared. A means of reconciling the differences between the models is presented. As part of this effort, a new user friendly code running both models under Microsoft-Windows was implemented. The second goal of this work was to create an experimental data set that allows a direct comparison of both flow models, and point out areas that require further work. 1.8. Thesis Organization The overall layout of the thesis is as follows: Chapter 2 reviews the previous work on AC/VD process models. Several relevant approaches and related experiments are described. 4 Chapter 3 details the inner workings of the sequential compaction flow model of Loos and Springer and the squeezed sponge flow model of Gutowski and Dave. Any additional assumptions required to implement these models are explicitly stated. As part of this work, the models are encoded in a computer program designated "LamCure" that runs under MS-Windows on a personal computer. Several extensions to these models are also discussed and implemented in LamCure. Chapter 4 reconciles the different theoretical bases of both flow models. By deriving some new expressions, it is shown that the squeezed sponge flow model can be made to mimic the sequential compaction flow model. Chapter 5 examines the effect of various lamina stress-strain relationships on the compaction behavior of a laminate. It also examines the effect of permeability on the compaction time of a laminate. Chapter 6 describes the experimental work. Laminate mass and thickness measurements at the end of the cure cycle are compared to the predictions of the flow models. In one case, for a cross-ply laminate, the laminae thicknesses are also compared. Qualitative resin pressure measurements as a function of position through the thickness of the laminate are made during the cure cycle and compared to those values predicted by the flow models. Chapter 7 summarizes the thesis and states the conclusions reached in this work as well as making several suggestions for future work. 5 1.9. Figures model and process control initial temperature, pressure, material -mass, etc thermal, Theological and kinetic properties signal analysis computer process model -, sensor signal temperature physical, mechanical and thermal properties actual process thermosetting resin and fiber polymer matrix composite material Figure 1-1: The relationship between the process model, the controller and the laminate for a closed loop system. vacuum bag release cloth breather caul plate (optional) release cloth non-porous film tape seal Figure 1-2: A typical lay-up for producing a composite panel for the AC/VD process. 6 Chapter 2: Literature Review This chapter reviews the literature on AC/VD process models and related experiments. The majority of the work to date has focused on flat laminates, although there has been some work on curved parts. The two main components of a process model are the heat transfer and the resin flow models. Whereas the heat transfer models are reasonably consistent, there are two major variations to resin flow models based on D'Arcy's Law. Furthermore, the experimental evidence is weak, as it is difficult to measured the pertinent parameters. 2.1. Heat Transfer Models Heat transfer models attempt to calculate the temperature distribution within the laminate. The temperature is an important processing parameter for two reasons. First, the resin requires an elevated temperature to cure, and second, the cure reaction is usually very exothermic. The extent of reaction is typically referred to as the degree of cure, (X, and is defined as where the HR is the total heat of reaction and H{t) is the heat released up to time t [5]. There are several resin qualities associated with temperature T and degree of cure a: viscosity fX, cure 7 da ~dt rate -^ - and the glass transition temperature Tg. These are usually given as a function of the degree of cure and the temperature [5-13] for example, V = f(a,T) (2-2) ^ = f ( « , r ) (2-3) dt All heat transfer models are based on the law of conservation of energy and include a term for the heat generated by the resin as it cures [1,7,13-16]. Thus d{pCPT) d (, dT^ - k dt dz dz + pHGR (2-4) where p = density Cp = specific heat k = thermal conductivity HGR = heat generation rate The physical properties of the laminate are functions of the resin and fiber properties. Typical functions are based on a rule of mixtures approach, being either a mass or volume weighted average of the constituent properties. The physical properties of the fiber and resin (other than the viscosity, degree of cure, cure rate and glass transition temperature), are usually taken to be independent of temperature. However, Mijovic and Wang [15,17,18] have measured the density, specific heat and thermal conductivity of epoxy resins and found them to be functions of the degree of cure and temperature. The temperature distribution of a laminate is typically calculated by a finite difference model, usually in an implicit formulation [1,7,16,19]. 8 2.2. Resin Flow Models Resin flow models attempt to calculate the fiber and resin distributions within a laminate. The resin distribution within the laminate is important for two reasons. First, the fibers must be in their correct positions, have the correct volume fractions and orientations to ensure the part will meet its strength requirements. Second, it is currently thought that the growth of voids is controlled by the hydrostatic pressure of the resin [20-22], and thus the determination of the pressure distribution is required. The pressure distribution is intimately coupled to the resin flow and therefore one requires realistic and accurate resin flow models to correctly calculate the pressure distribution. The majority of resin flow models are based on D'Arcy's Law, which describes the flow of viscous fluids through porous media1. The form of D'Arcy's Law here is for one dimensional flow and is given by Loos and Springer [1] as fj. dz where v = fluid velocity S = permeability of the medium fi = fluid viscosity pr = hydrostatic fluid pressure z = thickness There are two major implementations of D'Arcy's Law into resin flow models. The first is that of Loos and Springer [1] who used equation (2-5) in conjunction with the simple step-wise compaction model of Springer [25] to calculate a uniform value for the resin velocity through Lindt [23,24] has developed a resin flow models based on a squeeze flow starting with the basic principles of fluid flow. 9 the compacted zone of the laminate. The second implementation is that of Dave et al. [3,4] where they numerically solve the governing differential equation. 2.2.1. Sequential Compaction Model of Loos and Springer Springer [25] proposed a compaction model for laminae based on experimental analogy. A laminate was simulated by rafts of rods and porous plates placed at equal intervals in a constant viscosity oil. The bleeder was simulated by a perforated steel plate and the applied pressure by placing a weight on top of the plate. He observed that the raft closest to the "bleeder" moved towards the next raft until they were in contact. These two rafts now formed a "compacted zone" and began moving towards the raft directly beneath them (the third raft) and so on until all of the rafts were in contact with each other. Springer postulates that this same mechanism accounts for the compaction of a laminate. Further, he assumed that the fiber beds do not interact and all of the applied pressure is borne solely by the resin. Loos and Springer used this compaction model in their work [1]. They modelled a lamina as a layer with two sub-layers. One consists solely of resin and the other of fiber and resin containing all of the final resin mass of the lamina. There is some experimental evidence by Purslow and Childs [26] that shows thin layers of resin between the original prepreg plies in laminates that have been cured at low pressures. However, the Loos and Springer lamina model would require much thicker resin layers that shown in reference [26] for it to accurately represent a lamina. An interesting aspect of this compaction model is that the compaction occurs over a single lamina. The thicknesses of laminae in the compacted zone are uniform at the final thickness of the laminae while the uncompacted laminae have a uniform thickness at the initial thickness of the lamina. If this flow model were an accurate portrayal of the compaction process, one would expect to find a step-wise change in the thicknesses of the laminae in a laminate. 10 There is some suggestion of this step in lamina thickness mentioned in the work of Kim et al. [27] and Tang et al. [28]. Kim et al. [27] measured the thicknesses of each laminae for a 96 ply laminate at applied pressures of 0, 30, 60 and 90 psig to obtain laminae thickness as a function of position profiles. The profiles show three distinct regions. The first region consists of laminae having uniform thicknesses at the compacted thickness. The second region appears to be a compaction zone varying from 24 plies at 0 psig to 16 plies at 90 psig. In the third region the laminae again have uniform thicknesses at the initial thickness. This generally agrees with the predictions of Loos and Springer [1] but an important difference is that the thickness of the compaction zone is not a single ply but many plies. Tang et al. [28] examined the laminae thicknesses for a 60 ply laminate as a function of applied pressure for the data of Campbell et al. [29] . As in the case of Kim et al. [27] there were three distinct regions. However, the compaction zone for this data was over 2 or 3 laminae which is somewhat smaller than the results of Kim et al. [27] but much closer to the single ply process of Loos and Springer. Hanks et. al [30] give experimental laminae thicknesses for a 384 ply laminate that was cured with bleeder placed on both sides of the laminate. They found the lamina thickness distribution was symmetrical about the center of the laminate. Also, the laminae thicknesses for the first 140 plies from both laminate/bleeders interfaces increased gradually towards the center of the laminate. The thickness of the center 100 plies was fairly uniform. This gradual increase in thickness is quite different to what Loos and Springer [1] predict. Loos and Springer [1] consider the resin flow parallel to the plane of the composite to be decoupled from flow through the thickness of the composite. A simplifying assumption they make for resin flow in the plane of the laminate is that resin only flows parallel to the fiber direction. This is reasonably valid since the permeability in the fiber direction is orders of magnitude larger than the permeability transverse to the fiber direction. They model the resin flow in the fiber direction using an analogy to channel flow. The sub-layer of resin forms a channel that runs the length of the laminate. The pressure gradient between the resin pressure at 11 the midpoint of the channel and the resin pressure at the edge of the laminate is the driving force for flow. They assume the pressure gradient between the center of the laminate and its edge is linear. There are two problems with this formulation. First, decoupling the flows parallel and perpendicular to the composite leads to a solution where two different resin pressures are predicted by each sub-model at the same time and position in the laminate. Consider the resin pressure at the edge of the laminate in an uncompacted lamina. According to their compaction model, the resin pressure is equal to the applied pressure. For their channel flow model, the resin pressure is only slightly higher than the pressure in the vacuum bag. Hence, predicting the resin pressure at any point in the laminate when resin is flowing both parallel and perpendicular to the plane of the laminate is somewhat ambiguous. Second, as pointed out by Tang et al. [26], the true resin pressure distribution over the length of the channel is parabolic and not linear as assumed by Loos and Springer [1]. An implied assumption of Loos and Springer [1] is that the resin pressure at the toolplate is equal to the applied pressure. Roberts [31] gives experimental results that disagrees with this assumption. In his experiment, a pressure transducer that was mounted on the bottom of a toolplate to measure the resin hydrostatic pressure. It was found that the resin hydrostatic pressure was much lower than the applied pressure. In the case of a 64 ply laminate, the maximum resin pressure was only 18% of the applied pressure 10 minutes into the cure cycle. A second implied assumption of the Loos and Springer compaction model is that the resin pressure increases linearly from a value close to the vacuum pressure at the laminate/bleeder interface to the applied pressure at to the toolplate. Morrison and Bader [32] state that the hydrostatic resin pressure throughout the laminate is approximately equal the pressure in the vacuum bag once the laminate is consolidated. 12 2.2.2. Squeezed Sponge Model of Gutowski/Dave In contrast to the sequential compaction model of Loos and Springer [1] which explicitly states that the fibers do not interact, the Gutowski et al. and Dave et al. models are based on the applied pressure being shared by the resin and the fiber bed [2-4]. Gutowski [2] measured the load required to compress graphite fiber bundles impregnated with low viscosity oils. He found that the fiber beds behave as rapidly stiffening springs. The result was a relationship between the pressure in the fiber bed pf and the volume fraction of the fiber V-Pf = f{vf) (2-6) Gutowski assumes the resin is incompressible. Therefore, as pressure is applied to the lamina, resin flows out and the volume of the lamina decreases by a decrease in thickness. Dave et al. [3,4] consider this process to be directly analogous to the compression of liquid saturated soils [33] where the skeletal network of fibers is filled with resin. The hydrostatic pressure of the resin and the pressure in the fiber bed must balance the external pressure [2-4,34-36] as Pe« = Pr+Pf (2-7) Dave et al. [4] have presented results for a one dimensional consolidation model with one dimensional flow using the governing differential equation given by Terzaghi [33] as Dave et al. use the void ratio e , that is the ratio of resin volume to fiber volume, rather than Vf, as in Gutowski's work. 13 SP» d2pr _ dpr fumv d z dt where Sp = the permeability mv = the coefficient of volume change They predicted the resin pressures throughout the laminate and the degree of consolidation in each lamina. The coefficient of volume change is defined as the rate of change of porosity with fiber bed pressure, » . = - - £ . (2-9) dpf Solving equations (2-8) and (2-7) together results in the pressure distribution through the thickness of the laminate as a function of time and position. The method of solution for these equations is generally obtained via a finite difference scheme [4,19,36]. However, caution must be taken when using this approach since the distance between the nodes changes over time due to resin flow. This must be taken into account in the calculations or large errors may occur. In the above equations there are two properties that are properties of the lamina and not just dependent on the properties of the resin or the fiber. These are the permeability SP , equations (2-8) and (2-5), and the relationship between the fiber bed pressure pf and the fiber volume fraction Vf or porosity £ of the lamina, equation (2-6). The relationship between the fiber bed pressure and the resin content of the lamina has been measured by Gutowski et al. [2,34,35,37] for several different commercially available fibers and has been found to be characterized as a rapidly stiffening non-linear spring. The significance of obtaining reliable data for this property cannot be overstated. The resin content and fiber distribution of the laminate is dictated by this property. 14 Tang et al. [28] also state that the Loos and Springer model agrees with that of Gutowski [2,36] for fiber volume fractions below about 0.6 to 0.7. Above these fiber volume fractions the fibers contact each another and the toolplate and the fibers now bear some of the applied pressure and the assumptions of the Springer flow model [25] are no longer valid. The permeability of a lamina is generally considered to be a function of the fiber volume fraction or porosity of the lamina. Intuitively, this makes sense since the more tightly the fibers are packed the more difficult it should be for resin to flow through the fiber bed. However, the sequential compaction model of Loos and Springer [1] assumes the permeability of a lamina to be constant. This result does not agree with the work of Gutowski et al. [34,35] and Lam et al. [38] who have measured the permeability of prepreg and found that it decreases with resin content. Most other models use the well known Kozeny-Carman permeability equation [39]. A form of this equation is used by Dave et al. [3,4,38] and Gutowski et al. [37]. It includes the effect of the anisotropy of the prepreg as D2e3 ku= f- ^ (2-10) where Df is the fiber diameter, £ is the porosity and Ku is the Kozeny constant. Several researchers have measured the permeability of graphite fiber beds [34,35,38,40,41] and found the ratio of K22 to Kn to range from 17.9/0.7 [34] to 11/0.57 [38]. There is some debate as to whether the Kozeny-Carman equation with a constant Ku is representative of the true permeability of the fiber bed. Skartsis et al. [40,41] emphasize that Ku is not constant but a function of the porosity of the fiber bed. However they found equation (2-10) was a reasonable fit to their data over a porosity range of 0.5 to 0.25 (these porosities correspond to resin mass fractions of 0.40 to 0.18) which is close to the range of interest here. 15 Gutowski et al. [34] have developed a modified Kozeny-Carman expression that uses an additional constant in the form of the fiber volume fraction at which the permeability becomes zero. Not surprisingly, since they are using one more fitting constant, their expression shows better agreement with their experimental results. Lam and Kardos [38] have shown experimentally that the Kozeny constant is moderately affected by ply orientation. The K22 value for transverse flow through unidirectional fibers was 11 while for a cross-ply laminate it was 20. This conflicts with the work of Loos and Freeman [42] who obtained similar mass loss results, and hence similar permeability values, for 32 ply [0]32 and [0/90] 8s laminates. 16 Chapter 3: Modelling Details The goal of the mathematical modelling portion of this thesis is to simulate, on a computer, the physical conditions and changes that occur in flat composite laminates manufactured by the autoclave vacuum degassing (AC/VD) process. In this chapter the details of the implementation of a heat transfer model and two resin flow models are described. First, the heat and resin flow models of Loos & Springer [1], which predict the temperature and mass distributions within the laminate, are derived. Second, the resin flow model of Dave et al. [3,4] which has been extended to include the heat transfer model of Loos & Springer, is derived. The models have been encoded in a computer program designated "LamCure" which runs under MS-Windows. The mass transfer model of Loos & Springer is referred to as the sequential compaction model. The resin flow model of Dave et al. is referred to as the squeezed sponge model. The predictions of the resin flow models are compared in Chapter 4. 17 3.1. On the Use of an Object Oriented Programming Approach The computer simulation developed in this chapter was written in an object oriented programming (OOP) language1. The advantage of OOP is that the development of a computer model is very close to how one describes the physical world. Each component, or object, has knowledge of its own physical properties and functionality. For example, a fiber has mass, density, specific heat and thermal conductivity. A resin also has these properties but it also has some additional ones that a fiber does not have: viscosity, degree of cure and heat of reaction. Once a resin and a fiber are defined a lamina can be defined as a combination of resin and fiber. Additional information that more precisely defines a lamina are its width, length, volume fraction of fiber and the geometric arrangement of the fibers in the resin. The properties of the lamina are defined in terms of the properties of its constituents: fiber and resin. For example, the mass of the lamina is simply the sum of the its fiber and resin masses. Obviously, some lamina properties will only relate to one of its constituents. For example, the degree of cure or the viscosity of a lamina is the degree of cure or the viscosity of the lamina's resin. In a similar fashion, a laminate is constructed by combining a number of laminae. The physical properties of the laminate are derived in terms of the aggregate properties of its laminae. This method of building more complex objects out of simpler ones is the essence of OOP programming. The analogy is carried still further by defining an autoclave. An autoclave is a pressure vessel where the cure of a laminate is controlled by applying an external pressure, a vacuum pressure and temperature to the part. The components needed to cure a laminate via the AC/VD process are bleeder material (which absorbs resin as it bleeds off the laminate) and prescribed pressures and temperatures. The interactions between these components are described by the physical laws of heat and resin flow. Actor 3.0 from the Whitewater Group, Inc., 1800 Ridge Avenue, Evanston, Illinois 60201, USA. 18 The main reason why we used an OOP language for model development was the potential to reuse code. Once the components have been specified, different interaction models can be implemented by changing only that part of the code that defines how they interact. It is the use of the OOP approach that allows a relatively straight forward incorporation of two different resin flow models that can both use the same heat transfer model, into the simulation. 3.2. Comments on Nomenclature All variables appear in italics. For example, the mass of a lamina is identified by m . Each component of the autoclave is indicated by a subscript. Properties of the fiber, resin, bleeder and laminate are identified by the subscripts f,r,b and / respectively. Variables that denote lamina properties do not have subscripts. For properties where orientation is important the subscripts 11 and 22 refer to the x -direction (parallel to the fiber direction in the plane of the composite) and the z -direction (perpendicular to the fiber direction and the plane of the composite) respectively. 3.3. Common Material Input Parameters and Calculated Properties Key assumptions in the simulation are: 1. There are only two constituent materials in the lamina: resin and fiber. 2. The fiber is assumed to be stationary and only resin flows from lamina to lamina. 3. The mass of the fiber in any lamina is constant. 4. The existence and effect of voids or other foreign materials in the prepreg and laminate can be ignored. 5. The physical properties of the fiber, resin, prepreg and bleeder are assumed to be independent of temperature, the presence of resin or fiber, and the resin's degree of cure. 19 Since the density is assumed constant this implies that there is no volume change in the components due to temperature or pressure. The presence of the release cloth separating the laminate from the bleeder and from the toolplate is assumed to have a negligible effect on the heat and resin flow models. 3.3.1. Fiber Properties 3.3.1.1. Input Parameters The fiber is a relatively inert object. Five physical properties are required as input parameters. These are listed in Table (3-1). Table 3-1: Fiber Input Parameters. Input Parameter density specific heat thermal conductivity parallel to the fiber direction thermal conductivity perpendicular to the fiber direction fiber diameter Symbol Pf cP/ **, * * Df Units kg/m3 kJ/(kgK) kW/(m K) kW/(m K) m 3.3.1.2. Calculated Properties There are two properties for a fiber that are directly calculated from the input parameters. These are the heat accumulation rate coefficient and the volume. The heat accumulation rate coefficient, HARCf, is defined for convenience, to minimize the number of terms in the nodal equations for the heat transfer model. It is defined as the product of fiber specific heat and mass. HARCf = CP mf (3-D 20 The volume of the fiber, v,, is calculated as the ratio of its mass to its density. 3.3.2. Resin Properties 3.3.2.1. Input Parameters In contrast to the fiber, the resin is a relatively complex object. The resin initial state is a low molecular weight polymer that, during the cure, cross-links to form a single interlinked structure. Five physical properties and two relationships are required as input parameters. These are listed in Table 3-2. Table 3-2: Resin Input Parameters. Input Parameter density specific heat thermal conductivity heat of reaction degree of cure cure rate relationship viscosity relationship Symbol Pr cP{ K HR a %- = f(a,T) V = f(a,T) Units kg/m3 kJ/(kgK) kW/(m K) kW/(m K) none 1/s Pas Of these parameters the last four, namely heat of reaction, degree of cure, cure rate relationship and viscosity relationship, require some clarification and are explained in the next section. 3.3.2.2. Heat of Reaction, Degree of Cure, Cure Rate and Viscosity Calculations The heat of reaction, HR, is the total energy, in the form of heat, that is evolved per unit mass of fully cured resin. The degree of cure, a, of a resin is the fraction of the resin that has reacted. The degree of cure for a resin is defined as the ratio of the amount of heat evolved thus far to the total possible amount of heat evolution, i.e., Pf (3-2) 21 « - ^ (3-3, The rate at which a changes with respect to time is called the cure rate. The time derivative of the above expression is typically given by researchers as a function of the degree of cure and the temperature. The general expression is ^ T = f(a.T) (3-4) dt The degree of cure can be calculated form the cure rate by integrating the cure rate function over time It has been shown experimentally (see for example Lee, et al. [5]), from isothermal and dynamic differential scanning calorimetry, that the degree of cure is additive and adequately described by the above integral formulation. The viscosity of the resin, similar to the cure rate, is typically a function of the degree of cure and temperature. The general expression is fi = f(a,T) (3-6) The specific resin properties, cure rate and viscosity expressions for the resin of interest will be discussed in the section where simulation results are presented. Equations (3-5) and (3-6) imply that the cure rate and the degree of cure are functions of the resin's thermal history. The resin's degree of cure must be calculated and updated at each time step during the simulation. The simplest case is for a laminate with an isothermal temperature 22 distribution, where the thermal history and degree of cure at all points is the same and only a function of time. A more difficult case exists when the temperature distribution is not isothermal but a function of time and position. The degree of cure through the laminate becomes a function of position and time as well. If resin is allowed to flow from lamina to lamina, it becomes even more difficult to update the state of the cure throughout the laminate, as some method of tracking the resin movement is needed. A method for dealing with this problem is discussed in the next section. 3.3.3. Resin Collections in the Bleeder and Laminae The ability to track the resin movement from lamina to lamina requires some method of differentiating resin originally from different laminae. To do this the concept of the resin's origin, 0 r , was created. The origin of a resin is simply the index of the lamina that initially contained the resin. The laminae are numbered from 0 to n - 1 for a laminate of n laminae. An index of / indicates the resin was initially in the i'H lamina. Since the direction of resin flow is from the toolplate to the laminate/bleeder interface (the positive z -direction) and the amount of resin flowing in any one time step is small, it is possible for a lamina to contain resin that was initially contained in different laminae. Consider the /"* lamina. During consolidation, a mass of resin from the /'* +1 lamina flows into the i'h lamina, displacing some of its resin which in turn flows into the /"" - 1 lamina. Hence the Ith lamina will contain some of its original resin, Qr = i, and some resin from the lower lamina, Qr = i +1. The simulation moves the resin through the laminate in one of two ways. The first method assumes laminar flow where there is no mixing between resins from different laminae. The resin from each lamina moves as a slab through the laminate. The second method assumes perfect mixing where, as resin flows from lamina to lamina, the resin entering the lamina mixes perfectly with all the resin that is currently in the lamina. 23 The amount of resin is tracked by monitoring the mass of resin. The mass of a lamina is the mass of the fiber plus a collection of resin masses, each with a different origin. In the general non-isothermal case, each resin mass will have a different thermal history and resulting degree of cure. In the simulation code, the resins in a lamina or the bleeder exist as a collection of resin origins. However, the simulation does not directly operate on resins in a lamina or the bleeder but on resin collections. Therefore, the symbols for resin collection properties are identical to resin properties. 3.3.4. Calculation of Resin Collection Properties The simulation code was written with future expansion in mind. Since there may be a need to have more than one type of resin in a resin collection, the properties of a resin collection are an average of the properties of the resin masses in the resin collection. The expressions used to evaluate the various properties for a resin collection are given below. Where summations of resin mass properties are required it is assumed there arep resin masses, i.e., j = l,2,...,p. The mass of a resin collection, mr, is the sum of the individual masses of each resin mass in the collection; j=p mr = ^ mrj (3-7) The volume of a resin collection, vr, is the sum of the individual volumes of each resin mass in the collection; vr = £ — 0-8) 24 The density of a resin collection, pr, is the ratio of the mass of all the resin masses to their volume; pr=^ (3-9) The specific heat of a resin collection, CP, is determined by calculating the mass weighted average of the specific heat for all the resin masses in the lamina as J=P 5X,A CP, = ^  (3-10) 7=1 The thermal conductivity of a resin collection, kr, is determined by calculating the mass weighted average of the thermal conductivity for all the resin masses in the lamina as J=P XVo y=i m (3-11) 5X 7=1 The above assumption is most suitable for the case of perfect mixing of resin during flow. If laminar flow of resin is assumed, then an average thermal conductivity based on a series resistance analogy would be more suitable. The degree of cure for a resin collection, a, is defined as the ratio of the heat evolved by the collection to the total possible heat evolution; iz£ oc=^jr-p (3-12) 7=1 25 d(X The cure rate of a resin collection, ——, is defined as the time derivative of the degree of cure at expression; %=»K2 '' 0-13) at v u y'=i The heat of reaction of a resin collection, HR, is the ratio of the total possible heat evolution of the collection to its total mass. This is the mass weighted average of the heats of reaction and is calculated as i=p 1 H ^ HR=^jr-p (3-14) 5X 7=1 The viscosity of a resin collection, fj., is the mass weighted average of the viscosities of the individual resin masses and is calculated as J=P ^ = ^ (3-15) y=i This property is used only by the squeezed sponge resin flow model. 26 The heat generation rate, HGR, is defined to minimize the number of terms in the nodal equations for the heat transfer model. It is the sum of the individual heat produced by each resin mass in the lamina, i.e., J=p ( ?a \ HGR = J,[-^)HR1mri (3-16) The heat accumulation rate coefficient, HARCr, is also defined to minimize the number of terms in the nodal equations for the heat transfer model. It is the sum of the heat accumulation rate coefficients for the individual resin masses. HARCr = %Cpmr. (3-17) 3.3.5. Bleeder Properties Bleeder cloth, usually referred to as just bleeder, is a relatively inert object. It is essentially a sponge which absorbs resin as it bleeds off the laminate during the cure. In the simulation code the resin exists in the bleeder as a resin collection. The calculation of bleeder properties that only depend on its resin collection have already been given in sections 3.3.3. & 3.3.4.. 3.3.5.1. Input Parameters Only four physical properties are required as input parameters to specify bleeder. These are listed in Table (3-3). 27 Table 3-3: Bleeder Cloth Input Parameters. Input Parameter length width permeability porosity Symbol Lb wb sb £b Units m m m2 none The length and width of the bleeder are measured in the plane of the laminate and are usually the same as the adjacent laminate. In all cases it is assumed that there will be enough bleeder to absorb all the resin that bleeds off the laminate. The permeability of the bleeder is the resistance the material offers to liquids that flow through it. The porosity of the bleeder is dimensionless and is defined as the ratio of total volume of voids, vmids, to total volume of porous material, bleeder' £=K>ids_ ( 3 _ l g ) bleeder The permeability and porosity are assumed to be independent of applied pressure, temperature and degree of cure of any resin that it contains. The assumption of the permeability and porosity being independent of applied pressure is not particular good. The bleeder material, typically being somewhat spongy, decreases in volume upon application of pressure. Poursartip et al. [43] have shown that the pressure applied to the bleeder can have a considerable effect on the permeability, which decreases by an order of magnitude for an applied pressure of merely 6 psi. 3.3.5.2. Calculated Property The height of the resin in the bleeder, hb, is calculated assuming the bleeder material does not compress. This assumption is made to remain consistent with reference [1]. K=-^r— where Ab=LbWb (3-19) 28 3.4. Lamina Properties A lamina is a combination of fibers and resin. In the simulation code the resin exists in a lamina as a resin collection. The calculation of lamina properties that only depend on its fiber have already been given in sections 3.3.1.1. & 3.3.1.2. and for properties that only depend on its resin collection in section 3.3.4.. 3.4.1. Input Parameters The two flow models incorporated into the simulation require slightly different information to define a lamina. There are eight input parameters common to both flow models. These are listed in Table (3-4). Table 3-4: Lamina Input Parameters. Input Parameter length width initial total mass of the prepreg ply initial resin mass fraction permeability fiber packing initial degree of cure initial temperature Symbol L W mini, K.. 5* FP ««a T. . mil Units m m kg none m2 none none K Both the fiber and resin of a lamina have mass associated with them, mf and mr, respectively. The mass of each constituent is calculated from the input parameters in Table (3-4) as follows. mf=mJl-MrJ (3-20) mrm>=min«-mf (3-21) where Mr is the initial resin mass fraction, as defined in Table 3-4. The fiber always remains in its original lamina. However, the resin in a lamina moves due to resin flow in the x and z-29 directions. Therefore, whereas equation (3-20) is true at all times, equation (3-21) is only useful in determining the initial value, and new values of mr are evaluated at subsequent time steps. The lamina permeability can be set to be constant, so as to agree with the Loos & Springer approach, or variable, so as to agree with the Dave et al. approach, as described in section 3.9.. The geometric arrangement of the fibers is defined in a fiber packing parameter. This is used to calculate the thermal conductivity of the lamina, as described in the following section. 3.4.2. Calculation of Lamina Properties The area of a lamina, A, is simply the product of the length and the width of the lamina, A = LW (3-22) The total mass of the lamina, m, is the sum of its resin collection and fiber masses. m = mr + mf (3-23) The total volume of the lamina, v, is the sum of its resin collection and fiber volumes. v = vr + vf (3-24) The thickness is calculated assuming it is uniform over the whole area of the lamina. The lamina thickness, %, is the ratio of the volume to the area of the lamina. A = — (3-25) A The resin mass fraction, Mr, is the ratio of the resin collection mass to the total mass of the lamina. Mr=-f- (3-26) m 30 The fiber mass fraction, Mf, is the ratio of the fiber mass to the total mass of the lamina. Since a lamina only contains resin and fiber, the resin volume fraction is calculated from the resin mass fraction. Mf-\-Mr (3-27) The specific heat of the lamina, Cp, is calculated using the rule of mixtures (a mass weighted average) of the resin and fiber specific heats, based on mass. Cp = MrCPr + MfCPf (3-28) The heat accumulation rate coefficient of the lamina, HARC, is the sum of the heat generation rates for its resin collection and fiber. HARC = HARCr + HARCf (3-29) Loos & Springer [1] calculated the temperature distribution in the laminate via a one-dimensional heat transfer model in the z-direction (perpendicular to the plane of the laminate). The temperature in any horizontal plane of the laminate (xy-plane) is assumed to be constant. The justification for this assumption is that in practice, the length L and width W of a typical laminate are much larger than its thickness X,, (i.e. L, W » A,,). Since the heat transfer model only considers heat flow in the z-direction, the only thermal conductivity value required is for the z-direction, k22 2. The thermal conductivity of a lamina is a function of several factors: the ratio of resin to fiber (the fiber volume fraction, Vf, is a convenient value), the thermal conductivity of the resin, kr, and the fiber, kf , and the spatial arrangement of the fiber in the resin, Fp. i.e., k22=f(vf,kf,kr,FP) (3-30) Because of the transverse isotropy of a lamina, the 33 direction is equivalent to the 22 direction. 31 There are a number of different fiber packing options (Figure 3-1) and hence options for calculating the thermal conductivity of a lamina. These are tabulated in Table 3-5. Appendix C provides details of the derivation of the different equations, some of which are based on reference [44], and some which are our extensions to that work. Table 3-5: Equations Used to Calculate the Thermal Conductivity Values. Fiber Shape and Arrangement in the Lamina Appendix C Equation parallel C-3 series C-5 square fibers in a square packing array C-11 circular fibers in a square packing array C-19 circular fibers in a hexagonal packing array C-27 3.5. Laminate Input Parameters The input properties for a laminate are the same as for the laminae with the addition of one term that specifies the number of laminae in the laminate, n. Note that a laminate can be constructed of either identical or unique laminae. The details of how to specify each option are explained in detail in Appendix A, the User's Guide for LamCure. 3.6. Autoclave Input Parameters There are six input parameters for the autoclave. Five of these are boundary conditions for the heat and resin flow models and can vary arbitrarily over time. The final input parameter is the time step which is specified in arbitrary increments of time. The input parameters are listed in Table 3-6. 32 Table 3-6: Autoclave Input Parameters. Input Parameter laminate/bleeder interface temperature as a function of time toolplate temperature as a function of time bleeder temperature as a function of time external pressure as a function of time vacuum pressure as a function of time time step as a function of time Symbol TlF=TIF(t) iTp—iTp{t) Tb=Tb(t) Pe*=Pe*tt) Pvac=PVac(t) At = f(t) Units K K K Pa Pa s 3.7. integration of Heat Transfer and Resin Flow Models The simulation code consists of four modules: the heat transfer model and a sequential compaction resin flow model, both based on the work of Loos & Springer [1], an isothermal heat transfer model and a squeezed sponge resin flow model based on the work of Dave et al. [3,4]. Each of the models operates on the laminate/bleeder system within the autoclave. Since each sub-model is self contained, combining the heat transfer and resin flow models is quite straight forward. The particular combination of heat transfer and resin flow models to be used are specified as input parameters. The simulation's flowsheet is shown in Figure 3-2. There are four possible combinations of heat transfer and resin flow models. Depending on the resin flow model specified, the simulation either jumps directly into the heat transfer model or goes through an initialization step. Initialization of the laminate and the bleeder are required if the sequential compaction resin flow model is used in conjunction with the continuous compaction assumption. The necessity of initialization and meaning of the compaction model will be explain in section 3.8.2. which deals with the sequential compaction resin flow model. Next, the temperature distribution of the laminate is either calculated by the heat transfer model or an isothermal temperature distribution is imposed on the laminate where each lamina's temperature is set equal to the toolplate temperature. Once the temperature distribution has been 33 determined, there are two operations common to each heat transfer model that must be performed. First, the bleeder temperature is updated to the temperature at t + At specified in the bleeder temperature input parameter. Second, the degree of cure of the resin for each resin mass in the laminate and the bleeder must be incremented. The means of doing this is identical for both laminae and bleeder. The degrees of cure for the laminae's resin collections are updated by enumerating over each resin mass in their collections and calculating their degrees of cure by multiplying their current cure rate, -—, which depends on their previous degree of cure and at current temperature, by the current time step, At, and adding this value to their current degree of cure, a. Thus °'""= a'"+t*;, Ar (3-31) The resin is then redistributed in the laminate according to the selected resin flow model. Next the specified output values are written to either file(s) or DDE link(s)3. Finally, the time is updated by the current time step and the sequence of events is repeated until the stop time, tstop, specified by the user, is reached or exceeded. 3.8. Heat Transfer and Sequential Compaction Resin Flow Models The goals of the integrated model [1] relevant to this thesis are the following: 1. the temperature inside the laminate as a function of position and time, T = T{z,t) 2. the resin pressure inside the laminate as a function of position and time, pr = pr(z,t) 3. the degree of cure of the resin, a, in the laminate as a function of position and time, a =a(z,t) 3 Dynamic Data Exchange (DDE) Links allow separate programs to communicate in real time in the MS-Windows environment. 34 4. the resin viscosity, \l, as a function of position and time, fj. = fJ.(z,t) 5. the number of compacted lamina, nc, as a function of time 6. the thickness, X,, and mass, ml, of the laminate as a function of time Two of their models are implemented here: the thermochemical model and the sequential compaction resin flow model. The thermochemical model predicts the temperature distribution in the laminate as a function of position and time. Calculation of the distribution is based on conservation of heat in the laminate with an additional term to include the heat generated by the curing of the resin. They assumed that the only mode of heat transfer was by conduction. Thus d(pc„T)^d(kdry dt dz\ dz j + pHGR (3-32) where p, CP, k and HGR are the density, specific heat, thermal conductivity and heat generation rate (of the resin) at a given point. Further, these quantities are functions of the temperature, resin content and degree of cure at a given point. The calculations for these values have been given in previous sections. This heat transfer model has been implemented in our code using the finite difference method. The node network is shown schematically in Figure 3-3. The boundary nodes are located at the upper and lower surfaces of the laminate. The upper node is at the interface between the laminate and the bleeder and is referred to as the laminate/bleeder interface node. The lower node at the interface between the laminate and the toolplate is referred to as the toolplate node. All other nodes are at the midpoint of each lamina's thickness. Hence, a laminate of n laminae has n + 2 nodes. 35 3.8.1. Heat Transfer Model 3.8.1.1. Initial and Boundary Conditions for the Heat Transfer Model To develop a solution for the heat balance equation, equation (3-32), two initial conditions and two boundary conditions must be specified. The temperature at the top and bottom of the laminate are the thermal boundary conditions once their values are known as a function of time. If the sequential compaction resin flow model is used or if information on the state of the bleeder for the squeezed sponge resin flow model is required, the initial temperature of the bleeder and its temperature as a function of time are required as input parameters. The initial conditions for the laminate are the temperature and the degree of cure as a function of position. Hence for the laminate; initial conditions: T = Tinil(z)\ (3-33) > 0<z<L @ r = 0 «=a,«,(z)J boundary conditions: T = TIF{t) @ z = 0 ] ^ n (3-34) for f > 0 T = Tjj.it) @ z = L\ And, for the bleeder; (3-35) initial condition: Tb=Tt boundary condition: Tb=Tb{t) @ for t = 0 t>0 (3-36) 36 3.8.1.2. Nodal Equations In the derivation of the following equations the positive direction for heat flow is in the positive z-direction (i.e. from the toolplate to the laminate/bleeder interface). The fully implicit formulation of the nodal equation was used to ensure stability for arbitrarily large time steps. The heat balance results in a central difference equation, but, since the spacing between the nodes is not constant over time, the final equation looks somewhat different than a standard central difference formulation. To make the following equations more readable, the symbol k is used to denote the thermal conductivity of the lamina in the z -direction rather than k22. 3.8.1.2.1. Zeroth Node. Adjacent to the Laminate/Bleeder Interface The geometry of this node is shown in Figure 3-4. The heat balance for the 0'A node is; , . , N (nnt+At T-.r+Af\ f-rt+At rrt+Al\ 2 j r*o+M ° " (A0+A,) ( rpt+M rpt+At\ a -kA ° IF > ~2~ Qgen = I -^" j HRomr0 = HGR0 (rr-n) uAj?r (rr-u) 0 P° At At Qin-<lout+ 4\en = Qacc 0"37) (fpt+At _ rpl + At \ (rpl + At rj-,t+At \ {k0+kM ° }-2k0A{° ,F >+HGR0 / L 0 + Aj A j = HARC0 ( rpt + At r r ' \ ±o £o V Af J (3-38) 37 rearranging and collecting like terms, 2k0A HARC0 , {ko+kjA 4> Ar +- X0+ Xx j rnl+At , 10 + / I Q + AJ rpt+At HARC±T, +?MLT;+«+HGRO At ' 0 (3-39) 3.8.1.2.2. Interior Nodes The geometry of the nodes are shown in Figure 3-5. The heat balance for the i'h node is; A,+ A ;+i A.^+A,. (VAW) ^=1^}/^="^ <7occ = W . - C , (nnt+&t TH'^ Ar = //A/?C. (7;.'+A'-7;.') Ar (7J. '+ A , -T; . ' ) = /Mi?C. Ar (3-40) rearranging and collecting like terms, 38 rpt+At _ (k^+kM {ki+kijA , iM/?C,. v A M + A , - ^ , + >^,+i +- Ar <pt+At . (HARCirrt (3-41) t Ar ;.' + //G/?(] 3.8.1.2.3. Last Node. Adjacent to the Toolplate The geometry of the node adjacent the toolplate is shown in Figure 3-6. The heat balance for the n'h - 1 node is; (rpt+At _ rpt+At \ f rpt+At _ rpt+At \ h -IT A P . ""' - ? / ' 4- TP "_1 y™ - ^n- i 7 1 % — ZK-n-\n "n-\ K-r q°ut~{ 2 r(K-&K-A~{n~i -2] i (rpt+At _ft \ (rpt+At rpt \ s~< \Ln-\ 1n-\) „ . n / , \1n-l * n-\} Qacc = mH_yC. = HARCn_x (rpt+At _f'+At\ (rp t+At _ rp t+ At \ (rpt+At rp t+At \ 2k._AKTP . " "" ; - ( * - i + 0 * - n_1 " ""2 ' v n - l ' ^ /M/?C, A B _!+ A„_2 (rpt+At _qrt \ n-\ At -HGR. n-\ (3-42) rearranging and collecting like terms, 39 A (*-l+*-2M ^„_1+An_2 J f rpt+At _ 1n-l (k-x-K-i)* 2kn_,A t HARCn. v kn_-Xn_z -+ X-l +- Ar n-1 (3-43) 3.8.1.3. Method of Solution Figure 3-7 is the flowsheet for the heat transfer model. Calculation of the coefficients of the temperature matrix from the nodal finite difference equations results in a tridiagonal matrix. The matrix is used to calculate the new temperatures for the laminae via the Thomas algorithm. Methods for implementation of this algorithm are available in many books on numerical methods [45,46]. In the simulation code there is an option to inform the user if the temperature matrix is not diagonally dominant. This option is included because the diagonal dominance of a tridiagonal matrix is a property of a matrix that ensures the Thomas algorithm will succeed in finding a solution. If the matrix is not diagonally dominant this is an indication that something physically meaningless is occurring, i.e., heat is spontaneously flowing from cold regions to warm regions. 3.8.2. Sequential Compaction Resin Flow Model 3.8.2.1. Additional Input Parameters There are three additional input parameters required to completely specify a lamina for the sequential compaction resin flow model: the compacted resin mass fraction, Mr, the parallel flow coefficient, P, and the permeability in the z -direction, SP . The symbol and units for each parameters are given in Table 3-7. 40 Table 3-7: Additional Lamina Input Parameters for Sequential Compaction Flow Mode Input Parameter compacted resin mass fraction parallel flow coefficient permeability Symbol M-. P SP Units none none m2 The compacted resin mass fraction, Mr, is the final value for the resin mass fraction of the lamina. Once this value is reached the lamina is fully compacted and no further resin flow can occur. Loos & Springer [1] characterize the resin flow in the x -direction by an analogy to channel flow [47] using a fitted parallel flow coefficient, P. They also assume that the lamina permeability in the z -direction is constant and independent of the fiber volume fraction. 3.8.2.2. Calculation of Additional Lamina and Laminate Properties This section provides some additional properties required by the sequential compaction resin flow model. The compacted mass of a lamina, mc, is the mass of the lamina assuming there is no excess resin in the lamina. It is calculated as follows; m m. / 1-Af (3-44) The compacted resin volume of a lamina, vr, is calculated from its compacted mass as follows; mc — mf v. = -(3-45) The compacted resin volume fraction of a lamina, Vr, is calculated from its compacted resin volume as follows; ' \ +Vf (3-46) 41 The compacted fiber volume fraction of a lamina, Vf , is defined in terms of its compacted resin volume fraction as follows; Vf=l-Vr (3-47) Je c Since the mass of the fiber in a lamina is constant for all time, the compacted volume of a lamina, vc, can be calculated as follows; ve=^- (3-48) The excess resin volume of a lamina, vr , is calculated by taking the difference between its current resin volume and its compacted resin volume. The calculation is as follows; vr-vr (3-49) txctss The initial resin volume fraction for each lamina, V. , is used internally by the simulation code in some intermediate calculations. Its value is set at t = 0 before any resin flow has occurred and is calculated from the fiber volume fraction as V,M=l-Vf (3-50) The number of compacted lamina in the laminate, nc, is calculated as the number of laminae that satisfy M-Mr <1(T10 (3-51) A tolerance is required to eliminate the effects of round off errors. Its magnitude was determined through trial and error. 3.8.2.3. Perpendicular Resin Flow: Sequential Compaction Flow Model Once the temperature distribution has been calculated the next step is to move resin through the laminate. There are three possible directions for resin flow. Flow can occur through the 42 thickness of the laminate, mz, or parallel to the fiber direction in the plane of the composite, mx, see Figure 3-8. Flow in the y -direction, rhy, is assumed to be negligible because, if the length and width of the laminate are large compared to its thickness, then most of the resin flow will be in the z -direction. The problem is simplified by assuming that flow in one direction is independent of the other. Obviously this is not a particularly good assumption since flow in one direction depletes the resin supply available for flow in the other direction. The resin that flows out of the laminate in the z -direction flows into the bleeder and has a velocity, $out, that can be calculated from D'Arcy's law; *-~f# (3-52) where Sp is the permeability of the zone through which the resin, of viscosity \ir, is flowing. At this point some assumptions about a flow mechanism are required. Loos & Springer drew on earlier work [25] where rafts of thin plastic rods and porous plates were immersed in oil with a viscosity similar to that of epoxy resin during curing to simulate laminae in a composite. A schematic of this apparatus is shown in Figure 3-9. In this experiment a perforated steel plate was used to simulate the bleeder and a weight placed on top of it to simulate the applied pressure in an autoclave. Upon placing the weight on the steel plate, it was observed that the liquid between the raft of thin plastic rods or the porous plates was squeezed out sequentially. The behavior of the rafts of thin rods and the porous plates was the same. The liquid between the perforated steel plate and the raft of rods adjacent to it flowed through the steel plate until the raft contacted the steel plate. The liquid between the first and second raft then flowed through the first raft and the steel plate until all the liquid separating the two rafts had flowed out and the second raft contacted the first. The compacted layers then consisted of two rafts of rods. The liquid separating the compacted layers and the third raft started to flow through the compacted 43 layers and the steel plate. This process of the compacted layers moving towards the uncompacted layers continued until all the liquid separating the rafts of rods had flowed into the steel plate. Thus, as the process continued, the liquid in the lower layers flowed through an ever increasing thickness of compacted layers and the liquid saturated portion of the steel plate. Figure 3-10 is an illustration of this process. A second observation was the pressure measured in the liquid at the bottom of the container. This pressure was always constant and equal to the applied pressure. It was therefore suggested that the pressure drop, and hence resin flow, only occurs across those layers that have compacted and the pressure in the uncompacted layers is constant, since no flow occurs, and equal to the applied pressure. By analogy it was postulated that the same flow mechanism operates within a composite laminate for resin flow perpendicular to the plane of the laminate, the z-direction. The sequence of events for a laminate is as follows: 1. The resin starts to flow out of the lamina directly adjacent to the laminate/bleeder interface and into the bleeder. 2. The resin flows out of this lamina until it reaches some pre-determined final value for resin content at which point the second lamina starts to compact with its resin flowing into the first lamina and displacing the first lamina's resin, which in turn flows into the bleeder. 3. The process is repeated for each lamina in sequence until all the laminae are compacted. Further, the pressure gradient exists only across those laminae which are compacted and the resin saturated portion of the bleeder. Assume for the moment that the pressure in the resin at 44 the interface between the laminate and the bleeder, pIF, is known. Using D'Arcy's law the velocities of the resin out of the laminate, # , and into the bleeder, •&., can be written; ,« Sp(Pez,-PiF) Sp(Pe*,-PiF) , , „ . voul = 1 = — (3-53) f v i . I n,dz a _ ^b\P/F Pvac) _ ^b\PlF Pvac) ,~ CAS lb The integral of viscosity, VI, in the denominator of each expression is required since the viscosity can be a function of position in the laminate and the bleeder; the calculation of these integrals is given later in this section. Since mass is conserved the velocity of the resin flowing out of the laminate is equal to the velocity of the resin flowing into the bleeder. Hence the pressure in the resin at the interface between the laminate and the bleeder, plF, can be calculated as follows; SP(Pext ~PIF) _ Sb(PlF ~ Pvac) „ „ . VI, VIb Solving for the interface pressure; = P„+<PP„ w h e r e WA (3_56) l + <p v VIbSP Using this expression for the interface pressure the volume flow rate out of the laminate, Qoul, can be calculated by combining it with equation (3-53) and multiplying by the area of the laminate/bleeder interface, A, thus n - A A - ASP^Pext~PlF) ,~ „ s Qout ~ &vout — (3-57) 45 At this point the model for resin flow in the z -direction is essentially complete. The only values that remain to be defined are the viscosity integral of the bleeder, VIb, and of the laminate, VI,, in equations (3-56) & (3-57) respectively. Calculation of the viscosity integral for the bleeder was defined in equation (3-54) as; K VIb = ]iibdz (3-58) 0 where hb is the height of the resin saturated portion of the bleeder. The calculation of this value must take into account the different viscosities of the resin masses in the bleeder since they may have different degrees of cure due to different thermal histories. Numerically VIb can be calculated as the sum of the product of each resin mass's viscosity, fir, and its thickness, hT, for all resin masses in the bleeder, i.e., W> = 2 M S Whe1"e hrj=-T- <3"59) 7=1 £bAb The expressions developed for the viscosity integral of the laminate, VI,, depend on how one assumes that the individual laminae compact. In the original work on resin flow [1,25] a lamina is modelled as a layer that consists of two sub-layers. The first sub-layer contains all the fiber and has a resin content equal to the compacted resin mass fraction, Mr. The second sub-layer contains only resin and has a volume equal to the lamina's excess resin volume, vr . This is txctss shown schematically in Figure 3-11. The thickness of the first sub-layer is constant and equal to the lamina's final compacted thickness, Xc. The compacted thickness of the lamina can be calculated as K=^ (3-60) A 46 The thickness of the sub-layer containing only resin is the channel thickness, Xchan, and depends on the degree of compaction. The thickness of the channel is the difference between the lamina's thickness and its compacted thickness, i.e., K^-^c (3-61) In references [1,25] it is assumed that the number of lamina through which the resin must flow through increases incrementally, i.e., the number of compacted laminae increases by integer values. Initially, at t = 0, there are no compacted laminae in a laminate. However, since the lamina adjacent to the laminate/bleeder interface has its compacted sub-layer adjacent to the bleeder, the resin must flow through the compacted thickness before it can flow into the bleeder. Compaction starts when the excess resin in this lamina begins flowing into the bleeder. As resin flows from the lamina, the channel thickness Xchan decreases, eventually becoming zero, at which point the compacted thickness Xc equals the lamina thickness X, and the lamina is fully compacted. There is now one compacted lamina and resin starts to flow out of the second lamina. Figure 3-12 shows the first and second laminae at the moment when the first lamina is fully compacted. For the resin in the second lamina, the distance through which it must flow is the sum of the compacted thicknesses of the first and second laminae. Therefore, as the compaction of the laminate progresses, the compacted thickness of the laminate increases in a step-wise manner. The viscosity integral of the laminate, VI,, was defined in equation (3-53) as; h Vl^j^dz (3-62) 0 It can be calculated by summing the viscosity integral of each lamina, VI, in the laminate's compacted zone, i.e., W ; = f V / , (3-63) i=0 47 Since the compacted thickness, Xc, of a lamina does not depend on its degree of compaction the expression for the compacted thickness of a fully compacted lamina and a compacting lamina is the same, equation (3-60). For a lamina, the thickness of each resin mass, Xp is calculated assuming the volume of fiber in each resin mass is proportional to the resin volume of the lamina. Numerically, a lamina's integral of viscosity, VI, is calculated by summing the product of each resin mass's viscosity and its thickness. For a fully compacted lamina, the viscosity integral is calculated as (v. \ VI = ^ HjXj where X- - vvw v / + v o (3-64) For the compacting lamina only those resin masses in the compacted zone contribute to the viscosity integral. If there are k resin masses in the compacted zone, then the k +1 resin mass is only partially inside the compacted zone. Therefore the viscosity integral becomes i=k ( j=k \ A k+l (3-65) As an alternative to this incremental compaction assumption, a continuous compaction assumption was implemented. The concept of sequential compaction is extended to each lamina. Initially, before compaction starts, the compacted thickness Xc is zero. The fiber is distributed uniformly throughout the lamina. In the compacted sub-layer contains fiber at the compacted fiber mass fraction Mr and the uncompacted sub-layer contains fiber at the initial resin mass fraction Mr . Therefore the compacted thickness Xc can be calculated as follows; X=X V -V (3-66) 48 The compacted thickness of a compacting lamina varies continuously between its initial thickness and its final compacted thickness. The expressions for the lamina's viscosity integral are identical to the incremental lamina compaction model, equations (3-63) through (3-65). The viscosity integral of laminate V7, is calculated by summing the viscosity integral of the laminae in the compacted zone. Thus, if nc laminae have compacted, the compacting lamina is nc +1. Therefore the viscosity integral of the laminate is calculated; V/,= 5 > ; (3-67) <=0 Figure 3-13 is a plot of the compacted thickness of the laminate, A,, as a function of time for both options. Note that the continuous compaction assumption, after a short rapid initial increase from zero, is an average of the incremental compaction assumption which was used in the original work [1]. The overall effect of the using the continuous compaction model over the incremental model is to smooth, over time, any properties that use the compacted thickness Xc in their calculation. One important point about the sequential compaction flow model is that it explicitly calculates the resin flow out of the laminate. The consequence of this is that the calculated values for volumetric flow rate, Qout, can be unrealistically large for large time steps. This is especially true for the continuous lamina compaction model. Initially, the thickness of the compacted sub-layer in the lamina adjacent to the laminate/bleeder interface is zero and, since no resin flow has occurred, the height of the resin in the bleeder is also zero. Inspection of equation (3-53) reveals that in this case the viscosity integral for the laminate, VIn is zero which leads to an infinite velocity. Obviously this is not acceptable. Therefore, in the simulation code, the laminate/bleeder system is initialized by moving 1% of the excess resin in the 0lh lamina (adjacent to the laminateNbleeder interface) into the bleeder. This allows equation (3-53) to 49 return a reasonable value. However, it should be noted that using very small time steps during the initial compaction of the first lamina will ensure reasonable velocity values. A significant difference between the two compaction assumptions is the effect on the parallel resin flow model. This is explained in the next section, 3.8.2.4.. It is of considerable interest to derive an expression for the resin pressure as a function of position. This will allow a direct comparison of predicted pressure profiles between the sequential compaction and the squeezed sponge resin flow models. In the sequential compaction flow model, an overall flow rate of resin out of the laminate is calculated. Since there is no increase or decrease in the volume of resin contained within a lamina once it has fully compacted, the resin velocity through each lamina must be the same. The derivation of the resin pressure equation is similar to that for the interface pressure. Let p, be the resin pressure at the lower interface of the ith lamina (the interface between the i"1 and ith +1 laminae) at position z' from the laminate/bleeder interface within the compacted thickness of the laminate A, . Since mass is conserved, the velocity of the resin into and out of the ith lamina's lower interface must be equal. The velocity of resin out of the /"* interface can be calculated as 0 L = i . ( f c i Z P r ) = $ ! _ ( , ) (3.68) while the velocity of resin into the i'h interface is v, __SL{pext-pLi)= SL ^ = ^K J^ ^ = -^-(pe,-pJ (3-69) Let the integrals of viscosity in the above equations be represented by VIv=\lnudz and VIL=j'nLdz (3-70) 50 Setting the velocities equal to each other S, HPLi-P,F) = T^(PeXt-PLi) VI, ^U VIL [PU ~P,F)= •f-TTf-iPe* ~PL, ) fc±^ where $m^ (3.71) 1 + 0 S„ VI, U rAL Where pIF, is defined in equation (3-56). The assumptions used in the above derivation are that the permeability of the upper portion of the compacted zone, Sv, is equal to the permeability of the 0'h lamina and the permeability of the lower portion of the compacted zone, SL, is equal to the permeability of the i'h lamina. 3.8.2.4. Parallel Resin Flow: Sequential Compaction Model Loos & Springer [1] only considered resin flow in the x -direction (parallel to the fiber direction). They support this assumption by stating that in practical situations the resin flow perpendicular to the fibers is small due to the resistance of the individual fibers in this direction and restraints (dams) that are placed around the edges of the laminate. The first assumption is reasonable in light of Gutowski's work [35] where he measured the permeability of graphite fiber parallel to the fiber direction to be approximately 100 times greater than the permeability perpendicular to the fiber direction. Placing dams around the laminate physically prevents flow thereby increasing the resistance to flow in the y -direction. It is assumed that parallel flow can be characterized by viscous flow between parallel plates separated by a distance, Xchan, the channel thickness, see Figure 3-14. Resin properties within the channel are constant and a linear pressure gradient exists over the length of the channel. The average velocity of a liquid between parallel plates due to pressure gradient Ap over a length, x, is given by White [47] as 51 0 = AML, 1 48jc^r (3-72) Since the lamina is quasi-static, the pressure gradient can be estimated by a force balance on the lamina. Figure 3-15 is a schematic of the lamina model and shows the pressure distribution over the length of the channel. The pressure at the edges of the laminate is the pressure which surrounds it and is equal to the vacuum pressure, pvac. The maximum pressure, pmax, can be obtained by equating the force applied to the lamina to the force within the channel, i.e., 2[P-2P™^W = {pext -pjLW (3-73) Substituting for the pressure difference between pmax at the center of the midpoint of the channel length L from equation (3-73) and the pressure at the edge of the laminate pvac, into equation (3-72), the average resin velocity becomes o \rext t*vac /chart zo ^r \^ #_ = (3-74) \2lirL The expression given in reference [1] for the average resin velocity has an error in its derivation4. Their expression is a factor of 2 too small. The following expression is the corrected equation. or\pext— pvac)Achan vx= (3-75) 4 In reference [1], substitution of their equation (21) into their equation (19) produces their equation (22) which requires a 2 in the numerator on the right hand side of their equation (22). 52 where P is the parallel flow coefficient. The value of P is constant and determined experimentally. Comparison of equations (3-74) & (3-75) reveals that in reference [1] the factor of 96 was collected into the constant P . In an attempt to get agreement between values of P for the results in reference [1] and the simulation results, the same expression for average resin velocity, equation (3-75), is used in the simulation code. The total volume of resin flowing out of the lamina in the x -direction is the sum of the resin volume flowing out of both edges of the lamina. Hence, the volumetric flow rate of resin out of the lamina is 4=2t?Aa„W (3-76) Recall that for a fully compacted lamina the channel thickness, Xchan, is zero. Therefore, the volumetric flow rate is also zero; only uncompacted laminae contribute to resin flow in the x-direction. The volume of resin that flows out of the lamina is the product of the volumetric flow rate, Qx, and the time step, At. Recall that the resin in any lamina consists of a collection of p resin masses. When resin flows out of the lamina in the x -direction the volume of resin removed from each resin mass is in proportion to its share of the total resin volume of the lamina. Let vx be the volume of resin that flows out of the lamina in a given time step. The volume that flows out of each resin mass, vx , is calculated as v,, =-P!L-^ (3-77) 7=1 Consider the continuous lamina compaction case. The lamina geometry is shown in Figure 3-16. The same assumptions as for the incremental lamina compaction case are used with the exception that the resin in the channel contains some of the fiber in the lamina. Further, the 53 presence of fiber in the channel has no effect on the average resin velocity. The presence of fiber in the channel reduces the area through which resin can flow and is taken into account by adding an initial resin volume fraction term, Vr , to equation (3-76). i.e., ^ = lO^WV^ (3-78) The lamina compaction assumption has an effect on the volumetric flow rate, Qx. The value of Qx is lower for the incremental assumption than for the continuous assumption because the channel thickness Xchan is smaller for this case. Hence to get the same volume of resin flow as the continuous assumption, the values of the parallel flow coefficient P will be smaller for the continuous model than for the incremental model of Loos & Springer [1]. 3.8.2.5. Method of Solution The flowsheet for the sequential compaction resin flow model is shown in Figure 3-17. If parallel flow is specified then resin is removed from those laminae which are not fully compacted. If perpendicular flow is specified, the resin flows out of the laminate and into the bleeder. Note that for perpendicular flow, resin is only permitted to flow in the positive z -direction. Resin cannot be reabsorbed into the laminate from the bleeder. 3.9. Squeezed Sponge Resin Flow Model The squeezed sponge resin flow model is based on consolidation theory from soil mechanics [33]. i.e., - M ^ = % (3-79) where mv is the volume change coefficient and will be defined below, section 3.9.1.. Dave et al. [3,4] and Gutowski [2] consider a laminate as a liquid (resin) saturated porous bed (the fiber network of the laminate) where the fiber network is capable of supporting load. In 54 both the sequential compaction and the squeezed sponge resin flow models it is assumed that the resin in the laminate is an incompressible liquid and, since the system is consolidating very slowly, it is quasi-static. Using these assumptions an expression for the fiber bed pressure, pf, can be derived based on a force balance perpendicular to the plane of the composite. The external pressure, pext, applied to the laminate/bleeder system is opposed by the hydraulic pressure of the resin, pr, and the fiber bed pressure (the springiness of the fiber bed), pf. i.e. Pe« = Pr+Pf (2-7) •*• Pf = Pexl-Pr (3-80) Further, the fiber bed pressure of a lamina is related to its degree of consolidation, i.e. the less resin in the lamina the greater the compression of the fiber bed. Using the results of Gutowski for fiber bed pressure as a function of resin volume fraction [2], Dave et al. were able to relate the void ratio of the lamina, e, to its fiber bed pressure, pf. i.e., e = f(pf) (3-81) This relationship is an additional input parameter required by the squeezed sponge resin flow model. The details of how to specify this schedule are explained in Appendix A, the User's Guide for LamCure. Currently, the simulation is restricted to the same e = f{pf) relationship for all the laminae. The effect of this restriction is to force the initial resin mass fraction for each lamina to the same value. 3.9.1. Lamina Properties for the Squeezed Sponge Resin Flow Model The void ratio, e, is the ratio of the lamina's resin volume, vr, to its fiber volume, vf. The value of e is calculated from the lamina's fiber bed pressure, pf, defined in the input parameters. e = ^ - = f(pf) (3-82) 55 The compressibility coefficient, av, is the rate of change of the void ratio with respect to the fiber bed pressure. de a » = - - r - (3-83) dpf The volume change coefficient, mv, is the ratio of the axial (normal) strain to the axial (normal) stress for confined vertical compression with immovable vertical sides. Note that einit is the initial void ratio of the lamina, i.e., at pf = 0. m v = - ^ - (3-84) The compaction coefficient, CC, is defined to make the expressions for the nodal equations more readable. The compaction coefficient is calculated as; CC = ^ -^- (3-85) far 3.9.2. Specific Permeability The squeezed sponge resin flow model, Dave et al. [3,4] use an expression for permeability, SP„, that is dependent on the porosity of the lamina. Their work is based on the earlier work of Carman and Kozeny [39] with the values of the Kozeny constants taken from Williams et al. [48] and Sullivan and Hertel [49]. An interesting point in this approach is that the expressions for both the parallel and perpendicular permeability are the same except for the value of the Kozeny constant. The general expression is SP =^- (3-86) 56 where rH is the hydraulic radius of the lamina, defined as the cross-sectional area of the fiber, normal to the direction of flow, divided by the perimeter wetted by the resin. Thus Df€ DfVr rH = — 1 — = - ^ (3-87) " 4 ( l - e ) 4v, The term Ku is the Kozeny constant. The value for K22, perpendicular to the fiber direction, is 6 and for Kn, parallel to the fiber direction, is 0.3. Although the simulation does not currently use Ku it has been included for potential future expansion. With the current simulation code, it is possible to use either variable or constant permeability values for both the sequential compaction and the squeezed sponge resin flow models. 3.9.3. Resin Velocity for the Squeezed Sponge Resin Flow Model The velocity of resin flowing through the Ith lamina, t?22., can be calculated from D'Arcy's law as follows; /x,. dz ~\ where pressure gradient, —*-, can be calculated from the resin pressure at the lower interface of oz the i'h lamina, pLi, the resin pressure at the lower interface of the lamina above the ith lamina, pL , and the thickness of the /'* lamina, Xt, as follows; % = ^ f ^ L (3-89) oz A,. Hence the velocity of the resin flowing through the /"" lamina can be expressed as follows; 57 For the lamina adjacent to the laminate/bleeder interface, i = 0, the vacuum pressure pvac is used in place of p^ in equation (3-90) 3.9.4. Initial & Boundary Conditions for the Squeezed Sponge Resin Flow Model Since the sequential compaction resin flow model only calculates the resin pressure at the interfaces between adjacent laminae, p^, it is convenient to place the nodes of the finite difference network of the squeezed sponge resin flow model at the interfaces between adjacent laminae and at the center of the laminae. The physical properties associated with the nodes at the interfaces between the lamina are taken as an average of the two adjacent laminae. The initial and boundary conditions of the laminate for the squeezed sponge resin flow model [3,4] are; initial condition; (3-91) p = pinil(z) 0<z<L @ f = 0 boundary conditions; P = Pmc(t) @z = 0 | U 0 @z = L az (3-92) V for ?>0 v ' The boundary conditions are incorporated into the node network by placing nodes at the upper and lower surfaces of the laminate. Therefore a laminate of n laminae has 2n nodes. The node network is superimposed on the laminate geometry and shown in Figure 3-18. 3.9.5. Nodal Equations In the derivation of the following equations the positive direction for resin flow is in the positive z -direction (i.e., from the toolplate to the laminate/bleeder interface). A fully implicit formulation of the nodal equations was used to ensure stability for arbitrarily large time steps. 58 To make the following equations more readable, the symbol Sp is used to denote the permeability of the lamina in the z -direction rather than SP . 3.9.5.1. Laminate/Bleeder Interface Nodes The geometry of the 0'* node is shown in Figure 3-19(a). The mass balance for the 0"1 node is; _ Pr0ASP0 VA'+ ' Po+ ') _ ~ A r r ( ,+*t ,+M \ = 3 = 2ACC0[pl -p0 ) 2 ™ou, n AC ( n'+&l — n'+A'\ PrA^KP* Pa, )=2ACc0(Pr-Pr) Po ±1 2 K , (P'O+M-PO) mac. = — pr Amv acc 2 ° ° At K-^cu,=<cc (3-93) 2CC0{px -pQ )-2CC0[p0 -ptF j = (3-94) 2 Af rearranging and collecting like terms; f Kpr™^\^ , „ ^ _,+*_ (Kpr™*0 _, ^ 4CCQ + V ° 2At j Pr+2CCoPr=- ^ ° Po+2CC0p';P*! \ (3-95) The geometry of the V node is also shown in Figure 3-19(b). The mass balance for the l" node is; 59 m. Pr,ASP, \Pz ' -P[+ ') _ 0 . r r I <+A, ,+A, \ = — J = 2ACCl[p2 -px ) \k m„ Pr0ASPo \Pl ~P0 I 2 =2Acc0(Pr-pr) ™acc = A V A, Y *s + ™>,_ Y A* + P* YA'+A'-A' ^ Ar 2cc1(Pr" -Pr)-2cc0 (pr-p'r) Uo+ i^Xp^ + PnX^+mJ 16 Ar (3-96) rearranging and collecting like terms; 2cco Pr - 2CC0+2CC,+ I^ AT r (X0+X,)(pr + pr){m„ +m„ ) +2 CCX pT = - — i ^ - 2 — F r ' n v° ^ n,' 1 2 16Af l (3-97) 3.9.5.2. Interior Nodes The geometry of the interior node closest to the laminate/bleeder interface is shown in Figure 3-20(a). The mass balance for the 2i'h node is; 60 m.. m„ PrtASp KPli+l Pli ) _ n . n r I r+Ar „I+Af \ = 2 = 2ACC0{p2M - p 2 i ) 2 _ Pr,ASPs \Pzi '-Pltl) _ 0 A r r ( „<+Ar «+Ai \ - ; 5 - z / i L ( - o V / 7 2 < ~Pii-\} 2 macc--PrAmVi ^ 2 CC, \PIM - Pli ) - 2 CCi \Pn ~ Pu-i ) = ~ rearranging and collecting like terms; At (3-98) / 2CClP£Z- 4CC.+ h^\T+1cciPr=—^T7-p2i 2At (3-99) The geometry of the interior node closest to the toolplate is shown in Figure 3-20(b). The mass balance for the 2i'h +1 node is; PruA^Pu, \P2i Pti-\ ) _ ^ A n r , ( t+Lt ,+&,) ^ - Z / i L L / + , \P2i+2 — Pzi+l ) Hi +1 "l'+l PrASp{Pli+1 P2i ) / t+Al ,+A,\ meiu = —*— a = 2/4 CCL {p2M - p2i ) H-i 2 macc = A - / V**. Y"*, +m„ YP/i+ P*. Y/&?-*L-+.^ A? 61 2 CCM [P2M - P2M ) - 2 CCi \PlM ~ PT') __ ( V kM)(p, + p , , )(mVi + mVM) f p™ , ^ 16 Ar rearranging and collecting like terms; 2CC,pT- 2CCi+2CCM + (V^.>I)(P, + P ^ , ) K + ^ ) ' 1+Al 16Ar fti+i +2CC. + iP 2 . + 2 =-(A,+ Ai+1)( p + p )(mV| + mVw) , 16Ar Pa +1 (3-100) (3-101) 3.9.5.3. Toolplate Nodes The first of the toolplate nodes is essentially the same as the first of the interior nodes. The geometry of the first toolplate node is shown in Figure 3-21(a). The mass balance for the 2n'h - 1 node is; m. m„ _ Pr„.,ASP^ \P2n Pln-\) _ ^ A ^ ( t+At ,+&t\ ~ 2 _ Z / 1 L 'L ' n - lVP2/ 1 ~Pln-\) A / i - l Pn-n 4C I n'+A ' — n'+Al) X 2 A CCn-l \P2n-l ~ P2n-2 ) Pn-l " w - 1 X macc = ~JL± Pr Amv ace ~ • rn-\ "n-1 (PZ-I-PL-I) At 2 CCn_j (^2„ - p2 ) - 2 CCn_x \pln_, - p2n_2) = -i—^ 2 Ar rearranging and collecting like terms; 2CCn_Xn-2~ 4CCn_1 + Xn-\Pr™. 2At P2Z+2ccn_lP'2: t+At _ K-iP^*"^ 2At P2n-(3-102) (3-103) 62 The geometry of the node directly adjacent to the toolplate is shown in Figure 3-21(b). The mass balance for this node (the 2n'h node) requires use of the boundary condition at the toolplate; ^ = 0 for ?>0 (3-104) dz Therefore, from D'Arcy's law, the mass flow rate into this node, m,„, is always zero. The mass balance for the this node is; ra;„ = 0 'in — 5 ~ Z / 1 (-(-n-l KPln ~ Pln-\ I V-n-X ±>=L 2 macc = A^mv pr ^ ^ 4 v->yr-\ At J -1CCn-APln -P2n-l) = 777 \Pln ~ Pin) (3-105) 4Af rearranging and collecting like terms; 2CC + -iPr-m^ \p'2T=- n~lP;:y PT O-106) 4Ar J 4At 3.9.6. Method of Solution The flowsheet of the resin flow model is shown in Figure 3-22. The check for parallel flow (resin flow in the x -direction) is not currently used but included for potential future expansion. If parallel flow is inadvertently specified, an error message alerts the user that the option is not available. If perpendicular flow is specified, then resin is permitted to flow out of the laminate and into the bleeder; i.e. in the positive z -direction only. The calculation of the coefficients for the pressure matrix (using the nodal finite difference equations derived above) results in a 63 tridiagonal matrix, as did the heat transfer model. Checking the diagonal dominance of the pressure matrix and finding its solution is identical to the heat transfer model. If the matrix is successfully solved the pressures at the midpoint and the lower interface of each lamina is updated to the new pressures. The resin content of the i'h lamina is adjusted according to its mid-point resin pressure, pr. The fiber bed pressure in a lamina is calculated from a force balance for each lamina. Since consolidation of the laminate is a relatively slow process, the system is assumed to be quasi-static. Therefore the external pressure, pexl, applied to the laminate must be in equilibrium with the resin pressure, pr, and fiber bed pressure, pf, of each lamina, i.e., Pf=P~-Pn (3-107) Therefore, once pf, is determined, the void ratio as a function of fiber bed pressure relationship e = f(pf) is used to calculated the new volume of resin, v'r, in the /'* lamina, i.e., v' e = f(p) = ^ L => V'=vfe (3-108) vu The excess resin volume, v. , for the i'h lamina is calculated as the difference between the excess lamina's resin volume vr and the new resin volume v'r from equation (3-108); v =vr-ev, (3-109) txcessj i Ji The next step is to specify how the excess volume of each lamina flows through the laminate and eventually into the bleeder. As in the case of the sequential compaction resin flow model for perpendicular flow, the resin is moved from lamina to lamina either as laminar (no mixing) or perfectly mixed flow. The details of each mixing model were given earlier in section 3.3.3.. The sequence for moving resin through the laminate in the z -direction in a given time step is as follows: 64 1. The excess resin vr of the n' - 1 lamina, adjacent to the toolplate, is removed 'excess^ from this lamina and added to the n'h - 2 lamina. 2. The excess resin of the n'h — 2 lamina, which now contains its own excess resin vr and the excess resin of the n'h — 1 lamina vr is removed and added to the nth - 3 lamina. 3. This process of removing the excess resin volume from a lower lamina and adding it to the lamina directly above and adjacent to it, is repeated until all the excess resin has been moved into the 0"* lamina, adjacent to the laminate/bleeder interface. 4. The last step is to remove the excess resin of the 0rA lamina, which now contains all the excess resin in the laminate, and add it to the resin in the bleeder. The final step is to update the resin pressures in the laminate. This is necessary even if no resin flow occurs since the external pressure varies arbitrarily over time. Any increase or decrease in the external pressure, Ap, is reflected in each lamina as an increase or decrease of Ap in its resin pressure at the end of that time step. The updated resin pressure, p'*M, is calculated using the external pressure at the next time step, p'^'. PT^P'APT-P'J (3-no) where p'r is the intermediate resin pressure calculated from the conditions at time t. The effect of a varying vacuum pressure is taken into account in the mass balance equations for the lamina/bleeder interface where the vacuum pressure at the next time step, p'v^', is used. 3.10. Verification of Heat Transfer and Resin Flow Models In this section the implementation of the heat transfer model and both resin flow models is verified. The heat transfer model is compared to an analytic expression for temperature as a function of position and time. The sequential compaction resin flow model is compared to the 65 original work. The squeezed sponge resin flow model is compared to an analytic solution for pressure as a function of position and time as well as the original work. Tables 3-8 to 3-13 list the input parameters for the simulation runs. Some of the autoclave and resin input parameters differ depending which model is being considered and these are noted where appropriate. Table 3-8 : Fiber Input Parameters. Input Parameter density specific heat thermal conductivity parallel to the fiber direction thermal conductivity perpendicular to the fiber direction fiber diameter Symbol Pf C'< h, J22 Df Value 1799.0 0.721 26.0e-3 26.0e-3 8.4e-6 Units kg/m3 kJ/(kgK) kW/(m K) k W/(m K) m Table 3-9: Resin Input Parameters. Input Parameter density specific heat thermal conductivity heat of reaction cure rate relationship viscosity relationship Symbol Pr cPf K HR %=KaJ) fi = f(a,T) Value 1220.0 1.26 1.67e-4 474.0 ref[5] ref [5] Units kg/m3 kJ/(kgK) kW/(m K) kW/(m K) 1/s Pas Table 3-10: Bleeder Cloth Input Parameters. Input Parameter length width permeability porosity Symbol L W sb £b Value 0.3 0.3 5.6e-ll 0.57 Units m m m2 none 66 Table 3-11: Common Lamina Input Parameters. Input Parameter length width initial total mass of the prepreg ply initial resin mass fraction permeability fiber packing initial degree of cure initial temperature Symbol L W minU K.. s FP <*init T.. mil Value 0.3 0.3 0.0235 0.42 5.8e-16 circular 0 300.0 Units m m kg none m2 none none K Table 3-12: Additional Lamina Input Parameters for the Sequential Compaction Model. Input Parameter compacted resin mass fraction parallel flow coefficient Symbol Mrc P Value 0.2474 0.01 Units none none Table 3-13: Laminate and Autoclave Input Input Parameter number of laminae toolplate temperature as a function of time laminate/bleeder interface temperature as a function of time bleeder temperature as a function of time external pressure as a function of time vacuum pressure as a function of time Parameters. Symbol n i TP=i Tp(t) TIF=TlF(t) Tb=Tb(t) Pm=Pjf) PvGC=Pvac(f) Value 17 400 400 400 700 000 100 000 Units none K K K Pa Pa 3.10.1. Heat Transfer Model The heat transfer model is checked by comparing the results of the current simulation to the analytic solution for the temperature at different points through the thickness of the laminate at different times. The expression for the analytic solution is T(z,r) = 7 B C - X — e x p ' k22q7i \S pCPX t u sin ' qui (3-111) where TBC = TTP = T!F 67 Since the analytic solution is only correct for the case of constant coefficients, resin flow was disabled in both the x and z -directions. The initial temperature of the laminate was 300 K. The toolplate and laminate/bleeder interface temperatures were set to 400 K, independent of time. The initial degree of cure was set to 1.0 to eliminate heat generation by the resin during cure. The solution was generated using the first 301 terms of the series which was enough to reduce its oscillations near the boundary conditions to a sufficiently small amplitude. Figure 3-23 shows the results from the simulation run and equation (3-111) for the first 10 seconds for three values of At. As expected the agreement between the analytic and numerical solutions improves as the time step decreases. From the figure we can see that for a At value of 0.125 seconds there is excellent agreement between the analytic expression and the model. The experimental data for the temperature distributions in laminates in the work of Loos and Springer [1] was for laminates at low heating rates which results in near isothermal conditions in the laminate. Since good data where sufficient detail is given to simulate the experiment is not available, no comparison with the heat transfer model in the current simulation can be made. However, since the model agrees so well with the analytic solution, it should predict correctly the temperature distribution within a laminate. 3.10.2. Sequential Compaction Resin Flow Model Verification of this resin flow model is done by comparing the predictions of the current simulation to those of the original work by Loos and Springer [1]. In this section some modifications in the input parameters are required for checking the current simulation against the original work. The densities of the fiber and resin are 1790 and 1260 kg/m3, respectively. The initial mass of a lamina was 22.6 gm. The vacuum pressure is constant at 16.7 kPa for the case of flow in the z -direction only and 101.1 kPa for the case of flow in the x and z -directions. 68 Flow in the z -direction for the current simulation was compared to the simulation results in the original work5 using their value of 5.8(10"16) m2 for the permeability of the prepreg perpendicular to the fiber direction. Figures 3-24, 3-25 and 3-26 compare to the number of compacted laminae in 16, 32 and 64 ply laminates as a function of time for different applied pressures, respectively. Figure 3-24 shows that agreement between the current work and the original work is very good for the 586 kPa and 345 kPa results. The shape of the 172 kPa case is similar in shape to the current simulation and neither simulation predicts full compaction. However, the current case starts to diverge from the original work at 34 minutes and only 12 laminae fully compact compared to the original work which predicts 15 laminae fully compact. The reason for this discrepancy may be due to the difference in the size of the time steps. By inspecting the jumps in the original work for the 172 kPa case one can see that the time step is about 5 minutes whereas the time step for the current simulation is only 1 minute. Figure 3-25 shows the results for a 32 ply laminate. Excellent agreement is shown for an applied pressure of 586 kPa. At the higher pressures the current simulation is somewhat faster than the original work but the shape of each curve is essentially the same for both simulations. Looking back to Figure 3-24 we see that at lower pressures the current simulation is somewhat slower. This suggests that the current simulation is more sensitive to the external applied pressure than the simulation in the original work. This trend is continued in Figure 3-26 which shows the results for a 64 ply laminate. In this figure the agreement between both simulations is very good with the current simulation compacting slightly faster than the original work. Their Figure 15 in reference [1]. 69 The sequential compaction model also permits flow in the *-direction. In the original work the analogy to channel flow was used where the flow coefficient, P, was fitted to experiments on thick panels. The mass loss of a laminate is defined in the original work as the percentage change in its mass compared to its initial mass, i.e., mass loss = \ mi • j \ 'mi y 100% (3-112) The results for the current simulation are compared to their simulation results and the original experiments in Figure 3-27 for a 64 ply laminate6. Loos and Springer [1] reported a value of 170 for P in the original work. However, to get the agreement shown in Figure 3-27 it was necessary to use a value of 0.01 for P in the current simulation. Recall that it was shown earlier in section 3.8.2.4. that the derivation for the parallel flow equation, equation (3-75), was incorrectly stated in reference [1] as a factor of 2 too small. Also, it was shown the expression for the average resin velocity in the x -direction in the original work had collected a factor of 96 from the denominator into P, which is in the numerator. Hence if both the factor of 2 and 96 are incorrectly stated in reference [1], then the potential error in their work is a P value 192 times too large. The true value of P would be 0.89 instead of 170. However, the discrepancy between this value and the actual value of 0.01 cannot be explained. Figures 3-28 and 3-29 show the mass loss as a function of time for the 32 and 16 ply laminates, respectively. In both cases the agreement with the original simulation results is excellent. For the 16 ply case, Figure 3-29, the agreement between the experimental value and the current simulation for parallel flow is better than for the original work. The data and simulation results for the original work are from Figure 10 in reference [1]. 70 Based on the agreement between the current simulation and the original work, both the perpendicular resin flow model and parallel resin flow model (with the newly fitted value for the parallel flow coefficient) are taken to have been correctly encoded. 3.10.3. Squeezed Sponge Resin Flow Model The squeezed sponge resin flow model in the current simulation only allows resin flow in the z -direction. The pressure as a function of time and position through the thickness of the laminate is compared to the analytic solution. Using the boundary condition that the resin pressure gradient at the toolplate is always equal to zero, the expression for the analytic solution to this case is ^ - / ' v J v H r ' f(2q-l)7tz) ( {2q-\)2K2T^ pr(z,t) = 2 , -cos K ~{ 2q-\ I 2%, exp J V (3-113) tSP where Tv = —r Since the analytic solution is only correct for the case of constant coefficients, resin flow was disabled. The initial resin pressure in the laminate was 700 kPa. The solution was generated using the first 151 terms of the series which was enough to reduce its oscillations near the boundary conditions to a sufficiently small amplitude. For the rest of this section some modifications in the input parameters are required for checking the current simulation against the original work. The initial mass of a lamina was 25.2 gm for 15.2 cm square plies. The external and vacuum pressures are constant at 689.4 kPa and 101.1 kPa, respectively. Figure 3-30 shows the results from the current simulation and equation (3-113) for the first 1000 seconds for three values of At. The waviness of the analytic solution near the laminate/bleeder interface at t = 0 is due to the fact that a finite number of terms were used to approximate the 71 analytic solution. At 1000 seconds the simulation results for all time steps decreases to very close to the vacuum pressure, 100 kPa. The agreement between the simulation results and the analytic solution improves as the time steps decrease from 25 seconds to 1 second, as expected. From the figure we can see that a At value of 1 second gives excellent agreement. Flow in the z -direction for the current simulation was compared to the simulation results in the original work7. They generated plots of resin pressure, resin mass loss, laminate thickness and laminate resin mass fraction as a function of time for a 256 ply laminate with an initial resin mass fraction of 0.38. This corresponds to an initial void ratio of 0.904. However, it is explicitly stated that the laminate fully compacts. Hence all of the external applied pressure must finally be borne by the elastic response of the fiber bed. Given this, the final values for the resin mass loss, laminate thickness and laminate resin mass fraction must be dictated by the relationship between the void ratio and the fiber bed pressure. In the original work, the resin mass loss, laminate thickness and laminate resin mass fraction all have final values that correspond to a final void ratio of 0.59. The relationship between the void ratio and the fiber bed pressure is given in reference [4] as e=-1.542(10"6)/?,+0.81 for 0<pr <68707.6 f f (3-114) e = -0.10727 ln(pf) +1.8987 for 68707.6 < pf < 1030605.0 which gives an initial void ratio of 0.81 and a final void ratio value of 0.46. Obviously equations (3-114) could not have produced the results in the original work8. Let us assume that the void ratio of the second of equations (3-114) is correct at a fiber bed pressure of 68707.6 Pa and the void ratio at full compaction is 0.59. Then a new set of equations, of the form Their Figures 4, 5 and 6 in reference [4]. Communication with the author of reference [4] confirms that equations (3-114) are wrong [50] 72 e = -Apf + B i \ (3-115) e = -Cln{pf) + D can be derived. The coefficients for first of equations (3-115) can be calculated by using the void ratio of 0.90378 at 0.0 Pa fiber bed pressure as one point and the void ratio of 0.704 at 68707.6 Pa fiber bed pressure as the other point. The coefficients for second of equations (3-115) can be calculated by using the void ratio of 0.704 at 68707.6 Pa fiber bed pressure as one point and the final void ratio of 0.59 at a fiber bed pressure of 689473.3 Pa as the other point. Substituting these values into equation (3-115) produces a replacement set of equations for equations (3-114). Thus e = -2.9038(10^)/?, +0.90378 for 0<p,<68707.6 (3-116) e = -2.91l(l0^)p f +0.904 for 68707.6< pf < 1030605.0 Using this set of equations Figures 3-31, 3-32, 3-33 and 3-34 were generated for the resin pressure, resin mass loss, laminate thickness and laminate resin mass fraction as functions of time, respectively. resin mass loss = f \ m, —m, •hit _' m, 100% (3-117) In each case the current simulation produces results that are very close to those of the original work. This suggests that equations (3-116) are very close to the actual equations used in the original work for the squeezed sponge model. Further, since the agreement is good, we can conclude that the current simulation correctly reproduces the squeezed sponge resin flow model. 73 3.11. Summary In this chapter one heat transfer model and two resin flow models for predicting the transient and final state of flat composite panels were described. A heat transfer simulation based on the work of Loos and Springer was implemented such that it can work with either resin flow model. This model gave good agreement with the analytic solution. A sequential compaction resin flow simulation based on the work of Loos and Springer was implemented and its results compared to the original work with good agreement. The prepreg permeability value for the current simulation was identical to the value used in the original work. However, the value for the parallel flow coefficient was a factor of 17000 lower than in the original work for good agreement. A factor of 192 can be accounted for by errors in the original work, but the remaining factor is unexplained. A squeezed sponge resin flow simulation based on the work of Dave et al. and Gutowski et al. was implemented and its results compared to an analytic solution giving good agreement. This model was then compared to the results in the original work. Modification of the input values was required due to errors in their input parameters. After the required corrections, the current simulation produced results that agreed very well with the original work. 74 3.12. Figures (a) (b) (c) • • • • • • • • • • • • • • • I I • • • • • i p • • • • i I I I I resin • • fiber I y * (d) (e) Figure 3-1: The unit cubes for the fiber and resin geometries used for calculation of lamina thermal conductivity, k22. (a) parallel heat flow (b) series heat flow (c) square fibers in a square packing array (d) circular fibers in a square packing array (e) circular fibers in a hexagonal packing array. 75 no move 1% of excess resin mass from 0m lamina into bleeder set all laminae temperatures to the toolplate temperature temperature^>calculate mode l /— £_ heat transfer model set bleeder temperature to new temperature update degree of cure for all resin masses sponge squeezed sponge resin flow model sequential L sequential compaction resin flow model update time write output to file(s) and/or DDE link(s) Figure 3-2: Simulation flowsheet showing the integration of the heat transfer and resin flow models. 76 •i-1 'i+1 • n-2 'n-1 TP [aluminum toolplatel «-n-T X - * -Figure 3-3: The node network for the heat transfer model with the laminate/bleeder interface, TlF(t), and toolplate, TTP(t), boundary conditions. *„J_ U) \ 2 flu •r * , x-«-Figure 3-4: Geometry of the 0"1 node for the heat transfer model. 77 2 2 * r 4'' 4, 1+1 Figure 3-5: Geometry of the interior nodes for the heat transfer model. 2 2 T 1J ij \n-2 TTM^TP Figure 3-6: Geometry of the n'h - 1 node for the heat transfer model. calculate coefficients for the temperature matrix check diagonal \ y e s ^dominance. no confirm that matrix is diagonally dominant solve temperature matrix set lamina temperatures to the calculated values Figure 3-7: Heat transfer model flowsheet. z Figure 3-8: Possible direction for resin flow in a unidirectional lamina. - weight roller bearings , perforated steel plate i ^ « mm » m y-M<<//m?-wmmm/mmm0;mmMM |jqujd ^ i - - ^ . thin rods WZ^ or porous plates -viscous ' load sensor Figure 3-9: Schematic of the apparatus used to study compaction in reference [25]. pressure pressure ( f ( f f ( f pressure mini Figure 3-10: Compaction sequence during resin flow for the first three layers. 79 mz M.= M. Mr= 1 z A Figure 3-11: Geometry of a lamina for the sequential compaction resin flow model. M = M. M.= M, Mr=1 Figure 3-12: Geometry of the first and second laminae at the moment when the first lamina is fully compacted. 80 2.00E-03 1.80E-03 £ 1.60E-03 en 8 1.40E-03 | 1.20E-03 •a g 1.00E-03 a £ 8.00E-04 o u 2 6.00E-04 c I 4.00E-04 JO 2.00E-04 0.00E+00 '_ ': \ \ '• j ^ -^7X _ ^ V \ -L_,rf P ^ ^ r..,. incremental continuous i i i i 10 15 time (s) 20 25 Figure 3-13: The effect of the incremental and continuous lamina compaction assumptions on the compacted thickness of the laminate, Xl , over time. f M,= M,c M r = 1 m» Figure 3-14: The lamina geometry and fiber distribution for the incremental lamina compaction model. Note there are no fibers in the resin channel. 81 Figure 3-15: The applied pressures on a lamina and the pressure distribution in the resin channel for the incremental lamina compaction assumption. m.+ a Mr= Mfc Mr= M,„, x - * -Figure 3-16: The lamina geometry and fiber distribution for the continuous lamina compaction assumption. Note the presence of fiber in the resin channel. 82 move resin out of laminate by parallel flow mechanism move resin through laminate by perpendicular flow mechanism add resin collection flowing out of laminate (by perpendicular flow) into bleeder Figure 3-17: Flowsheet of the integration of perpendicular and parallel resin flow for the sequential compaction resin flow model. Figure 3-18: The node network for the squeezed sponge resin flow model. Nodes are placed at the interfaces between laminae and at the mid-point of the laminae. 83 ft/«--r ,1 m„\ * (a) m m I-2 (b) Figure 3-19: Geometry of the nodes adjacent to the laminate/bleeder interface for the squeezed sponge resin flow model. 12i-1 ma [2i mk 2i+1 (a) * 2i k2i+1 2i+2 (b) Aei Figure 3-20: Geometry of the interior nodes for the squeezed sponge resin flow model. 12/7-2 mL 'k2n-1 2n L i 2 2 2 k ft-t 2n-1 2n /77i,=0 (a) (b) Figure 3-21: Geometry of the nodes adjacent to the toolplate for the squeezed sponge resin flow model. parallelXves alert user that parallel flow for the squeezed sponge resin flow model is not available calculate the coefficients for the pressure matrix 'checib diagonal \yes sdominance^ noj solve the pressure matrix set lamina pressures to the calculated values move resin through laminate and into the bleeder via perpendicular flow JL increment resin pressure (accounts for changes in the boundary pressures) T i confirm that matrix is diagonally dominant Figure 3-22: The flowsheet for the squeezed sponge resin flow model. 85 400 280 0.0005 0.001 0.0015 0.002 position through thickness (m) 0.0025 0.003 Figure 3-23: Comparison of the temperature profiles through the thickness of the laminate at different times for the analytic solution and three values of the time step for the current simulation. 10 20 30 40 time (min) 50 60 70 80 Figure 3-24: Comparison of the number of compacted laminae for the current simulation and the original work on the sequential compaction model for a 16 ply laminate for perpendicular flow. 35 30 25 j» 15 10 • • : ; ; A_—,—,—_ (ST^T^ * " Af. i... 1379 kPa ^ J/jfs II I * li' /if * /» /*/ f V - ^ . V S . "l034kPa Y .• 1237 kPa /rr-^ 1 • • i — _ 586 kPa cuitenl sin - - - a- - - simulation-,. ulatton -refill • • * • • i • , •J i i -10 20 30 40 time (min) 50 60 70 80 Figure 3-25: Comparison of the number of compacted laminae for the current simulation and the original work on the sequential compaction model for a 32 ply laminate for perpendicular flow. 70 60 50 2 40 30 20 10 '• ' • D--• : • §~L~ «"-<J <"* - current simulation " simulation-ref(l) iti^^SS / <£^r<^-/ / / .--a ^/ ,c y' ^ _ ^ 7 ^ D • ' ,n'"' °~" i • • ' 3447 kPa I .VOkP' i Q586 kPa i — i 1 1 1— -J 1 1 10 20 30 40 time (min) 50 60 70 80 Figure 3-26: Comparison of the number of compacted laminae for the current simulation and the original work on the sequential compaction model for a 64 ply laminate for perpendicular flow. 87 Figure 3-27: Comparison of the mass loss of a 64 ply laminate as a function of time for the current simulation and the simulation and experiments of the original work. Figure 3-28: Comparison of the mass loss of a 32 ply laminate as a function of time for the current simulation and the simulation and experiments of the original work. 88 0 poooooo i 0 £ 10 15 20 25 30 35 40 45 50 55 60 time (min) 65 Figure 3-29: Comparison of the mass loss of a 16 ply laminate as a function of time for the current simulation and the simulation and experiments of the original work. 700000 T—•=> 600000 •* 500000 3 8 400000 Q. C £ 300000 200000 100000 : 0s •' '/ .' V • ' ' / : if • V : v ' ' 1 ' <7 • / f J 1 t y .it • "TVer faqe 5s ,x* r s 0 ^"J^ r **" "J**"*" ' 0 **** ' ^ ^ ^ ; ^^rrsn: 1000 s ••^ ; !J - i - - — - —' analytic d t = 1 s dt = 5s dt = 25 s toolDJate — 0.0005 0.001 0.0015 0.002 postion through thickness (m) 0.0025 0.003 Figure 3-30: Comparison of the squeezed sponge resin flow model with the analytical solution for different At values. 800000 700000 600000 £ 500000 « in a °- 400000 c *55 300000 200000 100000 '• \ \ 1 \ ' 1 1 1 ' V • simuloliot)—it? current Simula (4) t ion 50 100 150 time (min) 200 250 Figure 3-31: Comparison of the resin pressure at the toolplate as a function of time for the current simulation and the results from reference [4]. oo n -// V h /i li \i ji li (i i ji li li ji Si /' /' * ^ * i i i ' , - Simula lion~ref( current simulat 4) o n . . , 50 100 150 time (min) 200 250 Figure 3-32: Comparison of the resin mass loss as a function of time for the current simulation and the results from reference [4]. Note that the waviness in the simulation results is due to errors in digitizing the original data. 0.39 0.37 c 0.35 o 2 0.33 w vt to E 0.31 c 'in o> " 0.29 0.27 0.25 simulatbn-ref(4) current simulation 50 100 150 time (min) 200 250 Figure 3-33: Comparison of the laminate resin mass fraction as a function of time for the current simulation and the results from reference [4]. Note that the waviness in the simulation results is due to errors in digitizing the original data. 50 100 150 time (min) 200 250 Figure 3-34: Comparison of the laminate thickness as a function of time for the current simulation and the results from reference [4]. Note that the waviness in the simulation results is due to errors in digitizing the original data. 91 Chapter 4: Reconciliation of the Sequential Compaction and the Squeezed Sponge Resin Flow Models As discussed in Chapter 2, the sequential compaction model was developed using an apparently valid set of model experiments as its basis, whilst the squeezed sponge model was developed using different but also apparently valid experiments. The frameworks within which these models have been developed have never been compared, and, no attempt has yet been made to compare their predictions and reconcile them. This is the goal of this chapter. 4.1. Pressure Distribution in the Sequential Compaction Flow Model For the sequential compaction flow model, the pressure in the uncompacted portion of the laminate is equal to the external pressure, pext, and the pressure at the free surface of the resin in the bleeder is equal to the vacuum pressure, pvac. In Chapter 3 the resin pressure at the i'h lamina's lower interface is given as a function of position in the compacted zone by equation (3-71). Generalizing this equation to any height z in the compacted zone (0 < z' < X,), one has 92 />,(*')= ,,_<!>PeXI+PlF 1+0 U A SL VIU where <p — —- —-5i/ W L (4-1) To simplify a comparison of the two cases the following assumptions are made; 1. Constant resin viscosity. 2. Constant external pressure, pexl, and vacuum pressure, pvac. 3. Identical physical properties and laminae dimensions. 4. The resin pressure at the laminate/bleeder interface, pIF, can be approximated by the vacuum pressure, pvac. The last assumption is reasonable since the permeability of the bleeder cloth has been reported [1] to be approximately 105 m2 higher than the prepreg material; thus the resistance to resin flow in the bleeder is negligible. Using these assumptions equation (4-1) can be re-written as Pr (Z') = (Pe» ~ Pvac ) T ~ + P.ac (4"2) Klr Thus the resin pressure within the compacted zone is a linear function of position within the zone. A schematic of this resin pressure distribution is shown in Figure 4-1. 93 4.2. Force Balance in the Compacted Zone A key concept for the squeezed sponge flow model is that the fiber network in each lamina can support load. Thus for equilibrium Peil=Pr+Pf (2-7) where pf is the fiber bed pressure. A stated assumption of the sequential compaction flow model is that the fibers do not carry load. This is generally taken to mean that all the pressure applied to the laminate is borne by the resin. However, consider a force balance on the upper portion of the laminate/bleeder system in Figure 4-1 with a cut through the compacted zone as shown in Figure 4-2. Assume, as is the case for the low flow rates observed, that the system is quasi-static. Even though in the original work the laminae fiber beds are explicitly assumed not to interact, it can be seen from Figure 4-2 that the sequential compaction model implicitly defines a fiber bed pressure that must satisfy equilibrium in the form of equation (2-7). This results in a fiber bed pressure profile as shown in Figure 4-3. Once we define a fiber bed pressure profile for the sequential compaction model, it should be possible to define a relationship between the void ratio and the fiber bed pressure. This relationship can then be used in the squeezed sponge flow model to see if it will mimic the sequential compaction flow model. 4.3. Derivation of Void Ratio Vs Fiber Bed Pressure Relationship The goal is to derive an expression for the void ratio as a function of fiber bed pressure for the fiber bed pressure distribution of Figure 4-3. The details of the squeezed sponge flow model were given in Chapter 3. Briefly, the algorithm is as follows: First, the resin pressure distribution within the laminate is calculated. Second, the fiber bed pressure in each lamina is calculated using equation (2-7). Third, the resin volume of 94 each lamina is calculated from the relationship describing the void ratio as a function of fiber bed pressure. Finally, the excess resin in each lamina is moved through the laminate and into the bleeder. The following derivation assumes continuous lamina compaction, as opposed to the incremental compaction model, as described in Chapter 3. The compacted thickness of a lamina is expressed as (V.-V. A x=x r Tfa V -V (3-66) Figure 4-4 is a schematic of the resin pressure distribution over the compacted zone of the laminate. The compacted thickness of the laminate, X, , is the sum of the thickness of the compacted laminae, X} , plus the thickness of the compacted portion of the compacting lamina, X. . Thus c n c +l X, = X,+X where X,=neXe (4-3) Recall that the porosity, £, of a lamina is identical to its resin volume fraction, Vr. Therefore equation (3-66) can be written as X=X\£ £'"'' \£c-£initj (4-4) Substituting the void ratio e for the porosity and writing the lamina thickness X in terms of its initial mass minil and initial resin mass fraction Mr the compacting lamina's compacted thickness X, becomes 95 K =A l + e — £• • F —£. where X — -pfA (l + e) (4-5) which simplifies to "c+1 pAec-einit) K = ((l + ««v)«-«,«,) (4-6) Assuming that the resin pressure at the center of the compacted thickness of the compacting lamina p*r is representative of the resin pressure pr of the whole lamina, the resin pressure at this point can be written, from equation (4-2), as P'r={Pex,-Pvac) K +pv, (4-7) Substituting equations (2-7), (4-3) and (4-6) into equation (4-7) and rearranging, the void ratio of a compacting lamina can be expressed as a function of its fiber bed pressure pf and the number of compacted laminae nc as 1 + ( \PrA Pexl Pvac nl V Pf \£init J J m. M-MJ (4-8) This equation, which is in the form e = f(pf), should force the squeezed sponge flow model to mimic the sequential compaction flow model. 96 Equations (4-7) and (4-8) were used to generate the void ratio as a function of fiber bed pressure for each lamina, as shown in Figure 4-5. The void ratio of each lamina decreases until it fully compacts, at which point the lamina's void ratio corresponds to its compacted resin mass fraction. By observation, the fiber bed pressure at which a lamina becomes fully compacted, p*f, depends on the number of compacted laminae, nc, and can be written as; P*f = (Pex<-Pmc) < 1 ^ 2k+i) (4-9) This equation is evaluated for an eight ply laminate in Table 4-1. As can be seen from this table and equation 4-8, the relationship between a lamina's void ratio and its fiber bed pressure depends on the lamina's position within the laminate for the sequential compaction model. On the other hand, the squeezed sponge model requires this relationship to be identical for all laminae, which is reasonable since they are the same material. Therefore, the relationship for the fifth lamina (i.e. nc=A) is used as the average for input into the squeezed sponge model. Table 4-1: Lamina Fiber Bed Pressure at Full Compaction for the Sequential Compaction Flow Model. Compacted Lamina, 0 1 2 3 4 5 6 7 Fiber Bed Pressure, pf (Pa), atM,. 300000 150000 100000 75000 60000 50000 42857 37500 91 4.4. Idealized Stress Vs Strain Behavior A more conventional, material science oriented view of the lamina behavior can be obtained by interpreting the fiber bed pressure as the applied stress and plotting this against the lamina's strain, which we define as the change in thickness divided by original thickness. The stress-strain curves derived from the sequential compaction flow model are shown in Figure 4-6. Initially the laminae behave as softening materials (their modulus decreases) up to some value of strain, at which point they become perfectly rigid and incompressible. As compaction proceeds, the softening becomes less pronounced and the lower laminae behave more linearly before becoming rigid. However, for a real material one would expect the stress-strain behavior of a lamina to be a property of the material system and independent of the lamina's position in the laminate. Therefore the underlying assumption for lamina compaction in the sequential compaction model is not very realistic. Figure 4-7 shows the relationship between the void ratio and the fiber bed pressure used by Dave et al. [4] in their implementation of the squeezed sponge model. This is a fitted curve to the experimental results from Gutowski's earlier work [2]. This fiber stress-strain relationship is initially linear but becomes a hardening material (increasing modulus) at higher stress. This is a material relationship and is therefore the same for all laminae. Superimposed on Figure 4-7 are the stress-strain curves for the second and fifth laminae from Figure 4-6. It is interesting that they bound the Dave/Gutowski curve. 4.5. Comparison of the Flow Models This section will compare the simulation results for the sequential compaction model (SCM) and the squeezed sponge model (SSM) using the relationship between the void ratio and the fiber stress defined in equation (4-8) (which we will refer to as the SSM-SC case). 98 To test the validity of equation (4-8), the simulation was run for both flow models for an 8 ply laminate. The input parameters for both simulation runs are identical except where noted and are given in Tables 4-2 to 4-7. Table 4-2: Fiber Input Parameters. Input Parameter density specific heat thermal conductivity parallel to the fiber direction thermal conductivity perpendicular to the fiber direction fiber diameter Symbol Pf cPf kf« kfn Df Value 1799.0 0.721 26.0e-3 26.0e-3 8.4e-6 Units kg/m3 kJ/(kgK) kW/(m K) kW/(m K) m Table 4-3: Resin Input Parameters. Input Parameter density specific heat thermal conductivity heat of reaction degree of cure cure rate relationship viscosity relationship Symbol Pr c?f K HR a ^=f(«J) /z = f(a,r) Value 1220.0 1.26 1.67e-4 474.0 0 0 1.0 Units kg/m3 kJ/(kgK) kW/(m K) kW/(m K) none 1/s Pas Table 4-4: Bleeder Cloth Input Parameters. Input Parameter length width permeability porosity Symbol L W sb £b Value 0.3 0.3 5.6e-ll 0.57 Units m m m2 none 99 Table 4-5: Common Lamina Input Parameters. Input Parameter length width initial total mass of the prepreg ply initial resin mass fraction permeability fiber packing initial degree of cure initial temperature Symbol L W Mini, K.. S FP Uinit T. mil Value 0.3 0.3 0.0470 0.42 4.83e-14 circular 0 300.0 Units m m kg none m2 none none K Table 4-6: Additional Lamina Input Parameters for Loos & Springer Model. Input Parameter compacted resin mass fraction parallel flow coefficient Symbol Mrc P Value 0.2474 false Units none none Table 4-7: Laminate and Autoclave Input Parameters. Input Parameter number of laminae toolplate temperature as a function of time laminate/bleeder interface temperature as a function of time bleeder temperature as a function of time external pressure as a function of time vacuum pressure as a function of time Symbol n iTp=iTp(t) TIF=TIF(t) Tb=Tbit) Pe»=P<Xtt) Pvac=PyJf) Value 8 300 300 300 700 000 100 000 Units none K K K Pa Pa The resin has a constant viscosity set to 1.0 Pa s, independent of temperature or degree of cure, to ensure that only the resin flow models are being compared. Before we can compare the model predictions, we need to look more carefully at the relationship between void ratio and fiber bed pressure. Figure 4-8 shows the resin pressure at the lower surface of each lamina as a function of time for both cases. The resin pressure at the lower surface of the laminae rather than the center point is used for convenience. The SSM-SC case behaves similarly to the SCM case with the exception of the final resin pressures. For the SCM case the pressure gradient acts across the compacted 100 zone of the laminate. Once full compaction has been reached the pressure gradient is assumed to still exist, resulting in constant resin pressures greater than the vacuum pressure. For the SSM-SC case the resin pressure remains close to the SCM case up to 26 seconds, at which point its falls rapidly to the vacuum pressure as the laminate becomes fully compacted. Closer examination of Figure 4-8 shows two discrete jumps in resin pressure for the two laminae adjacent to the laminate/bleeder interface, i=0, at approximately 1.5 and 3.5 seconds respectively. The source of the decreases can be seen in Figure 4-9 where the resin pressure at the center of each lamina is plotted as a function of time. Recall that the fiber bed pressure p*f at which compaction stops is 60,000 Pa (nc = 4 from Table 4-1). The fiber bed pressure at which the first lamina is fully compacted should be 300,000 Pa. However, since an average fiber bed pressure of 60,000 Pa is used for all laminae, the first lamina compacts when the resin pressure in the first ply decreases to 640,000 Pa, at approximately 0.3 seconds. This is clearly seen in Figure 4-10 where the thickness of the first lamina becomes constant at approximately 0.25 mm. In fact, the resin pressure decreases abruptly for every lamina when its resin pressure drops to below 640,000 Pa. Figure 4-10 shows that the lamina compaction is sequential for both cases. However, in the SSM-SC case, since we are using an approximation to the true lamina stress-strain curve for the SCM, the compaction is not perfectly sequential. Recall that the fiber bed pressure is only correct for the compaction of the fifth lamina. Thus one would expect the first four laminae to compact slightly faster than the SCM case, the fifth lamina to compact in the same amount of time and the last three laminae to compact slightly slower. Figure 4-10 shows that the first lamina does compact slightly faster for the SCM case. The time to full compaction is very close for the next four laminae with no real difference between the two cases. The next two laminae compact slightly faster for the SCM case than for the SSM-SC case, as expected. Surprisingly the last lamina compacts faster for the SSM-SC case. This is contrary to the expected outcome but not large enough to be significant. 101 4.6. Resin Velocities Also of interest is a comparision of the resin velocities during compaction. For the sequential compaction model, a single average resin velocity is calculated for the entire compacted zone. Thus the resin velocity is uniform in the compacted zone and zero in the uncompacted laminae. For the squeezed sponge model the resin velocity in each lamina is calculated based on the pressure gradient across the lamina. Hence the resin velocity is not uniform through the laminate. A plot of the resin velocities for the SCM and SSM-SC cases was generated and is shown in Figure 4-11. It is obvious from this figure that the velocity predictions are very similar for both cases. However there are some important differences between the cases. The resin velocity for the SCM case is initially very large due to the initially very small thickness of the compacted zone. As mentioned above, the pressure gradient exists only over the compacted zone, hence the calculated resin velocity is very large, 2.3(10"3) m/s. In contrast, the initial velocity for the SSM-SC case is much smaller, 8.3(10~5) m/s. This value is much smaller than for the SCM case because the pressure gradient acts across the entire thickness of the laminate. This is a significant difference between the flow models. The sequential compaction model assumes that the resin pressure in an uncompacted lamina is equal to the external pressure. The squeezed sponge model allows the whole laminate to react to its boundary conditions. The laminae compact according to the pressure distribution within the laminate. Thus, the sequential compaction model does not permit a lamina to compact until all the laminae above it have become fully compacted. The squeezed sponge model allows all lamina to compact simultaneously at the rate dictated by the pressure gradient that exist across each lamina. This can be seen in Figure 4-11 where the non-zero resin velocities for the fourth, fifth and sixth laminae at approximately 11 seconds indicate that resin is simultaneously flowing out more than one lamina. 102 Note that the resin velocities for the SSM-SC case increase sequentially. As resin flows out of the first lamina its velocity increases until 0.1 seconds and then starts to decrease. The resin in the second lamina, i = 1, starts to flow and its velocity increases until it matches the first. Since the resin in the second lamina must flow through the first lamina, the resin velocity in the second lamina approaches the first as it depletes its excess resin. This process is repeated until the last lamina depletes its excess resin when it becomes fully compacted. Although the overall envelope for the two cases is very similar after the initial 0.1 seconds, their final values are quite different. The sequential compaction model predicts that the final resin velocity is a non-zero value. Obviously this cannot be correct since resin flow must cease when the laminate becomes fully compacted. The non-zero resin velocity is a result of the assumption that the pressure gradient continues to act across the compacted zone even when the laminate has become fully compacted. The squeezed sponge model, on the other hand, predicts that the resin pressure throughout the laminate decreases to the vacuum pressure. Thus eventually there is no pressure gradient between adjacent laminae, and the resin velocities are zero, as one would expect. From these observation, and the knowledge that the resin velocities cannot be infinite at the start and must be zero for a fully compacted lamina, we can conclude that the squeezed sponge flow model gives a better description of the resin velocities for the entire compaction time than does the sequential compaction model, even for the case where in broad terms they are the same. 4.7. Summary The sequential compaction flow model was shown to implicitly define a fiber bed pressure within each lamina. For a constant viscosity laminate it was shown that the relationship between the void ratio and the fiber bed pressure is a function of the number of compacted laminae and the position within the laminate. Using this relationship it is possible to get the squeezed sponge model to behave, in broad terms, as the sequential compaction model. However, the squeezed 103 sponge flow model more accurately describes the resin velocities over the whole range of compaction time. 4.8. Figures bleeder cloth compacted laminae uncompacted laminae tool pi ate ^^^^^H Figure 4-1: Schematic of a compacting laminate/bleeder system and the corresponding resin pressure distribution. bleeder cloth compacted laminae z Mmm X -Pr+Pf Figure 4-2: Free-body diagram of the pressures acting on the laminate/bleeder system at a cut through the compacted zone. 104 compacting zone Figure 4-3: The assumed resin pressure profile for the sequential compaction model and the corresponding implied fiber bed pressure profile required to satisfy equilibrium. laminate/bleeder interface toolplate Figure 4-4: Schematic of the resin pressure profile within the laminate for a constant viscosity condition. 1.1 0.9 £ 0.8 § 0.7 0.6 0.5 0.4 WW nc=7 \ \ \ • • i i f\\ \ \ \ > • ' » ' \nc=1 ' ' ' ' 50000 100000 150000 200000 fiber pressure (Pa) 250000 300000 Figure 4-5: Void ratio, e, as a function of fiber bed pressure, pf, for an 8 ply laminate. 300000 -a 3 k. a k. © 5 innnon -i 0 -• • • nc=1 i i • i nc=7 1 1 L. I 0.00 0.05 0.10 0.15 strain 0.20 0.25 0.30 Figure 4-6: Fiber bed pressure, pf, as a function of lamina strain for an 8 ply laminate. 600000 500000 •£• 400000 a. 300000 ~ 200000 100000 : • • -• ' -* ' ' ~ 1 1 1 1 1 )av6/Gutowski / nc=1_ — — i — i — i — i — — i i i i / / nc-4 / 0.00 0.05 0.10 0.15 strain 0.20 0.25 0.30 Figure 4-7: Fiber bed pressure, pf, as a function of lamina strain for the relationship used by Dave et. al. [4]. Also shown are two cases of the curves derived from the sequential compaction model. 700000 Q. e £ 300000 200000 100000 10 15 time (s) 20 25 30 Figure 4-8: Resin pressure at the lower interface of each lamina in an 8 ply laminate for the SCM and SSM-SC cases. Note that the resin pressure for the last lamina is, by definition, equal to the external pressure for the sequential compaction model. 107 600000 -sure (Pa) i 8 I 8 —^.,,^i,— Q. C 200000 -100000 -l\ o \-^_^ , 1 • • , L— - .J 1 1 | i i i i 1 1 1 1 i . i i 7 1 " ' • 10 15 time (s) 20 25 30 Figure 4-9: Resin pressure at the center of each lamina, pr, in an 8 ply laminate for the SSM-SC case. 3.50E-04 3.40E-04 3.30E-04 .-. 3.20E-04 (0 en « c O 3.10E-04 3.00E-04 2.90E-04 •2 2.80E-04 2.70E-O4 2.60E-04 2.50E-04 10 15 time (s) I \ '"'\ " •I 1 T 1 i 1\ : V 1' | \% \ | 4 \ X V \ \ ' ? l o l I - • ' ' ' ' • I ' * * \ """*''» A * V v. V. V-4 \ ' 3 \*. \ \' \ \ \ * * * » * " * * « • » % A x , V 5 \ \ \ \ v> \ \ * * * « \ "S-N \ 6 V. Y\ \ \ \ \ \ * i * \ l \ \ \ \ \ \ \ \ »\ \> % i — ^ * " SCM SSM-SC a A i i 20 25 30 Figure 4-10: Lamina thickness, A, for each lamina in an 8 ply laminate for the SCM and the SSM-SC cases. 108 1.40E-04 -2 00E-05 -O.OOE+00 -: j, - 0 - \\ • N^ / : V : 2 / y 3 / ' *^""**"=li;*-^ ^ 4 , - ' " t i j jmi -« .« t i - - i - i - ' i . .5;-/;;.".','."^, ' - > : : : : ; : : SCM SSM-SC ...7... - \ , , 10 15 time (s) 20 25 30 Figure 4-11: Resin velocities for the SCM case and for each lamina of the SSM-SC case. The integers denote the lamina number. 109 Chapter 5: Parametric Study of Lamina Stress-Strain Behavior and Permeability In Chapter 4 the compressibility of the fiber bed was related to the more familiar stress-strain plot. This chapter examines the effect of different lamina stress-strain relationships on the compaction behavior of a laminate for the squeezed sponge resin flow model. The effect of permeability on the compaction behavior of this flow model is also investigated and some generalizations are discussed. 5.1. Fiber Bed Stress-Strain Relationships The lamina stress-strain curves compared in this section are shown in Figure 5-1. The input parameters for the simulation runs have already been given in Tables 4-2 to 4-7 of Chapter 4. The different stress-strain curves used in this study result in large differences in the total time required for complete compaction of the laminate. We define a laminate to be fully compacted when the resin pressure in the lamina adjacent to the toolplate is within 1.0% of the vacuum pressure and its thickness is within 0.2% of its compacted thickness. The time taken to do this is the compaction time and is denoted tcomp. To facilitate comparisons between the different time scales the results are plotted as a function of normalized time, defined as t normalized time = (5-1) 110 The comparison of stress-strain curves begins with the plastic/rigid case in the lower right hand corner of Figure 5-1 and then proceeds toward the rigid/plastic case in the upper left hand corner. The results for each case will be discussed by comparing the laminae resin pressures and thicknesses as functions of normalized time. The first case is one of the extremes of lamina stress-strain behavior; the plastic/rigid case. The results for this case are shown in Figures 5-2 and 5-3. The most obvious feature of Figure 5-2 is the step-wise decrease in resin pressures over time. The linear pressure distribution through the compacted zone of the laminate is seen in this figure as the even spacing between the pressure plateaus at a given normalized time. In Figure 5-3 one can see that the first lamina (adjacent to the laminate/bleeder interface, J = 0 ) , compacts immediately and that the laminae compact sequentially. Note that the step in a lamina's resin pressure occurs at the precise moment when it becomes fully compacted and then remains relatively constant until the next lamina compacts. Finally, the point at which the last lamina starts to compact, it's "dwell time", is about 80% of tcomp and the laminate is essentially fully compacted by 98% of tcomp. Recall, from Chapter 3, the stated assumptions for lamina compaction of the sequential compaction flow model. It explicitly assumed that a lamina fiber bed offers no resistance to compaction until it becomes rigid. In Chapter 4, this was shown to be not true, as in fact a lamina stress-strain curve is implicitly defined by equilibrium requirements. However, as can be seen here, the compaction of the laminate will be sequential in both cases. The stress-strain curve for the sequential compaction flow model is also shown in Figure 5-1. Not surprisingly, it is similar, but not identical, to the plastic/rigid case. The results for the hardening stress-strain curve are shown in Figures 5-4 and 5-5. The shapes of the resin pressure profiles are quite different from those for the plastic/rigid case. The resin pressures decrease smoothly with no pressure plateaus. The spacing between the resin pressure profiles at a given normalized time is no longer uniform. In Figure 5-5 one can see that 111 compaction is no longer sequential. In fact the last lamina starts to compact before the first lamina has fully compacted. The first lamina does not immediately compact but its compaction is still very fast. At 40% of tcomp each lamina has some degree of compaction but none has fully compacted. The dwell time for the last lamina is about 20% of tcomp compared to 80% for the plastic/rigid case. The shape of the laminae thickness profiles over time is also different. The laminate is essentially compacted at about 80% of tcomp compared to the previous case where compaction was achieved in about 98% of tcomp. The results for the linear stress-strain curve are shown in Figures 5-6 and 5-7. Figure 5-6 shows the resin pressures decreasing smoothly and more rapidly as a fraction of the total time to compaction than in the hardening case. Figure 5-7 shows that the compaction of the laminae is becoming more uniform as the thickness of all laminae are decreasing concurrently instead of sequentially. The dwell time before the onset of compaction in the last lamina is much shorter for this case, about 3% of tcomp, than for the two previous cases. The results for the softening stress-strain curve are shown in Figures 5-8 and 5-9. The resin pressure decreases very rapidly with the pressure in the last lamina decreasing to 400,000 Pa in less than 4% of tcomp. Note that even though the resin pressure drop is initially very fast, decreasing to less than 110,000 Pa in 50% of tcomp, it still takes the remaining 50% of tcomp before the laminate is fully compacted. As in the linear case the laminae thicknesses decrease more uniformly, with virtually no dwell time before the last lamina starts to compact. The other extreme of lamina stress-strain behavior is the rigid/plastic stress-strain curve. Figure 5-10 shows the resin pressures decreasing even more quickly than the softening case. The pressure drops to a value very close to the vacuum pressure and the laminate is essentially compacted at less than 40% of tcomp. The value of tcomp is 1.0 seconds, much smaller than for any of the other cases. To get a better understanding of this case consider how the rigid/plastic nature of the stress-strain curve was implemented in the simulation input. The rigid/plastic behavior had to be approximated by a curve consisting of two straight line segments. The rigid 112 portion was a line with a very steep slope (rather than infinite) and the plastic portion was a line of very shallow slope (rather than zero). We believe that what we are seeing is an artifact of the numerical modelling procedure. If a lamina were truly rigid, all of the pressure would be borne solely by the fiber network. One would expect the resin pressure to drop to the vacuum pressure (100,000 Pa) instantaneously. Figure 5-11 shows the laminae thicknesses for this case. Only the first lamina even partially compacts. The rest of the laminae experience no compaction. Again, we believe this is an artifact of the numerical procedure. As before, if the laminae were truly rigid, none would ever compact. The final case examined here is the stress-strain behavior of a real lamina. Dave et al. [4] fit a curve to the earlier work of Gutowski [2], which was performed on prepreg with an initial resin mass fraction of about 0.38. Dave et al. used a system of two equations to fit Gutowski's experimental data. The first equation is a linear function of fiber bed pressure and describes the linear compression of the fiber bed. The second equation describes the primary compression where the fiber bed behaves as a rapidly stiffening spring. The lamina stress-strain curve used by Dave et al. [4] is shown in Figure 5-1. However, the input parameter required by the simulation is the void ratio as a function of fiber bed pressure. In this work, the experiments were performed on prepreg with an initial resin mass fraction of 0.42. The first equation has been modified slightly to allow for the higher resin content of the prepreg but retains the same void ratio value as the original equation at 68,707.6 Pa. Equation (5-3) is the same as in reference [4]. The equations are: e = -2.9038(10^)/^ + 1.0678 0 < pf < 68707.6 (5-2) e = -0.107271 I n ^ + 1.8987 68707.6<pf < 1030605.0 (5-3) Figures 5-12 and 5-13 show the laminae resin pressures and thicknesses for the stress-strain curve described by equations (5-2) and (5-3). The shape of the curves in these figures lies 113 between the linear case, Figures 5-6 and 5-7, and the hardening case, Figures 5-4 and 5-5. Given that this lamina stress-strain curve is a combination of a linear and hardening stress-strain curves, this is what one would expect. In Figure 5-13 one can see that the laminae compaction is neither uniform nor sequential but is somewhere between these extremes. There is a small dwell time for each lamina before the onset of compaction. For the last lamina the dwell time is about 14% of tcomp which is somewhat faster than the hardening case at 20% of tCBmp but slower that the linear case at 3% of tcomp. The laminate is essentially fully compacted at 70% of tcomp which is faster than the hardening case and, interestingly, about the same as the linear case. Now consider the actual compaction time required for each case. The last lamina, adjacent to the toolplate, is always the last to compact regardless of the stress-strain curve used. Hence the compaction of the last lamina corresponds to full compaction of the laminate. Figure 5-14 shows the thickness of the last lamina plotted as a function of time for all five cases. The results for the lamina stress-strain behavior based on the work of Dave et al. is not included because the compacted thickness is different for that case. An interesting feature of this plot can be seen if one considers the time required for the last lamina to compact, i.e., the difference between the time at the onset of compaction1 (dwell time) and full compaction, as shown in Table 5-1. 1 We define the onset of compaction in a lamina as the time at which its thickness has decreased by 0.2% of its compacted thickness. 114 Table 5-1: Time Required for Compaction of the Last Lamina as a Function of Laminate Compaction Time stress-strain curve rigid/plastic softening linear hardening plastic/rigid t comp (s) O O 400 150 36.5 950 onset of compaction in last lamina (s) O O 0.8 4 8.5 770 compaction time for the last lamina (s) O O 399.2 146 28 180 (%oftcomp) 100.0% 99.8% 97.3% 76.7% 18.9% The percentage of tcomp required for compaction of the last lamina decreases as the lamina stress-strain curve moves away from rigid/plastic behavior, which never compacts, towards plastic/rigid behavior, which compacts quite quickly. Recall that the plastic/rigid case led to sequential laminae compaction and that the softening case led to more uniform laminae compaction. Hence laminae that have plastic/rigid stress-strain behavior have short compaction times where all laminae compact sequentially and laminae with rigid/plastic stress-strain behavior have long compaction times where the laminae compact uniformly. However, in terms of the time required to reach full compaction, the laminae fastest to compact are those with a hardening stress-strain behavior, which turns out to be the behavior of real laminae. 5.2. Sensitivity of the Compaction Time to the Lamina Stress-Strain Curve Real laminae have a hardening stress-strain curve such as the curve marked Dave/Gutowski in Figure 5-1. It is therefore of interest to see how sensitive the resin pressures and thicknesses are to minor variations in the stress-strain curve. The family of stress-strain curves used to examine the sensitivity is shown in Figure 5-15. The base curve consists of two straight line segments. The first line goes from zero strain at zero fiber bed pressure to a strain of 0.25 at a fiber bed pressure of 200,000 Pa. The second line connects to the end point of the first, creating a knee in the curve, to a strain of 0.282 at a fiber bed pressure of 600,000 Pa. The effects of the minor variations are examined by inspecting the resin pressures and thicknesses of the first and last 115 laminae. The input parameters for the simulations are the same as in the previous section except the void ratio relationship which is redefined based on the above assumptions. The effects of minor variations in the stress-strain curve on the laminae resin pressures and thicknesses were obtained by shifting the intersection of the two line segments (the knee) by +/-10% in strain and fiber bed pressure as shown in Figure 5-15. The results for all four cases are shown in Figure 5-16 for the resin pressures and Figure 5-17 for the thicknesses. An obvious feature of both figures is the insensitivity of the first lamina's response to the changes in the stress-strain curve. The response of the last lamina however, is quite different. Moving the knee by +/-10% on the fiber bed pressure axis has almost no effect on either the resin pressures or thicknesses. However, moving the knee by +/-10% on the strain axis has a noticeable effect on both. It is of considerable interest to know the effect of small deviations in the lamina stress-strain curve on the laminae resin pressures and thicknesses over time. The experimentally measured value used to validate both the squeezed sponge [3,4] and the sequential compaction [1] flow models is the resin mass lost by the laminate as a function of time. This value is then compared to that predicted by each model. The present analysis shows that if the lamina stress-strain curve is not reasonably close to the actual curve, or if there is considerable variability, then the predictions for the squeezed sponge flow model can be in error by a significant amount. The same must be true of the sequential compaction model by analogy. Of practical interest is the effect these variations will have on the final compaction time. The compaction times for variations in strain were obtained using the same criteria as in the previous section. The compaction times for the base curve, -10% strain and +10% strain cases were 60, 75 and 45 seconds , respectively. Hence a change of +/-10% strain at the knee of the stress-strain curve leads to a change of 7+25% in the compaction times. The compaction times for the +/-10% changes in fiber bed pressure were unchanged and the same as the base curve. 116 5.3. Effect of Permeability In this section we examine the effect of permeability on the shape of the laminae resin pressure and thickness profiles and the compaction time of the laminate. The simulation input parameters for this section are the same as for the previous one with one exception; the permeability is not constant but a function of the resin content of the laminae. The expression for permeability through the thickness of the lamina, SP , is obtained by making the appropriate substitutions into equation (3-86) of Chapter 3. i.e.; e3D2 SP = . f x2 (5-4) * 16K22(l-ef where Df is the fiber diameter, K22 is the Kozeny constant and £ is the porosity. The Kozeny constant is the only variable in equation (5-4) that is not defined by the input parameters. In reference [4] the value of K22 was taken from Sullivan [49]. Sullivan measured K22 by forcing air through unidirectional fiber beds aligned perpendicular to the fiber direction. The value of K22 was found to be approximately equal to 6. The initial permeability in the z -direction, calculated by equation (5-4), is 4.33(10"13) m2. Loos & Springer [1] report 5 ^ to be 5.8(10~16) m2 for AS4/3501-6 graphite/epoxy prepreg, 103 m2 lower. Such a large difference for the value of SP suggests that the true value of K22 may actually be much higher than 6. Gutowski et al. [34] and Lam and Kardos [38] measured K22 for graphite fibers and found values for K22 ranging from 11 to 150. Hence the actual value of K22 may be higher than 6. To assess the effect of different permeability values on the shape of the laminae resin pressure and thickness profiles over time, a series of simulation runs were carried out for K22 values from 117 6 to 11002. Figures 5-18 and 5-19 show the resin pressure and laminae thicknesses plotted against normalized time for a K22 value of 6. The features of these plots are essentially the same as for Figures 5-12 and 5-13 in the previous section with the exception that current case compacts slightly faster3. In Figure 5-18 the resin pressure drops off quickly in all laminae and the laminate is essentially compacted in about 75% of tcomp. In Figure 5-19 we see that all laminae start to compact by about 3% of tcomp. Figure 5-20 and 5-21 show the resin pressures and thicknesses for different values of K22 (see Table 5-2) plotted as a function of normalized time for the first and last laminae. Table 5-2: Kozeny Constant Values, Time to Compaction and Permeabilities for Each Simulation Run. K22 6 275 550 875 1100 tcomp (s) 45 1680 3540 5220 6660 SP (m2) initial 4.33E-13 9.44E-15 4.72E-15 2.97E-15 2.36E-15 compacted 5.64E-14 1.23E-15 6.15E-16 3.87E-16 3.08E-16 Note the range of compaction times in Table 5-2 for each value of the Kozeny constant. Even though the compaction times vary over a large range, the laminae resin pressure and thickness profiles are, to a first order approximation, the same. This suggests that permeability only controls the time scale over which the laminate will compact and does not otherwise alter the compaction behavior of the laminate. Plotting the values of tcomp as a function of K22 from Table 5-2 generates Figure 5-22. The compaction time is directly proportional to the Kozeny constant. Intuitively this makes sense The rationale for choosing 1100 will be made clear in Chapter 6. The permeability of the current case, which is calculated from equation (5-4), is higher than the previous case, which had a constant permeability. Hence the current case compacts more quickly. 118 since the higher the value of K22 the lower the permeability and the greater the resistance to flow, resulting in longer compaction times. Fitting a straight line to the points in the figure and forcing the intercept to be zero ( ^ = 0 leads to infinite permeability, thus zero compaction time) produces the following equation; tcomp = 6.07K22 (5-5) By inspection we can see that equation (5-5) is a very good fit. 5.4. Summary In this chapter a series of simulation runs were performed for a number of different combinations of lamina stress-strain curves and permeabilities. It was found that the shape of the laminae resin pressure and thickness profiles are quite sensitive to the stress-strain behavior of the laminae. Laminae whose stress-strain curves are plastic/rigid in nature have shorter compaction times where the laminae compact sequentially. Laminae whose stress-strain curves are rigid/plastic in nature have longer compaction times where the laminae compact uniformly. The fastest compacting laminae are those whose stress-strain behavior is similar to real prepreg systems which are characterized as hardening curves. The time required for full compaction of the laminate was found to be affected by how rapidly the stress-strain curve hardens. Compressing the abscissa of the stress-strain curve 10% leads to a 25% reduction in the compaction time. Conversely, expanding the abscissa 10% leads to a 25% increase in the compaction time. Compressing and expanding the ordinate did not lead to any significant changes in laminae resin pressure, thickness profiles or compaction time. The permeability was found to affect only the time required to reach full compaction. The shape of the laminae resin pressure and thickness profiles were essentially the same for values of the Kozeny constant ranging from 6 to 1100. The profiles can be made to collapse onto each other by expanding or contracting the time scale by an appropriate factor. 119 5.5. Figures 0 0.05 0.1 0.15 0.2 0.25 0.3 strain Figure 5-1: The different lamina stress-strain curves used to generate the simulation results. 120 700000 600000 o 500000 a S 400000 V 4_ Q. c 2 300000 200000 100000 : 1 0 : • _ , j _ 1 J , 1 . \ 1= 1 X . . . . V 1 i —1 \ ) — » i— i,—t, i — . . . . 1= 1 T~ A 1 \ 1 \ 6 J \ 7 0.1 0.2 0.3 0.4 0.5 0.( normalized time 0.7 0.8 0.9 Figure 5-2: The laminae resin pressures as a function of normalized time for a plastic/rigid stress-strain curve. tcomp = 950 s. 3.50E-04 3.40E-O4 3.30E-04 •§• 3.20E-04 g 3.10E-04 c o 3.00E-04 J : £ 2.90E-04 E •2 2.80E-04 2.70E-04 2.60E-04 2.50E-04 \ ^ \ \ I l I I l 1 0 1 L \ \ \ V . . . 3 \ 4 \ 5 V6 y 0.1 0.2 0.3 0.4 0.5 0.6 normalized time 0.7 0.8 0.9 Figure 5-3: The laminae thicknesses as a function of normalized time for a plastic/rigid stress-strain curve, t = 950 s. /uuuuu -Q. w a. VI 200000 -100000 - • * • • — J i — J i__ * * • i .... _ i • • • • • ' L _ ™1 L_J L__ 1 ' • • 0.1 0.2 0.3 0.4 0.5 0.6 normalized time 0.7 0.8 0.9 Figure 5-4: The laminae resin pressures as a function of normalized time for a hardening stress-strain curve. tcomp = 36.5 s. 3.50E-04 3.40E-04 3.30E-04 ' 3.20E-04 3.10E-04 3.00E-04 2.90E-04 -2.80E-04 2.70E-04 2.60E-04 2.50E-04 : 0.1 0.2 0.3 0.4 0.5 0.6 normalized time 0.7 0.8 0.9 Figure 5-5: The laminae thicknesses as a function of normalized time for a hardening stress-strain curve, t = 36.5 s. 700000 600000 o 500000 £ jg 400000 Q. C S 300000 200000 100000 0.1 0.2 0.3 0.4 0.5 0.6 normalized time 0.7 0.8 0.9 Figure 5-8: The laminae resin pressures as a function of normalized time for a softening stress-strain curve. tcomp = 400 s. 3.50E-04 3.40E-04 3.30E-04 3.20E-04 3.10E-04 3.00E-04 2.90E-04 2.80E-04 2.70E-04 2.60E-04 2.50E-04 0.1 0.2 0.3 0.4 0.5 0.6 normalized time 0.7 0.8 0.9 Figure 5-9: The laminae thicknesses as a function of normalized time for a softening stress-strain curve, t = 400 s. 700000 600000 -o 500000 a) k_ 3 % 400000 Q. _C » 300000 200000 100000 0.1 0.2 0.3 0.4 0.5 0.6 normalized time 0.7 0.8 0.9 Figure 5-10: The laminae resin pressures as a function of normalized time for a rigid/plastic stress-strain curve. tcomp = 1 s. 0.00035 0.00034 0.00033 0.00032 0.00031 0.0003 0.00029 0.00028 0.00027 0.00026 0.00025 0.1 0.2 0.3 0.4 0.5 0.6 normalized time 0.7 0.8 0.9 Figure 5-11: The laminae thicknesses as a function of normalized time for a rigid/plastic stress-strain curve. / = 1 s. /uuuuu -600000 -o a. 3 0) C 5) 100000 -\\N 1— 1 1. 1 ... ' • • 1 — _ - l L—J 1 0.1 0.2 0.3 0.4 0.5 0.6 normalized time 0.7 0.8 0.9 Figure 5-12: The laminae resin pressures as a function of normalized time for stress-strain curve based on the work of Dave et al. [4] and Gutowski et al. [2]. tcomp = 70 s. 3.55E-04 -i £ O (0 ffl 2.55E-04 -c \ \ \ \ \ \ \ V V b. 0 "e=c~~ \ v ^ 1 0 2 0 3 0 4 0 normali 5 0 ted time 6 0 7 0 8 0 9 1 Figure 5-13: The laminae thicknesses as a function of normalized time for stress-strain curve based on the work of Dave et al. [4] and Gutowski et al. [2]. tcomp = 70 s. 126 3.50E-04 3.40E-04 3.30E-04 -g- 3.20E-04 g 3.10E-04 c o 3.00E-O4 2 2.90E-04 E •2 2.80E-04 2.70E-04 2.60E-04 2.50E-04 ', "" - - •^^SSiJ"" ' ' , : > ; ; ; \ \ \ ': \ \linear \ hardening 1 \ Y rigid/plastic J plastic/rigid V softening 10 100 1000 time (s) Figure 5-14: Comparison of the compaction times for the last lamina for the five stress-strain curves considered. CL fl> 3 0) Q. o -• (180000,0.25) (200000, 0. 25) 25) 225) (200000, 0.275) ^ i i S ^ S S ^ 5 £ -^-t i — i i ^ ^ ^ — i i i i / / ^^ -i^  . / / /i / 3 . - ""*  * • • • 0.05 0.1 0.15 strain 0.2 0.25 0.3 Figure 5-15: The family of stress-strain curves used for the sensitivity analysis of the compaction time. 700000 600000 o 500000 Q. m 400000 a. c £ 300000 200000 100000 V 0^***— ^Vr ^ AV-V V N > (180000,0.25) (200000, 0 .25) .25) .225) (200000, 0.275) 10 20 30 40 time (s) 50 60 70 Figure 5-16: The resin pressures as a function of time for the first and last laminae. Note that the line styles in this figure correspond to the line styles of the stress-strain curves in Figure 5-15. 0.00035 0.00034 0.00033 "§• 0.00032 « 0.00031 c « 0.0003 J 0.00029 E 2. 0.00028 0.00027 0.00026 0.00025 " " " ^ S : ; , o vTTtrr _ - • - . ^ . \ 7 \ j ' . ' T . ' C . T i \ \ A \ (180000,0.25) (200000, C (220000, C (200000, C .25) .25) .225) (200000, 0.275) r 10 20 30 40 time (s) 50 60 70 Figure 5-17: The laminae thicknesses as a function of time for the first and last laminae. Note that the line styles in this figure correspond to the line styles of the stress-strain curves in Figure 5-15. /UUULKJ * a c 100000 -° \ v s \ 7 0.1 0.2 0.3 0.4 0.5 0.6 normalized time 0.7 0.8 0.9 Figure 5-18: The laminae resin pressures as a function of normalized time for a Kozeny constant value of 6. tcomp = 45 s. 3.50E-04 3.40E-04 3.30E-04 _ 3.20E-04 E ^ 3.10E-04 (A J 3.00E-04 u £ 2.90E-04 a f 2.80E-04 ~ 2.70E-04 2.60E-04 2.50E-04 2.40E-04 \ \ \ \ \ 7 i : o 0.1 0.2 0.3 0.4 0.5 0.6 normalized time 0.7 0.8 0.9 Figure 5-19: The laminae thicknesses as a function of normalized time for a Kozeny constant value of 6. tcomp = 45 s. 700000 100000 0.1 0.2 0.3 0.4 0.5 0.6 normalized time 0.7 0.8 0.9 Figure 5-20: The lamina resin pressure as a function of normalized time for the first and last laminae for five values of the Kozeny constant. 3.50E-04 3.40E-04 3.30E-04 _ 3.20E-04 E "^ 3.10E-04 VI I 3.00E-04 o £ 2.90E-04 a 2.80E-04 2.70E-04 2.60E-04 2.50E-04 2.40E-04 ; ; ; ; ; : : N—£_ - '. 1—1 1—1 v N \ 7 V* \ > V\ \ \ \ • . i . x v ^ X^ "-"^S: . • • • <22 = 6 K22 = 550 K22 = 875 K22=1100 1 i 1 i , , , , i i i i 0.1 0.2 0.3 0.4 0.5 0.6 normalized time 0.7 0.8 0.9 Figure 5-21: The lamina thickness as a function of normalized time for the first and last laminae for five values of the Kozeny constant. /uuu -C o Q. E o o -" J '• '• * * " ^ . • i f J H ^ 1 _ , —1 1 L. _ _ • ^ * i J J i ^S* ^,m • • • • 200 400 600 kozeny constant, K22 800 1000 1200 Figure 5-22: Compaction time as a function of the Kozeny constant. 131 Chapter 6: Experiments and Results In this chapter we compare the predictions of the sequential compaction and squeezed sponge resin flow models to experimental results. The predicted resin pressures at various points through the thickness of the laminates are compared to those measured by a small postage stamp sized device during the cure cycle. The final mass and thickness of the laminates after curing are compared to the predictions of the flow models for three laminates. Finally, a detailed investigation of the thickness of each lamina for one of the laminates is compared to the predicted final laminae thicknesses for the flow models. 6.1. Sensor Construction and Response to Pressure This section describes the sensors used to measure the resin pressure within a laminate. A complete description of the attempts at calibrating these sensors is given in Appendix D. Given here is a brief description of how the sensors work. The sensors were purchased from Interlink Electronics of Santa Barbara California. At the beginning of this work the sensors had just been made available. During our investigation improved sensors with a better polymer for the substrate material became available. The 132 polymer material was Ultem1 which was said to have better high temperature characteristics. The sensors used in laminate #1 and laminate #2 were the original low temperature sensors. For laminate #3 the improved sensors were used. A typical prepreg lamina thickness is 0.125 mm. As mentioned by Cai and Gutowski [51], this places severe limits on the thickness of any sensor that can be placed in a laminate. The Interlink sensors are 0.34 mm thick, 15 mm wide and 20 mm long. Thus they can be inserted between prepreg layers without causing a large disturbance. A schematic of the sensor is shown in Figure 6-1. The sensors are formed by bonding two sheets of polymer film together; with the inner surface of one sheet coated with a solid semi-conducting layer and the other with interdigitated conducting fingers. A break in the adhesive forms a small channel leading to the interior of the device. Any pressure difference between the interior and the exterior of the device pushes the interdigitated fingers into contact with the semi-conducting material. This causes a change in the resistance which is inversely proportional to force and pressure. There is some question as to what the sensors actually measure. They are not truly force sensing devices since simple bench top experiments show that the resistance is a function of both the applied pressure and the area over which it is applied. In our case we use them to measure a pressure which should be constant over their surface area and we do not address this question any further. An artificial laminate was constructed to simulate, as closely as possible, the actual experimental conditions in which the sensors would ultimately be placed. The sensors were placed between two sheets of silicon rubber. This was placed on an aluminum toolplate and then covered by a sheet of breather cloth. A vacuum bag was formed by placing a sheet of nylon film over the I Ultem is a product of DuPont. 133 artificial laminate and sealing it to the toolplate with a sealant material. The sensor leads were passed through this material and connected to data logging equipment, see Figure 6-2. The rationale for using silicon rubber is to evenly distribute the pressure on the upper and lower sensor surfaces so that a uniform pressure field exists over both surfaces. The sensors were exposed to pressures between 0 and 100 psig by placing the artificial laminate in an autoclave. A vacuum was drawn on the laminate via a mechanical vacuum pump. The change in the sensors' resistance over time was measured at ambient temperature (approximately 20° C) at constant pressure. The external pressure in the autoclave was constant at atmospheric and the vacuum pressure constant at approximately 29" Hg. The resistance was measured at 1 minute intervals for a period of 1 hour. The sensor resistance as a function of time is shown in Figure 6-3. The resistance is constantly decreasing with time. Either the vacuum supplied by the pump improves with time or the sensors are creeping. The final resistance of each sensor was approximately 10 % lower than its initial resistance. Figure 6-3 shows that the majority of the change occurs within the first 10 minutes of the experiment. To minimize this effect, the vacuum was drawn for 10 minutes prior to the beginning of all subsequent experiments. The sensors were then exposed to a pressure cycle increasing from atmospheric to 80 psig and back to atmospheric. Figure 6-4 shows the resistance plotted as a function of external pressure for sensors si through s5. The initial pile-up of data points on the ordinate is due to the 10 minutes of vacuum pressure prior to the application of external pressure. Examination of Figure 6-4 shows there is a significant difference between the sensors. Their resistances after 10 minutes of vacuum pressure vary from just below 2750 Q. to over 4750 Q. Initially the resistance is an approximately linear function of pressure, up to about 10 psig. Between 10 and 80 psig the resistance is inversely proportional to pressure. The pressurization and de-pressurization paths are not the same. In all cases the resistance measured during 134 pressurization is higher than that measured during de-pressurization. In light of Figure 6-3 this result is not unexpected. Since a sensor's resistance constantly decreases with time, the separation between the pressurization/de-pressurization paths is time dependent. A very fast pressurization/de-pressurization cycle means that the shift due to this effect would be small; conversely, if the cycle was very long, the effect would be quite large. The effect of temperature on the sensors' response was evaluated by running the sensors though a series of isothermal pressure cycles. The pressure increased from atmospheric to approximately 100 psig and back to atmospheric over a 2 hour period. The temperatures for the isothermal cycles were 20, 40, 60, 80, 100 and 120°C. Figure 6-5 is a plot of the resistance as a function of pressure for sensor si for all of the isothermal runs and is typical of all sensors. An obvious aspect of Figure 6-5 is the decrease in resistance as the temperature increases. The 20°C and 40°C runs are very close together at 50 psig and above. This is similar to the response for the 100°C and 120°C runs where, over the course of the entire run, the two runs are within 50 W of each other. This suggests that between 20 and 40°C and 100 and 120°C the sensors are relatively insensitive to temperature. Between 40°C and 100°C the decrease in resistance at maximum pressure between these runs is approximately constant, and therefore the resistance is approximately linearly proportional to temperature. Ideally, the sensors would measure the hydrostatic resin pressure. However, it is not exactly clear what these sensors measure. Consider the schematic shown in Figure 6-6. At first glance it would appear the sensor must bear the hydrostatic pressure of the resin and the pressure from the fiber bed in the z -direction. However, since their is vacuum pressure inside the sensor, it will counteract part of the resin pressure. Hence the pressure borne by the sensor, psensor, can be expressed Psensor =Pr-P*ac+Pf (6"D 1 3 5 The above equation assumes that the vacuum pressure is always available inside the sensor. However, once the sensor is between prepreg plies, the interior of the device is no longer in contact with the pressure inside the vacuum bag. Hence the pressure inside these devices is now indeterminate. Coupling this problem with the high degree of variability in the sensors' response, it was concluded that the sensors should not be used as quantitative devices but only in a qualitative fashion. The sensors indicate whether the pressure in the local region is increasing, by a decrease in resistance, or if it is decreasing, by an increase in resistance. 6.2. Estimating the Permeability of the Prepreg Before the sequential compaction model can be compared to the experimental results, an estimate of the permeability of the prepreg perpendicular to the plane of the laminate, in the z -direction, SP , is required. The procedure for determining SP is described in reference [1]. The procedure is as follows: 1. A thin (4 to 8 ply) laminate is cured for a predetermined length of time. During cure, resin flows out of the laminate and into the bleeder. 2. At a specified time the cure cycle is stopped and the mass of the bleeder is determined from the difference between its initial and final mass. 3. The laminate mass is calculated by subtracting the resin mass in the bleeder from the laminate's original mass. 4. An initial value for SP is estimated and the cure cycle of the thin laminate is simulated by the flow model. The laminate mass predicted by the simulation is compared to the measured mass and the value of Sp is adjusted until the masses agree. 136 It is important to note that the cure must be interrupted before full compaction is reached. If the laminate is fully compacted then there is no way of determining the correct value for SP . One would only be able to get a lower bound on the actual value of SP . The prepreg used in this work is AS4/3501-62 with a resin mass fraction of 0.42. The determination of the initial resin mass fraction is given in Appendix E. We assume that the compacted resin mass fraction used by Springer et al. [1] of 0.2474 is applicable to the current prepreg, since their prepreg had the same initial resin mass fraction. Using the procedure above, we calculated SP to be 4.55(10~16) m2, which compares well with the value of 5.8(10"16) m2 calculated in reference [166] for their system. The measured and predicted laminate masses are shown in Figure 6-7. The simulation was also run for the squeezed sponge resin flow model with the Kozeny constant K22 equal to 6. This was the value used by Dave et al. in reference [4] which was obtained from the experimental work of Sullivan [49,52]. This result is also shown in Figure 6-7. The squeezed sponge model predicts the measured laminate mass is reached in approximately 1250 seconds, slightly less than half of the actual time. This suggest that the actual value of K22 for this prepreg may be higher than 6. Recall from section 5.3. of Chapter 5 that the permeability in the z -direction is expressed as SP = 7-L-TT (5-4) Supplied by Hercules Inc., Magna, Utah 84044. 137 where the variables were defined in Chapter 5. Thus K22 is inversely proportional to the permeability. The value of K22 required to get agreement with the experiment was found to be 1100 3. This result is also shown in Figure 6-7. In the remaining experimental work we will compare the experimental results to the predictions of the sequential compaction model using a permeability of 4.55(10)"16 m2 (denoted SCM) and two cases of the squeezed sponge model for K22 values of 6 (SSM-6) and 1100 (SSM-1100). The initial permeability of the SSM-6 and SSM-1100 cases are calculated by equation (5-4) as 3.94(10)"13 m2 and 2.15(10)~15 m2, respectively. Comparing the shapes of the SCM and the SSM-1100 cases in Figure 6-7 we see that they are quite similar. This implies that in practical situations, the difference in mass loss predictions between the sequential compaction and squeezed sponge resin flow models may be largely due to the large difference in the permeability values used, rather than some fundamental difference. In addition in Chapter 4 it was shown that the main difference between the formulations was the relationship between void ratio and fiber bed pressure. From the similarity between the SCM and SSM-1100 cases in Figure 6-7, we can conclude that any attempt to compare the flow models, based on comparing the predicted laminate mass, is not sensitive enough to offer a definitive answer. 6.3. Laminate Experiments The following sections describe three laminate experiments where sensors were placed between their laminae and then cured. In each case excess bleeder layers were provided and it was confirmed after the run that they were not saturated. The mass and thickness of each laminate is compared to the predictions of both the sequential compaction and the squeezed sponge resin Recall that in Chapter 5 the effect of permeability on the squeezed sponge resin flow model was examined for Kozeny constant values from 6 to 1100. 138 flow models. Finally the sensor data is compared to the resin pressures predicted by the flow models. 6.3.1. Laminate #1: A 24 Ply Unidirectional Laminate The next step in our investigation was to place sensors in a 24 ply unidirectional laminate to measure the pressure at various points through the laminate thickness. The first sensor was located between the toolplate and the laminate, the next three sensors were positioned every 6 plies though the laminate thickness and the final sensor between the laminate and the bleeder cloth, as shown in Figure 6-8. To eliminate any effect of pressure drop due to the distance from the center of the panel or interference between sensors, they were arranged horizontally in a pentagon, as shown in Figure 6-9. Four 22 gauge Teflon insulated copper leads were attached to the sensors, passed through a port in the autoclave and connected to a Solartron/Schlumberger Orion Delta Data Logger, model 3530D. The solder connections between the leads and the sensors were wrapped with 3 layers of 0.076 mm thick (by 12 mm wide) Eureka Teflon tape to prevent shorting by the carbon fibers across the sensor terminals. The manufacturer's recommended cure cycle is shown in Figure 6-10. Since the sensors are rated for a maximum temperature of 170°C and they cannot function without a vacuum pressure, the cure cycle was modified to that shown in Figure 6-11. A schematic of the laminate/bleeder assembly is shown in Figure 6-12. Note that dams have been placed around the laminate to prevent resin flow parallel to the plane of the composite. The assembly was then placed in an autoclave and cured. The cure cycle attained is shown in Figure 6-13. The maximum temperature difference between the upper and lower surfaces of the laminate was found to be less than 1.2°C, hence the laminate was essentially isothermal. Unfortunately, all of the sensors shorted out in the first few minutes of the run. Nonetheless, the laminate was subjected to the whole cure cycle. Inspection of the sensors revealed that the Teflon tape wrapped around the solder connections was not durable enough to prevent the connectors from punching through it and making contact with the carbon fibers. To avoid this 139 problem in subsequent experiments, a sheet of 0.11 mm nylon film, approximately 15 mm by 10 mm, was placed on either side of the sensor solder connections and wrapped with Teflon film to hold the nylon film in place. Figure 6-14 shows the predicted laminate mass as a function of time. The initial laminate mass was measured to be 568.5 gm of which 21.0% was lost for a final mass of 449.1 gm. The measured final laminate mass is also shown in Figure 6-14. All three predictions are quite close with the SSM-1100 case being the worst. The laminate thickness was measured from the microphotograph shown in Figure 6-15 and found to be 2.88 mm. The location of the laminate section used to measure the thickness is shown in Figure 6-16. Since the fibers in a unidirectional laminate all run the same direction, there is no way of distinguishing individual laminae, and so we cannot measure their individual thicknesses. The final value for the laminate thickness and the predictions are shown in Figure 6-17. Since the thickness of the laminate is proportional to its mass, one would expect the shapes of the laminate mass and thickness profiles to be similar except for a scaling factor. As was the case for laminate mass, none of the predictions is clearly better than any other. Despite the sensors shorting out early in the cure cycle, it is still possible to use some of the resistance data collected during the experiment. Since the sensors were exposed to cure cycles in previous calibration experiments, it is possible to estimate minimum values of resistance that would be expected for each sensor during the cure cycle. By discarding resistance data in the current experiment that was less than one-half of the lowest previously observed resistances, the resistance values should be reasonably free of data that is corrupted by shorting out. Since the original attempts to calibrate the sensors proved to be inadequate in quantifying their response (see Appendix D), a second normalizing procedure was developed based on two assumptions: 1. The sensor resistance is approximately inversely proportional to the pressure. 140 2. A normalized inverse resistance can be calculated such that it has values between zero and 1 where zero corresponds to the lowest pressure and 1 corresponds to the highest pressure. The following equation satisfies the above criteria and is referred to as the second normalized inverse resistance, RN , expression; R = * ( * ) - * - , ( 6 . 2 ) R — R min max Figure 6-18 shows RN as a function of time for all five sensors. All sensors show a jump at 600 seconds that corresponds to the application of external pressure. The first 600 seconds of the run corresponds to the 10 minutes of vacuum pressure. The response of sensor si , at the toolplate, is a relatively constant increase in RN over the time prior to shorting out at approximately 3100 seconds. The response of sensor s5, the interface sensor, closely follows the toolplate sensor up to just over 900 seconds where its response begins to fluctuate before shorting out at 1400 seconds. Sensor s2, 6 plies above the toolplate, and sensor s4, 6 plies below the laminate/bleeder interface, have initial RN values much higher that sensors si or s5. Sensor s2 undergoes dramatic fluctuations starting at 1400 seconds until it shorts out at 1700 seconds. The life of sensor s4 was just under 800 seconds before shorting out. Finally sensor s3, at the mid-plane of the laminate, had a very erratic response with rapid fluctuations during the first 600 seconds when only the vacuum pressure is applied. This suggests the sensor was shorting out from the beginning of the run. Sensors s3, s4 and s5 begin to fluctuate at approximately 600, 650 and 900 seconds respectively. This suggests that the resin pressure in these vicinities is decreasing and that resin flow has begun. Since sensor si is adjacent to the toolplate, no resin can flow past it, and hence one would not expect to see fluctuations. Examining Figure 6-18 we see that this is the case. 141 Figure 6-19 is a plot of the resin pressure predictions for the three flow model cases. Note that the SSM-6 case, with its relatively high permeability, compacts very quickly at 2200 seconds, just after the resin pressure 6 plies down from the laminate/bleeder interface for the other two cases begins to decrease. The SCM and SSM-1100 cases, with their much lower values of permeability, never fully compact. The shape of the resin pressure profiles for these cases is generally the same but the magnitude of the pressure decrease is greater for the SSM-1100 case than the SCM case. The predicted resin pressure in the lamina adjacent to the toolplate is different for these cases. The resin pressure at this point decreases over time for the SSM-1100 case whereas the resin pressure for the SCM case does not and is, by definition, equal to the external pressure. It should be emphasized that it is possible that the fluctuations are due to intermittent shorting out of the sensors. However, we shall see similar results for the next laminate, which implies that these results are valid. Also, upon removing the laminate/bleeder interface and toolplate sensors from the laminate after curing, they were found to still function normally. 6.3.2. Laminate #2: A 24 Ply Unidirectional Laminate A second laminate with the same geometry and sensor placement as laminate #1, but using sensors s6 to slO, was constructed with small patches of nylon film on either side of the sensor connections to prevent shorting. The attained cure cycle for laminate #2 is shown in Figure 6-20. Note that the cure cycle has been modified by decreasing the pressure in discrete steps at the end of the cycle to examine the sensor response. Figure 6-21 shows the experimental and simulation results for the laminate mass as a function of time. The initial laminate mass was measured to be 570.1 gm of which 23.4% was lost for a final mass of 436.5 gm. The best agreement with the measured final mass is for the SSM-6 case. The thickness of the laminate was measured from a microphotograph similar to that of laminate #1 and found to be 3.04 mm. The position in the cross-section in laminate #2 is shown in 142 Figure 6-16. The simulation predictions and the measured value for laminate thickness as a function of time for the three flow model cases are shown in Figure 6-22. Again, the best agreement is for the SSM-6 case, although the agreement is not as good as for the laminate mass. Figure 6-23 shows the normalized inverse resistance as a function of time. As for laminate #1, the application of external pressure corresponds to a jump in RN for all sensors. The response of sensors s6, at the toolplate, and slO, at the laminate/bleeder interface, are relatively constant values of RN over the whole cure cycle. Sensors s7, 6 plies above the toolplate, and s9, 6 pUes below the laminate/bleeder interface, have initial RN values much higher than the other sensors. This was observed for laminate #1 as well and suggests that this behavior may be due to their symmetric position within the laminate. For the three interior sensors, s7, s8 and s9, their RN values begin to decrease just as the external pressure reaches its maximum pressure at 750 seconds. For sensor s7, RN decreases rapidly between 750 and 850 seconds at which point its resistance was higher than the measuring capability of the data logging equipment4. At 1900 seconds its resistance becomes measurable and RN increases rapidly to a value close to that of s6 and slO and is stable for the rest of the cure cycle. The sensor at the mid-plane of the laminate, s8, closely follows the behavior of the interface sensor, slO, up to 650 seconds at which point its response began to decrease with its smallest RN value just below 0.8. This was a much smaller decrease than the decreases for sensors s7 and s9 which reached values of zero. Sensor s9, 6 plies below the laminate/bleeder interface, had a response similar to sensor s7 but its normalized inverse resistance value never quite reached zero. The most striking feature of this sensor is its rapid fluctuations over very short time periods between approximately 1050 and 1800 seconds. After 1800 seconds the sensor's resistance was measured to be very low, about 3 Q, indicating that the sensor had likely shorted out. In excess of 130 kQ.. 143 The final observation of this figure is that four out of the five sensors have become stable and have approximately the same value past 2000 seconds. Sensors s8 and s9 become stable at values of RN similar to the sensors at the toolplate and laminate/bleeder interface after periods of decreasing and fluctuating RN values. This kind of response was also observed for laminate #1 and suggests that the decreases and fluctuations in RN values are real phenomena and not malfunctions of the sensors. As was the case for laminate #1, the laminate/bleeder interface and the toolplate sensors functioned normally when removed from the laminate after the cure. Again this is further evidence that the fluctuations are a real effect. Thus the sensors indicate that the pressure within the laminate changes very rapidly, and that resin is flowing, after application of the full pressure. However, the pressure changes, and hence the resin flow, are essentially complete by 2000 seconds. The above observations can be used as a check on the flow models of the simulation. Figure 6-24 is a plot of the resin pressure predicted for each of the three flow models for this laminate. The general shapes of these curves are the same as for laminate #1. The predictions for the SSM-6 case agree quite well with the normalized inverse resistance results of Figure 6-23. The SSM-6 case predicts a rapid decrease in resin pressure and that the laminate is fully compacted by 2200 seconds, which is consistent with the normalized inverse resistance results. An interesting response of the flow models to the discrete pressure drops at the end of the cure cycle can been seen in Figure 6-23. The resin pressures for SCM and SSM-1100 cases drop in discrete steps in sync with the decreases in external pressure. As the pressure drops, the SCM case decreases in the resin pressure at each point through its thickness proportionately. The resin pressure predictions for the sequential compaction model never become negative. In contrast, for the SSM-1100 case the differences in pressure between points through the thickness of the laminate remain constant. The resin pressure profiles are only shifted down by the same amount as the decrease in external pressure and the resin pressure 12 plies below the laminate/bleeder interface for the SSM-1100 case actually becomes negative. The SSM-6 case also exhibits these 144 effects but because the resin pressure has already decreased to a value close to the vacuum pressure, they are off the scale of the figure. Since the resin has become a solid at the end of the cure cycle, these negative resin pressures may not be fallacious but actually indicate residual stresses in the fiber beds of the laminae. Experimental evidence of residual stresses in laminae is given by Gutowski [2] where thermoplastic laminates deconsolidate when heated to sufficient temperature to allow their resins to flow. Overall, the best agreement with the observed normalized inverse resistance behavior is the SSM-6 case. The measured laminate mass and thickness shown in Figures 6-21 and 6-22 agree quite well for this case, adding additional support. 6.3.3. Laminate #3: A 48 Ply Cross-Ply Laminate Laminate #3 was a 48 ply laminate with a [0/902/0]12 cross-ply lay-up. The sensors placed in this laminate were made of the new high temperature material and had 31 gauge 4 conductor ribbon cable leads. The placement of the sensors is shown in Figure 6-25 and a schematic of the laminate/bleeder assembly in Figure 6-26. An additional sensor was placed beside the laminate on the same toolplate between two sheets of silicon rubber as a control. The purpose of the control sensor is to differentiate between those changes in sensor response due to temperature and pressure and the changes of the sensors inside the laminate which are due to conditions within the laminate. The cure cycle attained for this run is shown in Figure 6-27. Figure 6-28 shows the experimental and predicted results for the laminate mass as a function of time. The laminate's initial mass was measured to be 1200.6 gm and lost 19.8% of its mass for a final mass of 962.9 gm. As was the case for laminate #2, the best agreement is for the SSM-6 case. The SCM case is not a particularly good fit to the data. The experimental results of Loos and Springer [1] obtained good agreement between their SCM model and the laminate mass predictions on thick, 64 ply, laminates. In their experiments they allowed resin to flow in both the x and z -directions. Our experiments only allowed flow in the 145 z-direction and result in our SCM prediction underestimating the laminate mass loss. We cannot explain the differing results in terms of the sequential compaction model. However, if we consider the compaction behavior of the squeezed sponge model, a probable explanation is obtained. Full compaction is reached for the squeezed sponge flow model when equilibrium between the fiber bed pressure in the laminae and the applied pressure is reached. Assuming the laminates in their work and the current work had the same amount of time for resin flow, the Loos and Springer model predicts more resin flow out of their laminates due to resin flow in two direction, whereas our prediction only permitted flow in the z -direction. Since this laminate is a cross-ply laminate, one can examine the thicknesses of the individual laminae on a ply by ply basis [30,53,54]. In this case the lay-up is [0/902/0]12, hence only the laminae adjacent to either interface are single laminae whereas the interior laminae can only be differentiated as 2 laminae units. In this work three cross-sections of laminate #3 are examined, labeled A, F and H, and their locations shown in Figure 6-29. Figure 6-30 is a microphotograph of cross-section A, and is typical of all three cross-sections. Note that the laminae layers can easily be distinguished and that the lamina adjacent to the laminate/bleeder interface is missing. The laminate was particularly resin poor in this layer and it sloughed off while sectioning the laminate. The thickness of each lamina or laminae pair was measured for 6 traverses of each cross-section. The thickness of a individual lamina was taken as half of the thickness of a laminae pair for the interior laminae. The average values for the laminae thicknesses and the predictions for each of the flow model cases are shown in Figures 6-31, 6-32 and 6-33 for cross-sections A, F and H, respectively. Consider cross-section A. Overall, the laminate appears to be fully compacted. The laminae thicknesses range from 0.12 mm to 0.158 mm. The average lamina thickness is 0.138 mm. Interestingly, the first few lamina adjacent to the laminate/bleeder interface are somewhat thicker than the average value, contrary to what one would intuitively expect. 146 The SCM case predicts the laminae compact sequentially. If this were the case one would expect the to see a distinct change in the laminae thicknesses over a single lamina. Comparing the SCM results to the average laminae thicknesses in this figure clearly shows that this assumption is not valid. This case predicts that the first 17 laminae are fully compacted, the 18th lamina is partially compacted and the remaining are uncompacted at the initial thickness of 0.183 mm. The SSM-1100 case predicts there is a smooth lamina thickness gradient from the lamina adjacent to laminate/bleeder interface, at 0.132 mm, to the lamina adjacent to the toolplate, at the initial thickness of 0.183 mm. Between these extremes, the laminae thicknesses increase as the distance from the laminate/bleeder interface increases. Our work has used the same assumption as in the original work on the squeezed sponge model [4], where the relationship between void ratio and fiber bed pressure was taken from Gutowski [2]. In his experimental work Gutowski used a prepreg with an initial resin mass fraction of 0.38; this work uses prepreg with a value of 0.42. Since the current prepreg has a higher resin content than that of Gutowski, one would expect it to compact more than a lower resin content prepreg. The measured average lamina thickness is slightly higher than predicted by the SSM-6 case but the laminate appears to be fully compacted. This is contrary to the expected result. However the likely cause of the smaller predicted thickness may be due to a difference between the assumed and actual relationships between the void ratio and fiber bed pressure. The SSM-6 case predicts the laminate to be fully compacted with a final uniform lamina thickness of 0.132 mm. Examining the Figure 6-31 we see that this thickness is somewhat smaller than the average observed lamina thickness. However this case gives the best overall agreement with the data. Examining the other two cross-sections of the laminate, F and H, in Figures 6-32 and 6-33 respectively, we see that the shapes are generally the same as for cross-section A. Both cross-147 sections have high values of thickness for the laminae adjacent to the toolplate and for several lamina adjacent to the laminate/bleeder interface. The average lamina thickness for cross-sections F and H are 0.139 mm and 0.143 mm respectively. Both values are quite close to that of A at 0.138 mm. In Figure 6-32 the lamina thickness decreases as one moves away from the toolplate. Interestingly, this is the same trend one would expect if the SCM or SSM-1100 cases were correct. However, note the location of the cross-section in the laminate, see Figure 6-29; F is on the edge of the laminate while both A and H are near the center of the laminate. It is possible that extra resin was able to bleed off and not be absorbed by the bleeder. This would result in slightly thinner laminae thicknesses near the laminate/bleeder interface. The features of cross-section H in Figure 6-33 are the same as cross-section A. It should be stressed at this point that it is difficult to pin down the exact cause of the difference between the observed results and the predictions of the squeezed sponge flow model. This flow model has two parts which essentially control how compaction proceeds. The first part is the relationship between the void ratio and the fiber bed pressure. In Chapter 5 this relationship was investigated and it was seen that the final compacted thickness is controlled by it. If the laminate has enough time to fully compact, before the resin gels, then the lamina thickness is uniform through the laminate. The second part is the permeability relationship, and, for this work, the value of the Kozeny constant. The permeability was found to only affect the time required to reach full compaction; the higher the permeability the longer the time required to reach compaction. Since the average final compacted thickness for the squeezed sponge flow model was 0.132 mm and the measured final thickness of the laminates was between 0.138 and 0.143 mm, it suggests, assuming the laminate is fully compacted, that the relationship between the void ratio and fiber bed pressure is slightly in error. Also, since the average laminae thicknesses for each of the cross-sections did not show any significant trends, except for cross-section F which is slightly suspect, it is reasonable to assume that the laminate was fully compacted. 148 The normalized inverse resistance response of the sensors is shown in Figures 6-34 to 6-37. In these figures the control sensor and the applied pressure (the difference between the external and vacuum pressures) is shown for comparison. Figure 6-34 shows the response of the sensors at the laminate/bleeder interface and adjacent to the toolplate. The RN response of the control sensor has the same shape as the pressure applied and changes in sync with it. The response of the toolplate sensor decreases quickly between approximately 2500 and 2800 seconds and then more gradually after 3300 seconds. The first decrease probably represents a pressure change inside the laminate whereas the second decrease, because it is relatively slow, is a creep effect. The response of the interface sensor, s8, is essentially the same as the control sensor except for its RN values prior to application of external pressure. Figure 6-35 shows the response of sensors s2 and s3, 12 plies above the toolplate. The shape of their responses are basically the same. There is a sharp decrease in their RN responses at approximately 2200 seconds. The response of both sensors continues decreasing until just before 2400 seconds at which point their responses increase for the next 1000 seconds. During the increase in RN both sensors experience fluctuations. After this point their responses peak at 3100 seconds and then decrease slowly for the rest of the cure cycle. Again, it is thought that the initial decrease and fluctuations are real whereas the second decrease is due to creep. Figure 6-36 shows the response for the sensors at the mid-plane of the laminate, 24 plies above the toolplate. Sensor s5 has a shape that is very similar to that seen in Figure 6-35. Sensor s4 increases and decreases linearly at several points through the cure cycle. Since this type of response has not been observed before, it may indicate a either malfunction in the sensor, the leads or the measuring equipment. However, ignoring the first increase between 1900 and 2300 seconds, we see that the general shape of this response is the same as sensor s5. A key difference between Figures 6-35 and 6-36 is the time at which the initial decreases in RN start. 149 For the sensors 12 plies above the toolplate their response begins decreasing at approximately 2200 seconds. For the sensors at the mid-plane of the laminate their response begins decreasing about 200 seconds later at approximately 2400 seconds. A lag in the initial response of each sensor, increasing as one moves away from the laminate/bleeder interface towards the toolplate, is predicted by both the sequential compaction and squeezed sponge flow models. Figure 6-37 shows the response for the sensors 12 plies below the laminate/bleeder interface. One of the sensors, s6, has a response of the same general shape as seen in Figure 6-35. The response of the other sensor, s7, is essentially the same as the control sensor. The predictions for the SCM, SSM-6 and SSM-1100 cases are shown in Figure 6-38. We compare the response of all sensors, where the changes in normalized inverse resistance values occur between approximately 1000 to 3500 seconds, to the predictions. The SCM and SSM-1100 cases both predict changes in the resin pressures to be much later in the cure cycle than the sensor results imply. In both cases the resin pressure 12 plies below the laminate/bleeder interface does not decrease until just under 3000 seconds for the SSM-1100 case and not before 4500 seconds for the SCM case. The SSM-6 case predicts the decrease in resin pressures should begin at approximately 1500 seconds for the sensor 12 plies below the laminate/bleeder interface and to be over by 3000 seconds for the all sensors. Comparing the predictions to the sensor response strongly suggests that the SSM-6 case closely approximates the actual changes in resin pressure within the laminate. 6.4. Summary In summary, the SCM and SSM-1100 cases, which are essentially the same, do not agree with the laminate mass and thickness data beyond the 8 ply laminate. Conversely, the SSM-6 case agrees with the laminate mass and thickness data, except for the 8 ply laminate. The normalized inverse resistance response indicates the resin pressure begins decreasing almost immediately with the application of external pressure for the sensors closest to the laminate/bleeder interface; 150 6 plies and 12 plies below the laminate bleeder interface for the 24 and 48 ply laminates respectively. As one moves towards the toolplate the lag increases. The sensor response also indicates that most of the resin flow occurs in the first 2000 seconds of the cure cycle for the 24 ply laminates and the first 3500 seconds for the 48 ply laminate. The flow model case for the best overall agreement for the sensor results of all three laminates is the SSM-6 case. 6.5. Figures Figure 6-1: Schematic of the Force Sensing Resistor. The channel that leads to the interior of the device is formed by the break in the adhesive layer. Figure 6-2: Schematic of the lay-up used to calibrate the sensors. The sensor lead passes through the tape seal. 151 5000 -i 4500 -4000 -3000 -2500 -2000 -» •-•-•-•. c. • " " • - . . • i i i • s1 —o s2 • s3 ° s4 * s5 • I . • • • I • • • • 10 20 30 time (min) 40 50 60 Figure 6-3: Sensor resistance as a function of time for sensors si to s5 at ambient temperature (approximately 20°C ) and 0 psig. The vacuum pressure was approximately 29" Hg. The majority of the decrease in resistance occurs in the initial 10 minutes. 30 40 pressure (psig) Figure 6-4: Sensor resistance as a function of external pressure for sensors si to s5. The temperature increased linearly from 22°C to 38°C in 82 minutes (corresponding to the maximum pressure). The temperature returned to 24°C upon returning to atmospheric pressure for a total run time of 156 minutes. 152 0 10 20 30 40 50 60 70 80 90 100 pressure (psig) Figure 6-5: The resistance as a function of pressure of sensor si for all the isothermal runs. Figure 6-6: Schematic of the interaction between sensor, resin and fiber. The sensor bears the load carried by the fibers, which is transmitted to the sensor at the points where they come into contact, and the hydrostatic pressure of the liquid resin which surrounds it. 4.80E-O2 4.60E-02 4.40E-02 E 4.20E-O2 | | 4.00E-02 3.80E-02 3.60E-02 V -: . • • • • * , , - ~ ^ ^ sequential compaction squeezed sponge measured mass ' i i ' \ \ • ' ' ' * " ' « . N K22=1100 ^ K22=6 1 1 1 1 xs Nx \ I I ' • 1 1 1 I 500 1000 1500 time (s) 2000 2500 3000 Figure 6-7: Fitting the permeability for the sequential resin flow model to the laminate mass data of an 15 cm square 8 ply laminate. Also shown are the laminate mass predictions for the squeezed sponge resin flow model for two values of the Kozeny constant. I aluminum toolplate I Figure 6-8: The positions of the sensors through the laminate thickness for a 24 ply laminate. fiber direction s2 SV s5 dimensions in cm « 11-4 ^»s3 s4 sensor si s2 s3 s4 s5 plies above toolplate 0 6 12 18 24 Figure 6-9: The sensor positions in the plane of the laminate for laminates #1 and #2, 24 ply unidirectional laminates. 180 160 140 ^ 120 p £ 100 3 g. 80 £ - 60 40 20 0 " external f ; / / / / / ; / • : vacuum | iressure ^m ^ B M I ^ ^ m >ressure 1 1 t 1 — * • / ' / / / - _/ 1 • . ' ' ' ' 1 1 1 1 V \ \ \ \ \ 1 ' • ' ; : — — : : : 100 50 100 150 200 time (min) 250 300 :- 60 _ t 40 2 •o, 20 & -20 350 Figure 6-10: The manufacturer's recommended cure cycle for AS4-3501-6 graphite/epoxy prepreg. 140 120 ~ 100 o K 40 -20 -' -• • • external pressure 1 1 / / '*~7 1 1 / . - / , , , f vacuum pressure i i i i 1 1 1 1 i i i i \ —^~ ; \ ' . _ _ _ . ; ' • : ^ 1 1 1 L i 100 50 100 150 time (min) 200 60 ! 20 :- 0 2 a. -20 250 Figure 6-11: The cure cycle used for the 24 and 48 ply laminates. nylon vacuum bag v / breather porousTeflon release cloth aluminum caul plate tape seal porous Teflon release cloth non-porous nylon film Figure 6-12: Schematic of the lay-up assembly for laminates #1 and #2, 24 ply unidirectional laminates. 420 400 380 g 360 to |- 340 320 300 280 ] '• \ \ : " q—J — * - / : IS»-*-•**0 : : : : ; 2000 4000 6000 time (s) 8000 10000 800000 700000 600000 500000 £. v 3 400000 s 300000 200000 100000 12000 Figure 6-13: The cure cycle for Laminate #1, a 24 ply unidirectional laminate. The symbols superimposed on the curves are the discretized boundary conditions used in the simulation. 0.58 0.56 0.54 oi i-0.52 0) CO cs £ 0.5 n | 0.48 <o 0.46 0.44 0.42 : ^*-«^ ^<***^ k ; \ \ \ \ \ \ -\ \ \ \ \ • * , K22=6 ^ 1 1 1 1 \K22=1100 i i • ' • sequential compaction squeezed sponge measured mass • ! • < ' i • ' * 2000 4000 6000 time (s) 8000 10000 12000 Figure 6-14: The measured laminate mass and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #1, a 24 ply unidirectional laminate. 1 mm Figure 6-15: A cross-section of laminate #1 in the xy-plane near the center of the laminate. The position in the laminate is indicated in Figure 6-16. Note that since this is a unidirectional laminate the original prepreg laminae cannot be identified. •30-1 fiber direction " • s 5 # dimensions in cm s2 laminate #1 • / / • s 3 t *s4 , laminate #2 30 sensor si s2 s3 s4 s5 plies above tool plate 0 6 12 18 24 Figure 6-16: The position of the laminate thickness measurements for laminates #1 and #2. 158 4.20E-03 4.00E-O3 •g- 3.80E-O3 w £ 3.60E-03 o S 3.40E-03 a 2 3.20E-O3 3.00E-03 2.80E-03 \ ^ \ \ \ \ : \ \ \ X ,K22=6 ^(22=1100 1 1 sequential compact ion squeezed sponge • measured thickness • 2000 4000 6000 time (s) 8000 10000 12000 Figure 6-17: The measured laminate thickness and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #1, a 24 ply unidirectional laminate. 0.9 » 0.8 I °7 £ 0.6 | 0.5 I 0.4 | 0.3 = 0.2 ; • : s4 \t •'•'.> '• \ \ \ ; -_^r—T' , " - -'"'•V^ /s j 1:** " ."* t 1 * * 1 ' 1 • « VXf 'M'f. ?:• /: "•' —A' *• — • _ — i i i — ^J& **—>^*~^?f $ 3; V-s£ —' i i i i [ • i ! r-,7T- - - ^ 1 n lit! S Hi* i 1 gj , il * » i i si 1 1 i i - r r rT , , , . _^/< ] \ si - s2 • s3 - s4 • s5 "" pres [ ': ': \ ': ': , l I 500 1000 1500 2000 time (s) 2500 3000 100 90 80 70 60 50 2 40 « 30 20 10 3500 Figure 6-18: The normalized inverse resistances as a function of time for Laminate #1, a 24 ply unidirectional laminate. 700000 600000 500000 a a. 400000 a- 300000 200000 100000 2000 4000 6000 time (s) 8000 10000 12000 Figure 6-19: The resin pressure predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #1, a 24 ply unidirectional laminate. 420 2000 4000 6000 time (s) 8000 10000 800000 700000 600000 500000 9^  400000 I 300000 200000 00000 12000 Figure 6-20: The cure cycle for Laminate #2, a 24 ply unidirectional laminate. Note the pressure steps at the end of the cure cycle. The symbols superimposed on the curves are the discretized boundary conditions used in the simulation. 0.58 0.56 0.54 5-0.52 0> V) (0 £ 0.5 S a | 0.48 CO 0.46 0.44 0.42 ; ^",'---»v ^f^cAt*,*^  ; \ \ \ \ \ ' '• \ \ \ N \ , K22=6 V ^K22=1100 - . . • sequential compact ion squeezed sponge measured mass . . . . T . . • • 2000 4000 6000 time (s) 8000 10000 12000 Figure 6-21: The measured laminate mass and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #2, a 24 ply unidirectional laminate. 4.20E-03 4.00E-03 •g- 3.80E-03 CO £ 3.60E-O3 o « 3.40E-03 E 2 3.20E-03 3.00E-03 2.80E-03 ^ ~ — * ^ * » ^ \ \ \ \ 1 • ' ' \ \ \ \ \ ^K22=6 1 1 1 1 J<22=1100 --. . ' ' ' • 1 1 sequential compact ion squeezed sponge • measured thickness 1 1 1 1 • — i — i — i — i — 2000 4000 6000 time (s) 8000 10000 12000 Figure 6-22: The measured laminate thickness and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #2, a 24 ply unidirectional laminate. 500 1000 1500 time (s) 2000 2500 3000 Figure 6-23: The normalized inverse resistance as a function of time for Laminate #2, a 24 ply unidirectional laminate. 700000 600000 500000 400000 o. 300000 200000 100000 2000 4000 6000 time (s) 8000 10000 12000 Figure 6-24: The resin pressure predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #2, a 24 ply unidirectional laminate. -30-0° direction i s5 s7 s 2 # - # # # s 8 s6 s4 dimensions in cm 30 sensor si s2 s3 s4 s5 s6 s7 s8 plies above toolplate 0 12 12 24 24 36 36 48 Figure 6-25: The sensor position in the plane of the laminate for Laminate #3, a 48 ply cross ply laminate. nylon vacuum bag 2 sheets perforated Teflon film porous Teflon release cloth breather tape seal aluminum caul plate porous Teflon release cloth non-porous tape seal nylon film Figure 6-26: A schematic of the lay-up assembly for laminate #3, a 48 ply cross ply laminate. 163 420 700000 600000 100000 280 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 time (seconds) Figure 6-27: The cure cycle for Laminate #3, a 48 ply cross-ply laminate. The temperature cycles at the toolplate and the laminate/bleeder interface are different. The symbols superimposed on the curves are the discretized boundary conditions used in the simulation. 1.25 1.2 1.15 g 1.1 to E a> | 1.05 I 1 0.95 0.9 : : ^ v . " '• '• '• 1 1 1 1 1 1 1 1 1 "T**»>-^_ 'K22=6 1 1 1 1 /K22=l / . , , , 100 ' i 1 • 1 1 sequen squeez • 1 1 1 1 measur • • • i tial compact ion 3d sponge ed mass 1 1 I i i * ' i i , 2000 4000 6000 8000 10000 time (s) 12000 14000 16000 18000 Figure 6-28: The measured laminate mass and the predictions for the SCM, SSM-6 and SSM-1100 flow models for Laminate #3, a 48 ply cross-ply laminate. ?o-0° direction s i . s5 | s7 • I • • s3 H 1 A dimensions in cm s 2 * • s6 • • s S s4 50 sensor si s2 s3 s4 s5 s6 s7 s8 plies above toolplate 0 12 12 24 24 36 36 48 Figure 6-29: The position of parts A, F and H in laminate #3. KH i mm Figure 6-30: A cross-section of laminate #3 part A near the center of the laminate. The position in the laminate is indicated in Figure 6-29. Note that since this is a cross-ply laminate the original prepreg laminae can be easily identified. 165 0.2 0.19 0.18 "g 0.17 e "ST o.i6 (0 « 0.14 | 0.13 0.12 0.11 0.1 initial •—• experimental SSM-6 ^—t- -t- -t- I ' • - l -12 18 24 30 36 position through thickness (lamina number) 42 Figure 6-31: The measured laminae thicknesses and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #3, pan A. 0.2 0.19 : initial 0.18 -E f O . 1 7 E, » 0.16 <n I 0.15 £ a 0.14 c | 0.13 0.12 0.11 0.1 SSM-1100 sS\l V^  SSM-6 experimental _J i i i i j i i -+-j i i I i ' ' • •+--+- -+-12 18 24 30 36 position through thickness (lamina number) 42 Figure 6-32: The measured laminae thicknesses and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #3, part F. 0.2 0.19 -E 0.18 0.17 -f 0.16 -; 0.15 -t 0.14 0.13 -F 0.12 -; 0.11 -E 0 initial SCM ' SSM-1100 V./" experimental SSM-6 -t- -f- 1 I 42 0 12 18 24 30 36 position through thickness (lamina number) Figure 6-33: The measured laminae thicknesses and the predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #3, part H. 1 0.9 8 °-8 « 0.7 I 0 ' 6 § 0.5 "g 0.4 f 0.3 ° 0.2 0.1 0 : ;' i ¥~ >'J i i i v \ il P J ] -l^-^ f I \ —-L :— * = interface — > ".^ ^ 1 control oolplate s1 interface s8 control s9 ^ . , . )res . , . . \ [ '•_ -^toolplate ^^~^!' ; • \ \ \ 1000 2000 3000 time (s) 4000 5000 110 100 90 80 70 « 60 J 50 ? a 40 30 20 10 6000 Figure 6-34: The normalized inverse resistance and the pressure applied to the laminate as a function of time for the sensors at the toolplate and the laminate/bleeder interface and the control sensor for Laminate #3. 167 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 ;• • i • I r • ; : • -f • • • ' • ^ *—U . —f-\ VM|| • I ' ; : S. „ . i-; 3/4 S2 3/4 S3 control s9 - i i — i — pres ; ; ; " ; . . , _ — i — " _ 1000 2000 3000 time (s) 4000 5000 110 100 90 80 "a 70 « 60 £ 50 £> 40 30 20 10 6000 Figure 6-35: The normalized inverse resistance and the pressure applied to the laminate as a function of time for the sensors 12 plies above the toolplate and the control sensor for Laminate #3. 1 0.9 v 08 J " § 0.5 0.3 0.2 0.1 0 1000 i f ; r i J \r \ • \ • • : f—! y> V *-KJ^ci~ \ L / n * i \ *s j \ \ \ — i — 1 _ . , - j — i -/ • ^ v ^ • • • • ; ; ; 1/2 s4 1/2 s5 control s9 1 ' • • pres ; ; ; ; ; • • • • _ 2000 3000 time (s) 4000 5000 110 100 90 80 70 S, 60 50 40 30 3 W V> « a. 20 10 6000 Figure 6-36: The normalized inverse resistance and the pressure applied to the laminate as a function of time for the sensors 24 plies above the toolplate and the control sensor for Laminate #3. 1 0.9 g 0.8 r i S 0.5 "g 0.4 | 0.3 c 0.2 0.1 0 • J : I t i i i ! ; r i / r •• i \ -i+-^ s 6 \ 3 7 ^ ; | ; ': 1/4 s6 1/4 s7 control s9 . . . . pres \ [ \ \ ; : , . . 1 _ — _ 1000 2000 3000 time (s) 4000 5000 110 100 90 80 "m 70 « a. 60 | (A </) 50 £ a. 40 30 20 10 6000 Figure 6-37: The normalized inverse resistance and the pressure applied to the laminate as a function of time for the sensors 36 plies above the toolplate and the control sensor for Laminate #3. 700000 600000 500000 400000 Q- 300000 200000 100000 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 time (s) Figure 6-38: The resin pressure predictions for the SCM, SSM-6 and SSM-1100 cases for Laminate #3, a 48 ply unidirectional laminate. Chapter 7: Conclusions and Recommendations for Future Work In this chapter we examine the conclusions of the thesis and list some recommendations for future work 7.1. Conclusions The squeezed sponge flow model of Dave/Gutowski can be made to mimic the sequential compaction model of Loos and Springer by two different methods. The first method makes appropriate changes to the lamina stress-strain curve of the squeezed sponge and forces the squeezed sponge flow model to produce laminae thickness and resin pressure profiles that are similar to the sequential compaction model. The required lamina stress-strain curve is an approximation of the implicitly assumed lamina stress-strain curve of the sequential compaction model. The second method uses similar values of permeability for both flow models and produces essentially the same laminate mass and thickness profiles. For the squeezed sponge flow model the shape of the lamina stress-strain curve is very important. It affects greatly the compaction behavior of a laminate. Laminates where the lamina stress-strain curves that are plastic/rigid in nature compact very quickly with the laminae compacting sequentially. Conversely, laminates where the lamina stress-strain curves are rigid/plastic in nature compact very slowly with the laminae compacting uniformly. 170 The permeability only affects the time required for a laminate to compact but not the laminae thickness or resin pressure profiles as a function of time. The Interlink sensors cannot be used as quantitative pressure measuring devices due to the amount of creep in the polymer substrate of these sensors at the temperatures, pressures and time scales of interest here. However, they can be used as qualitative devices indicating if the pressure is increasing or decreasing. The experiments suggest that for 24 ply and 48 ply laminates the resin flow begins shortly after the application of the applied pressure and stops 40 minutes into the cure cycle for the 24 ply laminates and 1 hour into the cure cycle for the 48 ply laminate. Comparison of the flow model mass and thickness predictions to the measured values was the best for the squeezed sponge flow model. 7.2. Recommendations for Future Work To gain a better understanding of the mechanisms of resin flow in laminates, more experiments are needed that directly measure quantities of interest for laminae and laminates. For the laminae, more experiments are required that measure the permeability as a function of the resin content in all three principal directions and measure the deformation of the fiber bed in response to an applied stress. For the laminate, more experiments and techniques are required that can measure the resin pressure, mass and thickness of the laminate during the cure in real time. Finally, experiments using curved shapes are required to gain a better understanding of the interaction between resin flow and laminae deformation for the more complex shapes of real structures. 171 References [1] Loos, A.C. & Springer, G.S., "Curing of Epoxy Matrix Composites," Journal of Composite Materials, Vol. 17, No. 2,1983, pp. 135-169. [2] Gutowski, T.G., "A Resin Flow/Fiber Deformation Model for Composites," SAMPE Quarterly, Vol. 16, No. 4,1985, pp. 58-64. [3] Dave, R., Kardos, J.L. & Dudukovic, M.P., "A Model for Resin Flow During Composite Processing: Part 1 - General Mathematical Development," Polymer Composites, Vol. 8, No. 1,1987, pp. 29-38. [4] Dave, R., Kardos, J.L. & Dudukovic, M.P., "A Model for Resin Flow During Composite Processing: Part 2 - Numerical Analysis for Unidirectional Graphite/Epoxy Laminates," Polymer Composites, Vol. 8, No. 2, 1987, pp. 123-132. [5] Lee, W.L, Loos, A.C. & Springer, G.S., "Heat of Reaction, Degree of Cure, and Viscosity of Hercules 3501-6 Resin," Journal of Composite Materials, Vol. 16, No. 6, 1982, pp. 510-520. [6] Dusi, M.R., Lee, W.L, Ciriscioli, P.R. & Springer, G.S., "Cure Kinetics and Viscosity of Fiberite 976 Resin," Journal of Composite Materials, Vol. 21, No. 3,1987, pp. 243-261. [7] Kenny, J.M., Apicella, A. & Nicolais, L., "A Model for the Thermal and Chemorheological Behavior of Thermoset. I: Processing of Epoxy Based Composites," Polymer Engineering and Science, Vol. 29, No. 15, 1989, pp. 973-983. [8] Kenny, J.M., Maffezzoli, A. & Nicolais, L., "A Model for the Thermal and Chemorheological Behavior of Thermoset Processing: (II) Unsaturated Polyester Based Composites," Composites Science and Technology, Vol. 38, No. 4, 1990, pp. 339-358. [9] Kenny, J.M., Trivisano, A. & Berglund, L.A., "Chemorheological and Dielectric Behavior of the Epoxy Matrix in a Carbon Fiber Prepreg," SAMPE Journal, Vol. 27, No. 2, 1991, pp. 39-45. 172 [10] Mijovic, J. & Kim, J., & Slaby, J., "Cure Kinetics of Epoxy Formulations of the Type Used in Advanced Composites," Journal of Applied Polymer Science, Vol. 29, No. 4, 1989, pp. 1449-1462. [11] Mijovic, J. & Lee, C.H., "A Comparison of Chemorheological Models for Thermoset Cure," Journal of Applied Polymer Science, Vol. 38, No. 2, 1989, pp. 2155-2170. [12] Mijovic, J. & Wang, H.T., "Cure Kinetics of Neat and Graphite-Fiber-Reinforced Epoxy Formulations," Journal of Applied Polymer Science, Vol. 37, No. 9, 1989, pp. 2661-2673. [13] Mallow, A.R., Muncaster, F.R. & Campbell, F.C., "Science Based Cure Model for Composites," Proceedings of the American Society for Composites, Third Technical Conference, Technomic Publishing Co. Inc., 1986, pp. 171-186. [14] Halpin, J.C., Kardos, J.L. & Dudukovic, M.P., "Processing Science: An Approach for Prepreg Composites Systems," Pure & Appl. Chem., Vol. 55, No. 5,1983, pp. 893-906. [15] Mijovic, J. & Wang, H.T., "Modelling of Processing of Composites Part II - Temperature Distribution During Cure," SAMPE Journal, Vol. 24, No. 2, 1988, pp. 42-55. [16] Mijovic, J. & Wijaya, J., "Effects of Graphite Fiber and Epoxy Matrix Physical Properties on the Temperature Profile inside their Composite During Cure," SAMPE Journal, Vol. 25, No. 2,1989, pp. 35-39. [17] Mijovic, J. & Wang, H.T., "Processing-Property Relationships in Autoclave-Cured Graphite/Epoxy Composites—Part I," 43rd Society of Plastic Engineers, Annual Technical Conference Proceedings, 1985, pp. 1266-1268. [18] Mijovic, J. & Wang, H.T., "Processing-Property Relationships in Autoclave-Cured Graphite/Epoxy Composites—Part II," 44th Society of Plastic Engineers Annual Technical Conference Proceedings, 1986, pp. 554-556. [19] Chen, P.C. & Ramkumar, R.L., "RAMPC - An Integrated Three-Dimensional Design Tool for Processing Composites," 33rd International SAMPE Symposium, March 7-10, 1988, pp. 1697-1708. [20] Dave, R., Mallow, A., Kardos, J.L. & Dudukovic, M.P., "Science—Based Guidelines for the Autoclave Process for Composites Manufacturing," SAMPE Journal, Vol. 26, No. 3, 1990, pp. 31-38. [21] Kardos, J.L., Dudukovic, E.L., McKague, E.L. & Lehman, M.W., "Void Formation and Transport During Composite Laminate Processing: An Initial Model Framework," Composite Materials: Quality Assurance and Processing, ASTM STP 797, C.E. Browning, Ed., American Society for Testing and Materials, pp. 96-109. 173 [22] Kardos, J.L. Dudukovic, M.P. & Dave, R., , "Void Growth and Resin Transport During Processing of Thermosetting — Matrix Composites," Advances in Polymer Science Vol. 80 Epoxy Resins and Composites IV, Springer Verlag, 1986, pp. 101-123. [23] Lindt, J.T., "Engineering Principles of the Formation of Epoxy Resin Composites" Subtitle: "Mathematical Model of the Fluid Flow," SAMPE Quarterly, Vol. 14, Oct 1982, pp. 14-19. [24] Lindt, J.T., "Consolidation of Cylinders in Newtonian Fluid. I. Simple Cubic Configuration," The Society of Rheology, Vol. 30, No. 2,1986, pp. 251-269. [25] Springer, G.S., "Resin Flow During the Cure of Fiber Reinforced Composites," Journal of Composite Materials, Vol. 16, No. 5,1982, pp. 400-410. [26] Purslow, D., & Childs, R., "Autoclave Moulding of Carbon Fibre-Reinforced Epoxies," Composites, Vol. 17, No. 2, 1984, pp. 127-136. [27] Kim, T.W., Yoon, K.J., Jun, J.E. & Lee, W.I., "Compaction Behavior of Composite Laminates During Cure," SAMPE Journal, Vol. 24, No. 5, 1988, pp. 33-36. [28] Tang, J. & Springer, G.S., "Effects of Cure and Moisture on the Properties of Fiberite 976 Resin," Journal of Composite Materials, Vol. 22, No. 1,1988, pp. 2-14. [29] Campbell, F.C., Mallow, A.R. & Amuedo, K.C., "Computer Aided Curing of Composites," McDonnel-Douglas Corp., St. Louis, Missouri, Report No. IR-0355-4, 1985. [30] Hanks, D.E., Lee, M.C., Young, R.C. & Tajima, Y.A., "Processing Science of Thick-Section Composites," SAMPE Quarterly, Vol. 19, No. 2, 1988, pp. 19-28. [31] Roberts, R.W., "Cure Quality Control," in Engineered Materials Handbook, Vol. 1, Dostral, C , Woods, M., Ronke, A., Eds., ASM International, Metals Park, Ohio, 1987, pp. 745-760. [32] Morrison, C.E. & Badger, M.G., "Computer Modelling of Resin Flow During Laminate Cure," Composites, Vol. 20, No. 1, 1989, pp. 9-13. [33] Terzaghi, K., Theoretical Soil Mechanics, John Wiley and Sons, Inc., 1943, pp. 265-296. [34] Gutowski, T.G., Cai, Z., Bauer, S., Boucher, D., Kingery, J. & Wineman, S.J., "Consolidation Experiments for Laminate Composites," Journal of Composite Materials, Vol. 21, No. 7, 1987, pp. 650-669. [35] Gutowski, T.G., Cai, Z., Soil, W. & Bonhomme, L., "The Mechanics of Composites Deformation During Manufacturing Processes," 1st Technical Conference, Technomic Publishing Co., Lancaster, PA, 1986, pp. 154-170. 174 [36] Gutowski, T.G., Morigaki, T. & Cai, Z., "The Consolidation of Laminate Composites," Journal of Composite Materials, Vol. 21, No. 2, 1987, pp. 172-188. [37] Gutowski, T.G., Cai, Z., D., Kingery, J. & Wineman, S.J., "Resin Flow/Fiber Deformation Experiments," SAMPE Quarterly, Vol. 17, No. 4,1986, pp. 54-58. [38] Lam, R.C. & Kardos, J.L., "The Permeability of Aligned and Cross-Plied Fiber Beds During Processing of Continuous Fiber Composites," Proceedings of the American Society for Composites, Third Technical Conference, Technomic Publishing Co. Inc., 1986, pp. 13-22. [39] Carman, P.C., Flow of Gases Through Porous Media, Butterworth, London, 1956. [40] Skartsis, L., Kardos, J.L. & Khomami, B., "Resin Flow Through Fiber Beds During Composite Manufacturing Processes. Part I: Review of Newtonian Flow Through Fiber Beds," Polymer Engineering and Science, Vol. 32, No. 4,1992, pp. 221-230. [41] Skartsis, L., Khomami, B. & Kardos, J.L., "Resin Flow Through Fiber Beds During Composite Manufacturing Processes. Part II: Numerical and Experimental Studies of Newtonian Flow Through Ideal and Actual Fiber Beds," Polymer Engineering and Science, Vol. 32, No. 4,1992, pp. 231-239. [42] Loos, A.C. & Freeman, W.T., "Resin Flow During Autoclave Cure of Graphite Epoxy Composites," High Modulus Fiber Composites in Ground Transportation and High Volume Applications, ASTM STP 873, D.W. Wilson, Ed., American Society for Testing and Materials, 1985, pp. 119-130. [43] Poursartip, A. & Riahi, G., unpublished results, 1989. [44] Springer, G.S. & Tsai, S.W., "Thermal Conductivities of Unidirectional Materials," Journal of Composite Materials, Vol. 1, No. 2,1967, pp. 166-173. [45] Charpa, S.C. & Canale, R.P., Numerical Methods for Engineers, 2nd Ed., McGraw-Hill, Inc., 1988, pp. 285-288. [46] Press, W.H., Flannery, B.P., Teukolsky, S.A. & Vetterling, W.T., Numerical Recipes, Cambridge University Press, 1986, pp. 40-41, 635-642. [47] White, F.M., Viscous Fluid Flow, McGraw-Hill, Inc., 1974, pp. 336-338. [48] Williams, J.G., Morris, C.E.M., & Ennis, B.C., "Liquid Flow Through Aligned Fiber Beds," Polymer Engineering and Science, Vol. 14, No. 6, 1974, pp. 413-419. [49] Sullivan, R.R. & Hertel, K.L., "The Flow of Air Through Porous Media," Journal of Applied Physics, Vol. 11,1940, pp. 761-765. [50] Private communication with Dr. R. Dave, summer, 1992. 175 [51] Cai, Z. & Gutowski, T., "Fiber Distribution and Resin Flow in the Laminate Molding Process," Proceedings of the Seventh International Conference on Composite Materials, Wu, Y., Gu, Z. & Wu, R., Eds., November 22- 24, 1989, pp. 76-82. [52] Sullivan, R.R. "Specific Surface Measurements on Compact Bundles of Parallel Fibers," Journal of Applied Physics, Vol. 13, 1942, pp. 725-730. [53] Kim, T.W., Jun, J.E. & Lee, W.I., "Compaction Behavior of Thick Composite Laminates During Cure," 34th International SAMPE Symposium, May 8-11, 1989, pp. 15-19. [54] Meade, L.E., "Fabrication of Thick Graphite/Epoxy Wing Surface Structure," 24th National SAMPE Symposium, May 8-11, 1979, pp. 252-259. [55] Progelhof, R.C., Throne, J.L. & Ruetsch, R.R., "Methods for Predicting the Thermal Conductivity of Composite Systems: A Review," Polymer Engineering and Science, Vol. 16, No. 9,1976, pp. 615-625. [56] Halpin, J.C., Primer on Composite Materials: Analysis, Technomic Publishing Co., Inc., 1984, pp. 143-145. [57] Grove, S.M., "A Model of Transverse Thermal Conductivity in Unidirectional Fibre-Reinforced Composites," Composites Science and Technology, Vol. 38, No. 3, 1990, pp. 199-209. [58] Thornburg, J.D. and Pears, CD., "Predictions of the Thermal Conductivity of Filled and Reinforced Plastics," ASME, Paper 65-WA/HT-4,1965. 176 Appendix A: User's Guide for LamCure This appendix describes how to use the simulation code and specifies the format and syntax for the input files. A.1. Requirements for LamCure LamCure requires MS-Windows 3.x and about 224 kilobytes of memory to run with the default setting for its static and dynamic memory settings. LamCure also uses about 150 kilobytes of free disk space for *.TMP files while running. Users are strongly urged to look at the two example input files to get a good understanding of how to implement an input file. A.2. Installing LamCure Installation of LamCure consists of copying the LamCure files, LAMCURE.EXE and LAMCURE.IMA, into the same directory. Both of these files must be in the same directory for LamCure to run. Also, copy the *.INP files on the LamCure floppy disk to either this directory or their own so they are accessible to LamCure. A.2.1. Adding the LamCure Icon to the Program Manager If you add the LamCure icon to the Program Manager, LamCure can be started by double clicking its icon. To add the icon for LamCure: Start MS-Windows and activate the Program Manager. - Select the program group where the icon is be to add. Choose File Open from the Program Manager's menu. This opens the program group. - Choose File New from the Program Manager's menu. This opens the New Program Object dialog. 177 Select Program Item in the New field, then double click the dialog's OK button to accept the setting. This opens the Program Item Properties dialog. - In the dialog's Description field, type LAMCURE. This is the name that will be displayed under the icon. In the dialog's Command Line field, type; <path>LAMCURE.EXE This will start LamCure without any input file associated with it. If desired, a input file may also be specified at this time to make LamCure automatically open that file each time LamCure is started. In the Command Line dialog type; <path>LAMCURE.EXE <pathxfileName>.INP - Click the dialog's OK button. This adds the icon to the active program group. - To start LamCure double click on its icon. A.2.2. Using WIN.INI to control Memory Allocation The only options for the settings of the WIN.INI file are for the static and dynamic memory requirements of LamCure. The default settings are; [LamCure] Static=64 Dynamic=80 The above lines are not required if the default settings are found to be adequate. If very large laminates are specified, in excess of 100 laminae, a higher dynamic memory setting may be required. A.2.3. Updating the WIN.INI File Adding the following line under the [Extensions] heading of the WIN.INI file will allow access to input files with an INP extension. INP=LAMCURE.EXE MNP A.2.4. Associating MNP Files with LAMCURE.EXE To associate MNP files with LAMCURE.EXE: Select an MNP file in the Windows File Manager. - Choose File Associate from the File Manager. This opens the Associate dialog. - Enter a path followed by filename LAMCURE.EXE. Assuming LamCure was installed in the directory C:\LAMCURE, the complete entry would be; 178 C:\LAMCURE\LAMCURE.EXE Click the dialog's OK button. This associates *.INP files with LamCure's executable file, LAMCURE.EXE. A.2.5. Starting an Input File from the Program Manager To start an input file from the Program Manager: Start Windows and activate the Program Manager. - Choose File Run from the menu. This opens the Run dialog. - In the Run dialog, type; <path>LAMCURE.EXE <pathxfileName>.INP where <name> is the file that LamCure will use as its input file. Note that this is optional. If no <fileName> is given LamCure will start and the user will have to open the input file. - Click the dialog's OK button. A.2.6. Starting an Input File from the File Manager To start an input file from the Windows File Manager, open the directory that contains the input file, then double click the file name. This only works if you have updated the WIN.INI file or associated *.INP files with the executable file LAMCURE.EXE. A.3. Running LamCure--the Menu Options Figure A-1 shows the main window of LamCure. Menu choices followed directly by an exclamation mark means that immediate action is taken, i.e such as stepping through time, Step!, or resetting the model, Reset!. Menu options without exclamation marks prompt the user for more information. A.3.1. File A.3.1.1. Open This option opens the open dialog box, shown in Figure A-2. The default extension for the listed files is *.INP. The user may navigate through the directories and disk drives to find the appropriate input file. Note that input files generally have a INP extension. However any file with an extension other than INP can be read into LamCure, provided the file is in the standard format described in this manual. A.3.1.2. Close This option closes the currently selected input file and returns LamCure to the state prior to opening an input file. Before closing the current input file, and any output files or dynamic data exchange links, LamCure asks for confirmation with the close dialog shown in Figure A-3. 179 A.3.2. Run!/Resume! This option steps the simulation through time until the model time is greater than or equal to that specified in the RunUntil menu option. If the user has not specified a time to RunUntil the default is one time step as specified in the current input file's [timeStep] schedule. Note that Run! is the menu selection if the simulation time is zero, i.e., the input file has just loaded and the user has not yet clicked Run!. Resume! becomes the menu selection once the user has either clicked Run! or Step!. The logic is that both Run! and Step! result in LamCure stepping through time, and therefore the current time can no longer be zero. Hence one resumes stepping through time in LamCure. A.3.3. Stop! This option stops LamCure at the current time step and sets RunUntil to the current time. There may be a noticeable delay since the Stop! procedure must wait until after LamCure has stepped through the current time step. Stop! is useful for stopping the simulation execution when a RunUntil time that is too long has been specified. A.3.4. RunUntil This option specifies how long LamCure will step through time. For example, if the user specifies 1000 seconds LamCure will step through time until the sum of its time steps (as specified in the input file [timeStep] schedule) is equal to or exceeds 1000 seconds. The RunUntil dialog is shown in Figure A-4. As mention in section A.3.2. the initial RunUntil time is set to the time for the first time step specified in the input file. A.3.5. Step! This option makes LamCure take a single time step (as specified in the input file [timeStep] schedule). A.3.6. Reset! This option re-initializes LamCure back to the initial conditions specified in the input file by re-reading the current file from the disk. The simulation time is equal to zero. Before resetting the current input file LamCure asks for confirmation with the reset dialog shown in Figure A-5. A.3.7. DDE A.3.7.1. Display Server(s) This option is a toggle switch. A display window(s) that show(s) the messages sent back and forth for any dynamic data exchange links are either displayed or hidden. Hiding or displaying the messages has no effect on the data exchange itself. It is merely a debugging tool for DDE problems. A.3.7.2. Terminate DDE Links The user has the option to terminate the DDE links. This option will notify the other applications in the DDE links that they are being terminated. As with the File Close and Reset! 180 menu selections, LamCure asks for confirmation before terminating the DDE links. The dialog is shown in Figure A-6. A.4. The Input File Structure A.4.1. Symbols and Units for the Input Values All input values for the model are assumed to be in the meter-kilogram-seconds system of measurements. The units required for each model input are given in Table A-3 in section A.8.. A.4.2. General Layout of the file: Headings and Sub-sections The lay out of the input file has been left as general as possible to allow for future improvements and expansion. Any text editor can be used to create or edit these input files. Notepad for Windows, for example, works adequately. The overall layout of an input file is a collection of 15 possible headings, each with its own data and format that can be read by LamCure. These 15 headings have the same syntax and format and are listed in Table A-l below. Table A-l: Possible Headings for the Input [toolplateTemp] [interfaceTemp] [bleederTemp] [vacuumPressure] [externalPressure] [timeStep] [execution] [voidRatioVsFiberPressure] File. [bleeder] [fiber] [resin] [laminate] [dynamicDataExchange] [fileOutput] rend] Note that each heading is bracketed by square braces, [ ], and that the first character is always lowercase and the first letter of each word in a heading is capitalized (with the exception of the first letter). The spelling of each heading must be exactly as given in Table A-l. If not, LamCure will read in as much of the file as it can and then bring up an error dialog stating that the mis-spelt heading is not a valid heading. Note that the last heading for every input file must be [end]. Comments are delimited by the'%' character on either side of the comment. There are no restrictions on the characters that can be placed in a comment. An example is % The laminate specification for a 16 ply laminate for the squeezed sponge flow model % A.4.3. Order is Important for Certain Headings The order of most of the headings is not significant. For example, all of the schedule headings can be placed in any order. Typically, headings that must precede other heading are ones that are used in the construction of following items. For example, the laminate heading, [laminate], must come after the [fiber], [resin], [execution] and [voidRatioVsFiberPressure]1 headings. This Which is only required if the squeezed sponge flow model is specified in ihe [execution] heading, i.e., the flowModel entry of the [execution] heading is equal to sponge. 181 is reasonable as the laminate is composed of laminae of resin and fibers. Therefore knowledge of resin, fiber, etc., is necessary before the model can create the laminate. A.4.4. Representation of Boundary Conditions This simulation, although general, requires the specification of certain boundary conditions to obtain the desired solutions. These are: - the temperature of the toolplate - the temperature of the bleeder2 - the temperature of the interface between the bleeder and the laminate as a function of time - the pressure inside the vacuum bag - the external pressure inside the autoclave, exerted on the laminate, as a function of time The representation of these boundary conditions is in the form of ordered pairs (time-item objects) and a collection of these ordered pairs over time (schedules) specifies the boundary conditions. The terms in parentheses are fully explained in the following two sections. A.4.5. Time-Item Objects Each of the boundary conditions is expressed in the form of an ordered pair. The first number of the ordered pair is the magnitude of the quantity, the second number of the ordered pair is the duration of time over which the magnitude of that value is valid. For example, to specify a temperature of 352°K for 1 hour the ordered pair would be (352, 3600). This time-item object would be a valid entry for any of the schedule headings that end with Temp. Suppose this time-item object was under the [toolplateTemp] heading. LamCure would interpret this single time-item object as meaning the toolplate temperature is 352°K for that entire 3600 seconds period. This works well for values that are constant over time but nearly any case of interest requires varying boundary conditions. To handle these cases one needs to understand how the LamCure interprets more than one time-item object. Collections of time-item objects are called schedules and are the topic of the next section. A.4.6. Schedules: Collections of Time-Item Objects that Change Over Time A schedule, as defined for the purposes of LamCure, is a collection of one or more time-item objects. A schedule must contain more than one time-item object if the schedule's value is to be a function of time. For example, consider how to describe a constant temperature of 352°K for 900 seconds, increasing the temperature over 1800 seconds to a new temperature of 452°K, holding this temperature constant for 2700 seconds followed by a ramp down to 352°K over Only required if the sequential compaction flow model is specified in the [execution] heading, i.e., the flowModel entry of the [execution] heading is equal to sequential. A bleeder temperature may be specified for the squeezed sponge flow model if one wants information on the state of the bleeder but it is not necessary for this model. 182 3600 seconds and finally ending after 4500 seconds at 352°K. The schedule that would describe this is the following sequence of time-item objects: (352, 900), (352, 1800), (452, 2700), (452, 3600), (352, 4500) The plot of the temperature as a function of time for the above schedule is given in Figure A-7. The following equation describes how LamCure interprets schedules. Suppose there are i = 1,2,3,...,« time-item objects, (V^^D^), (V ,^Z>) and (Vi+1,DM) and the current time is within the duration of the i'h time-item object. The current value of Vt is, n = \ Vt=Vx for all t (A-l) n>\ V,= rvM-ViV j~ ^ v A+, j #=«-i t-^Dj +Vt for t<^Dj V M J M y=»- l where ^T Dj, = 0 for / = 1 y=i V = v Y t r n 2 for X A ^ ^ S A «=I A.4.7. Schedule Objects within the Model A.4.7.1. [toolplateTemp] This temperature cycle is the thermal boundary condition at the toolplate for the heat transfer model in LamCure. This temperature cycle along with the laminate/bleeder interface temperature cycle makes up the thermal boundary conditions of the laminate during cure. A.4.7.2. [interfaceTemp] This temperature cycle is the thermal boundary condition at the laminate/bleeder interface for the heat transfer model in LamCure. A.4.7.3. [bleederTemp] This is the temperature of the bleeder during the cure. This schedule is only required if one is using the sequential compaction flow model. LamCure assumes that the bleeder never completely saturates with resin during curing. In line with the original work on the sequential compaction model [1], LamCure further assumes that all the resin in the bleeder is at the same 183 temperature. Thus LamCure assumes the resin in the bleeder is isothermal at the temperature specified by the [bleederTemp] schedule. A.4.7.4. [vacuumPressure] This is the pressure inside the vacuum bag during the cure. A word of warning is required for this point. Although any arbitrary pressure cycle can be specified, LamCure assumes the external pressure applied to the laminate is greater than the vacuum pressure inside the bag. Obviously, if the vacuum pressure is greater than the external pressure, the vacuum bag would inflate and no resin flow would occur. A.4.7.5. [externalPressure] This is the pressure outside of the vacuum bag and inside the interior of the autoclave during cure. Note the warning under the [vacuumPressure] heading of this section. A.4.7.6. [timeStep] This specifies how LamCure steps through time. One of the requirements for any solution to numerical heat or resin flow models is a time increment. Since this is typically specified by the writer of the given numerical model, it is necessary that this information be supplied beforehand. LamCure is written to allow variable time increments (referred to as time steps hereafter). Variable time steps are useful in solving diffusive (parabolic) differential equations. Variable time steps have the advantage of coping with the changing state of the system (i.e., smaller time steps are used when the state of the system is changing rapidly and longer time steps when the system is relatively stable) instead of having to use a small time step throughout the simulation to cope with the one event that rapidly changes the state of the system. This schedule is just like the other schedules for the boundary conditions but the magnitude value of each time-item object is an increment of time. The algorithm for interpolation between time-item objects is given by equation (A-l). Note that the smallest permissible time step is 0.01 seconds. Smaller values will generate an error. A.4.7.7. [voidRatioVsFiberPressure] This schedule is only meaningful if one is using the squeezed sponge flow model. The void ratio as a function of fiber bed pressure refers to the degree of compaction a given lamina will undergo when a pressure of known magnitude is applied to it. For example, when a pressure is applied to a laminate stack, the stack will tend to compress. The amount of compression will be a function of the amount of pressure that the fiber network of the prepreg layers can support (assuming the resin carries none of the applied pressure). Obviously the greater the applied pressure the greater the amount of compression experienced by the laminae stack. This is explained in detail in the work of Dave et al. [3,4] and Gutowski [2]. This schedule is just like the other schedules for the boundary conditions except that instead of a time-item object, this is an ordered pair of the void ratio and the applied pressure. The void ratio is the magnitude value and the applied pressure is the duration value. 184 A.4.8. Controlling execution of the program This section deals with those sub-sections that fall under the [execution] heading. Each of the following keywords may be present under this heading. If a keyword is not specified then the default value is used, see Table A-2. Table A-2: The Default Settings for the [execution] Heading. Keyword flowModel parallelFlow perpendicular flow sequentialCompactionModel mixing permeabilityModel thermalModel checkDiagonalDominance Default Value sponge false false incremental laminar variable isothermal true It is the responsibility of the user to ensure that each sub-section is specified. If a sub-section heading is included that does not correspond to the listed sub-sections, or if the sub-section heading is mis-spelt, LamCure will bring up an error dialog stating that the mis-spelt sub-section heading is not a valid heading. LamCure will continue to read the input file after the user clicks OK. The syntax of each entry under this heading is as follows. Currently, each entry is either a logical value (true for true or false for false) or an alphabetic string associated with that keyword. The entry line follows the same capitalization rule as the headings. There must be an equals sign '=' after the entry word to separate it from its state (spaces before or after the equals sign have no effect on LamCure's interpretation of that line). Note that each entry line must be terminated by a semi-colon ';' character. For example, a valid line for the flowModel entry is; flowModel=sponge; All of the headings have the same format and their meaning is discussed in the following sections. A.4.8.1. flowModel The states are either sequential or sponge, sequential specifies that the resin flow model based on the work of Springer and his co-workers [1] will be used and sponge specifies that the resin flow model based on the work of Dave and co-workers [3,4] will be used. A.4.8.2. sequentialCompactionModel The states are either incremental or continuous, incremental specifies that the lamina compaction model based on the work of Springer and his co-workers [1] will be used and continuous specifies that the lamina compaction model as described in chapter 3 will be used. 185 A.4.8.3. parallelFIow This indicates whether or not resin will be allowed to flow parallel to the fiber direction. The states are either true of false. This can only be true if flowModel is set to sequential since LamCure currently does not incorporate parallel resin flow for the squeezed sponge resin flow model. If flowModel is set to sponge an error dialog box alerts the user to this condition. A.4.8.4. perpend icularFlow Indicates whether or not resin will flow perpendicular to the fiber direction. The states are either true of false. There are only two cases for which specifying false would make sense. The first is when one is only using the parallel flow model of the sequential compaction flow model. The second is to have no flow in either the parallel or perpendicular directions in which case the simulation would be for a zero bleed system. Note that in the latter case each lamina's original resin would stay in that lamina and the resin is effectively stagnant. A.4.8.5. mixing LamCure was written with the ability to trace the movement of resin through the laminate. The resin is always identified with the index of the lamina from which it originated. The ordering convention is that the lamina adjacent to the laminate/bleeder interface is 0 (zero) and the lamina adjacent to the toolplate is n-1 where the laminate is composed of n laminae. The states are either laminar or mix. laminar indicates that the resins do not mix as resin from laminae closer to the toolplate moves towards the laminate/bleeder interface. Consider three laminae, a, b and c. Lamina a is closest to the toolplate. Lamina c is closest to the laminate/bleeder interface. Lamina b is between laminae a and c. If a given volume of resin from lamina a flows into lamina b, an equal volume of resin from lamina b flows out of b and into lamina c. This process can continue as long as the lower lamina (c), has excess resin to flow into the upper laminae (b and c). Eventually, depending on the laminae location relative to the toolplate and laminate/bleeder interface, it is possible that all of a lamina's original resin may be displaced by resin from laminae beneath it. In a similar manner, mix indicates that any resin flowing into the lamina mixes perfectly with all resins present in that lamina before an equal volume of resin is displaced and forced out of that lamina into laminae closer to the laminate/bleeder interface. The algorithm LamCure uses for determining the composition of the resin displaced from the lamina is based on a mass weighted averaging scheme. LamCure determines the volume of resin that must be removed from the lamina in question based on the mass fraction of each resin in the lamina. For example, consider three resins in the given lamina. The resins present in the volume of resin displaced from that lamina will be in mass ratios exactly the same as the original lamina from which they came. In the same manner as the laminar case, resin can be moved through the laminate from the toolplate to the laminate/bleeder interface. A.4.8.6. permeabilityModel Specifies whether the permeability of the laminae is constant (independent of the fiber volume fraction) or variable (a function of the fiber volume fraction). The states are either constant or 186 variable. For the case of variable permeability the formulae used in LamCure are given by Dave and co-workers [3,4]. A.4.8.7. thermalModel This specifies if the heat transfer model will calculate the temperature of each lamina as a function of position and time or if an isothermal condition will be imposed on the laminate. The states are either calculate or isothermal. The heat transfer model used in LamCure is based on the model used by Springer and co-workers [1]. If the thermalModel is set to calculate the heat transfer model will calculate each lamina's temperature for each time step. If the thermalModel is set to isothermal the whole laminate will have the same temperature as the toolplate at each time step. A.4.8.8. checkDiagonalDominace This specifies if LamCure will, at each time step, check to see if the matrices that are used for the heat and resin flow models are diagonally dominant. Typically, the kinds of differential equations that represent physical phenomena result in matrices that are diagonally dominant. If a given matrix is not diagonally dominant it usually means that something physically meaningless is occurring, i.e., heat is spontaneously moving from a cold region to a warm region. In cases where non-diagonally dominant matrices are encountered, LamCure alerts the user through a dialog box. When this entry is equal to true, LamCure checks the appropriate matrices (depending on which thermalModel and flowModel is specified) to see if they are diagonally dominant. Note that this means a considerable number of additional calculations are required for each time step. If one is completely sure that the input file is debugged and runs as expected, then one may consider setting this entry equal to false. However, if there are any problems with the output it is a recommended that this option be set to true. A.4.9. Specifying Autoclave Geometry and Physical Properties This section describes the representation of the components that are placed inside an autoclave to make a laminate: the fiber and resin, the laminate and the bleeder. The first three, bleeder, fiber and resin can have two possible forms for their entry lines in the input file. Within LamCure are some known physical properties of each type of component; i.e., the physical properties of Hercules AS4 fibers are incorporated into LamCure. LamCure does not limit users to this one choice of the materials but allows generic components to be specified as a means of dealing with materials systems different from those incorporated within LamCure. Each sub-section identifies the required physical properties for the various components. The general form of each entry is a sequence of words and/or numbers delimited by the comma character ',' and terminated by the semi-colon character ';'• For the bleeder, fiber and resin headings the first item in the entry line is the type of component that is being specified. The choices are either their respective built-in values, Hercules AS4 fibers, Hercules 3501-6 resin and Mochburg bleeder, or their generic counterparts. The sequence of items is given in the following sub-sections. 187 A.4.9.1. Specifying the bleeder There are two possible choices for the bleeder. Either Mochburg bleeder, as used in the original work [1] on the sequential compaction flow model, or a generic bleeder can be specified. In the first case the word MochburgBleeder followed by the bleeder's length (L) and width (W) are required. In the second case the word is GenBIeeder and the permeability (Sb) and porosity (eb) of that bleeder material must also be given. The units for each physical property are given in section A.8. The syntax for each entry is given in the two sub-sections below. A.4.9.1.1. Case 1: Mochburg bleeder cloth MochburgBleeder, L, W; A.4.9.1.2. Case 2: Generic bleeders GenBIeeder, L, W, Sb, eb; An example with the appropriate numerical constants for the Mochburg bleeder is; GenBIeeder, 0.3, 0.3, 5.6e-ll, 0.57; A.4.9.2. Specifying the fiber There are two possible choices for the fiber. Either Hercules AS4 fibers, as used in the original work [1] on the sequential compaction flow model, or a generic fiber can be specified. In the first case all that needs to be specified is the word AS4. In the second case the word is GenFibers followed by the density (pf), the specific heat (CP ), the thermal conductivity parallel to the fiber direction (kfn), the thermal conductivity perpendicular to the fiber direction (kf ) and the fiber diameter (Df). The units for each physical property are given in section A.8. The syntax for each entry is given below. A.4.9.2.1. Case 1: Hercules AS4 fibers AS4; A.4.9.2.2. Case 2: Generic fibers GenFibers, pf,CP, kfn , kf , Df; An example with the appropriate numerical constants for the Hercules AS4 fiber is; GenFibers, 1799.0, 0.721, 26.0e-3, 26.0e-3, 8.4e-6; 188 A.4.9.3. Specifying the resin There are two possible choices for the resin. Either Hercules 3501-6 resin, as used in the original work [1] on the sequential compaction flow model, or a generic resin can be specified. In the first case all that needs be specified is the word R35016. In the second case the word is GenThermoset followed by the density (pr), the specific heat (CP), the thermal conductivity (kr), the heat of reaction (HR), the constants in the equations that describe the cure rate reaction kinetics (given in the Generic resin sub-section below) and the constants in the equations that describe the rheological behavior of the resin (also given in the Generic resin sub-section below). LamCure currently requires that the general form of the equations for the cure rate reaction kinetics and the viscosity of the generic resin be in an analogous form to those equations given by Springer and his co-workers [l]3 . The units for each physical property is given in section A.8. of this guide. The syntax for each entry is given below. A.4.9.3.1. Casel: Hercules 3501-6resin R35016; A.4.9.3.2. Case 2: Generic resin LamCure allows the user to change the physical parameters that describe both the cure rate reaction kinetics and the rheological behavior, subject to the form of each set of equations remaining constant. Within these restrictions the user may change parameters arbitrarily. The form of both the cure rate and rheological equations are those used by Sringer and co-workers for Hercules 3501-6 resin [5]. The following equations represent the cure rate relationship used within LamCure. — = (Kl -K2a)(l-a)(B-a) a<C 3 ( A " 2 ) ^ = K3(l-a) a>C at In future versions, it is intended that any general equation may be parsed into LamCure. 189 where; * i = A e x p | - - ^ j -K2 = A, exp| - | | | (A-3) K3 = A3 expl — RT The parameters that may be changed are: A1, \ , A3, AEj, AE2, AE3, B and C. For the rheological model the viscosity can be described by the following equations; I* = /*- expf—+Ka J (A-4) where the above parameters that may be changed are: /i„, U and K. The form of the entry for generic resin is: GenThermoset, pr, Cp , kr, HR, AEl9 AE2, AE3, U, Ax, A2, A3, / /„, B, K, C; An example with the appropriate numerical constants for the Hercules 3501-6 resin is; GenThermoset, 1220.0, 1.26, 1.67e-4, 474.0, 80.7,77.8, 56.6,90.8, 3.5017e7, -3.3567e7, 3.2667e3,7.93e-14, 0.47,14.1, 0.3; A.4.9.4. Specifying the laminate There are two options for describing the laminate. It is important that the resin, bleeder and fiber headings be placed before this heading in the input file. The required information for the first case of n identical laminae is essentially the same as the case where all laminae are unique. They only differ by an extra parameter that specifies how many laminae are in the laminate. In both cases the common information for defining a prepreg ply is length (L), width (W), initial mass of the fiber and resin that compose the prepreg ply (minit), initial resin mass fraction (Mr ), compacted resin mass fraction (Mr), permeability (Sp), parallel flow coefficient (P), fiber packing (Fp), initial degree of cure (aMl) and initial temperature (Tinit). Note that some of these values will be equal to either a numerical value or false depending on the entries under the [execution] heading. If the flowModel entry is equal to sequential then numerical values for a compacted resin mass fraction are required and, if resin flow parallel to the fiber direction is 190 wanted, a parallel flow coefficient will be required. If flowModel is equal to sponge then these entries will be ignored, and internally set to false. The value for permeability depends on the permeability entry in the [execution] heading. If permeability is equal to constant under the execution heading then the permeability given the laminate definition will be a numerical value; if permeability under the [execution] heading is equal to variable then the permeability given in the laminate definition will be ignored, and internally set to false. The value for the fiber packing specifies how the thermal conductivity of each prepreg layer, perpendicular to the fiber direction, will be calculate. The parameter can be one of five choices: parallel the fibers and resin are assumed to behave as two thermal resistances in parallel. series the fibers and resin are assumed to behave as two thermal resistances in series. - square the fibers have a square cross-section and are arranged in a square packing array. circular the fibers have a circular cross-section and are arranged in a square packing array. hexagonal the fibers have a circular cross-section and are arranged in a hexagonal packing array. A complete explanation of these terms and the formulae that are used to calculate these values is given in Appendix C. The syntax of the laminate's definition is the same for both cases. In each case the parameters are separated be the comma character, ',', and terminated by the semi-colon character, ';'• The first case of n identical laminae has 11 parameters in its definition line. For the second case of unique laminae, each lamina has its own definition line and hence has only 10 parameters in the definition line. The first lamina defined is adjacent to the laminate/bleeder interface. The last lamina defined is adjacent to the toolplate. A.4.9.4.1. Case 1: identical lamina n,L, W,mMl,M. ,M, ,SP,P,FP,ainit,7]m,; 191 A.4.9.4.2. Case 2: unique laminae The superscripts refer to the laminae positions in the laminate. The laminae are numbered from 0 (zero) for the lamina adjacent to the laminate/bleeder interface to n - 1 for the lamina adjacent to the toolplate. L° W° m° Ml Ml Si P° FP° «L 7* ii w m' MI M:C s'P P< F; O L TU L"-1 wn-1 m"-' M;_1 M;_1 S;-1 pn~l F;_1 a";1 T":1 "wit rc r r Mil mil A.5. Specifying Output LamCure currently supports two forms of output. The first is a text file on the specified drive (This is an ASCII file that can be read by any text editor or spreadsheet. MS-Notepad or MS-Excel will work). The second is much more elegant; it uses the dynamic data exchange (DDE) feature built into MS-Windows. This option provides LamCure with the ability to send the data it generates, immediately after calculation, directly to another application such as MS-Excel, where the data can be manipulated and plotted using MS-Excel's plotting routines. One can have both forms of output (files and DDE links) specified in the same input file thereby providing a raw data file as well as the processed data file within another application. As many output files and/or DDE links as desired can be specified subject to certain naming criteria described below. The heading syntax for both a output file and a DDE link is identical to the other headings. For an output file the heading is [fileOutput] and for a DDE link the heading is [dynamicDataExchange]. Regardless of which type or combination of output is to be used, the means of specifying the properties to be output are very similar. Immediately following the heading on the next line is the name of the particular output enclosed in curly brackets, {}. For output files this is the complete path name and file name of the output file, i.e. {d:\lamcure\testl .opt} This line instructs LamCure to write the output data to a file called TESTl.OPT in the directory LAMCURE on drive D. Note that two files cannot have a combination of path and file names that are identical; if so an error will be reported and an error dialog will appear. Mixing upper and lower case letters does not result in an error for a file. The equivalent for a DDE link is 2 names separated by a comma character ','• The first name is the application name for the client application that is to respond to messages from LamCure. The second name is the topic name for the data being exchanged between LamCure and the client application, i.e., {LamCure, CureData} 192 This instructs LamCure to write the output data to all DDE conversations that have their application names set to LamCure and their topic name set to CureData. Note that it is necessary that the combination of application name and topic name be unique for each link. Internally LamCure uses the topic name for the item name. Therefore only the topic name needs to be specified. DDE links differentiate between upper and lower case letters, therefore, before the client application can interact with LamCure, both names must be correctly spelt including the case of each letter. LamCure can communicate via two links to the same application if two different topic names are used. Note that LamCure automatically makes the topicName and itemName for each DDE link the same. Hence, in applications with links to LamCure, use the same topicName as the itemName. The next line of the output definition specifies how often output is to be sent to the particular output location. For output files this is the key word printEvery followed by an equals sign '=' and the intervals at which to send the output to the file. For DDE links the key word is ddeEvery and it behaves identically to printEvery. For example printEvery=60; instructs LamCure to send data to the file every 60 seconds of simulation time. LamCure determines if it is time to send output by taking the modulus of the current time with the printEvery time and sending output if the result of this operation is zero. Therefore for the above setting of printEvery=60, output would be written at a simulation time of 360 seconds but not at a simulation time of 361 seconds. Thus if the time steps specified in the [timeStep] schedule never results in zero remainder when combined with printEvery or ddeEvery then no output will be written during the simulation. The smallest value the user can specifiy for printEvery and ddeEvery is 0.01 seconds. Therefore, to write the output at every time step, specifiy the printEvery or ddeEvery to 0.01 seconds. The entries that follow the output headings are the particular properties that can be sent to the appropriate output location. These properties are the subject of the next section. A.5.1. Specifying Autoclave, Bleeder, Laminate and Lamina Properties Specifying which of the various properties of the simulation to output is achieved by providing a list of the desired properties. The syntax of the list is essentially the same as for the entries under the [execution] heading. There are four parts of the simulation to which the user may attach a list of properties. A keyword instructs LamCure to assign the list of properties following that key word to one of the four components of the simulation: the autoclave, the bleeder, the laminate or a lamina. Each entry line consists of a key word, followed by an equal sign character '=' the list of properties, separated by comma characters ',' and is terminated by a semi-colon character ';'• The various available properties for each component are given in section A.9. Each property's argument type, if applicable, is also given. An example of a completed properties list for a 24 ply laminate using the sequential flowModel in the correct format is; autoclave= interfacePressure, resinPressureSequential, 6, 193 resinPressureSequential, 12, resinPressureSequential, 18, externalPressure; bleeder= temperature, viscosity; laminate= viscosity, min, viscosity, max, degreeOfCure, min, degreeOfCure, max, mass, thickness; lamina= resinVolumeFraction, 0, resinVolumeFraction, 6, resin VolumeFraction, 12, resin VolumeFraction, 18, resin VolumeFraction, 23; Of the four types of properties that can be specified for output, arguments are needed by one autoclave property, about half the laminate properties and all the lamina properties. Bleeder properties and most autoclave properties do not require any arguments. There are two types of property arguments; integer index arguments and statistical arguments (the latter are only required for laminate properties). Integer index arguments identify a particular lamina (by its location in the laminate). The laminae are numbered from 0 (zero) for the lamina adjacent to the laminate/bleeder interface to n-1 for the lamina adjacent to the toolplate. Statistical arguments describe how LamCure calculates the desired property of the laminate. There are three statistical arguments: min for minimum, max for maximum and avg for average, min and max return the minimum and maximum value for that property from all of the laminae in the laminate, avg returns the number average of that property. Note that the property and the argument must be separated by a comma character ','• Also note that some properties are only defined for a particular flow model; either the sequential compaction or the squeezed sponge flow model. The tables in section A.9. indicate for which flow model a particular property is defined. A.6. Example Input Files The following two input files are meant to be examples of the file structure described in the previous sections. The first file is for the sequential compaction flow model and the second for the squeezed sponge flow model. In the next section is sample output for each output file defined in their input files for the first four time steps. Each file describes a 16 ply laminate where the material system is Hercules AS4 fibers and 3501-6 resin with Mochburg bleeder cured using the manufacturers recommended cure cycle. Figures 8, 9 and 10 are plots of the schedules specifies in each of the input files. Note that the temperature cycle, pressure cycle and material systems for both input files are identical. The file SEQUENT.INP uses the built in keywords for these materials whereas SPONGE.INP uses their generic counterparts to achieve the same end. 194 Figure A-8 is a plot of the toolplate, interface and bleeder temperature (these schedules are exactly the same) as a function of time. Figure A-9 is a plot of the external pressure and vacuum pressure as a function of time. Figure A-10 is a plot of the void ratio as a function of the fiber bed pressure. This last figure is only used in the SPONGE.INP file since the [voidRatioVsFiberPressure] schedule has no meaning for the sequential compaction flow model. SEQUENT.INP [toolplateTemp] 294.1,2550; 388.5,4200; 388.5, 1650; 449.7, 7200; 449.7, 1800; 366.3, 3600; [interfaceTemp] 294.1,2550; 388.5,4200; 388.5,1650; 449.7, 7200; 449.7, 1800; 366.3, 3600; [bleederTemp] 294.1,2550; 388.5,4200; 388.5,1650; 449.7, 7200; 449.7, 1800; 366.3, 3600; [vacuumPressure] 3387.0, 6749; 3387.0, 1; 101300, 14250; [externalPressure] 687400.0, 6749; 687400.0, 1; 790800.0, 10649; 790800.0, 1; 101300.0, 3600; [timeStep] 15, 59; 15,1; 60, 20940; 195 [execution] parallelFlow=false; perpendicularFlow=true; flowModel=sequential; sequentialCompactionModel=continuous mixing=laminar; checkDiagonalDominance=true; permeabilityModel=constant; thermalModel=isothermal; [bleeder] MochburgBleeder, 0.3, 0.3; [fiber] AS4; [resin] R35016; [laminate] 16, 0.3, 0.3, 0.0235, 0.42, 0.2474, 5.8e-16, 0.01, circular, 0, 294.1; [fileOutput] {116spri.opt} printEvery=60; autoclave= interfacePressure, interfaceResinPressureSequential, 4, interfaceResinPressureSequential, 8, interfaceResinPressureSequential, 12, externalPressure; bleeder= resinMass; laminate= viscosity, min, viscosity, max, degreeOfCure, min, degreeOfCure, max, mass, thickness; [dynamicDataExchange] {LamCure,CureData} ddeEvery=60; autoclave= interfacePressure, interfaceResinPressureSequential, 4, interfaceResinPressureSequential, 8, interfaceResinPressureSequential, 12, extemalPressure; bleeder= resinMass; laminate= viscosity, min, viscosity, max, degreeOfCure, min, degreeOfCure, max, mass, thickness; [end] SPONGE.INP [toolplateTemp] 294.1,2550; 388.5, 4200; 388.5, 1650; 449.7, 7200; 449.7, 1800; 366.3, 3600; [interfaceTemp] 294.1,2550; 388.5, 4200; 388.5, 1650; 449.7, 7200; 449.7, 1800; 366.3, 3600; [bleederTemp] 294.1,2550; 388.5, 4200; 388.5, 1650; 449.7, 7200; 449.7, 1800; 366.3, 3600; [vacuumPressure] 3387.0, 6749; 3387.0, 1; 101300, 14250; [extemalPressure] 687400.0, 6749; 687400.0, 1; 790800.0, 10649; S-'S o 2. 1 3 o 3/ £**< ii. if 00 Q -O CD l=variabl thermal; o 3 -O 5 P Crq o 3 P 5 O 3. 5 P 3 O II 3 B. 3' CTP II p * 3 3 3 a ^ o 2 * >3 2S dicu odel II sr 3 o p. 1 c X3 fi? p £ i CD II X"? p 00 ct> o c k ^ > o 3 ^ o 45. i—» LO H-* H-* ~ ' o 45k 1—» OS o Lrt LO l o 45k 45k ON LO bo O 45k to O L/l • — » 0 0 l o 45k 45k O N 45k to o 45k to 45. 45k •—* V O l o 45k 45k ON 45-lo O 45. to 0 0 45k ON l o 4^ 4^ ON 45. lo o 45k LO to Ov - J l o 45k 45k O N 45. Lo O 4*. LO ^ 1 o u> l o *>. 45k ON 4*. lo o 4^. 4^ K-* ON h-» JO l o 45k 45k ON 45. lo O 4*. 45. O N LO 0 0 - J l o 45k 4*. O N _ * > > Lo O '•P>. L/l k—1 LO • - J oo "Lo 45k 45k ON 45k to O 45k LA ON ON k—» l o 45k 45k ON 45k Lo o o o o o o 45k 45k *45k 45k 45k 45k O N ON - J Oo 0 0 vO t o - J 45k O ^ 1 45k H-> v o O L/i 45k o o H-i >-» 45k 45k ~ J ~-J **• y y> „°° ^  i ° l o LO LO LO l o LO 45k 45k 45k J5k 45k 45k 45k 45k 45k 45k 45k 45k ON ON ON ON ON ON 45k 45k 45k 45k 45k 45k Lo Lo Lo Lo Lo to O L/i o to oo to LO 45k 45k O N 45k Lo O L/i 1—1 k—* 45k O 9° LO 45k 45k O N 45k Lo O U\ to o - J 45k to LO 45k 45k O N 45k Lo p p p p p p l/i l/i L/i L/i {j\ ON L O 45k U i O N o o o O t o 45k v o L/» Lfi VO ( O VO t o ~ J LO ON ON O t o ON t o ON 0 0 LO - O LO H-> LO LO LO LO LO LO 45k 45k 45k 45k 45k 4 > 45k 45k 45k 45k 45k S ON ON ON ON ON ON 45k 45k 45. 45. 4S. j ^ Lo Lo Lo to Lo Lo o O N to vo to u> LO 45k 45k O N 45k Lo O P O N < 1 ON O P LO • - ON h - * J—* LO LO 45k 45k 45k 45k O N O N 45k _45k Lo to o o O N - • *-J £; oo 70 k_4 P tioV ,689 00 T1 In cr erPressure 7; O N >—• O L/i l; 20940; • — » L/I L/l r—i 3 o> 0 0 ft k—» P L O o p b L O ON o o - J vO o 0 0 o p p k _ _ ^ 0) c VO oo [bleeder] GenBleeder, 0.3, 0.3, 5.6e-ll, 0.57; [fiber] GenFibers, 1799.0, 0.721, 26.0e-3, 26.0e-3, 8.4e-6; [resin] GenThermoset, 1220.0, 1.26, 1.67e-4, 474.0, 80.7, 77.8, 56.6, 90.8, 3.5017e7, -3.3567e7, 3.2667e3, 7.93e-14, 0.47, 14.1, 0.3; [laminate] 16, 0.3, 0.3, 0.0235,0.42, false, false, false, circular, 0, 294.1; [fileOutput] {116spong.opt} printEvery=60; autoclave= vacuumPressure; bleeder= resinMass; laminate= viscosity, min, viscosity, max, degreeOfCure, min, degreeOfCure, max, mass, thickness; lamina= interfaceResinPressure, 3, interfaceResinPressure, 7, interfaceResinPressure, 11, interfaceResinPressure, 15; [dynamicDataExchange] {LamCure,CureData} ddeEvery=60; autoclave= vacuumPressure; bleeder^ resinMass; laminate= viscosity, min, viscosity, max, degreeOfCure, min, degreeOfCure, max, mass, thickness; lamina= interfaceResinPressure, 3, interfaceResinPressure, 7, interfaceResinPressure, 11, interfaceResinPressure, 15; [end] A.7. Example Output Files and a Comment on DDE Links The following two output files are the results for the first four time steps of the SEQUENT.INP and SPONGE.INP input files of the previous section. Note that the first line is always an alphabetic string that contains the properties sent to the file. The names of the individual properties are delimited by commas and the string is terminated by a carriage return. The order of the property names in the alphabetic string is identical to the order of the properties in the numeric strings, also delimited by commas. Note that if the property specified in the input file required an argument, that argument immediately follows the property name and is enclosed in square brackets, [ ]. This type of output can be read into a spreadsheet such as MS-Excel where it can be parsed into columns for analysis. DDE links follow the same format as output files. The properties to output to the DDE links specified in the SEQUENT.INP and SPONGE.INP files are exactly the same as for the output files. The first string passed to the DDE client is the alphabetic string and each string thereafter is a numeric one. If the DDE link is terminated and later re-established, LamCure sends the alphabetic string again. Once the client has received the DDE data it is up to the client application to process it. L16SPRI.OPT time,interfacePressure,interfaceResinPressureSequential[4],interfaceResinPressureSequential[8], interfaceResinPressureSequential[12],externalPressure,resinMass,viscosity[min],viscosity[max], degreeOfCure[min],degreeOfCure[max],mass,thickness 0.,3405.93399,687400.,687400.,687400.,687400.,9.87e-005,1063.275814,1063.275814,0.,0.,0.3759013,2.784273588e-003 60.,3405.93399,687400.,687400.,687400.,687400.,1.661033739e-003,862.6586382,862.6586382,4.806390426e-006,4.806390426e-006,0.3743389663,2.770044683e-003 120.,3405.93399,687400.,687400.,687400.,687400.,2.042824235e-003,805.0464307,805.0464307,1.032372675e-005,1.032372675e-005,0.3739571758,2.766567539e-003 180.,3405.93399,687400.,687400.,687400.,687400.,2.376971052e-003,612.0226092,612.0226092,1.619082929e-005,1.619082929e-005,0.3736230289,2.763524307e-003 200 L16SPONG.OPT time,vacuumPressure,resinMass,viscosity[min],viscosity[max],degreeOfCure[min],degreeOfCur e[max],mass,thickness,interfaceResinPressure[3],interfaceResinPressure[7],interfaceResinPressu re[ 11 ] interface ResinPressuref 15] 0.,3387.,0.,1063.41141,1063.41141,0.,0.,0.376,2.785172495e-003,687400.,687400.,687400.,687400. 60.,3387.,7.156735807e-003,862.76803,862.76803,4.806390426e-006,4.806390426e-006,0.3688432642,2.719992752e-003,685426.4629,687399.3673,687399.9999,687400. 120.,3387.,1.050179189e-002,805.1483255,805.1483255,1.032372675e-005,1.032372675e-005,0.3654982081,2.68952776e-003,676776.2285,687346.0515,687399.8066,687399.9992 180.,3387.,1.30685565e-002,612.0994965,612.0994965,1.619082929e-005,1.619082929e-005,0.3629314435,2.666151033e-003,665273.4463,687067.6292,687397.6184,687399.983 A.8. Nomenclature and Units of the Input Quantities The following table gives the units for the quantities required as inputs for LamCure. nable A-3: Units of the Input Quantities. Quantity laminate/bleeder interface temperature as a function of time toolplate temperature as a function of time bleeder temperature as a function of time external pressure as a function of time vacuum pressure as a function of time void ratio as a function of applied stress time step as a function of time length of bleeder width of bleeder apparent permeability of bleeder porosity of bleeder fiber specific heat fiber density fiber thermal conductivity parallel to fiber direction Symbol TIF=TIF(t) *TP TP^ ' Tb=T„{t) Pext=PexX0 P.ac=PVatf) e = f(pf) At=f(t) Lb wb sb zb c,, Pf Ju Units K K K Pa Pa none s m m m2 none U/(kgK) kg/m3 kW/(m K) Valid Flow Model(s) both both both both both Sponge both Sequential Sequential Sequential Sequential both both both fiber thermal conductivity perpendicular to fiber direction fiber diameter resin specific heat resin thermal conductivity resin density resin heat of reaction viscosity equation activation energy viscosity equation constant viscosity equation pre-exponential factor cure rate equation activation energy cure rate equation constant cure rate equation pre-exponential factor cure rate equation transition degree of cure number of plies length width ply mass initial resin mass fraction compacted resin mass fraction apparent permeability of prepreg parallel flow coefficient of prepreg fiber packing initial degree of cure initial temperature * * Df cPr K Pr HR u K V-AEi,i = 1,2,3 B V = 1,2,3 C n L W m 'uut Mrc sP P FP <Xinit T. mit kW/(m K) m kJ/(kgK) kW/(m K) kg/meter3 kJ/kg kJ/mole none Pas kJ/mole none 1/s none none m m kg none none m2 none parallel, series, square, circular, hexagonal none K both Sponge both both both both both both both both both both both both both both both both Sequential both Sequential both both both A.9. Output Properties Tables A-4 to A-9 give all properties for each component to which the simulation may respond. Some properties only apply to one of the flow models and this is indicated where appropriate and any arguments they may require. A.9.1. Autoclave Properties Table A-4: Autoclave Properties, Required Arguments, Units and Valid Flow Model(s). Property interfaceTemp toolplateTemp bleederTemp externalPressure vacuumPressure interfacePressure interfaceResinPressureSequential resin VelocitySponge resinVelocitySequential Symbols T T Tb Pex, irvac PlF ^ tf22/ ^ Units K K K Pa Pa Pa Pa m/s m/s Valid Arguments none none none none none none 0 -^ n-1 0 - > « - l none Valid Flow Model (s) both both both both both Sequential Sequential4 Sponge Sequential A.9.2. Bleeder Properties ' ^able A-5: Bleeder Properties, Units and Valid Flow Model(s). Property area cureRate degreeOfCure length permeability2 porosity resinDensity resinHeight resinMass resinOrigin Symbols \ \<k)b ab h s„ £b Pr K mr ©, Units m2 1/s none m m 2 none kg/m3 m kg none Valid Arguments none none none none none none none none none none Valid Flow Model(s) both both both both both both both both both both interface resin pressure for the squeezed sponge flow model is a lamina property. 203 resinType resinVolume temperature viscosity viscositylntegral width vr Tb Vb vrb wb none m3 K Pas P a s m m none none none none none none both both both both Sequential both A.9.3. Laminate Properties Table A-6: Laminate Properties, L Property compactedLamina compactedVolume compactedResinVolume compactedThickness cureRate degreeOfCure density excessResinMassSequential fiberMass fiberMassFraction fiberPacking fiberType fiberVolumeFraction length mass permeability25 resin Mass resin MassFraction resinType fnits, Required Arguments and Valid Flow Model(s). Symbols " c \ \ K ( * ) , a, Pi mr txctssl it Mf, F», Vf, A ml \> Mn Units none m3 m3 m 1/s none kg/m3 kg kg none none m kg m2 kg none Valid Arguments none none none none min,max,avg min,max,avg none none none min,max,avg none none min,max,avg none none none none min,max,avg none Valid Flow Model(s) Sequential Sequential Sequential Sequential both both both Sequential both both both both both both both both both both both 5 permeability perpendicular to the fiber direction. resin VolumeFraction temperature thickness viscosity viscositylntegral volume width Vn T, I, Hi vi, vi w, none K m P a s P a s m m 3 m min,max,avg min,max,avg none min,max,avg none none none both both both both Sequential both both A.9.4. Lamina Properties 'able A-7: Lamina Resin Collection Properties, Property cureRate degreeOfCure heatOfReaction heatGenerationRate resinDensity resinMass resinOrigin resinSpecificHeat resinThermalCond resinType resinVolume viscosity Symbols da dt a H« HGR Pr mr © r c>. K Vr V Units, Arguments and Valid Units 1/s none kJ/kg kJ/s kg/m3 kg none kJ/(kgK) kW/(m K) m 3 P a s Valid Arguments 0—»« —1 0 - > « - l 0—>«—1 0 - » w - l 0 - > » - l 0 - » « - l 0 - » » - l 0—>« —1 0 - » « - l 0 - » » - l 0 - » » - l 0 - » 7 l - l Flow Model (s) Valid Flow Model(s) both both both both both both both both both both both both 'able A-8: Lamina Fiber Properties Property fiberDensity fiberDiameter fiberMass fiberSpecificHeat fiberThermalCondl fiberThermalCond2 fiberType , Units, Arg Symbols Pf Df mf cPf */„ kf» uments and Valid Flow Moc Units kg/m3 m kg kJ/(kg K) kW/(m K) kW/(m K) Valid Arguments 0 - » w - l 0 - » « - l 0 - » « - l 0 —» /7 - 1 0 - » « - l 0 - » w - l 0 - » / / - ! el(s). Valid Flow Model(s) both both both both both both both fiberVolume vf m3 0 - * / ? - ! Table A-9: Lamina Properties, Units, Arguments and Valid Flow Model(s). Property area channelThickness compactedFiberMassFraction compactedFiberVolumeFraction compactedMass compactedResinMassFraction compactedResinVolumeFraction compactedThickness compactedVolume compressibilityCoeff compactCoeff density excessResinMassSequential excessResinVolumeSequential fiberMass fiberMassFraction fiberPacking fiberPressure fiberVolumeFraction hydraulicRadius initialResinVolumeFraction interfaceResinPressure kozenyl kozeny2 length mass parallelFlowCoeff parallelResinVelocity permeability 1 Symbols A \han ML \ ™c Mh \ K vc fl. CC P mr excess vr excess mf M, FP Pf Vf rH 'uta PL, Ku ^ 2 2 L m P A sP Units m2 m none none kg none none m m3 1/Pa m kg/m3 kg m3 kg none Pa none m none Pa none none m kg none m/s m2 Valid Arguments 0-^n-l 0-*«- l 0->«-l 0->w-l 0->«-l 0->«-l 0->«-l 0->w-l 0->«-l 0->«- l 0->«-l 0->»-l 0->«-l 0->«-l 0->«-l 0 - W 7 - 1 0-^n-l 0-^n-l 0->«-l 0->«-l 0-»w-l 0-»w-l 0->»-l 0->«-l 0-»«- l 0 - * / ? - l 0 - » w - l 0 - » w - l 0 - » w - l both Valid Flow Model (s) both Sequential Sequential Sequential Sequential Sequential Sequential Sequential Sequential Sponge Sponge both Sequential Sequential both both both Sponge both Sponge Sequential Sponge Sponge Sponge both both Sequential Sequential both permeability2 porosity resinMassFraction resinPressure resinRatio resin VolumeFraction specificHeat temperature thermalCondl thermalCond2 thickness viscositylntegral voidRatio volume volumeChangeCoeff width e Mr Pr er vr cP T Ku K22 X VI e V mv W m2 none none Pa none none kJ/(kgK) K kW/(m K) kW/(m K) m P a s m none m 3 1/Pa m 0 - » « - l 0 - > « - l 0 - > w - l 0 - > « - l 0-^n-l 0-^rt-l 0 - > 7 7 - l 0->77~l 0->«-l 0 - > 7 7 - l 0-^n-l 0 - > 7 7 - l 0 - ^ 7 7 - 1 0 —> « — 1 0 - > 7 7 - l 0 - ^ 7 7 - 1 both both both Sponge both both both both both both both Sequential Sponge both Sponge both A.10. List of Errors Errors that will occur while running LamCure, which we call user errors, will typically be due to incorrect syntax in the input file. Figure A-11 shows a typical error dialog box. The caption consists of three parts. The first part is "LamCure:" which identifies the error as a LamCure error. The second and third parts of the caption, in this example "CurelnputFile Error:" and "buildBleederProp" respectively, provide the names of the routines where the error occured. The message displayed in the main dialog box is what will be informative to the user. The message describes, in general terms, what error has occurred. This will usually be a DOS error telling the user that the file was not found on the specified drive or something to that effect. All error messages are listed below with a comment on what each specific error means. The second kind of error that will sometimes occur is what we call an internal error. This type of error should not occur, but debugging code has been left in the code to ensure that the program is functioning as expected. If an internal error occurs, a serious and unexpected condition has been encountered. The list below specifies whether an error is a user error or an internal error. Autoclave boundCondOK user The <aSchedule> is not defined for times longer than <aTime> seconds.The model time during the next time step will be <aTime + aTimeStep> seconds. One of the boundary conditions, temperature or pressure schedules does not have a value for the next simulation time. 207 calculateNextResinPressure user The pressureMatrix matrix is not diagonally dominant. Check the boundary conditions and the size of the time steps. Something physically meaningless is occurring in the resin flow model. If the state of the system is changing rapidly smaller time steps may be required. calculateNextTemperature user The temperatureMatrix matrix is not diagonally dominant. Check the boundary conditions and the size of the time steps. Something physically meaningless is occurring in the heat transfer model. If the state of the system is changing rapidly smaller time steps may be required. massFlowRateOutOfLaminate internal There is no resin in the bleeder to prevent division by zero in the massOutOfLaminate method moveResinMassSponge user/internal I haven't figured out how to do parallel flow for the squeezed sponge resin flow model yet. The parallelFlow keyword under the [execution] heading is equal to true and theflowModel keyword is equal to sponge. LamCure currently does not support parallel resin flow for the Sponge flow model. CurelnputFile buildAutoclaveProp user The <aArgument> argument of the autoclave property <aProperty> is not a valid argument. The argument type for that property is incorrect. buildAutoclaveProp user <aProperty> is not a valid autoclave property. The property specified for output is not an autoclave property. buildBleederProp user <aProperty> is not a valid bleeder property. The property specified for output is not a bleeder property. buildCureOutputFile user <aComponent> is not a valid [fileOutput] sub-heading in the input file. There are five possible sub-headings under the [fileOutput] heading in the input file: printEvery, autoclave, bleeder, laminate and lamina. buildDDEServer user <aComponent> is not a valid [dynamicDataExchange] sub-heading in the input file. There are five possible sub-headings under the [fileOutput] heading in the input file: ddeEvery, autoclave, bleeder, laminate and lamina. buildExecution user <aState> is not a valid flowModel choice. Valid choices are sponge and sequential. There are two states for flowModel: sponge and sequential. 208 buildExecution user <aState> is not a valid mixing choice. Valid choices are mix and laminar. There are two states for mixing: mix and laminar. buildExecution user <aState> is not a valid thermalModel choice. Valid choices are calculate and isothermal. There are two states for thermalModel: calculate and isothermal. buildExecution user <aControl> is not a valid [execution] entry in the input file. There are seven sub-headings under the [execution] heading in the input file. All are required. These are: parallelFlow, perpendicularFlow.flowModel, mixing, checkDiagonalDominance, permeabilityModel and thermalModel. buildLaminaProp user The <aArgument> argument of the lamina property <aProperty> is not a valid argument. The argument type for that property is incorrect. buildLaminaProp user <aProperty> is not a valid lamina property. The property specified for output is not a bleeder property. buildLaminateProp user <aArgument> is not a valid laminate property. The list of acceptable laminate properties is given in Section A8. buildLaminateProp user The <aArgument> argument of the laminate property <aProperty> is not a valid argument. Valid arguments are: min, max and avg. The list of acceptable arguments for laminate properties is given in Section A.8. buildLaminate user Wrong number of parameters in laminate description. There are 11 parameters to describe a laminate of identical laminae and 10 parameters to describe a laminate of unique laminae. parselnputText user <aHeading> is not a valid heading in the input file. The <aHeading> is not one of the possible headings. There are fifteen possible headings for an input file. These are: [toolplateTemp], [interfaceTemp], [bleederTemp], [vacuumPressure], externalPressure], [timeStep], execution], [voidRatioVsFiberPressure], [bleeder], [fiber], [resin], [laminate], [dynamicDataExchange], fileOutput] and [end]. readFileStrm user Cannot read <aFile> Something is wrong with the input file <aFile> that does not allow LamCure to read its contents. 209 dosError user CureOutputFile("<aPath\aFile>") reported DOS error#<aNumber>, <aCondition>. The messages that are displayed in <aCondition> state what is wrong with the file I/O. CureOutputFile dosError user CureInputFile("<aPath\aFile>") reported DOS error#<aNumber>, <aCondition>. See CurelnputFile dosError above. Cure Window nextState user flowModel is not a possible value in the nextState method. TheflowModel keyword under the [execution] heading is not equal to sponge or sequential. Lamina fiberPressure user/internal The calculated aFiberPressure is less than zero. Check the void ratio as a function of fiber pressure schedule. Or make the time steps smaller for this time. tc2CircularFiberHexagonalPack user/internal The fiber volume fraction of the lamina is greater than 68.02%.Physically this Vf is meaningless for circular fibers in a hexagonal packing array. Either, for the Sponge flow model, the void ratio as a function of fiber pressure defines a void ratio that is too low or, for the sequential compaction flow model, a compacted resin mass fraction that is too low has been specified that results in a fiber volume fraction that is too high than the theoretical maximum for this fiber packing geometry. tc2CircularFiberSquarePack user/internal The fiber volume fraction of the lamina is greater than 78.54%.Physically this Vf is meaningless for circular fibers in a square packing array. Either, for the Sponge flow model, the void ratio as a function of fiber pressure defines a void ratio that is too low or, for the sequential compaction flow model, a compacted resin mass fraction that is too low has been specified that results in a fiber volume fraction that is too high than the theoretical maximum for this fiber packing geometry. voidRatio user/internal The calculated aFiberPressure is less than zero Pa. Check the void ratio as a function of fiber pressure schedule. Or make the time steps smaller for this time. 210 Laminate compactedThickness Specified lamina is not in the compacted zone lamFlowThruCompactedLamina A resin collection was not passed to the lamFlowThruCompactedLamina method. partialViscositylntegral The specified interface is not within the compacted zone. perpendicularResinFlowSponge The pGrad temporary variable is negative. The idx is equal to "an index of a lamina" perpendicularResinFlowSequential A negative value for mass was passed to this method Object error An internal error has occurred. The user should never see this error. If it appears, examine the input file closely to see if the error might be caused by an unrealistic physical condition. Failing that please send the authors a copy of the input file and the simulation time that this error occurred so we can try to figure out what happened. fail Make corrections to the input file and re-open it LamCure is unable to interpret the input file due to a previous error. primError A primitive internal error has occurred. See Object error above. PermeableLayer removeResin There is not enough resin mass in the layer to remove massFlow. ResinCollection addResin rColl class is not ResinCollection. removeResinLaminar aMass does not have the Number class as an ancestor. removeResinMix aMass does not have the Number class as an ancestor. internal internal internal internal internal internal user/internal internal internal internal internal internal 211 subtract resinOrigin does not contain any resin. The resinOrigin resinCoUection is empty. Schedule durations um Method tried to sum over more regions than are available in the schedule. The sum of all durations is: <aTime> seconds. linearlnterpolate The schedule does not contain 2 timeltem objects to interpolate between. findDuration A negative time was passed. A negative value for the time duration of a time-item object in a schedule is specified in one of the schedules. Thermoset setDegreeOfCure Degree of cure greater than 1 or less than zero. Old degree of cure = <oldDegreeOfCure> New degree of cure = <newDegreeOfCure> Check the constants that describe the cure rate as a function of degree of cure and temperature. A.11. Figures internal internal internal user/internal user Figure A-1: LamCure's main window. 212 Directory: e:\lamcure Files: Directories: E3fT7Ts^ dave.inp dde2dave.in dde2spri.inp Iam2dave.in Iam2spri.inp springer.inp Op*™ Figure A-2: The File Open dialog. Close current output file(s) and DDE serverfs]? :0&W$MMW Figure A-3: The File Close dialog. stopTime = | g j iM""mj"~"'-4 ", -Cancel Figure A-4: The RunUntil dialog. Resetting means that any output file(s) and/or DDE conversations will be over-written and re-initiated. mtwwm .:-;:x<^;:*±:±-3 fmsmssm mmm*4 Figure A-5: The Reset! dialog. Terminate DDE conversation(s) of any and all serverjsj? Figure A-6: The Terminate DDE Links dialog. 500 480 460 440 •.£ 420 400 E 380 360 340 320 300 2000 4000 6000 8000 time (sec.) 10000 12000 14000 Figure A-7: The temperature schedule represented by the time-item objects. 214 500 450 400 350 I 300 250 5000 10000 15000 time (seconds) 20000 25000 Figure A-8: The toolplate, interface and bleeder temperature cycle. 800000 700000 600000 500000 400000 300000 200000 100000 \ *| \ \ '• ] \ . . . .i external vacuum / / ^ 1 1 1 1 1 1 I 1 5000 10000 15000 time (seconds) 20000 25000 Figure A-9: The external and vacuum pressure cycle. 215 1.1 0.9 0.8 0.7 0.6 0.5 0.4 200000 400000 600000 800000 fiber pressure (Pascal) 1000000 1200000 Figure A-10: The void ratio as a function of fiber bed pressure. > 7( LamCure: iCureltiptRJe ErroribmiBi&^MrBiop^ mass is not a valid bleeder property. >^MM Figure A-11: The Error dialog. 216 A.12. Conversion Formulae The following formulae are relationships that are useful for converting between the various quantities used in defining the input for LamCure. e = '[ Pr V " r 1-V, e = V. V, 1-K V , 1-e e = V=-1 + e v .+v, M = • 1-K 1 + V, - 1 A/r \-Mr 52L + 5 P/ •^ where v,mV = W L A ^ M = • Pr Vf = , P Mf 1+ Hr  PfMr Vf=- l + e Pressure l a t m = 101325 Pa 1 arm = 14.696 psia 1 atm = 760 mm Hg 1 atm = 29.92 in Hg 1 psia = 6894.733 Pa 1 in Hg = 3386.5 Pa 100 psig = 689473.3 Pa 0 psig = 1 atm Temperature C = 5/9*(F - 32) K = C + 273.15 Gas Constant R = 8.314 J/(moleK) R = 8.314(10-3) kJ/(moleK) R = 1.987 cal/(moleK) 217 Appendix B: Calculation of Additional Properties This appendix lists the derivations for those properties that are not key to the description of the models in Chapter 3. B.1. Material Properties The assumptions in the derivations of the properties are: 1. There are only two constituent materials in the lamina: resin and fiber. 2. The fiber is assumed to be stationary and only resin flows from lamina to lamina, i.e., the mass of the fiber in any lamina is constant. 3. The existence and effect of voids or other foreign materials in the prepreg and laminate are negligible. 4. The physical properties of the fiber, resin, prepreg and bleeder are not affected by temperature, pressure, the resin's degree of cure or the presence of resin and/or fiber. B.2. Calculation of Lamina Properties In the context of the simulation, a lamina consists of a resin collection, where the specific resin masses in the collection may change over time due to resin flow, and fiber. The density of the lamina, p, is the ratio of its volume to its mass. m P = - (B-l) v The fiber volume fraction of a lamina, Vf, is the ratio of the fiber volume to its total volume. Vf=^- (B-2) v 218 The resin volume fraction of a lamina, Vr, is the ratio of the resin collection volume to its total volume. Since a lamina only contains resin and fiber, the resin volume fraction is calculated from its fiber volume fraction. Vr=l-Vf (B-3) The fiber mass fraction of the lamina, Mf, is the ratio of its fiber mass to its total mass. Since a lamina only contains resin and fiber, the resin volume fraction is calculated from its resin mass fraction. Mf = \-Mr (B-4) The porosity of the lamina, £, is the same as its resin volume fraction, Vr. £ = Vr=^- (B-5) v Traditionally, porosity is a term used by soil scientists to denote the volume fraction of a material. Both terms are include since the squeezed sponge resin flow model [3,4] defines some of its parameters in terms of porosity. The resin ratio of a lamina, er, is the ratio of its resin collection volume to its fiber volume. er=-^ (B-6) v / Comparing this equation to the calculation of the void ratio, e, in Chapter 3, equation (3-86), one can see that they are the same. The reason for defining the resin ratio is because the squeezed sponge resin flow model requires, as an input parameter, its void ratio as a function of its fiber bed pressure, e = f(pf). Since the sequential compaction resin flow model does not use such a function, the resin ratio was defined as a means of calculating the void ratio of a lamina for the sequential compaction model. The following three properties are only defined for the sequential compaction resin flow model and are not applicable to the squeezed sponge resin flow model. The compacted fiber mass fraction of a lamina, Mf , is calculated as follows; Mft=l-Mrc (B-7) The compacted mass of a lamina, mc, is calculated as follows; 219 mc=-f- (B-8) Mf< The excess resin mass of a lamina, mr , is the difference between its current resin mass and its compacted resin mass. The calculation is as follows; mr =m— mr (B-9) 'excess B.3. Calculation of Laminate Properties With the exception of the density, each laminate property is calculated by one of four formulae given in the following sub-sections. The properties that are calculated by each formula are given in each sub-section in an accompanying table. The formula used to evaluate each property is obtained by replacing the < aProperty > symbol in the formula with the specific property symbol from the table. In the formulae that show the method the calculation of these laminate properties the subscript i refers to the lamina and a / refers to the laminate. A laminate is composed of n laminae which are numbered from 0 to n — 1. The 0"" lamina is adjacent to the laminate/bleeder interface and rih - 1 lamina is adjacent to the toolplate. B.3.1. density The density of the laminate, p,, is calculated by taking the ratio of its mass to its volume, i.e. P / = — (B-10) B.3.2. Properties that Return a Statistical Value: Minimum, Maximum and Average Statistical arguments describe how the simulation calculates the laminate property. In the equations that follow, the symbol < aProperty >, refers to any of the properties listed in Tables B-1 & B-2 below. There are three statistical arguments: min for minimum, max for maximum and avg for average, min and max return the minimum and maximum value for the property evaluated over all of the laminae in the laminate, avg returns the number average value for the property evaluated over all of the laminate's laminae as follows; < aProperty >, = — ^ < aProperty > • (B-11) n =0 The given property is calculated for each lamina and the appropriate value returned. Table B-1 lists all the properties of a laminate that require statistical arguments. 220 Table B-l: Laminate Properties that Require a Statistical Argument. Property cureRate degreeOfCure fiberMassFraction fiberVolumeFraction heatGenerationRate Symbol ( * ) , a, Mf, Vf, HGR, Property permeability2 resinMassFraction resin VolumeFraction temperature viscosity Symbol \> Mn \ T, P-i There are three other types of properties that do not require an argument. These properties are calculated by evaluating the property for all the lamina in the laminate and then calculating either their sum, their maximum value or a set of the lamina values. B.3.3. Properties that Return a Sum of Values Laminate properties that are calculated by summing the value of the property evaluated for each lamina in the laminate are listed in Table B-2. For example, the mass of the laminate is the sum of the individual masses of the lamina. The general formula is as follows; i=n-l <aP roper ty>,= ^j<aProperty>i (B-12) (=0 "able B-2: Laminate Prop Property compactedVolume excessResinMass Springer1 fiberMass mass erties that are Sums of Lamina Properties. Symbol \ mr mf ii ml Property resinMass thickness volume Symbol mn *>, vi B.3.4. Properties that Return a Set of Values These are laminate properties for which the only important characteristic is membership in a set. For example, the fiber packing of a lamina, Fp, has one of five possible values: series, parallel, square, circular and hexagonal. Initially the set is empty. The fiber packing of each lamina is evaluated. If the value is already in the set it is not added to the set. However, if the value is not in the set it is added to the set. The set which is returned has all possible values of fiberPacking for the laminate. Each set is constructed for the various properties in a similar manner. The properties are listed in Table B-3 below. Note that excessResinMassSpringer is only defines for the sequential compaction resin flow model. 221 '. 'able B-3: Laminate Properties that are Sets of Lamina Properties. Property fiberPacking fiberType Symbol F, Property resinType Symbol B.3.5. Properties that Return the Maximum Value There are three properties of a laminate that return the maximum value for the laminate. The properties are listed in Table B-4 below. The property is evaluated for each lamina in the laminate and only the maximum value is returned. Table B-4: Laminate Properties that Return a Property length width Symbol A w, Maximum Value. Property area Symbol 4 222 Appendix C: Derivation of Lamina Thermal Conductivity Expressions This appendix gives the details of the derivations of the expressions used to calculate the thermal conductivity of a lamina. The parallel and series expressions are straight forward derivations based on series and parallel resistive heat flow, as given by many authors [44,55,56]. The expressions for square and circular fibers in a square packing array are based on reference [44]. Their expression for circular fibers in a square packing array was stated incorrectly but the correct expression is given in this work. The final expression is for circular fibers in a hexagonal packing array which is our extension to the work in reference [44]. The thermal conductivity of a lamina is a function of several factors: the direction of interest (11,22,33), the ratio of resin to fiber (the fiber volume fraction, Vf, is a convenient measure), the thermal conductivity of the resin, kr, and the fiber, kf__, and the geometric arrangement of the fiber in the resin, Fp. i.e., ku = f(vf,kfu,kr,FP) (C-l) The expressions used for calculating ku are taken from Springer & Tsai [44]. Their basic assumption is that the combined conductivity of resin and fiber is adequately described by a combination of thermal resistances in series and/or parallel. Their additional assumptions are: 1. The composite is macroscopically homogeneous. 2. Locally the matrix is homogeneous and isotropic and the fiber is homogeneous and transversely isotropic with respect to the fiber direction. 3. The thermal contact resistance between the fiber and the matrix is negligible in the direction of heat flow and infinite perpendicular to the direction of heat flow. 4. The heat transfer problem is two dimensional, i.e., the temperature distribution is independent of x, see Figure C-l. 223 5. The packing array is symmetrical. An implication of assumption 4 is that the resin and fiber volume fractions are equal to their area fractions. Figure C-1 shows is a schematic of a lamina. There are three principal directions for heat transfer, each parallel to the x, y and z axes. The thermal conductivities in each of these directions are denoted kn, k22 and k33. Closer examination of the figure shows that the arrangement of matrix and fiber in y and z-directions is symmetric about the x-axis, therefore, the thermal conductivity in the y and z-directions are equal, k22 = k33. Figure C-2 shows the combinations of fiber shapes and packing arrangements encoded in the simulation. In Figure C-2 it assumed there is a temperature gradient of AT across the height of each cube and that area of fiber and resin in theyz-plane is Af and Ar. C.1. Thermal Conductivity Parallel to the Fiber Direction Loos & Springer use a parallel thermal model to calculate the thermal conductivities parallel to the fiber direction. The unit cube for this geometry is shown in Figure C-2(a). Since no heat can flow transversely between the fiber and resin, the total heat flux across the unit cube, <7toto,, is the sum of the individual heat fluxes across the fiber, qf, and the resin, qr. Therefore the effective thermal conductivity parallel to the fiber direction, kn can be written; , ..AJT AT A T 11 L fn f L L kn=kAyf+krvr (c-2) Grove [57] has pointed out that for the case of continuous fiber reinforcement, excellent agreement between measured and predicted thermal conductivities using equation (C-2) is achieved, due to the continuous path for heat conduction in the fiber and resin. However, Springer & Tsai [44] state that this can only be viewed as an upper bound for the true value since Thornburg and Pears [58] have measured kn and found their values to be lower than predicted by equation (C-2). C.2. Thermal Conductivity Perpendicular to the Fiber Direction The following derivations are based on specific assumptions about the geometric arrangement of the fiber and the resin. Hence the expressions obtained are limited to particular maximum values for the fiber volume fraction, Vf. The limiting Vf for each expression is listed in Table C-1. 224 Table C-l: Maximum Fiber Volume Fractions for Each Packing arrangement and/or fiber shape assumptions fiber and resin in parallel fiber and resin in series square fiber in a square packing array circular fibers in a square packing array circular fibers in a hexagonal packing array Packing Arrangement. Equation C-3 C-5 C-11 C-19 C-27 Maximum Vf 1.0 1.0 1.0 0.7854 0.6802 C.2.1. Parallel Thermal Resistances The derivation for the thermal conductivity perpendicular to the fiber direction using a parallel model and the unit cube for this geometry is the same as that given for the thermal conductivity perpendicular to the fiber direction with the exception that the fiber thermal conductivity in now kf . Thus the effective thermal conductivity can be written; k22 = kf2Vf+KK (C-3) Intuitively, one would not expect this equation to yield good values for the thermal conductivity but it is included because it is used by Mijovic and Wang [16] who found that this equation provided the best fit to their data [15]. C.2.2. Series Thermal Resistances The unit cube for this geometry is shown in Figure C-2(b). The direction of heat flow is in the positive z-direction as indicated by the arrows. Let Tv and TL be the temperatures of the upper and lower edges of the cube, respectively. Let Tinterface be the temperature of the interface between the fiber and the resin. Since there is no thermal contact resistance between the fiber and the resin the heat flux through the fiber and matrix must be equal. Since the area perpendicular to the direction of heat flow is equal for both constituents, the length fraction of fiber, Lf, and resin, Lr, are equal to their volume fractions, Vf, and , Vr, respectively. Therefore the interface temperature, Tinterface, can be calculated; k A i \ kf A , , j \ l L L interface) j \ x interface 1U) T. interface 1 + (|) kf Vr where (l) = - z s — (C-4) Noting that the total heat flow through the cube is equal to the heat flux through either the fiber or the resin portions of the cube, the expression for Tinterface can be substituted into the heat flux equation for heat flow through the fiber. 225 Qrnal = <?/ k22A (1) Canceling common terms and rearranging; " L Lf \ 1 + 0 £22= — (C-5) kV +k V C.2.3. Square Fibers in a Square Packing Array The individual fibers are assumed to have a square cross-section of length s on edge. The unit cube for this geometry is shown in Figure C-2(c). The heat flow is modelled by breaking the unit cube into three parts, as shown in Figure C-3. On either side of the center section of the cube are two identical sections of resin. The center section is broken into three sub-sections: a sub-section of resin, a square sub-section of fiber and a second sub-section of resin identical to the first. The three sections act as parallel heat conductors, therefore, the heat flux through the two resin sections, qr, can be calculated through application of Fourier's law. Ml-^AT (1) The three sub-sections of the center section act as series heat conductors. Therefore, the thermal conductivity of the center section, k22, can be calculated by equation (C-5) in the previous section. kk=—;f, (c-7) skr +{l-s)kf Hence the heat flux through the center section, q^, can be written; . ^ = ^ l ) A r ( C 8 ) The total heat flux through the cube is the sum of the heat flux through the two resin sections, qr, and the heat flux through the center section containing resin and fiber, q^. 226 = kn(l)iAT = kr(l-s)(l)AT q,otal (1) (1) hiatal ^Qr+Qrf krkf s(l)AT skr + (1 - s)kf j Hence the effective thermal conductivity, k22, can be written; (1) k22= .?,f\. +(ls)kr (C-9) skr+{l-s)kf The edge length of the fiber, s , in equation (C-9) can be expressed in terms of the fiber volume fraction of whole unit cube, Vf, as follows; V,=^p- = s2 =» s = JV; (C-10) Substituting the value for s in the above equation into equation (C-9) gives the thermal conductivity of the lamina in terms of the fiber volume fraction. C.2.4. Circular Fibers in a Square Packing Array The individual fibers are assumed to have a circular cross-section of length s on edge. The unit cube for this geometry is shown in Figure C-2(d). As in the previous case of square fibers in a square packing array, the heat flow is modelled by breaking the unit cube into three parts, as shown in Figure C-4. On either side of the center section of the cube are two identical sections of resin. The center section is broken into three sub-sections: a sub-section of resin, a disk sub-section of fiber and a second sub-section of resin identical to the first. The three sections act as parallel heat conductors and the heat flux through the two resin sections, qr, is identical to the previous case, therefore qr is given by equation(C-6). The three sub-sections of the center section act as series heat conductors. The calculation of the effective thermal conductivity for the center section, k22, is somewhat more difficult than the previous case since it requires integrating the series thermal conductivity expression, equation (C-5), for an infinitesimal slice, dy, over its width, s, as shown in Figure C-5. The fiber volume fraction, Vf, and resin volume fraction, Vr, for the slice can be written; Vf=h Vr = l - h (C-12) 227 Therefore, the effective thermal conductivity of the center section, kc22, can be expressed using equation(C-5) as; bc - . '•a r v22 hkr+(l-h)kf (C-13) The height of fiber, h, in the infinitesimal slice, dy, can be expressed as a function of y; hW=2J^-y (C-14) By exploiting the symmetry of the center section, substituting equation (C-14) into equation (C-13) and integrating the infinitesimal slices, dy, over the width of the fiber, s, the effective thermal conductivity of the center section, k22, can be written; y=2 *22 — *• J Kf K 122 r "WT-»'* V ' J •dy (C-15) Solving the integral yields; nk 4t c22 B B!-^ •tan" 2„2 1- Bzs 1 - Bs where 5 = 2 \kf J (C-16) The total heat flux through the cube is the sum of the heat flux through the two resin sections, qr, and the heat flux through the center section containing resin and fiber, q^. Q,o,al=Rr+ilrf k22(l)2AT kr(l-s)(l)AT , ,c s(l)AT (total (1) (1) + k 22 (1) Hence the effective thermal conductivity, k22, can be written; /Co^ — K y 1 o ) * K,jjb (C-17) 228 An expression for the fiber volume fraction, Vf, in terms of s , the fiber diameter, for the whole cube is required for substitution into the above equation. Examination of Figure C-2(d) reveals that the fiber volume fraction can be written; Vf = KS (C-18) Canceling the common terms and substituting equation (C-18) into equation (C-17) gives the effective thermal conductivity, k22, for the lamina in terms of the fiber volume fraction. K - tan where B = 2 Kkf J 1-5 2YL K w (C-19) In reference [44], their final equation for k22, their equation (10), is incorrect. The B term in the denominator of the tan-1 term of the above equation must be outside of the square root sign, since vB2 = B only for positive values of B. In reference [44] the B term was incorrectly placed inside the square root sign as B2. In a review paper of thermal conductivity models by Progelhof, et al. [55] they quote the incorrect equation, now their equation (32), thus making the same error. C.2.5. Circular Fibers in a Hexagonal Packing Array The individual fibers are assumed to have a circular cross-section of length s on edge in a hexagonal packing array. The unit cube for this geometry is shown in Figure C-2(e). The heat flow is modelled by breaking the unit cell into three parts, as shown in Figure C-7 (note the unit cell is not a unit cube). On either side of the center section, which only consists of resin, are two edge sections that consist of resin and fiber. The three sections act as parallel heat conductors. Consider a single edge section. The resin and fiber are assumed to act as series heat conductors. The calculation of the effective thermal conductivity for the edge section, kc22, requires integrating the series thermal conductivity expression, equation (C-5), for an infinitesimal slice, dy, over its width, s/2, as shown in Figure C-7. The fiber volume fraction, Vf, and resin volume fraction, Vr, for the slice can be written; V,=h V. = l - h (C-20) 229 Therefore, the effective thermal conductivity of the edge section, k21, can be expressed using equation(C-5) as; khJcr v22 hkr + (l-h)kf (C-21) The height of fiber, h, in the infinitesimal slice, dy, can be expressed as a function of y; m-Jj-y1 (C-22) Substituting equation (C-22) into equation (C-21) and integrating the infinitesimal slices, dy, over the width of the fiber, s/2, the effective thermal conductivity of the edge section, ke22, can be written: v=— " 2 K,= j kfkr J22 r v=0 , \S 2 , ~ ^ -dy i-V7-> (C-23) Solving the integral yields; nk 4k B & , - ^ ^tan" 1 -B2s2 1- Bs where B: (C-24) The total heat flux through the cube is the sum of the heat flux through the edge section containing only resin, qr, and the heat flux through the two edge sections, qrf. k22j3(l)AT _ kr{j3-s)(l)AT e s(i)AT itotal (1) (1) • + k. 22 (1) Hence the effective thermal conductivity, k22, can be written; 230 t 2 2 &r(V3- s) + k22s s (C-25) An expression for the fiber volume fraction, Vf, in terms of s , the fiber diameter, for the whole cube is required for substitution into the above equation. Examination of Figure C-2(e) reveals that the fiber volume fraction can be written; 7ZS ..,MV, K (C-26) Canceling the common terms and substituting equation (C-26) into equation (C-25) gives the effective thermal conductivity, k22, for the lamina in terms of the fiber volume fraction. ( k -k "•22 Kr f -\ 1-2,M-K B it-V W -tan" 1 - 5 K K J) (C-27) where B = —- - 1 C.3. Comparison of Thermal Conductivity Expressions The expressions for thermal conductivity perpendicular to the fiber direction are compared in Figure C-8. The normalized thermal conductivity, defined as the ratio of fiber to resin thermal conductivity, is plotted as a function of the fiber volume fraction. The values for the thermal conductivity of the fiber and resin are taken from reference [1]; 26 W/(m K) for the fiber and 0.167 W/(m K) for the resin. 231 C.4. Figures Figure C-l: A schematic of a lamina showing the three principal directions. (a) (b) (c) • • • • • • • • • • I I resin • 1 fiber z 11 (d) (e) Figure C-2: The unit cubes for the fiber and resin geometries used for calculation of lamina thermal conductivity, k22. (a) parallel heat flow (b) series heat flow (c) square fibers in a square packing array (d) circular fibers in a square packing array (e) circular fibers in a hexagonal packing array. 232 1-s 1-s 2 s 2 1-s s I fiber • resin z ,j t t q, q„ Figure C-3: The thermal model for square fibers in a square packing array. 1-s 1-s 1 -H^h i i 2 s 2 1-s s = # + = •H • i fiber CD resin " A a <L Figure C-4: The thermal model for circular fibers in a square packing array. m h(y) dy Figure C-5: The geometry of the infinitesimal slice, dy, for the center section for circular fibers in a square packing array. WHS qrf a a. Figure C-6: The thermal model for circular fibers in a hexagonal packing array. Figure C-7: The geometry of the infinitesimal slice, dy, for the center section for circular fibers in a hexagonal packing array. 234 ^ n • / : / : / 7 j puiu seri, C cfsp, C-3 :-5 M l C-19 hfsp. C-27 — i — i — i — i — r r -c - t r^ r — i 1—I i 1 kf/km= 155.7 1 1 i j i /' « • • * ' • ' ' ' ' —1 ' ' • L—l i—l ....... s s • • : i i i i J — i — i — i — i — / ; / t 1 L_J 1 — 0.1 0.2 0.3 0.4 0.5 0.6 fiber volume fraction 0.7 0.8 0.9 Figure C-8: A comparison of the normalized thermal conductivity as a function of the fiber volume fraction for the five derived expressions. The values for the resin and fiber thermal conductivities were taken from reference [1]. Appendix D: Sensor Experiments This appendix details the experiments performed to calibrate the Interlink FSR sensors. Some general conclusions about the usefulness of these sensors are made. The sensors were purchased from Interlink Electronics of Santa Barbara California. At the beginning of this work the sensors had just been made available. During our investigation improved sensors with a better polymer for the substrate material became available. The polymer material was Ultem1 which has better high temperature characteristics than the original material. The manufacturer claims the sensors have a maximum use temperature of 170°C. The part to part variability is quoted at 15% with a sensor cycle to cycle variability of 2%. The optimal pressure range is given as 0.01 to 100 psi with a maximum pressure of 500 psi. This is an ideal pressure range since the pressure in our experiments ranges from 0 to 100 psig. Since reusing a sensor that has been placed inside a composite laminate is not really an option, these sensors have the advantage of being relatively inexpensive (approximately $30). A typical prepreg lamina thickness is 0.125 mm. As mentioned by Cai and Gutowski [51], this places severe limits on the thickness of any sensor that can be placed in a laminate. The Interlink sensors are 0.34 mm thick, 15 mm wide and 20 mm long. Thus they can be inserted between prepreg layers without causing a large disturbance. D.1. Sensor Construction and Principle of Operatfon A schematic of the sensor is shown in Figure D-l. The sensors are formed by bonding two sheets of polymer film together; with the inner surface of one sheet coated with a solid semi-conducting layer and the other with interdigitated conducting fingers. A break in the adhesive forms a small channel leading to the interior of the device. Any pressure difference between the interior and the exterior of the device pushes the interdigitated fingers into contact with the semi-conducting material. This causes a change in the resistance which is inversely proportional to force and pressure. I Ultem is a product of DuPont. 236 There is some question as to what the sensors actually measure. They are not truly force sensing devices since simple bench top experiments show that the resistance is a function of both the applied pressure and the area over which it is applied. In our case we use them to measure a pressure which should be constant over their surface area and we do not address this question any further. D.2. Calibration Apparatus To determine if these sensors could be turned into an analytic device for measuring the resin pressure between laminae, an artificial laminate was constructed to simulate, as closely as possible, the actual experimental conditions in which the sensors would ultimately be placed. The sensors were placed between two sheets of silicone rubber. This was placed on an aluminum toolplate and then covered by a sheet of breather cloth. A vacuum bag was formed by placing a sheet of nylon film over the artificial laminate and sealing it to the toolplate with a sealant material. The sensor leads were passed through this material and connected to data logging equipment, see Figure D-2. The rationale for using silicone rubber is to evenly distribute the pressure on the upper and lower sensor surfaces so that a uniform pressure field exists over both surfaces. The sensors were exposed to pressures between 0 and 100 psig by placing the lay-up in an autoclave. A vacuum was drawn on the lay-up via a mechanical vacuum pump. D.3. What Do the Sensors Measure? Ideally, the sensors would measure the hydrostatic resin pressure. However, it is not clear what they actually measure. Consider the schematic shown in Figure D-3. The sensor must bear the hydrostatic pressure of the resin and the pressure from the fiber bed in the z -direction. Hence the pressure borne by the sensor, psensor, can be expressed as PsenSor=Pr+Pf ( D " D The above equation assumes that the vacuum pressure, if present, is always sufficiently small that it can be neglected. D.4. Autoclave Testing of Sensors 1 to 5 All of the experiments in this section were performed inside the autoclave using the test panel containing the sensors as described above. The first batch of 5 sensors were made of the low temperature material and are identified as si through s5. D.4.1. Change in Resistance at Ambient Pressure and Temperature The change in sensor resistance over time was measured at ambient temperature (approximately 20°C) at constant pressure. The external pressure in the autoclave was constant at atmospheric and the vacuum pressure constant at approximately 29" Hg. The resistance was measured at 1 minute intervals for a period of 1 hour. The sensor resistance as a function of time is shown in Figure D-4. The resistance is constantly decreasing with time. Either the vacuum supplied by the pump improves with time or the sensors are creeping. The final resistance of each sensor 237 was approximately 10 % lower than its initial resistance. Figure D-4 shows that the majority of the change occurs within the first 10 minutes of the experiment. To minimize this effect, the vacuum was drawn for 10 minutes prior to the beginning of all subsequent experiments. D.4.2. Typical Sensor Response The sensors were then exposed to a pressure cycle increasing from atmospheric to 80 psig and back to atmospheric. Figure D-5 shows the resistance plotted as a function of external pressure for sensors si through s5. The initial pile-up of data points on the ordinate is due to the 10 minutes of vacuum pressure prior to the application of external pressure. Examination of Figure D-5 shows there is a significant difference between the sensors. Their resistances after 10 minutes of vacuum pressure varies from just below 2750 Q, to over 4750 Q. The variability from sensor to sensor is much higher than the 15% claimed by the manufacturer. Initially, the resistance is approximately a linear function of pressure, up to about 10 psig. Between 10 and 80 psig the resistance is inversely proportional to pressure. The pressurization and de-pressurization paths are not the same. In all cases the resistance measured during pressurization is higher than that measured during de-pressurization. In light of Figure D-4 this result is not unexpected. Since a sensor's resistance constantly decreases with time, the separation between the pressurization/de-pressurization paths is time dependent. A very fast pressurization/de-pressurization cycle means that the shift due to this effect would be small; conversely, if the cycle were very long, the effect would be quite large. The sensors are less sensitive to pressure changes in the 30 to 80 psig range than they are in the lower pressure regions. The sensor resistance at the end of the de-pressurization portion of the experiment varied from the lowest value of 1992 Q. to the highest value of 3148 Q. This variability was similar to the variability of the sensor resistance at the end of the initial 10 minutes of vacuum pressure. From this one can conclude that the sensors are significandy different from each other thus requiring the individual calibration of each sensor, rather than some general calibration scheme that can be applied to well-behaved devices like thermocouples. D.4.3. Reproducibility of Sensor Response The reproducibility of the sensors' response was investigated by comparing the resistance as a function of time for two pressure cycles at the same temperature (approximately 20°C). This is shown in Figure D-6 for sensor 1 and is typical of all sensors. The most apparent feature of the figure is that the resistances at a given pressure during pressurization are significantly different. For the first run, labeled slcl , the initial resistance is approximately 3400 Q. at the end of 10 minutes of vacuum pressure while for the second run it is about 3000 Q. The difference between these two resistances decreases as the pressure increases hence the final compressed state of the sensor is more reproducible than the initial state. The initial point of the sensor for the second run is lower than the first run indicating that the sensor was changed by the application of pressure and that its physical shape may have been changed. The de-pressurization curves are very similar above 5 psig. At pressures lower than 5 psig the differences between the two runs start to reappear, with the resistances of the first run being larger that of the second run. However, the difference is quite small and can most likely be attributed to experimental error rather than another change in the shape of the sensors. 238 Regardless of the cause of this variation in initial position, the sensors seem to be reasonably reproducible at pressures above 50 psig. Comparing the resistances for both runs at 50 psig shows that the deviation in pressurization and de-pressurization to be about 100 Q. in 1500 Q or less than 7 % of the sensor's resistance. This is significantly larger then the manufacturer's quoted variability of 2% from cycle to cycle. D.4.4. Effect of Temperature on Resistance The effect of temperature on the sensors' response was evaluated by running the sensors though a series of isothermal pressure cycles. The pressure increased from atmospheric to 100 psig and returned to atmospheric over a 2 hour period. The temperatures for the isothermal cycles were 20, 40, 60, 80, 100 or 120°C. Figure D-7 is a plot of the resistance as a function of pressure for sensor si for the isothermal runs. This plot is typical of each sensor with a small exception of one run for sensor 2 that will be discussed later. An obvious aspect of Figure D-7 is the decrease in resistance as the temperature increases. The 20° and 40°C runs are very close together at 50 psig and higher pressures. This is similar to the response for the 100°C and 120°C runs where, over the course of the entire run, the two runs are within 50 Q of each other. This suggests that between 20 and 40°C and 100 and 120°C the sensors are relatively insensitive to temperature. This is quite different from the 40, 60, 80 and 100"C runs where, since the decrease in resistance at maximum pressure is approximately constant, one can conclude that the resistance is approximately linearly proportional to resistance between 40 and 100°C. The exception for sensor s2 is the 60°C run. The pressurization portion of the pressure cycle has a higher resistance than the de-pressurization cycle near 0 psig. The pressurization curve starts about 100 Q lower than where the de-pressurization curve stops. This is a small difference and, since all other sensors do not exhibit this feature, it is believed that this is just an artifact of the 60°C run and not deemed to be significant. D.4.5. Reproducibility of Sensor Initial Resistance Figure D-8 is the resistance of sensor 2 plotted against pressure. Most of its features are identical to sensor 1 and the discussion is not repeated. The point of interest on this plot is the 120°C run where the initial resistance is about 3600 Q and the final resistance after the pressure cycle is about 2300 Q.. Compared to other sensors this is a very high initial resistance. By 50 psig, the difference between the pressurization and de-pressurization curves is about 100 Q. out of 1300 Q. or about 8%. Hence the sensors may exhibit highly variable initial resistances. However, after the pressure has reached 50 psig the resistance settles down to a value within 8% of that measured during de-pressurization. Nevertheless, the variation is larger than the manufacturer's quoted values. D.4.6. First Normalization Procedure Many different types of normalization procedures were tried. The following relationship was found to adequately describe the behavior of the sensors. The approach is as follows: The sensor response is assumed to be only a function of applied external pressure and temperature. This ignores any creep and assumes that only a negligible amount will occur after the standard 239 10 minute period of vacuum pressure. It was thought that if each sensor could be compared to its own resistance measured at two different points, an expression for a normalized inverse resistance as a function of pressure might be found. The equation is R R(p,T)-R(Pmin,T) "' R(p^T)-R(Pimn,T) The R(p„in,T) and R(pmax,T) terms are the resistance measured by the sensor at two constant pressures at the same temperature as the R(p,T) term. Originally the pressures used for R(Pmax>T) and Rip^^T) were the initial and final resistance values for the sensors during the isothermal pressure runs. The problem with using the sensor initial resistance for R(p„i„,T) is the variability that the sensors exhibit. To minimize this effect, it was thought that more reproducible results could be obtained by using the resistance at 20 psig from each test, denoted R(pmin,T). The lowest resistance for each isothermal run depends on the highest pressure achieved during the run. To eliminate this effect, the resistance at 80 psig, reached in all of the isothermal runs, was used as the R{pmax,T) value. Figure D-9 shows the normalized inverse resistance of all the isothermal runs plotted together. An artifact of this type of normalization procedure, equation (D-2), is that all the results will collapse onto one point at 20 psig and 80 psig, and any resistance measured between 20 and 80 psig will have a RN value between 1 and zero respectively. Pressures over 80 psig will result in normalized inverse resistance values lower than zero and pressures lower than 20 psig will give normalized inverse resistances greater than 1. Examining Figure D-9 it is reasonable to conclude that the data from each sensor can be made to collapse onto a single master curve by use of the normalization procedure above. The advantage of being able to describe any sensor in terms of normalized inverse resistance is a great simplification in the testing procedure to convert a resistance measurement to a pressure. Only 2 tests need to be performed before a sensor can be used. These tests are the temperature cycles over the temperature range of interest at constant pressures of 20 and 80 psig. The previously described series of isothermal tests need not be performed since we know that any sensor can be described by the master curve. The resistance as a function of temperature is shown in Figures D-10 and D-11 for the constant pressure runs at 20 psig and 80 psig respectively. Figure D-10 shows the 20 psig isobaric run. The slight change in slope of the data at 65°C is due to a brief pressure increase from the lowest pressure of the test, just below 19.7 psig, to the higher pressure of 20.4 psig. Figure D-11 shows the 80 psig isobaric run. Examination of both figures shows that resistance can be described very well by a straight line. A linear fit to the data was done and then used to generate a second master curve of normalized inverse resistance versus pressure to remove any effects of the earlier non-linear isothermal runs. 240 D.5. Autoclave Testing of Sensors 6 to 10 As a check on how well the normalizing procedure worked, the calibration of a second batch of sensors, made of the low temperature material, denoted s6 to slO, was begun. The first run on the new sensors was at a 20 psig constant pressure to obtain the R{p„in,T) resistances. The results of the first run are shown in Figure D-12, where the resistance is a fairly linear function of pressure as noted previously. At the end of this run the sensors were allowed to cool with only the vacuum pressure applied to the testing lay-up. The resistance of each sensors was now a measurable value and not in excess of 130 kQ. 2. This suggests that the sensors took on some permanent set while cooling. An estimate of how large the change is over time can be obtained from the end of this run where the temperature and pressure were held constant at a pressure of 20 psig and a temperature of 125°C for 30 minutes. The resistance and percentage change in resistance over the 30 minute period are shown in Figure D-13 . Since the pressure and temperature are constant, this indicates that the sensors have a viscoelastic component to their behavior. In 30 minutes the sensor resistance is typically lower by about 5%. Note that the temperature is only 125"C, 45°C below the maximum temperature specified by the manufacturer. It is reasonable to conclude that the change in sensor resistance for the R(pmin,T) curves in Figure D-12 was caused by viscoelastic behavior of the sensors at elevated temperatures. D.6. New Material and Curite Software In the course of this experimental work, the manufacturer of the sensors began producing sensors made of the new high temperature polymeric material as mentioned previously. These sensors came bundled with software that is supposed to determine the pressure applied to the sensor based on the resistance and temperature of the sensor. In the course of evaluating these new sensors it was discovered that they were only slightly less susceptible to creep that the earlier material. The software was found to be based on a simple exponential relationship between resistance and pressure and did not take into account the effect of creep. The result was that it was no better at calculating the applied pressure than the normalizing procedure outline above. Indeed, the manufacturer warns that the sensors can only be used once. Given these conditions, it was concluded that the sensors cannot be calibrated accurately and that they can only be used as qualitative devices. D.7. Summary This appendix detailed the experiments that were performed to calibrate the Interlink pressure sensors. Due to the considerable amount of creep the sensors experience over the time span of a cure cycle, the absolute value of the pressure cannot be reliably found by the normalizing For all of the earlier runs, the resistance of the initial sensors, before the application of the vacuum pressure, was in excess of 130 kQ, the limits of the testing equipment. 241 procedure presented here or by the Curite software supplied by the manufacturer. Coupling this problem with the high degree of variability in the sensors' response, it was concluded that the sensors can not be used quantitatively but only qualitatively. The sensors indicate whether the pressure in the local region is increasing, by a decrease in resistance, or if it is decreasing, by an increase in resistance. D.8. Figures Figure D-l: Schematic of the Force Sensing Resistor. Note the channel formed by the break in the adhesive layer that leads to the interior of the device. Figure D-2: Schematic of the lay-up used to calibrate the sensors. Note the sensor lead passes through the tape seal. 242 semi-conductor silver adhesive Pr + P, interdigitated fingers l l l l t lH IUHK/ t l l l t MHMMIM fiber resin " " vac Figure D-3: Schematic of the interaction between sensor, resin and fiber. The sensor bears the load carried by the fibers, which is transmitted to the sensor at the points where they come into contact, and the hydrostatic pressure of the liquid resin which surrounds it. 5000 4500 4000 3500 3000 2500 2000 30 time (min) Figure D-4: Sensor resistance as a function of time for sensors si to s5 at ambient temperature of approximately 20°C at 0 psig. The vacuum pressure was approximately 29" Hg. The majority of decrease in resistance occurs in the initial 10 minutes. 5000 1000 10 20 30 40 pressure (psig) 50 60 70 80 Figure D-5: Sensor resistance as a function of external pressurefor sensors si to s5. The temperature increased linearly from 22°C to 38°C (corresponding to the maximum pressure) in 82 minutes . The temperature returned to 240C upon returning to atmospheric pressure for a total run time of 156 minutes. 3500 -t 3000 2500 2000 1500 1000 90 100 Figure D-6: The resistance of sensor 1 as a function of pressure for the first two runs. 10 20 30 40 50 60 pressure (psig) 70 100 Figure D-7: The resistance as a function of pressure of sensor 1 for all the isothermal runs. 40 50 60 pressure (psig) 90 100 Figure D-8: The resistance as a function of pressure of sensor 2 for all the isothermal runs. o 4 £ 2 a £ o c 1 t • • • » JUK •^ Hl ^ • • ^ ^ p*4w»(hL« 3!sSJefSf«'5( •fc*3*>F«*«:fjai«»Q]axsSS^3 10 20 30 40 50 60 pressure (psig) 70 80 90 100 Figure D-9: Normalized inverse resistance as a function of pressure for all isothermal runs. 2400 600 20 30 40 50 60 70 80 temperature ('C) 90 100 110 120 Figure D-10: Resistance as a function of temperature for sensors 1 to 5 at 20.1 psig +/- 0.4 psia. 1800 2 0 3° 40 50 60 70 80 temperature ("C) 90 100 110 120 Figure D-11: Resistance as a function of temperature for sensors 1 to 5 at 80.2 psig +/- 0.4 psia. 2000 1800 •p- 1600 E .c _o o> o c <o c 1200 1000 1400 20 40 60 80 temperature ('C) 100 120 140 Figure D-12: The resistance as a function of temperature for runs 1 and 2 for the second batch of sensors, s6 to s 10 at a constant pressure of 20 psig +/- 0.2 psia for both runs. 247 130 135 140 145 150 time (s) 155 160 165 Figure D-13: The resistance and percent change in resistance of the sensors 6 to 10 as a function of time. Note the temperature and pressure are constant at 125 C and 20 psig. The results were obtained from the same run as Figure D-12. 248 Appendix E: Determination of Prepreg Resin Mass Fraction The following procedure was used for determining the initial resin mass fraction of the prepreg: 1. Cut a sample from the roll, approximately 300 by 50 mm, and weigh it. 2. Weigh the filter paper. 3. Place the sample in a breaker and add enough acetone to completely immerse the sample. Stir occasionally for a 10 minute period. 4. Decant the acteone from the beaker being careful not to allow any fibers to escape. Repeat step 3 two more times. 5. Filter the contents of the beaker. 6. Dry the filter paper and fibers at 110°C for one hour. 7. Weight the filter paper and fibers. 8. Calculate the resin mass fraction as follows 1-m, M. =• ms where mf = fiber mass ms - sample mass (E-l) 249 

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