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Viscoelastic constitutive models for evaluation of residual stresses in thermoset composites during cure Zobeiry, Nima 2006

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V I S C O E L A S T I C C O N S T I T U T I V E M O D E L S F O R E V A L U A T I O N OF R E S I D U A L STRESSES IN T H E R M O S E T COMPOSITES DURING C U R E by NIMA ZOBEIRY B.Sc. (Civil Engineering), University of Tehran, 1997 M . S c . (Civil Engineering/Structural Engineering), University of Tehran, 1999  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR T H E D E G R E E OF  D O C T O R OF PHILOSOPHY in T H E F A C U L T Y OF G R A D U A T E STUDIES (Civil Engineering)  The University of British Columbia July 2006 © N i m a Z o b e i r y , 2006  Abstract A particularly important aspect in the behaviour of thermoset matrix composite materials during the manufacturing process is the development of mechanical properties o f the matrix and the resulting buildup of stresses. The behaviour of the matrix is generally acknowledged to be viscoelastic, and as both temperature and degree of cure vary with time, the characterization and representation of the behaviour is both critical and complex. Different approaches have been suggested for modeling this behaviour. The common approaches that invoke the simple linear elastic cure hardening model have been shown to provide good predictions but have not been studied for their accuracy and applicability. More sophisticated representations of viscoelastic behaviour are the Prony series of Maxwell elements implemented in finite element codes in 3D hereditary integral forms. In this thesis, different constitutive models are considered and their suitability for representing the behaviour of composite materials during cure is studied. The presented models provide the user with a range of options depending on whether costs or accuracy of solutions are of primary concern. For elastic hardening models, it is shown that the full viscoelastic formulations can be progressively simplified, and that these simplifications are valid for the typical cure cycles. It is shown that in general i f these models are properly calibrated they are valid and efficient pseudo-viscoelastic models. It is also noted that these models are not always applicable and an efficient viscoelastic model is needed. For such cases, viscoelastic behaviour of the polymer is represented using a differential form approach. It is shown that this form is equivalent to the more common integral form, but has significant benefits in terms of extension to more general descriptions, ease of coding and implementation, and computer runtimes. This formulation is extended to composite materials, using an appropriate micromechanical approach, and to 3D behaviour with finite element implementation such that it can be used with an existing code. Some important features are included, such as time-variability of all material properties, methods for calculating polymer and fibre stresses, and considering thermoelastic effects. Several case studies are presented for verification/validation purposes and to highlight various features of the models.  ii  Table of Contents  Table of Contents Abstract  »  Table of Contents  "i  List of Tables  .  vii  List of Figures  ix  List of Symbols  xvi  Acknowledgements  xxi  Chapter 1.  Introduction and Background  1  1.1.  Processing and Process Modeling  1  1.2.  Research Objectives and Thesis Outline  4  Chapter 2.  Review of Constitutive Models  8  2.1.  Mechanical Behaviour of Thermoset Polymers during Cure  8  2.2.  Elastic Models  9  2.3.  C H I L E Models  9  2.4.  Viscoelastic Models  10  2.4.1.  Integral Form of Viscoelasticity  12  2.4.2.  Fractional models  17  2.4.3.  Differential Form of Viscoelasticity  17  2.5.  Summary  Chapter 3. 3.1.  19  Pseudo-Viscoelastic Models  Review o f Two Viscoelastic Models  24 24  3.1.1.  Material 1  24  3.1.2.  Material II.  26  3.2.  Pseudo-Viscoelastic Model Derivation  3.2.1.  Simplification of Viscoelastic Formulations  iii  28  29  Table of Contents  3.2.2. 3.3.  Different Forms ofPVE  35  Case Studies  37  3.3.1.  Material 1  37  3.3.2.  Material II.  39  3.3.3.  Comparison of Efficiency  41  3.4.  Summary and Discussion  Chapter 4. 4.1.  42  Differential Form of Viscoelasticity in ID  Governing Equations in I D  56 56  4.1.1.  Differential Equations for Non-Curing Materials  56  4.1.2.  Differential Equations for Curing Materials  59  4.1.3.  Numerical Solution of Differential Equations  65  4.1.4.  Comparison of Differential Form and Integral Form  66  4.2.  Summary  Chapter 5.  67  Micromechanics  73  5.1.  Fundamental Equations  73  5.2.  Micromechanical Model for Elastic Materials  74  5.3.  Micromechanical Model for Viscoelastic Materials  77  5.3.1.  Correspondence Principle  5.3.2.  Virtual Material Characterization  80  5.3.3.  Material Properties  84  5.3.4.  Thermoelastic effects  5.3.5.  Summary of the proposed approach  5.4.  78  85  Numerical examples  90  5.4.1.  Isothermal Case  5.4.2.  Different Temperatures and Degrees of Cure  Chapter 6. 6.1.  89 90  Differential Form of Viscoelasticity in 3D  Governing Equations in 3D  92 103 103  6.1.1.  Choice of Material Properties and Development of the Differential Form  103  6.1.2.  Equations for Thermal Expansion and Cure Shrinkage  112  6.2.  Finite Element Formulation  114  iv  Table of Contents  6.2.1.  Numerical Solution of Differential Equations  114  6.2.2.  Development of the FE Formulation  116  6.2.3.  Summary and Discussion  120  6.3.  Reverse Micromechanics  6.3.1. 6.4.  Proposed Method  123  Summary  Chapter 7. 7.1.  123  125  Implementation and Verification  Implementation in a code  126 126  7.1.1.  Code AIgorithm andPseudo Code  127  7.1.2.  Code Features  128  7.2.  Verification Problems  128  7.2.7.  Transversely Isotropic Cases  7.2.2.  Isotropic Cases  131  7.2.3.  Reverse Micromechanics Examples  135  7.2.4.  Thermoelastic behaviour  137  7.2.5.  Summary  139  Numerical Applications  156  Chapter 8.  129  8.1.  Material Properties  157  8.2.  Single Element Cases  160  8.2.1.  Standard Cure Cycles  160  8.2.2.  Other Cure Cycles  165  8.3.  More Complex Cases  168  8.3.1.  Standard Cure Cycles  168  8.3.2.  Non-Standard Cure Cycles  173  8.4.  Summary  Chapter 9.  177  Conclusions and Future Work  211  9.1.  Summary  211  9.2.  Conclusions  211  9.3.  Future work  212  9.4.  Contributions  214  v  Table of Contents  References  216  Appendix A .  Mathematica Notebooks  232  Appendix B.  P s e u d o - C o d e for the Differential F o r m I m p l e m e n t a t i o n  245  Appendix C .  A B A Q U S Viscoelastic M o d e l  251  vi  List of Tables  List of Tables Table 3.1 Relaxation times and weight factors for 3501-6 resin for a° = 0.98  44  Table 3.2 Relaxation times and weight factors for 8551-7 resin..  44  Table 3.3 Comparison of run times for integral form V E and PVE for a cured resin  44  Table 3.4 Comparison of run times for integral form V E and PVE for a curing resin in a one-hold cure cycle 45 Table 3.5 Comparison of the features of the different models  45  Table 4.1 Comparison of Run-times for Thermoplastic Matrix (or constant degree of cure in athermoset matrix) 69 Table 4.2 Comparison of Run-times for Thermoset Matrix (increasing degree of cure)  69  Table 4.3 Comparison of DF and IF models  70  Table 5.1 Material properties of fibre and resin for the numerical example  93  Table 5.2 Relaxation times and weight factors used in the numerical example  93  Table 5.3 Relaxation times and weight factors for different material properties in the example  93  Table 5.4 Values of the unrelaxed and relaxed moduli of the composite obtained from the V E micromechanics in Mathematica  94  Table 7.1 Comparison of the features of the added U M A T and the A B A Q U S built-in V E model  140  Table 7.2 Summary of examples to be analyzed  140  Table 8.1 Mechanical properties of fibre and resin used for case studies  180  Table 8.2 Thermal and cure shrinkage coefficients obtained for 3501-6 resin for the case studies  180  Table 8.3 Unrelaxed and relaxed values of different material properties obtained for AS4/3501-6  180  Table 8.4 Relaxation times and corresponding weight factors for different material properties of AS4/3501-6  180  Table 8.5 Thermo-chemical properties of AS4/3501-6  181  Table 8.6 Properties of the tooling materials used in the case studies  181  Table 8.7 Summary of single-element cases to be analyzed  181  vii  List of Tables  Table 8.8 Results o f spring-in angle in degrees for an L-shaped part on different tooling, analyzed with the Differential Form of Viscoelasticity and Pseudo-Viscoelasticity 182 Table 8.9 Comparison o f runtimes in a standard cure cycle for an L-shaped composite on different toolingl82 Table 8.10 Spring-in and runtimes from the analysis on a unidirectional part shown in Figure 8.29  182  Table 8.11 Spring-in from the analysis on the part shown in Figure 8.29 with different lay-ups  182  Table 8.12 Spring-in results for a C-shaped composite part under a cure cycle with a post cure heat-up 183 Table 8.13 Spring-in results for a C-shaped composite part under a cure cycle with secondary cure of a partially cured material 183 Table 8.14 Results of spring-in for cycles in the form of secondary cure of partially cured parts  viii  183  List of Figures  List of Figures Figure 1.1 Integrated sub-model approach in process modeling (adapted from Johnston et al., 2001)  7  Figure 2.1 A typical temperature/pressure cycle used to cure thermoset polymers in an autoclave  21  Figure 2.2 Joining of molecules as a result of curing in a thermoset polymer  21  ;  Figure 2.3 Polymer changes form as it goes through the cure cycle  :  21  Figure 2.4 Undeformed and deformed shapes in a typical U-shaped composite part after cure  22  Figure 2.5 Schematic of time-temperature superposition  22  Figure 2.6 A Maxwell element  22  Figure 2.7 A K e l v i n element  22  Figure 2.8 A generalized Maxwell element  23  Figure 2.9 Different levels of constitutive modeling  23  Figure 3.1 Comparison of shift factors for 8551-7 resin using two different equations (with ot=0.99)  46  Figure 3.2 Comparison of reduced times for P V E and V E models for 3501-6 resin (with a=0.99) for different values of cool-down rates  46  Figure 3.3 Comparison of moduli for P V E and V E models for 3501-6 resin (with a=0.99) for different values of cool-down rates 47 Figure 3.4 Schematic of a typical variation of shift factor as a function of time during a hold  47  Figure 3.5 Temperature and degree of cure for the cure cycle used for 3501-6 resin  48  Figure 3.6 Comparison of two definitions of modulus, based on storage modulus and relaxation modulus, for 3501-6 resin 48 Figure 3.7 Comparison o f two definitions of modulus, based on storage modulus and relaxation modulus, for 8551-7 resin 49 Figure 3.8 Elastic modulus profile based on a variable time Pseudo-Viscoelastic model for the one- cure cycle in Figure 3.5 (3501-6 resin) 49 Figure 3.9 Comparison of stress profiles for variable time Pseudo-Viscoelastic and Viscoelastic models for the cure cycle in Figure 3.5 (3501-6 resin) 50 Figure 3.10 Temperature and degree of cure profile for the two-hold cycle used in the case study for 3501-6 resin  ix  50  List of Figures Figure 3.11 Elastic modulus profiles for different constant frequency Pseudo-Viscoelastic models for the cure cycle in Figure 3.10 (3501-6 resin) 51 Figure 3.12 Comparison of stress profiles from the constant frequency Pseudo-Viscoelastic models and the Viscoelastic model for the cure cycle in Figure 3.10 (3501-6 resin) 51 Figure 3.13 Residual stress as a function of frequency for the constant-frequency P V E model, the cure cycle in Figure 3.10 (3501-6 resin). Also shown is the V E model prediction 52 Figure 3.14 Predicted residual stress for variable time Pseudo-Viscoelastic and Viscoelastic models for different cooling rates for a one-hold cycle with T h i d 1 8 0 °C (3501-6 resin) 52 =  o  Figure 3.15 Predicted residual stress for variable time Pseudo-Viscoelastic and Viscoelastic models for different cooling rates, m , for a one-hold cycle with T i d 1 6 0 °C (3501-6 resin) . . 53 =  h o  v  Figure 3.16 Predicted residual stress for constant time Pseudo-Viscoelastic and Viscoelastic models for different cooling rates, m , (and corresponding times) for a one-hold cycle with T i d 1 8 0 °C (35016 resin) 53 =  ho  Figure 3.17 Predicted residual stress for constant time Pseudo-Viscoelastic and Viscoelastic models for different cooling rates (and corresponding times) for a one-hold cycle with T i d 1 6 0 °C (3501-6 resin) 54 =  ho  Figure 3.18 Variable time Pseudo-Viscoelastic and Viscoelastic stress predictions for a wide range of different cure cycles (8551-7 resin)  54  Figure 3.19 Constant frequency Pseudo-Viscoelastic and Viscoelastic stress predictions for a wide range of different cure cycles (8551-7 resin) 55 Figure 4.1 (a) Maxwell element (b) Generalized Maxwell model  71  Figure 4.2 Cure cycles used to compare IF and D F stress predictions  71  Figure 4.3 Comparison of predicted stress histories using IF and D F approaches. Note that the two approaches give identical results for each case and therefore cannot be distinguished  72  Figure 5.1 Concentric Cylinder Assembly ( C C A ) model  95  Figure 5.2 Schematic illustration of the micromechanical approach  95  Figure 5.3 Vertical shift factor for thermorheologically complex behaviour of 3502 resin  95  Figure 5.4 Flowchart of the micromechanical analysis used  97  Figure 5.5 Comparison of the value of composite modulus from different methods: micromechanics, Prony series fit to micromechanics, and Prony series with the resin weight factors 98 Figure 5.6 Comparison o f the value of composite Poisson's ratio from different methods: micromechanics, Prony series fit to micromechanics, and Prony series with the resin weight factors98  x  List of Figures Figure 5.7 Comparison of the value of composite modulus from different methods: micromechanics, Prony series fit to micromechanics, and Prony series with the resin weight factors  99  Figure 5.8 Comparison of the value of composite modulus from different methods: micromechanics, Prony series fit to micromechanics, and Prony series with the resin weight factors  99  Figure 5.9 Comparison of the value of composite modulus from different methods: micromechanics, Prony series fit to micromechanics, and Prony series with the resin weight factors  100  Figure 5.10 Comparison of the value of composite Poisson's ratio from different methods: micromechanics, Prony series fit to micromechanics, and Prony series with the resin weight factorslOO Figure 5.11 Comparison of the value of composite coefficient of thermal expansion different methods: micromechanics and Prony series fit to micromechanics 101 Figure 5.12 Comparison of the value of composite coefficient of thermal expansion different methods: micromechanics and Prony series fit to micromechanics 101 Figure 5.13 Comparison of the values of composite modulus for different shift factors  102  Figure 5.14 Comparison of the values of composite modulus for different shift factors  102  Figure 7.1 Schematic form of the relationship between the DF code and an available commercial code 141 Figure 7.2 Algorithm of the Differential Form code for a given time step  142  Figure 7.3 Specifications for Example 1  143  Figure 7.4 Comparison of a  as predicted by the Differential Form U M A T and analytically for Example  n  1 Figure 7.5 Comparison of c  144 as predicted by the Differential Form U M A T and analytically for Example  2 2  1  144  Figure 7.6 Comparison of a  as predicted by the Differential Form U M A T and analytically for Example  u  2  145  Figure 7.7 Comparison of <x as predicted by the Differential Form U M A T and analytically for Example 22  2  145  Figure 7.8 Specifications for Example 3  146  Figure 7.9 Comparison of £" as predicted by the Differential Form, ABAQUS, and analytically for 22  Example 3  146  Figure 7.10 Comparison of <r as predicted by the Differential Form U M A T and analytically for n  Example 4  147  xi  List of Figures Figure 7.11 Comparison of the exact cure shrinkage and the simplified linear model, used in Example 5147 Figure 7.12 Temperature cycle and the resulting degree of cure used in Example 5  148  Figure 7.13 Comparison of stresses as predicted by the Differential Form U M A T and the previously verified I D code in Example 5  148  Figure 7.14 Geometry of Example 6  149  Figure 7.15 Standard mesh used in Example 6  150  Figure 7.16 Comparison of radial stresses at different times t=0.1 to 10 as predicted by the Differential Form U M A T (dots) and analytically (lines) using a standard mesh in Example 6 150 Figure 7.17 Refined mesh, used for Example 6  151  Figure 7.18 Comparison of radial stresses at different times t=0.1 to 10 as predicted by the Differential Form U M A T (dots) and analytically (lines) for a refined mesh in Example 6 151 Figure 7.19 Comparison of matrix stresses as predicted by the Differential Form and analytically for Example 7  152  Figure 7.20 Comparison of matrix stresses as predicted by the Differential Form and analytically for Example 7  152  Figure 7.21 Comparison of matrix stresses as predicted by the Differential Form and analytically for Example 8  153  Figure 7.22 Comparison of matrix stresses as predicted by the Differential Form and analytically for Example 8  153  Figure 7.23 Comparison of strains as predicted by the Differential Form U M A T and analytically for Example 9  154  Figure 7.24 Comparison of strains as predicted by the Differential Form U M A T and analytically for Example 9  154  Figure 7.25 Comparison of strains as predicted with and without thermoelastic effects by the Differential Form U M A T for Example 9 155 Figure 7.26 Comparison o f strains as predicted with and without thermoelastic effects by the Differential Form U M A T for Example 9 155 Figure 8.1 Evolution of the volume change for 3501-6 resin in a cure cycle, used to find thermal and cure shrinkage properties 184 Figure 8.2 Temperature cycle and the resulting degree of cure used in Case 1 to Case 6  184  Figure 8.3 Comparison of <J as predicted by the Differential Form U M A T and P V E for Case 1  185  U  xn  List of Figures Figure 8.4 Comparison of <x as predicted by the Differential Form U M A T and P V E for Case 1 22  185  Figure 8.5 Comparison of axial stresses as predicted by the Differential Form U M A T and P V E for Case 2186 Figure 8.6 Comparison of axial stresses as predicted by the Differential Form, with and without thermoelastic behaviour for Case 3  186  Figure 8.7 Comparison of axial forces as predicted by the Differential Form U M A T and P V E for Case 4187 Figure 8.8 Comparison of axial forces as predicted by the Differential Form U M A T and P V E for Case 5187 Figure 8.9 Comparison of axial forces as predicted by the Differential Form U M A T with different element types for Case 5  188  Figure 8.10 Schematic of the boundary conditions in Case 6 to avoid rigid body movements  189  Figure 8.11 Comparison of axial forces as predicted by the Differential Form U M A T and P V E for Case 6189 Figure 8.12 Undeformed (solid line) and deformed (dashed line) shapes, using a layered element in Case 6 190 Figure 8.13 Undeformed (unshaded) and deformed (shaded) shapes, using single element per layer in Case 6  190  Figure 8.14 Cure cycle used in post cure heat-up of the material in Case 7  191  Figure 8.15 Temperature cycle and the degree of cure used to model a post-cure heat-up in Case 7  191  Figure 8.16 Comparison of cr,, as predicted by the Differential Form U M A T and P V E for Case 7  192  Figure 8.17 Comparison o f <x as predicted by the Differential Form U M A T and P V E for Case 7  192  Figure 8.18 Complete cure cycle in secondary cure of a partially-cured material in Case 8  193  22  Figure 8.19 Temperature cycle and the degree of cure used to model a secondary cure of partially-cured material in Case 8 193 Figure 8.20 Comparison o f cr as predicted by the Differential Form U M A T and P V E for Case 8  194  Figure 8.21 Comparison of cr  194  n  22  as predicted by the Differential Form U M A T and P V E for Case 8  Figure 8.22 Mesh of an L-shaped composite part on a convex mould used in case studies  195  Figure 8.23 Mesh of an L-shaped composite part on a concave mould used in case studies  195  Figure 8.24 Cure cycle used for case studies on AS4/3501-6 composite material  196  Figure 8.25 Distribution of longitudinal stresses for an L-shaped unidirectional composite on a convex steel mould from Differential Form of Viscoelasticity 196  xiii  List of Figures  Figure 8.26 Distribution of longitudinal stresses for an L-shaped unidirectional composite on a convex steel mould from Pseudo-Viscoelasticity 197 Figure 8.27 Distribution of transverse stresses for an L-shaped unidirectional composite on a convex steel mould from Differential Form of Viscoelasticity 197 Figure 8.28 Distribution of transverse stresses for an L-shaped unidirectional composite on a convex steel mould from Pseudo-Viscoelasticity 198 Figure 8.29 F E mesh o f an L-shaped AS4/3501-6 part on an aluminium tool used in the case studies ... 198 Figure 8.30 Deformed part after tool removal in a cure cycle, along with the undeformed shape  199  Figure 8.31 Evolution of temperature and residual stresses at the point shown in Figure 8.29  199  Figure 8.32 Evolution of moments in an L-shaped composite part for a cross-section passing through the point shown in Figure 8.29 200 Figure 8.33 Evolution of axial forces in an L-shaped composite part for a cross-section passing through the point shown in Figure 8.29 200 Figure 8.34 Distribution of final axial residual stresses through the thickness for a cross-section passing through the point shown in Figure 8.29 201 Figure 8.35 Distribution of moments along the length of the part shown in Figure 8.29  201  Figure 8.36 Evolution of composite and matrix stresses in the fibre direction at the point shown in Figure 8.29 202 Figure 8.37 Evolution of composite and matrix stresses perpendicular to the fibre direction (in-plane) at the point shown in Figure 8.29 202 Figure 8.38 Distribution of moments along the length of the part shown in Figure 8.29for two analyses with and without thermoelastic effects 203 Figure 8.39 F E mesh of a C-shaped AS4/3501-6 part on an aluminium tool used in the case studies  204  Figure 8.40 Cure cycle applied to a C-shaped composite shown in Figure 8.29  204  Figure 8.41 Evolution of temperature and residual stresses at the point for a C-shaped part shown in Figure 8.29  205  Figure 8.42 Distribution of moments along the web of the C-shaped part before tool removal, as in Figure 8.40 205 Figure 8.43 Distribution of moments along the web of the C-shaped part shown in Figure 8.29 at the end of post cure heat-up 206 Figure 8.44 Evolution of composite and matrix stresses in the fibre direction for a C-shaped composite at the point shown in Figure 8.29 under the cure cycle in Figure 8.40 206  xiv  List of Figures Figure 8.45 Evolution of composite and matrix stresses perpendicular to the fibre direction for a C-shaped composite at the point shown in Figure 8.29 under the cure cycle in Figure 8.40 207 Figure 8.46 Cure cycle applied to a C-shaped composite shown in Figure 8.29  207  Figure 8.47 Evolution of temperature and residual stresses at the point for a C-shaped part shown in Figure 8.29  208  Figure 8.48 Distribution of moments along the web of the C-shaped part before tool removal, cure cycle shown in Figure 8.46 208 Figure 8.49 Distribution o f moments along the web of the C-shaped part at the end of cycle, cure cycle shown in Figure 8.46 209 Figure 8.50 Evolution of composite and matrix stresses in the fibre direction for a C-shaped composite at the point shown in Figure 8.39 209 Figure 8.51 Evolution of composite and matrix stresses perpendicular to the fibre direction for a C-shaped composite at the point shown in Figure 8.39 210 Figure 8.52 Different cure cycles in the form of secondary cure of partially cured parts  xv  210  List of Symbols Parameter used to define the shift factor for 3501-6 resin Parameter used to define the shift factor for 3501-6 resin Vertical shift factor Horizontal Shift factor Material stiffness Thermal expansion coefficient Longitudinal coefficient of thermal expansion Transverse coefficient of thermal expansion Transverse coefficient of thermal expansion Fiber coefficient of thermal expansion Glassy coefficient of thermal expansion Matrix coefficient of thermal expansion Rubbery coefficient of thermal expansion Thermal expansion coefficient in the radial direction Thermal expansion coefficient in the circumferential direction Creep compliance Material stiffness matrix Material stiffness matrix for fibres Material stiffness matrix for matrix material Material stiffness matrix, for a viscoelastic material Initial value of creep compliance Young's modulus Laplace transformed modulus Longitudinal Young's modulus in an orthotropic media Transverse Young's modulus in an orthotropic media Transverse Young's modulus in an orthotropic media  xvi  '  j  List of Symbols E™  Relaxed Young's modulus  E  Young's modulus of the matrix  E,  Young's modulus of the fibre  E  Relaxed Young's modulus  m  r  E  Unrelaxed Young's modulus  F  Load vector  G  Shear Modulus  u  G,  2  In-plane shear modulus in an orthotropic media In-plane shear modulus in an orthotropic media  2 3  G  Transverse shear modulus in an orthotropic media  G  Storage shear Modulus  G"  Loss shear Modulus  Si  Weight factors, to define bulk modulus  I  Number of Maxwell elements to define relaxation modulus  k  Spring stiffness in an analogue element, e.g. Maxwell element  k  Spring stiffness rate, for thermoelastic effects  k,  Spring stiffness of any of the Maxwell elements in a model  k"  Unrelaxed modulus  K  Bulk modulus  K  T  Global stiffness matrix  K  Plane strain bulk modulus in a transversely-isotropic media  K  Glassy bulk modulus  2  g  Rubbery bulk modulus  m  Cool-down rate in a cure cycle  n  Time step number  N  Number of Maxwell elements to define relaxation modulus  N  Number of Maxwell elements for material property p  P  A number, associated with one the material properties  P  Denoting any material property  xvii  List of Symbols t  Time, current time  t'  Dummy time (for time integration)  t  A measure o f the variable time, used in the pseudo-viscoelastic model  t  A parameter, used to evaluate the variable time in the pseudo-viscoelastic model  t  Variable time, used in the pseudo-viscoelastic model  t  Time at the onset of cool-down in a cure cycle  T  Temperature  T°  Reference temperature  T  Glass transition temperature  T  Glass transition temperature of the monomer  x  2  e  f  g0  T  Glass transition temperature of the fully cured polymer  u  Displacement vector  v  Fibre volume fraction  V  Volume of the composite  Vj  Volume of the fibres  V  Volume of the matrix  w  j  Weight factors, to define relaxation modulus  w  Weight factors, to define relaxation modulus  a  Degree of cure  a  Reference degree of cure  a  Value of the degree of cure at the onset of cool-down in a cure cycle  P  A small number, used to evaluate the variable time in the pseudo-viscoelastic model  ga>  f  m  0  f  Force vector increment AF_  Force vector increment, from thermo-chemical strains  AF_  Force vector increment, from internal stresses  As  Strain increment  ASf  Free strain increment  Ad  Change o f angle in a composite part (spring-in)  f  a  xviii  List of Symbols  At  Time increment  AT  Temperature increment  A C T  Stress increment  £  Strain  £  Strain rate  £  Laplace transformed strain Volume-averaged strain of the composite  — G  Deviatoric strain vector Strain, in tensor form  H  £  Volumetric strain vector  £K  Volume-averaged strain of the fibres  IL  Volume-averaged strain of the matrix Strain, associated with material property p Free strain, associated with material property p  s'  Thermo-chemical strain  ,o,a,  Total strain  r  Modulus, in Laplace space  c  £  Dashpot viscosity in an analogue element, e.g. Maxwell element  n,  Dashpot viscosity o f any of the Maxwell elements in a model  i  A material property in a transversely-isotropic materials  V  Poisson's ratio  X2  V  "13  Major Poisson's ratio in an orthotropic media Major Poisson's ratio in an orthotropic media Major Poisson's ratio  (9  Initial angle in a composite part  cr  Stress  fj  Stress rate  a  Laplace transformed stress Volume-averaged stress of the composite  xix  List of Symbols  a  Volume-averaged stress of the fibres  f  _  Deviatoric stress vector  G  cr.  Stress in any of the Maxwell elements in a model  ov  Stress, in tensor form  q_  Volumetric stress vector  a  Volume-averaged stress of the matrix  K  —tn  CT  a  Stress in Maxwell element i, associated with material property p  T  Dummy time (for time integration)  pl  r  Reference relaxation time  0  T  Equivalent relaxation time, for thermoelastic effects  T  Relaxation times, associated with shear behaviour  r,  Relaxation times, to define relaxation modulus  r  Relaxation times, associated with bulk behaviour  CI  Ki  T  Peak relaxation time, used for 3501-6 resin  r .  Relaxation times, associated with material property p  T  Relaxation times, to define relaxation modulus  co  Frequency of a dynamic test  £  Reduced time, at current time (t)  P  A  Reduced time, at dummy time ( r )  xx  Ac know ledgements  Acknowledgements I would like to acknowledge several individuals who truly made this work possible. First, I would like to thank my supervisors Dr. Reza Vaziri and Dr. Anoush Poursartip. I really appreciate the guidance and help they gave me during the course of my PhD. Many thanks to several past and present U B C Composites Group Members for their friendship and help. Especially, I would like to thank M r . Ahamed Arafath and M r . A m i r Osooly for our fruitful discussions and their assistance on many occasions during the course of my degree. I truly value the friendship of several close friends that I found in Vancouver. I appreciate their enthusiasm, support, and encouragements during the highs and lows of my degree. Last but not least, I would like to thank my mother, father, and brother who were with me every step of the way and their love and affection was always what gave me the strength to continue. I thank Hiva, who in the past several months was always by my side and her words of kindness and support were my constant source of motivation.  xxi  Chapter 1: Introduction and Background  Chapter 1. 1.1.  INTRODUCTION AND B A C K G R O U N D  PROCESSING AND PROCESS MODELING  Fibre reinforced polymer matrix composite materials are increasingly finding new applications in industry. The growing popularity of these materials has been due to their advantages to other available materials, e.g. their high strength to weight ratio, high stiffness to weight ratio, and durability. A t the same time, in using these materials in the industry it has been critical to optimize their manufacturing for the purpose of reducing the cost of production, an important and sometimes prohibitive proposition, which i f done properly can work as an advantage for these materials. Also, it has been essential to have a proper understanding of the manufacturing process to be able to control the final shape of the material. For this reason, modeling of the manufacturing process of composite materials or process modeling has gained popularity as a tool able to meet the above demands, in addition to its ability to predict residual stresses generated in the material during the manufacturing, an important factor in designing composite structures. Process modeling has been the topic of many research works in the literature. One of the first models presented is the work done by Hahn and Pagano (1975). This work is simplified in several aspects, e.g. it uses a linear elastic constitutive model and assumes a stress-free state prior to cool-down. Since this work, many studies have been carried out on this topic, which have taken various complexities of the process into account and have used different approaches. Due to the complexity of process modeling, composite materials have generally been modelled using the 'integrated sub-model' approach.  Based on this approach, a complex process model is divided into  several simpler sub-models, which can be studied more or less independently, as shown schematically in Figure 1.1, presented by Johnston (2001) and used in the development of the composite process modeling code, C O M P R O . According to the schematic of the sub-model approach, the modeling procedure  Chapter 1: Introduction and Background includes several modules, each one responsible for one aspect of the material behaviour. The module that is of interest in this work is the stress module, i.e. the module responsible for calculation of the residual stresses and deformations. In most of this thesis we restrict our attention to this module in the process model. This approach was first applied to process modeling by Loos and Springer (1983). Since then, many other researchers have used this methodology more comprehensively and for different processes. Some examples for autoclave processing are Bogetti and Gillespie (1991, 1992), White and Hahn (1992a, 1992b), Kiasat (2000), Johnston et al. (2001), K i m et al. (2002), Z h u et al. (2003), Wijskamp et al. (2003), Antonucci et al. (2006), and Clifford et al. (2006) and for other processing methods are K i m and White (1998) for filament winding, Yang et el. (2004a) for map-mould process, and Douven et al. (1995) for injection moulding. In earlier works on process modeling, the researchers modeled only the final cool-down section of the curing process. This was due to the fact that (according to A d o l f and Martin, 1996) the majority of the studies assumed that the only source of residual stress in the material was thermal strains (e.g. Griffin, 1983; Loos and Springer, 1983). For this reason, a stress-free temperature was considered and it was assumed that stresses generated by cooling down from this temperature. Some examples, using elastic constitutive models include Flaggs and Crossman (1981), Hahn (1984); Hahn and Pagano (1975), and with viscoelastic constitutive models are Weitsman (1980) and Harper and Weitsman (1981). The recent works model the entire cure cycle for the calculation of stresses and take into account other sources of residual stress generation, such as cure shrinkage strains or tool-part interaction, thus making for more complex and accurate models. More information on the sources of residual stress can be found in Johnston etal. (2001). Another aspect o f the models is their dimensionality. The majority of the studies have used onedimensional models in their analyses (e.g. Loos and Springer, 1983; Prasatya et al., 2001; Djokic et al., 2001; A d o l f and Martin, 1996; Lange et al., 1995, 1997). Some others have assumed that the effect of one  Chapter 1: Introduction and Background of the directions does not significantly change the behaviour of the material and modeled the material by 2-D analyses (e.g. Bogetti and Gillespie, 1991, 1992; Johnston et al., 2001). A t the same time, more recent studies have modeled the material in 3-D (e.g. Zhu et al, 2003; Poon et al., 1998) to be able to analyze composite parts of any shape. These models offer a better representation of the geometry and the state of stress in the material and therefore are more desirable. It should be noted, however, that due to the complexity of the constitutive models employed in the available research works these approaches have normally been very time-consuming. A n important consideration in process modeling is the method used to solve the relevant equations over the domain of interest. Some people have used Laminated Plate Theory ( L P T ) based models for stress analysis. LPT-based models are simpler methods and, as clear from their names, are based on the theory of thin plates. This approach can be used to model different lay-ups and material behaviour, but is not able to model complex shapes or boundary conditions. Hahn and Pagano (1975) applied this approach to calculate the stresses in cool-down only. Bogetti and Gillespie (1992), however, modeled the whole cure cycle for this purpose. Some other examples of this approach are Loos and Springer (1983) and White and Hahn (1992). Finite difference has also been used to analyze fairly complex cases (Bogetti and Gillespie, 1992). The method of choice for most studies done recently on composite materials, however, is the finite elements (FE) method. This method enables modeling of more complex cases and is not limited to the assumptions made by L P T . Several researchers have used elastic F E analysis to calculate the cool-down stresses (e.g. Griffin, 1983; Stango and Wang, 1984). However, modeling the whole cure cycle has been more limited in the literature. This is perhaps due to the complexity and requirement of larger computation of FE-based methods. Some examples of this approach include Bogetti (1989), Johnston et al. (2001), and Svanberg and Holmberg (2004a, b). Different aspects of residual stress calculation in process modeling were discussed above and some of the complexities of the problem were mentioned. Taking such complexities of the behaviour into account, we  Chapter 1: Introduction and Background need a suitable approach to properly simulate the response of composite materials during processing. This underlines the importance of the accuracy of the employed models and their ability to take into account different complexities of the process in a comprehensive fashion. In addition, process models usually involve significant numerical runs, consistent with the size and complexity of the industrially relevant composite parts. This makes efficiency of the model a crucial factor in controlling the modeling costs.  1.2.  RESEARCH OBJECTIVES AND THESIS OUTLINE  The objective of this thesis is to develop accurate yet more efficient approaches for modeling residual stresses in thermoset polymer matrix composite materials during cure. This is accomplished by studying the available modeling options and further expanding these capabilities by: •  Studying simpler constitutive models available in the literature, as efficient models  •  Development of a more accurate constitutive model that is also efficient  •  Improving upon other features available in existing constitutive models  •  Implementing the constitutive models in a functional form with Finite Element method  •  Evaluating different models studied for their benefits, limitations, and costs  If these objectives are achieved, a variety of constitutive models for composite materials during cure will be available. This w i l l provide different modeling options in cost vs. accuracy decisions in process modeling of composites. It would also offer new modeling features not currently available in the literature. Based on these objectives, this thesis is organized as follows: Chapter 2 - Review of Constitutive models:  Chapter 1: Introduction and Background A review of constitutive models available in the literature for polymers and composites is presented. Elastic, instantaneously elastic, and viscoelastic models are discussed. Chapter 3 - Pseudo-Viscoelastic  Model:  Instantaneously elastic models are studied as efficient and simple models. The bound of validity and accuracy of these models is discussed and the concept of pseudo-viscoelastic model is introduced. Chapter 4 — Differential Form of Viscoelasticity in ID: A viscoelastic approach based on solving governing differential equations is introduced and the equations for polymer materials in one-dimensional form are derived. The effects of temperature and degree of cure on material behaviour are considered. Also, a comparison with other models in the literature is conducted. Chapter 5 -  Micromechanics:  Micromechanics is used to combine the polymer and fibre mechanical properties in order to obtain the composite properties. The micromechanical approach employed in this thesis for viscoelastic materials is described. Chapter 6 - Differential Form of Viscoelasticity in 3D: Having established the constitutive equations in I D along with a micromechanical model, the equations for a three-dimensional state of stress are derived. The relevant finite element equations are developed. Chapter 7 - Implementation and Verification: The procedure to implement the derived formulation in a code and the coding itself are discussed. Some simple examples to verify the implementation are presented. Chapter 8 - Numerical  Applications:  Chapter 1: Introduction and Background  '  Some case studies are presented to show the capability of different constitutive models, their validity, and their efficiency. First some simple examples are analyzed to show the behaviour of the models more clearly. These are then confirmed by looking at more complex cases. Appendix A — Mathematica  Notebooks:  The Mathematica notebooks, used to obtain analytical solutions for a number of examples are presented. Appendix B - Pseudo Code for the Differential Form Implementation: The detailed algorithm for implementation of this constitutive model in a code is presented. Appendix C - ABAQUS  Viscoelastic Model:  The viscoelastic model available in the commercial code A B A Q U S is discussed. It is also compared to the model implemented in this thesis.  Chapter I: Introduction and Background  Chapter 2: Review of Constitutive models  Chapter 2.  R E V I E W O F CONSTITUTIVE M O D E L S  A suitable constitutive model for the material is an important part of any process model. In this chapter, constitutive models used in the literature in residual stress modeling for polymers and polymer composite during cure are reviewed. Also, since some of the details related to these models are needed for the future chapters, some background information is also presented. First, the mechanical behaviour of the polymer during a curing process is discussed. Then, a review of different models used in the literature to model the polymer and/or composite materials is presented.  2.1. MECHANICAL BEHAVIOUR OF THERMOSET POLYMERS DURING CURE As mentioned in the previous chapter, polymers are taken through a curing process with elevated temperature and pressure, e.g. Figure 2.1 for an autoclave. A s a result, the molecules join and form a 3-D network, i.e. they cure, as shown schematically in Figure 2.2. During processing, the material transforms from a liquid state at the beginning of the process to a gel type material (gelation) as cure advances and subsequently to a solid state (vitrification) by the end of the cure; Figure 2.3 shows these phases in a typical cure cycle. As a result of changes taking place in the polymer, mechanical properties of polymers vary as a function of temperature and the advancement of cure. In addition, there are ample indications in the literature that polymers show viscoelastic behaviour, which is even more pronounced at higher temperatures. There are numerous studies in the literature that show the viscoelastic properties o f polymers; examples of some prominent works are Tobolsky (1958), Leaderman (1958), and Ferry (1980). Taking into consideration the complexities of the materials, a proper constitutive model is needed in residual stress modeling of the composite. A s discussed in the previous chapter, such a model needs to be accurate in taking into account different aspects of the material behaviour and also efficient, due to the size of the problems normally encountered in real world applications.  Chapter 2: Review of Constitutive models Different models in the literature have been used to describe the behaviour of thermoset polymers during cure. A review of the common models available in the literature is presented below.  2.2.  ELASTIC MODELS  In earlier process models, some researchers used purely elastic models to describe the behaviour of the material (e.g. Harper and Weitsman, 1981; Loos and Springer, 1983; Stango and Wang, 1984; Nelson and Cairns, 1989). Although a purely geometric approach, the work by Nelson and Cairns is noteworthy here. They showed that when a curved composite part undergoes a temperature change it would deform due to the difference between through-thickness and in-plane properties. This phenomenon, shown in Figure 2.4, is called "spring-in". For this simple geometry, they showed that the spring-in can be calculated as follows:  A6 = G  (CTE - CTE ) AT e  R  (2.1)  1 + CTE AT R  where A6 is the change in the included angle, 9 is the original angle, CTE  g  and CTE  R  are the laminate  thermal expansion coefficients in the circumferential and radial directions, and AT is the temperature change. B y assuming a stress-free temperature in a cure cycle, the residual deformations are calculated by cooling down from this temperature and taking advantage of the above equation. Elastic models offer good insight in general, but are not sophisticated enough to capture the complexity o f the problem or give quantitatively good results in real cases.  2.3.  C H I L E MODELS  Some researchers have used C H I L E models (this term was coined by Johnston et al., 1996, to denote Cure Hardening Instantaneously Linear Elastic) to analyse the polymer as it goes through the curing process (e.g. Bogetti and Gillespie, 1992; Lange et al., 1995; Johnston et al., 2001; Fernlund, 2002a,  Chapter 2: Review of Constitutive models 200b, 2003; Svanberg and Holmberg, 2004a, b; Antonucci et al, 2006). These models are essentially elastic, in which the modulus of elasticity changes as a function of temperature and degree of cure. In other words:  Aa = E'(T,a)Ae  (2.2)  Or in integral form:  (2.3)  In which, a is the stress, E' is the instantaneous elastic modulus, s is the strain, t is the current time, x is the time integration variable, T is the temperature, and a is the degree of cure. C H I L E models have been popular in the literature due to the ease of their characterization, and finite element implementation. A s mentioned, polymers generally exhibit viscoelastic ( V E ) behaviour. Considering the behaviour of the material it is expected that only using a V E formulation for polymers provide accurate prediction for deformations and stresses. However, the accuracy of the C H I L E model has not been comprehensively investigated before nor have there been enough experimental studies done in the literature to verify the results of either of the models. In general, the C H I L E constitutive models for thermoset polymers have been shown to provide good predictions in analysing the behaviour of composite materials (Fernlund et al., 2002a, 2002b, 2003). Nevertheless, the fundamental validity and applicability of these models have not been proven. A n investigation of this issue, by Zobeiry et al. (2006), w i l l be presented in detail in this thesis.  2.4.  VISCOELASTIC M O D E L S  It is known that polymers in general show viscoelastic behaviour (e.g. Ferry, 1980). This behaviour is especially pronounced for partially cured polymers at high temperatures in a cure cycle. For this reason,  - 10-  Chapter 2: Review of Constitutive models the trend of the literature for residual stress modeling of polymers and polymer composites has been towards employing V E constitutive equations (e.g. Douven et al., 1995; K i m and White, 1996; A d o l f and Martin, 1996; Poon et al., 1998; Wiersma et al., 1998; Prasatya et al., 2001; O'Brien et al., 2001; Zhu, et al., 2003). The modeling of V E materials with Finite Element Method ( F E M ) spans around four decades. The research on this area has not been limited to only polymers or polymer composites. For a review of modeling V E materials using F E M , the reader can refer to Zocher et al. (1997). Early works on composite materials, such as Morland and Lee (1960), established the fundamental equations for composite V E behaviour with temperature change. Most V E works on composites in the past focused on the behaviour of the material with or without change of temperature and sometimes with environmental effects. Some examples of the earlier works are Schapery (1974), Flaggs and Crossman (1981), Nakamura et al. (1988), and L i n and Hwang (1989) and some recent studies include Chen et al. (2000) , Y i et al. (2002), and Ernst et al. (2003). However, it was not until recently that process modeling of composites considered V E behaviour (taking the effects of cure into account as well), perhaps due to the complexity of the analysis. The analyses of composites during cure started with such works as Martin and A d o l f (1990) and White and Hahn (1992a, 1992b). Since then, other researchers have performed V E modeling of composites during processing. Some important references are Douven et al. (1995), A d o l f and Martin (1996), Kokan (1997), K i m and White (1997, 1998), Lange et al. (1997), White and K i m (1998), Poon et al. (1998), Wiersma et al. (1998), Park (2000), Prasatya et al. (2001), O'Brien et al. (2001) , Djokic et al. (2001), K i m et al. (2002), Zhu, et al. (2003), Wijskamp et al. (2003), Lahtinen (2003), Svanberg and Holmberg (2004a, b), Yang et al. (2004a), and Clifford et al. (2006). Most of these studies on composite materials have assumed a linear V E behaviour. In some works, however, the behaviour of materials has been considered to be nonlinear (e.g. Qiao et al., 1994; Chen et al., 2001; Y i et al., 2002) and there is experimental evidence to support this (Schapery, 1969; Qiao et al.,  Chapter 2: Review of Constitutive models 1994; Xiao et al., 1994; X i a et al., 1998). This assumption results in a complex constitutive model, which can be very time consuming both in computer implementation and runtime. In addition, most of these models have not modeled the composite behaviour during cure. Alternatively, a linear V E model is easier when modeling the behaviour of composite materials during cure. In the following sections, a review of different mathematical representations of the V E behaviour used in the literature will be presented.  2.4.1. Integral Form of Viscoelasticity In almost all V E models in the literature, the constitutive equations have been based on the integral form of viscoelasticity, the so-called Integral Form (IF). This includes almost all the references given above that use V E models for composite process modeling. To describe the behaviour of a linear V E material in 1-D with IF, the Boltzmann superposition principle (see e.g. Ferry, 1980) is used. A s a result, the constitutive equation for such a material under isothermal conditions can be written as follows:  (2.4)  In which, E is the relaxation modulus. This equation can be used in homogeneous, non-ageing materials, with no change in temperature. Alternatively, in 3-D we can write:  (2.5)  where, C  iJkl  is the material stiffness tensor.  Under non-isothermal conditions, the best way to incorporate the temperature changes in the constitutive equation is by using time-temperature superposition (see e.g. Ferry). This principle, used in almost all of the recent research works on viscoelasticity, is based on the idea that the effects of temperature and time  - 12-  Chapter 2: Review of Constitutive models on the polymers are similar. In other words, the behaviour of the material, for instance, at increased temperature is equivalent to that after a longer time with the original temperature. In effect, a change in temperature can be replaced by a shift in time. Considering a relaxation modulus vs. time graph, this idea can be shown more clearly by showing the effect of different temperatures, T ,on the modulus, which is a shift in time marked by the shift factor a , as seen in Figure 2.5. In this figure, E" is the unrelaxed r  modulus, and E" is the relaxed modulus. This principle has been extensively used in the literature, especially for the characterization of materials. The behaviour of polymers can span through 10, 15, or 20 decades of time (Tschoegl, 2002b), which can make the characterization difficult or impractical. However, by applying increased or decreased temperature, the behaviour of material for different time scales can be characterized. Examples of this approach are ample, but some include Yee and Takemori (1982), A d o l f and Martin (1990), Douven et al. (1995), Simon et al. (2000), O'Brien et al. (2001), Sane and Knauss (2001b), Ernst et al. (2003), Svanberg and Holmberg (2004a, b), Melo and Radford (2004), and Hojjati et al. (2004). There are cases, such as Plazek (1991, 1996) and He et al. (2004), where time-temperature superposition has not been used. In this reference, creep and recovery data have been combined to construct the entire creep curve. Also, Heymans (2003) has dealt with this using hierarchical models. Most recent studies on V E residual stresses have applied this established principle to combine the degree of cure with time and temperature (time-temperature-cure superposition). However, the proof of validity of this approach does not seem to be present in the literature. Some researchers (e.g. A d o l f and Martin, 1996; Lange et al., 1997) have not taken advantage of time-temperature-cure superposition and have made other assumptions in their analyses. The constitutive equation in 1-D for a polymer in a cure cycle is:  - 13 -  Chapter 2: Review of Constitutive models  cr(f)=  \E(t-T,T,a)—dr  (2.6)  Another important assumption made on the behaviour of polymers in most studies is that the material is thermorheologically simple (see e.g. Schapery, 1974). This means that the constitutive model presented above will become:  (2.7)  In this equation, £ and  are called the reduced time and are calculated as follows:  (2.8)  Put differently, this assumption in effect means that only a horizontal shift in modulus vs. time graph is sufficient to superpose the material properties under different temperatures, which also means that the unrelaxed and relaxed moduli are constant with temperature and degree and cure, as shown in Figure 2.5. Many studies in the literature have used this assumption on different materials (e.g. K i m and White, 1996, Prasatya et al., 2001). For a review of literature on the effects of temperature (and pressure) on thermorheologically simple polymers, one can refer to Tschoegl et al. (2002b). It should also be noted that there are studies in the literature showing thermorheologically complex behaviour for the resin material. Even though most of the work in this thesis will be done on thermorheologically simple, spme discussion on the other types of materials will be presented, e.g. refer to Sections 4.1.2 and 5.3.4. The conventional and most commonly used method of representing the V E material behaviour is to use linear spring and dashpot elements. B y combining springs and dashpots virtually unlimited forms of material behaviour can be modelled. These rheological analogue models have been widely used in the literature to model the curing behaviour of polymer resins. Examples of simple forms of these elements are given in Figure 2.6, Maxwell element, and Figure 2.7, K e l v i n element.  - 14-  Chapter 2: Review of Constitutive models In most available V E models (e.g. K i m and White, 1996; Prasatya et al., 2001), a number of Maxwell elements in parallel are used to model the behaviour of the material, i.e. Figure 2.8. The relaxation modulus of such a material is of the following form, i.e. Prony series:  E(t) = E +(E -E )JTw e^ r  U  r  f  (2.9)  ;=1  In which, w are the weight factors, and r are the relaxation times and N is the number of Maxwell t  i  elements. Another form of modulus commonly used in the literature called the power law (e.g. Djokic, 2001), is as follows:  E(t) = E +(E -E )e~^ ' r  2.4.1.1.  Integration  U  r  y  (2.10)  techniques  Looking at the IF constitutive model, clearly an integration scheme is needed in order to calculate the stresses from strains. Harper (1981), White and Hahn (1992a, 1992b), and Klasztomy and Wilczynski (2004) simply used a direct integration between time zero and the current time, / . This method, when applied to the hereditary integral shown above, would require a re-calculation of this integral at every point in time from time zero. Evidently, this approach, i f applied to process modeling, would be very time consuming and impractical for real world examples. For this reason, some researchers have developed other, mostly recursive, integration techniques. In these approaches, knowledge of the values of the parameters (e.g. stresses) in only one or two previous steps is required for the calculation of their new values, i.e. at current time, during the analysis. This eliminates the need for the history dependent integration. The most common of these approaches was developed by Taylor et al. (1970) and has been used in numerous works, such as L i n and Hwang (1989) and Zhu et al. (2001).  - 15 -  Chapter 2: Review of Constitutive models Zak (1967) and White (1968) presented two other integration methods. According to Hammerand and Kapania (1999), the integration method developed by Taylor et al. is more accurate than these two, due to the way the integrals are evaluated in the approach. Even though these methods are much more efficient than the direct integration approaches mentioned above, they are still very time-consuming. More importantly, looking at these formulations and considering the complexity o f these approaches it is clear that their implementation in a finite element code is very difficult. Another approach was used by Zocher et al. (1997) and also similarly by Kaliske and Rothert (1997), Poon and Ahmad (1998), and Kiasat (2000). The application of this method, sometimes called the internal variable approach, in different studies has been reviewed by Johnson (1999). In this approach, some internal variables, say stresses, are chosen and their values are stored at each time step. These values are then used to calculate the stresses in the next time step. According to Poon et al. (1998), this approach is easier to implement in an available finite element code, since it only requires the manipulation of the code at the integration point level, unlike other established methods such as by Y i et al. (1995) and K i m and White (1997), which 'requires manipulation of the finite element equations at the global level'. It is noted that not all models in this category have written the equations in this way; for example, Zocher et al. appear to have developed the formulation such that it requires manipulation at the global level. Even though these techniques are more efficient than the others mentioned above, there are simplifications involved in most derivations. For example, in these works the reduced time, instead of real time, is discretized. This would require assumptions on the form of the shift factor. This will be discussed more later on. It is important to note that these methods have been developed for polymers modeled with Prony series. According to Simo and Hughes (1998) it appears that similar method for power law or other forms of modulus have not been developed, which would require the use of history dependent integration.  - 16-  Chapter 2: Review of Constitutive models  2.4.2. Fractional models A n alternative approach to V E modeling as reported in the literature is to use fractional models (e.g. Alcoutlabi and Martinez-Vega, 1998; Schmidt and Gaul, 2002). Although these models have not been extensively applied to process modeling their popularity has been on the rise. These models are based on fractional calculus and they are able to model the behaviour of materials using fewer material parameters. Therefore, the characterization of the model parameters is easier. However, it is acknowledged by Schmidt and Gaul that such methods, however elegant, are not numerically efficient in implementation.  2.4.3. Differential Form of Viscoelasticity Another approach to viscoelastic modeling is to use the governing differential equations  as the  constitutive model; this approach will be called the Differential Form (DF) of V E . This form of V E is in fact fundamentally the same as the IF, with a different mathematical representation. A n example of these equations for a Maxwell element (Figure 2.6) is as follows:  +  k  (2-11)  £  n  where, k and n are the spring stiffness and the dashpot viscosity, respectively. It appears that this form o f viscoelastic modeling has been used for modeling the flow of polymers in the field of rheology (see, e.g. Bird et al., 1987 and Macosko, 1994). However, studying these models is outside the scope o f this thesis, limited to modeling V E materials in a solid mechanics framework, and they will not be discussed here. It is noted that when the number of spring/dashpot elements in the analog model are increased to better represent the real behaviour of material, the order of the differential equation grows and can make implementation of this formulation very difficult or even impractical.  - 17-  For the widely-used model  Chapter 2: Review of Constitutive models consisting of N  Maxwell elements in parallel, e.g. Figure 2.8, the differential equation takes the  following general form:  da d'a d"a , ds , d's , d"s a a + a, +... + a, —- +... + a = b s + b, — +... + h —- +... + b ' dt ' dt' " dt" dt ' dt' " dt" n 0  n  0  (2.12)  1  where, all a. and b are constants, with i = l,...,n. i  Even though there are examples of the D F present in the literature, this form of constitutive modeling has not been widely used for calculation of stresses in V E solids; the rapidly increasing complexity of the above equation (as for instance used by Wiersma et al., 1998 and Mokeyev, 2001) is the apparent reason. Zienkiewicz (1968), Bazant (1972, 1974a, 1974b), and Carpenter (1972) first looked at D F for analysis of the behaviour of viscoelastic materials. Some more recent references for general use of the D F include Jurkiewiez (1999), Idesman et al. (2000, 2001), Mesquita et al. (2001), and Mesquita and Coda (2003). Application of this approach for residual stress modeling of composite materials during processing is even more limited. The studies that could be found in the literature are those by E l l y i n et al. (e.g. X i a and Ellyin, 1998; Chen et al., 2001) on non-curing composites, Kokan (1997) on the filament winding process, and Wiersma et al. (1998). Another model, by Zobeiry et al. (Submitted), w i l l be discussed in the coming sections. Ellyin et al. analyzed the material with a multiple-Kelvin nonlinear V E model in cool-down, assuming a stress-free temperature. Wiersma et al. modeled the material properties with a simple combination of a single Maxwell element and a spring. Kokan analyzed a composite cylinder under cure. He assumed multiple Kelvin elements for representation of the transverse behaviour of the material. The material parameters in this study were assumed to be dependent on the degree of cure, but not on temperature.  - 18-  Chapter 2: Review of Constitutive models  2.4.3.1.  Solution techniques  In a similar way to the IF, a solution to the constitutive model is required to relate stresses and strains at each point in time. Since the equations are in the form of differential equations the solutions available in the literature for ordinary differential equations can be used (a review of these solutions can be found in Butcher, 2000). Bazant (1972) and Carpenter (1972) used Runge-Kutta solutions in their V E analysis (see e.g. Butcher, 1996, 2003). However, the method of choice for the majority of studies on D F has been the finite difference method (see Hughes, 1987).  The studies mentioned above have also used different finite  difference schemes, e.g. Kokan, and Mesquita and Coda used a backward Euler method, while Zienkiewicz, Bazant, Jurkiewiez, and Idesman et al. used an exponential form o f solution. In general, the finite difference schemes, no matter how different they may seem, in effect update the stress based on the values of stress at the previous time step. This is in fact very similar to the internal variable approach used in some IF studies, as mentioned in Section 2.4.1.1. These approaches are easier to implement in an available code, since they operate at the integration point level (Poon et al., 1998). Also, due to the nature of finite difference schemes, only knowledge of variables at the previous time-step is needed, unlike the prevalent IF methods, such as Taylor et al. (1970), where knowledge of variables from the previous two time steps is required.  2.5.  SUMMARY  Different constitutive models available in the literature for modeling the behaviour of composite materials during cure were discussed in this chapter. The simplest formulations, i.e. elastic models, are not complex enough to provide accurate results for our applications. The simplest model that seems to be adequate in process modeling is the C H I L E model. It was mentioned that the C H I L E models have been shown to provide good predictions, and are fast and easy to implement.  - 19-  i Chapter 2: Review of Constitutive models  In addition, characterization of material behaviour for input in the codes using this approach, such as C O M P R O (Johnston, 2001), is fairly easy and runtimes are short even on personal computers. However, the formulations based on this model have not been studied carefully for their general validity and accuracy. A s a result, there is a need to reconcile what we might call the pragmatic and economical C H I L E approach with the fundamentally sound viscoelastic approach. This can prove valuable, when efficiency of the model is important. For this reason, a study of this model and its bounds of accuracy and validity w i l l be performed in the next chapter. A t the same time, when a more accurate model is needed or when an analysis falls outside the bounds of validity o f the C H I L E models, a viscoelastic formulation is needed. Most models available in the literature are very time consuming and complex for implementation in finite element codes. Differential form of viscoelasticity has a simpler form than the integral form commonly used in residual stress modeling of composite materials during processing. In addition, these models can be easily implemented in F E codes, since they only need manipulations at the integration point level. There is an apparent lack of models based on differential form of viscoelasticity in this field, but due to their promise of efficiency and simple form they seem to be the best choice for residual stress modeling of composite materials during cure, especially when accuracy of the model is of importance. Study of different models in subsequent chapters will demonstrate a range of constitutive relations that can be used in modeling and demonstrates that there is a continuum of trade-off of investment versus accuracy as we go from a full viscoelastic approach to the simplest elastic approach. A schematic relationship between different models, i.e. elastic, C H I L E , and V E , in general terms is illustrated in Figure 2.9.  -20-  Chapter 2: Review of Constitutive models  2  3  4  5  6  C u r e cycle time (hours)  - Autoclave temperature  Autoclave pressure |  Figure 2.1 A typical temperature/pressure cycle used to cure thermoset polymers in an autoclave  -  s  \ 1  \ /  .  ,  1 /_  Cure Figure 2.2 Joining of molecules as a result of curing in a thermoset polymer  200  100  150  200  250  Time (min)  Figure 2.3 Schematic of changes in the state of a polymer as it goes through a typical cure cycle  -21 -  Chapter 2: Review of Constitutive models  Dash = Initial shape Solid = Final shape LU-I Figure 2.4 Undeformed and deformed shapes in a typical U-shaped composite part after cure  log(time)  Figure 2.5 Schematic of time-temperature superposition  1  k  — W W -  Figure 2.6 A Maxwell element  — D — M Figure 2.7 A Kelvin element  -22-  Chapter 2: Review of Constitutive models  •A/WV-,  —0  WW-  1  K  Tin  Figure 2.8 A generalized Maxwell element  Figure 2.9 Different levels of constitutive modeling  -23 -  Chapter 3: Pseudo-Viscoelastic Model  Chapter 3.  PSEUDO-VISCOELASTIC MODELS  In this chapter we compare the CHILE approach to the 'gold standard' of the viscoelastic approach, discuss why and when CHILE can work well, and determine the bounds of its validity and accuracy. In order to do this, it will be shown that there are approximations that allow the full viscoelastic approach to be simplified to a form of CHILE model, and that these approximations are appropriate for the typical cure cycle that a thermoset polymer undergoes. First, a summary of the viscoelastic models for two different materials will be presented. Then, the necessary equations will be derived and finally, some case studies will be presented. This study can also be found in the reference by Zobeiry et el. (Submitted).  3.1.  R E V I E W OF T w o VISCOELASTIC M O D E L S  In this section, we present a summary of two similar viscoelastic models that have recently been developed for curing thermoset polymers. These formulations are based on the works of Kim and White (1996) and Prasatya et al. (2001).  3.1.1. Material I Recall the integral form of a viscoelastic constitutive model as follows:  (3.1) 0  Where e""" and s'  c  are the total strain and free thermo-chemical strains, respectively.  Kim and White proposed a V E constitutive model for 3501-6 epoxy resin that is formally identical to this equation, in which the relaxation modulus is expressed as:  £ ( « , £ ) = ET (a) + [E  u  (a) - ET ( « ) ] £ W„ (a) exp  -24-  (3.2)  Chapter 3: Pseudo-Viscoelastic Model  where E°° and E" are the relaxed and unrelaxed modulus, respectively, and W  a  and r  a  are the weight  factors and discrete stress relaxation times for the co' Maxwell element (in an assembly of / parallel h  Maxwell elements), respectively. Based on the experimental results for this material, it is assumed that E°° and E" are constants and independent of the degree of cure and temperature. Also, the weight factors are assumed to be constant throughout the cure process. The relaxation times, however, change with the degree of cure according to the following equation:  log(r (a)) = log(r (a )) + [ / ' ( a ) - ( a - a°)logfX)] 0  ffl  where, a  0  ffl  (3.3)  is the reference degree of cure (0.98 in this case) and:  f\a)  = -9.3694 + 0.6089a + 9.1347a  2  (3.4)  (3.5)  where, r is the peak relaxation time shown in Table 3.1 along with other relaxation times and weight p  factors for the reference degree of cure. The reduced time, cf , is calculated as follows:  (3.6)  where, a is the shift function calculated from the following equation: T  log(a ) = c,(a)T + c ( a ) r  2  in which:  -25-  (3.7)  Chapter 3: Pseudo-Viscoelastic Model  c, (a) = -a exp(-^—) - a x  (3.8)  2  a-\  c (a) = -T (a)  (3.9)  0  2  C]  In these equations, for 3501-6 epoxy resin, a =lA/°C,  a = 0 . 0 7 1 2 / ° C , and T° is the reference  l  2  temperature (in this case T° = 3 0 ° C ) .  3.1.2. Material II Simon, McKenna, and co-workers (Simon et al., 2000 and Prasatya et al. 2001) have presented a similar V E model for the bulk behaviour of Hexcel 8551-7 epoxy resin. The form o f the bulk relaxation modulus is as follows:  K(,T,t) =  (3.10)  K +[K -K ]£ exp r  where, K is the bulk modulus, K and K r  g  g  r  g!  are the rubbery and glassy values of the bulk modulus, both  of which are assumed to be independent of temperature and degree o f cure, T, is the i' relaxation time h  and g is the corresponding weight factor, such that _ ^ g , =1.0. Both parameters are assumed to be t  constant throughout the cure cycle. The values for these parameters are listed in Table 3.2. In this model, the shift factor, a  Ta  , is also used to account for the dependence of the relaxation time r, on  temperature and degree of cure and is evaluated as follows:  T-T^  T (a)-T„ g  in which, C and T are constants and T is the glass transition temperature taken to be a function of the m  g  degree of cure as follows:  -26-  Chapter 3: Pseudo-Viscoelastic Model  T.-To  Aa  T „-T  1-(1-A)«  g  where T  g0  is the T  g  ga  of the monomer, T  gai  is the T  g  (3.12)  of the fully cured material, and X is a material  constant. The major difference between this model and the previous one lies in the form of the shift factors (Equation (3.7) vs. Equation (3.11)). For the purpose of deriving the equations in the next section, however, it is desirable to represent the shift factor in the form of Equation (3.7). It can be shown that the difference between these two forms in representing the results of the experiments is minimal (This comparison is only made to show that the two material models are similar and the same derivation procedure can be used for both materials; the actual formulation will not be changed). The graph that is used in Prasatya et al. to derive the shift factor parameters is reproduced here and shown in Figure 3.1. It is clear that Equation (3.7) is a good fit to the results, and thus the same approximation approach will be valid for both materials. For this material the parameters for a fully cured resin are c (a) = -0.2298 and x  T =160°C. 0  There is another difference between the two models. In the reference paper, the stress is calculated from the following formulation: 7  OO  o- = -\K(t-t',T,a)—dt'  (3.13)  v< This equation implicitly assumes a simple definition of reduced time, namely the ratio of time to shift factor, — — , as opposed to the integral in Equation (3.6). Such a definition does not correspond to the  widely used form of Equation (3.6) (see e.g. Schapery, 1974 and also Section 4.1.2 for mathematical derivation). It is unclear how much difference such a definition would make in the results (for examples of some sample cure cycles and a discussion, see Section 3.3.2) but there is nothing to prevent us from  -27-  Chapter 3: Pseudo-Viscoelastic Model using the modulus in Equations (3.10) and (3.11), with the shift factor of Equation (3.6), in the constitutive equation, i.e. Equation (3.1), instead. This approach is adopted here in order to keep the viscoelastic formulation consistent (and correct) for both materials.  3.2.  PSEUDO-VISCOELASTIC M O D E L DERIVATION  As stated in Chapter 2, a C H I L E model is essentially an elastic model in which the modulus is assumed to be constant at each instant of time, but changing as a function of temperature and degree of cure. However, since the real behaviour o f the material is viscoelastic, in order to use an essentially elastic model one needs to have a proper definition of the elastic modulus. In order to do this, one can perform dynamic tests at some pre-determined frequency (or static tests at a certain time), and the resulting modulus w i l l be a function o f this frequency (or time). Note that for simplicity, where appropriate the term frequency or time w i l l be used interchangeably from here on. To date, there has been little or no rationalization on how this frequency should be selected; and typically a low frequency, say OA Hz, is used.  Although such values are intuitively valid their selection is relatively arbitrary (e.g. Johnston,  2001). Here, it will be shown that one can define, a priori, an elastic modulus for a C H I L E model as being the viscoelastic relaxation modulus at a specific time, or the viscoelastic storage modulus at a certain frequency, as obtained from a cyclic loading test, e.g., a dynamic mechanical analyzer ( D M A ) . With this elastic modulus as input to a C H I L E model, the results will be approximately equivalent to the viscoelastic approach. Thus this special case of an appropriately calibrated C H I L E model will be called a "Pseudo-ViscoElastic ( P V E ) " model.  Recalling the C H I L E constitutive equation, we can write:  (3.14)  In this equation, E' is the instantaneous elastic modulus evaluated at some frequency.  -28-  Chapter 3: Pseudo-Viscoelastic Model  Since temperature and degree o f cure can be described as functions o f time, E' can be thought of as an implicit function of time and therefore Equation (3.14) may be rewritten as:  c r ( 0 = W > ^ r  (3.15)  3.2.1. Simplification of Viscoelastic Formulations We will now determine the conditions under which the V E formulation simplifies to a corresponding P V E formulation for a wide range of conventional cure cycles. In a viscoelastic material, rewriting Equation (3.1) here in a more convenient form, stress can be calculated as follows:  <T(t)=lE(£-?)^-dT  dr  (3.16)  in which,  ' 1 1 cf = \—dt' and cf' = \—dt' r  0 T  (3.17)  0 T  a  a  The objective here is to determine how Equation (3.16) can be simplified so that the stress can be calculated from Equation (3.15). Now, i f the following identity:  E'(r) = E(£-?)  (3.18)  can be enforced, the resulting stress from Equations (3.15) and (3.16) w i l l necessarily be equal. This equality may seem odd, since the right-hand side is a viscoelastic modulus that is a function of both t (current time) and r (integration time), while the left-hand side is an elastic modulus that is only a function o f a specific instant o f time r . For this purpose, E' is defined such that it coincides with the  -29-  Chapter 3: Pseudo-Viscoelastic Model value of the relaxation modulus, E , at a suitable and as yet undetermined time, say t , that is appropriate e  for the temperature and degree of cure at time r , i.e. T(r)  and a(x).  According to this definition, we  can write:  E'(T) = E(t ,T(r),a(T))  .  e  (3.19)  Invoking the time-temperature superposition and noting that in this case r represents a fixed instant of  time and not a dummy integration variable, the reduced time becomes  [—-— = — - — , and the relaxation  a (r)  J 0  r  a (r) T  modulus can be expressed as:  E(t ,T,a) = E(-^-) e  (3.20)  a (r) T  From Equations (3.18) - (3.20), we have:  £(-4-r) =  (3.21)  a (r) T  This equation can only be valid i f the arguments of E are equal, i.e  (3.22)  a (r) T  Substituting Equation (3.17) here, we obtain:  J ^ J f - J f - J f L a  r( ) T  o r a  oT a  (3  .23)  tT a  In order to find suitable values of t , simplifications in the V E formulation are made by considering two e  distinct temperature regimes within the cure cycle: the cool-down, in which the temperature varies (reduces) linearly with time, and the hold, in which the temperature is constant. These temperature regimes are characteristic of all conventional cure cycles.  -30-  Chapter 3: Pseudo-Viscoelastic Model  3.2.1.1.  Simplification during cool-down  The temperature-time relationship in the cool-down segment can be written in a generic form as follows:  T(t') = -m(t'-p)  +q  (3.24)  in which, m is the cooling rate, t' is the time, and p and q are constants. Now, using Equation (3.7) for the shift factor we can write:  a =  w r + c  T  >  = exp(  w  ) log(e)  C | r +  (3.25)  C 2  B y substituting these two equations into Equation (3.23), assuming that the degree of cure is constant during the cool-down (a valid assumption), say a ,  performing the integral, and simplifying, we obtain:  f  1  1  a (r)  a (r)  (3.26)  a (t)  T  r  T  where  t  ]  =--^L c^(a )m  (3.27)  f  Note that c,(a) is a negative number. N o w , since the shift factor changes exponentially, its final value is  much greater than its value at any other instant during the cool-down. In other words, —-— « — - — and a (t) a (r) T  T  consequently, t a t . e  l  The assumption that the shift factor at t is much greater than that at r is particularly true when t is the duration of the cure cycle and the final residual stress is being calculated. When r approaches t in the cool-down regime the error resulting from such an assumption increases. However, this will not cause a significant error in the calculations, because when r approaches t, the value of  -31 -  tends to zero and  Chapter 3: Pseudo-Viscoelastic Model the relaxation modulus at time t will be equal to the unrelaxed modulus. Thus, an error in evaluating the e  shift factor will not affect the results significantly, as the modulus will almost be equal to the unrelaxed modulus. Thus using the following simple equation will yield stresses that are equivalent to those predicted by the full V E model for the cool-down:  (3.28)  To show the effect of this simplification, the reduced times and moduli from the P V E and full V E methods are compared for a cure temperature of 1 8 0 ° C , a * l , and cool-down to room temperature at three different cooling rates, namely for m = \, 3, and 5 ° C / m i n . The results for  and moduli  development are plotted versus r (for t equal to the time at the end of the cure cycle) in Figure 3.2 and Figure 3.3, respectively. It can be seen that the reduced times obtained from the two methods are very close. Some errors occur, as r approaches t. But, as explained before and shown in Figure 3.3, this error in evaluating the reduced time does not result in a large difference in the predicted modulus. These figures clearly indicate that by using the P V E method in cool-down the predicted stresses should closely match those calculated by the far more computationally intensive V E model. It is also noted that the slight waviness in Figure 3.3 is due to discrete relaxation spectrum of the modulus, i.e. the relaxation modulus consists of several M a x w e l l elements that relax in different time frames and cause the waviness. A n important observation is that the value of t  e  is a function of the cooling rate m . This means that for  any given cure cycle, the appropriate time should be calculated and used.  3.2.1.2.  Simplification during temperature hold  On the hold, Equation (3.23) can be written as:  -32-  Chapter 3: Pseudo-Viscoelastic Model  J___ f___ ,  T  f  f__  +  f__  ,  \a  ia  a (r) where, t  , /  r  (3.29)  Ja  r  T  denotes the time at the end o f the hold and beginning o f the cool-down regime. The second  integral pertaining to the cool-down has already been evaluated in the previous section. Therefore, we merely need to simplify the first integral. We note that during the hold the variation o f shift factor as a function time is at the limit either the form shown in Figure 3.4, or constant with time. To evaluate the integral, which is the area under the curve, we define a time t , prior to which — decreases linearly with time, and thereafter it is constant: 2  a  T  1  a (t ) T  =P—!— + ( ! - / * ) — ? —  a (r)  2  (3.30)  a (t )  r  T  f  The form o f the above equation captures both limits of the variation o f — , i f (5 is chosen to be an  a  T  appropriately small number. Generally, the observed behaviour is closer to the curve in Figure 3.4, where ^ time • after t , we have  1  «  2  a {t')  1  .  a (r)  T  r  The area under the curve can then be found as: /  f^ i(, _ )|_^ _^] (, =  a  T  2  2  r  +  V  67 (r)  +  a (t ) j T  r  2  /  ?  2  )_^ a (t ) T  (3.3D  f  In the cases where — is not constant past t , there will be some error in evaluating the integral. 2  a  r  However, this error is not significant since the first term is dominant. N o w , using Equations (3.29) and (3.31) we have:  -33 -  Chapter 3: Pseudo-Viscoelastic Model  ' - - V ' > fVi ?a£(tV) ( < , - ' . )a?(t£) « , +  T  2  T  <" > 2  f  This gives us the value of time to evaluate the modulus during the hold. We note that /? is chosen to be a  small number, but it should not be so small that the approximately linear form o f — between times r  a  r  and t is undermined. A value of 0.1 has been observed to work well. 2  It should be noted that in order to use the above equations it is assumed that the degree of cure is known at every time step during the hold. These values are needed in order to compute t from Equation (3.30). 2  A s a result, one needs to perform the thermo-chemical analysis on the model first before the residual stress analysis. If the values o f the degree o f cure cannot be calculated beforehand, one can assume t = T and therefore 2  use a simpler form o f the above equation as follows:  0.33) In this equation, only the knowledge of the degree of cure at time /  / ;  i.e. the onset o f cool-down, is  sufficient.  3.2.1.3.  Summary of equations  Here, a summary of the P V E equations is presented. The stresses are calculated as follows:  er(0 = )E(t ,T(T),a(T))^dT e  Where there are two regimes to calculate t , during cool-down and during temperature hold. e  During cool-down:  -34-  (3.34)  Chapter 3: Pseudo-Viscoelastic Model  r . = / , = - ^ f c, {af )m Where a  f  (3.35)  is the degree of cure at the onset of cool-down and m is the cool-down rate.  During temperature hold:  , = I(, - ) #  2  r  1  +  «rM  (, -  +  a (t ) r  +  ,  (3  .36)  f  Where t is the time at the onset of cool-down and t is a time, defined as follows: f  2  1  T  =P— — + (\-p)-^— {  a (t ) 2  a (r)  a (t )  T  r  (3.37)  f  P is chosen to be a small number. If the values of the degree of cure cannot be determined beforehand, the following can be used during the temperature hold:  3.2.2.  Different Forms of P V E  The above simplification assumes that the material is fully characterized viscoelastically, but that we wish to be computationally efficient. From now on, this simplification procedure will be called the 'variable time' P V E method, where the instant of time at which the elastic modulus is calculated varies throughout the cure cycle. This value of time clearly comes from a direct simplification of the V E constitutive model. Alternatively, i f one wishes to be even more efficient and simple, a single, constant time can be defined at which the modulus is calculated at all points in the cure cycle. One could then save significantly on characterization costs. This can also be done using a constant value o f frequency. The incentive for the latter is to be able to use the results o f cyclic loading tests, such as D M A (Dynamic Mechanical  -35 -  Chapter 3: Pseudo-Viscoelastic Model Analyzer), used for material characterization. These additionally simplified procedures will be called the 'constant time' or 'constant frequency' P V E method, noting that the added simplification comes at a cost of reduced accuracy. The best estimate for this constant value of time is the one that applies to the cool-down, as calculated by Equation (3.27). If use of a constant frequency is desired we note from Ferry (1980) that:  G(t) = G'{(o) - 0.40G"(0.40<y) + 0.014G"(10<y)  (3.39)  in which G is the relaxation modulus, G' is the storage modulus, and G" is the loss modulus when co , the circular frequency, is equated to j/^. This shows the relationship between the time- and frequencydependent moduli. This equation can be simplified even more. According to A d o l f and Martin (1996) for most thermoset polymers G" is small and the above equation can be converted to the following simple form:  G(t) = G'(o) = ^)  (3.40)  Therefore, the relaxation modulus evaluated at a specific time is equivalent to the storage modulus at a relevant frequency. In other words, the following value of frequency  1  c.(a )m f  a> = - = - ' h log(e) 1  (3.41)  To check the validity of this equation, the values of the moduli on both sides o f the equation for both materials presented previously were compared, using the sample cure cycle shown in Figure 3.5. The results are shown in Figure 3.6 and Figure 3.7. It can be seen that the difference between the two moduli after vitrification is less than 15%, and the two sides gradually converge as the polymer is cooled down. As a result, the difference between the stresses calculated from the two definitions of modulus will be minimal.  -36-  Chapter 3: Pseudo-Viscoelastic Model It should be noted that i f Equation (3.27) is not used, and an intuitive or arbitrary value of frequency is used instead, then we have the classical C H I L E model availabel in the literature (e.g. Johnston et al. 2001).  3.3.  C A S E STUDIES  To evaluate the validity of the simplification procedure introduced in the previous section and the different forms of P V E some case studies are presented here. For simplicity, a fully constrained block of polymer undergoing a given cure cycle is considered. Neglecting the influence of external mechanical loads (e.g. autoclave pressure) the induced strain will consist only of thermal and cure shrinkage contributions. The advancement of cure must be taken into account using an appropriate thermo-chemical model. To perform the analysis, a simple F O R T A N code was written. This program analyses a 1-D structure (i.e. a bar) using a P V E and/or a full V E formulation as it undergoes the cure cycle and calculates the residual stress at each instant. It is noted that the bar is held at fixed length and the end effects are not examined.  3.3.1. Material I The temperature profile for a conventional one-hold cure cycle and the resulting degree of cure are shown in Figure 3.5. The thermo-chemical model is based on the work by Lee et al. (1982). The time histories of the elastic modulus of the P V E , and stress resulting from a variable time P V E and V E are presented in Figure 3.8 and Figure 3.9, respectively. The agreement between the stress predictions of variable-time P V E and the full V E methods is excellent. It is noted that the relaxed modulus is assumed to be present throughout the cure cycle and hence it also exists before gelation. This is necessary, as the material is assumed to be a viscoelastic solid, which has a non-zero relaxed modulus. The temperature profile for a two-hold cure cycle and the resulting degree of cure are shown in Figure 3.10. A constant frequency P V E analysis at frequencies of 0.1 Hz,  -37-  0.001 Hz,  and 0.0001 Hz  was  Chapter 3: Pseudo-Viscoelastic Model performed and the results are presented in Figure 3.11 and Figure 3.12. A l s o shown in Figure 3.12 are the stresses predicted by the V E model. Clearly an accurate prediction results i f the selected frequency is somewhere between 0.001 Hz and 0.0001 Hz . To investigate the effect of frequency on residual stress prediction, the P V E predictions are plotted versus frequency in Figure 3.13 and compared with the unique (frequency independent) value of the stress obtained from the V E model. A frequency of around 2.7 x l O ^ T / z yields a residual stress that closely matches the V E predicted stress. Use of Equations (3.27) and (3.40) gives 9.0 x 10" Hz, which leads to a 4  P V E prediction of only about 1 MP a higher than the V E prediction, thus suggesting that Equation (3.40) is a good estimate of the appropriate frequency to characterize a material. It should be noted that these frequencies are quite low, and thus higher frequencies, as used in a large number o f cases in the literature, will result in slightly higher stresses. This is perhaps unavoidable since the required frequencies are experimentally impractical. To evaluate the effect of the cooling rate on the residual stresses, several one-hold cure cycles with varying cooling rates have been analyzed. Figure 3.14 and Figure 3.15 show the variable time P V E and V E predicted residual stress as a function of cooling rate for these cycles at two different hold temperatures. Note that in each case the temperature hold continues until complete cure prior to cooldown. In all cases, very good agreement is obtained between the V E and P V E results. Furthermore, these results show that the effect of cooling rate on the residual stress is only significant at very low rates o f cooling (between 0.1 to 1 ° C / m i n ) . Constant time P V E predictions for these case studies are presented in Figure 3.16 and Figure 3.17. The appropriate time for each cooling rate is calculated using Equation (3.40). While the correlation between these P V E predictions and the V E predictions is not as good, with a maximum error of 8%, it is still quite acceptable. This suggests that when we use an appropriately calculated constant time to evaluate the modulus in a P V E model, a reasonable estimate of residual stresses is obtained. Note that the biggest  -38-  Chapter 3: Pseudo-Viscoelastic Model contributor to error is the stress build up during hold. In fact, the stresses generated during cool-down are observed to match closely.  3.3.2.  M a t e r i a l II  In Prasatya et al. (2001), V E analyses were performed for a wide range of cure cycles to obtain a large variation in predicted residual stress: This set of cure cycles is therefore particularly useful in determining the robustness of the P V E approach. Figure 3.18 compares the variable time P V E predictions with the V E predictions. Note that because of the discussion presented in Section 3.1.2, the definition of the reduced time was changed and the V E results for all the cure cycles (27 in total) presented in the reference were obtained from the I D code. It is noteworthy here that the results were also obtained for a simple definition of reduced time (Section 3.1.2) to evaluate the difference that this definition can make. It was observed that even though the definitions of the reduced time are fundamentally different the difference in the results were not significant, e.g. stresses range from -7.1 to 37.4 M P a at the cure temperature and from 69.7 to 98.5 M P a at room temperature for this definition of reduced time, while these ranges for the integral definition of reduced time are -7.3 to 39.6 M P a and 73 to 102.6 M P a , respectively. To explain the reason why the difference between the results is not large, consider the following thought experiment. Assume that a unit of strain is applied to a thermoplastic (no cure) at time / = 0 and the resulting stress is monitored. The temperature is held at 7], except that for a period it is increased to T . T h i s is shown 2  schematically in Figure 3.19. Consider the difference in predictions from the two definitions of reduced time at times t , A  •  At t , A  t , and t : B  c  both definitions will give the same relaxation modulus and the correct answer. This is  obviously the classical (and trivial) constant temperature stress relaxation problem.  -39-  Chapter 3: Pseudo-Viscoelastic Model  •  A t t , the non-integral reduced time will over-estimate the relaxation, as the temperature will be B  assumed to have been T all the time. 2  •  A t t , the non-integral reduced time will under-estimate the relaxation, as the temperature will be c  assumed to have been 7j all the time.  Given the above discussion, it seems the reason why the two definitions give relatively close answers is  the same as why the P V E approach works. In general, — tends to zero for cure problems, as either cure  progresses and/or temperature decreases. Therefore, the relevant relaxation for a unit of strain that is introduced at a given time occurs within a short window after that. So long as that is captured reasonably well, the error is small. Even though the difference between the two definitions of reduced time is not large, the integral definition will be used from now on. It should also be noted that for all other parameters (e.g. shift factor and glass transition temperature) were calculated from the formulations presented in the reference, i.e. as in Section 3.1.2. In Figure 3.18, excellent agreement between V E and variable time P V E can be observed over a very wide range of predicted stresses, corresponding to very different cure cycles. In similar fashion, Figure 3.20 compares the constant frequency P V E predictions with the V E predictions. The appropriate frequency can be calculated from Equations (3.27) and (3.40), i.e. :  1  a =- =h  c,(a )m f  ' log(e) f  (3.42)  which for this case it is 1.33xl0~ /fe. A s before, the agreement is not as good, but it is still very 2  acceptable. In addition, it is observed that a few points deviate noticeably from the line. A l l but one of these stresses represent cure cycles where the hold temperature is low ( 1 3 0 ' C - 1 4 0 ° C ) and the P V E  -40-  Chapter 3: Pseudo-Viscoelastic Model stresses at the end of hold show considerable error in prediction. In one case, the deviation is due to the fact that the material is heated-up after it is sufficiently solidified, which is against the assumption of cool-down in the simplification of V E to P V E . In all these cases, however, the P V E predictions for the stresses generated in cool-down are very close to those from the V E analysis.  3.3.3. Comparison of Efficiency A s mentioned previously, one of the advantages anticipated for the P V E model in comparison to the integral form of V E is efficiency. In order to quantity this, the run-times for both methods can be measured and compared. For this purpose, a one-hold cure cycle has been analyzed with the I D code mentioned previously, using different number of time steps. To make an objective comparison, the run times have been measured for the stress calculation core of the code only. Also, to get tangible runtime numbers, the core section has been repeated multiple times. First, to discount the effect of thermo-chemical calculations on the run times and to compare only the stress constitutive models, a completely cured resin has been analyzed. The cure cycle has been analysed with different number of time steps and for each the core of the code has been repeated a specific number of time. The results for this case are shown in Table 3.3. It is observed that P V E runs approximately 15 times faster than the full V E formulation. Now, i f the runs are repeated with the thermo-chemical model included, since calculation of cure takes significant time the absolute benefit of the P V E approach becomes smaller in this case. Nevertheless, the P V E model is still significantly faster (approximately 5 times) than the integral form. The results for this case are presented in Table 3.4. These examples show a large difference in the run times for the two models even for the 1-D system considered here. Clearly, this difference will be magnified considerably when analysing larger problems requiring 2-D or 3-D numerical solutions.  -41 -  Chapter 3: Pseudo-Viscoelastic Model  3.4.  S U M M A R Y AND DISCUSSION  In this chapter, based on a 1-D closed form analysis and simplification procedure, it has been shown that for conventional cycles consisting of temperature holds and cool-down ramps: •  A Viscoelastic formulation for the cure of thermoset polymer composite material can be simplified to a variable time Pseudo-Viscoelastic formulation, with excellent agreement in the predicted residual stress, for a wide range of cure cycles. A variable time P V E formulation requires a full V E material characterization, but permits a much more efficient numerical implementation.  •  A V E formulation can be simplified even further to a constant time/frequency P V E formulation, with slightly reduced agreement, but still within acceptable limits.  The major benefit of this  additional simplification is that characterization tests can be performed at a single frequency or time, depending on the cure cycle being used.  This leads to efficiencies in both material  characterization and numerical analysis. •  The C H I L E models in the literature are clearly constant time/frequency P V E formulations, except that the frequency or time at which the material is characterized is relatively arbitrary. However, so long as a relatively low test frequency has been used, the results should be reasonably accurate.  Table 3.5 provides a summary of the trade-offs  between the different options. Note that P V E  approximations for temperature holds and cool-down ramps have been presented.  Similar P V E  approximations can be derived for heat-up ramps or other permutations and variations of temperature cycles for even more generalized processing cases. It must be noted that the simplifications and the relevant conclusions presented in this chapter are valid for typical cure cycles. Therefore, i f unconventional cure cycles are to be analyzed or when the accuracy  -42-  Chapter 3: Pseudo-Viscoelastic Model of the model is of importance, a V E analysis is necessary. A s mentioned before, one the main problems in using the available V E models in the literature is that they are very inefficient in runtimes. Therefore, i f an efficient V E model can be developed it will be a very desirable modeling option that can complement the P V E approaches in presenting modellers with different choices of analysis tools. The next chapter will therefore deal with developing a viscoelastic model in differential form, which looks promising in this regard.  -43 -  Chapter 3: Pseudo-Viscoelastic Model  Table 3.1 Relaxation times and weight factors for 3501-6 resin for a = 0.98 0  (0  T<»,(min)  1 2 3 4 5  2.92E+01 2.92E+03 1.82E+05 140E+07 2.83E+08 7.94E+09  0.059 0.066 0.083 0.112 0.154 0.262  1.95E+11 3.32E+12 4.92E+14  0.184 0.049 0.025  6(t ) 7 8 9 p  Table 3.2 Relaxation times and weight factors for 8551-7 resin logfr/s) -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5  gi 0.0215 0.0215 0.0215 0.0215 0.0267 0.0267 0.0375 0.0405 0.0630 0.0630 0.1054 0.1160  log(ii/s)  gi 0.1160 0.1653 0.0561 0.0561 0.0199 0.0119 0.0055 0.0028 0.0008 0.0002 0.0003 0.0003  -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 5.0  Table 3.3 Comparison of run times for integral form V E and P V E for a cured resin  Integral fa rm  Pseudo Viscoelastic Repetitions  Steps  Total core run-time  Core run-time  Total core run-time  Core run-time  200  500  0.13  0.00065  1.95  0.00975  15.00  100  5000  0.64  0.0064  9.8  0.098  15.31  20  20000  0.51  0.0255  7.9  0.395  15.49  20  40000  1.04  0.052  15.8  0.79  15.19  -44-  IF/PVE  Chapter 3: Pseudo-Viscoelastic Model Table 3.4 C o m p a r i s o n of r u n times for integral form V E and P V E for a curing resin in a one-hold cure cycle  Pseudo Viscoelastic  Integral form  Repetitions  Steps  Total core run-time  Core run-time  Total core run-time  Core run-time  IF/PVE  100  500  0.27  0.0027  1.29  0.0129  4.78  100  5000  2.67  0.0267  12.89  0.1289  4.83  20  20000  2.15  0.1075  10.3  0.515  4.79  20  40000  4.29  0.2145  20.56  1.028  4.79  Table 3.5 Comparison of the features of the different models Characterization Run-time Costs  Constitutive Model  Material Characterization  Accuracy  Integral form V E  Viscoelastic characterization of materials using methods such as relaxation tests  Baseline  High  Baseline  Variable time PVE based on simplification of V E  Same as above  Excellent, especially for predicting the final residual stresses  High  15 times less*  Good  Low  15 times less  Good, given a proper selection of the test frequency  Lowest  15 times less  Constant time/frequency PVE Dynamic tests at a constant frequency based on further simplification calculated from shift factor behaviour ofVE CHILE model  Dynamic tests at a constant frequency selected essentially arbitrarily  * numbers for a fully cured thermoset  -45-  Chapter 3: Pseudo-Viscoelastic Model  8 1.E+05 3 0 1  1.E+03 1.E+01 H 1.E-01 «  60  80  100  120  140  160  180  Cool-down time from 180°C (min)  Figure 3.2 Comparison of reduced times for PVE and VE models for 3501^6 resin (with a=0.99) for different values of cool-down rates  -46-  Chapter 3: Pseudo-Viscoelastic Model  3500 -i  180 Cool-down Time from 180°C(min)  Figure 3.3 Comparison of moduli for P V E and V E models for 3501-6 resin (with a=0.99) for different values of cool-down rates  1/Uy  Figure 3.4 Schematic of a typical variation of shift factor as a function of time during a hold  -47-  Chapter 3: Pseudo-Viscoelastic Model  t(min) Figure 3.5 Temperature and degree of cure for the cure cycle used for 3501-6 resin  -- 0.55 1 -I 0  -n  50  1  — i  100  150  1 200  1- 0.5 250  Time(min)  Figure 3.6 Comparison of two definitions of modulus, based on storage modulus and relaxation modulus, for 3501-6 resin  -48-  Chapter 3: Pseudo-Viscoelastic Model  1  10000  0.95 0.9 0.85  a. E  = •a  0.8 0.75  1000  o E  0.7  •G"fc>) E(t=1/ ) E/G' ffi  0.65 0.6 0.55 0.5  100 20  60  40  80  Time(min)  Figure 3.7 Comparison of two definitions of modulus, based on storage modulus and relaxation modulus, for 8551-7 resin  -49-  Chapter 3: Pseudo-Viscoelastic Model  25 20  /  ~ 15 H ra a.  PVE  VE  50  150  100  200  250  t(min)  Figure 3.9 Comparison of stress profiles for variable time Pseudo-Viscoelastic and Viscoelastic models for the cure cycle in Figure 3.5 (3501-6 resin)  t(min) Figure 3.10 Temperature and degree of cure profile for the two-hold cycle used in the case study for 3501-6 resin  -50-  Chapter 3: Pseudo-Viscoelastic Model  3500 n  0  50  100  150  200  250  300  350  t(min) Figure 3.11 Elastic modulus profiles for different constant frequency Pseudo-Viscoelastic models for the cure cycle in Figure 3.10 (3501-6 resin)  Figure 3.12 Comparison of stress profiles from the constant frequency Pseudo-Viscoelastic models and the Viscoelastic model for the cure cycle in Figure 3.10 (3501-6 resin)  -51 -  Chapter 3: Pseudo-Viscoelastic Model  2.7x10-  4  -3  -2  Log(Frequency) Figure 3.13 Residual stress as a function of frequency for the constant-frequency P V E model, the cure cycle in Figure 3.10 (3501-6 resin). Also shown is the V E model prediction  25 20 • P V E g T(room)  15  • V E @ , T(room)  w  A P V E @ T(hold)  2 10  W E < § T(hold)  5 0 p. X 4  6  8  10  12  m (C/min) Figure 3.14 Predicted residual stress for variable time Pseudo-Viscoelastic and Viscoelastic models for different cooling rates for a one-hold cycle with T i =180 ° C (3501-6 resin) ho  -52-  d  Chapter 3: Pseudo-Viscoelastic Model  25 20 in  • PVE @ T(room) • VE @ T(room) A PVE @ T(hold) VE @ T(hold)  15 01 2 10 H 5 0  UA  A  4  6  8  10  12  m (C/min) Figure 3.15 Predicted residual stress for variable time Pseudo-Viscoelastic and Viscoelastic models for different cooling rates, m, for a one-hold cycle with T i =160 °C (3501-6 resin) ho  d  Constant Time (min) 3.05  25  1.53  1.02  0.76  0.61  0.51  :  20  • P V E @ T(room)  ro a. 15  • VE @ T(room) A P V E @ T(hold) XVE@T(hold)  (0 £ 10  • P V E for cooldown  V)  + VE for cooldown  5 0 aat  A 6  8  10  12  m (C/min)  Figure 3.16 Predicted residual stress for constant time Pseudo-Viscoelastic and Viscoelastic models for different cooling rates, m , for a one-hold cycle with T id=180 °C (3501-6 resin) ho  -53-  Chapter 3: Pseudo-Viscoelastic Model  Constant Time (min) 25  3.05  1.02  0.76  0.61  0.51  •  •  20  k  1.53  a  • P V E @ T(room) • V E @ T(room)  15  • P V E @ T(hold) X VE @ T(hold)  10  • P V E for cooldown i V E for cooldown  5 0  ~i—  —i—  2  4  - 1 —  6  12  10  m (C/min)  Figure 3.17 Predicted residual stress for constant time Pseudo-Viscoelastic and Viscoelastic models for different cooling rates, m, for a one-hold cycle with T id 160 °C (3501-6 resin) =  hO  PVE Stresses (MPa) Figure 3.18 Variable time Pseudo-Viscoelastic and Viscoelastic stress predictions for a wide range of different cure cycles (8551-7 resin)  -54-  Chapter 3: Pseudo-Viscoelastic Model  T  t Figure 3.19 Schematic of a temperature cycle for the discussion in Section 3.3.2.  PVE Stresses (MPa) Figure 3.20 Constant frequency Pseudo-Viscoelastic and Viscoelastic stress predictions for a wide range of different cure cycles (8551-7 resin)  -55-  Chapter 4: Differential Form of Viscoelasticity in ID  Chapter 4.  DIFFERENTIAL FORM OF VISCOELASTICITY IN I D  In this.chapter, the differential form (DF) o f V E is presented as an efficient and accurate alternative method to the common integral form (IF) in modeling the viscoelastic behaviour o f composites during cure. For this purpose, only the one-dimensional behaviour, effectively the longitudinal behaviour of a polymer, is considered in order to develop the model in a simple form and be able to compare the two approaches. First, the governing differential equations for a linear viscoelastic material in 1-dimensional (ID) cases, using spring and dashpot elements and in particular with a number o f Maxwell elements in parallel, during cure are derived. Then, the constitutive model is compared to the IF of viscoelasticity. This is achieved by using some numerical examples in I D to compare the methods' results and efficiency.  4.1. GOVERNING EQUATIONS IN ID In this section, spring and dashpot elements are used to derive the governing V E differential equations in I D for isothermal and non-isothermal (and curing) materials.  4.1.1. Differential Equations for Non-Curing Materials 4.1.1.1.  Single Maxwell element  For a single Maxwell element, the constitutive equation is:  £  =  —  k  +  —  n  (4.1)  where, k is the spring stiffness and rj is the dashpot viscosity. Written differently, the equation is:  ke = & H — r  -56-  (4.2)  Chapter 4: Differential Form of Viscoelasticity in ID It is generally understood in the literature that the D F and IF formulations are equivalent and are merely two different forms to represent the same behaviour. This can be shown for a single Maxwell element in the above equation. Recall that in the IF:  a(t)=\E(t-t')^dt' „ dt'  (4.3)  J  where for a single Maxwell element the relaxation modulus is:  E(t) = kexp(--)  (4.4)  T  In order to show that the two forms are equivalent, we take the Laplace transform of both sides of Equation (4.2). Taking advantage of the Laplace transform of a convolution integral and using the Laplace transformation of derivative of a function, we will have: oo  CO  k \se-"dt = J(<T + -)e~ 'dt  (4.5)  s  0  0  T  Or:  k[ss(s) - s(0)] = [sa(s) - o-(O)] +  (4.6) r  where  A  over a parameter denotes a transformed function in the Laplace space.  B y putting <r(0) = E(0) = 0 and rearranging, we obtain:  '  •  &(s) = s^ji(s) s+— r  Taking an inverse Laplace transform, the result is:  -57-  (4.7)  Chapter 4: Differential Form of Viscoelasticity in ID  k  a{f) = £1-1  * r ' (ss(s))  1 V  (4.8)  s + TJ -  where I '() is the inverse Laplace transform of ()and * represents a convolution integral Taking advantage of some common rules for Laplace transform and noting that s(Q) = 0 , we will have:  dt  •  dt'  (4.9)  v  '  which is equivalent to the IF in Equation (4.4).  4.1.1.2.  Generalized Maxwell model  As mentioned before, to represent the real behaviour of materials, several Maxwell elements in parallel are used. B y adding more Maxwell elements, it is evident that the order of the differential equation grows and can make implementation of this formulation very difficult or even impractical (see Chapter 2). However, the alternative approach is to write multiple, simpler differential equations. For the mechanical analog model presented in Figure 4.1 (b), the generalized Maxwell model, it is obvious that:  s =e =... l  2  = s =e ll  (4.10)  Recalling Equation (4.2), we can write the following equation for each Maxwell element:  o-,=M-—  (4.11)  r,  in which, the subscript / denotes the Maxwell element number and:  (4.12)  where r, 's are the relaxation times.  -58-  Chapter 4: Differential Form of Viscoelasticity in ID Now, since we have:  * =X>,  (4.13)  The governing differential equation becomes:  (4.14)  where k" - ^k.  is the unrelaxed modulus of the material. Equations (4.13) and (4.14) constitute the V E  constitutive equations for non-curing materials. Note that since the material is a viscoelastic solid, the relaxed modulus is non-zero and as a result, one of the elements in Figure 4.1 is merely a spring, representing the relaxed modulus, in parallel to the Maxwell elements. Since this element is equivalent to a Maxwell element with a dashpot constant of infinity, the general form of the above equation will not change. It should only be kept in mind that one of the relaxation times would be considered infinitely large to account for the lone spring element.  4.1.2. Differential Equations for Curing Materials In this section, the constitutive model for a spring and a dashpot with changes o f temperature and cure are studied separately.  4.1.2.1.  Thermoelastic effects  Assume that a spring, essentially an elastic material, with a temperature-dependent stiffness is undergoing a cycle of variable load and temperature and the constitutive model o f the material is to be found. Change of temperature in elastic materials is a reversible process and as such only knowledge of the current value of strain and the temperature-dependent modulus is needed, independent of the history. This is due to the  -59-  Chapter 4: Differential Form of Viscoelasticity in ID fact that a reversible process path independent of and depends only on the initial and final values of the parameters. In other words the constitutive equation for this case can be written as follows:  a(t,T) = k(t,T).8(t,T)  (4.15)  In which, k and s are the final values for the spring stiffness and the strain at the current temperature, regardless of their intermediate values. To explain this equation, an example for a special hypothetical case is presented here: Consider an elastic bar compressed by a strain of E . Assume that the stiffness of the bar is k . So, we 0  0  have:  a =k .£ 0  0  (4.16)  0  Now assume that the bar undergoes a change in temperature, during which its mechanical strain is kept constant at e . Further assume that as a result of this process the modulus of elasticity becomes zero. 0  Now, i f there is any stress remaining in the bar and the bar is released, since there is no modulus it will stretch to infinity, which does not agree with common sense. The real behaviour is that of Equation (4.15) according to which the stress will be zero (the expected value). The above constitutive equation can also be presented in incremental form. B y differentiating Equation (4.15) with respect to time, we obtain:  d ,, dt  .  , ds dt  dk dT , . = k£ + k£ dT dt  cr = —(k.e) = k.— + s.—.—  .. ,  (4.17)  For the case of a dashpot, though a change in temperature is not a reversible process for strains it can be said that it is reversible for the strain rate, £ . This argument, although may not be physically meaningful, will help us establish the constitutive model. Thus:  <r = n(T).e  -60-  (4.18)  Chapter 4: Differential Form of Viscoelasticity in ID in which, all the values are the current values and independent of the history. Having the behaviour of a spring and dashpot, the differential equation for a single Maxwell element, according to the parameters in Figure 4.1, can be written as:  E  ~  +  ~ d r k  )  n'^k  +  (4.19)  k  2  )  +  Thus, the constitutive equation in this case is:  & = k(T(t))E  1  £(7X0)  r(7/(0)  k(T(t))  •  (4.20)  Defining an equivalent relaxation time, r , as follows:  r  r  (4.21)  k  The constitutive equation (Equation (4.20)) can now be written in a more convenient form:  a = ks  (4.22)  This is very similar to Equation (4.2) for a single Maxwell element, but both k and r are functions of temperature. A t the same time, as mentioned before, most researchers have assumed that the unrelaxed and relaxed moduli are constant with temperature. That translates to assuming a constant k, which will convert the above equation to the same form to those for a non-curing material, with the exception that now r = r will vary with temperature. For the case of a constant k, we will show here that the D F will result in a similar modulus to the one used for the IF in the literature. Recall that for generalized Maxwell model, the relaxation modulus is:  E(t) = £» + (E - E ) u  x  J  W, exp  -61 -  (4.23)  Chapter 4: Differential Form of Viscoelasticity in ID The definition of relaxation modulus is the stress resulting from a constant applied unit strain. Assuming a constant strain, Equation (4.22) becomes:  &+  a  T(T(t))  =0  (4.24)  The solution to this differential equation is:  -if a = E(t) = Ce "  (4.25)  in which, C is the integration constant and t' is a dummy integration variable. Since at time? = 0, the stress should be equal to k , then C = k and: -if  a = E(t) = ke  (4.26)  0  Now, in a thermorheologically simple material, the relaxation times can be found using the timetemperature superposition: z(T) = a (T).r T  where a  T  is the shift factor and r  0  (4.27)  0  is the reference relaxation time at a reference temperature T . Using 0  this equation we w i l l have:  1 f  dt'  E(t) = k.e < T  aAm)  Using the definition of the reduced time:  the relaxation modulus then becomes:  -62-  '  (4.28)  Chapter 4: Differential Form of Viscoelasticity in ID  E(t) = k.e " r  (4.30)  This is the relaxation modulus in Equation (4.23) for a single Maxwell element.  4.1.2.2.  Effect of cure  Cure in elastic materials, unlike change of temperature, is an irreversible process and as a result the material behaviour during cure is history dependent. To demonstrate the effect of cure on the constitutive equation, consider the example used in the previous section: an elastic bar under compression. This time, suppose that it starts to cure under constant temperature and its modulus changes in the meantime. Although the modulus is changing the change is due to a transformation in the material itself and the new modulus will not contribute to bearing the load. In fact, it is similar to adding another bar to the first one, such that the structure will be stiffer, but the new material will not help in carrying any of the current stress. Therefore, the constitutive equation in this case in differential from will be:  & = k.s  (4.31)  The obvious difference between cure and temperature change is whether or not a second term is added to it: changes in elastic modulus due to temperature change contribute to stress but not the ones resulting from cure. This understanding of the material behaviour has also been realized in molecular network theories, e.g. Wineman and Shaw (2002), where network changes as a result of scission and healing (similar to cure) have been represented by similar equations. It should emphasized that this discussion applies only to the changes of modulus and its effects on the stresses and does not suggest that thermochemical strains themselves do not lead to stress development. In fact, cure shrinkage strains generate stress in the same manner as thermal strains. Combining both temperature and cure effects, the constitutive equation for a spring can be written as follows:  -63-  Chapter 4: Differential Form of Viscoelasticity in ID .  , .  dk dT  cr-k.s + £  dT dt  d „ , dk da = —(ke) - s dt  (4.32)  da dt  For the case of a dashpot, unlike that of a spring, there is no need to distinguish between the effects of temperature and cure. Consider the bar example: i f the bar, subjected to a constant rate of strain, sees a viscosity change due to cure it is the new viscosity that will govern the behaviour of the bar. Therefore, the effect of cure on the behaviour of the dashpot is similar to that of temperature and the performance of the dashpot is only a function of the final value of the viscosity. The constitutive model in this case can be written as:  a = n(a).£  (4.33)  To obtain the constitutive equation for a Maxwell element, Equations (4.32) and (4.33) can be combined. The result is:  ( • a . d d "l a £ + — —£ = —+ — — — + — dt 7 dt r [k  (4.34)  where,  1  dk dT  k dT  dt  (4.35)  It can be seen that the constitutive equation for a variable stiffness is quite complicated. A n important observation is that when the stiffness is constant throughout cure, the equations available in the literature for temperature-dependent material processes can be extended to include cure as well, as done by K i m and White (1996) and Zhu et al. (2001). However, i f the stiffness is not constant, this generalization cannot be made directly and a more rigorous scheme has to be adopted. It is noted that in such a case, using IF may be impractical, but the D F can still be used to solve the constitutive equation.  -64-  Chapter 4: Differential Form of Viscoelasticity in ID  4.1.3. Numerical Solution of Differential Equations To implement the D F in its current I D form, differential equations need to be solved to relate the stresses and strains at each point in time. This will be done here for a model where the moduli are constant. For this purpose, the unconditionally stable, 2  n d  order accurate Crank-Nicholson or central difference scheme  (Hughes, 1987) will be employed. A s a result, the stress in Maxwell elements / , through solving Equation (4.22), can be obtained as follows:  *r=  l  1  where, cr"  +1  and s"  +l  +  A  {  "  (4-36)  2^r  are the new stresses and strains, cr" and s" are the current stresses and strains, and  At is the increment of time. Having all the element stresses, the stress in the material can be obtained from Equation (4.13).  4.1.3.1.  Remarks  Note that in the D F approach all the Maxwell stresses from the previous time-step need to be stored. These values effectively account for the loading history. This solution for the differential equations is in fact similar to the internal variable technique (Johnson, 1999) used in the IF, as mentioned before. Although the internal variable method, which can have different forms, and a finite difference solution, such as the one used here, are not identical the fundamental concept behind them is the same; both save some internal variables equal in the number to number of Maxwell element to account for the loading history. However, it is observed that arriving at such an equation is more intuitive and easier to obtain with the D F . Also, it appears that all internal variable approaches are developed for discretization of the reduced time, i.e. in a simple form as follows:  -65 -  Chapter 4: Differential Form of Viscoelasticity in ID  cr(r ) = cr(r) + AtT +1  (4.37)  This means that either the reduced time ( £ ) instead of the real time (t) has to be discretized, which is very inconvenient in process modeling, or some simplifying assumption on the relationship between the reduced time and the real time has to be made, which can generate errors in the results. Overall, it appears that despite similarities between the two approaches, the D F has clear advantages to the internal variable approach.  4.1.4. Comparison of Differential Form and Integral Form In this section, the D F and IF approaches w i l l be compared based on their I D forms. Though simple cases, this w i l l give us a better understanding of the similarities and differences between the two methods.  4.1.4.1.  Comparison of the stresses  To compare the D F approach to the IF and evaluate its efficiency, the simple F O R T R A N code used in Chapter 3 was expanded. The program therefore calculates the stresses using the D F and IF approaches from Equations (4.14) and (4.3). The material model presented by K i m and White (1996) is again used here for the examples. For the purpose of comparing the stresses resulting from the two methods, two practical and one theoretical cure cycles are used. The temperature cycles are shown in Figure 4.2 and the resulting stresstime histories from the IF and D F approaches are shown in Figure 4.3. A s expected, the results from the two methods are identical: it is not possible to distinguish the results from each other.  4.1.4.2.  Comparison of efficiency  To compare the efficiency of the D F to the IF, similar to the comparison of P V E and IF in Chapter 3, two different cases have been considered: a fully cured/constant  cure thermoset matrix (essentially a  thermoplastic), where there is no computation of degree of cure, and a thermoset matrix, where cure is  -66-  Chapter 4: Differential Form of Viscoelasticity in ID computed at every time step. In each case, a sample cure cycle is run with different number of time steps, using D F , IF, and P V E approaches. The results are presented in Table 4.1 and Table 4.2. The results for the fully cured (thermoplastic) matrix are shown in Table 4.1. It is observed that the D F is 13 times faster than the IF. A t the same time, interestingly the D F approach is only slightly slower than the variable-time P V E approach (only around 20%). Table 4.2 shows the results for a curing thermoset matrix. In this case, calculation of cure takes significant time, and thus the absolute benefit of the D F approach is smaller. Nevertheless, even for this simple I D case, the D F approach is approximately 5 times faster than the IF approach. Again, the D F is only nominally slower than the P V E .  4.2. SUMMARY The D F of viscoelasticity for a linear viscoelastic polymer during cure is developed and studied in this chapter. The D F and IF are in fact equivalent and merely two different forms to describe the same behaviour, as discussed previously and shown in the numerical examples above. Despite similarities, the two approaches have major differences as well. Note that using internal variable approaches for implementation of the IF is advantageous to the simplified integration techniques of the IF, e.g. Taylor et al. (1970), in terms of ease of implementation, and runtimes. However, as discussed in this chapter there are assumptions needed in order to use these approaches in process modeling. A s a result, the comparison of D F to IF will be performed for simplified integration techniques. Differences of the IF and D F are explained as follows: The D F approach appears to provide a better framework for understanding the fundamental assumptions made regarding the behaviour of the material, such as thermorheologically simple behaviour and the concept of time-temperature superposition. A s mentioned before, due to the complexity of the IF approach, it is also more difficult to expand the IF formulation to more complex cases, such as modeling thermoelastic effects, unlike the D F approach which was extended to more general cases sections.  -67-  Chapter 4: Differential Form of Viscoelasticity in ID The IF approach is a more complex formulation and therefore is more difficult to implement in a numerical code even using the common integration techniques used in the literature, the most common of which is by Taylor et al. (1970). Furthermore, due to the form of the integral equation the IF is much more time consuming in analysis than the D F approach. When using available simplified methods, it was shown that the D F approach is much faster when implemented in a Finite Elements (FE) code. The main reason is the existence o f multiple integrations in those methods. Moreover, the efficiency of the D F is more evident in comparison of run times with P V E , where the D F is only marginally slower. Regarding computational storage requirements, while both models need to store intermediate values for each Maxwell element, the IF needs to keep data for two time steps. One possible drawback to using the D F approach could be that it is bound to the differential equation that results from the spring/dashpot model. Therefore, in the case that a relaxation modulus is characterized using experiments that do not make use of these elements, it may not be easy to use the D F , since finding the differential equation may not be possible.  O f course, in such a case the simplified integration  methods cannot be applied to the IF approach either, forcing history-dependent integration and making the IF run times prohibitively long. A summary of the different aspects of the two models is provided in Table 4.3. Having the D F for I D polymers, the next step is to expand this formulation to 3 D behaviour of composite materials.  For this purpose, a proper micromechanical approach is needed in order to combine the  polymer and fibre mechanical properties to obtain the composite properties. The procedure to find and apply such a micromechanical approach will be presented in the next chapter.  -68-  T a b l e 4.1 C o m p a r i s o n o f R u n - t i m e s f o r T h e r m o p l a s t i c M a t r i x ( o r c o n s t a n t d e g r e e o f c u r e i n a t h e r m o s e t m a t r i x )  Pseudo Viscoelastic Repetitions  Steps  200 100 20 20 20  500 5000 20000 40000 5000  Differential F o r m  T o t a l core  Core run-  Total core  run-time  time  run-time  0.13 0.64 0.51 1.04 0.12  0.00065 0.0064 0.0255 0.052 0.006  0.15 0.74 0.63 1.23 0.15  C o r e run-time  0.00075 0.0074 0.0315 0.0615 0.0075  Integral F o r m T o t a l core  Core run-  IF/DF  DF/PVE  run-time  time  Ratio  Ratio  1.95 9.80 7.90 15.80 2.00  0.00975 0.098 0.395 0.79 0.1  13 13.24 12.54 12.85 13.33  1.15 1.16 1.24 1.18 1.25  T a b l e 4.2 C o m p a r i s o n o f R u n - t i m e s f o r T h e r m o s e t M a t r i x ( i n c r e a s i n g d e g r e e o f c u r e )  Pseudo Viscoelastic Repetitions  Steps  100 100 20 20  500 5000 20000 40000  Differential F o r m  Core run-  T o t a l core  run-time  time  run-time  0.27 2.67 2.15 4.29  0.0027 0.0267 0.1075 0.2145  0.27 2.78 2.24 4.47  Total core  C o r e run-time  0.0027 0.0278 0.112 0.2235  Integral F o r m Core run-  IF/DF  run-time  time  Ratio  Ratio  1.29 12.89 10.3 20.56  0.0129 0.1289 0.515 1.028  4.78 4.64 4.6 4.6  1 1.04 1.04 1.04  T o t a l core  DF/PV1  Chapter 4: Differential Form of Viscoelasticity in ID  Table 4.3 Comparison of D F and IF models Item  Integral Form (IF)  Differential Form (DF)  Formulation  Fairly easy in a variety of cases  May be difficult in some cases; e.g. thermoelastic effects  Implementation  Relatively easy  Difficult  Solution  Needs to equations  Storage (for a multipleMaxwell model)  Needs to store the stresses in each Maxwell element  For Taylor et al. (1970) needs to store a set of parameters which correspond to each Maxwell element and some parameters from the previous time-step  Speed  Fast. Even comparable to that of CHILE  Very time consuming even with the available simplified integration method  solve the  differential  -70-  Uses simplified integration methods  Chapter 4: Differential Form of Viscoelasticity in ID  ^  k  W W -  H]  (a)  H]—WVV^ a  {]—wvvH ^  k  n  n  (b) Figure 4.1 (a) Maxwell element (b) Generalized Maxwell model  Cycle 2  50  100  150  200  250  300  Time(Min)  Figure 4.2 Cure cycles used to compare IF and DF stress predictions  -71 -  350  Chapter 4: Differential Form of Viscoelasticity in ID  350 -5  J  Time(min)  Figure 4.3 Comparison of predicted stress histories using IF and D F approaches. Note that the two approaches give identical results for each case and therefore cannot be distinguished.  -72-  Chapter 5: Micromechanics  Chapter 5.  MICROMECHANICS  Micromechanical models are used to combine the mechanical and physical properties of the phases at the micro level to model the response of a composite at the macro level as a homogeneous material. In doing so, different aspects of the material behaviour need to be considered; such as the fact that the resin is a curing viscoelastic material and that the fibres are not isotropic. These issues and others will be dealt with in this chapter to obtain a (virtual) characterization of the V E behaviour of composite materials with a micromechanics ( M M ) approach, which will be an essential part o f process modeling o f the composite materials. First the fundamental equations of micromechanics are presented. Next, micromechanical equations for elastic materials are discussed. Based on these, V E material properties are derived and presented along with discussions on the different issues concerning the material behaviour, such as thermoelastic effects. A t the end, numerical examples are presented to clarify the approach.  5.1.  FUNDAMENTAL EQUATIONS  In this section, some fundamental equations for heterogeneous materials are presented. These equations are general and do not depend on any specific micromechanics model (Hashin, 1972). These equations can be written for homogeneous and multi-phase materials, but w i l l be presented for the case of twophase bodies that are of interest here. The volume average stress q_ of <r in the composite is written as: c  (5.1) v where the stress vector, a, for the purpose of F E implementation is as follows:  (5.2)  22'  -73-  Chapter 5: Micromechanics Rewriting this equation in terms of the constituent volumes, we will have:  v =j(j<LdV+  \°dV)  c  m  (5-3)  f  V  V  where V is the total volume of the composite, and V and V are the volumes o f the matrix and fibre, m  f  respectively. Thus,  = ^(V ° m  + V g_j) = (1 - v )a  m  s  f  m  + va f  f  (5.4)  where v denotes fibre volume fraction. f  In a similar fashion, the strains can be written as follows:  l c  = ^ { K l  m  + V lf) f  =  (1 - v )L„ + v , £ , f  (5-5)  Equations (5.4) and (5.5) are two fundamental equations of M M , which are derived from definitions of stress and strain and can be universally used in any M M model for two-phase bodies. These two equations, along with the constitutive equations for the two phases w i l l be used in the analysis of composite materials. In addition to these, another equation is needed, which should describe the stressstrain relationship in the material, i.e. its constitutive model. These, along with the finite elements equation will provide adequate number of equations to find the stresses and strains in the material, as will be shown in the following sections.  5.2.  MICROMECHANICAL M O D E L FOR ELASTIC MATERIALS  Micromechanics has been the subject of many research studies in the literature and numerous approaches have been proposed. Micromechanical models are normally developed for cases where both phases are  -74-  Chapter 5:. Micromechanics  elastic. Some of these models are mentioned here for reference. M o r i & Tanaka (1973) developed a model that is named after them. Christensen (1990) worked on a generalized self-consistent method. Homogenization method has been studied by many researches. Suquet (1987) used this approach for both elastic and viscoelastic materials. Aboudi (2001, 2002) used the Method of Cells or the Generalized Method of Cells, which analyze a repeating unit of a structure (unit cell). Some other examples of this method are Banks-Sills et al. (1997) and Guinovart-Diaz et al. (2002), who used an asymptotic homogenization method. Some people have also used finite element-based approaches to model the properties of composites, e.g. Ellyin et al. (2002). Another elastic method that has been widely used in the literature has been developed by Hashin & Rosen (1964) and, also discussed by Hashin (1972) and H i l l (1964). This model is based on modeling a representative volume element of the structure, which in the case of unidirectional fibre reinforced polymers takes the shape of concentric cylinders representing resin and fibre with their volumes proportional to the actual volume fraction of each phase (Figure 5.1). In this case, the model is also called the Concentric Cylinder Assembly ( C C A ) . Choosing a representative volume element in this model is valid for a statistically homogeneous material (for a comprehensive discussion on this issue see Hashin, 1972). Some researchers have compared different micromechanical methods, notably Christensen (1990), Hollister and Kikuchi (1992), Nguyen Viet et al. (1995), Brinson and L i n (1998), and Herakovich (1998). The understanding from these works is that although there are arguments for or against different methods, most methods show good representation of the behaviour of the material for the standard case of fibrereinforced polymers. Since we will be dealing with unidirectional long-fibre laminae and due to relative simplicity of the model developed by Hashin and Rosen it will be used here. It should be noted that even though a specific  -75 -  Chapter 5: Micromechanics  model will be used here, the proposed approach is not dependent on its specific form and using it does not preclude us from using a different elastic M M model. By applying unit strains to this model and taking into account the isotropic behaviour o f polymer and transversely isotropic behaviour o f fibres, one can obtain the transversely isotropic elastic mechanical properties o f the composite, using elasticity equations. The corresponding moduli o f the composite are presented for reference as follows (Bogetti and Gillespie, 1992). In these equations, the properties of the composite, fibre, and matrix have subscripts c, / , and m, respectively. Note that a typing error existed in the reference in calculation of the longitudinal Young's modulus, which has been corrected here. •  The longitudinal Young's modulus:  £  l c  =£  1 /  v +£ /  l m  40  (l-v ) + /  -v  m  m  )k k G (\-v )v m  2f  m  f  }  (5.6)  (K + G )k +(k - k )G v f  where k  12f  m  m  2f  m  m  f  is the plane strain bulk modulus o f polymer and is calculated as follows:  k„ =• 2(\-v -2vl)  (5.7)  m  •  The major Poisson's ratio:  i, m  ~ l2f  V  Vnc= nc=Vn v +V Q-v )  +  v  f  f  m  f  V  X*„  -  2f  k  )m  0  G  (k +G )k +(k -k )G v 2f  m  m  2/  m  m  ~ f V  >/  (5.8)  f  The inplane shear modulus:  {G +G )HG -G )v aI  G  l2c  =  G  U c  =  G  m  (G,  m  2 /  x2f  m  +GJ-(G  -76-  f  1 2 /  -GJv  (5.9) /  Chapter 5: Micromechanics  The transverse shear modulus:  =  CJ ,_ 9  G [k (G G ) m  m  n+  K(G  21/  +G  m  2 3 /  2G G k (G -G )v ]  +  ) + 2G  23/  2 3 /  m+  m  23/  G -(k + 2G )(G m  m  m  23f  m  p.lu)  f  -G )v m  f  The plane strain bulk modulus:  JK +G )K (K -KJG v  K  2/  m  m+  2/  m  f  (K G )-(K -K )v 2/+  •  m  2f  m  f  The transverse Young's modulus:  E  2 c =  E  (\lAK ) 2c  •  +  (\lAG , ) 2 c  -( - )  n —  3 c = — , +  5  (v,\jE )  12  Xc  The transverse Poisson's ratio:  2E K -E E -4vf K E lc  2c  lc  2c  2c  2c  2i  (5.13)  2E K lc  •  2c  The coefficients of thermal expansion (CTE's) in the three principal directions can be written as:  CTE E v CTE E (l-v ) lf  lf  f+  uf  E  CTE  2c  5.3.  = CTE  3c  v  m  +  m  f  ~\}- f)  E  (5-14)  = (CTE  V  2f  + v CTE ) l2/  if  v + CTE ( l + v ) ( l - v ) - [v v f  m  m  f  X2f  f  + v (\- v, )]CTE m  ic  MICROMECHANICAL M O D E L FOR VISCOELASTIC MATERIALS  In contrast to the case o f elastic materials, there are not many studies done in the literature on micromechanics of viscoelastic materials. In some works, this is dealt with using approximations, e.g. White and K i m (1998). However, in general, most researchers who have modeled the viscoelastic materials have done so using the Correspondence Principle. According to this principle (to be discussed more below), a Laplace transformation converts a  -77-  Chapter 5: Micromechanics linear viscoelastic problem to a linear elastic problem in Laplace space, which then enables one to use the elastic micromechanical models. After obtaining the material properties in this space, applying an inverse Laplace transformation brings the material properties back to the time domain, i.e. the required result (e.g. see Matzenmiller and Gerlach; 2001). Some examples of this approach are: Schapery (1967), Brinson and Knauss (1992), Barbero and Luciano (1995), Y i et al. (1998), Harris and Barbero (1998), Kumar and Singh (2001), N o h and Whitcomb (2003), Matzenmiller and Gerlach (2004), Zocher et al. (2004), L i u et al. (2004), and Yang et al. (2004a). In general, there does not seem to be a comprehensive work on this subject that includes all the required aspects of the material behaviour without unnecessary simplifications. Most researchers have used approximate inverse Laplace transformations, e.g. Schapery, N o h and Whitcomb, and Zocher et al. Consideration of temperature changes and cure has not been widely done. There are few works, in which this process has been carried out completely, but even in those cases some other assumptions have been made. For example, Barbero and Luciano, and Y i et al. made the incorrect assumption that the Poisson's ratio is constant with time (see Tschoegl et al. (2002a) and Hilton (2003)). Taking advantage of the available models in the literature and taking into account all necessary aspects of the material behaviour, including consideration of temperature changes and cure, variability of all resin properties with time, and allowing the coefficients of thermal expansion of the composite to be time dependent even i f they are constant in the phases, we can obtain the material properties of the composite. This will be explained in the subsequent sections.  5.3.1.  Correspondence Principle  This principle can be used to find the stresses and strains in V E materials, by taking advantage of Laplace transforms. For a simple description of this principle, consider the I D integral form of V E in a material, as follows:  -78-  Chapter 5: Micromechanics  a(t)=JE(t-T)—dr  o  (5.15)  dr  If we apply the Laplace transformation to this equation, i.e. a convolution integral, the result will be:  £(a(t)) = &(s) = t(E * ^f) = E(s) • I A dt dt  (5.16)  6(s) = E(s)-[si(s)-s(0)]  (5.17)  We will then have:  And by putting e(0) = 0 , we obtain:  &(s) = sE(s)-s(s)  (5.18)  This describes an elastic behaviour in Laplace space, in which the modulus of elasticity is:  f(s) = sE(s)  (5.19)  This is the basis for the Correspondence Principle, which states that a V E problem is equivalent to a corresponding elastic problem, in which the boundary conditions are the Laplace transformation of those in the V E problem (with the assumption that boundaries are independent o f time) and the moduli are found from the above equation. This principle is also sometimes called "Static Correspondence Principle", as opposed to a similar principle for dynamic loading (see e.g. Brinson and Knauss, 1992) Having the corresponding elastic problem, it can be solved in Laplace space to find the stresses and strains. Then, the stresses and strains in time space can be obtained by applying inverse Laplace transformations to these values as follows:  * 0 = r'(*(-))  -79-  Chapter 5: Micromechanics  5.3.2. V i r t u a l M a t e r i a l Characterization 5.3.2.1.  Isothermal Conditions  Taking advantage of the C C A approach for elastic materials and the Correspondence Principle we will be able to find the V E moduli o f the composite material as functions o f time. The step-by-step procedure is explained below. First, the moduli o f the constituents are obtained in Laplace space. Showing the longitudinal relaxation moduli of the fibre and matrix as an example, we have:  For  fibre:  f (s)  For matrix:  f  = sE (s) = s£(E )  (5.21)  f (s)  = sE (s) = s£(E (t))  (5.22)  m  f  f  m  m  Assume that the elastic Micromechanical ( M M ) equations are written in the following generic form:  E = MM{E ,E ,v ) c  f  m  (5.23)  f  By substituting the corresponding elastic moduli o f Equations (5.21) and (5.22) in the M M model, Equation (5.23), we will obtain the corresponding elastic modulus of the composite:  t (s) c  = MM{t (s),t (s),v ) f  m  f  (5.24)  B y applying the inverse Laplace transformation to this modulus, we will find the composite modulus in time domain:  -.,r (*)  E (t) = C \ - c^ - ) c  (5.25)  5  A proper method o f applying the inverse should be used here. Both approximate inversions (e.g. N o h and Whitcomb, 2003; Zocher et al., 2004) and exact algorithms (e.g. Barbero and Luciano, 1995; Y i et al., 1998) have been used in the literature.  -80-  Chapter 5: Micromechanics Having the V E modulus of the composite material as a function of time, it can then be described using a mathematical function, such as the Prony series:  (5.26)  In order to do this, a regression algorithm can be used to fit the results to the above equation. In general, one should assume all the parameters in the above equation, i.e. relaxation times, weight factors, and unrelaxed and relaxed moduli, as variable (see e.g. Baumgaertel and Winter, 1989). This will result in a nonlinear regression problem (see Bates and Watts, 1988). However, a more common approach in the literature, e.g. Y i et al., 1998; Simon et al., 2000; Arzoumanidis and Liechti, 2003, is to choose the relaxation times, typically spaced equally at one per each decade of logarithmic scale of time, and then perform a linear regression analysis to find the other unknowns. Having these values one can then test for a goodness of fit (e.g. Simon et al.) or compare the fit and actual moduli (e.g. K i m and White, 1996). It is noted, however, that since the material properties are smooth functions rigorous statistical tests are not required. Some other notable references on finding these relaxation modulus variables are Maztenmiller and Gerlach (2001) and especially Emri and Tschoegl (1995). The approach explained above is also used in experimental characterization of polymers. In a way, by combining the properties of the phases into the composite we are characterizing the composite materials in a virtual environment. For this reason, this procedure is here referred to as "virtual material characterization". Summary of the proposed M M approach is shown schematically in Figure 5.2.  5.3.2.2.  Non-isothermal conditions  The procedure presented above can be used to characterize a composite material for constant temperature and degree of cure. This is due to the fact that in fitting a Prony series to the composite moduli the relaxation times, r , in Equation (5.26) need to be constant with time. c i  -81 -  Chapter 5: Micromechanics To include temperature changes and cure in the M M approach, time-temperature-cure superposition, as described in Chapter 2, can be utilized. In other words, the goal here w i l l be to find the composite shift factor, which may be different from that of the polymer. The relation between these two factors is discussed below. Consider the I D V E integral form for a thermorheologically simple polymer as follows:  <T{t)='\E(g-?)^dT  (5.27)  in which the reduced times are:  '  1  T  1  <f = j— dt' and ? = j—dt' 0  0  T  Q  a  (5.28)  T  Clearly, Equation (5.27) can be written in an alternative form as follows:  <7(0=  W-O-TJ^'  (5-29)  This equation has also been mentioned in the literature, e.g. Kumar and Talreja (2003). The above equation means that the form of the governing equation for composite materials under nonisothermal condition remains the same as the isothermal condition, with the stipulation that time should be replaced with reduced time, £ . A s a result, to find the properties of composite as a function of temperature and degree of cure one can use the same process described in previous section, with the difference that all the calculations should be performed in 'reduced time space' instead of 'time space', e.g. Laplace transformation should be applied with respect to £ . The result will be the moduli of composite under non-isothermal conditions, which are identical in form to that of the isothermal conditions, but a function of the reduced time. Symbolically:  £  n o n  - , o = £,,,(#)  -82-  (5-30)  Chapter 5: Micromechanics This is the same relationship that is written for the polymer and it effectively means that the definition of the reduced time in composite and polymer are the same. Then, it follows that the corresponding shift factors are identical. In other words, to model behaviour of composite as a function of time and temperature one needs to employ the time-temperature-cure superposition using the polymer shift factor. Symbolically:  ( T a  ) composite  =  (T a  ) matrix  (5-31)  The underlying assumptions here are that: •  The material is thermorheologically simple  •  Relaxation times are constant (i.e. assuming that temperature and cure dependence is captured only through the shift factor)  •  The shift factor for all the material properties in the matrix is unique, e.g. longitudinal and shear relaxation moduli can be described with the same shift factor.  The above approximations cover most typical cases. However, there are cases where the above assumptions are not valid, e.g. Deng and Knauss (1997) and Sane and Knauss (2001) have reported more than one shift factor. For the cases where any of these conditions does not hold, the following discussion is presented to show how the material can be characterized. The issue of thermorheologically complex materials will be discussed in the next section. If any of the above assumptions is not valid, the problem that we will face is only in fitting the Prony series to the moduli as functions of time. A t that point, the problem becomes a characterization problem where the shift factor is unknown. In other words, it is similar to the characterization of the material from experimental results and as such, similar approaches can be taken. The details of the approaches can be found in such works as K i m and White (1996) and Simon et al. (2000). Put simply, one can proceed as follows: a pair of constant temperature and degree of cure is assumed. For these values, the material can  -83-  Chapter 5: Micromechanics be characterized as explained before for the isothermal case. A s a result, relaxation times, weight factors, unrelaxed and relaxed moduli will be obtained. Then, this procedure will be repeated for other temperatures and degrees of cure. This does not have to be done for all possible values of temperature and degree of cure; we only need enough values to construct the master curve for the material, similar to experimental procedures. Then, based on these values the shift factor and the relaxation times can be obtained. The exact details of the procedures will depend on the specific material under study and one can refer to the references mentioned above.  5.3.3.  Material Properties  The characterization procedure explained above showed only the longitudinal modulus for simplicity. However, knowledge of other mechanical properties of the phases is needed and other composite properties should be characterized. The fibres are elastic and normally transversely isotropic, which means that five material properties are needed to describe them; these properties are known for a given fibre. The polymers, on the other hand, are normally isotropic and therefore two material properties are required to describe them, which are chosen out of the four common values of Young's modulus (E), modulus ( G ) , bulk modulus (K),  and Poisson's ratio (v).  shear  Due to the complexity o f characterizing two  of these properties at the same time some researchers have characterized one and made assumptions on another, e.g. K i m and White (1997) who measured G and assumed that v has a constant value. In several studies a time independent Poisson's ratio has been assumed for the resin, e.g. Poon et al. (1998), Y i et al. (1998), K i m et al. (2002), and Melo and Radford (2003). However, there is enough evidence in the literature to the contrary, e.g. Hilton (2003) makes the argument that such an assumption is invalid. There are many other works in the literature on this topic. A review of the literature on this issue can be found in Hilton (2001) and Tschoegl et al. (2002a). Some other researchers have assumed a constant bulk behaviour with time, e.g. Garnich and Hansen (1997) and Idesman et al. (2000).  -84-  Chapter 5: Micromechanics There are several works in the literature, where time dependent bulk and shear behaviour have been considered, e.g. A d o l f and Martin (1996), Holzapfel and Simo (1996), and characterized with experiments, e.g. Yee and Takemori (1982), Park (2000), Arzoumanidis and Liechti (2003), and Yang et al. (2004 a,b); a comprehensive study o f P M M A resin on this topic has been done by Sane and Knauss (2001a, 2001b). In summary, although some researchers have assumed time-independent bulk behaviour or Poisson's ratio, it seems to be more because o f the complexities involved in their measurement than the actual material behaviour. Based on the evidence in the literature, all material properties should be assumed time dependent. Here, the micromechanical calculations will be done as such, with a Prony series representing each material property (modulus or Poisson's ratio). Clearly, modeling a material with a constant material property for resin w i l l be quite easy, by simply assuming a constant value for that property. Another issue to keep in mind is the form of the coefficients of thermal expansion and cure shrinkage for the composite. According to Equation (5.14) these two are functions o f the material properties and as such are time dependent. A n assumption of time independency on these parameters would therefore be a mere simplification without insight into the material behaviour. It will be seen in the following sections that in fact the variability o f these parameters is not negligible. It should be noted that this dependence on time is the result of mechanical interaction between phases, and as a result is similar in form to that of the moduli. Therefore, the relationship between thermal strains o f the composite and temperature changes is similar in form to that o f the stress-strain relationship, where thermal strains act as stresses and temperature as strains in this analogy. Consequently, the coefficients o f thermal expansion and cure shrinkage can be represented by Prony series. This will be considered in the fllowing sections.  5.3.4. Thermoelastic effects In most o f the discussions in this and the previous chapter, a thermorheologically simple behaviour for the polymer was assumed. This implies that the unrelaxed and relaxed values o f the relaxation modulus  -85-  Chapter 5: Micromechanics are temperature and cure independent. This is based on some of the works in the literature, in which these values were measured experimentally, e.g. K i m and White (1996) and Simon et al. (2000). However, there is evidence in the literature that some materials do not behave this way. The experiments done separately and on different materials by McCrum and Pogany (1970), Weitsman (1982), Harper and Weitsman (1985), and A d o l f and Martin (1990) all showed that thermorheologically simple assumptions for the material of their study is not valid and a simple horizontal shift is not adequate to construct a master curve. A review of some prominent works in this area is presented here to show how thermoelastic effects have been modeled in the literature and to get some ideas on specific values that may be assumed in representing the material properties. Schapery (1974) used the term "thermorheologically complex materials" ( T C M ) for the materials that did not follow the thermorheologically simple ( T S M ) assumption. Obviously, this is a very general term and the material properties can be modeled in different forms. To formulate such behaviour, Schapery defined T C M - 2 material, with T C M - 1 being a composite consisting of different T S M materials, which has the following constitutive equation in integral form:  (5.32)  where D refers to the creep compliance, D, is the initial value of creep compliance, and £ and the reduced times, similar to the values defined in Chapter 2, i.e.:  (5.33)  Schapery showed that i f the initial value of creep compliance is assumed to be as follows:  D,=D,(T )la (T) 0  G  where, T is a reference temperature, then the constitutive model will take the following form: 0  -86-  (5.34)  are  Chapter 5: Micromechanics  a = a,  o  (5.35)  ^'  which means that the temperature dependent relaxation modulus (E ) in this case is: T  E =a (T)E(£) r  (5.36)  G  In other words, these equations define a vertical shift factor (i.e. a ), which in addition to the horizontal a  shift factors can be used to construct a master curve. A s shown by M c C r u m and Pogany, different forms of initial creep compliance and vertical shift factors can be defined to shift the data points to a master curve. A more general form of T C M - 2 , obtained experimentally by Harper and Weitsman for 3502 resin, was later simplified and used by Hashin et al. (1987), and Sadkin and Aboudi (1989). The latter approach is consistent with T C M - 2 and Equation (5.32) and values of both shift factors have been obtained in these works. It will be shown here that assuming the spring stiffness in the M a x w e l l model as a function of temperature, as shown in Chapter 4, is consistent with T C M - 2 and the form used by Hashin et al., and Sadkin and Aboudi. Recall from Equation (4.22) that the D F for a Maxwell element with variable spring stiffness is as follows:  & = ks - — x  (5.37)  Then, by applying a constant strain for a relaxation test, we need to solve the following equation:  (5.38)  T  where  -87-  Chapter 5: Micromechanics  r  r  (5.39)  k  The solution, similar to Equations (4.26) and (4.28), is:  o-(0 = Cexp(-j4) o  r  r  ' dt'  V,i flfit  = C.exp -  = Cexp[ - J — + J( V o  r  o  - + ln[£(7X0)]  (5.40)  Or:  cr(0 = Cexp(ln[/t(T)])exp  (5.41)  = c*(r)£(£)  Since at time ? = 0, the stress should be equal to the spring stiffness. Thus,  k(T ) = Ck(T )k(T )->C = a  0  a  (5.42)  k(T ) 0  where, k(T ) is the spring stiffness at time zero. Q  Since the stress for an applied unit strain is equal to the relaxation modulus, we get:  k(T) KT )  (5.43)  0  k(T) The above equation is consistent with Equation (5.36), where the ratio  acts as the vertical shift  k(T ) 0  factor. This shows that the assumption of temperature variability o f the spring stiffness is consistent with the form of T C M - 2 discussed before. The experiments done by Harper and Weitsman (1985) and later used by Hashin et al. (1987) were performed on 3502 resin. Even though this is different from the materials that have been discussed in this thesis, due to lack of information in this area, the vertical shift factor function provided by Hashin et al. will be used here as a guide. Then, some reasonable assumptions on the unrelaxed and relaxed values of  Chapter 5: Micromechanics moduli as functions o f temperature can be made here. These values will then be used in modeling thermoelastic effects in Chapter 8, as needed. The horizontal and vertical shift factor for 3502 resin are as follows (Hashin et al.):  (5.44)  The vertical shift factor has been represented by an exponential function, but can easily be represented by a linear function, as shown in Figure 5.3. The linear function will be used here, due to its simpler form. The linear vertical shift function can be shown as follows:  a  G  =1.0591-0.00227  (5.45)  This equation defines a shift factor that has a value of unity around 30° C , the reference temperature, and reduces the unrelaxed and relaxed moduli to 0.66 of this value at temperature of 180'C . Again, although this has been obtained for a specific resin, due to lack of other experimental results, it will be used in Chapter 8.  5.3.5.  S u m m a r y of the proposed approach  The procedure to find the mechanical properties of composite materials was discussed in the previous sections. Through this approach one can combine the properties of the fibre and the polymer under changes of temperature and advancement of cure. Discussion on each part of this process along with characteristics and limitation of each was given before. The approach is shown schematically in Figure 5.4, in conjunction with a summary of comments regarding each part of the process.  -89-  Chapter 5: Micromechanics  5.4. NUMERICAL EXAMPLES In this section, some examples will be presented to show the virtual characterization procedure used to find the material properties of the composite under isothermal and non-isothermal conditions. For this purpose, the mathematical package Mathematica® has been used to perform the calculations. A l l the calculations needed for M M analysis are performed using the internal functions of Mathematica, except for the inverse Laplace transformation, where an add-on package called "numerical inversion" has been used. This package is capable of applying the inverse Laplace with five different methods. Here, the method developed by Stehfest (1970) has been used.  5.4.1.  Isothermal Case  This example involves virtual characterization of a composite with an isotropic matrix and transversely isotropic fibres under isothermal conditions. The material properties are listed in Table 5.1 and Table 5.2. The properties of the fibre are those of A S 4 carbon, and shear properties of the polymer are taken from the work of White and K i m (1998) for 3501-6 resin and fiber volume fraction (v ) f  is taken to be 0.6. Due  to lack of knowledge of the bulk behaviour, some reasonable values of unrelaxed and relaxed moduli were assumed. Also, the relaxation times and weight factors for bulk modulus were assumed to be equal to those of the shear behaviour. The shift factor for this example is assumed to be a nominal value of 10" . The form of the moduli are as follows: s  K (a,T,t) = K  r  m  G (a,T,t) = m  m+  [K"  m  -/C]£g„  exp  G [G" -G ]}Zg ^P r  r  m+  m  m  a  -t  -t  (5.46)  (5.47)  The micromechanical model of choice here is the C C A . The Mathematica notebook is provided in Appendix A . l . In this example, the number of the Maxwell elements and the relaxation times of the  -90-  Chapter 5: Micromechanics composite have been assumed to be the same as those of the resin and based on these, the weight factors have been obtained. The results are shown in Table 5.3. To check the goodness o f fit the coefficient of determination has been calculated and shown in the notebook. The value of this coefficient for all properties is very close to one and therefore excellent fit is achieved. To show the results, first the values of unrelaxed and relaxed modulus of the composite, presented in Table 5.4, are noted. These values, obtained from the viscoelastic micromechanical method in Mathematica are identical to those obtained directly by combining the properties of the fibre and unrelaxed and relaxed properties of the polymer through elastic micromechanics [Equations (5.6)-(5.14)], which shows consistency of the results. To illustrate the goodness of fit the mechanical properties based on the micromechanical analysis and the fitted Prony series versus reduced time, £ (in this case f = — ) , have been compared in Figure 5.5-  Figure 5.12. To show the effects of micromechanics on the weight factors of the material, another curve has been added to these figures, which in each case has the same unrelaxed and relaxed values as the other two, but uses the resin weight factors instead. It is observed that in all cases the Prony series fit matches the functions it is fitted to, i.e. the result of the micromechanical analysis, exactly. It is also evident from the graphs that using resin weight factors for composite has a significant effect on the results (expect for the longitudinal modulus, where dominance of fibres results in a practically constant modulus). Although some of the values used here are only assumed and do not represent experimental values, the results show that using the resin properties for composite can be a source of considerable error. Using the method presented here, the parameters for the composite can be easily obtained and no simplifying assumptions, as done by White and K i m (1998), need to be made.  -91 -  Chapter 5: Micromechanics 5.4.2. Different Temperatures and Degrees of Cure It was shown in Section 5.3.2 that the polymer and composite have the same shift factor (Equation (5.31) ). In this section, an example is used to show the validity of this equation. For this purpose, the same material properties as in the previous example are used. In addition to the shift factor of 1CT used in that example, two other shift factors of values 10" and 10~ (which correspond to 6  2  10  a different set of temperatures and degrees of cure) for the polymer are assumed. For each shift factor, the micromechanical procedure presented before is repeated and the composite material properties are obtained. The results for two of these material properties, G  n  and E  2  (the others are omitted to avoid  repetition), are presented in Figure 5.13 and Figure 5.14. The figures show that the values needed to shift a curve in order to superpose it with another curve (i.e. the relative value of the composite shift factors for the two curves) is equal to the relative values of the resin shift factors for those curves. This means that the same factors as the resin can be used for the composite.  -92-  Chapter 5: Micromechanics  Table 5.1 Material properties of fibre and resin for the numerical example AS4 fibre  Property  3501-6 resin  0.2  G (Gpa)  1.185  0.25  G (Gpa)  0.018  210  K (Gpa)  3.556  E (Gpa)  17.2  K (Gpa)  0.80  G =G (Gpa)  27.6  CTE(ue/°C)  58  CTE! (1/°C)  -0.9  CTE (1/°C)  -7.2  Property vi2=v  v  13  23  E,(Gpa) 2  12  13  2  u  r  u  r  Table 5.2 Relaxation times and weight factors used in the numerical example CO  xjmin)  Sat  1  2.92E+01  0.059  2  2.92E+03  0.066  3  1.82E+05  0.083  4  1.10E+07  0.112  5  2.83E+08  0.154  6  7.94E+09  0.262  7  1.95E+11  0.184  8  3.32E+12  0.049  9  4.92E+14  0.025  Table 5.3 Relaxation times and weight factors for different material properties in the example Maxwell element 1 2 3 4 5 6 7 8 9  Relaxation time 2.92E+01 2.92E+03 1.82E+05 1.10E+07 2.83E+08 7.94E+09 1.95E+11 3.32E+12 4.92E+14  E, 0.057 0.064 0.080 0.109 0.150 0.259 0.193 0.058 0.030  Vl2  G3  0.009 0.012 0.017 0.029 0.048 0.150 0.347 0.255 0.133  0.041 0.048 0.063 0.090 0.132 0.266 0.240 0.080 0.041  2  -93-  G 0.052 0.059 0.076 0.105 0.148 0.270 0.206 0.057 0.029 1 2  E 0.038 0.045 0.059 0.084 0.124 0.254 0.245 0.098 0.053 2  V 3 2  0.009 0.011 0.016 0.026 0.044 0.122 0.301 0.294 0.176  a. 0.058 0.065 0.081 0.110 0.152 0.262 0.190 0.054 0.028  a 0.014 0.018 0.025 0.039 0.063 0.177 0.339 0.216 0.109 2  Chapter 5: Micromechanics  Table 5.4 Values of the unrelaxed and relaxed moduli of the composite obtained from the V E micromechanics in Mathematica Property  E^MPa)  v  Unrelaxed Relaxed  127410 126014  0.253 0.314  t2  G (MPa) K (MPa) G (MPa) E (MPa) 12  4068.0 71.8  2  6405.5 1831.1  23  2726.2 70.2  -94-  2  7619.8 270.3  v  23  0.398 0.926  CTE^u/C)  CTE (uVC)  -0.311 -0.890  27.0 30.4  2  Chapter 5: Micromechanics  Resin  Figure 5.1 Concentric Cylinder Assembly (CCA) model  Figure 5.2 Schematic illustration of the micromechanical approach  aG • • • Linear fit  1.2 i  S 0.60.4 0.2 0  -I  1  1  1  1  0  20  40  60  80  1—  1  1  1  1  120  140  160  180  :  100  1 200  Temperature (C)  Figure 5.3 Vertical shift factor for thermorheologically complex behaviour of 3502 resin  -95-  P r o p e r t i e s  C h a r a c t e r i s t i c s :  n e e d e d :  - It models two behaviour (e.g. shear and bulk )-> Poisson's ratio functions o f  - Unrelaxed and relaxed moduli (constant moduli)  time; so are coefficients o f thermal expansion and cure shrinkage  - Relaxation times (r ), weight factors (w;), and number o f t  M a x w e l l elements (A^  - The material behavior is modeled with multiple Maxwell elements  can be different for shear and  moduli  are Prony series  bulk behaviour, except r 's that are chosen equal (  - The temperature (7) and degree o f cure (a) change  - Coefficient o f thermal expansion and cure shrinkage - Dependence on a and T  \ ^ P o s s i b l e dependence o f constant moduli on T  time-temperature-cure  superposition  shift factor  - The constant moduli, can be constant (thermorheologically simple) or variable  y  v^with T (Thermoelastic behavior, thermorheo logically complex)  Fiber properties  'Laplace Transform (LT)^ a.T constant  Characteristics:  P r o p e r t i e s  - Orthotropic behavior  - Material properties  - C a n model thermoelastic behavior  - Fiber volume fraction  n e e d e d :  • Using Correspondence Principle • Converts both phases to elastic materials, with all parameters functions o f the variables • In Laplace space, other matrix parameters (such as E, v) can be calculated  O S  Combining the using MM  - A suitable micromechanics (MM) model should be chosen -> can be virtually any model - The result will be the material properties in all directions (assuming orthotropic behaviour)  • A method o f numerical inversion o f LT is chosen • The result of the inversion is the material properties as functions o f time • The accuracy o f the inversion should be verified  J  Fit Prony series to all^ properties  -(Test for goodness o V ^ f i t  The test can be statistical or  • In general, a nonlinear regression is  only include comparison of data: the moduli are smooth  required to find r, and u^s - r 's can be assumed  and rigorous statistical tests are  (  \ ^ not normally required  • Properties will be functions of a,T • The procedure is similar to what is  - It can be shown that the shift factor is not affected through M M  done for experimental data  Result: V E constitutive model of ] \Composite material  Figure 5.4 Flowchart of the micromechanical analysis used  a linear  regression will be required  Chapter 5: Micromechanics  127600 i  125800 -I 1E-11  1  1  1E-08 0.00001  1  1  1  1  0.01  10  10000  1E+07  1  1  1E+10 1E+13  Reduced time (min)  Figure 5.5 Comparison of the value of composite modulus from different methods: micromechanics, Prony series fit to micromechanics, and Prony series with the resin weight factors  0.35 0.3 0.25 -\ 0.2 Micromech. Fit Resin  0.15 0.1 0.05 0 1E-11  1E-08 0.00001  0.01  10  10000  1E+07  1E+10 1E+13  Reduced time (min) Figure 5.6 Comparison of the value of composite Poisson's ratio from different methods: micromechanics, Prony series fit to micromechanics, and Prony series with the resin weight factors  -98-  Chapter 5: Micromechanics  4500 4000 3500 3000 J-2500 Is 2000 H 1500 1000 500 0 1E 11  1E-08  0.00001  0.01  10  10000  1E+07  1E+10  1E+13  Reduced time (min)  Figure 5.7 Comparison of the value of composite modulus from different methods: micromechanics, Prony series fit to micromechanics, and Prony series with the resin weight factors  3000 i  0 "I 1E-11  1 1E-08  1 0.00001  1  1  1  1  1  0.01  10  10000  1E+07  1E+10  1 1E+13  Reduced time (min)  Figure 5.8 Comparison of the value of composite modulus from different methods: micromechanics, Prony series fit to micromechanics, and Prony series with the resin weight factors  -99-  Chapter 5: Micromechanics  9000 -1 8000 -  0 -I 1E-11  . 1E-08  1  0.00001  1  0.01  -i 10  1  1  10000  1E+07  :  1  1  1E+10  1E+13  Reduced time (min)  Figure 5.9 Comparison of the value of composite modulus from different methods: micromechanics, Prony series fit to micromechanics, and Prony series with the resin weight factors  1 1  0.3 0.2 0.1 0 -I 1E-11  1  .  1E-08  0.00001  1  0.01  1  .  1  1  10  10000  1000000 0  1E+10  1  1E+13  Reduced time (min)  Figure 5.10 Comparison of the value of composite Poisson's ratio from different methods: micromechanics, Prony series fit to micromechanics, and Prony series with the resin weight factors  - 100-  Chapter 5: Micromechanics  O.E+00  -1.E-06  J  Reduced time (min)  Figure 5.11 Comparison of the value of composite coefficient of thermal expansion different methods: micromechanics and Prony series fit to micromechanics  3.1E-05 -j 3.1E-05 -  2.7E-05 -I 1E-11  1  1  1  1  1  1  1E-08  1E-05  0.01  10  10000  1E+07  *  1  1E+10 1E+13  Reduced time (min)  Figure 5.12 Comparison of the value of composite coefficient of thermal expansion different methods: micromechanics and Prony series fit to micromechanics  - 101 -  Chapter 5: Micromechanics  % 2500 2 2000  1E-08  0.00001  10000  0.01  1E+07  1E+10  1E+13  Reduced time (min) Figure 5.13 C o m p a r i s o n of the values of composite modulus for different shift factors  9000 -i 8000 7000 6000 Q .  5 CM I D  5000 4000 -  *  10  4  \  *r*  10  —aT=10M0 --aT=10A6 aT=10A2  4  3000 2000 1000 0 1E-11  0.0000001  0.001  10  100000  1E+09  1E+13  Reduced time (min) Figure 5.14 Comparison of the values of composite modulus for different shift factors  - 102-  Chapter 6: Differential Form of Viscoelasticity in 3D  Chapter 6.  DIFFERENTIAL F O R M OF VISCOELASTICITY IN 3D  Having the differential form (DF) of viscoelasticity in one-dimensional form and the composite materials characterization from previous chapters, we can develop the D F in a general 3 D form for composite materials during cure. The development of the constitutive model and the finite element equations are presented in this chapter. First, the constitutive equations for 3 D behaviour are derived. Then, based on these the finite element equations are developed. In the last section, the reverse micromechanics equations, i.e. the formulation to obtain the phase stresses and strains from the composite stresses and strains, are presented.  6.1. GOVERNING EQUATIONS IN 3 D 6.1.1. Choice of Material Properties and Development of the Differential Form In this section, the governing differential equations for composite materials in 3D are derived. In order to explain the procedure better, first the isotropic form o f V E materials are derived and then the D F equations for transversely isotropic V E materials are developed. A s it will be explained, a proper choice of material properties in representing the material behaviour is important in both cases and will be discussed as follows.  6.1.1.1.  Isotropic viscoelastic materials  The stress-strain relationship for an elastic isotropic material, considering only infinitesimal strains, is written as follows:  a = Ds  (6.1)  where matrices are designated with double underlines and vectors with a single underline and:  D = EA~  l  - 103 -  Chapter 6: Differential Form of Viscoelasticity in 3D where E is the elastic modulus and A includes the effect of the Poisson's ratio, v, as follows:  1  -v  -v  0  0  0  -V  0  0  0  -V  1  -V  —V  1  0  0  0  0  0  0  0  0  0  2(1 + v) 0  0  0  0  0  0  0  0  2(1 + v) 0  2(1 +  Also the following stress and strain vectors are introduced for the purposes of F E implementation:  u  £  a  £  0-33  £33  22  22  < <? = <  •,e = •  (6.3)  >  CT  12  2^12  <7  |3  2 13 £  =  =  /l2  X\3  2s = y  23.  CT  23  23  However, i f the material is viscoelastic E and v will both be functions of time and the above equations cannot be directly generalized. For example, for a material modeled with a single Maxwell element we cannot simply write:  &=  —  (6.4)  £>£-—a T  because the fact that v is time dependent, like a Prony series, is not taken into account in this equation, e.g. the relaxation time associated with the Poisson's ratio, say x , is not included in the equations. v  A s will be shown shortly, the volumetric-deviatoric form of the equations can be used to overcome this problem. For an elastic material, the equations in tensor form can be written as follows:  a = 2Ge =  =G  +Ks =0 =K  =G  +a =K  where G and K are the shear and bulk moduli, respectively, and:  - 104-  (6.5)  Chapter 6: Differential Form of Viscoelasticity in 3D  e  =e-=*-I = 3 =  =o  in which, s  and s  (6.6)  are the deviatoric and volumetric strain tensors, respectively, s  kk  is the trace of the  strain tensor and / is the identity matrix.  In other words, our constitutive equation in tensor form becomes:  cr =2Gs  | =  G  =  \a  (6.7)  G  =Ke  where indices indicate the material property the stress or strain is associated with, e.g. £  G  is the stress  associated with the shear modulus, G. Therefore, in this case we will have two equations that are similar in form to, but simpler than, Equation (6.1). We can write these in vector form as follows:  2La = 2L = K  -c  Ge  (6.8)  1K  k  Now, using the above concept it will be easy to write the equations for a V E material. Since both shear and bulk behaviour are represented by Prony series (Chapter 5), similar to Equation (6.4) and generalizing to multiple Maxwell elements (Chapter 4), we can write:  /=1 Gi T  (6.9)  <=1 Ki T  where r  Gi  and r  Kj  are sets of relaxation times and N  G  and N  K  are the number of Maxwell elements,  associated with moduli G and K , respectively. The idea of decomposing the V E behaviour into two,  - 105-  Chapter 6: Differential Form of Viscoelasticity in 3D normally shear and bulk, behaviour has been used previously in such works as Adolf and Martin (1996) and Wijskamp (2005). Similar to Equation (6.5), we have:  (6.10) Therefore:  N  1  1  a  T - i  & = (G» e +K"e )G  £—a  K  -  G i  i=l  i  £—<r =  i  (6.11)  T  Ki  Due to the fact that the terms in the brackets are equivalent to the stress in an elastic material, as in Equation (6.5), we can write:  £ B" £ =  Z '=1  —Gi ~~  Gi  Z  (6.12)  ^Ki  i=\  Ki  l  where as is the case in any elastic material:  K" - f G "  K" -jG"  0  0  0  ±G"+K"  K"-±G"  0  0  0  0  0  0  G"  0  0  G"  0  ±G"+K  U  D"  Sym.  (6.13)  G" The above equation can be written in the following more simplified form:  (6.14)  where p is a number representing the material property G or K, and r . and a  pj  are the relaxation  times and stresses associated with that material property for Maxwell element / . This is the governing  - 106-  Chapter 6: Differential Form of Viscoelasticity in 3D differential equation for the isotropic V E material. The differential equation for each Maxwell element in the assembly is:  (6.15)  in which P is the spring stiffness in Maxwell element i for either of the material properties G or K . t  Note that the stress and strain vectors associated with material properties, i.e. a , pi  are complete vectors,  i.e. 6 x 1 for 3 D stresses as in Equation (6.3). However, due to the fact that there will be some repetitions, e.g. the bulk strains are the same, we do not need to keep all the stresses and strains. Here, we only need to save five strains and five stresses for the shear behaviour, because the other strain and stress components can be calculated as follows:  (^33)0 (°"3 )G 3  ~  (^11  =-(  <  J  2I)G  ll  (6.16)  £  +  +  ° " 2 2 ) G  A similar argument is valid for the bulk modulus, where only one strain (bulk strain) and stress (pressure) is needed to describe the behaviour. Using Equations (6.5) and (6.6), the stresses and strains associated with different material properties can be related as follows:  s  \(2s  £33)  22  u  j(2f — f — £ 3 3 ) n  22  2£  12  2£,  (6.17)  3  2s  n  ~  { £U  +  ^22  +  ^33}  ->  ° K  ={ il Cr  =  0  " 2 2  =  0  " 3 3 }  A  :  Note that from now on, the above definitions for isotropic stresses and strains will be used.  - 107-  Chapter 6: Differential Form of Viscoelasticity in 3D Having the material associated stresses from above, the stresses in any direction are obtained by adding all the stresses associated with different material properties in that direction. In other words:  [K) +K)K G  (6.18)  <7  XN  6.1.1.2.  Transversely isotropic materials  Similar to isotropic materials, first elastic materials will be considered. We know that in order to describe the behaviour of elastic transversely isotropic materials five properties are needed. Similar to isotropic materials, these properties should be chosen properly. In other words, the choice should be such that in the stress-strain relationship those moduli become decoupled, e.g. Equation (6.5). Hashin (1972) presents such a decoupled form of stress-strain relationship for use in micromechanics, which can be utilized here. The properties to be used in this case are the plane strain bulk modulus K , shear modulus G , and shear 2  modulus G  2 3  1 2  , where plane 2 - 3 is considered to be the plane of isotropy. Two new moduli n and £ are  introduced as follows:  I^'  +  4  \l = 2K v 2  where £, and v  12  ^  (6.20)  n  are the longitudinal modulus and the major Poisson's ratio.  Note: These two moduli can be evaluated by applying a unidirectional strain to a cylinder of the material, with its axis along the 1 direction, as in the C C A approach by Hashin, i.e.  - 108-  Chapter 6: Differential Form of Viscoelasticity in 3D  s,, ^ 0, all other e. = 0  (6.21)  The moduli can then be calculated as follows:  (6.22) £  a =  2 2  33  a =  The stress-strain relationship can now be written as:  a = ne+ £g + K g  +G s  (6.23)  a = a +a +a +o~ +<r = Vcr  (6.24)  (  +G e  2  x2  2i  Or:  =  =  = t  n  = K  =  2  G  1  =  2  G  2  * - ' = P  3  In which the indices associate each stress or strain with a material property. In expanded form, the above equations are:  0 0"  a =n 0  23  0  0  0 22  £  ~  23  2s  23  k  f  22  0  0  0  +K 0 2  +  —f ) 33  0 22  E  +  0  0  0  ^ 3 3  2s  0  33  £  0  0 0  "0  +G 0  0 +£  0  0  22  £  G  12  2E ;  2e  2s  0  0  0  0  X2  ]2  _2s  n  0  ^ 3 3  0  " 0  0 £  22  + fjj  (6.25)  X  For V E materials, similar to the isotropic case, we can write a differential equation for each material property:  - 109-  Chapter 6: Differential Form of Viscoelasticity in 3D  '=1  i  T n  < 1  N  <=i 2  K l  =Kls  K 2  ti  T  -±—a >=1  c /=1  ,  (6.26)  Ki 2  "on  w  K i  T  !  2  Z  G  I  ;  2  where the superscript u denotes the unrelaxed value of the modulus. These equations can be written in following generic form:  2 =P"l -tz-2 P  p  (6-27)  pi  where, P" represents any of the unrelaxed moduli and N  is the number of Maxwell elements associated  with material property . In order to find the overall governing differential equation, we combine Equations (6.24) and (6.26):  tt = ife •  n  + fs  t  + K\e  Ki  + G' \s 2  -  +  Gn  '  <  r  „  (- ) 6  28  ,=1/=!^  The terms covered in (*) represent an elastic behaviour with the unrelaxed moduli. Therefore, they can be replaced by D" i, where D" is the unrelaxed elastic material stiffness matrix. In other words:  ^==2^-ZZ-^ P=\  /=i  i  r P  where:  - 110-  (- ) 6  29  Chapter 6: Differential Form of Viscoelasticity in 3D  r 2  f  + 23  K  G  K  ~ 23 G  +  D=  G"  23  0  0  0  0  0  0  0  0  0  0  0  G"  2  G"  Sym.  (6.30)  0  2  23  G  The differential equation for a Maxwell element is:  1  £ ,=P,£ -—Z. P  P  pl  (6.31)  Pi  where P, is the stiffness of Maxwell element i corresponding to the material property P. The above three equations (Equations (6.29) to (6.31)) describe the behaviour of the material completely and are needed in the implementation of the D F . The stresses and strains associated with different material properties are also calculated from Equation (6.25). Similar to the isotropic case, only some of the stresses and strains for each material property are needed to describe the behaviour of the material. The strains and the resulting stresses are as follows:  • " 1 =  [ 22 (T  ^K,  —  \ 22 £  +  £  3 3 }  &K  ={ 22 a  2  '22  a  33 J  r  \2s  ]:  - G  ]  2  (6.32)  n }  K  t  2  "33  —  f cr„ = - c r  2fiV  2e„  =  CT  23  1 r  J c ,  "1  cr..  and for stresses, we have:  - Ill-  Chapter 6: Differential Form of Viscoelasticity in 3D  (o- ) +(a ) 22  (  22  +(a \  K2  22  (6.33)  (J = <  According to these equations, for a transversely isotropic material there are 8 equations for each Maxwell element. Therefore, i f the material properties all have equal numbers o f Maxwell elements, designated as N, we will have %N equations. Note: To account for thermoelastic effect, i.e. i f the Maxwell stiffness changes with temperature as discussed in Section 4.1.2, we need to replace \jx  pi  by \jx , everywhere in the equations where: pj  1 pi  1 p>  (34) '  Here P = P^TQ)) in a cure cycle. j  6.1.2.  Equations for T h e r m a l Expansion and C u r e Shrinkage  A s discussed in Chapter 5, the thermal expansion and cure shrinkage coefficients change with time in a similar way to the other material properties and therefore their form is represented by a Prony series just like the moduli. In a way, the relationship between thermal (or cure shrinkage) strain and temperature (or degree of cure) is the same as between stress and strain. In this analogy, thermal strain acts like stress and temperature is like strain. The D F for calculation o f the thermal expansion will then be identical in form to the other material properties, as follows:  - 112-  Chapter 6: Differential Form of Viscoelasticity in 3D  s = CTE T  Where, e  T  (6.35)  dt  is the thermal strain vector, CTE" includes the values of C T E at time zero (the "unrelaxed  values"), and r  Tj  are the relaxation times in each Maxwell element for thermal strains, obtained from  micromechanics characterization. These values for a transversely isotropic material are:  CTE,' £  f  CTE  2  2 2  22  <  0  CTE  2  ;CTE"=-  0  0  0  0  0  (6.36)  Note that only two values in the above vectors are needed in the calculations and the entire vectors do not have to be saved or calculated. A similar approach can be taken for other types of free strains generated in the composite (e.g. cure shrinkage). Thus, the equation governing free strains can be written as the following generic differential equation:  df  1  u  a  Where, s  fl  l  f  i  (6.37)  ~fl  are the free strains in each Maxwell element, c" are the unrelaxed values of the coefficients  (such as a"), and /  are the parameters, such as temperature or the degree of cure and we have:  (6.38)  -113 -  Chapter 6: Differential Form of Viscoelasticity in 3D Effectively, this form means that in an F E code it is required to calculate the free strain in each time-step by solving the above differential equation, i.e. identical to what is done for the stresses and strains, and then save the values for the next time-step.  6.2.  FINITE E L E M E N T F O R M U L A T I O N  The formulation developed in the previous sections can be implemented in a Finite Element (FE) code. For this purpose the F E equations will be developed here, keeping in mind the efficiency and convenience of the implementation.  6.2.1. Numerical Solution of Differential Equations Similar to the I D case, we need to solve the governing differential equation to calculate the stresses in a time step. Rewriting Equation (6.31) for a Maxwell element and including the thermoelastic effects,  <L ,=m -e r)-±<L P  where, s  pf  p  B  pi  (6.40)  is the free strain rate associated with material property p and can be obtained from the free  strains using Equation (6.32). Using the finite difference method, e.g. as presented by Hughes (1987), with a Crank-Nicholson (central difference) scheme we can write:  • rt+K 1 n+1 -£ )--^gL V  ./i+l °L = P,(e pi  °L  pi  p  pf  pi  At, . n+1 . n =v +—(°L +2L ) pi  pi  pi  where the first two equations are the differential equations at time steps n and n +1 and the third comes from the finite difference method.  - 114-  Chapter 6: Differential Form of Viscoelasticity in 3D Substituting the first two equations in the third and rearranging, we get:  1  0+T  At  (6.42)  A t this point, we need to distinguish between the cases with thermoelastic effects and no-thermoelastic effects. The reason is that this effect makes the equations more complex and there is no need to include those complexities in the cases where the effect is not present.  6.2.1.1.  No thermoelastic effects  In this case, the stiffness remains constant. Therefore, the last terms on the right hand side of Equation (6.42), become:  ycr'C+fee'+£)=/5(sr'-o  (6.43)  where the last equation follows from the finite difference scheme similar to the third equation in (6.41). Therefore the stresses in each Maxwell element can be calculated from Equations (6.42) and (6.43) as follows:  r  /  n+l  n \  ,  /i+l  n  x  n  (6.44)  The total stress at time step n + l can be calculated by summing the above:  n+l  =LL°•pi  2i p  6.2.1.2.  i  With thermoelastic effects  For a general function / , from the definition of the central difference method we can write:  - 115 -  (6.45)  Chapter 6: Differential Form of Viscoelasticity in 3D  f  n  (6.46)  r -f x  n  2At  Using this and the third equation in (6.41), the terms on the right hand side of Equation (6.42) become:  ( At  { £ p  ~  p)  {  '  i  n+l  £  n-\ \  —£  2At  )  )  (6.47)  } iipr +*pr)(C-z )-(pr-p:){c-*r)] I 4' n  t  P  A  As"  -p  Thus,  1 At 2 r ; _„ ..  n+l =  1+  1 At ^*  2F;  .  (j/>"' + {7f )(A ", - A s £  n p f  )-\AP»{As";  1 A? 1+-  +  2F;  + I  -As" } ) }  p  (6.48)  +  Again, the total stresses are calculated from Equation (6.45).  6.2.2. Development of the F E Formulation In this part, the differential equations presented in the previous sections will be used to develop the FE equations. The general F E equation can be written as:  JB adV T  =F  (6.49)  v  In which V is the volume of the material, F is the external load, and:  s = Bu  (6.50)  Rewriting the equation in incremental form for time step n:  \l?Aa dV n  = AF"  - 116-  (6.51)  Chapter 6: Differential Form of Viscoelasticity in 3D where from Equation (6.45) we have:  A f f W ' - ^  (6.52) P  >  Now, similar to the previous section the F E equations will be derived for two different cases, depending on the presence of thermoelastic effects.  6.2.2.1.  No thermoelastic effects  Taking advantage o f the solution of the differential equations, Equation (6.44), and the above equations we can write:  1 Af  P  i  P  1 +  1 Af  1+ 2 r "pi  i  2 r "pi  + 1  + 1  The first term on the right hand side is equivalent to the calculation of stresses in an elastic material, with a modified unrelaxed modulus. Therefore, similar to Equation (6.28) we can write this as follows:  ZK—fV ^ ^ A  P  i  1+  ) =  ( i ? A  ^  ) a s  ^ " A £  (6  -  54)  P  2 r "pi  + 1  Where, I? is a form o f material stiffness matrix closely related to an elastic material stiffness matrix based on the unrelaxed values o f the moduli, i.e. P], modified according to current time-step size and relaxation times, i.e. P " . r  Therefore the stresses, after including the free strains, will be:  ACT" = ^ ( A £ " - t e  f  ) - Y L ± P i p  where,  - 117-  i  (- ) 6  55  Chapter 6: Differential Form of Viscoelasticity in 3D \_At_  1 At  2 T" pi ,  2 pir  n+l  +  (6.56)  "  1 At  -  p i  2 ^  1 +  Substituting in the F E equation, i.e. Equation (6.51), we obtain:  JB D" (BAu" T  r  -As" )dV-  (6.57)  \ Y L ^ dV = AF"  \B  T  f  V P  •  J  Thus,  \}f [f BdV T  \v  \AU" = AF" + JB D Ae" dV r  T  J  +  u  f  dV  \g  v  v  V P  '  (6.58)  J  Therefore, the governing F E equation at a given time step is:  K Au = AF +  AF +AF  T  f  a  (6.59)  where,  K = JB r? BdV T  T  T  v  AF  f  =  JB lf A£ dV r  T  f  dV  v  p  •  (6.60)  J  where D", is formed as an elastic stiffness matrix, using the following material properties:  At 1+—  =  2r"  - 118-  T  (6.61)  Chapter 6: Differential Form of Viscoelasticity in 3D  6.2.2.2.  With thermoelastic effects  To develop the F E equations for this case, thermoelastic effects in an elastic material are considered first for simplicity. In this case, according to the discussion presented in Section 4.1.2 the constitutive equation is:  A C T = EAs + AEs  (6.62)  Direct implementation of this equation would result in an F E equation in the following generic form:  KAu + AKu = F  (6.63)  However, the standard form of the F E equation used in commercial codes such as A B A Q U S , is as follows:  KAu = F  (6.64)  Given that this is an approximation, unless sufficiently small time steps are used in the analysis, an iterative scheme is needed to analyze a case with thermoelastic effects. We know that in iterative schemes using the exact tangent matrix (in this case, the stiffness matrix) in iterations is not a necessity. In other words, i f a proper stiffness matrix combined with the correct constitutive equation, i.e. Equation (6.62), is used to calculate the stresses the iterative scheme would converge to the correct results. Clearly, the key in successful convergence of the results is the choice of the stiffness matrix. For V E materials, a similar argument is valid. Looking at Equation (6.48), a direct implementation would result in a term, similar to AEs  in Equation (6.62) due to presence of the term AP" in the equation.  However, making use of the discussion presented above and following a procedure similar to the one in the previous section, we can write:  K Au r  = AF + AF +AF f  -119-  rr  (6.65)  Chapter 6: Differential Form of Viscoelasticity in 3D where all the parameters are the same as what was presented previously, except that r should be replaced by T and the material stiffness matrix is calculated as follows:  (i P  n+1  V  ,  1 +  ±ip"l y<  UD"  +  1 At  2 r pi"  +1  r  =T  (6.66) V  '  It should be noted, however, that the convergence of the iterative scheme with this equation has not been studied completely and the convergence of the results needs to be carefully monitored. Having said that, since the temporal solution scheme is based on a stable and accurate finite difference technique it seems reasonable to expect convergence in most standard cases.  6.2.3.  Summary and Discussion  The equations for the D F in 3D were developed in this section. From these equations, the stresses and strains in the composite can be obtained as follows: solving the F E equation (Equation (6.60)) gives the values of displacements, from which the strains are evaluated (Equation (6.50)). Then, the solution of the D F (Equation (6.42) or (6.48)) is utilized to give the stresses. Some relevant remarks on this procedure are as follows: •  The form of Equation (6.59) is fairly similar to that of an elastic material. A s a result, it is expected that the V E formulation can be easily implemented in an available elastic code. A s mentioned in Chapter 2, it is very desirable to write the formulations in a way that they can be implemented at the integration point level, so that no knowledge of the mesh and other space-related parameters is needed. Here, we will discuss how an elastic code can be used to model V E materials with minimal modifications, i.e. by applying changes only at integration point level. For this purpose, it is noted that the form of the stiffness matrix here and in an elastic code are identical. Therefore, to form the V E stiffness matrix the material stiffness matrix in an elastic code can be changed to that of Equation (6.59); i.e. merely a change of the definition of the material  - 120-  Chapter 6: Differential Form of Viscoelasticity in 3D stiffness matrix is sufficient. A similar argument applies to the calculation of both the stresses, and AF  f  for which only the definition of the free strains need to be changed and assembly of values  for the elements in the model can be done by the original code. However, calculation of AF_ , a  Equation (6.60), cannot be directly carried out at the integration point level, since integration over the volume of the material is required. To overcome this problem, we re-write this equation as follows:  (6.67)  Then, we can define an equivalent free strain increment:  (6.68) p  This means that by adding the above equivalent strains to the free strains, the force vector that the code calculates will in fact be the sum of the forces AF_  f  and AF . a  In this manner, all the  calculations for the V E modeling can be done at the integration point level and no knowledge of the meshing or other global attributes of the model is necessary. •  The dependence on the value of t/r  j  is evident from the equations presented in the previous  section. These values clearly play an important role in the accuracy of the solution of the differential equation. One very simple way of discussing the accuracy of the solution is to consider that in each time step these values are constant. Then, using the actual values of these variables and the value of time steps, the ratio can be calculated. Examining the values presented in different works (e.g. Prasatya et el., 2001; K i m and White, 1996) shows that this ratio is often a very small number.  - 121 -  Chapter 6: Differential Form of Viscoelasticity in 3D Although the stiffness matrix changes through the cure cycle depending on the material properties, it can be seen that the changes are not dramatic and the values fluctuate around a constant value. Considering the fact that the ratio t/r  is generally a small number, the material stiffness matrix  i  from Equation (6.61) w i l l be:  I—Y-Tr»Y.r,=r ^tf u  ' 1+ 2 rp> "  +1  r  (6.69)  '  where, P" is the unrelaxed value of modulus. Although there is no need to use this simplification, this shows that the stiffness matrix will be close to an elastic stiffness matrix corresponding to the unrelaxed value of modulus. The term AF_ clearly corresponds to such forces that arise from internal stresses, but are applied a  as external loads. In addition, this term in a way accounts for the difference between elastic and viscoelastic material, especially that as mentioned the material stiffness matrix is more or less constant. The behaviour of V E materials can be explained further as follows: In a V E material with existing stresses and strains, when a new load is applied to the material it reacts by showing two distinct behaviour: 1.  Increments of stress and strain will be generated based on material's elastic behaviour, the basis of which will be a parameter close to the unrelaxed value o f modulus.  2.  The current stresses and strains change due to V E relaxation and/or creep. This part is represented by the term AF_ . a  This understanding of the material behaviour is also reflected by material's constitutive equation. Recall that for a Maxwell element:  - 122-  Chapter 6: Differential Form of Viscoelasticity in 3D  k  /i  Considering the increment of strain as external load, the first term represents the elastic behaviour and the second one includes the current stresses.  6.3.  REVERSE MICROMECHANICS  In Chapter 5 the micromechanics approach for characterization of composite materials was presented. Based on this approach and the formulations derived in this chapter, composite material is modeled using combined properties of the phases, without any knowledge of the stresses and strains in those phases. However, evaluating these values can be very important in other applications, especially in the design of composite materials where knowledge of the stresses in the constituents is necessary. In this section, a procedure w i l l be developed to calculate stresses and strains of the phases from the stresses and strains in the composite, as obtained from the F E model. This process will be called 'reverse micromechanics'.  6.3.1. Proposed Method The composite stresses and strains, a  and s , are calculated in an F E code from the equations presented  c  c  in the previous sections.(strain from (6.50) and stress from (6.44) or (6.48)). We now need to calculate these values for the phases, namely q_ , s , m  m  a , and s , f  f  where m and /  indices indicate the values in  matrix and fibre, respectively. Having the constitutive equations for both phases and recalling phase average equations of micromechanics (Equations (5.4) and (5.5)) we will have four equations with four unknowns to evaluate. Rewriting equations, we will have:  - 123 -  Chapter 6: Differential Form of Viscoelasticity in 3D  P  <  pi  T  (6.71)  where, p and / range from (p = 2or5)  and i = 1,...,N . Also, s p  mf  and s_ denote the free strains in the ff  matrix and fibre, respectively. Note that the over bars have been omitted for simplicity. We start by combining the last three equations:  2c = ( - ^ / ) o : + v o : =(\-v )a +v D (e -s )  = (\-v )a  1  m  /  /  f  m  f  f  f  ff  f  m  +D  [s -v £ -(\-v )s ]  f  c  f  ff  f  m  (6.72) Using the incremental form of this equation at time step n:  A o : : = (1 - v )Acr" + D [As" -v Ae" f  m  f  c  f  - (1 - v )Ae" ]  ff  f  m  (6.73)  A t the same time, recalling Equation (6.55) for the matrix, the first equation written in incremental form is:  (6.74) p  >  Combining these two equations, we obtain the increment of strains in the matrix as:  (6.75)  where all the values on the right hand side are either given as input or have been calculated in the previous time step. In addition, despite the size of the above equation, the calculation is relatively straight forward, since it mostly consists of simple matrix manipulation on integration point level.  - 124-  Chapter 6: Differential Form of Viscoelasticity in 3D Having the value of strain in the matrix, the matrix stress can be found by solving the first equation in (6.71), following a similar procedure to the one used for composites, i.e. Equations (6.42) and (6.45). Also, the fibre strain can be easily calculated as follows:  (6.76)  It should be noted that since these calculations are not needed in the analysis of the composite, calculation of the phase stresses and strains can be controlled with a switch in an F E code to decide i f and when they are needed.  6.4.  SUMMARY  In this chapter, the equations for D F of viscoelasticity in 3D were developed and the finite element equations for the analysis of composite materials during cure were obtained. The advantage of such a formulation is that it applies only at the integration point level. Therefore, it can be implemented in any available F E code. The formulation is developed for both isotropic and transversely isotropic materials, where all material properties have been assumed time-dependent; thermal expansion and cure shrinkage coefficients are also assumed to vary with time, similar to other material properties. In addition, reverse micromechanics equations were developed to calculate the phase stresses and strains from those of the composite materials. Furthermore, the formulation allows for modeling thermoelastic effects of the materials. In the next chapter, the discussion on implementation of these formulations in an available F E code and verification of this implementation will be presented.  - 125 -  Chapter 7: Implementation and Verification  Chapter 7.  IMPLEMENTATION AND VERIFICATION  The formulations for the Differential Form (DF) of viscoelastic ( V E ) material behaviour were developed in the previous chapter. In this chapter, the discussion on implementing this formulation in an available code and some examples to verify this implementation will be presented. For this purpose, first the details of the implementation procedure, including the algorithm and the "pseudo code" written for the implementation, are discussed and the features covered in the code are listed. Then, some simple examples are shown to verify the code implementation of the differential form of viscoelasticity.  7.1.  I M P L E M E N T A T I O N IN A C O D E  The D F model developed in the previous chapter can be easily implemented in an available finite element (FE) code. This is due to the fact that the D F formulation developed here only applies at the integration point level. This way, an available code can be expanded to include the capability to model V E materials with few changes. Furthermore, theses changes can be carefully structured such that the general nature of the formulation can be maintained. This way, the formulation would not bind the user to any specific code and in principle any commercial code which allows for insertion o f external (user) subroutines can be used for implementation of the D F . In order for the implementation to be general, it will consist of a number of separate subroutines, each responsible for an aspect of the calculations for a V E material, to be called " D F code". These subroutines can then be linked to the "main" code using an interface. This interface w i l l transfer information between the main F E code and the D F code and will normally consist of one (or more) subroutine(s). This relationship is schematically shown in Figure 7.1. This structure will allow us to use the D F code in virtually any code. If a code does not allow user routines, since the subroutines are modular they can be called internally from the main code,.  - 126-  Chapter 7: Implementation and Verification In the next sections the details of the DF implementation will be discussed. It should be noted that for the implementation model in this thesis, the commercial F E software A B A Q U S has been used. As a result, the discussions on implementation, naming of the interface subroutines, etc., in the following sections will be specific to ABAQUS.  7.1.1.  Code Algorithm and Pseudo Code  In ABAQUS, user subroutines, U M A T and UEXPAN, allow for defining new constitutive models and user-defined thermal strain calculations, respectively, and will act as the interface between ABAQUS and the DF code. The DF subroutines are therefore called through these user subroutines. In addition to calling the subroutines, these user subroutines are utilized to translate the variables to what the main code expects and to save the parameters required in calculations in each time step, e.g. the internal stresses in the Maxwell elements. A schematic of the algorithm, developed for A B A Q U S , is shown in Figure 7.2. A brief description of different subroutines is given below: In subroutine U E X P A N , CURE_MOD is called to calculate the degree of cure based on thermochemical model of choice. Based on this parameter and temperature value at the integration point CalcEFree is called to calculate the free strains. Then CalcESigma is called to calculate the equivalent free strains, representing the effect of the current stresses. These are then passed on to A B A Q U S for the calculation of the global force vector. It should be noted that since the calculation of the equivalent free strain, i.e. calculation of &F_ , is carried out in subroutine a  U E X P A N this subroutine needs to be called, even if there is no change in temperature or cure. The subroutine U M A T serves two functions. First, the material stiffness matrix is calculated by calling CalcDT and passed on to the main code for calculation of the global stiffness matrix. Then, after the ABAQUS engine solves the FE equations, i.e. it calculates the displacements and strain, and subsequently passes them on to U M A T subroutine, the routine SolveStress is called to  - 127-  Chapter 7: Implementation and Verification solve the differential equations and calculate stresses in the Maxwell elements and consequently in the composite. Also, i f needed, ReverseMM is called in this routine for the reverse micromechanics computations. Based on the written algorithm, a so-called 'pseudo code' was written to give a better insight into the code structure. This in fact is a detailed written form of the code algorithm and provides a more comprehensive description for it. The pseudo code written for the D F code is presented in Appendix B .  7.1.2. Code Features Based on the algorithms presented in the previous section, the D F subroutines have been written in F O R T R A N language and implemented through U M A T and U E X P A N subroutines in A B A Q U S . The D F code is capable of modeling an isotropic or transversely isotropic viscoelastic composite material in 3D under temperature change and cure. It models all the mechanical material properties, and thermal expansion and cure shrinkage coefficients as being time-dependent with Prony series representation. The viscoelastic behaviour can be thermorheologically simple or complex. It is also capable of using reverse micromechanics to calculate the average matrix and fibre stresses and strains. The features available in the D F code and internal V E model o f A B A Q U S have been compared in Table 7.1. A more detailed look at the internal V E model of A B A Q U S and its similarities and differences with the D F code is presented in Appendix C .  7.2.  VERIFICATION PROBLEMS  In this section, some examples are used to debug and verify the implementation o f the D F code. These are simple examples for which there are known, mostly analytical, solutions. In order to verify different parts of the code, 9 examples have been designed to verify different aspects of the implementation. These examples along with the results from both the code and the exact solution will be presented here. A summary of the examples is presented in Table 7.2.  - 128-  Chapter 7: Implementation and Verification  7.2.1.  Transversely Isotropic Cases  Example 1.  Rod under uniaxial strain  General Description  In this example, a single solid element with a transversely isotropic behaviour is modeled. The loading is a simple relaxation test, with a longitudinal strain o f 1, i.e. strains in the two other directions are zero. The dimensions o f the element are shown in Figure 7.3.  Properties  Recalling the stress-strain relationship from Chapter 6, shown below, the only parameters needed are n and  t, 0  0"  g = T] 0  0  0 +£  0  0  0  0  0  ~n £  "0  +G 0 23  0  22  £  0 22  £  +  33  £  0  0  n  0  +K 0  •  -  [f  22  — £33)  1 2  S  22  0  lK  + G  £  2  s  0  0  ~~ 33  2E  £  0  0  0  + £33  0  0  " 0  2e  2£, "  2e  0  0  0  0 ^  l2  n  _2*  13  £  2 2  + £33  (7.1)  3  These properties have been assumed as follows:  77 = 1260 + 10exp(-0  (7.2)  * = 22 + 44exp(-f/1.5) Note:  The properties assumed above come from a consistent set o f fibre and resin properties and the resulting composite properties are within physical bounds, e.g. the Poisson's ratio is 0 < v < 0.5 .  Analytical Solution  According to Equation (7.1), we have:  - 129-  Chapter 7: Implementation and Verification  cr a  n  2i  =ns =1260 + 10exp(-0 n  =  C  3 3  R  = w =22 + 44exp(-f/1.5)  (7.3)  l£  Results Variation of the stresses cr,, and a  22  versus time, resulting from the user material and the analytical  solution are compared in Figure 7.4 and Figure 7.5. It can be seen that the results match very well and the curves are indistinguishable.  Example 2.  Fully constrained rod under temperature change  General Description The solid element from the previous example is completely confined and subjected to a constant change of temperature, equal to 10". Here, the U M A T is tested in modeling a transversely isotropic material under temperature changes.  Properties According to Equation (7.1), in addition to the properties provided in the previous example, we need  K, 2  which is assumed to be:  K =18 + 46 e x p ( - 0 2  (7.4)  C T E is also assumed to follow a Prony series as:  (7.5)  Analytical Solution The C T E is assumed constant for the analytical solution, since this assumption will not change the results significantly, while a variable C T E modeled in the code verifies the implemented formulation. According to Equation (7.1), we will have:  - 130-  Chapter 7: Implementation and Verification  =ns + £(s + £ ) = 130.4 +1 Oexp(-f) +8.4 exp(-f/1.5) a = te + K (e + £ 3 3 ) = 5.8 + 9.2exp(-/) + 4.2exp(-r/1.5) cr,, 22  n  22  u  2  33  (7.6)  22  Results The principal stresses from the user material and the analytical solution are compared in Figure 7.6 and Figure 7.7. Again, it is seen that the results match perfectly.  7.2.2.  Isotropic Cases  Example 3.  Rod under uniaxial stress  General Description This example is taken from the ABAQUS Benchmark Manual Version 6.4, Section 3.1.1. Similar to previous examples, a single solid element is modeled, but this time with isotropic behaviour. The dimensions of the element are 10*1*0.5 and it is subjected to a creep test with a longitudinal stress of 100, as shown in Figure 7.8. Note that the longitudinal direction in this example is 2.  Properties The material properties are as follows:  (7.7)  Analytical Solution The solution, according to the ABAQUS Benchmark Manual is:  e  22  =0.1(1 -0.9 exp(-r/10))  Results The element was analyzed through the user material. Also, to compare the DF to the formulation available in ABAQUS, time step sizes were not assumed in ABAQUS, which means the code itself would  - 131 -  Chapter 7: Implementation and Verification  choose the appropriate size of the time steps. The resulting longitudinal strains are compared in Figure 7.9. It can be seen that even though both sets of results are very close to the analytical solution, only the result from the user material can be said to match the exact solution. A look at the two sets of results shows that the internal V E model in ABAQUS uses more time steps (14 vs. 9 time steps) but the results are less accurate. In addition, U M A T is about 20% more efficient in runtime than the internal engine.  Example 4.  Fully constrained rod under temperature change  General Description For this example, a single isotropic solid element is completely confined and subjected to a change of temperature of 10°. Properties The material properties are assumed to be:  K = 7000 + 3000 exp(-f/l) ' CTE = 0.01 + 0.0001 exp(-f / 2)  (7.8)  K V  Analytical Solution The stress-strain relationship for an isotropic material, as derived in the previous chapter, is as follows:  a = Gg + Ke =G(g-^-IJ G  K  + K(s r)  (7.9)  a  Therefore in this example:  <7„ =a  22  =o- = 3 ^ , , =2100 + 900exp(-f/l). 33  •  (7.10)  Similar to Example 2, changes in free strains with respect to time are not significant and for the analytical solution a constant C T E has been assumed. In the numerical modeling of this example, the actual C T E is assumed.  - 132-  Chapter 7: Implementation and Verification Results The principal stresses resulting from the user material and the analytical solution are compared in Figure 7.10. Clearly, the two sets of stresses are identical.  Example 5. Fully constrained polymer rod in a cure cycle General Description This example is designed to compare the results from the code to the type of results generated by the I D code in Chapter 3. In includes a single isotropic solid element, completely confined and subjected to temperature changes and advancement of cure. The dimensions are 10*1*1.  Properties The material properties are chosen such that the resulting longitudinal relaxation modulus corresponds to the data provided by K i m and White [1996] for 3501-6 resin. For this reason, the same set of relaxation times and weight factors are assumed for the shear and bulk modulus. The resulting modulus along with the C T E are as follows:  £ = 31 + 186.97 exp(-r / 29.2) + 209.15 exp(-f / 2 . 9 2 £ 3 ) + 263.03 exp(-f /1.8255) + 354.93 exp(-r /1.\ET) + 488.03 exp(-r / 2.83£8) + 830.28 exp(-f / 7 . 9 4 £ 9 ) +  (7.11)  583.1 exp(-r /1.95£11) +155.28 exp(-r / 3.32£12) + 79.23 exp(-f / 4 . 9 2 £ 1 4 ) CTE = 5.E-5  (7.12)  It must be noted that in the above reference a linear relationship between volumetric cure shrinkage and cure is assumed, which corresponds to a cubic relationship between the cure shrinkage strain and cure parameters. However, to be able to use the current Prony series model in the code, we need a linear relationship between these two parameters. For this reason, the model provided is plotted vs. degree of cure and compared to a line in Figure 7.11. It is seen that the difference between the two models is  - 133 -  Chapter 7: Implementation and Verification negligible and the linear relationship can therefore be used in the user material. The slope of the line, i.e. the coefficient of cure shrinkage to be used in the D F code, is -0.1695. For this example, the cure cycle and the resulting degree of cure used in the calculations are shown in Figure 7.12.  Results The results of the user material and the I D code are compared in Figure 7.13 and they match perfectly.  Example 6.  Reinforced cylinder under internal pressure  General Description This example is taken from Zienkiewicz et al. [1968]. A hollow isotropic viscoelastic cylinder, reinforced by an isotropic elastic material on the outside is subjected to internal pressure. The structure is assumed to be in a state of plane strain.  Properties The V E cylinder has an inner ( a ) and outer (b) radii of 2" and 4 " , respectively. The elastic material has a thickness of  , as shown in Figure 7.14. The internal pressure ( p) is 1.  The properties of the elastic material are:  £ = 3 . £ 7 ; v = 0.3015  (7.13)  The properties of the V E material are:  G = 37500 e x p ( - r / l )  K = 1E5  - 134-  (7.14)  Chapter 7: Implementation and Verification Analytical Solution The exact solution for this problem was given by Lee et al. [1959]. In this reference, the problem was first solved for elastic materials and then the correspondence principle was used to find the solution for the V E problem. Since neither of the references provides the final result, Mathematica was used to obtain the analytical solution. The Mathematica notebook is provided in Appendix A . 2 .  Results In modeling the material, a quarter of the cylinder was meshed using solid elements. To be able to model the cylinders with 3 D element, a nominal out-of-plane thickness was assumed. Also, to make the structure plane strain the corresponding nodes on the two sides of the cylinder (through the thickness) were constrained together. The mesh, plotted by A B A Q U S , is shown in Figure 7.15. The normalized radial stresses at different time intervals resulting from the user material and the exact solution are compared in Figure 7.16. In the figure the lines represent the analytical solution and the dots are the A B A Q U S results. The results show excellent agreement. There is a slight difference between the two sets of results. This difference is due to the element sizes chosen for the example; a similar trend can be seen in the results given by Zienkiewicz et al. To show that refining the mesh improves the results, a finer mesh is made that has 4 times the number of elements as the previous one and is illustrated in Figure 7.17. The stresses resulting from this mesh are shown in Figure 7.18. These results show better agreement, which indicates convergence of the stresses to the analytical solution.  7.2.3. Reverse Micromechanics Examples In this part, two examples are provided to verify the reverse micromechanics model implemented in the user material code.  - 135-  Chapter 7: Implementation and Verification  Example 7.  Transversely isotropic rod under uniaxial strain  General Description To verify this aspect of the code, the solid element modeled in Example 1 is again used here to calculate the matrix stresses and strains.  Properties The composite properties are the same as in Example 1. The constituent properties, which are consistent with the composite properties, are assumed as follows: Matrix properties:  G = 0.18 + 11.67 e x p ( - f / l ) m  K = 8 + 27.56exp(-//1.5)  (7.15)  m  Fibre properties:  (7.16)  The fibre volume fraction is assumed to be 0.6.  Analytical Solution The analytical solution has been obtained using the correspondence principle in Mathematica. The Mathematica notebook is presented in Appendix A.3, with the resulting stresses and strains at two points in time shown as example.  Results The results of the matrix principal stresses (in directions 1 and 2) are compared to the analytical results in Figure 7.19 and Figure 7.20. The results match perfectly.  - 136-  Chapter 7: Implementation and Verification  Example 8.  Transversely isotropic fully constrained rod under temperature change  General Description In this example, the second on reverse micromechanics, matrix stresses and strains for Example 2 are calculated and compared to the analytical solution. Properties The composite properties are the same as in Example 1, and those o f the constituents are the same as Example 7. The C T E of the resin and fibre are as follows:  (7.17)  and:  (7.18)  Analytical Solution Again, using correspondence principle in Mathematica the analytical solution is obtained and is shown in Appendix A . 4 . The sections containing the constituent properties are omitted to avoid repetition.  Results The results of matrix stresses in directions 1 and 2 are compared in Figure 7.21 and Figure 7.22 and are identical.  7.2.4. Thermoelastic behaviour In this part, an example is used to verify modeling of thermoelastic behaviour in the D F user material. The formulation for this model was derived in Chapter 6.  - 137-  Chapter 7: Implementation and Verification  Example 9.  Isotropic rod under uniaxial strain  General Description To verify this part of the code, the solid element modeled in Example 3 is used here again to calculate the stresses and strains. Due to the difficulty in finding an analytical solution for a transversely isotropic material the material is assumed to be isotropic.  Properties For the material properties some nominal, but consistent, values were assumed. The material properties are as follows:  G = 375 + 3375exp(-//l) K = 7000 + 3000 exp(-f /1.5)  (7.19)  Thermoelastic effects are assumed to affect the moduli by a linear function of temperature, as discussed in Section 5.3.4, as follows:  P" (T) = P  u0  (1.0591 - 0.0022 T)  (7.20)  Where, P represents both K and G, i.e. a single function of temperature is assumed to govern the changes in the unrelaxed bulk and shear moduli. The temperature is held constant throughout the creep test; a varying temperature makes it difficult to solve the problem analytically, while an isothermal creep test is a suitable problem for verifying the implementation of the code.  Analytical Solution To obtain the analytical solution, we note that according to Schapery (1974) for T C M - 2 , the isothermal creep compliance is as follows:  D =D(t)/a (T) T  G  - 138-  (7.21)  Chapter 7: Implementation and Verification where, D  r  is the compliance for T C M - 2 , £>(£) is the compliance for T S M , and a  G  is the vertical shift  factor. The solution, using Mathematica, is shown in Appendix A . 5 .  Results The strains in 1 and 2 directions are compared in Figure 7.23 and Figure 7.24. The longitudinal direction in this example is 2. Again, the results from D F code and analytical solution match. One may want to compare these results to strains from a non-thermoelastic analysis. For this purpose, a second analysis has been performed, assuming that the values of the unrelaxed moduli provided above do not change, i.e. a value of P = 1. The resulting strains from the two models are compared in Figure 7.25 and Figure 7.26. Some clear differences in the results can be observed. In thermoelastic analysis, the unrelaxed and relaxed moduli at this temperature  are less than the non-thermoelastic one by  approximately 20% and that is consistent with the difference between the two sets o f results throughout the test at this temperature.  7.2.5.  Summary  In this chapter, it was discussed how the differential form of viscoelasticity can be implemented in a code and subsequently its implementation in A B A Q U S was examined. Then, through the examples presented, different aspects of the differential form code were verified. These aspects include the transversely isotropic and isotropic behaviour of materials with time-dependent material properties, with temperature changes and advancement of cure. In addition, by using these examples implementation of the formulations developed previously for reverse micromechanics and thermoelastic effects were verified. A s a result, we have a working implementation of the differential form of viscoelasticity in A B A Q U S that is capable of modeling composite materials during cure. Taking advantage of this tool, some numerical applications w i l l be studied and presented in the next chapter.  - 139-  Chapter 7: Implementation and Verification  Table 7.1 Comparison of the features of the added U M A T and the A B A Q U S built-in V E model  UMAT  Features  A B A Q U S built-in V E model  Isotropic behaviour Temperature-dependent behaviour Transversely isotropic behaviour Cure kinetics  </  Time-dependent CTE's  •/  Reverse Micromechanics Thermoelastic effects  Table 7.2 Summary of examples to be analyzed Example  Part  L o a d Type  1  Transversely-isotropic rod  Axial strain  2  Transversely-isotropic rod  Temperature  Note Fully constrained  3  Isotropic rod  A x i a l strain  4  Isotropic rod  Temperature  Fully constrained Fully constrained  5  Isotropic rod  Cure cycle  6  Reinforced cylinder  Internal pressure  7  Transversely-isotropic rod  Axial strain  Example 1 with Reverse Micromechanics  8  Transversely-isotropic rod  Temperature  Example 2 with Reverse Micromechanics  9  Isotropic rod  Strain/Temperature Example 3 with thermoelastic effects  - 140-  Chapter 7: Implementation and Verification  Available Commercial Code Interface  DF Subroutines Figure 7.1 Schematic form of the relationship between the D F code and an available commercial code  - 141 -  Call CURE_MOD Calculate degree o f cure Call CalcESigma Calculate equivalent free strain increment, add to free strains  Call CalcEFree Calculate free strain increment  UMAT  Call CalcDT Calculate material stiffness matrix  Call SolveStress Calculate Maxwell stresses  ^  Parts of ABAQUS  Call ReverseMM Main subroutine for reverse micromechanics.  The DF Code added  Figure 7.2 Algorithm of the Differential Form code for a given time step  Chapter 7: Implementation and Verification  3  2  j  1 0  Figure 7.3 Specifications for Example 1  - 143-  Chapter 7: Implementation and Verification  130 125 -j 120 in in £ 115 tn 110  — S11(UMAT) - » - s11 (Analytical)  H  H  105 100  6  12  10  Time  Figure 7.4 Comparison of a  as predicted by the Differential Form U M A T and analytically for  u  Example 1  -^s22(UMAT) -»-s22(Analytical)  6  10  12  Time  Figure 7.5 Comparison of cr  22  as predicted by the Differential Form U M A T and analytically for Example 1  -  144-  Chapter 7: Implementation and Verification  145  -j  140 ' 135 130 m in  £  125 -  -s11(UMAT) -s11 (Analytical)  120 115 110 105 100 6  10  12  Time Figure 7.6 Comparison of <r, , as predicted by the Differential Form U M A T and analytically for Example 2  25  n  -— s22(UMAT) s22(Analytical)  6  10  12  Time Figure 7.7 Comparison of <x as predicted by the Differential Form U M A T and analytically for 22  Example 2  - 145 -  Chapter 7: Implementation and Verification  '22  100  1 3 0.5 10  Figure 7.8 Specifications for Example 3  0.12  e22(UMAT) e22(Analytical) ••- -e22(ABAQUS)  10  20  30  40  50  60  Time Figure 7.9 Comparison of s  22  as predicted by the Differential Form, A B A Q U S , and analytically for Example 3  - 146-  Chapter 7: Implementation and Verification  -s11(UMAT) -s11 (Analytical)  6  10  12  Time Figure 7.10 Comparison of <J  U  as predicted by the Differential Form U M A T and analytically for Example 4  Degree of cure Figure 7.11 Comparison of the exact cure shrinkage and the simplified linear model, used in Example 5  - 147-  Chapter 7: Implementation and Verification  -s11(UMAT)  -s11(1Dcode)  w  50  100  150  200  250  T i m e (min)  Figure 7.13 Comparison of stresses as predicted by the Differential Form U M A T and the previously verified ID code in Example 5  - 148-  Chapter 7: Implementation and Verification  Chapter 7: Implementation and Verification  \ \ V ' \\ r  l  -l  T  *  4  i-i  Figure 7.15 Standard mesh used in Example 6  0.4 0.3 0.2 0.1 0  I  >  0.5  0.6  1 0.7  1 0.8  1 0.9  1 1  r/b  Figure 7.16 Comparison of radial stresses at different times t=0.1 to 10 as predicted by the Differential Form U M A T (dots) and analytically (lines) using a standard mesh in Example 6  - 150-  Chapter 7: Implementation and Verification  Figure 7.17 Refined mesh, used for Example 6  0.4 0.3 0.2 0.1 0 I 0.5  ,  1  0.6  0.7  1  1  1  0.8  0.9  1  r/b  Figure 7.18 Comparison of radial stresses at different times t=0.1 to 10 as predicted by the Differential Form U M A T (dots) and analytically (lines) for a refined mesh in Example 6  - 151 -  Chapter 7: Implementation and Verification  -s11(UMAT) • s11 (Analytical)  10  6  12  Time Figure 7.19 Comparison of matrix stresses as predicted by the Differential Form and analytically for Example 7  Figure 7.20 Comparison of matrix stresses as predicted by the Differential Form and analytically for Example 7  - 152-  Chapter 7: Implementation and Verification  25  n  in m  -s11(UMAT)  £  -s11 (Analytical)  6  12  10  Time  Figure 7.21 Comparison of matrix stresses as predicted by the Differential Form and analytically for Example 8  - » - -s22(UMAT) — -s22(Analytical)  6  10  12  Time  Figure 7.22 Comparison of matrix stresses as predicted by the Differential Form and analytically for Example 8  - 153 -  Chapter 7: Implementation and Verification  0 10  20  30  40  50  60  -0.01 -0.02 in  |  -e11(UMAT) • e11 (Analytical)  -0.03  x  -0.04 -0.05 -0.06  Time  Figure 7.23 Comparison of strains as predicted by the Differential Form U M A T and analytically for Example 9  0.12 0.1 0.08 in  c  - e22(UMAT) - e22(Analyticai;  2 0.06 CO  0.04 0.02 0  10  20  30  40  50  60  Time  Figure 7.24 Comparison of strains as predicted by the Differential Form U M A T and analytically for Example 9  - 154-  Chapter 7: Implementation and Verification  e11(With Thermoelastic) -°— e11 (no Thermoelastic)  Time  Figure 7.25 Comparison of strains as predicted with and without thermoelastic effects by the Differential Form U M A T for Example 9  0.12  e22(With Thermoelastic)  - e22(no Thermoelastic)  10  20  30  40  50  60  Time  Figure 7.26 Comparison of strains as predicted with and without thermoelastic effects by the Differential Form U M A T for Example 9  - 155 -  Chapter 8: Numerical Applications  Chapter 8.  N U M E R I C A L APPLICATIONS  The formulation to model the material with a Differential Form (DF) of Viscoelasticity (VE) or PseudoViscoelastic (PVE) formulation was developed and implemented in the previous chapters. These tools can be utilised for process modeling of composite materials to find the stresses and deformations. Here, we will show the capabilities of these models and how they compare to each other and also other existing results in the literature. It is noted that there are only a few V E models in the literature whose results can be used for comparison here. There are also very few experimental data in the literature measuring V E residual stresses to validate these models. Furthermore, comparing the residual stresses from the models developed in this thesis to other models or experimental results may not yield a meaningful evaluation of the material models, since a difference in the results may be due to inaccurate representation of aspects of the model other than the material model, e.g. behaviour of the tool-part interface. For these reasons, in addition to the comparison of DF, PVE, and IF made in Chapter 3 and Chapter 4 a general validation of the models will be carried out by looking at trends and qualitative comparison of the results. Overall, this chapter outlines some case studies, where the goals are: To show that the predictions made by the models developed in this thesis agree with general understanding of the material behaviour and compare well qualitatively with existing models in the literature. To show cases where different models give similar predictions and studying in detail their similarities and differences in simple and more complex cases, with a general comparison to elastic models. To show cases where simpler P V E model cannot give accurate results and a V E model would be necessary.  - 156-  Chapter 8: Numerical Applications To showcase capabilities developed for the models in this work and not available or achievable through other models in the literature, e.g. R M M , thermoelastic effects, and efficiency. To discuss the advantage of these models in their ability to represent the material behaviour more accurately. To achieve these, first some simple examples consisting of only one element w i l l be analyzed in order to gain a better understanding of the material behaviour. Then, some more realistic cases will be analyzed to confirm these findings.  8.1.  MATERIAL PROPERTIES  The formulations presented in the previous chapters are general and not limited to a specific material. However, in analyzing the following examples a choice of a specific material is needed. For this purpose, the material properties for the resin and fibre provided by K i m and White [1996] combined with the micromechanical model of Bogetti and Gillespie [1992] will be used here. For the P V E analysis, the definition of the elastic modulus based on the formulations derived in Chapter 3 is used. To model the P V E in 3 D the A B A Q U S user material routine written by Arafath (2004) was used. For this purpose, the code that was originally written with a C H I L E constitutive model was slightly modified to allow for a P V E models. For the D F analysis, the mechanical properties derived by K i m and White for the composite materials can be used here. However, in that work some assumptions on the relaxation times and the weight factors were made (the difference arising from such assumption has been Uustrated in Section 5.3.2). In addition, some inconsistencies were observed in their material properties; specifically, the values of relaxed and unrelaxed moduli provided in the article do not match the properties obtained by combining the fibre/resin properties through the elastic micromechanics, i.e. the approach used in the work. For these reasons and due to simplicity and accuracy of the micromechanical approach presented in Chapter 5, this  - 157-  Chapter 8: Numerical Applications approach can be used to obtain the material properties for AS4/3 501-6 composite. For this purpose, the fibre and resin properties are taken from K i m and White's work, and listed in Table 8.1 and the constants for the viscoelastic model o f K i m and White have already been listed before in Table 5.2. We also note that polymers are known to have two regimes of thermal expansion behaviour: above and below the glass transition temperature  (T ). g  However, in the aforementioned reference a single  coefficient of thermal expansion ( C T E ) has been assumed. In order to have a more realistic material model the C T E of the material has been assumed to have two different, but constant, values below and above T . To have a proper estimate of these values, the work of Russell et al. (2000) was used. In this g  work, the total volume change of the 3501-6 resin in a cure cycle was experimentally measured. In order to obtain the values of the required parameters, C T E (two values) and the coefficient of cure shrinkage (one value) were assumed as unknowns and the total volume change of the polymer in the given cure cycle was calculated by adding the volume change due to variation of temperature and advancement of cure. Then by using the Solver Add-in in Microsoft Excel®, the unknowns were found such that the overall shape of the volume change vs. time corresponded closely to that of the experiment. The fitted result is shown in Figure 8.1. The C T E ' s and coefficient of cure shrinkage are listed in Table 8.2. B y combining the material properties of the phases through micromechanics, the viscoelastic properties of the composite was obtained. The process is identical to the one presented in Section 5.4.1, except that the calculations for the C T E have to be done twice to obtain the two composite C T E ' s in the rubbery and glassy regimes. Note that as in Section 5.4.1, the relaxation times have been taken to be the same for all material properties and the weight factors have been obtained based on regression analyses. Also, the variability of the following material properties with time is minimal (see Table 8.3) and therefore they have been assumed to be constant in time: n, the first modulus, CTE the transverse C T E , and cs , the 2  2  transverse coefficient of cure shrinkage. The results of the unrelaxed and relaxed values of different material properties are presented in Table 8.3. The relaxation times and corresponding weight factors for  - 158-  Chapter 8: Numerical Applications those material properties variable with time are listed in Table 8.4. Note that CTE  g  and CTE are the r  glassy and rubbery C T E ' s . Thermoelastic behaviour of the polymer material was discussed in Section 5.3.4. It was also explained in the same chapter how the composite material properties in such a case can be obtained. Since here we do not have the exact material property for thermoelastic effect and only a reasonable assumption was made, the same will be done for the composite properties. It is expected that each material property of the composite, i.e. five moduli, two C T E ' s and two cure shrinkage coefficients, have a vertical shift factor. It seems reasonable to expect that the material properties in the matrix-dominated directions behave more or less the same as the matrix, while fibre-dominated direction are affected to a lesser degree. In other words, material properties that are clearly viscoelastic will have a vertical shift factor close to the matrix, while the other properties that are not time-dependent will need much smaller vertical shifts. To confirm this, the matrix moduli at a specific temperature, e.g. at 120°C where the polymer vertical shift factor from Section 5.3.4 is 0.8, were used to find the unrelaxed and relaxed values of all material properties. It was observed that the time dependent properties needed a vertical shift factor of around 0.8, while the other ones needed a value closer to 0.95. Based on these, in the examples that thermoelastic effects are modeled in this chapter, the vertical shift factor for the time-dependent properties will be assumed to be 0.8 at a temperature of 120°C and 0.95 for the time-independent properties at the same temperature. Clearly, the unit value of the shift factor occurs at approximately 27° C , as before. The composite cure kinetics equation is taken from Lee et al. (1982) and is as follows:  da —  = (k +k a)(l-a)(0Al-a) l  2  («<0.3)  f  (8.-1)  — = Ul-a) dt  (a>0.3)  3  where:  - 159-  Chapter 8: Numerical Applications  k, =A sxp(-AE /RT) i  (8.2)  i  where R is the universal gas constant, A. the frequency factors, and AE the activation energies. The i  thermo-chemical properties of AS4/3501-6 composite are listed in Table 8.5 and thermal properties of the tooling materials are given in Table 8.6.  8.2.  SINGLE ELEMENT CASES  In this section, some simple examples will be modeled using the D F and P V E models in a single element. This way, a better understanding of the similarities and differences between the two models can be gained before investigating larger scale cases, where dimensional complexity o f the problems makes it difficult to recognize these effects. For this purpose, a single 3D composite brick element with different lay-ups and cure cycles w i l l be analyzed. For each case, the analysis will be done using the A B A Q U S  UMAT  developed here for the D F and the modified user material routine, written by Arafath (2004) for A B A Q U S , for the P V E . A summary of the cases is provided in Table 8.7 for reference.  8.2.1.  Standard C u r e Cycles  Case 1.  Fully constrained unidirectional laminate  General description In this example, a single unidirectional laminate that is completely confined is taken through a cure cycle. The cure cycle and the resulting degree of cure are shown in Figure 8.2. The dimensions of the element are 10*1*1.  - 160-  Chapter 8: Numerical Applications  Results The material is analyzed and the resulting residual stresses in longitudinal (<x ) and transverse directions n  (°~22  =  °"33)  n  a  v  e  been calculated using both P V E and D F . The results are shown in Figure 8.3 and Figure  8.4. The figures show excellent agreement between the two approaches, with a maximum difference of 5% at the end of cure.  Case 2.  Axially constrained unidirectional laminate  General description The previous example is analyzed here, with the only difference that the rod is now only axially constrained, i.e. it is free to move in the local 2 and 3 directions. The cure cycle and the other properties remain the same.  Results The D F and P V E residual stresses for the axial direction are illustrated in Figure 8.5. Again, the two approaches compare very well. The slight difference between the two cases mostly comes from the prediction of stresses on the heat-up. A s mentioned in Chapter 3, P V E estimate of modulus in a heat-up is not accurate, which in this case results in a slight error, as compared to the D F . Regardless of the reason, this difference is very small and does not affect the assessment of the results.  Case 3.  Axially constrained unidirectional laminate with thermoelastic effects  General description The previous example is analyzed here with the assumption of thermoelastic effects. The cure cycle and the other properties remain the same.  - 161 -  Chapter 8: Numerical Applications Results The D F residual stresses in the axial direction with and without the thermoelastic effects are illustrated in Figure 8.6. The difference between the two sets of results, and therefore the effect of thermoelastic behaviour is clear from the figure. The figure shows reduced slopes for both stress increase on heat-up and stress decrease in cool-down, for the thermoelastic run. This is obviously due to the reduced modulus, as a result of thermoelastic behaviour.  Case 4.  Fully constrained [30 °] laminate  General description For this example, a fully constrained laminate with a lay-up of [ 3 0 ° ] is modeled. The other parameters used in the model are the same as the previous examples. This example w i l l show the response of an angle ply laminate in a cure cycle.  Results It is noted that in cases where a local orientation, i.e. the fibre direction, is defined A B A Q U S outputs the residual stresses only in the local direction. Therefore, to compare the D F and P V E method the resultant axial forces (N ) x  are calculated. This way, the stress resultants, which are better measures of the laminate  behaviour especially in complex lay-ups, are compared. The two sets of results are shown in Figure 8.7 and agree very well.  Case 5.  Axially constrained cross-ply laminate  General description For this example, a [0° I90"\ axially constrained laminate is modeled. It should be noted that in a s  completely confined laminate the layers do not interact. A s a result, the layers behave identically to a unidirectional laminate in their local direction. Evidently, this w i l l not provide us with any new  - 162-  Chapter 8: Numerical Applications information. On the other hand, axial constraint allows interaction of the layers. For this reason, only axial boundary conditions are applied. Another issue to note is the type of element used in modeling the composite materials in ABAQUS. The approach normally employed is to use a single multi-layered solid element, as opposed to a single solid element per layer of the composite. This is advantageous, especially in complex cases, since it eases the mesh generation and simplifies the mesh itself. However, the inherent difference between the two elements should be taken into consideration. Single layered element approximates the displacements through the thickness of the layer using shape functions, as opposed to the single element per layer model, where obviously a node is considered for each layer. Also, it should be kept in mind that in ABAQUS the layered elements use Simpson's rule to integrate through the thickness using an odd number of integration points per layer, while an 8-noded brick element uses Gauss integration through the thickness with 2 integration points. These differences can generate some differences in the results and should be kept in mind when modeling. Here, both types of elements will be used to model the material in this and the next example to show the similarities and differences between them.  Results Based on the analysis with a single layered element, the resulting axial forces from the DF and PVE have been compared in Figure 8.8. The results from the two methods compare very well. Also, due to the fact that the laminate is symmetric no bending moment is generated. The resulting axial forces for two element types using the DF model have been compared in Figure 8.9. A small difference in the final residual stresses can be observed. This is most probably due to the fact that the model with multiple solid elements is more flexible, due to having more nodes. Therefore, its residual stresses are slightly lower than those of the layered composite model.  - 163 -  Chapter 8: Numerical Applications  Case 6.  Axially constrained quasi-isotropic laminate  General description A symmetric quasi-isotropic laminate is used for this example. Again, the rod is only axially constrained. Due to the fact that the number of plies is twice as the previous example the thickness of the rod is increased from 1 to 2. Again, the material is first modeled using a layered element to compare D F and P V E . Then, it is modeled using single solid elements per layer to compare the D F results for the two element types. For reference, the same boundary conditions have been applied to both cases: A l l longitudinal displacements have been constrained. To avoid rigid body motions, node 1 in all directions and node number 9 in direction 2 have been constrained, as illustrated in Figure 8.10.  Results The axial forces obtained from the D F and P V E approaches have been compared in Figure 8.11. Again, the results agree very well. In comparing the results of the two different element types, very different behaviour is observed. This is clear by looking at their deformed shapes as shown in Figure 8.12 and Figure 8.13. It is seen in the above figures that the layered element shows shrinkage in transverse directions. Note that the shrinkage in the direction 1 is small, due to presence of fibres, especially as compared to the thickness direction that is dominated by the polymer. However, in the single-element-per-layer approach in addition to the shrinkage a torsion effect is seen. This does not seem unreasonable; while the laminate does not bend under temperature changes because of symmetry through the thickness it could go under torsion when subjected to temperature changes, since it is not symmetric for torsional effects. O n the other hand, the layered element is modeled with a single shape function through the thickness and does not show torsional effects with change of temperature.  - 164-  Chapter 8: Numerical Applications This example underscores more of the differences between these two element types. Although most of the time the two approaches are expected to show the same behaviour, it is not always the case. A s a result, choosing the proper modeling approach should be done in tandem with consideration for the properties of each approach.  8.2.1.1.  Summary  In this section, some simple examples were presented to compare the results from the D F and P V E . The overall conclusion, consistent with what was discussed in Chapter 3, is that the two approaches give very close results for the standard cure cycle that was used for these examples.  In addition, it is noted that there are two element types in modeling the composite materials, i.e. layered elements and single element per layer. The results from these two are not always the same and care should be taken in modeling to choose the proper approach.  8.2.2.  Other C u r e Cycles  In this section, two examples are chosen to show cases where the P V E approach does not give answers close to those of the D F .  Case 7.  Post-cure heating of completely confined unidirectional laminate  General description This case consists o f a single unidirectional composite modeled as a completely confined solid element, e.g. similar to Case 1. The material is taken through a cure cycle and is then subjected to post-cure heating, i.e. as shown in Figure 8.14. In reality, i f a composite part is to go through such a temperature cycle, before being subjected to the heat-up, i.e. while at room temperature, it will be removed from the tool. This means that the residual stresses in the material will be partly released. In order to have a single element model that closely  - 165 -  Chapter 8: Numerical Applications corresponds to this behaviour the stresses need to be released at room temperature. This will cause the stresses to reduce to zero. This, in effect is equivalent to subjecting the cured material to a heat-up from time zero. In other words, the heat-up cycle can be modeled by applying the cycle in Figure 8.15 to the material, with the resulting degree of cure shown in the same figure. Note that after releasing the stresses, the element will be completely confined again.  Results The resulting stresses from the D F and P V E approaches in two principal directions 1 and 2 are shown in Figure 8.16 and Figure 8.17. The results show that even though the two models show a similar trend of stresses at the very beginning, they start to diverge quickly. This was expected, due to the fact that the P V E formulation has been designed with a cool-down in mind and it does not give reasonable results in heat-up.  Notes The D F stresses show two different regimes in the figures. First, the compressive stresses increase, due to the fact that the temperature is increasing. In the second part, the stresses start to decrease dramatically and in the case of a , it will go into tension. The reasons for this are as u  follows: First, the relaxation in the resin causes the stresses in the material to decrease. This has more effect in the resin-dominated directions, i.e. c r . Second, the relaxation of the resin affects 22  the C T E of the composite. Looking at the material properties in Table 8.3, it is seen that the C T E of the composite in the fibre direction, i.e. CTE , is negative and also changes dramatically with ]  time. When the resin relaxes, the C T E has a larger negative value, as evident in the first region of the stresses. However, when resin relaxes the value of CTE  X  becomes even a smaller negative  number and will put the material into tension, even though temperature is increasing.  - 166-  Chapter 8: Numerical Applications It was mentioned above that two phenomena affect the behaviour of a viscoelastic material in heat-up: relaxation o f stresses and changes o f C T E . In the P V E model, it is observed that the model is able to show the variability of C T E at least qualitatively, e.g. in Figure 8.16 the stresses reach a maximum value and then start to decrease from this point on. This is expected, since the C T E of the material is calculated through micromechanics and this will in part ensure that the relaxation of the resin is taken into account in this aspect of the behaviour. However, as we know no stress reduction can be predicted by P V E , as also observed in the figures.  Case 8.  Curing partially-cured and completely confined, unidirectional laminate  General description The geometry and material properties for this example are identical to the previous case. A different cure cycle is chosen here to investigate more of the difference between the D F and P V E models. In practical applications, it is sometimes desirable to perform cure in two parts (e.g. George, 2004). First, the temperature cycle is applied until the material gels and can hold itself. After that, it is cooled down and the tool is removed. The remainder of the cure cycle is applied off the tool. This secondary cure of a partially cured material can be useful in reducing residual stresses. A sample temperature cycle and the resulting degree of cure for this case are shown in Figure 8.18. Recalling the discussion given for the previous case and keeping in mind that the stresses are relaxed between the primary and secondary cycles we can model the material as it goes through only the secondary cure cycle by merely changing the initial degree of cure to its value at the beginning of the secondary cycle, which is 0.79. This equivalent cycle and the resulting degree of cure are presented in Figure 8.19.  Results The stresses predicted by the two approaches are illustrated in Figure 8.20 and Figure 8.21.  - 167-  Chapter 8: Numerical Applications Again, large differences between the results of the two models are seen, which points to failure of the P V E model in predicting the V E stresses. It is also clear that the difference in the results mostly comes from heat-up. If the difference in stresses at the end of heat-up is set aside, the trend of the P V E results would be closer to the D F stresses.  8.2.2.1.  Summary  Through the two cases presented above, it is obvious that although P V E has worked very well in many cases (e.g. standard cure cycles) it cannot predict the V E stresses in every possible case. Specifically, in the cure cycles where some heating occurs after gelation point of the resin the P V E model cannot correctly predict the residual stresses. It must be noted here that P V E can conceivably be generalized to a formulation that can cope with such conditions, a simple example o f which was presented by Svanberg and Holmberg (2004 a, b). However, such a formulation is not available at this point.  8.3.  M O R E COMPLEX CASES  In this part, examples of some more realistic composite parts on tooling materials have been chosen and the result of applying the curing process will be studied. First, the results for standard cure cycles are studied, followed by non-standard cure cycles.  8.3.1. Standard C u r e Cycles  8.3.1.1.  Small thin L-shaped composite part  This part was originally modeled by Wiersma et al. (1998) and later by Zhu et al. (2001). It is noted that because of the differences in material properties and aspects o f modeling other than the composite constitutive model, a direct comparison of the results with theirs is not possible. However, a qualitative comparison of the results will be made in order to gain more confidence in the-validity of the present approach.  - 168-  Chapter 8: Numerical Applications The part is a unidirectional L-shaped composite and is placed on two different tool shapes, i.e. convex and concave as shown in Figure 8.22 and Figure 8.23, respectively and different tool materials, i.e. steel, aluminium, and invar. Due to symmetry, only half of the part has been modeled. The length of the straight part is 16mm , the thickness of the part is 4mm, and the corner radius is 5mm . This part is taken through a cure cycle, shown in Figure 8.24, and analyzed with both the D F and P V E . After the completion of the cure cycle the tool is removed and the spring-in angle is measured with a similar procedure to the one used by Albert and Fernlund (2002). The results from the D F analysis and comparison to P V E are presented in Table 8.8. It is observed that in both convex and concave mould shapes the final spring-in is a function of the C T E of the mould material. A specific trend is seen in the results, where the values of spring-ins for the two moulds converge as the mould C T E reduces. Furthermore, spring-ins for the convex mould increase with increasing mould C T E , while for the concave mould they decrease. This is consistent with the results reported by Albert and Fernlund. In that work, it is shown that the total spring-in in an L-shaped part can be broken into two parts: warpage of the straight arm and corner spring-in. The warpage of the straight section of the part is mostly due to tool-part interaction (see Twigg et al., 2004). The corner component o f the spring-in arises from the mismatch between the longitudinal and transverse C T E of the composite part. This was also discussed by Nelson and Cairns (1989) and mentioned in Chapter 2, where the spring-in for an L-shaped part can be calculated as follows:  A9 = 9  (CTE - CTE ) AT g  R  1 + CTE AT  (8.3)  R  If the part is modeled as a plane-strain problem, then the above equation would have to be modified as follows:  - 169-  Chapter 8: Numerical Applications  ^-v CTE eR  A9 = 9  Where E  R  and E  g  R  v )CTE AT  + CTE -(1 + e  l + (l +  eR  R  v )CTE AT eR  —  (8-4)  R  are the modulus of elasticity in the radial and hoop directions, and v  gR  is the  Poisson's ratio. From this equation, the corner part of the spring-in for this case is 0.60°. This value is consistent with the value of spring-in that the convex and concave tools converge to. In other words, by reducing the tool C T E to zero we eliminate the warpage and the value of the spring-in will only consist of the corner effect. These results are also consistent with the findings by Wiersma et al. and Z h u et al. The spring-in results of P V E in Table 8.8 closely agree with the D F prediction; the maximum difference between the results being 5%. Even though the difference between the two models changes between the runs, due to their small size, these differences do not seem to be attributable to a meaningful trend. The runtimes of the two models have also been compared in Table 8.9 and they show that the D F is only approximately 1.7 times slower than the P V E in these cases. This is consistent with the comparisons presented in previous chapters and confirms the efficiency of the D F . The final residual stresses, i.e. just before tool removal, for the P V E and D F can be compared through their contours, as shown in Figure 8.25 to Figure 8.28 for the longitudinal and transverse stresses for a convex steel mould. Examination of the figures shows that the stress distribution for the two methods is very similar. The maximum longitudinal stresses for D F and P V E are 194.3 and 189.5 MPa, respectively, which is a 2.5% difference between the two models. The maximum transverse stress predicted by D F and P V E are 89AMPa  and 87.7MPa, respectively (1.5% error for P V E ) and they both occur at the inner  side of the curved section. This shows that the two predictions made by the models are very close, although some differences are seen.  - 170-  Chapter 8: Numerical Applications side of the curved section. This shows that the two predictions made by the models are very close, although some differences are seen.  8.3.1.2.  Long L-shaped composite part  In this section, a different geometry will be analyzed to perform a more detailed comparison of the D F and P V E . The length o f the straight sections for this part is 100mm, the thickness is 4mm, and the corner radius is 5mm . The cure cycle is the same as the previous example, i.e. shown in Figure 8.24. The part is a unidirectional composite and the tool is aluminium. F E mesh of this part is shown in Figure 8.29, consisting of 8-noded solid elements under plane-strain condition. A test of convergence of the results with mesh refinement was not performed, however the number of elements used in this model seems adequate for the finite element code to give accurate results. Therefore, the results may not be most accurate, but due to the fact they are only being used for comparison of the D F and P V E and no other quantitative comparison is being made this will not be o f concern. Instead, we will avoid very long runtimes associated with models containing large number of elements. The deformation of the part after the tool removal is shown in Figure 8.30. The spring-in and runtimes for the D F and P V E have been shown in Table 8.10. The results are very close, with negligible differences. The relatively large spring-in values predicted for this run are noteworthy. This is due to the fact that complete bonding between the part and the tool has been assumed and this greatly increases the tool-part interaction, which results in high warpage values, especially for longer parts such as this (see Twigg et al., 2004). The corner components of the spring-ins have also been presented in the table and show reasonable values. The runtimes also compare very well, with the D F being 1.8 times slower than P V E . In order to gain a better understanding of the stresses predicted by the two approaches, the values have been compared in several ways. First, the history of stresses at a specific point, i.e. the point indicated by a dot in Figure 8.29, are compared in Figure 8.31, along with air and the local temperature at the point.  - 171 -  Chapter 8: Numerical Applications Note that these are stresses before tool removal. The two approaches follow the same trend, although the values show some differences and are 6% different at the end of cure. We can also look at the moments and axial forces in cross-sections of the part. For this purpose, first the evolution of moments and axial forces acting on a cross-section, passing through the reference point, throughout the cure are compared in Figure 8.32 and Figure 8.33. The results from D F and P V E are very close in this case and the differences are minimal, i.e. 3% for final residual moments and 2% for forces. To show the stresses in the cross section, the distribution of axial stresses at the end of cure (before tool removal) are plotted and compared in Figure 8.34. The figure confirms the findings that D F and P V E results agree well. We know that the deformations after the tool removal come from the residual stresses in the material, mainly in the form of moments in the cross section. For this reason, a comparison of the distribution of moments along the length of the part has been presented in Figure 8.35. In this figure, the origin of the length is the middle of the curved corner (plane of symmetry of the part). The results again show a very good comparison of the two models. The difference in the maximum moments shown in this figure is 4%. In order to investigate the results for other lay-ups, the same part has also been modeled with cross-ply and quasi-isotropic lay-ups. The spring-in results for these cases are shown in Table 8.11. The results show a good comparison of the D F and P V E . It is also seen that the spring-in values for the two lay-ups, while different from the unidirectional result, are not very different from each other.  8.3.1.3.  Reverse micromechanics and thermoelastic behaviour  A t this point, the reverse micromechanics ( R M M ) procedure is used to calculate the phase stresses for the unidirectional laminate on aluminium tool. The evolution of matrix and composite stresses at the point shown in Figure 8.29 along two directions are compared in Figure 8.36 and Figure 8.37. For the stresses in the fibre direction, i.e. Figure 8.36, the fibres take most of the load and therefore the stresses in the  - 172-  Chapter 8: Numerical Applications matrix are close to zero. This changes near the end of the cure cycle (in cool-down), where the matrix has hardened and takes some of the total stress. The value of matrix stress at the end of cure is 213MPa . The stresses perpendicular to the fibre direction and out of plane, shown in Figure 8.37 as S , 3i  however,  behave differently. It is observed that the stresses in the matrix approximately follow the same trends as the composite. The values of the stresses, unlike the previous case, are comparable to those in the composite and reach a maximum of 27MPa at the end of cure. This is consistent with the fact that values of fibre transverse modulus ( E ) and apparent modulus of the matrix are comparable. 2  In order to see the results of analysis with thermoelastic effects, this case is analyzed again with relevant material properties. The spring-in result for this case is 3.50°, i.e. 5% less than the model without thermoelastic effect. It is seen that the predictions from the two different analyses are not very different, even though the input moduli are not close. This can also be observed in Figure 8.38 where the distributions of moments along the length of the part for the two analyses are compared.  8.3.2. Non-Standard C u r e Cycles As discussed in the previous chapters and shown in Section 8.2.2, the P V E cannot be used for all cure cycles. In this part, some composite parts are taken through non-standard cure cycles in order to show a distinct difference between the results predicted by the D F and P V E . For this purpose, some C-shaped composite parts are chosen. Due to the symmetry of a C-shaped part, only half of it needs to be modeled. Therefore, the modeled geometry will be similar to an L-shaped part, with the only difference being in the boundary conditions; in this case the symmetry boundary conditions are applied. The geometry used for the examples here is the same as the L-shaped part in Figure 8.39.  8.3.2.1.  Post cure heat-up  In some applications, a heat-up of the part after cure may be needed. For this purpose, a cure cycle reflecting such a case will be modeled here. The cure cycle is shown in Figure 8.40. In this cycle, after  - 173-  Chapter 8: Numerical Applications applying a standard cycle and removing the tool, the part is heated up again. The results of the D F and P V E deformations are then compared in Table 8.12. It is seen that the results o f the two methods are in close agreement right after the tool removal. With the results at the end of heat-up, it may appear that the two methods give comparable results, since the difference in the spring-ins are only 2%. However, the proper way of evaluating the two constitutive models is to consider the deformations generated during the heat-up only. Then it is seen that as expected the results from the two methods are in fact very different and the difference in their predictions for this case grows to 14%. Again, the difference between the runtimes shows a favourable confirmation of the D F efficiency; the D F is only 1.9 times slower than PVE. In order to a get a better understanding of the differences between the two models, some comparison of stresses, similar to the L-shaped part in the previous section, w i l l be made. First the history of stress in the fibre direction for the point shown in Figure 8.29 is presented in Figure 8.41. The comparison shows that the D F and P V E are fairly similar and a maximum difference of 6% between the stresses is observed. After the tool removal, i.e. during the secondary heat-up, the D F stress at this point is relatively constant, but as the temperature increases it starts to drop. However, no such drop in the P V E stresses is observed. The distributions of moments along the length of the web just before tool removal have been shown in Figure 8.42. Similar to the L-shaped part, the two models predict fairly similar values and the distributions are very close, i.e. maximum error is 8%. This emphasizes the good correlation between the two models for a standard cycle. The moment distribution at the end of heat-up is shown in Figure 8.43. The curves for D F and P V E look rather similar in this case as well. However, the absolute values reveal more difference; the maximum difference in this case being 24%. A n interesting point observed in these graphs is that even though after the tool removal P V E is greater than the D F at all points (a less negative value), at the end of heat-up this monotonic difference changes; the moments predicted by D F and P V E switch places in being the higher value along the length. This is consistent with the fact that P V E does not  - 174-  Chapter 8: Numerical Applications relax the material and its reduction of stresses, and therefore leads to moments that are higher than the DF. Again, by performing the R M M analysis the stresses in the phases are obtained. The composite and matrix stresses for the point shown in Figure 8.29 along and perpendicular to the fibre direction are presented in Figure 8.44 and Figure 8.45, respectively. For the stresses along the fibre direction, the trend is similar to the L-shaped part, where the matrix stress remains close to zero relative to the composite stress until cool-down, and then starts to increase. It is also observed that the matrix stress in the heat-up decreases, even though the composite stress remains relatively constant. This is not surprising, as during the heat-up the composite can move freely, but the matrix and fibre expand differently and hence internal stresses are generated. Perpendicular to the fibres, the matrix stresses are comparable to those of the composite, similar to the L-shaped part. Again, after tool removal the matrix stress decreases with increasing temperature, while the composite stress does not change significantly. Note that these transverse stresses are lower in value (7.6MPa)  than the fibre direction stresses (\0.2MPa).  It is also  observed that near the end of heat-up both matrix stresses taper off, which is a result o f relaxation of the resin.  8.3.2.2.  Secondary cure of a partially cured composite  It has recently become common practice in the industry (e.g. George, 2004) to partially cure the material until it develops sufficient stiffness, after which it is removed from the tool, and the cure is completed off the tool. This is primarily done to reduce the cost, by enabling re-use of the tools and the autoclave. To study this type of cycle, an example of such a non-standard cure cycle, shown in Figure 8.46, is analyzed here. The results of spring-in for the C-shaped part under this cycle are presented in Table 8.13. It is seen that the predictions from the D F and P V E models are identical after the tool removal. However, when the results at the complete cure are considered, no correspondence between P V E and D F is observed. This  - 175-  Chapter 8: Numerical Applications shows the difference between the D F and P V E in a non-standard cure cycle more clearly. It also emphasizes that the P V E model fails to give reasonable results when the part is heated-up after the polymer has hardened enough to carry loads, i.e. it has gelled. The stress history plotted for a specific point is shown in Figure 8.47. It is seen that even though the P V E model predicts higher stresses on the first temperature hold it gives a higher stress drop in the first cooldown and the results from the two models reach a similar value, as also reflected by the spring-in results. In other words, close correspondence between the D F and P V E results at tool removal happens more or less accidentally. In the second part of the cycle, it is seen that P V E does not behave in anyway similarly to the D F and the results are very different. The main difference in the figure is that the D F stresses decrease, i.e. the D F relaxes the stresses, while in the P V E they stay constant, since P V E is not able to relax the stresses in its current form. Similar conclusions can be drawn for the distribution of moment along the length of the part web. The distributions for the two models before tool removal are presented in Figure 8.48. The D F and P V E moments are similar in trend, with an error of 9% for the P V E . However, after removing the tool and applying the second part of the cure cycle the D F will relax the values to give a zero value for spring-in, while P V E will not, as illustrated in Figure 8.49. Using R M M procedure, the composite and matrix stresses for the point shown in Figure 8.29 along and perpendicular to the fibre direction are computed and shown in Figure 8.50 and Figure 8.51, respectively. For the fibre direction, the matrix stresses remain close to zero until after the tool removal and beginning of the second part of the cure cycle. In the second part, one can see that the matrix stresses increase and then decrease, in fact following the temperature. A t the end, even though the stress in the composite is negligible (close to zero), the matrix stress is not small (\6.SMPa)  and should be taken into account. The  transverse direction stresses have a more complex behaviour. In this case, the matrix stress follows the trend of the composite stress fairly closely until the end of the first part of the cycle and tool removal. In  - 176-  Chapter 8: Numerical Applications the second part, the composite stress remains close to zero, as a free standing material under changes of temperature. The changes of the matrix stress can be explained as follows: during heat-up, the cure shrinkage and thermal expansion act in different directions and the result, as appears in the figure, is that they cancel each other out and the stress does not change considerably. O n the hold, however, only cure shrinkage acts on the matrix and generates stresses. This is partially countered by the effect o f thermal shrinkage in the cool-down, which results in a residual stress in the matrix. It is worth noting that the matrix stress in this direction (5AMPa) is much smaller than the one in the fibre direction (16.8MPa ). The interesting fact seen in the results for this cycle is the near zero residual spring-in o f the part at the end of the cycle. In other words, such a cure cycle is clearly able to reduce the spring-in value. This is consistent with the results of available experiments done by George (2004). To test the capability of similar cure cycles to reduce the spring-in to the same degree, two other temperature cycles were analyzed. These cycles, shown in Figure 8.52, all consist of two parts, i.e. a first part to bring the part to gelation, and a second part to complete the cure off the tool. The difference between these cycles is the hold temperature in the first part of the cycle, where temperatures of 1 0 0 ° C , 120°C , 150°C have been used. The result of predicted spring-ins for these cycles is shown in Table 8.14. It is seen that the springin at tool removal differs significantly depending on the hold temperature. However, all three cycles predict the same spring-in at the end of cure and the first hold temperature does not have any effect on the final deformation. This is perhaps due to the fact that during the heat-up in the second part of the cycle the part stresses are relaxed and the final deformation is generated only during the hold and cool-down in the second part of the cycle.  8.4.  SUMMARY  In this chapter, some numerical examples were presented where different constitutive models developed in this thesis, i.e. the D F of V E and P V E models, were used to analyze the material as they go through different cure cycles. First, some simple examples, consisting of only a single element, were shown to  - 177-  Chapter 8: Numerical Applications facilitate the study of available constitutive models. Then, some more realistic cases were presented to confirm these findings. Due to unavailability of experimental results in these cases, where both a V E material model and residual stress measurements are available, a qualitative validation of the results was carried out. Then, the results from available constitutive models were compared to give a better understanding of their characteristics. Those aspects of these constitutive models developed in this thesis, and not available in other works in the literature, were also showcased and the results were studied for qualitative validation of the procedures. Overall, the following conclusions can be drawn from the examples illustrated in this chapter: •  The material properties used for the examples were chosen such that they represent more of the real material behaviour; all material properties were assumed to be time-dependent. This included the moduli and the Poisson's ratios as well as the cure shrinkage coefficient of the composite. In addition, the resin material was assumed to have two different regimes of thermal expansion behaviour, each of which caused separate time-dependent thermal expansion behaviour for the composite material. The material properties can be found and handled easily, by taking advantage of the micromechanical and the differential form modeling.  •  Small L-shaped parts were used to show that the predictions by both the DF and PVE models are consistent with experiments and real physical behaviour. The spring-in values could be broken into two components, i.e. corner spring-in and warpage. The corner component is the value that the spring-in of the part on convex and concave tools converges to, as the warpage value is reduced for tools of lower C T E value. This is consistent with experiments available in the literature (e.g. Albert and Fernlund, 2002).  •  The P V E model predicts results close to the DF baseline in the standard cure cycles. This confirms the validity of the PVE for use in a majority of cure cycles. However, the PVE does not  - 178-  Chapter 8: Numerical Applications work for the non-standard cure cycles presented in this chapter. In cases like these, where the material underwent heating after the polymer had gelled, the PVE cannot be used. •  The DF model is very efficient, as compared to the PVE. This compares favourably with the much longer times that the common integral form models consume, as discussed in Chapter 3. In addition, in cases where the PVE does not work or a more, albeit slightly, accurate model is needed, the DF is the only viable choice.  •  Reverse micromechanics model used for these examples is a necessary tool in the design of composite materials, since it gives insight into the stresses and strains of the phases, not previously available in the literature. The results of such analyses were consistent with common sense and show that in the cure cycle large stresses may be generated in the polymer.  •  Using the DF, a set of thermoelastic material properties was assumed and the material was analyzed with this effect. This can be a useful tool, if such effects in the material are deemed significant. In the cases studied here, the effect of thermoelastic properties was variable, i.e. where in one case they resulted in significant changes to the results, in another case this effect was diminished. However, it must be noted that without such an analysis it may not be possible to know a priori how much effect thermoelasticity would have.  - 179-  Chapter 8: Numerical Applications  Table 8.1 Mechanical properties of fibre and resin used for case studies Property v =v 12  AS4  fibre  0.2  13  V 3  0.3  E,(Gpa)  207  E (Gpa)  20.7  G =G (Gpa)  27.6  CTE! (u/°C)  -0.9  C T E (u/°C)  7.2  2  2  12  Property E (Gpa) E (Gpa)  13  2  3501-6 resin  u  3.2  r  0.031  V  0.35  Vf  0.58  Table 8.2 Thermal and cure shrinkage coefficients obtained for 3501-6 resin for the case studies Parameter  Value  „ « , , • / Coeri. thermal expansion (ue CI  Glassy(T<T )  57.6  Rubbery(T>T )  185.4  g  g  Coeff. cure shrinkage (us)  18639  Table 8.3 Unrelaxed and relaxed values of different material properties obtained for AS4/3501-6 Property ri(MPa) i(MPa) K (MPa) G (MPa) Gi (MPa) C T E ( n / C ) C T E ( n / C ) C T E ( n / C ) C T E ( n / C ] 2  Unrelaxed 123300 3483.0 Relaxed 120100 49.3  6817.8 97.6  Z3  2  2744.5 36.8  1n  3861.6 43.2  1r  -0.26 -0.89  2B  1.15 -0.88  2r  37.0 36.8  105.7 105.1  CS,(M/C) CS (H/C) 2  202.4 2.0  10568 10516  Table 8.4 Relaxation times and corresponding weight factors for different material properties of AS4/3501-6 Maxwell element 1 2 3 4 5 6 7 8 9  Relaxation time 2.92E+01 2.92E+03 1.82E+05 1.10E+07 2.83E+08 7.94E+09 1.95E+11 3.32E+12 4.92E+14  t 0.041 0.048 0.063 0.091 0.133 0.271 0.242 0.074 0.037  K 0.041 0.047 0.062 0.090 0.132 0.272 0.244 0.075 0.037 2  - 180-  G"23  0.043 0.050 0.066 0.094 0.137 0.272 0.234 0.070 0.035  G 0.052 0.059 0.075 0.105 0.148 0.270 0.206 0.057 0.029 1 2  CTE  l g  0.052 0.059 0.075 0.105 0.148 0.270 0.206 0.057 0.029  CTE  lr  0.059 0.066 0.083 0.112 0.154 0.264 0.187 0.050 0.025  CSj  0.059 0.066 0.083 0.112 0.154 0,264 0.187 0.050 0.025  Chapter 8: Numerical Applications Table 8.5 Thermo-chemical properties of AS4/3501-6  x 10  AT (min" )  2.102  A (min" )  -2.014 x 1  A (min" )  x 10 8.07 x 10" 7.78 x 10 5.66 x 10"  1  1  2  1.960  1  3  AEi (J/mol)  a  2  AE (J/mol) 3  1578  C (J/Kg.K)  862  k (W/m.K)  0.4135  k (W/m.K)  12.83  H (J/Kg)  198.6  R (J/mol.K)  8.13  a  10  p  5  T  L  4  AE (J/mol)  P (Kg/m )  r  x 10  3  Table 8.6 Properties of the tooling materials used in the case studies  Tool Material E (GPa)  C T E ( | J / C ) p(kg/m ) C ( J / K g . K ) k(W/m.K)  v  0  3  p  210  0.300  12.0  7833  434  60.5  Aluminum  69  0.327  25.0  2707  896  167.0  Invar  150  0.280  0.0  8000  515  110.0  Steel  Table 8.7 Summary of single-element cases to be analyzed Case Boundary Conditions  Lay-up  1  Fully constrained  [0]  2  Axially constrained  [0]  3  Axially constrained  4  Fully constrained  [0] [30]  5  Axially constrained  [0/90]  6  Axially constrained  7  Fully constrained  8  Fully constrained  Note  With thermoelastic effects Two different A B A Q U S elements used for comparison  s  [0/45/90/-45] Two different A B A Q U S elements used for comparison s  [0] [01  Heat-up after complete cure Secondary cure of partially-cured material  - 181 -  Chapter 8: Numerical Applications Table 8.8 Results of spring-in angle in degrees for an L-shaped part on different tooling, analyzed with the Differential Form of Viscoelasticity and Pseudo-Viscoelasticity CTE Convex mold  (u£/°C)  DF  PVE  0 12 25 CTE (u£/°C)  0.65 0.74 0.89 DF  0.62 0.71 0.90  0 12 25  0.61 0.41 0.16  0.61 0.41 0.18  moId  mold  Concave mold  PVE  Table 8.9 Comparison of runtimes in a standard cure cycle for an L-shaped composite on different tooling CTE  mold  (u£/°C)  DF (Sec)  0 12 25 0 12 25  Convex mold  Concave mold  PVE (Sec) 496 526 500 513 464 530  907 839 899 805 820 884  Ratio (DF/PVE) 1.83 1.60 1.80 1.57 1.77 1.67  Table 8.10 Spring-in and runtimes from the analysis on a unidirectional part shown in Figure 8.29 Description  DF  PVE  Total spring-in (°)  3.69  3.66  Corner spring-in (°)  0.58  0.58  10551  5787  Runtime (Sec)  Table 8.11 Spring-in from the analysis on the part shown in Figure 8.29 with different lay-ups  [[0/90] ] 4  Spring-in(PVE)  P V E error  4.29  4.24  -1.2%  4.17  4.09  -1.8%  Spring-in(DF)  Lay-up s  [[0/45/90/-45] ] 2  s  - 182-  Chapter 8: Numerical Applications Table 8.12 Spring-in results for a C-shaped composite part under a cure cycle with a post cure heat-up Spring-in (°)  DF  PVE  P V E Error  After tool removal  3.75  3.74  -0.3%  End of heat-up  3.32  3.25  -2.1%  Only heat-up  0.43  0.49  14%  Table 8.13 Spring-in results for a C-shaped composite part under a cure cycle with secondary cure of a partially cured material Spring-in (°)  DF  PVE  After tool removal  0.70  0.70  End of cycle  -0.08  0.42  Table 8.14 Results of spring-in for cycles in the form of secondary cure of partially cured parts Spring-in(End of cycle)  First Hold Temperature  Spring-in(Tool removal)  100  0.70  -0.08  120  0.76  -0.08  150  1.03  -0.07  - 183 -  Chapter 8: Numerical Applications  — TEMP. -"-Vol. Change  200 T  _ O  6  150 O)  2  2 100  S O O  a> Q .  E  0  I o  >  50  — i — i — i — i — i — i —  100  200  300  1  400  r-  500  Time (Min) Figure 8.1 Evolution of the volume change for 3501-6 resin in a cure cycle, used to find thermal and cure shrinkage properties  - 184-  Chapter 8: Numerical Applications  50  -1  40 -  (Mpa)  30 20 -  -s11(DF) -s11(PVE)  in  to 10 0)  CO  250  -10 -20 Time (Min)  Figure 8.3 Comparison of er as predicted by the Differential Form U M A T and P V E for Case 1 u  80 70 60 50 |  8  40 1 — S22(DF) — s22(PVE)  3 0  20 10 0 -10  + 1 —  4  50  100  150  200  250  -20 Time (Min) Figure 8.4 Comparison of cr as predicted by the Differential Form U M A T and P V E for Case 1 22  - 185 -  Chapter 8: Numerical Applications  20 -i  Time (Min)  Figure 8.5 Comparison of axial stresses as predicted by the Differential Form U M A T and P V E for Case 2  — s11 (Thermoelastic) — s11 (No Thermoelastic)  20  -i  /  \  // A  -  / /  /  0  i  i  50  100  150  200  250  Time (Min)  Figure 8.6 Comparison of axial stresses as predicted by the Differential Form, with and without thermoelastic behaviour for Case 3  - 186-  Chapter 8: Numerical Applications  — Nx(DF) —- Nx(PVE)  250  Time (Min)  Figure 8.7 Comparison of axial forces as predicted by the Differential Form U M A T and P V E for Case 4  — Nx(DF) — Nx(PVE)  250  Time (Min)  Figure 8.8 Comparison of axial forces as predicted by the Differential Form U M A T and P V E for Case 5  - 187-  Chapter 8: Numerical Applications  — Nx(DF, layered solid elements)  100  Nx(DF, single element per layer)  150  200  250  Time (Min) Figure 8.9 Comparison of axial forces as predicted by the Differential Form U M A T with different element types for Case 5  - 188-  Chapter 8: Numerical Applications  32  \L  Figure 8.10 Schematic of the boundary conditions in Case 6 to avoid rigid body movements  25 20 „ 15 n  Q.  8  — Nx(DF) Nx(PVE)  1 0  0)  *•* CO  100  150  200  250  Time (Min) Figure 8.11 Comparison of axial forces as predicted by the Differential Form U M A T and P V E for Case 6  - 189-  Chapter 8: Numerical Applications  Figure 8.12 Undeformed (solid line) and deformed (dashed line) shapes, using a layered element in Case 6  Figure 8.13 Undeformed (unshaded) and deformed (shaded) shapes, using single element per layer in Case 6  - 190-  Chapter 8: Numerical Applications  Figure 8.14 Cure cycle used in post cure heat-up of the material in Case 7  Figure 8.15 Temperature cycle and the degree of cure used to model a post-cure heat-up in Case 7  - 191 -  Chapter 8: Numerical Applications  - 192-  Chapter 8: Numerical Applications  - 193 -  Chapter 8: Numerical Applications  — s11(DF) — s11(FVE)  Time (Min)  Figure 8.20 Comparison of cr,, as predicted by the Differential Form UMAT and PVE for Case 8  —-s22(DF) — s22(PVE)  Time (Min)  Figure 8.21 Comparison of a  n  as predicted by the Differential Form UMAT and PVE for Case 8  - 194 -  Chapter 8: Numerical Applications  Chapter 8: Numerical Applications  0  50  100  150  200  250  300  350  time (min)  Figure 8.24 C u r e cycle used for case studies on A S 4 / 3 5 0 1 - 6 composite material  Figure 8.25 Distribution of longitudinal stresses for an L-shaped unidirectional composite on a convex steel mould from Differential F o r m of Viscoelasticity  -  196-  Chapter 8: Numerical Applications  SS,  311  CAve. G r i t . : 75*) +8.766«+01 46.475*401 41.889*401 -4.045*400 -Z.698*401 -4.991*401 -7.284e+01 -9.S7?e401 -1.187e+02 -1.416*402 -1.646e402 -1.875*402  Figure 8.26 Distribution of longitudinal stresses for an L-shaped unidirectional composite on a convex steel mould from Pseudo-Viscoelasticity  - 197-  Chapter 8: Numerical Applications  Figure 8.28 Distribution of transverse stresses for an L-shaped unidirectional composite on a convex steel mould from Pseudo-Viscoelasticity  M  Figure 8.30 Deformed part after tool removal in a cure cycle, along with the undeformed shape  •S11 (DF)  50  100  S11 (PVE)  150  Air Temp.  200  250  Node Temp.  300  350  400  Time (Min)  Figure 8.31 Evolution of temperature and residual stresses at the point shown in Figure 8.29  - 199-  Chapter 8: Numerical Applications  - Moment (DF) — Moment (PVE) — Air Temperature  E E c 0>  E o  50  100  150  200  250  300  350  400  Time (Min) Figure 8.32 Evolution of moments in an L-shaped composite part for a cross-section passing through the point shown in Figure 8.29  — Force (DF) —Force (PVE) —Air Temperature  200 180 160 140  00  150  200  250, \ 300  350  4io 20 1  £  100 «  8 -500  o  80  | 0)  60 *" 40 20 0 Time (Min) Figure 8.33 Evolution of axial forces in an L-shaped composite part for a cross-section passing through the point-shown in Figure 8.29  -200-  Chapter 8: Numerical Applications  — DF  — PVE  S11 Stress (MPa) Figure 8.34 Distribution of final axial residual stresses through the thickness for a cross-section passing through the point shown in Figure 8.29  -M (DF) — M (PVE)  3000 2000 1000 ? E z  0 -1000  | -2000 S -3000 ^1000 -5000 4 -6000  Distance from the part plane of symmetry (mm)  Figure 8.35 Distribution of moments along the length of the part shown in Figure 8.29  -201 -  Chapter 8: Numerical Applications  — S11 (Composite) —S11 (Matrix)  500 400 ~ 300 D.  s  in (A d)  200  1_  10  100 0 50  100  150  200  250  300  350 400  -100 Time (Min) Figure 8.36 Evolution of composite and matrix stresses in the fibre direction at the point shown in Figure 8.29  - S33 (Composite) — S33 (Matrix)  50 40 30 ^ re  Q. 20  sCO  in  £ 10 to 0  - t e e - ^ 150  200  0/  300  350 400  -10 -20  Time (Min)  Figure 8.37 Evolution of composite and matrix stresses perpendicular to the fibre direction (inplane) at the point shown in Figure 8.29  -202-  Chapter 8: Numerical Applications  — M (Thermoelastic) — M (No Thermoelastic)  3000 2000  ^  1000 0 20  i 40  60//  \ 80  100  I 120  § -2000 -3000 -4000  J  -5000 -6000 Distance from the part plane of symmetry (mm)  Figure 8.38 Distribution of moments along the length of the part shown in Figure 8.29for two analyses with and without thermoelastic effects  -203 -  Chapter 8: Numerical Applications  Symmetry  Figure 8.39 F E mesh of a C-shaped AS4/3501-6 part on an aluminium tool used in the case studies  Temperature — DoC  +— 0  ,  ,  ,  100  200  300  •  V  400  500  0  Time (Min) Figure 8.40 Cure cycle applied to a C-shaped composite shown in Figure 8.29  -204-  Chapter 8: Numerical Applications  -S11 (DF)  100  S11 (PVE)  Air Temp.  Node Temp.  300  400  200  500  Time (Min) Figure 8.41 Evolution of temperature and residual stresses at the point for a C-shaped part shown in Figure 8.29  — M (DF) — M(PVE)  -6000  J  Distance from part comer along web (mm) Figure 8.42 Distribution of moments along the web of the C-shaped part before tool removal, as in Figure 8.40  -205 -  Chapter 8: Numerical Applications  •M (DF) - - M (PVE)  400 i 300 ? 200 E -  100  cu  E o S  0  20  40  60  80  100  120  -100 -200 Distance from part corner along web (mm)  Figure 8.43 Distribution of moments along the web of the C-shaped part shown in Figure 8.29 at the end of post cure heat-up  — S11 (Composite) — S11 (Matrix)  500  500  Figure 8.44 Evolution of composite and matrix stresses in the fibre direction for a C-shaped composite at the point shown in Figure 8.29 under the cure cycle in Figure 8.40  -206-  Chapter 8: Numerical Applications  — S33 (Composite) — S33 (Matrix)  500  Time (Min) Figure 8.45 Evolution of composite and matrix stresses perpendicular to the fibre direction for a C shaped composite at the point shown in Figure 8.29 under the cure cycle in Figure 8.40  Figure 8.46 Cure cycle applied to a C-shaped composite shown in Figure 8.29  -207-  Chapter 8: Numerical Applications •S11 (DF)  S11 (PVE)  Air Temperature  Node Temperature  220  250 200  170  o  ~ 150 ns  120 £  3 +-> RJ  (o 100  L_  in o  CO  70  50  d>  |  20  0 100  200  300  400  500  600  700  800 -30  -50 Time (Min)  Figure 8.47 Evolution of temperature and residual stresses at the point for a C-shaped part shown in Figure 8.29  — M (DF) — M (PVE)  100  -1200  120  J  Distance from part corner along web (mm) Figure 8.48 Distribution of moments along the web of the C-shaped part before tool removal, cure cycle shown in Figure 8.46  -208-  Chapter 8: Numerical Applications  — M (DF)  M (PVE)  80 60 ^ 40 E  I  20  |  0  o £  20  40  60  80  100  120  -20 -40 -60  Distance from part corner along web (mm)  Figure 8.49 Distribution of moments along the web of the C-shaped part at the end of cycle, cure cycle shown in Figure 8.46  [ — S11 (Comp) —S11 (Matrix) —Air Temperature  250 i  Time (Min)  Figure 8.50 Evolution of composite and matrix stresses in the fibre direction for a C-shaped composite at the point shown in Figure 8.39  -209-  Chapter 8: Numerical Applications  — S33(Comp) —S33 (Matrix) —AirTemperature  re 2 </> (A  £  V)  8  200  6  150  4  100  2  50  0 -2  100  200  300  800  400  ,0 50  -4 -6  100  -8  150  -10  200  o  o i_ 3  re a> a.  E  Time (Min)  Figure 8.51 Evolution of composite and matrix stresses perpendicular to the fibre direction for a Cshaped composite at the point shown in Figure 8.39  — Cycle 1 — Cycle 2 — Cycle 3  100  200  300  400  500  600  700  800  Time (Min)  Figure 8.52 Different cure cycles in the form of secondary cure of partially cured parts  -210-  Chapter 9: Conclusions and Future Work  Chapter 9. 9.1.  CONCLUSIONS AND FUTURE W O R K  SUMMARY  Different constitutive models for calculation of residual stresses in composite materials during cure were developed in this work. For this purpose, the models were first derived in simple I D form and their validity was demonstrated. It was shown that the Pseudo-Viscoelastic model ( P V E ) and the Differential Form (DF) of viscoelasticity have different characteristics and may be used according to the objectives of the modeller. In order to have a working model for practical cases, a micromechanical approach was presented to combine the properties of the viscoelastic (VE) polymer and the fibres into a V E model for the composite. Then, this was utilized to develop the Finite Element (FE) solution for the D F formulation. Some specific capabilities were deemed important in material modeling and were built into this approach. These are: modeling all the material properties as functions of time; ability to simulate the thermoelastic behaviour in the material; and the option to back calculate the phase stresses from the composite stresses, i.e. reverse micromechanics. This model was implemented in the commercial general-purpose F E code, A B A Q U S , and the validity of this implementation was verified. Some case studies were presented to show the capability of the models developed in this work and highlight the similarities and differences between them in order to help with the choice of a model to achieve either accuracy or efficiency of the solution.  9.2.  CONCLUSIONS  The conclusions based on the work presented in this thesis are as follows:  -211 -  Chapter 9: Conclusions and Future Work •  The P V E is a valid simplification of the available V E models in the literature for the cure of thermoset composite material for a wide range of cure cycles and can be used as an efficient alternative to V E models in many cases.  •  The micromechanical approach presented here can be used for obtaining the V E mechanical properties of the composite from those of the phases, especially by using available mathematical computer packages.  •  The D F is an efficient modeling form of viscoelasticity that is easier to develop than the more common integral form models currently used in the literature. This approach provides a better framework  for  extending the  constitutive  model to  include aspects  such  as  reverse  micromechanics. The D F is also easy to implement, due to the fact that the developed formulation operates at the integration point level of an element. •  A l l material properties were assumed to be time dependent. It was shown that modeling all the material properties as time-dependent, including coefficients o f thermal expansion and cure shrinkage, is failry straightforward with the D F .  •  The reverse micromechanics model developed in this thesis is a helpful tool for calculation of stresses in the constituents. Calculation of these stresses can be essential in monitoring the state of the polymer when designing composite materials.  •  Thermoelastic behaviour of the polymer can be an important factor in the response of the composite material and its cure induced residual stresses. One form of this behaviour was implemented and verified here.  9.3.  FUTURE WORK  The following are areas in which the work in this thesis can be advanced:  -212-  Chapter 9: Conclusions and Future Work •  To be practical, the D F needs a proper characterization of the material. There are only a few materials characterized for their V E behaviour and characterization o f more materials is needed. In addition, in these experiments the possibility of the nonlinear behaviour of the material should be explored.  •  Experiments to measure the cure induced residual stresses in the composite materials are also currently lacking in the literature. The measurements in complex cases provide useful validation of the finite element models used in simulating the overall behaviour of the composite structure; however, simpler forms of the experiments may be designed to only verify the mechanical behaviour model.  •  The D F was developed based on the assumption that the material is solid. A general model that spans the whole cure cycle and takes into account the transition of the polymer from liquid to gel to solid needs to be developed to properly predict the final shape and stresses of the composite material.  •  The P V E model was developed with the specific assumptions that only cool-down after gelation was considered. The possibility of generalizing the current P V E model to other cases, such as heat-up after cure, needs to be explored.  •  Experimental investigation o f the thermoelastic behaviour o f available polymers would be useful. In the experiments, possible dependence of the unrelaxed and relaxed moduli on the degree of cure should be checked. Based on these, the model development and implementation of the formulation presented in this thesis may need to be modified. A l s o , the procedure for finding the composite properties through micromechanics model was described in this thesis, however, the results should be validated with corresponding experimental measurements.  -213 -  Chapter 9: Conclusions and Future Work  9.4.  CONTRIBUTIONS  The following contributions are made in this thesis: •  Different forms of the P V E model, as efficient alternatives to V E models, were developed. This approach shows how a C H I L E formulation can be properly calibrated to be a true P V E model. These provide the user with modeling options with a characterization cost versus accuracy tradeoff.  •  A micromechanical approach was presented that can be used to obtain the V E mechanical properties of the composite from those of the phases. Although based on the concepts available in the literature, this provides the user with a consistent and modular approach where few assumptions on the specific form of elastic micromechanics model are made.  •  A D F of viscoelasticity was developed that is easier to develop and more efficient than the integral models in the literature that are most commonly used. This provides the user with an approach that is easier to understand and generalize to more complex cases. The algorithm for this implementation was designed in such a way that it can be used with virtually any finite element code.  •  In this thesis, material properties are modeled in a more general way, (e.g. all properties including coefficients of thermal expansion and cure shrinkage are assumed to be time-dependent), than previous works in the literature. A s a result, no simplifications are needed in either the model development or numerical runs.  •  A reverse micromechanics approach, calculation of stresses in the constituents from those in the composite, for viscoelastic materials was developed and implemented in the F E framework. The formulations that exist in the literature only apply to elastic materials.  -214-  Chapter 9: Conclusions and Future Work •  A model that accounts for thermoelastic effects was developed and implemented within an F E framework. Although previous researchers have developed equations that incorporate these effects, they generally do not go as far as F E implementation and application of such formulations.  -215 -  References  REFERENCES A B A Q U S Theory Manual, Version 6.4, A B A Q U S Inc., 2003. ABAQUS/Standard User's Manual, Version 6.4, A B A Q U S Inc. 2003. 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Y i , S . , Hilton,H.H. and Ahmad,M.F., "Curing Process Induced Viscoelastic Residual Stresses in Polymer Matrix Laminated Composites", A S M E International Mechanical Engineering Congress, Proceedings of the A S M E Materials Division, pp. 65-76, 1995. Y i , S . , Chian,K.S., and Hilton,H.H.,  "Nonlinear Viscoelastic Finite Element Analyses of Thermosetting  Polymeric Composites During Cool-Down After Curing", Journal of Composite Materials, 36(1), pp. 3-17,2002. Y i , Y . M . , Park,S.H., and Youn,S.K.,  "Asymptotic Homogenization o f Viscoelastic Composites With  Periodic Microstructures", International  Journal of Solids and Structures,  35(17),  pp. 2039-  2055,1998. Zhu,Q., Geubelle,P.H., L i , M . , et al,  "Dimensional Accuracy of Thermoset Composites: Simulation of  Process-Induced Residual Stresses", Journal of Composite Materials, 35(24), pp. 2171-2205,2001. 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Zocher,M.A., A l l e n , D . H . , and Groves,S.E.,  "Stress Analysis of a Matrix-Cracked Viscoelastic  Laminate", International Journal of Solids and Structures, 34(25), pp. 3235-3257,2004.  -231 -  Appendix A: Mathematica Notebooks  Appendix A . A.l  MATHEMATICA NOTEBOOKS  Notebook for the example in Section 5.4.1.  Defining the properties in time domain • Matrix properties The Matrix properties are defined with two properties, each in Prony series. Shift factor is defined az an input.  Off[General::spell] Off[General::unfl] G = 1185; G" = 18; M = 9; a = 10 -2; <x„ = 5.8 10 -5; To = {2.92 10 l , 2.92 10 3, 1.82 10 5, 1.10 10 7, 2.83 10 8, 7.94 10 9 , 1.95 10 11, 3.32 10 12, 4.92 10 14} ; w = {0.059, 0.066, 0.083, 0.112, 0.154, 0.262 , 0.184, 0.049, 0.025} ; u  A  m  m  A  T  A  A  A  A  A  A  A  A  A  G  H  G [t_] :=G + (G -G ) 2 r  m  u  m  r  m  '  &" H  1 ,  i  m  .  E 5 i p [  -  ••  .  t / I | 3 t [ i ] ] / a T : 1  1=1  K \ = 3556;K = 800;P = 9; i = {2.92 10 l , 2.92 10 3, 1.82 10 5, 1.10 10 7, 2.83 10 8 , 7.94 10 9 , 1.95 10 11, 3. 32 10 12, 4.92 10 14} ; w = {0.059, 0.066, 0.083, 0.112, 0.154, 0.262, 0.184, 0.049, 0.025} ; v = 0.6; I  l  m  A  A  A  A  A  A  A  K  A  A  K  KnltJ  f  :=K  (K^-K^)  r m +  2  Wk[[±11  Exp[-t/r [[i])/a ] K  T  i =l • Fiber properties  Eif = 210000; v = 0.2; G^f = 27600; v a = -7.2 10 -6; 1 2f  = 0.25; E = 17200; a = -0.9 10 -6; A  23f  2 f  A  2f  <hic = E  2 £  / (2 (l + v  23f  K = 1/ ( 4 / E - l / G 2f  2 f  )) ; 2 3 f  -4v  A 1 2 f  2/E ); l f  Transformation to Laplace space • Transforming moduli and calculating other parameters :  •' i • The other material parameters are calculated in Laplace space. ;  : : :  G [s_] : = s LaplaceTransform[G [ t] , t, s] m  m  K [s_] : = sLaplaceTransformCK^[t] , t, s] m  E„[s_] v [s_] := {l¥^\s\ m  G [s]/ m  (3^(8]  + G [s]) m  -2G [s]) / (ei^Is] +2G [s]) m  m  -232-  i f  Appendix A: Mathematica Notebooks  Applying micromechanics  t  • Calculation of parameters using micromechanics The CCA model is used in the Laplace space. Any other model can be used as well.  Ei[s_] := EifV +E [s] ( l - v ) + (4 (v [sj - v ) K K J s ] G [s] ( l - v ) v ) / ((K + § [s]) K,„[s] + ( K - l U s n v) m  f  f  m  m  2f  Vi2[s_] : = v i  1 2 f  m  2 f  2 f  f  £  f  v +v [s] ( l - v ) +  2f  £  m  f  (<v [s] - v ) (K»[s] - K ) G [s] <l-v ) v )/((K G [s])K»[s] + (K -iiW[s]) G [s] v ) m  u f  G [s_] := G [s] ( ( G 12  m  2£  m  12f  £  f  m  2£+  + G [s]) + (G -G [s]) v ) / ( (G m  12f  m  m  2f  f  12 £  + G [s] ) - ( G m  1 2 f  f  - G [s]) v ) m  f  '  := ( ( K G [s]) K»[s] * ( K - K „ [ s ] ) »f) / ((«2 f + G [sJ) - ( K - K ^ s ] ) v ) 6 [s_] := (G [s] [K^IB] feft§,[!])*2C ^[!]*K„[s] (G - G [s ]) v )) / ( i U s ] (G j + G [s]) + 2 G G [ s ] - (K^s] + 2G [sJ) (G - G„ [s]) v )  '  E [s_] := 1/ (1/ (4K [i|) + 1/ (4 G [s]) + (v [s] 2 / E [ s ] ) )  '  m  2 £ +  23  m  2f  m  2  !Jf  £  m  23£  2 f  23 f  m  m  m  f  f  23 £  £  A  2  23  2  v [s_] := (2Ei[s]K [s] - E [s] E [s] - 4 2  23  x  2  12  ±  [s] » 2 K [s] E [s]) / (2 Ei[s] K [s]) 2  2  2  «i[s_] := (aifEirVf + o^E^rs] (1 - v )) / (E v + E [s] ( l - v ) ) £  1£  f  m  « [s_] := (a + <*i v ) v *a„, (l+v [s]) (1 - v ) 2  2£  £  u£  £  m  £  *  £  (vi2  £  v + v [s] ( l - v ) ) S i [ s ] £  m  £  Inverse Laplace transform • Applying the inverse Laplace transform A number of available models can be used to transform the parameters to time domain. The Stehfest method is used here to give the "exact" values of moduli. The "Numerical Inversion" package was obtained from Wolfram's website.  Remove[Stehfest] Remove[f] << HumericalHath Humericallnversion' Ei[t_] : = Stehfest[Ei[s]/s, s, t] V]2[t_] : = S t e h f e s t [ s ] / s , s, t] G12 [t_] : = Stehfest [612 [s] }s, s, t] K [t_] : = Stehfest[K [s] fs, s, t] 2  2  G [t_] : = Stehfest[S [s]/s, s, t] 23  23  E [t_] : = Stehfest[E [s] / s , s, t] 2  2  v [t_] : = Stehfest [v [s] / s, s, t] 23  23  ai[t_] : = Stehfest[&i[s] Is, s, t] a [t ] : = Stehfest[o [s] Is, s, t] 2  2  -233 -  '  Appendix A: Mathematica Notebooks • Evaluation of the constant moduli zero = 10 -12; inf = 10 12; A  A  E  X t  = Ei[inf] ;  E  l u  =E![zero] ;  E m =Gi2 [inf ] ; Gi2u = &12 [zero] ; K = K |inf] ; 2 l  K  2 u  2  = K [zero] ; 2  <523t =G2 [inf ].;. 3  G?3u =<»23 [zero] ; E =E [inf] ; E = E [zero] ; 2 r  2  2 u  2  vi2r = [inf]; : : V]2„ = V J J [zero] v = v [inf ] ; v 3u = v [zero] ; «ir = <»i[inf]; «iu = <«i[zero]; a = a [inf]: 2 3 r  23  2  23  2 r  2  a „ = a [zero] ; : 2  2  Curve fitting • Data generation Assuming theresinrelaxationtimes as give, the weight factors are calculated using a linearregressionanalysis. In this part, a number of points for each material parameter profile are generated to be used in curve fitting. Tin»[i_] := 10 <-11 + 0.5i) A  »[m_, n_] : = If [Time[m]7i [[n]] < -Log[SliinHui*er], Ej5>[-Tiiiie[m] / z [[n]] / a ] , 0] K  K  X= Table[A[m, n], {m, 0, 45}, {n, 9}]; dataEi = TaJ)le[(E [Time[i] ] - E x  datav^ =Table[(v  ) / (E  l t  [Time[i] ] - v ^ , ) / ( v  i2  -E  l u  1 2 u  l r  ) , { i , 0, 45}] ;  - V i ) , { i , 0, 45}] ; 2r  dataGu = Table[(G [Time[i] ] - G ^ ) / (G „ - G 12  tlataKj =Table[(K [Time[i] ] - K 2  12  ) / (K  2 l  dataG^ = Table[(G [Time[i] ] - G 23  dataE =Table[(E [Time[i] ] - E 2  2  2 l  23  dataoi! = T a b l e [ ( « ! [Time[i] ] - a ) lz  2 3 t  2  2£  2r  23 u  - G  2 3r  ) , { i , 0, 45}] ;  - E ) , {i , 0, 45} ] ;  2u  2r  ) / (v „ - v 2 3  2 3  , {i, 0,45}];:  I ( a i - a ) , {i , 0 , 45} ] ;  dataa = Table [(a [Time [i] ] - a ) / (a 2  ) , { i , 0, 45}] ;  - K ) , { i , 0, 45} ] ;  2u  ) / (G  ) / (E  datav = Table[(v [Time[i] ] - v 23  2 3 t  1 2r  u  2 u  ± r  - a ) , {i , 0, 45} ] ; 2 r  -234-  T  Appendix A: Mathematica Notebooks • Curve fitting : The curve fitting is done: using a linearregressionanalysis for each material parameter. '  :  f i t E i = Inverse [Transpose [X]. X] . Transpose [X] :dataEi;; fitvi2 = Inverse [Transpose [X] .X]. Transpose [X] ^ datavi^; fitGu .'=Inverse [Transpose [X]. X]. Transpose [X]. dataG^ r f itK = Inverse [Transpose [X].X]:. Transpose [X].dataK ; .: 2  2  fitG^j = Inverse [Transpose [X] .X]. Transpose [X]. dataG23; f i t E - Inverse [Transpose [X]. X] . Transpose [XI. dataE ; 2  fitv  23  2  = Inverse [Transpose [X]. X]. Transpose [X]. datav ; 23  f i t a i = Inverse [Transpose [X] . X] . Transpose [X] ^dataax;:  :  f i t a = Inverse [Transpose [X] . X] . Transpose [X] . dataa ; 2  2  • Checking goodness of fit The coefficient of deternunaticn is calculated fci each parameter to check the goodness of fit.  t o t a l d i f f = d a t a E i - Table [Mean[dataEi]:,- { i , 0> 45}:] ; sst = t o t a l d i f f . t o t a l d i f f ; sse = ( d a t a E i - X . f i t E . J . (dataE - X . f i t E ) ; ±  ±  Rsquared = 1 - (sse / ( 4 6 - (P + 1) )) / (sst / ( 4 6 - 1 ) )  totaldiff = datavii - Table [lfcanldatavu], {i, 0,45}];: : sst = totaldiff . t o t a l d i f f ; .: sse = (datavu-X. f itvi2). (datavu - X. f itvi2); Rsguared=l- (sse/(46:- (P + l))):/:(sst 7(46 -1))  totaldiff = dataGx2 - Table[Mean[dataG :]v{i> 0, 45}]; 12  sst=totaldiff.totaldiff; sse = (data&u - X. fitfr^) . (dataG^- X. fit&tj) ; : :: Rscjuared =1- (sse / (46 - (P * 1))) / (sst / (46 - 1)) 0.999999  totaldiff = dataK - Table [Mean[dataK ] , {i, 0, 45}] ; 2  2  sst = t o t a l d i f f . t o t a l d i f f ; sse = (dataK - X. fitK ) . (dataK - X. fitK ) ; 2  2  2  2  Rsquared = 1- (sse / (46 - (P + 1))) / (sst / (46 - 1))  :i:i:Lp:?^998?;;;:i:;:v : :  totaldiff = dataG^ - Table[Kean[dataG ], {i, 0, 45}]; 23  sst = t o t a l d i f f . t o t a l d i f f ; sse = (dataGis - X.fitG? ). (dataG^ - X. f i t G ^ ) ; 3  Rsquared =1- (sse / (46 - (P + 1))) / (sst / (46 - 1)) 0.999989  -235 -  Appendix A: Mathematica Notebooks totaldif f .= dataE - Table[Hean[dataE ], {1, 0, 45}]; sst = totaldiff.totaldiff sse = (dataE - X. fitE ) • (dataE - X. fitE ); Rsquarecl= 1- (sse / (46 - (P + 1))) / (sst/ (46 - 1)) 2  2  2  2  2  2  0.999981  totaldif £ = datav - Table[Hean[datav ]; {1; 0, 45}]; sst = totaldiff.totaldiff; sse = (datav - X. f itv ). (datav - X. f i t v ) ; ; Rs(juared = 1- (sse:/(46 - (P + 1)))/(sst / (46 - 1)) 23  23  23  23  23  23  totaldiff = dataa - Table[Mean[dataai]> {i;: 0> 45}]; sst=totaldiff.totaldiff; sse = (dataai - X. fitax) . (dataox - X. fitoi);: Rsquared = 1-(sse / (46 - (P + 1))) / (sst / (46 - 1)) i:  totaldif f = dataa - Table[Mean[dataoc ], {i, 0, 45}]; 2  2  sst =: totaldiff .totaldif f; sse = (dataa - X; fita ). (dataa - X . f itotj); Rscjuared = 1- (sse / (46 - (P + 1))) / (sst / (46 - 1)) 2  2  2  0.999708  Output • Calculating the values of the moduli in time In this part, the "exact" moduli profiles vs. time are generated, which can be used for comparison.  resultsEi = Table[{Time[i] , Ei[Time[i]]} , {i, 0, 45} ] ; MatrixForm[resultsEi] ; resultsv^ = Table [{Time [i] ••; Vi :[Time [i] ]}, 2  r e s u l t sGu = Table [{Time [i],  {i, 0 , 45} ] ^MatrixFormliesultsvi^];  [Time[i]] }, {i , 0, 45} ] ; MatrixFormJresultsGi^];  resultsK = Table [{Tine [i] , K [Time[i]]} , {i, 0, 45}] ; MatrixForm[resultsK ]; 2  2  2  resultsG^ = Table [{Time [i], G^ [Time[i]]}, {i, 0, 45}] ; MatrixFoxm [ r e s u l t s ^ ]; : r e s u l t sa^ = Table[{Time[i], y.a± [Time [i]]} ,:{i •> 0, 45} ] ; MatrixForm[resultso<i]; :  : : result sa = Table[{Time[i], a [Time [i] ]}, 2  2  {i , 0 > 45}] ; MatrixForm[resultsa ]; 2  -236-  Appendix A: Mathematica Notebooks  • Exporting the values (* Exporting the f i t parameters *) Export["fitel.txt", f i t E i , "Table']; Export [" fitnl2.txt ', f i t v ^ , "Table"]; 1  Export['fitgl2.txt",  fitGr^,  "Table"];  Export["fitk2.txt", fitK ,"Table"]; 2  Export["fitg23.txt", fitG2 , "Table"]; 3  E x p o r t [ " f i t a l . t x t " y f i t a i , "Table"]; Export!"fita2.txt", £ i t a , "Table"]; 2  (* Exporting tne:exact values  *)^^^::  Export [" resultse l . t x t " , re s u i t s E , "Table"]; x  Export["resultsnl2.txt" >resultsvi2,"Table" ]; Export["resultsgl2.txt", r e s u l t s G u , "Table"]; Export["resultsk2.txt" , r e s u l t s K , "Table" ]; 2  :  Export["resultsg23.txt", resultsG2 ,"Table"]; 3  Export["resultsal.txt" / resultsai, "Table"]; Export["resultsa2.txt" , resultsa , "Table"]; 2  -237-  Appendix A: Mathematica Notebooks  A.2  Notebook for the example in Section 7.2.2.  a= 2; b =4; h = 4 / 3 3 ; AA = 4 / 3 * 10" ; BB = 4 / 3 * 10" ; CC = 3 * 1 0 ; 5  E  5  5  = 3 * 1 0 ; v = .3015; 7  a  E  a- E * h / ( 1 - V E 2 ) / b ; a  a  »rr[r_s_] :=  V:;.";  •/.';  ••-PV'S'*;"'.:'; ': ;V:'•'•.^ V' ;;. -V: ;  ;  V  ''  :  :  :  (((l-b 2/r 2-aAA)(2AACC+l)-3a»Ab 2/r 2)s 2 + A  A  A  A  A  ((1-1> 2 / r 2 ) 2 B B C C - a B B ( 3 b 2 / r 2 + 4 A A C C + l ) ) 8 - 2 a BB 2 CC) / A  A  A  A  A  ( ( ( l - b 2 / a 2 - a A A ) (2AACC + l ) - 3 a A A b 2 / a 2 ) s 2 + A  ( (1 - b  A  A  A  2/a  A  5 » [ t _ s J := -P/s *  2) 2 BB CC - o BB (3 b 2 / a A  A  A  2 •'+ 4 AA CC + 1) ) s - 2 a BB 2 CC)  A  A  '"X'?. ';:-. : -. :' :':V.:':'vV.7:'^ '.v: ': • ;  : ;  ;  :• ; - '•; '•'/••:••'•;-:•:  ;  v.\v  :  ;  ( ( ( l + b 2 / r 2 - a A A ) (2AACC+1) + 3 a B A b 2 / r 2 ) s 2 + A  A  A  A  A  ( (1 + b 2 / r 2) 2 BB CC - a BB (-3 b 2 / r 2 + 4 AA CC + 1) ) S - 2 a BB 2 CC) / A  A  A  A  A  ( ( ( 1 - b 2 / a 2 - a A A ) (2AACC+1) - 3 a AAb A  ((I-b  A  A  A  2 / a 2) s 2 + A  A  2 / a 2 ) 2BBCC - « B B ( 3 b 2 / a 2 + 4 A A C C + l ) ) s - 2 a B B 2 C C ) A  A  5rr[rs]  A  •.: "\-\. -~ :  '; 1. ( - 1 0 6 . 6 6 6 ^ ( 9 ( 1 - i f ) -  A  '• '-"-V•'.v".•  :  /'•'•.•'•'.':'•'•'  :  13,3332 (17 + 4 ) ) 3+;(9 (-12/3332 - if-) - 1 2 L « i ) « ) g  3 (-106.666 - 410.664 3 -306.998 3')  1. (-106.666+ (-13.3332 (17 - **-)  +  8 (l V ^ ) ) s  +  (9 (-12. 3332  .  +  if)';+ m^ll.)  V)  3 (-106.666 - 4 1 0 . 6 6 4 3 - 306.998 s ) J  InverseLaplaceTransform [ d -1.  (l.  + e  I C  [r s ] , s , t ]  - ° ^ ^ ( - 0 . 0 0 5 3 6 2 9 2 , °" ^  1  7  ),  (-0. 633074  ))  InverseLaplaceTrans form [free[r s ] , s , t ] -1.  -0.3S£7VSt  1. +e  (-0.633074-  2.5323 v j, cUgot t. t' J + e *: * (-0.00536292 -  -238-  0.0214517 JJ n  Appendix A: Mathematica Notebooks  A.3  Notebook for the example in Section 7.2.3. Defining the properties in time domain • Matrix properties :: the Matrix properties are defined with two properties;each in Prony series. Shift factor is defined ez an input  •  :  Off[General::spell] Off [General: :tinfl] G"„ - 11.85; G„ .- .18; is - 1.: <x„ - 5.8 10"-S: r  K"„ = 35.56;K' . = 8.,- I = 1.5: 1  K  H , [ t J :=G„.» (G"„-G„.) Ex»[-t/7 ] r  r  S  H»[t_] : = K„.* (K"„-K'„) Ex»[-t/7 J r  K  E"„ = 9 K"„ G\, / (3 K"„ • 0".);  . .  v"„= |3K\-Z6*,)/ [ S K \ . 2 t M ;  ! ' . = !lf.l?./ PIC^GM; v».= (3K-.-2C",|/(6lf„.2i;V); ; • Fiber properties ..  .  Eic = 2100. ; V12 = 0.2; On, = 276-. ; v = 0.25; E = 172. ; a c = -0.9 10--6; a = -7.2 10"-6; f  v  c  2JC  2C  X  2£  •* 0.6;  G2 r=E / (2 (l+v )); K = 1/ (4 / E - 1 / Gj,, - 4 v 2 / E ) ; : :*:-El* •'.4.:KlfVta(».2) ::• • f = 4. K f ^r 5  2r  23r  A  2f  2f  12f  lr  2  Composite variables  e[t_]  126. t 1. Exv|-t /l.] 2.2*4.2Exp[-t/1.3] 2.2*4.2Ex»[-t/1.5]  : =  0  Transformation to Laplace space Gjn[s_] ' • LaplaceTransforri[G^ [ t ] , t,flJ s  fe[s_j]: := s LaplaceTzansf oxm[K [ t ] , t, a] II  .:.:.::: c[s:]:: - LaplaoeTxans£orjn[e [ t ] t, :a\ als-J : - LaplaoeTxansf oxm[o~[ t ] •, t a].-.  Stiffness moduli  D [s_] :m  4G„[s]/3*K„Is] feMfe[s|i-2 §„[fl]/3 :n:.:;. K»|s] -2G^[s]/3:4G„[s]/3*Ki,[B| fe[sJ - 2 §„[s]/3 0 fel«l -2G„[s]/3 fe[s]-!C»[»]/3:4e„[s]/!.;K,|»| : 0  0 0 1  It  s s  t  K i t + G^f. K K  0 0 : 0:  2 C  - G23f  0 0 0  K  2 f  - G2 •  2 C  + G23£  3£  0 0 0  0 0 0: 0 0 0 Guf 0: 0 0 0  0 0 0: 0 0::  Gwf  -239-  ::: 0. .' 0 G„[s] 0 0  Appendix A: Mathematica Notebooks  Reverse micromechanics Dinc[s_] : = Df-D„[s] e [s^] i:= (1/(1 -v )) Inverse [Dine [s]].(D . e[s] -S[s]) m  c  f  5m[s_] : = D [s].e [s] m  E [s_] f  m  := (l/v ) <e[s] -<l-v ) i [s]) r  r  m  9[s_] : = (l/vr) <8[s] - (1 - vd 8 [s]) f  m  Inverse Laplace transform • Applying the inverse Laplace transform A number of available models;can be used to transform the parameters to time domain. The Stehfest method is used here.  : Remove [Stehfest]::. Remove [Durbin] . Remuve[t] •  . •  .. .  .  .. .  < < HumericalHath •' Humerlcallnversion' «m[t_] :•= Stehfest [ i [ s ] , s, t] m  »»[t_] : = Stehfest[S„[s], s, t] ef[t_] : = Stehfest [if [sj,s,t] "t[t_] : = Stehfest[Sf[s], s, t] e„[.001] <r„[.001] {{0.102413}, {0.0032590!.)}, {(i.C~32S909), {0.}, {0.}, {0.}}  !3i  {{S.4366S}, (3.089CS}, (3.089CS), ( 0 . ) , {0.}, {0.}}  i *3T  e [1.3 m  {{0.1Qp333} :{0.0280328}> .{0.0280328}, /  :  {0. }, {0. }, {0. }}  ((4.03746), {3.52594}, {3.52594}, ( 0 . ) , {0.}, {0.}}  -240-  3 1  Appendix A: Mathematica Notebooks  A.4  Notebook for the example in Section 7.2.3. Composite variables  E[t  131.7 • 1.01Expt-t/l.] * 8.484 Exp[-t/ 1.51 ' 5.858 • 4.242 Expt-t/1.5] +9.292 E j p p l - t / l . ] : : o [ t _ ] : = 5.858 . 4 . 242ExrJ[-t/l:5] +9.292ExrJ[-t71.] 0 0  ]:=  Transformation to Laplace space C „ [ H _ ] : = s Lai)laceTransform[G„ [ t ] , t , •:]::.• : K^ls^] : = s L a p l a c e T x a n s f orm[K„|t) , t , s) i[8_] : = L a p l a c e T r a n s f o r m [ e | t ] , t ; s]: 8[«jj]:::. L a p l a c e T r a n s f o r m [ o [ t ] t , s]  Stiffness moduli f4C^[s] /3 + K „ [ s ] i L l s ] - 2 k [ ! ] / 3 2G [s]/3  4G„r8]/3.K»Is)  m  i L l s ] -2G„ts]/3 i^(s| - 2 B„[s]/3  0 0  •2C,[s]/3 K„[s) - J G „ [ s ] / 3 4 G ^ s ] / 3 . K„ |s |  0  :::0: :::.;;::0:;;:: 0  0  e [s] 0 0 0 e„[s] 0 0 0 e^tsi m  If Df •  s  7 K  2  f  0 0 0  r G23 r K 1 0 .:: 0 0 2  + G23 f 0 0 ' 0  n 0 0  0 0 0 0.  0 0:  Gl°f  0  0 0 0 0 ; 0;.: &23f  <>2£ Err =  iT  0 0 0  Reverse micromechanics Dinc[s_] : = D - D „ [ s ] £  e^[s_] : = Inverse[Diiic[s]].<(l/(I-v )) c  ( D . ( E [ S ] - v e ) - 8 [ s ] > - D [s]. e^) c  r  5r,[s_] : = D „ [ s ] . (S„,[s] - e ) rf  8 Cs_] : = ( l / v > (e[s] - (1 - v ) e [s]) f  f  f  m  -241 -  ff  w  Appendix A: Mathematica Notebooks  Inverse Laplace transform • Applying the inverse Laplace transform A number of available models can be used to transform the parameters to time domain. The Stehfest method is used here. Remove[Stehfest] Remove[Durbin] Remove[f] << Himcricalllath Numerical Inversion En[t_]  : = Stehfest[e [s], s, t] m  " [t_] : = Stehfest[S [s], s, t] m  m  • e [t_]: = Stehfest [e [ s ] , s , t ] : f  f  Of[t_] : = Stehfest[Of [s], s, t] EmtOOl]  o^[001] {{-0.154928), {{-21.8338),  {-0.251698),  (-0.251698), ( 0 . ) , {0.),  (-24.1193), (-24.1193), ( 0 . ) , {0.),  (0.))  ] ]  {0.))  e [1.001] m  •o [1..001] m  {{-0.15684}, ( - 0 . 0 8 2 6 9 7 7 ) , {-0.0826977), {{-6.51169),  ( - 5 . 0 2 3 2 5 ) , ( - 5 . 0 2 3 2 5 ) , {0.},  ( 0 . ) , {0.}, (0.),  {0.})  -242-  {0.})  1  Appendix A: Mathematica Notebooks  A.5  Notebook for the example in Section 7.2.4.  Defining the properties in time domain  I  • Material properties The Matrix properties are defined with two properties, each in Prony series. Shift factor is defined az an input.  Off[General::spell] Off[General::unfl] T[t_] :=120;P[t_] : = 1. 0591 - 0. 0591T[t] / 26. 86 G\ = 3 7 5 0 . ; G " i  = 3 7 5 . ; T = 1.;  f m  g  cin = 0.0001; K „ = 10000. ; K " u  i  £ m  = 7000. ; i  G^t] : = G^ * (G - G f  u  m  K„[t_]  : = lf" * f  m  inf  m  (k -ie u  nf  m  m  m  K  = 1.5;  *  ) Exp[-t / 7 ]  ]  G  ) E*p[-t/ ]  ^  lK  External Load variables f 100. 0 0 "m[t_l := 0 0 0  ; E  R  F  T[t] - 120 ' T[t] - 120 T[t] - 120 [t_] : = etn 0 0 0  Transformation to Laplace space G [s_] : = s LaplaceTraj\sform[G [t ] , t, s] m  m  K [s_] : = s LaplaceTransform[K,[t] , t, s] m  5 [s_] : = LaplaceTransform[ff [t] , t, s] m  m  Stiffness moduli 4G [s]/3 K [s] K»[s] -2G [s]/3 K»[s] -2G [s]/3 0 0 K»[s] -2G [s]/3 4G [s]/3 + K [s] i^Js] - 2 G [s]/3 0 0 Kls] -2G [!]/3 K„[s] -2G„[s]/3 4 G»[s]/3 + |s] 0 0 D [s ] : = 0 0 0 G^[s] 0 0 0 0 0 G [s] m  +  m  m  m  m  m  ri  m  m  m  m  Calculating the strains E»[s ] := Inverse [D [s]] Is m  -243 -  0 0 0 0 0 G„[s]  Appendix A: Mathematica Notebooks  Inverse Laplace transform • Applying the inverse Laplace transform A number of available models can be used to transform the parameters to time domain. The Stehfest method is used here. Remove[Stehfest] Remove [Durbin] Remove[f] << HumericalHath Humericallnversion E [ t _ ] : = Stehfest [e„[s], s, t ] . o [ t ] /P[t] e [ t ] m  m  +  r f  e [.001] m  -244-  Appendix B: Pseudo-Code for the Differential Form Implementation  Appendix  B. PSEUDO-CODE FOR THE DIFFERENTIAL F O R M IMPLEMENTATION  In this Appendix, the algorithm for the implementation of the Differential Form (DF) of viscoelasticity in a finite element code will be presented. This will consist of text description of the command lines of the code, i.e. what is called the 'pseudo-code', and will be presented for the main subroutines of the D F code. Each includes an explanation of the function of the subroutine, followed by the actual pseudo-code.  Subroutine C a i c D T :  This subroutine calculates the material stiffness matrix, D " , at a single integration point. The subroutine is currently written for both transversely isotropic and isotropic V E materials. The possibility of additional models has also been provided. 1) Check the material type; i f not viscoelastic, R E T U R N 2)  Call Relaxation to calculate the current value of relaxation times, T"^ , e.g. from Equation (4.27)  3)  Check whether thermoelastic effect is present; i f it is, a.  Call Thermoelastic to calculate the current and previous values of the moduli based on the temperature, e.g. Equation (5.45)  b. 4) 5)  6) 7)  Calculate Maxwell stiffness by combining the above values, i.e. Equation (6.48)  Loop over the number of material properties, 2 for isotropic and 5 for transversely isotropic Loop over the number of Maxwell elements for this material property, N  Add contribution of the material stiffness, i.e. P". in Equation (6.54) Check the material behaviour type, e.g. transversely isotropic or isotropic  -245 -  Appendix B: Pseudo-Code for the Differential Form Implementation  a.  Calculate the arrays of the stiffness matrix, D", from material properties  Subroutine CalcEfree: This subroutine calculates the free strain increment vector, As  f  , for an integration point. This subroutine  is written for the case where thermal expansion and cure shrinkage exist and that behaviour of both is characterized by a Prony series. 1) Check the material type; i f not viscoelastic, R E T U R N 2)  Call Relaxation to calculate thermal expansion relaxation times for the current and previous temperature (T"  +]  3)  and T") and degree of cure ( a "  + 1  and a"), e.g. from Equation (4.27)  Check whether thermoelastic effect is present; i f it is a.  Call Thermoelastic to calculate the current and previous values of the thermal expansion moduli, e.g. Equation (5.45)  4)  Loop over the size of the thermal strain vector, i.e. 1 for isotropic and 2 for transversely isotropic  5)  Loop over all Maxwell elements of the thermal expansion strain model  6)  Call DiffSolve to solve for the thermal expansion strain  7)  Calculate the thermal expansion strain increment vector, As  8)  Call Relaxation to calculate cure shrinkage relaxation times for the current and previous  T  temperature (T"  +i  9)  and T") and degree of cure (a  n+i  similar to stress in Equation (6.45)  and a"), e.g. from Equation (4.27)  Check whether thermoelastic effect is present; i f it is a.  Call Thermoelastic to calculate the current and previous values of the moduli for cure shrinkage, e.g. Equation (5.45)  -246-  Appendix B: Pseudo-Code for the Differential Form Implementation 10) Loop over the size of the cure shrinkage vector, i.e. 1 for isotropic and 2 for transversely isotropic 11)  Loop over all Maxwell elements of the cure shrinkage strain model  12)  Call DiffSolve to solve for the cure shrinkage strain  13) Calculate the cure shrinkage strain increment vector,  As  a  14) Call RearrFree function to rearrange the strain increments into vector form 15) Add the strain vector to calculate the free strain increment vector  Subroutine CalcEsigma: This subroutine calculates an equivalent free strain increment associated with internal stresses, Ae  a  , for a  single integration point. In this subroutine; calculation of the material compliance matrix depends on the material behaviour, but the rest of the subroutine is generic for any material. 1) Call Relaxation to calculate the current and previous values of relaxation times, r" f and r" , p  pj  respectively, e.g. from Equation (4.27) 2)  Check whether thermoelastic effect is present; i f it is, a.  Call Thermoelastic to calculate the current and previous values of the moduli based on the temperature, e.g. Equation (5.45)  b.  Calculate M a x w e l l stiffness by combining the above values  3)  Loop over the number of material properties, 2 for isotropic and 5 for transversely isotropic  4)  Loop over the number of Maxwell elements for this material property, N  5)  p  Add contribution of the material stiffness, Pj  -247-  Appendix B: Pseudo-Code for the Differential Form Implementation  6)  Calculate the arrays of the material compliance matrix, C" , from material properties. T  7) Loop over the number of material properties, 2 for isotropic and 5 for transversely isotropic 8)  Loop over the size of the stress vector for this material property, from Equation (6.17) or (6.32)  9)  Loop over the number of Maxwell elements for this material property, N  p  10)  Calculate  A,,  from Equation (6.56)  11)  Add contribution of the above value to X X ! — / "  p  •  12) Call RearrangeStress to rearrange the above values into a vector, from Equation (6.18) or (6.33) 13) Multiply matrices from steps 6) and 12) to calculate the equivalent free strain increment Subroutine SolveStress: This subroutine calculates the new total stress and also stresses for all the Maxwell elements. This is done for an integration point, using the current strain increment and stress values. This routine is generic, but depends on whether thermoelastic behaviour exists or not. 1) Call Relaxation to calculate the current and previous values o f relaxation times, r" f and r" , p  pj  respectively, e.g. from Equation (4.27) 2)  Call RearrangeStrain  to calculate the current value of strain increments associated with  different material properties by rearranging the arrays of the mechanical strain matrix, from Equation (6.17) or (6.32) 3)  Check whether thermoelastic effect is present; i f it is,  -248-  Appendix B: Pseudo-Code for the Differential Form Implementation a.  Call Thermoelastic to calculate the current and previous values of the moduli, e.g. Equation (5.45)  b.  Call RearrangeStrain to calculate the previous value of strain increments associated with different material properties, from Equation (6.17) or (6.32)  4) 5)  Loop over the number of material properties, 2 for isotropic and 5 for transversely isotropic Loop over the size of the stress vector for this material property, from Equation (6.17) or (6.32)  6)  Loop over the number of Maxwell elements for this material property, N  p  7)  Call DiffSolve to solve the differential constitutive equation to get <r , pj  Equation (6.44) or (6.48) 8)  9)  Add contribution of the above value to g_  p  Call RearrangeStress to calculate the composite stresses by rearranging elements of q_ , from Equation (6.18) or (6.33)  Subroutine R e v e r s e M M : This subroutine is the main subroutine for reverse micromechanics. It calculates the matrix strains and stresses. Currently, this subroutine works based on the assumption that no thermoelastic effect is present. 1) Check whether thermoelastic effect is present. If it is R E T U R N 2)  Check the material type; i f not viscoelastic, R E T U R N  3)  Call CalcDT to calculate the matrix material stiffness matrix, i.e. D" from Equation (6.61)  -249-  Appendix B: Pseudo-Code for the Differential Form Implementation 4)  Call RMMStiffF to calculate the fibre material stiffness matrix, i.e. D  5)  Call RMMInverse to calculate the inverse of D - D"  6)  Call RMMFree to calculate the free strains in matrix and fibre, s  7)  Call Relaxation to calculate the current and previous values o f relaxation times for matrix  8)  Loop over the number of material properties of the matrix (2)  9) 10)  T  mf  and s  ff  , respectively  Loop over the size o f the stress vector for this material property, from Equation (6.17) Loop over the number of Maxwell elements for this material property,  11)  Calculate A ,, from Equation (6.56)  12)  Add contribution of the above value to  N  /» P  1  13) Call RearrangeStress to rearrange the above values into a vector, from Equation (6.18) 14) Calculate the other variables needed for the matrix strain increment in Equation (6.75) 15) Calculate the matrix strain from 13) and 14) and strain increment, i.e. Equation (6.75) 16) Call SolveStress to calculate the value of stress in matrix, from Equation (6.44)  -250-  p  Appendix C: ABAQUS Viscoelastic Model  Appendix C . A B A Q U S  VISCOELASTIC M O D E L  In this Appendix, the available Viscoelastic (VE) model in the commercial code ABAQUS will be described and its characteristics will be discussed. This will be done to compare the Differential Form (DF) developed in this thesis to another available model in the literature. For this purpose, first the formulations developed by ABAQUS in its Theory Manual are presented. After that, the relations between these and the DF,' with their similarities and differences, are discussed. Note that in explaining the ABAQUS V E model the notation used in the Theory Manual Version 6.4is adopted. The basic V E integral equation is:  where e and ^ are the mechanical deviatoric and volumetric strains, K and G are the bulk and shear moduli, dot represents derivative with respect to t', and r is the reduced time, defined as follows:  -I  dt' A(9(t'))  Here 6 is the temperature and A is the shift factor, calculated from the WLF equation: g  -\ogA =h(9) e  =  cf(e-e ) g  ci+(0-0 ) g  where C f and C f are constants and 9 is the glass transition temperature. g  The moduli are assumed as equivalents of multiple Maxwell elements in parallel:  K(T) = K„ Y K,e +  j  G(T) = G f G e tD+ j  i  -251 -  Appendix C: ABAQUS Viscoelastic Model In A B A Q U S , it is assumed that the relaxation times for both moduli are the same, but the number of Maxwell elements can be different. To develop the finite element implementation, the deviatoric behaviour is considered. Bulk behaviour can be modeled in a similar way. The deviatoric stresses can be written as:  ft JO  I l  G  1  ("T  /  i=\  j  xi  l  N  G  N  ,•_]  Note: The integration parameter has been changed. With this manipulation, from this point on all the calculations are performed in the reduced time space ( r ) and not the real time (t). The above equation is rewritten in the following form:  "a  S = 2G„  where G is the instantaneous shear modulus, 0  l-e  v  J  /=•  V  =  , and  / !  J  dt'  (C.l)  is the viscous (creep) strain in each Maxwell element. Now, assume that in a given time step this strain varies linearly with the reduced time, i.e.  de/dr' = A e / A r . With this assumption and some manipulation on the above equation the recursive equation for the calculation of the viscous stresses is obtained:  -252 -  Appendix C: ABAQUS Viscoelastic Model  Ae,=-<-  Ar'  •+e  Ar  -Ar/r, _  j  Ae + ( l - e -  A r A  ')(e"-e;)  (C.2)  Having these values, the increment of stresses can be calculated from the following equation:  AS = 2G  (C.3)  Ae-X«,(e; '-e;) +  0  The combination of these two equations provides us with stress in a time step.  Differentiating this equation with respect to the deviatoric strain increment ( ^ ^ - ) will give the tangent dAe modulus:  G =G  n  i-E«/—  AT  + e-  Arh  -1  (C.4)  tT AT Ar  Note that A B A Q U S simplifies these equations for — « 1 and considers it separately. But, this will not  be shown here for simplicity. Now, since all these calculations are done in reduced time space and because it is the real time that will be eventually discretized, a relationship between the two time scales is needed. In A B A Q U S , the function h(0) is assumed to vary linearly as a function of temperature. Note that, albeit a good approximation for many cases, this is merely a simplification. From this assumption, it can be proven that:  h(0 )-h(0") n+]  -253 -  (C.5)  Appendix C: ABAQUS Viscoelastic Model  Discussion 1. We claim that the solution to the integral equation in reduced time space presented above is in fact equivalent to a form of finite difference solution to the governing differential equation. Looking at the viscous strain in Equation (C.l), we can rewrite it as:  The integral on the right-hand side is in fact nothing but a form of stress in Maxwell element i. In other words:  Substituting this value of viscous strain in the recursive equation strains (Equation (C.2)) we will get:  Rearranging this equation:  AS, = 2G, -^-(l - e~ ') Ae - (l - e' AT Ar/r  ') S,  AT/r  This is in fact a recursive equation for the calculation of Maxwell stresses, i.e. similar to the equations derived in Chapter 4, e.g. Equation (4.36). This equation is not similar to the classic cases of the finite difference normally used to solve the differential equations. However, it shows that the method used by ABAQUS is practically equivalent to a differential form solution, although using an exponential form of finite difference discretization. This approach is in fact similar to the formulation developed by Bazant and Wu (1974) adopted for the DF. The difference between the A B A Q U S approach and the one used in  -254-  Appendix C: ABAQUS Viscoelastic Model  the reference to a large degree is the choice of the internal variable used; A B A Q U S uses the Maxwell viscous strains for this purpose while the one by Bazant and W u employs the Maxwell stresses. 2. The similarity between the A B A Q U S approach and the present differential form is obviously valid when the problem is solved in the reduced time space. Put differently, these two formulations are similar for isothermal cases. However, when temperature changes are present the A B A Q U S relates the two times using an approximation (Equation (C.5)), while the differential form does not make any assumptions. Due to the similarity between the two approaches, it might be possible to modify the A B A Q U S formulation for non-isothermal cases to avoid any assumption on the solution. However, the complexity of the integral formulation as the basis for the A B A Q U S approach w i l l complicate this development. 3. The A B A Q U S formulation modifies the tangent modulus with Equation (C.4), in a similar way to the DF formulation. However, no modification to the force vector in the F E equations is done, unlike the D F approach (e.g. Equation (6.59)). The current A B A Q U S implementation w i l l clearly generate errors, the magnitude of which w i l l depend on the time step sizes and conditions of the case under study.  Summary A comparison of features available in the two forms of V E models, namely the present D F and A B A Q U S internal model, was shown in Table 7.1. In this appendix, a comparison of the details of the V E models for A B A Q U S and the D F was shown. Through the discussion presented above, it is clear that the viscoelastic model present in A B A Q U S has some similarities to the D F o f viscoelasticity in the calculation of stresses and modification of the stiffness matrix. However, due to the simplifications made in A B A Q U S the results obtained through this code w i l l lead to inaccuracies. In addition, due to lack of modification of the force vector in A B A Q U S , this method w i l l need smaller time-steps for the same level of accuracy as the D F , as shown in Section 7.2.2.  -255 -  

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