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Efficient multi-scale modelling of viscoelastic composites with different microstructures Malekmohammadi, Sardar 2014

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EFFICIENT MULTI-SCALE MODELLING OF VISCOELASTIC COMPOSITES WITH DIFFERENT MICROSTRUCTURES  by Sardar Malekmohammadi  B.Sc., Amirkabir University of Technology, 2007 M.A.Sc., The University of British Columbia, 2009  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  July 2014  © Sardar Malekmohammadi, 2014 ii  Abstract  Modern composites such as advanced fibre-reinforced and strand-based wood composites are increasingly being used in the new generation of aerospace and civil engineering structures. The structural analysis of such composites often requires knowledge of their effective (homogenized) properties. Several micromechanical models have been developed and are available in the literature for predicting the effective elastic properties of fibre-reinforced solid composites. However, the underlying assumptions in these models somewhat limit their application in solving some practical problems related to the viscoelastic behaviour of composite materials.   Two seemingly different classes of composites, i.e. thermoset fibre-reinforced composites and strand-based wood composites with distinct viscoelastic properties are considered in this work due to their wide application in aerospace and construction industry. For viscoelastic analysis of such materials, aspects which require further investigations at the micro-scale are identified first. Specifically, available analytical micromechanics models are extended to predict the shear properties of thermoset fibre-reinforced composites during cure where the resin evolves from a viscous fluid to a viscoelastic solid. For strand-based composites consisting of high volume fraction of orthotropic wood strands, analytical micromechanics models are developed. These models are employed for predicting the effective elastic and viscoelastic properties of strand-based composites. The validity ranges of these models are then examined using experimental data or numerical reference solutions that employ the computational homogenization technique.  iii  To enable viscoelastic analysis of large scale composite structures with generally orthotropic properties, an efficient and easy-to-implement approach in the context of 3-D multi-scale modelling, is presented. A multi-scale modelling framework involving analyses at different scales for composites with two difference microstructures is developed and implemented in a general purpose finite element code, ABAQUS®. The accuracy of the developed multi-scale approach is demonstrated for some practical applications involving MOE (apparent modulus of elasticity in bending) prediction of strand-based wood composites. Using this approach, the effect of microstructural parameters (e.g. fibre geometry, orientation, waviness, volume fraction, etc.) on the time-dependent macroscopic response of orthotropic composite structures can be investigated, quantitatively. iv  Preface  This thesis presents the research performed by Sardar Malekmohammadi. The research was supervised by Dr. Reza Vaziri and co-supervised by Drs. Anoush Poursartip and Fernand Ellyin at The University of British Columbia and Dr. Carol Nadot-Martin at Institut P', CNRS -Université de Poitiers – ENSMA.  Parts of the numerical approach in Chapter 2, Appendix A, Appendix B (Section B.3 ), Chapter 4 (Section 4.2.2), have been published. Gereke T., Malekmohammadi S., Nadot-Martin C., Dai C., Ellyin F., Vaziri R. (2012) Multi-scale stochastic modeling of the elastic properties of strand-based wood composites. J  Eng  Mech 138:791-799. Most parts of this paper were written by Dr. Thomas Gereke (a post-doctoral fellow, now a team lead at Technische Universität Dresden). In a collaborative work with Dr. Thomas Gereke, I contributed to the introduction, background and the unit cell model in this paper.  In Chapter 2, Mr. Benjamin Tressou (a visiting scholar from France) wrote the Python codes for constructing the unit cell models under my supervision. He also assisted me in running the Python codes and comparing the numerical and analytical results in this chapter. The experimental data of thermoset composites during cure in Chapter 2 were provided by a previous MASc student, Mr. Ryan Thorpe. I acknowledge both Mr. Benjamin Tressou and Mr. Ryan Thorpe for their help and providing their numerical results and experimental data.  v  A version of Chapter 3 has been submitted for publication. Zobeiry N., Malekmohammadi, S., Vaziri, R. and Poursartip, A. (2014) A Differential Approach to Finite Element Modelling of Viscoelastic Materials. In this Chapter, I extended the differential approach developed by a previous PhD Student (Dr. Nima Zobeiry) for viscoelastic modelling of orthotropic materials. The submitted paper is mostly based on the previous work of Dr. Nima Zobeiry for transversely isotropic materails. I contributed to this paper by revising a draft of the paper which was originally written by Dr. Nima Zobeiry. I completed the introduction, the literature review part of the paper by exploring and writing the state of the art in viscoelastic modelling of composites. Additionally, by revisiting the numerical examples and adding two new examples to the paper, I demonstrated the capability of the differential approach in viscoelastic modelling of composites. For this purpose I wrote new Mathematica® codes. I employed the FE formulation in Zobeiry for transversely isotropic materials and with some modifications I wrote a UMAT for orthotropic materials. The extended version of Zobeiry’s DF approach (performed by myself) to orthotropic composites is presented in Chapter 3.  A version of Chapter 4 has been published. Malekmohammadi S., Tressou B., Nadot-Martin C., Ellyin F., Vaziri R. (2013) Analytical micromechanics equations for elastic and viscoelastic properties of strand-based composites. J Composite Mater. doi: 10.1177/0021998313490977. I wrote the paper and the coauthors provided useful comments. The analytical section is my main contribution in the paper. Mr. Benjamin Tressou also assisted me by writing the Python codes for obtaining the numerical results used for comparison purposes.  vi  A version of Chapter 5 has been submitted for publication. Malekmohammadi, S., Zobeiry N., Gereke, T., Tressou, B., and Vaziri, R. (2013) A Comprehensive Multi-Scale Analytical Modelling Framework for Predicting the Mechanical Properties of Strand-Based Composites. I wrote the paper and the coauthors provided me with useful comments. Dr. Navid Zobeiry (a post-doctoral fellow) assisted me in connecting the multi-scale steps together and incorporating the mathematical equations in Microsoft Excel by writing Macro codes. He also contributed to the optimization methodology presented in this chapter. Figure 5-14 is generated based on his work. The numerical parts in Sections 5.3 and 5.4 of this thesis are the contributions of Dr. Thomas Gereke (a post-doctoral fellow, now a Professor at Technische Universität Dresden) during our collaborative work. He has played a major role in implementing the multi-scale model into the finite element software used for comparison here. The analytical version of multi-scale framework presented in this chapter was developed by myself based on its numerical version which was previously developed in collaboration with Dr. Thomas Gereke. Also Figure 5-2 has been provided by Dr. Thomas Gereke. Numerical results were generated by Mr. Benjamin Tressou using the Python codes he wrote under my supervision.  vii  Table of Contents  Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iv Table of Contents ........................................................................................................................ vii List of Tables ............................................................................................................................... xii List of Figures ...............................................................................................................................xv Acknowledgements .................................................................................................................. xxiv Dedication ................................................................................................................................. xxvi Chapter 1: Introduction ................................................................................................................1 1.1 Motivation ....................................................................................................................... 4 1.2 Research Objectives ........................................................................................................ 9 1.3 Research Scope and Goals .............................................................................................. 9 1.4 Thesis Structure ............................................................................................................ 13 Chapter 2: Micro-Mechanical Modelling of Circular Fibre Composites ...............................17 2.1 Elastic Analysis ............................................................................................................. 19 2.2 Thermo-elastic Analysis ............................................................................................... 25 2.3 Viscoelastic Analysis .................................................................................................... 26 2.3.1 Analytical Approach ................................................................................................. 27 2.3.2 Numerical Approach ................................................................................................. 28 2.3.3 Validations and Comparisons ................................................................................... 30 2.4 Analytical Model for Viscoelastic Composites During cure ........................................ 33 2.4.1 Modelling Approach ................................................................................................. 34 viii  2.4.2 Validation .................................................................................................................. 37 2.4.2.1 Experimental data ............................................................................................. 37 2.4.2.2 Comparison ....................................................................................................... 40 2.5 Summary and Conclusions ........................................................................................... 42 Chapter 3: Macro-Mechanical Modelling of Generally Orthotropic Viscoelastic  Composites ....................................................................................................................................46 3.1 Introduction ................................................................................................................... 47 3.2 Review of Literature on Viscoelastic Models and Solution Techniques ...................... 51 3.2.1 Integral Form (IF) of Viscoelasticity ........................................................................ 52 3.2.2 Differential Form (DF) of Viscoelasticity ................................................................ 57 3.3 Development of Viscoelastic Constitutive Equations for Orthotropic Materials ......... 60 3.3.1 Formulation ............................................................................................................... 60 3.3.2 Numerical Implementation ....................................................................................... 66 3.4 Verification Examples .................................................................................................. 68 3.4.1 Isotropic Material under Shear Stress ....................................................................... 68 3.4.2 Isotropic Material under Uniaxial Stress .................................................................. 72 3.4.3 Transversely Isotropic Material under Thermal Loading ......................................... 74 3.4.4 Orthotropic Material under Uniaxial Stress .............................................................. 76 3.5 Reverse Micromechanics and Application to Process Modelling ................................ 78 3.6 Summary and Conclusions ........................................................................................... 79 Chapter 4: Micro-Mechanical Modelling of Strand-Based Composites .................................81 4.1 Background ................................................................................................................... 81 4.2 Elastic analysis .............................................................................................................. 85 ix  4.2.1 Analytical Approach ................................................................................................. 85 4.2.1.1 Full Resin Coverage .......................................................................................... 88 4.2.1.2 Partial Resin Coverage ...................................................................................... 99 4.2.2 Numerical Approach ............................................................................................... 103 4.2.3 Comparison of Analytical and Numerical Predictions ........................................... 107 4.3 Viscoelastic Analysis .................................................................................................. 114 4.3.1 Analytical Approach ............................................................................................... 114 4.3.2 Numerical Approach ............................................................................................... 115 4.3.3 Comparison of Analytical and Numerical Predictions ........................................... 117 4.4 Summary and Conclusions ......................................................................................... 120 Chapter 5: Multi-Scale Modelling of Strand-Based Composites ..........................................122 5.1 Development of the Multi-scale Modelling Framework ............................................ 122 5.2 Preprocessing Step ...................................................................................................... 124 5.3 Micro-mechanical Step ............................................................................................... 128 5.3.1 Numerical Approach ............................................................................................... 128 5.3.2 Analytical Approach ............................................................................................... 130 5.3.3 Results ..................................................................................................................... 131 5.3.3.1 Validation and Comparison between Experimental Data and Predictions ..... 131 5.3.3.2 Comparisons between Numerical and Analytical Predictions ........................ 133 5.4 Meso-mechanical Step ................................................................................................ 137 5.5 Macro-mechanical Step .............................................................................................. 139 5.5.1 Strand Orientation Distribution (in-plane and through-thickness) ......................... 139 5.5.2 Flexural Modulus Calculation................................................................................. 142 x  5.6 Validation .................................................................................................................... 145 5.6.1 Experimental Data from Literature ......................................................................... 145 5.6.2 MOE Predictions and Comparisons with Experimental Data ................................. 148 5.7 Applications ................................................................................................................ 150 5.7.1 Optimization of OSB Panels ................................................................................... 150 5.7.2 Reliability Analysis of PSL Beams......................................................................... 152 5.7.3 Creep Modelling of PSL Beams ............................................................................. 154 5.8 Summary and Conclusions ......................................................................................... 162 Chapter 6: Conclusions and Future Work ..............................................................................163 6.1 Summary ..................................................................................................................... 163 6.2 Conclusions ................................................................................................................. 165 6.3 Contributions............................................................................................................... 168 6.4 Future Areas of Research ............................................................................................ 169 Bibliography ...............................................................................................................................172 Appendices ..................................................................................................................................189 Appendix A Multi Scale Modelling Approach and Computational Homogenization ............ 189 A.1 Brief Review of Multi-scale Modelling Approach for Composite Materials ......... 189 A.2 Computational Homogenization Technique ........................................................... 194 A.3 Stiffness Matrix Computation ................................................................................. 195 A.4 Periodic Boundary Condition (PBC) ...................................................................... 196 A.5 Implementation in ABAQUS® using Python Scripting ......................................... 198 Appendix B Micro-Mechanical Elastic Analysis of Circular Fibre Composites .................... 202 B.1 Introduction ............................................................................................................. 203 xi  B.2 Analytical Approach ............................................................................................... 213 B.3 Numerical Approach ............................................................................................... 228 Unidirectional (Hexagonal and Square packing) ........................................................ 229 Unidirectional with Waviness (Hexagonal packing) .................................................. 233 B.4 Validations and Comparisons ................................................................................. 239 Appendix C Measured Elastic Properties of Different Composites ....................................... 255 Appendix D Finite Element Formulation Based on (Zobeiry 2006) ...................................... 260 Appendix E Integral Form (IF) of Viscoelasticity in ABAQUS® ......................................... 263  xii  List of Tables  Table 2-1 Summary of well-known micromechanical models which have been the basis for other developed models in the literature. ............................................................................................... 21 Table 2-2 Constituents’ properties for AS4/3501-6 composite as reported in (White and Kim 1998). ............................................................................................................................................ 30 Table 2-3 Viscoelastic properties of 3501-6 epoxy (Prony series parameters for the shear and bulk moduli) obtained from neat resin testing (Kim and White 1996). ........................................ 30 Table 2-4 Material properties of fibre, resin and fibre bed used for comparison. ........................ 41 Table 3-1 Viscoelastic properties of the transversely isotropic material given in (Zobeiry 2006) ........................................................................................................................................................ 74 Table 3-2 Viscoelastic properties of the transversely isotropic material employed by Zobeiry (Zobeiry 2006). ............................................................................................................................. 75 Table 3-3 Viscoelastic properties of the orthotropic material taken from (Poon and Ahmad 1998)........................................................................................................................................................ 77 Table 4-1 Parameter   used in calculating the elastic moduli of rectangular shaped short fibre composites..................................................................................................................................... 86 Table 4-2 Constituents’ elastic properties of wood and resin. .................................................... 108 Table 4-3 Resin viscoelastic properties (Prony series parameters for the shear and bulk moduli)..................................................................................................................................................... 116 Table 5-1 Longitudinal Young’s modulus of wood species in Figure 5-4. Other elastic moduli may be calculated using the coefficients representing their value with respect to the longitudinal Young’s modulus (Forest Products Laboratory 1999) ................................................................ 136 xiii  Table 5-2 Panel composition as given in (Chen et al. 2010). Note that the values are the overall value and vary through the layers depending on the layer compaction ratio.............................. 146 Table 5-3 Unrelaxed and relaxed values of components of the relaxation matrix for a PSL unit cell. .............................................................................................................................................. 158 Table 5-4 Prony series parameters obtained from curve fitting of the numerical results ........... 159 Table B-1 Fibre geometry parameter in the Halpin-Tsai equations for circular short fibre composites as suggested by Halpin and Kardos (Halpin and Kardos 1976) and Tucker and Liang (Tucker and Liang 1999). ........................................................................................................... 224 Table B-2 Engineering constants for AS4/3501-6 composite (fibre volume fraction of 0.60). Experimental data and predictions of micromechanical models (Numerical and Analytical) are given for comparison. Constituents’ properties are given in Appendix C. ................................. 241 Table B-3 Engineering constants for Boron/Aluminum composite (fibre volume fraction of 0.47). Experimental data and predictions of micromechanical models (Numerical and Analytical) are given for comparison. Constituents’ properties are given in Appendix C. ........................... 242 Table B-4 Engineering constants for E-Glass/Epoxy composite (fV 0.54). Experimental data and predictions of micromechanical models (Numerical and Analytical) are given for comparison. Constituents’ properties are given in Appendix C. ................................................ 242 Table B-5 Numerical predictions of engineering constants for AS4/3501-6 composite at different volume fraction assuming hexagonal packing of fibres. Constituents’ properties are those of Garnich and Karami (Garnich and Karami 2004) and  given in Appendix C. ........................... 244 Table B-6 Numerical predictions of engineering constants for AS4/3501-6 composite with different waviness ratios assuming hexagonal packing of fibres and fibre volume fraction of 0.66 xiv  (Garnich and Karami 2004). Predictions using two different unit cells are compared. Constituents’ properties are given in Appendix C. ..................................................................... 249 Table B-7 Numerical predictions of engineering constants for AS4/3501-6 composite at different waviness assuming hexagonal packing of fibres and fibre volume fraction of 0.66 (Present Work). Predictions using two different unit cells are compared. Constituents’ properties are given in Appendix C. .................................................................................................................. 250 Table B-8 Constituents’ elastic properties for a typical thermoplastic composite reported by Tucker and Liang (1999) ............................................................................................................ 252  xv  List of Figures  Figure 1-1 Schematic of the proposed multi-scale modelling approach for viscoelastic modelling of orthotropic composites ............................................................................................................. 10 Figure 2-1 Various micromechanics models at a glance. Note that this figure only depicts a schematic overview of the best known micromechanics models and it is not a comprehensive categorization. The complexity and computational time of different models may vary. ............. 22 Figure 2-2 Comparison between numerical (hexagonal packing) and analytical predictions of viscoelastic properties for AS4/3501-6 composite at different volume fractions. (a) Young’s moduli; (b) Shear moduli; (c) Poisson’s ratios. Constituents’ elastic properties and viscoelastic properties of the resin are taken from (White and Kim 1998) and are given in Table 2-2 and Table 2-3, respectively. Young’s and shear moduli of the composite are normalized by the glassy (unrelaxed) values of the resin. ..................................................................................................... 32 Figure 2-3 Comparison between numerical and analytical predictions of the effective coefficient of thermal expansion for AS4/3501-6 composite at different volume fraction. (a) Longitudinal direction; (b) Transverse direction. Constituents’ properties are taken from (White and Kim 1998) and are given in Table 2-2. ................................................................................................. 33 Figure 2-4 Prepreg deformation under shear stress. (a) Deformation of fibre bed along with resin. (b) .................................................................................................................................................. 35 Figure 2-5 Proposed analog representation for modelling thermoset composites during cure ..... 36 Figure 2-6 Modulus development during cure for MTM45-1 resin and prepreg. Only high confidence data are shown. The storage modulus is plotted on a logarithmic scale. ................... 39 xvi  Figure 2-7 Comparisons of experimental data with different models’ predictions. DMA data are magnified on the top left. Note that both axes are logarithmic scales. ......................................... 41 Figure 3-1 (a) Maxwell model. (b) Kelvin model......................................................................... 48 Figure 3-2 Generalized Maxwell elements. .................................................................................. 49 Figure 3-3 Comparison between DF and IF solutions for the shear stress in a 1-D problem (pure shear loading) under 3 different applied strain-time functions: (i), (ii) and (iii). The viscoelastic behaviour of the material was characterized by (a) Case I, single element Prony series given in (ABAQUS Inc. 2010), and (b) Case II, multiple Prony series given in (Kim and White 1996). The predicted stress (shear stress) at t = π/2 sec and 109 min, for cases I and II, respectively is shown as a function of the time step Δt used in the numerical computation. ............................... 70 Figure 3-4 Isotropic rod under a constant uniaxial stress (creep condition). (a) Comparison of the longitudinal strain, as predicted numerically (with both DF and IF), and analytically using ABAQUS® automatic time stepping algorithm. (b) Comparison of the longitudinal strain predictions at t = 30 sec using fixed time steps (strains are normalized by the exact value,   ). . 73 Figure 3-5 (a) Transversely isotropic rod under temperature change. Comparison of numerical predictions (using DF) and analytical solution for (b) longitudinal stress, and (c) transverse stress. A constant time step size of 1 sec is used. ......................................................................... 76 Figure 3-6 An orthotropic rod subjected to an applied uniaxial stress in the longitudinal direction, and free of stress in other directions (Poon and Ahmad 1998). Comparison of numerical predictions (using DF) and analytical solution for strains in the longitudinal and transverse directions. ...................................................................................................................................... 78 Figure 4-1 Geometrical characteristics of wood strands. .............................................................. 82 Figure 4-2 (a) Small sample of PSL (Parallel Strand Lumber) beam and (b) PSL cross-section. 83 xvii  Figure 4-3 A unit cell of a strand-based composite. For clarity, the resin blocks are shown detached from the strand. .............................................................................................................. 87 Figure 4-4 Micromechanical model used for normal properties. (a) Isostress condition between a strand and two resin blocks in front and back and (b) Isostrain condition between the homogenized new strand and the four resin blocks surrounding it. ............................................. 89 Figure 4-5 Micromechanical model used for shear properties. (a) Isostress condition between embedded strand and resin surrounding it. (b) Isostrain condition between the homogenized new strand and two resin blocks in front and back. .............................................................................. 92 Figure 4-6 Stress-strain analysis in a material unit cell for determining     and    . (a) A unit cell under normal between the resin and the strand inside; and (c) Isostress condition between the front and rear resin blocks and the homogenized new strand in the middle.stress; (b) Isostrain condition in the longitudinal direction and isostress condition in the transverse directions ........ 94 Figure 4-7 Stress-strain analysis in a material unit cell for determining    . (a) A unit cell under transverse stress; (b) Isostress condition between top and bottom resin blocks and the strand inside; and (c) Isostrain condition between the side resin blocks and the homogenized new strand in the middle. ................................................................................................................................ 98 Figure 4-8 Incorporating interface properties for modelling the effect of partial resin coverage. (a) Shear stress; and (b) Normal stress. ...................................................................................... 101 Figure 4-9 Identifying the material unit cell  (a) PSL meso-structure; (b) idealized meso-structure; (c) material unit cell; (d) discretized unit cell. ............................................................ 104 Figure 4-10 Resin area coverage as a linear function of resin content for partial resin coverage, scenario 1 (constant resin thickness)........................................................................................... 105 xviii  Figure 4-11 Resin area coverage as a function of resin content according to Dai’s model (Dai et al. 2007) for partial resin coverage, scenario 2 (variable resin thickness). ................................. 106 Figure 4-12 Numerical results for longitudinal Young’s modulus under partial resin coverage (constant and variable resin thicknesses) using void elements versus full coverage. ................. 109 Figure 4-13 Numerical results for partial resin coverage (Dai’s model) using void elements showing void type effect on longitudinal modulus at different resin contents. Voids Type I have a volume range of 0.1–1mm3 while the volume of Voids type II varies in the range of 1–10mm3 depending on the resin content of the unit cell. The solid curve shows the resin area coverage variation with resin content. ........................................................................................................ 110 Figure 4-14 Effect of resin content on the longitudinal Young’s modulus considering two scenarios (constant and variable thicknesses) for partial coverage: (a) Using void elements. (b) Using equivalent resin properties. Finite element results are shown with symbols. Solid lines refer to estimates using equations (4.6) and (4.36). Dashed lines refer to estimates using equations (4.11) and (4.36). ........................................................................................................ 111 Figure 4-15 Comparisons between analytical and numerical results assuming constant resin thickness and using equivalent resin properties (equations (4.31) and (4.36)) for partial coverage. (a) Transverse Young’s Moduli; and (b) Shear moduli. Finite element results are shown with symbols. Solid lines refer to estimates using a two-step approach with successive isostress and isostrain conditions equations (4.6) and (4.9). Dashed lines refer to estimates using a two-step approach with successive isostrain and isostress conditions (equations (4.11) and (4.12)). ...... 112 Figure 4-16 Comparisons between analytical and numerical results assuming variable resin thickness (Dai’s model) and using equivalent resin properties (equations (4.31) and (4.36)) for xix  partial coverage. (a) Transverse Young’s moduli. (b) Shear moduli. Symbols, solid and dashed lines have the same significances as those in Figure 4-15. ......................................................... 113 Figure 4-17 Comparison between analytical (lines) and numerical (symbols) longitudinal relaxation moduli for full resin coverage, using resin viscoelastic properties. Resin modulus    and numerical reference transverse relaxation moduli (   and   ) are given for comparison. Halpin-Tsai prediction for    (dotted dashed line) overlaps with current estimate (Equation (4.6)). .................................................................................................................................................. 118 Figure 4-18 Comparisons between analytical (lines) and numerical (symbols) relaxation moduli for full resin coverage, using resin viscoelastic properties. (a) Transverse Young’s moduli; and (b) Shear moduli. Resin relaxation shear modulus    is added for comparison. Halpin-Tsai predictions are presented with dotted and dashed lines. ............................................................. 119 Figure 4-19 Comparisons between analytical predictions (solid lines) and numerical results (symbols) for full resin coverage, using resin viscoelastic properties; Poisson’s ratios. ............ 119 Figure 5-1 Steps involved in the multi-scale analysis framework showing the input and output parameters for each step.............................................................................................................. 124 Figure 5-2 Unit cell of softwood: a) schematic representation, b) cell wall parameters, c) discretized unit cell ..................................................................................................................... 124 Figure 5-3 Comparisons between the model predictions (both analytical (Gibson and Ashby 1999) and numerical (Gereke et al. 2011)) and experimental data (Forest Products Laboratory 1999) for longitudinal modulus of different wood species: (a) softwoods and (b) hardwoods .. 132 Figure 5-4 Comparisons between the model predictions (both analytical (Gibson and Ashby 1999) and numerical (Gereke et al. 2011)) and experimental data  (Forest Products Laboratory xx  1999) for Young’s modulus in (a) radial direction,  (b) tangential direction and shear modulus in (c) LT plane, and (d) RT plane for different softwoods ............................................................. 133 Figure 5-5 Elastic moduli of a hypothetical wood strand with regular hexagonal cellular microstructure (  L cwE = 35 GPa, cw = 1500 kg/m3,  = 30°) under various level of compaction. (a) Young’s moduli (b) Shear moduli ......................................................................................... 135 Figure 5-6 (a) Cross-section of the material, and (b) Material unit cell at the meso-scale ......... 137 Figure 5-7 Schematic representation of the various input and output quantities in the meso-mechanical step ........................................................................................................................... 138 Figure 5-8 Dividing the panel into multiple layers (a) real meso-structure, and (b) idealized meso-structure. Each layer has a different density as denoted by colors .................................... 140 Figure 5-9 Incorporation of strand orientation in the macro-mechanical step of the framework (a) top view of the real panel, (b) top view of the idealized panel, and (c) cross-section (side view) of the idealized panel....................................................................................................................... 141 Figure 5-10 Vertical panel density profiles for five panels with different overall densities. Data are extracted from (Chen et al. 2010) ......................................................................................... 147 Figure 5-11 Three-point-bending test on the panels. Strands in face layers are (a) parallel, and (b) perpendicular to the beam longitudinal axis ............................................................................... 148 Figure 5-12 Comparison between current predictions (both analytical and numerical) and experimental results (filled triangles). Predictions using the numerical approach (at micro- and meso-scale) and the analytical approach (at all three scales) are shown with hollow squares and filled circles, respectively ........................................................................................................... 149 Figure 5-13 Schematic of Young’s and shear modulus vs. layer density of OSB panels obtained from analytical equations. ........................................................................................................... 151 xxi  Figure 5-14 Vertical Density Profile (VDP) of an OSB panel. Panel with the optimized density profile has an overall density that is 12% lower than the as-manufactured one while maintaining the same MOE of 8.05 GPa ........................................................................................................ 152 Figure 5-15 Schematic of the procedure used in reliability analysis of a PSL beam ................. 153 Figure 5-16 MOE Probability density function of a PSL beam (a) Case I (b) Case II ............... 154 Figure 5-17 Variation of the effective relaxation matrix components ijC versus time obtained from numerical analysis of PSL unit cell. Symbols denote numerical results and lines are the associated fitted curves. .............................................................................................................. 157 Figure 5-18 Strand orientation in the PSL beam consisting of random strands according to (Gereke et al. 2012)..................................................................................................................... 160 Figure 5-19 Creep curve for PSL beams consisting of aligned strands and random strands...... 161 Figure A-1 Directions and global coordinate systems at different resolution levels; (a) Structure (macro-scale) and (b) Repeating unit cell (micro-scale) representing the periodic micro-structure of the material. For more details of the unit cell and the assumed periodic micro-structure of different composites see Chapter 2 and Chapter 4...................................................................... 197 Figure A-2 Schematic overview of the implementation of numerical approach for predicting the unit cell effective properties in ABAQUS®. Python scripts denoted by “.py” extension. ......... 200 Figure B-3 Idealized cross-sections of unidirectional long fibre reinforced composites and their smallest repeating unit cells assuming: (a) square packing (SQR), and (b) hexagonal packing (HEX) of fibres. .......................................................................................................................... 230 Figure B-4 Unidirectional long fibre composite. Discretized unit cell of (a) Square (b) Hexagonal packing at fibre volume fraction of 0.6. ...................................................................................... 232 xxii  Figure B-5 Wavy unidirectional long fibre composite. (a) idealized periodic microstructure, and (b) repeating unit cell .................................................................................................................. 233 Figure B-6 Wavy unidirectional long fibre composite. Discretized wavy unit cell assuming hexagonal packing of fibres. ....................................................................................................... 234 Figure B-7 Wavy unidirectional long fibre composite. Discretized straight unit cell with incorporated material orientation assuming hexagonal packing of fibres. Local coordinates are defined to represent the material orientation inside the straight unit cell. .................................. 236 Figure B-8 Schematic of short fibre composite unit cell. ........................................................... 237 Figure B-9 Discretized unit cell of a short fibre composite. ....................................................... 238 Figure B-10 Unidirectional long fibre composite. Comparison between numerical and analytical predictions of effective elastic properties for AS4/3501-6 composite at different volume fractions. (a) Young’s moduli; (b) Shear moduli; (c) Poisson’s ratios. Constituents’ properties are those employed by (Garnich and Karami (Garnich and Karami 2004) and are given in Appendix C. ................................................................................................................................................. 245 Figure B-11 Unidirectional long fibre composite. Comparison between numerical and analytical predictions of effective transverse modulus for AS4/3501-6 composite using resin relaxed modulus. Constituent’s properties are those reported in (White and Kim 1998). ...................... 247 Figure B-12 Short fibre composite. Comparison between numerical and analytical predictions of effective elastic properties for the composite at different fibre aspect ratios and at a constant volume fraction of 0.2. (a) Young’s moduli; (b) Shear moduli. Constituents’ properties are typical of fibre reinforced thermoplastics which are employed by Tucker and Liang (1999) and are given in Table 6..................................................................................................................... 252 xxiii  Figure B-13 Short fibre composite. Comparison between numerical and analytical predictions of effective elastic properties for the composite at different fibre aspect ratios and fibre volume fractions. (a) Young’s moduli; (b) Shear moduli. Constituents’ properties are typical of fibre reinforced thermoplastic composites which are employed by Tucker and Liang (1999) and are given in Table B-8. ..................................................................................................................... 254  xxiv  Acknowledgements  My first and sincere thanks go to my supervisor, Professor Reza Vaziri for his enormous amount of time and support in the past 4 years. Without his direction, this work couldn’t have been possible. I also thank my co-supervisors Professor Anoush Poursartip and Professor Fernand Ellyin whose penetrating questions taught me to question more deeply. I owe particular thanks to Professor Carole Nadot-Martin at Pprime Institute for all her support and the encouragement she gave me during these years both in Vancouver and in Poitiers, France. Special thanks to Professor Ricardo Foschi in the supervisory committee for his feedback on strand-based wood composites modelling.  I acknowledge NSERC for providing me with CGSD scholarships, and Michael Smith Foreign Study Supplements in France. Additionally, I acknowledge The University of British Columbia’s for supporting me financially in the Four Year Fellowship (4YF) program.  Furthermore, I should thank Dr. Thomas Gereke (TU Dresden), Dr. Navid Zobeiry, Mr. Benjamin Tressou and Mr. Ryan Thorpe for their help in the past 4 years. Without their support this work wouldn’t have been possible. I offer my enduring gratitude to the faculty, staff and my fellow students at UBC, and CRN who have inspired me to continue my work in this field. Many thanks to my lovely sister, Nazanin, my great brother, Salar, and many dear friends including Alireza Forghani, Kamyar Gordnian, Leyla Farhang, Mehdi Haghshenas, Saeed Allahdadian, Ehsan Vahedi, Pooya Taheri, Sarmad Mehrbod, Sara Hormozi, Hamed Ghasvari Jahromi and many others for their encouragement during my PhD program.   xxv  My deepest thanks go to my beloved parents, who have always supported me throughout my life, and encouraged me to follow my passions. I am immensely grateful for the love and support of my best friend of all times, Fahime, who has believed in me, listened to my concerns, helped me through hardships, and constantly encouraged me in the past year. Finally, I dedicate this work to my father who passed away in 2012. His sacrifices paved the way for my success in life.   xxvi  Dedication  To the loving memory of my father “Moossa”  1  Chapter 1: Introduction  New composites are increasingly being used in different engineering applications. This is due to the need of some industries to use materials with tailorable properties in their design, e.g. high specific strength or specific stiffness at reasonably low cost compared to conventional engineering materials. For instance, woven composites and strand-based wood composites are increasingly being used for structural applications in aerospace and construction industries due to their unique characteristics such as higher impact, stiffness and strength properties than other competing materials.  Despite these unique characteristics, such issues as durability of these materials and poor dimensional control in manufacturing large composite structural components present challenges in their industrial applications. These issues can be treated back to the microstructural interactions between constituent phases of composites. In fact, a composite material consists of at least two distinctly different phase materials with different properties. Polymers are often used as one of the two (or more) constituent phase materials, in composites. Due to time-dependent properties of polymers, composite parts exhibit distinct time-dependent or viscoelastic behaviour during their manufacturing process as well as during their service life. The viscoelastic behaviour of composites requires special attention and care when these materials are employed in structural applications.  Several tools have been developed to analyse the viscoelastic nature of composites at their various stages from manufacturing to in-service. Researchers have developed different process 2  models to simulate the behaviour of composites during their manufacturing process. Accurate prediction of the behaviour of the composite part during its manufacturing process is very important in applications where the final shape of the manufactured part should meet certain criteria on shape or dimensional tolerances. This is crucial for automotive and aerospace industries. During manufacturing of thermoset composite parts, the resin evolves from a low viscosity fluid to a highly cross-linked viscoelastic solid. In this transition, residual stresses develop gradually. Manufactured composite parts that contain residual stresses may lead to undesired curvature or warping, poor dimensional control, or degraded mechanical load capacity during the service life of the structure. Advancements in process modelling have helped to minimize these effects through accurate predictions of the final shape and mechanical properties of large composite parts (Johnston et al. 2001). However, there are still limitations and gaps in using current process models for composites with complex microstructures which are addressed in this thesis.   Once the composite part has been manufactured, it is very important to analyse its response under complex loading conditions as a structural member. Depending on the application, different types of loading such as impact, quasi-static load (e.g. snow load) or thermal load (heat) may be applied to a structural member during its service life. Predictive tools can be very helpful in estimating the behaviour of a structure under critical situations such as hail impact on an aeroplane wing, heavy snow load on a composite beam or occurrence of repetitive heating-cooling cycles during the life of an oriented strand board in a building.  3  Although the viscoelastic behaviour of a composite part may be quite different during or after its manufacturing, still the knowledge of the effective (homogenized) properties is required in determining the overall response of the composite structure (e.g. an aeroplane wing or a simple beam made of engineered wood composite) at each stage. Measuring such properties can be quite tedious and time-consuming. Moreover, there are several variables and combination of materials which should be considered before manufacturing and testing a composite part for a specific purpose. In other words, designing a new composite material may require significant effort and experimental characterizations.  Many researchers have attempted to develop analytical micromechanics models for the purpose of predicting the effective properties of heterogeneous materials such as composites. Generally, there are some approximations involved in these micromechanical models and they have certain inherent limitations such as the degree of mismatch between constituents’ properties, the special geometry and volume fraction of fibres, etc. Most analytical closed-form solutions are available for solid composites with “elastic” isotropic or transversely isotropic phase properties, and they provide good predictions of the effective elastic properties at “low to medium” volume fraction of fibres with prescribed geometries. Therefore, their application to new composites (e.g. advanced fibre-reinforced composite with an additional toughening phase layer or wood composites with high volume fraction of orthotropic strands that are partially covered with resin) or even in manufacturing conventional unidirectional fibre composites which exhibit significant time-dependent (viscoelastic) behaviour requires revisiting the accuracy range of the available analytical micromechanics equations. Some of the most well-known micromechanical models 4  employed for fibre-reinforced composites and their historical development are reviewed in Chapter 2 and Appendix B.  As mentioned above, predicting the effective or homogenized properties of composites at the micro-scale (scale of constituents) during their manufacturing process as well as their service life is very important. For some industries such as the construction industry, accurate predictions at the first stage (e.g. manufacturing beams with accurate dimensions) may not be as important as the second stage (in-service behaviour). However, the effective properties are still required to determine the structure’s response, i.e. macro-scale analysis, under complex loads.  For problems involving nonlinear effects and time-dependent properties, analysing the structure’s response (macro-scale analysis) is more complicated than linear elastic problems. Using results of micro-scale analysis of macro-scale analysis for generally orthotropic composites requires specific treatments at both the micro-scale and the macro-scale which will be addressed in this thesis.   1.1 Motivation The motivation of the present work comes primarily from the needs of the aerospace and wood composites industries where large, composite structural members are manufactured. Composite materials with various microstructures and properties are increasingly being developed to meet the needs of these industries. With the move towards knowledge based manufacturing of composite structures, having predictive tools which are capable of relating the material properties at different scales to structural response of composites is desirable. 5   In aerospace industry, process modeling has been an increasingly successful strategy for understanding and predicting the behaviour of composite structures during manufacturing. In current generation of process models, the flow response of the resin is dealt with separately from the solid response (stress development) phase. During the pre-gelation stage, the resin flows in the presence of pressure gradients and is simulated using flow models based on Darcy’s Law. During the post-gelation stage where crosslinking of the resin occurs, resin flow stops and stresses begin to develop as a result of property mismatch between the fibre and the resin. At the latter stage the solid behaviour becomes important and the material response is simulated using modulus and stress-development models (Hubert et al. 1999; Johnson 1999; Johnston et al. 2001).   There are a few problems associated with the above approach such as inherent inefficiency and inability to capture the interaction of the resin flow with the stress development in the composite laminate, which have been addressed by Haghshenas (Haghshenas 2012). The next generation of integrated process models (e.g. (Haghshenas 2012)) aim to cover the full range of resin behaviour. Having an integrated simulation capability enables the analysis of the composite behaviour continuously during the complete cure cycle. One of the key issues in an integrated process modelling approach is the use of a correct and consistent formulation for all material properties from an uncured to cured state. As discussed in Chapter 2 and shown in Appendix B, analytical micromechanics equations have been developed and used with great success for predicting the effective elastic properties of the solid unidirectional circular fibre composites. However, their application in predicting the effective viscoelastic properties of composite or cases were one 6  phase is much softer than the other phase has not been investigated fully. As shown in Chapter 2, these equations are unable to predict the shear modulus of the prepreg in the early stages of cure where most of the deformation occurs. Therefore, the application of these equations in predicting the effective viscoelastic properties of circular fibre composites is investigated in Chapter 2.  As new composite materials with more complex microstructures (compared to conventional unidirectional circular fibre composites) are being developed, the application of analytical micromechanics models becomes even more questionable. Woven composites, composites with a toughening phase layer as well as strand-based composites where the fibre (strand) has a rectangular cross-section are examples of such composites. Most of these composite materials have complex microstructure and show directionally-dependent (orthotropic) properties which cannot be dealt with using classical micromechanics models. To deal with these new materials either available analytical micromechanics need to be modified or new micromechanics models (numerical or analytical) should be developed. Moreover, to predict the structural response of composites exhibiting time-dependent characteristics special treatment of constitutive models is required.  Here, as an example, we consider strand-based wood composite as a non-conventional composite due to its (i) complex hierarchical microstructure, (ii) significant time-dependent and moisture dependent characteristics, and (iii) increasingly wide application in building constructions. Strand-based wood composites including Parallel Strand Lumber (PSL), oriented strand board (OSB) and oriented strand lumber (OSL) are increasingly being used in structural applications.   7  Strand-based composites are structural materials manufactured through combining orthotropic wood strands with small amount of adhesive (below 5%). The adhesive (e.g. a thermoset resin) glues the strands together. There are advantages associated with strand-based wood composites (e.g PSL) over solid wood. The structural members made from these composites are often considered to be stronger, more reliable and more dimensionally stable than solid sawn lumber (Forest Products Laboratory 1999; Fridley 2002). The use of such structural composite lumber products as construction materials demands certain requirements of their mechanical properties be known such as stiffness and strength as well as some serviceability requirement such as the allowable amount of creep due to moisture, heat or long duration of loads (e.g. snow loads).  There are many parameters and variables that affect the mechanical properties of strand-based wood composites, such as compression ratio, strand size and orientation distribution, wood species, resin type and content, void content, moisture content, etc. It is desirable to have tools that enable analysis of these wood composites under typical loading conditions considering all these parameters. Currently, due to large variability of parameters, the wood composite industry mostly relies on empirical models (e.g. (Barnes 2000)) rather than mechanistic models for this purpose. To calibrate these models, extensive experimental tests are required. Furthermore, for each new type of composite a new set of tests should be conducted.   A viable alternative to time-consuming and costly experiments is to develop a multi-scale modelling strategy which relates microstructural parameters (e.g. strand size) to the time-dependent response (e.g. creep) of such composite structures. A comprehensive multi-scale modelling approach can also serve as a tool to control and optimize the macroscopic material 8  behaviour considering costs. As an example, manufacturing costs of a wood composite part is a very important issue for wood composite industry. The resin employed in the manufacturing of wood composite products is quite costly compared to wood itself. It has also been realized by industry that the amount of resin affects the structural properties of these materials nonlinearly at low range of resin application. Therefore minimizing the manufacturing costs, while maximizing the performance of a wood composite structural member, can be quite beneficial for the construction industry.  To date there has been no mechanistic model capable of predicting even elastic properties of wood composite structural members based on the properties of both the wood and resin phases. A model that can serve as a tool for investigating the effects of several parameters and their variation on the overall response of a structural component made of these materials would be desirable.   Several multi-scale models have been developed to take into account the hierarchical nature of composites and solving problems arising from the complex microstructure of new composites. The concept of multi-scale modelling, its origins, recent developments and current challenges in this field have been discussed in Appendix A. Currently, multi-scale modelling is a well-established approach in literature for analysing composite materials. However, a multi-scale approach still needs to be efficient enough to be employed for analysing the behaviour of large composite structures in practice. Efficiency becomes a particularly important issue in those applications where time-dependent (viscoelastic) response of the structure is of interest such as process modelling or predicting the long-term behaviour (creep) of composites. In addition to the efficiency requirement for a multi-scale modelling tool, accuracy is also important for new 9  composites with directionally dependent (orthotropic) properties that are being developed for aerospace and construction applications as discussed in the previous section.  1.2 Research Objectives To address the current challenges in composite industry, development of an efficient and easy-to-implement multi-scale modelling approach which considers the orthotropic nature of new viscoelastic composites is the primary goal of this thesis. In the development of this approach special attention has been paid to the techniques employed at each step so that the presented multi-scale approach becomes more general and versatile. Therefore, it can be employed not only for unidirectional fibre-reinforced composites or strand-based wood composites, but for the efficient structural analysis of new composites that possess more complex microstructures.   1.3 Research Scope and Goals The multi-scale approach presented in this thesis is based on other well-established approaches and models and involves analyses at two major resolution levels; namely, micro-level and macro-level as depicted in Figure 1-1. It should be noted that throughout this thesis, the macro-scale is the scale at which we are interested in determining the structural response. Scales below the macro-scale such as the meso- and micro-scale may be identified for different composites. Lower scales, in general, are referred to as micro-scale throughout this thesis unless stated otherwise.    10   Figure 1-1 Schematic of the proposed multi-scale modelling approach for viscoelastic modelling of orthotropic composites Micro-structural Parameters (e.g. )Constituents’ Properties (e.g. )**Generate Unit CellDiscretize Unit CellApply Strain andPBCs ***Solve Microscopic ProblemExtract Average StressesCompute  Effective Stiffness Matrix (time domain)Fit Prony Series To All Components of Stiffness MatrixGenerate Structure GeometryApply Loads and BCsExtract Structure’s responseSolve Macroscopic ProblemDiscretize StructureApply inverse LaplaceApply Laplace TransformUse Analytical Closed-form Equations (s-space)Obtain Composite’s Stiffness Matrix (s-space)Macro-ScaleMicro-Scale** For multi-level composites such as wood composites, Micro-Scale may refer to the scale of wood                                cellular structure and this step is called Meso-Scale (see Chapter 5)** Constituent properties may be obtained either from experiments or another lower-scale analysis (e.g. Nano-Scale analysis)*** Periodic Boundary ConditionsNumerical Approach (Performed with Python® scripts in Abaqus®)Analytical Approach(Performed in Mathematica®)(Transition)InputsStrainsStressesOrthotropic UMATCalculate Maxwell StressesCalculate Material Stiffness Matrix11  To address the efficiency of viscoelastic analysis and filling the gaps of available models at the two main scales (i.e. micro- and macro-scale), the following tasks have been undertaken:   Reviewing well-known analytical micromechanics approaches and examining the validity range of some of these models in determining the effective elastic and viscoelastic properties of the material at the micro-scale. This can be achieved by comparing the predictions of analytical closed-form equations with numerical reference solutions obtained based on computational homogenization technique.   Implementing the computational homogenization technique in a finite element code (ABAQUS®) to obtain the micro-scale effective elastic and viscoelastic properties of composites consisting of reinforcements with circular and rectangular cross-sections.   Wherever possible, replacing the computationally intensive finite element simulations at the micro-scale with analytical closed-form equations for predicting the effective material properties at this scale.   Modifying classical solid micromechanics models for the purpose of predicting the effective shear properties of thermoset matrix composites during cure. As will be demonstrated, during cure of thermoset composites, the resin evolves from a viscous fluid to a viscoelastic solid and classical micromechanics is unable to predict the effective shear modulus of the composite before gelation.  12   Developing analytical micromechanics equations for highly filled strand-based composites consisting of orthotropic wood strands covered fully with a thin layer of resin. Using the interface parameters, the application of these equations has been extended to strand-based composites consisting of partially covered strands with resin.   Examining the validity of the developed analytical micromechanics equations for predicting the effective properties of strand-based wood composites considering full and partial coverage of strands with resin.   Extending the differential form of viscoelasticity, previously developed for viscoelastic analysis of transversely isotropic materials, to orthotropic composites. As will be discussed, this is a major step in efficient multi-scale viscoelastic modelling of new composites with orthotropic properties.   Implementing the differential form of viscoelasticity for orthotropic composites in a user material subroutine (UMAT) developed for viscoelastic analysis of orthotropic composites in a general purpose finite element code (ABAQUS®).   Evaluating the accuracy of the differential form of viscoelasticity compared to its integral form through some examples. Finite element results obtained using the UMAT and the ABAQUS ® built-in material model (based on the integral form of viscoelasticity) are compared with exact solutions for this purpose.  13   Developing an analytical multi-scale modelling framework for strand-based wood composites. This framework is based on closed-form analytical models at three different resolution levels; micro-, meso- and macro-levels.   Validating the accuracy of this analytical framework in predicting an important structural parameter (MOE) of oriented strand boards using experimental data obtained from the literature.   Creep modelling of a strand-based wood composite beam (PSL) using the multi-scale modelling approach presented in Figure 1-1.  To demonstrate the capabilities of the multi-scale modelling approach for industrial applications, two classes of composites are considered in this work; namely, fibre-reinforced thermoset matrix composites and strand-based wood composites. These two classes of composites have wide range of applications in aerospace and construction industries. If the above goals are achieved, the multi-scale modelling approach presented in Figure 1-1 can be employed for the analysis of structures made from these composites.   1.4 Thesis Structure The thesis is organized into six independent chapters. The current challenges in composite industry and the underlying motivation for the current work are presented in Chapter 1.   14  To establish an efficient multi-scale approach for composites, employing closed-form analytical micromechanics equations for predicting the effective properties of composites can be very beneficial. However, there are numerous models available in the literature and the choice of the best analytical model for the composites that we are interested in this thesis can be very challenging. Therefore, reviewing the bases of such analytical models and the approximations involved for the purpose of employing them in efficient multi-scale modelling of viscoelastic composite structures seems necessary. In Chapter 2, analytical models with the intention of finding the best analytical models that can be employed for predicting the effective elastic, thermo-elastic and viscoelastic properties of circular fibre composites at the micro-scale are studied.   To examine the accuracy of analytical models, available experimental data as well as numerical reference solutions are employed. The numerical reference solutions are obtained using the computational homogenization technique based on full-field microstructural simulations. This technique, together with its implementation in a finite element code and the procedure to obtain the numerical reference solutions are presented in Appendix A.   Using numerical reference solutions, a comprehensive study to investigate the accuracy of selected well-known analytical micromechanics models in predicting the elastic properties of solid composites with circular, long or short, fibres is performed in Appendix B. Extension of these analytical models, aspects which need further investigation and the necessary tasks for predicting effective viscoelastic properties of circular fibre composites are also pursued in Chapter 2. 15    Once the effective micro-scale properties of the composites have been determined, they can be employed in structural analysis of composites at the macro-scale. However, for large structures such as aircraft or buildings and in the case of orthotropic viscoelastic composites, this analysis is not trivial. Viscoelastic analysis of composite structures has been a dynamic research topic for many years. Therefore, in Chapter 3, various aspects in analysing the viscoelastic behaviour of composites will be addressed. Several approaches based on the differential form or the integral form of viscoelasticity and their solution techniques are reviewed. An efficient and easy-to-implement numerical approach based on the differential form of viscoelasticity is presented for viscoelastic orthotropic materials. To illustrate how this approach can be implemented in a general purpose finite element code (e.g. ABAQUS®), a user material subroutine (UMAT), was developed. Some numerical examples performed using this UMAT are included for verification purposes.  In addition to fibre-reinforced composites consisting of circular long or short fibres, the behaviour of a special class of composites, namely, strand-based wood composite, is studied in Chapter 4. This class of composites is increasingly being used for structural applications in mid-rise buildings. An analytical approach based on principles of classical micromechanics is developed for strand-based composites, consisting of fully or partially covered strands with resin. Additionally, the accuracy of this approach in predicting the effective properties of strand-based composites is evaluated using numerical reference solutions.  16  Based on the analytical mathematical equations developed in Chapter 4, a comprehensive analytical multi-scale modelling framework has been developed for strand-based wood composite products such as Oriented Strand Board (OSB) and Parallel Strand Lumber (PSL). This multi-scale modelling framework is presented at three different resolution scales in Chapter 5. The structure of the developed framework, modelling steps and the framework validation are described. Some applications of the multi-scale approach in manufacturing process of these products are presented. Moreover, using the multi-scale modelling approach presented in this thesis, creep of a wood composite structure (a PSL beam) is presented.  Finally, in Chapter 6, contributions, limitations, suggested future works, possible applications and areas to which the presented multi-scale approach can be extended are briefly discussed.  It should be noted that the numerical reference solutions employed in Chapters 2, 4 and 5 were obtained using the computational homogenization technique based on full-field microstructural simulations. This technique, together with its implementation in a finite element code and the procedure to obtain the numerical reference solutions are presented in Appendix A.         17  Chapter 2: Micro-Mechanical Modelling of Circular Fibre Composites  As a first step in multi-scale modelling of viscoelastic composites, the effective viscoelastic properties of composites through a micro-scale analysis is required. Such an analysis is usually referred to as micro-mechanical modelling and involves analysing stress and strain micro-fields which can be quite challenging, especially if they are included within a multi-scale modelling framework. The purpose of this chapter is to explore the most accurate and efficient ways to determine the effective properties of viscoelastic composites with circular fibres so they can be readily used within a multi-scale modelling framework such as that used in process simulation of composite structures employed in aerospace industry.  To analyse composite materials in general, analytical models have been developed by several researchers for predicting their effective elastic properties. The bases of these models and the main assumptions in deriving the analytical micromechanics expressions for predicting the effective elastic moduli and thermal properties (coefficients of thermal expansion) of composites are reviewed in Appendix B (Section B.1). Although numerous models have been developed and been in existence, revisiting their bases and the approximations involved for the purpose of employing them in efficient viscoelastic analysis of composite structures seems necessary. Therefore, a comprehensive study to investigate the accuracy of selected well-known analytical micromechanics models in predicting the elastic properties of solid composites with circular, long or short, fibres has been presented in Appendix B. Some of the findings from this comprehensive study on elastic properties of composites are discussed in Sections 2.1 and 2.2.  18  As discussed in Section B.1 of Appendix B, for UD composites with isotropic or transversely isotropic circular fibres, the application of analytical micromechanics equations in predicting their effective elastic properties has been investigated extensively by several researchers. However, the application and accuracy of such analytical equations for predicting the effective viscoelastic properties of composites have not been examined thoroughly in the literature. Moreover, the ability of these equations in predicting the effective properties of thermoset matrix fibre-reinforced composites at the early stage of cure where the resin is in a fluid state has not been previously investigated. The use of analytical micromechanics equations to represent the smeared response of fibre and matrix is common in current generation residual stress and deformation models that focus on post-gelation resin behaviour. However, the next generation of integrated process models aim to cover the full range of resin behaviour, and thus the utility of the current micromechanics equations must be evaluated in this range. Therefore, in Sections 2.3 and 2.4 two other aspects in micro-scale analysis of UD composites which are very important in process modelling of fibre-reinforced composites will be addressed.  First, the accuracy of selected closed-form analytical equations in predicting the effective viscoelastic and thermal properties of solid viscoelastic composites consisting of UD circular fibres at different fibre volume fractions are examined using numerical reference solutions in Sections 2.3. The employed numerical technique is described in more detail in Appendix A  while its predictive capability is verified in Appendix B. To the author’s knowledge, such a comprehensive examination of these analytical equations for the effective viscoelastic properties of UD composites has not been reported in the literature.  19  Next, for thermoset composites during cure, these equations are modified by incorporation of the fibre bed behaviour in Section 2.4.  This approach and its validity for predicting the effective shear modulus of a thermoset composite during cure are examined using available experimental data (Section 2.4.2). Although the concept of fibre bed has been introduced in the literature, its incorporation in analytical micromechanics equations has not been reported. This is very important in process modelling of thermoset matrix composites in which continuous prediction of shear modulus is desirable.  The outcome of the study presented in this chapter and the companion Appendix B is identification of the most effective analytical micromechanics models that can be employed to replace the computationally intensive numerical models for determining the effective viscoelastic and thermal properties of UD composites consisting of circular fibres. As will be shown in Chapter 3, these effective properties can be employed in higher scale (e.g. meso- or macro-scale) analyses aimed at predicting the viscoelastic response of composite structures efficiently within a multi-scale modelling framework.  2.1 Elastic Analysis Due to the large body of work available on elastic analysis of composites, a comprehensive review of the literature was carried out and reported in Appendix B. Among the different micromechanical models reviewed there, some well-known models developed for elastic analysis of composite materials are summarized in Table 2-1 and shown schematically in Figure 2-1. As more morphological features are included in the model, the computational time increases. It should be noted that in most cases, specific phase geometries and microstructures 20  are assumed. On the far right are models which consider the complexity of the microstructure. While on the far left are simple closed-form equations which are easy to work with and can be implemented in higher level, computationally intensive models. Depending on the application and the required accuracy some of these models can be used interchangeably.                    21  Table 2-1 Summary of well-known micromechanical models which have been the basis for other developed models in the literature.    Voigt arbitrary isotropic isostrain N/A 1889Reuss arbitrary isotropic isostress N/A 1929elasticity/ diluteenergy(Vf < 1%)[1]Composite Sphere Assemblage (CSA)spherical isotropic elasticity< 40% [2]1962Hashin &ShtrikamanComposite Cylinder Assemblage (CCA)continuous long fibrestransversely Isotropicelasticity< 70% [2]1964spherical/ellipsoidal(long & short fibre)spherical ,cylindrical,rectangularMori-TanakaSchemeGeneralized Self-Consistent Schemespherical, (GSC) cylindricalMethod of Cells square short fibrestransversely isotropic equilibrium conditions at the interface< 30% [4]1983[2] Hashin (1983)[4] Aboudi (1983)Eshelby equivalent inclusion method< 30%[1] 1973[3] Halpin & Kardos (1976)  YearSelf-Consistent Scheme (SC)transversely isotropicEshelby equivalent inclusion methodN/A 1965arbitrary isotropicvariational principlesN/A 1963Inclusion GeometryModelMaterial PropertiesBasisVolume Fraction                   Range[1] Tucker & Liang (1999)  Eshelby ellipsoidal isotropic 1957transversely isotropicEshelby equivalent inclusion methodN/A 1979Halpin-Tsaitransversely isotropicSelf-Consistent< 70% [3]1969ellipsoidal anisotropic22   Figure 2-1 Various micromechanics models at a glance. Note that this figure only depicts a schematic overview of the best known micromechanics models and it is not a comprehensive categorization. The complexity and computational time of different models may vary. Micromechanics ModelsMorphological Approach3Numerical Models3Simple Models1Voigt (isostrain) Reuss (isostress)Method of Cells3Finite ElementFinite Difference1Isotropic 2 Transversely Isotropic3 OrthotropicType of Composite:SphericalLong cylindricalLong/short ellipticalFibre Geometry:LV Low (<10%)MV Moderate (<80%) HV High (>80%)AV ArbitraryFibre Volume Fraction:Morphological DetailsComposite Assemblage ModelsCCA2LV-MVCSA1Eshelby Based ModelsHalpin-Tsai2(3)Mori-Tanaka Scheme3SC2LV-MVGSC2HVAVSimple Complex23  Although some researchers (e.g. (Bogetti and Gillespie 1992; White and Kim 1998; Zobeiry 2006)) have employed Hermans’ Generalized Self-Consistent (GSC) model for the purpose of predicting the effective elastic properties of UD composites, according to the study reported in Appendix B, Equations (B.13), (B.14),  (B.17) (based on Hashin’s CCA model) complemented by Equations (B.24)-(B.28) (based on Christensen and Lo’s GSC model) seems to provide better predictions over a wide range of fibre volume fractions. This is very important for the analysis of advanced fibre-reinforced composite structures used in aerospace industry. Due to the importance of viscoelastic behaviour of such structures, the accuracy of these analytical equations for predicting the effective viscoelastic properties of UD composites is investigated in Section 2.3. The application of these effective properties for macro-scale analysis of viscoelastic composites will be later discussed in Chapter 3.  For circular short fibre composites with low fibre volume fraction, the Halpin-Tsai equations (see Equation  (B.29) and (B.30)) are widely being used to predict the effective moduli with reasonable accuracy. Note that for calculating the parameter   in these equations, the expressions presented in Table B-1 was suggested by some researchers. However, the validity of these expressions for predicting the effective properties of short fibre composites at a wide range of fibre aspect ratios and fibre volume fractions has not been examined. Only Tucker and Liang (Tucker and Liang 1999) performed such a study for short fibre composites with fV < 0.3. For this purpose, they compared the predictions of Halpin-Tsai equations with numerical reference solutions obtained for the effective properties of short fibre composites with periodic microstructures. Although they considered periodic microstructures in their numerical analysis, periodic boundary conditions were not prescribed and the validity of numerical reference 24  solutions for such a comparison is questionable. Therefore, a comprehensive study similar to (Tucker and Liang 1999) but with wider range of variables (i.e. aspect ratios and volume fractions) and using numerical reference solutions obtained with appropriate periodic boundary conditions is performed in Appendix B (Section B.4.2). For this study, the constituents’ elastic properties of a typical thermoplastic composite reported in (Tucker and Liang 1999) are employed. According to the results (Figure B-13), the fibre aspect ratio effect for all effective moduli (except 1E ) is negligible at low volume fractions of fibres ( fV < 0.3) and Halpin-Tsai equations can predict the effective moduli of short fibre composites with wide range of fibre aspect ratios quite accurately at low volume fraction of fibres (fV< 0.3). Employing Halpin-Tsai equations for predicting the effective properties of short fibre composites with rectangular shape reinforcements has not been examined in the literature. In Chapter 4, the accuracy of these equations for predicting the effective properties of strand-based composites is examined.  The above equations can be used for composites with isotropic or transversely isotropic fibre properties. However, for composites with orthotropic phase properties or geometries that violate the transversely isotropic assumption, the use of analytical models is limited. For these composites, Mori-Tanaka model may be considered to be the best choice if the fibre shape can be approximated as being elliptical in geometry and for low to moderate fibre volume fractions of reinforcement (the low volume fraction is largely due to the limitations in the geometry). However, it should be noted that the Mori-Tanaka model may result in unsymmetrical effective stiffness matrix predictions (Böhm 2014; Ferrari 1991), which can cause numerical problems in macro-model simulations. 25   Composites with high volume fractions of orthotropic fibres have had rare applications in industry. Only recently, strand-based wood composites, consisting of highly filled partially covered orthotropic strands with resin, have found applications in construction of mid-rise buildings. For this class of composites the available models cannot be used. Therefore, new analytical equations are developed for predicting their effective properties and they are presented in Chapter 4 of this thesis.  2.2 Thermo-elastic Analysis Knowing the effective elastic properties of a two phase composite with arbitrary phase properties and geometries, its effective coefficient of thermal expansion can be calculated using the Rosen and Hashin’s general approach which lead to Equation (B.35). It should be noted that the components of the effective compliance tensor (obtained from elastic micromechanical analysis) are required as input for determining the effective coefficients of thermal expansion.   Compared to mechanical properties of composites, their thermal properties such as coefficients of thermal expansion (CTE) and thermal conductivities have not been analysed in the literature as extensively. Although some models such as Schapery’s equations are very popular in the literature and have been extensively used by researchers (e.g. (Bogetti and Gillespie 1992; White and Kim 1998; Zobeiry 2006)), a survey of the literature revealed that all available theoretical predictions for CTE except Rosen and Hahsin’s approach, do not account for Poisson’s ratio effects in the transverse direction accurately (see Appendix B).  26  As shown by Bowles and Tompkins (Bowles and Tompkins 1989), the analytical solutions presented by Rosen and Hashin (Rosen and Hashin 1970) which can be employed for predicting the effective coefficients of thermal expansion of any two-phase material regardless of its microstructural geometry, give the closest agreement with experimental data as well as numerical results. Therefore, in this thesis, the analytical expressions described in (Rosen and Hashin 1970) are proposed for predicting the effective coefficients of thermal expansion of composites, especially for composites with irregular geometries and orthotropic phases.  2.3 Viscoelastic Analysis  In the previous two sections, the best analytical micromechanics models for predicting the effective elastic and thermo-elastic properties of circular fibre composites were presented. However, as discussed in Appendix B and shown in Figure B-11, by increasing the mismatch between phase properties, the discrepancies between analytical and numerical predictions can be exaggerated at high fibre volume fractions. This can occur in viscoelastic composites where the mismatch between the phase properties increases when one phase is in a relaxed state. Therefore, for composites in which at least one phase exhibits time-dependent (viscoelastic) properties, the applicability of such analytical models needs to be investigated.  In the work by Brinson and Lin (Brinson and Lin 1998) and Yancey and Pindera (Yancey and Pindera 1990) they examined the use of Mori-Tanka model and Method of Cells for predicting the effective properties of viscoelastic composites. However, extension of the best analytical models described in Sections 2.1 and 2.2 to viscoelastic composites with circular fibres has not been examined fully. 27   Therefore, in this section, we examine the use of  Equations (B.13), (B.14) and (B.17) for predicting longitudinal properties, and  Equations (B.24)-(B.28) for predicting transverse properties of a viscoelastic solid composite (AS4/3501-6) by comparing their predictions with the UD unit cell finite element simulation results at different volume fractions. The numerical approach established in Appendix A and verified in Appendix B has been used for such a comprehensive study.  The use of these micromechanics equations for composites undergoing cure in which the resin evolves from a viscous fluid to a viscoelastic solid is investigated in Section 2.4.   2.3.1 Analytical Approach The viscoelastic properties of the composite can be estimated using the correspondence principle as employed by most researchers, e.g. see (Christensen 1979; Matzenmiller and Gerlach 2004; Zobeiry 2006; Brinson and Lin 1998). Using the correspondence principle, the linear viscoelastic heterogeneous problem in the real time domain is first transformed to a virtual linear elastic problem in Laplace space. The latter is then solved using linear micromechanical schemes. Finally, the effective viscoelastic properties are obtained using numerical inversion to time domain. This classical approach is applied here using the analytical closed-form expressions presented in Section B.2.1. The software Mathematica® has been used to perform the analytical calculations. An add-on package called “Numerical Inversion” has been employed for the inverse Laplace transformation. This package is capable of applying the inverse Laplace with different methods. The method developed by Stehfest (Stehfest (1970)) and previously used by 28  Zobeiry (Zobeiry 2006) is employed as the numerical inversion method to obtain the results reported here.  2.3.2 Numerical Approach In order to examine the accuracy of the analytical approach presented in Section 2.3.1, the viscoelastic behaviour of a unit cell of the UD fibre-reinforced composites, assuming hexagonal packing of fibres, is modelled using the ABAQUS® finite element software. Similar to the elastic analysis, the same unit cell of the material (see Section B.3.1) is subjected to six elementary loads which are held constant during time. Components of the effective stiffness tensor are determined during time using the volume average stress and strain tensors at each time step. Therefore, the stiffness tensor components are functions of time and each component can be expressed as a set of continuous mathematical functions, known as Prony series. Note that the method has already been well-established and employed for modelling different composite materials by various researchers (e.g. (Abadi 2009; Abolfathi et al. 2009; Garnich and Karami 2004; Karami and Garnich 2005; Naik et al. 2008; Xia et al. 2003)). In this thesis we employ the numerical approach as a tool to evaluate the accuracy of analytical approaches for predicting effective viscoelastic properties at the micro-scale.   For the analysis, properties of at least one phase need to be time-dependent which can be expressed using Prony series. Here, a viscoelastic thermoset composite is considered for the analysis. The properties of the fibre (AS4 fibre) and the resin (3501-6 epoxy) are taken from (White and Kim 1998) for AS4/3501-6 composite and are listed in Table 2-2 and 29  Table 2-3. The CTE of the phases were assumed to be constant at different temperatures, although this may not be a realistic assumption. However, this allows us to investigate the validity of the analytical approach carefully and demonstrate the CTE variation during relaxation time due to resin relaxation. The resin phase shear and bulk moduli (G  and K , respectively) are defined by Prony series expansions (Equation (2.1) and (2.2)) following the formulation available in ABAQUS® (ABAQUS Inc. 2010).     011 1 expnkk ktG t G g                (2.1)  01( ) 1 1 expnkk ktK t K k                (2.2) The characteristic parameters of the Prony laws, kg  and kk , are the weight factors defined as  0 0,k kk kG Kg kG K   (2.3) where kG  and kK  are the shear and bulk moduli associated with the relaxation times k , and 0G  and 0K  represent the instantaneous glassy shear and bulk moduli, respectively.  The above series for the shear and bulk moduli are distributed over nine relaxation times (see  Table 2-3). Relaxation loading paths of time duration 1510 min for each of the six elementary loadings are here prescribed to the hexagonal unit cell using kinematical periodic conditions. This allows computing the effective stiffness tensor at each time step in order to finally deduce the time evolution of the Young’s moduli, shear moduli and Poisson’s ratios.   30  Table 2-2 Constituents’ properties for AS4/3501-6 composite as reported in (White and Kim 1998).   Table 2-3 Viscoelastic properties of 3501-6 epoxy (Prony series parameters for the shear and bulk moduli) obtained from neat resin testing (Kim and White 1996).   2.3.3 Validations and Comparisons Unit cell relaxation Young’s and shear moduli are shown in Figure 2-2(a) and (b), respectively. In order to investigate the volume fraction effect, results are presented for four different volume fractions. Numerical results are shown with symbols while analytical results (using the CCA model (Hashin 1972) for longitudinal and the GSC model (Christensen and Lo 1979) for transverse properties) are plotted with lines. As previously shown in the elastic analysis (Section B.4.1), estimates using equations  (B.13), (B.14),  (B.17) and (B.24)-(B.28) are in very good E1 (GPa) 207.00 3.20E2 (GPa) 20.70G12 (GPa) 27.60 1.19G23 (GPa) 6.89ν120.20 0.35ν230.30 0.35α1 (με/°C) -0.90 57.60α2 (με/°C) 7.20 57.60Property AS4 carbon fiber 3501-6 epoxy Maxwell Element1 0.059 0.059 29.22 0.066 0.066 2.92E+033 0.083 0.083 1.82E+054 0.112 0.112 1.10E+075 0.154 0.154 2.83E+096 0.262 0.262 7.94E+097 0.184 0.184 1.95E+118 0.049 0.049 3.32E+129 0.025 0.025 4.92E+14   (min)    31  agreement with numerical reference data. For Poisson’s ratios (Figure 2-2 (c)), while the analytical approach is able to estimate 12  accurately during the entire relaxation period, it is able to estimate 23  accurately only at the beginning of the relaxation period.  32   Figure 2-2 Comparison between numerical (hexagonal packing) and analytical predictions of viscoelastic properties for AS4/3501-6 composite at different volume fractions. (a) Young’s moduli; (b) Shear moduli; (c) Poisson’s ratios. Constituents’ elastic properties and viscoelastic properties of the resin are taken from (White and Kim 1998) and are given in Table 2-2 and Table 2-3, respectively. Young’s and shear moduli of the composite are normalized by the glassy (unrelaxed) values of the resin. 33   Estimates of time variation of coefficients of thermal expansion are also compared with numerical results. Equation (B.35) has been employed with the correspondence principle for this purpose. As shown in Figure 2-3, the presented analytical approach is able to estimate the CTE in both longitudinal (1 ) and transverse directions ( 2 ) accurately during the entire relaxation period. According to Figure 2-2 and Figure 2-3, the analytical approach can be used effectively to predict viscoelastic properties of unidirectional composites with reasonable accuracy.   Figure 2-3 Comparison between numerical and analytical predictions of the effective coefficient of thermal expansion for AS4/3501-6 composite at different volume fraction. (a) Longitudinal direction; (b) Transverse direction. Constituents’ properties are taken from (White and Kim 1998) and are given in Table 2-2.  2.4 Analytical Model for Viscoelastic Composites During cure During processing of thermoset composites, the resin evolves from a low viscosity fluid to a highly cross-linked viscoelastic solid. Predicting the behaviour of thermoset composites during cure is one of the key issues in process modelling of thermoset composites. While classical solid (a) (b)   = 0.550.650.750.85             34  micromechanics equations provide us with a tool to predict the overall effective properties of composites, their accuracy in predicting the effective properties of the prepregs during curing as the resin transforms from a viscous fluid to a viscoelastic solid has not been fully investigated. Therefore, in this section the behaviour of thermoset composites during cure is investigated.   2.4.1 Modelling Approach As noted in Section B.3.1, fibres in long fibre composites are not perfectly straight. It was shown by Gutowski et al. (Gutowski et al. 1987) that at high volume fractions of fibre (fV  > 0.5) the fibre bed, i.e. the slight waviness of the fibres in prepreg, plays an important role in carrying the load in the transverse direction due to fibre waviness.  Building on this, (Cai and Gutowski 1992) developed a 3-D fibre bundle model to predict the deformation behaviour of the fibre bed. In a similar approach to the one used by Cai and Gutowski (Cai and Gutowski 1992), the deformation of fibre bundles under transverse loading is studied here.  Figure 2-4 schematically depicts an impregnated fibre under shear stress. A wavy fibre is idealized as a beam with irregular shape. The waviness in the fibre bed enhances the overall stiffness of the system by bending of its vertical sections. Figure 2-4 (a) illustrates that in vertical sections, the resin surrounding the beam is deformed with approximately the same strain as the fibre bed, while in the horizontal section (Figure 2-4 (b)) the beam is deformed under the total stress carried by both the resin and the fibre bed. This load transfer mechanism can be described simply using an analog representation shown in Figure 2-5. The contribution of each element to the overall stiffness of the material is denoted by k . As explained in (Malekmohammadi et al. 35  2011b), FBG  depends on fibre volume fraction. When  fV approaches its maximum possible value (78.5% and 90.7%, in square or hexagonal packing, see Section B.3.1), the number of contacts between fibres increases and FBG  increases accordingly. Hence, the fibre element in Figure 2-5 dominates the prepreg behaviour. At the other extreme, as the resin volume fraction (rV ) increases, fewer contacts are made possible and FBG  diminishes. In this case, the contribution of the resin element becomes more important and the prepreg shear modulus may be simply expressed by classical solid micromechanics equations (e.g. Equation (B.17)).   Figure 2-4 Prepreg deformation under shear stress. (a) Deformation of fibre bed along with resin. (b) Deformation of fibre under the overall shear stress  (b) Isostressr FBȖ Ȗr f (a) IsostrainWavy FibreResinResinWavy Fibre36   Figure 2-5 Proposed analog representation for modelling thermoset composites during cure  According to Figure 2-5, the fibre bed stiffness acts in parallel to the resin stiffness. In other words, the fibre bed enhances the stiffness of the resin phase. Therefore, it is assumed that the fibre bed waviness perturbs the resin shear modulus rG , by FBG  (fibre bed shear modulus). Extending this approach, if any solid micromechanical model is represented by a function ,  ,   ,  r f fMM G G V , the prepreg shear modulus would be estimated as:   Prep f r FB fG MM G ,G G ,V  (2.4) Experimental measurements on uncured prepregs have shown that the initial fibre bed stiffness is approximately 100 kPa during compaction (Hubert and Poursartip 2001). Assuming rV   0.4, FBG would be in the range of 40-50 kPa. In the following, predictions using Equation (2.4) with FBG  = 45 kPa are compared with experimental results throughout a wide range of resin shear moduli.    37  2.4.2 Validation In order to evaluate the validity of the presented modified micromechanical approach, experiments were performed on Advanced Composites Group MTM45-1, an out-of-autoclave material. Both resin film and unidirectional prepreg shear properties were characterized using rheological and dynamic mechanical tests at a constant frequency of 1 Hz and a temperature ramp rate of 1°C/min. The experimental procedure has been described in detail in (Thorpe 2013). The neat resin results are employed as input to the micromechanics models and the predictions compared to the prepreg results.  It should be mentioned that the Cure Hardening Instantaneously Linear Elastic (CHILE) model (Johnston et al. 2001) has been employed to describe the behaviour of the prepreg continuously during cure. By assuming that the modulus changes only as a function of temperature and degree of cure, an approximate comparison can be made. The accuracy of this approach for thermoset resins is shown to be satisfactory in conventional cycles consisting of temperature holds and cool-down ramps. The validity and applicability of this model has been investigated elsewhere (see (Zobeiry et al. 2010)). A true comparison requires full characterization of both resin and prepreg at different frequencies and degrees of cure. Once both the resin and the prepreg have been fully characterized, the analytical approach described in Section 2.3.1 can be applied.   2.4.2.1 Experimental data Oscillatory shear tests were performed with a TA Instruments AR2000 rheometer to characterize the pre-gelation behaviour.  Measurements were performed from the uncured state, starting at 50°C, to the gelled state at ~165°C, with a temperature ramp rate of 1°C /min and constant 38  frequency of 1 Hz. It should be noted that a rheometer measures a combination of  12G  and 23G  values. However, as discussed in (Malekmohammadi et al. 2011b), the prepreg shear moduli in 1-2 and 2-3 directions are very close for the range of resin properties used here.  Post-gelation behaviour of both resin and prepreg were characterized using a TA Instruments Q800 DMTA (Dynamic Mechanical and Thermal Analyzer). During the early stage of cure, the resin modulus is very low and conducting tests with the DMTA is quite challenging. In order to overcome this problem a steel shim was used to support the uncured resin and prepreg samples. Each sample was adhered to the steel shim and then trimmed to make a Bi-Material-Beam (BMB). Tests were performed under 3-point bending on BMB samples at a constant frequency. Uncured BMB samples were heated in the DMA up to 180°C at the same ramp rate used in the rheological tests (1°C /min). Once at 180°C, the samples were cured at constant temperature for 120 minutes.  The data extracted followed the procedure described in (Curiel and Fernlund 2008) and has been explained in (Thorpe 2013).  For each data point, the transverse storage modulus (2E ) was extracted from the DMTA data and the transverse shear storage modulus (23G ) was computed using the following relation:   223232 1EG    (2.5)  During the evolution of the resin from a liquid to a viscoelastic solid, the resin Poisson’s ratio varies. However, Bogetti and Gillespie (Bogetti and Gillespie 1992) reported no significant sensitivity in macroscopic composite properties by assuming a constant Poisson’s ratio for the 39  resin. Therefore a constant Poisson’s ratio (23 ) of 0.59 for the composite was chosen based on available data in the literature (Herakovich 1998) to extract the shear modulus. In order to evaluate the accuracy of BMB tests, 3-point bending tests were also performed on Homogeneous Beams (HB), i.e. prepreg without a steel shim.  High confidence DMA and rheological results for both resin and prepreg are overlaid in Figure 2-6. Due to the fact that only experimental results with high accuracy are shown in this figure, a gap between DMA and rheological data is observed. However, it is clear that there is a continuous behaviour over this gap for both the resin and the prepreg that could not be captured fully due to current measurement limitations.  Figure 2-6 Modulus development during cure for MTM45-1 resin and prepreg. Only high confidence data are shown. The storage modulus is plotted on a logarithmic scale.   -500501001502001.E+011.E+041.E+071.E+1050 150 250 350 450Temp (°C)/DOC %t (min)G', PrepregG', ResinTempDegree of Cure (DOC)14    107 1010 DMA HighConfidence DataRheological Data101 G' (Pa)040  2.4.2.2 Comparison The experimental results for the prepreg and resin modulus over the full range are cross-plotted in Figure 2-7 on a logarithmic scale. The predicted values using different micromechanical models for 12G  and 23G  of the composite are presented on this figure for comparison. The mechanical properties of the resin, fibre and fibre bed that have been used for predictions are listed in Table 2-4. For lack of better choice, the fibre properties were estimated as those of AS4 graphite fibre (White and Kim 1998). Based on the manufacturer’s data sheet, a fibre volume fraction of 0.58 was used and was assumed to be constant over the entire temperature history.  The value for the fibre bed shear property (FBG ) was selected such that it matched the plateau region observed in the rheometry experiments as shown in Figure 2-7. It was found that any value between 40 kPa and 50 kPa could match the experimental results before gelation. Therefore, an average value of 45 kPa was used in this figure. This value is also consistent with experimental values measured elsewhere (Hubert and Poursartip 2001). The temperature variation of fibre properties was neglected since that variation is negligible when compared to variation in resin properties over the same temperature excursion. For resin bulk modulus (rK ), the glassy bulk modulus of epoxy was used from (Prasatya et al. 2001). Although the bulk modulus of epoxy system changes from rubbery state (0.62 GPa) to glassy state (3.2 GPa), its effect on predicted composite modulus was found to be negligible. Finally, the presence of fibres was assumed to have an insignificant effect on the cure kinetics of the resin.    41   Table 2-4 Material properties of fibre, resin and fibre bed used for comparison.    Figure 2-7 Comparisons of experimental data with different models’ predictions. DMA data are magnified on the top left. Note that both axes are logarithmic scales.  E1 (GPa) 207.00E2 (GPa) 20.70G12 (GPa) 27.60 0-1 0.045G23 (GPa) 6.89ν120.20ν230.30K  (GPa) 3.20(a)  White and Kim (1998)(b) Prasatya et al. (2001)Property Fibre (a) Resin (b) Fibre bed 1.E+011.E+031.E+051.E+071.E+091.E+111.E+01 1.E+03 1.E+05 1.E+07 1.E+09G'Prep(Pa)G'r (Pa)Experiments (DMA/BMB & Rheometry)Experiments (DMA/HB) Isostress MMModified Isostress MMModified CCA (G12)Modified GSC (G23)103    105    107   109   101   101103      105107109101   1.E+071.E+091.E+111.E+07 1.E+0942  Figure 2-7 demonstrates that by incorporating fibre bed properties into classical solid micromechanics (e.g. isostress, CCA or GSC), the predicted shear modulus of a prepreg is a continuous function of the resin properties. In order to highlight the contribution of fibre bed to the prepreg shear properties, the isostress micromechanics prediction for 12G  is also shown. A region corresponding to the cured state is magnified to better illustrate the predictions using different analytical micromechanics equations. As would be expected, DMA results of fully cured samples (HB) follow the 23G  values predicted by Christensen and Lo (Christensen and Lo 1979) GSC model with higher accuracy than other micromechanical models.  2.5 Summary and Conclusions Although there are several analytical micromechanics models available in the literature for determining the effective elastic and thermo-elastic properties of composites with certain microstructures, there are still microstructures of interest that cannot be handled by these analytical models. Moreover, the accuracy of some of these models for predicting the effective viscoelastic properties of composites has not been fully investigated in the literature. Using computational homogenization, a numerical tool for predicting the effective properties of periodic microstructures was developed in Appendix A. This tool is versatile and can be used for composites with variety of microstructures and constituents’ properties. It can also be used for examining the validity range of available analytical micromechanics models.  The goal of this thesis is to present an efficient multi-scale model for viscoelastic composites. To establish an efficient multi-scale approach for viscoelastic composites, the analytical approach (if available) can be used as an alternative to the numerical approach. Therefore, available closed-43  form analytical micromechanics equations in the literature were reviewed and their application for the purpose of predicting the effective properties of viscoelastic composites were examined in this chapter. For this purpose, comprehensive studies were carried out on effective elastic, thermo-elastic and viscoelastic properties using numerical reference solutions. Some of the findings based on these comprehensive studies presented in this chapter and Appendix B are highlighted below:   Although some researchers (e.g. (Bogetti and Gillespie 1992; White and Kim 1998; Zobeiry 2006)) have employed Hermans’ GSC model for the purpose of predicting the effective elastic properties of UD solid composites, Equations (B.13), (B.14) and  (B.17) (based on Hashin’s CCA model) complemented by Equations (B.24)-(B.28) (based on Christensen and Lo’s GSC model) form a system of equations that seem to provide better predictions of effective properties of transversely isotropic UD composites over a wide range of fibre volume fractions.  For short fibre composites, the effect of fibre aspect ratio for all effective moduli (except 1E ) is negligible at low volume fractions of fibres ( fV  < 0.3) and Halpin-Tsai equations can predict the effective elastic moduli of short fibre solid composites with a wide range of fibre aspect ratios quite accurately at low volume fraction of fibres (fV  < 0.3).  For predicting the effective coefficients of thermal expansion of any two-phase material regardless of its microstructural geometry, the analytical solutions presented by Rosen and Hashin (Rosen and Hashin 1970) are the most accurate ones for thermo-elastic analysis of any two-phase composite. 44   Using the correspondence principle, application of the best analytical models for elastic properties (see the first bullet item above) can be extended with confidence to predict the effective properties of viscoelastic solid composites with higher accuracy than previous models employed for process modelling of composites (e.g. (Bogetti and Gillespie 1992; White and Kim 1998; Zobeiry 2006)).   Analytical micromechanics equations developed for composites with solid constituents are unable to predict the shear modulus of thermoset composites in the early stage of cure, where the resin is in a fluid state. Therefore, for accurate prediction of the shear modulus throughout cure, solid micromechanical models were modified by incorporating the fibre bed behaviour for the first time. Predictions using modified micromechanical equations were compared with experimental data and promising agreement was found.  The results for circular fibre composites in this chapter suggested that once their validity range has been verified with numerical reference solutions, analytical micromechanics approaches can be used for micro-mechanical modelling of viscoelastic composites with confidence. Note that for composites with rectangular, orthotropic fibres (e.g. strand-based wood composites) involving high volume fraction of strands, the available analytical models are not applicable. Therefore, an analytical approach for strand-based composites has been developed as a part of this thesis and presented in Chapter 4. In order to evaluate the accuracy of the developed analytical approach for such orthotropic composites, the numerical tool (developed based on the computational homogenization technique presented in Appendix A and verified in Appendix B) is employed for the purpose of generating reference solutions.  45  The next chapter (Chapter 3) of this thesis involves efficient macro-scale modelling of generally orthotropic viscoelastic composites. It will be shown how the results of micro-scale analysis (e.g. for circular fibre composites or strand-based composites) can provide the necessary inputs for macro-scale analysis of viscoelastic composite structures.         46  Chapter 3: Macro-Mechanical Modelling of Generally Orthotropic Viscoelastic Composites  A differential form (DF) of viscoelasticity has recently been presented by Zobeiry (Zobeiry et al. 2006; Zobeiry 2006; Zobeiry et al. 2014) as an efficient approach for modelling the response of polymer composite materials. However, the formulations in (Zobeiry et al. 2006; Zobeiry 2006; Zobeiry et al. 2014) were developed for transversely isotropic materials and cannot be employed for modelling viscoelastic behaviour of orthotropic composites. Here, as a last step in multi-scale modelling of general viscoelastic composite structures, the formulation in (Zobeiry 2006)  is extended and employed in macro-scale modelling the viscoelastic behaviour of orthotropic composites. The motivation for considering this material behaviour comes from new multi-level materials such as strand-based wood composites which show distinct orthotropic viscoelastic behaviour (see Chapters 4 and 5). However, it can also be employed in modelling the viscoelastic behaviour of advanced UD fibre-reinforced composites consisting of fibre, resin and an additional toughening phase layer or woven composites for which the assumption of transversely isotropy breaks down.  First, the challenges in other available approaches for modelling viscoelastic materials and advantages of the differential formulation will be discussed. Next, the 3-D formulation for orthotropic viscoelastic materials will be presented. The finite element formulation is developed in a very similar way to (Zobeiry 2006; Zobeiry et al. 2014), see Appendix D. Therefore, it can be easily implemented in virtually any existing code. To illustrate this, a user material subroutine 47  (UMAT) has been developed and implemented in the ABAQUS® software. The implementation of the formulation is then verified through some benchmark examples. The capabilities of this approach in comparison to other available models, in terms of accuracy and efficiency, will also be discussed and demonstrated.  3.1 Introduction Viscoelastic behaviour of various solid materials and their modelling has been the subject of many publications and review papers (e.g. (Schapery 1974; Ferry 1980; Christensen 1982; Zocher 1995; Tschoegl 1997; Klasztorny and Wilczyñski 2002)). In these publications, different aspects of the material behaviour are considered. Different approaches have also been proposed by many others for modelling the behaviour of these materials. Some researchers have considered linear viscoelasticity of isotropic materials (Kim and White 1998; White and Hahn 1992; Chazal and Pitti 2009; Taylor et al. 1970) while some others have considered nonlinear viscoelasticity of isotropic (e.g. (Ellyin and Xia 2006; Muliana and Khan 2008; Ellyin et al. 2007; Haj-Ali and Muliana 2004)), orthotropic (Sawant and Muliana 2008) and anisotropic materials (Poon and Ahmad 1998; Poon and Ahmad 1999) in their modelling approaches and their formulations.   The motivation of the current work is to predict the response of complex, three-dimensional, large composite structures made of orthotropic viscoelastic materials efficiently. It is intended to demonstrate how this approach can be used within a multi-scale modelling strategy to simulate the response of composite structures exhibiting viscoelastic characteristics.  48  In modelling the structural response of composites under a set of assumptions, e.g. linear viscoelastic behaviour of the material, the efficiency and accuracy of the models are very important. The importance of efficient viscoelastic modelling of polymer composite material has been shown in (Zobeiry et al. 2006; Zobeiry et al. 2010; Zobeiry 2006). Although we only consider linear viscoelastic material behaviour, the differential approach presented can also be extended to non-linear viscoelastic problems as shown in (Ellyin et al. 2007).  Among the earliest and simplest models for representing the linear viscoelastic behaviour are Maxwell and Kelvin analog models. These models consist of linear spring and dashpot elements as shown in Figure 3-1.   Figure 3-1 (a) Maxwell model. (b) Kelvin model  In most applications, a simple model, such as Figure 3-1(a) is not sufficient to describe the real behaviour of viscoelastic materials. Hence, in many models for viscoelastic solids  (Taylor et al. 1970; Tran et al. 2011; Kim and White 1996; Jurkiewiez et al. 1999; Bažant and Wu 1974; Kaliske 2000; Ellyin et al. 2007; Sawant and Muliana 2008; ABAQUS Inc. 2010; Zobeiry et al. (a) (b)kk49  2006), a number of Maxwell elements in parallel, as shown in Figure 3-2 is used. The relaxation shear modulus of such a material is represented by Prony series as:  1( ) ( ) iNiir u r tG t G G G w e     (3.1) in which, N  is the number of Maxwell elements, iw  are the weight factors, i  are the relaxation times defined as:  iiiG   (3.2) where, iG  and i  are the stiffness of spring and the viscosity of the dashpot in the Maxwell element  “ i ”, respectively (see Figure 3-2). Note that for non-ageing materials the spring stiffness and dashpot viscosity are assumed to be constant while for ageing materials they are functions of time.  Figure 3-2 Generalized Maxwell elements.  The unrelaxed (glassy) and relaxed (rubbery) moduli are represented by superscripts u  and r  in Equation (3.1), respectively. Assuming that 11niiw , the unrelaxed modulus can be written as: rG11 2G2nGn 50   1 1( )n ni ii iu r r u rG G G G G G w        (3.3) In this work, we restrict our attention to this representation of the material behaviour.  Equation (3.1) was written for shear modulus. It is noted that most other material properties are also observed to be time-dependent; for example, time-variability of the Poisson’s ratio is established, as discussed in (Hilton 2001; Tschoegl et al. 2002). Thus, all material properties, e.g. two for an isotropic, five for a transversely isotropic, and nine for an orthotropic material are assumed to be time-dependent here. Furthermore, all these material properties are represented by Prony series.  In previous works, a DF of viscoelasticity was compared to some other models for polymer composite materials in one-dimensional (Zobeiry et al. 2006) and three-dimensional (Zobeiry 2006) cases. It was shown that the DF has distinct advantages over the other common models, with the current Prony series representation of the material behaviour. In this chapter, we expand the viscoelastic formulation presented in (Zobeiry 2006; Zobeiry et al. 2014) to obtain a formulation suitable for finite element analysis of orthotropic viscoelastic materials in general, which can be used in efficient macroscopic modelling of composite structures.  For this purpose, first different viscoelastic models and strategies described in the literature are reviewed briefly and their characteristics are discussed. Afterwards, the proposed three-dimensional DF of viscoelasticity for orthotropic materials and its finite element implementation is described. The generalized Maxwell representation with an infinite number of Maxwell 51  elements, which is one of the most general representations of linear viscoelasticity (Tran et al. 2011), is employed for this purpose. It is shown that this approach is flexible and can be used in conjunction with virtually any available code. Finally, the procedure is verified and some numerical case studies are presented.   3.2 Review of Literature on Viscoelastic Models and Solution Techniques The solution of linear viscoelastic problems may be obtained by the application of the correspondence principle to the analytical equations of elasticity (Webber 1969; Christensen 1982; Tschoegl 1997). This approach is restricted to problems for which it is possible to find an explicit solution of the associated equations of elasticity. In order to obtain solutions of more complicated problems, it is necessary to develop numerical, rather than analytical, techniques (Nerantzaki and Babouskos 2011; Zienkiewicz et al. 1968; White and Kim 1998). These numerical methods avoid the retention of the entire history of stress and strain and enable us to deal with complex viscoelastic structures involving complicated boundary conditions. The key to such methods is to incrementalize the constitutive equations of viscoelasticity by means of analytical techniques.   The constitutive equations of viscoelastic materials may be expressed in either integral form (IF) or DF (Xia and Ellyin 1998; Xia et al. 2003; Ellyin et al. 2007; Nerantzaki and Babouskos 2011; Marques and Creus 2012). The IF of constitutive equations is derived based on principles of superposition, while the DF is not a superposition type model (Xia et al. 2003). There are certain advantages associated with each form of constitutive model. For instance, the formulation in DF is physically based and can easily be incorporated in finite element programs for analysis of 52  complex problems (Xia et al. 2003; Bažant and Wu 1974). On the other hand, the IF is a general (no Prony series employed for representing the viscoelastic behaviour of the material), thermodynamically-based form which may lead to complex mathematical expressions.    As explained in Section 3.2.2, the two forms may be converted to one another in cases where Prony series expansion is employed to describe the relaxation or creep functions. Based on these two forms, several viscoelastic modelling techniques have been proposed in the literature. A brief summary of these models and a review of their solution techniques are presented in Section 3.2.1 and Section 3.2.2. For a more detailed review of these models, the reader is referred to (Zocher 1995; Chazal and Pitti 2011).  3.2.1 Integral Form (IF) of Viscoelasticity Several researchers have employed the constitutive equations for viscoelastic materials using the hereditary integral, or the so-called Integral Form (IF) in the time domain. Based on Boltzmann’s superposition principle, the constitutive relationship between the components of stress tensor, ij, and components of strain tensor, kl , of a linear non-ageing viscoelastic material, in a general three-dimensional analysis, may be written in the following IF in time domain:  0( )( ) ( )t klij ijklt C t d      (3.4) where ijklC  are the components of the material relaxation modulus tensor, t  is the current time and   is a dummy integration variable representing previous times. The above equation expresses the time dependent stress-strain relationship which can be used in simulation of a viscoelastic structure. The above equation in a simple 1-D case can be written as follows: 53    0( ) ( )t dt E t dd     (3.5) in which,  E t  is the relaxation modulus. It should be noted that the stress and strain are assumed to be zero for 0t   in the above formulation, i.e. the material is undisturbed before time zero. Equation (3.5) can be used for homogeneous, non-ageing materials, with no change in temperature. The temperature effect has not been included in the above equation. Under non-isothermal conditions, by assuming that the material exhibits thermo-rheologically simple behaviour, time-temperature superposition can be applied. Hence, by introducing the concept of reduced time (see (Zocher et al. 1997; Taylor et al. 1970)) Equation (3.5) can be written as:        0t dt E t dd       (3.6) In the above equation,   and    are called the “reduced time” or ”effective time” and are calculated as:         0 01 1;tT Tt dt dta T a T       (3.7) where, Ta  is called the shift factor which is a material property (Zocher et al. 1997) and t  is a dummy integration variable.  The IF of linear viscoelasticity has been generalized to non-linear viscoelasticity by Schapery (Schapery 1969; Schapery 1997). Schapery’s non-linear model is derived based on principles of irreversible thermodynamics. In this thesis, we focus on linear behaviour of viscoelastic materials. Details on implementation of non-linear viscoelastic constitutive models can be found elsewhere (Xia et al. 2005; Poon and Ahmad 1999; Crochon et al. 2010). 54   For simple problems, direct integration schemes have been employed by some researchers to calculate stress and strain at each point (e.g. (White and Hahn 1992; Harper and Weitsman 1981; Hopkins and Hamming 1957; Lee and Rogers 1963). However, the viscoelastic constitutive equation in its IF (Equation (3.4) in 3-D or Equation (3.6) in 1-D), is not suitable for numerical computations of complex structures (Carol and Bazant 1993) due to storage problem.  In order to address the storage problem for large structures, most researchers have developed recursive integration techniques (e.g. (Simo and Hughes 1998; Taylor et al. 1970; Zak 1968; Muliana and Khan 2008; Haj-Ali and Muliana 2004; Kaliske 2000; Zocher et al. 1997)). The recursive relationship was established through representation of relaxation moduli by a set of Prony series (e.g. Equation (3.1)). This eliminated the need to store all previous time-step results in order to find stresses at the current time step, which existed in other integration techniques (e.g. (Hopkins and Hamming 1957)).  Taylor et al. (Taylor et al. 1970) were among the first who employed a recursive technique in a computer code for the analysis of viscoelastic solids. They proposed a computational algorithm for an isotropic, homogeneous, linear viscoelastic solid based on the finite element discretization of the boundary value problem. Kim and White (Kim and White 1998) employed Taylor’s recursive formulation to analyse residual stresses in filament-wound composite cylinders in a finite element model developed for this purpose.   55  Even though Taylor’s recursive approach is more efficient than the direct integration approaches mentioned earlier, small time steps are required due to the assumption of constant strain during a time step. The algorithm proposed by Taylor et al. (Taylor et al. 1970) was later extended in (Simo and Hughes 1998) by introducing the concept of internal state variables and assuming constant strain rate (contrarily to the constant strain in (Taylor et al. 1970)) within a time step to increase the efficiency. In their proposed algorithm, the value of a state variable is computed from its value at the previous time step and the stress or strain determined in the previous time step. A similar approach for an isotropic viscoelastic material has been implemented in the ABAQUS® software (see (Marques and Creus 2012; ABAQUS Inc. 2010) ). This approach is being used for comparison with the differential approach in this chapter.   For orthotropic viscoelastic materials, Zocher (Zocher 1995; Zocher et al. 1997) and other researchers (Kiasat 2000; Kaliske 2000; Poon et al. 1998) employed internal state variable approach. For this purpose, Zocher converted the constitutive equations in the IF into an incremental algebraic form (using a Finite Difference (FD) procedure), which resulted in a recursive relationship, and then solved the resulting set of linear algebraic equations (Zocher 1995; Zocher et al. 1997). This facilitated the implementation of the incremental form into the finite element formulation. In a very similar approach, Chazal and Pitti used integral constitutive equations in terms of creep compliance, instead of relaxation moduli, to derive incremental constitutive equations for linear, non-ageing viscoelastic material (Chazal and Pitti 2011). This approach was applied in modelling the behaviour of ageing viscoelastic materials in the finite element context as well (Chazal and Pitti 2009; Chazal and Pitti 2010).  56  In a different approach, Chazal and Pitti converted the IF of viscoelasticity to a set of differential equations (Chazal and Pitti 2012; Chazal and Pitti 2010). By taking the derivative of the hereditary integral equation of each element with respect to time, the differential equations governing the behaviour of linear, non-ageing, viscoelastic materials was derived. Using finite difference discretization, the solution of the differential equations was obtained. In order to do that, the time derivative of stress or strain during each time increment was assumed to be constant in this procedure. This led to incremental constitutive equations which were later introduced in a finite element discretization in order to obtain solutions to complex viscoelastic problems. It should be noted that although the incremental constitutive equations were derived from the IF of viscoelasticity, the approach was called a differential approach by its authors, probably due to the application of differential operators in discretizing the hereditary integral equations.  One of the main differences between the work by Zocher (Zocher et al. 1997), (also in (Chazal and Pitti 2011)) and other similar internal variable approaches is the assumption of constant strain rate (constant stress rate in (Chazal and Pitti 2011)) instead of constant strain or constant stress over a time step. This enables better predictions of creep and relaxation behaviour of the material.   Poon and Ahmad (Poon and Ahmad 1998) presented an integration point algorithm, very similar to (Zocher et al. 1997) but with a directionally dependent reduced time for anisotropic materials. Later, they extended their approach to anisotropic, small strain, thermo-rheologically simple, non-linear viscoelastic materials employing Schapery's non-linear model (Poon and Ahmad 57  1999). For nonlinear, thermo-rheologically complex, viscoelastic materials, similar recursive models have been developed (e.g. (Sawant and Muliana 2008; Muliana and Khan 2008)). However, they fall outside the scope of this chapter.   3.2.2 Differential Form (DF) of Viscoelasticity The IF of viscoelasticity can be converted into a set of linear differential equations through a Prony series expansion of relaxation or creep functions (Carol and Bazant 1993; Zobeiry et al. 2006). In fact, it can be shown that the resulting linear differential equations are the governing differential equations of generalized Kelvin or Maxwell rheological models (see (Zobeiry et al. 2006)). These generalized rheological models were shown to converge to the same results as the number of Kelvin or Maxwell elements in the representation of non-ageing viscoelastic materials increases (Carol and Bazant 1993). In other words, both of these rheological models (in their generalized form) represent the same phenomenon for non-ageing materials.  The DF of viscoelasticity has been investigated less than the IF of viscoelasticity in the literature, probably due to the rapidly increasing complexity of governing differential equations (Zobeiry 2006). The resulting higher order differential equations in DF of viscoelasticity require special solving techniques, while in the IF of viscoelasticity integro-differential field equations may be solved using several available numerical techniques (Nerantzaki and Babouskos 2011) described in Section 3.2.1. However, the DF provides more insight into the physics of the viscoelastic problem (Marques and Creus 2012). Therefore, effects of temperature and degree of cure can be implemented more easily in a finite element program (see (Zobeiry 2006)).   58  Webber, Zienkiewicz, Carpenter, Bazant and Wu (Webber 1969; Zienkiewicz et al. 1968; Bazant and Wu 1974; Carpenter 1972; Bažant and Wu 1974) were among the first who employed the DF of viscoelasticity in the analysis of viscoelastic materials. More recent works can be found in (Idesman et al. 2000; Jurkiewiez et al. 1999; Mesquita et al. 2001; Xia and Ellyin 1998; Ellyin et al. 2007; Chazal and Pitti 2009). Some of the main solutions technique employed to solve the differential equations in these works have been briefly reviewed in (Zobeiry 2006). More detailed review is give below.  In DF of viscoelasticity, the constitutive equations that relate stresses to strains at each point are in the form of differential equations. To solve these equations, researchers have used FD schemes. Among the different FD schemes, some researchers (Zienkiewicz et al. 1968; Greenbau and Rubinste 1968; Bažant and Wu 1974; Idesman et al. 2000; Jurkiewiez et al. 1999) used first-order numerical methods such as Euler methods (backward and exponential) while Bazant and Carpenter used higher-order numerical methods such as Runge-Kutta methods (Bazant 1972; Carpenter 1972). Zienkiewicz and coworkers (Zienkiewicz et al. 1968) assumed that the material behaviour may be represented through a combination of Kelvin models and solved a set of ordinary differential equations by the Euler method. Due to the assumption of constant stress during each time step in (Zienkiewicz et al. 1968), small time steps were required.  Similar to Zienkiewicz et al. (Zienkiewicz et al. 1968), Bazant used both the Kelvin (Bazant 1972) and Maxwell (Bažant and Wu 1974) models first for non-ageing and later for ageing (Carol and Bazant 1993) viscoelastic materials. In (Bažant and Wu 1974), a set of linear differential equations were employed to represent the constitutive law of the viscoelastic 59  material. The solution of these equations was represented in terms of incremental stress and strain in a time interval. Carpenter (Carpenter 1972) also used the material representation of (Zienkiewicz et al. 1968) to formulate linear, homogeneous, isotropic viscoelastic problems using DF.   Contrary to Zienkiewicz et al. (Zienkiewicz et al. 1968), both Bazant (Bazant 1972) and Carpenter (Carpenter 1972) employed higher order numerical procedure (e.g. Runge-Kutta method) to solve the differential equations. As noted in (Carpenter 1972; Bazant 1972), Runge-Kutta methods are more stable, converge faster and allow larger time steps compared to Euler method. Moreover, the maximum allowable time step size can be determined using an error estimation process. However, they require more data storage and computational effort which sometimes may be compensated by employing larger time steps (see (Carpenter 1972)).   Similar to Carpenter and Bazant (Carpenter 1972; Bazant 1972),  Chazal and Pitti (Chazal and Pitti 2009) used a generalized Kelvin-Voigt model to establish constitutive viscoelastic equations in an incremental form. The incremental formulation in (Chazal and Pitti 2009) was derived for three-dimensional finite element analysis of structures made of linear, ageing viscoelastic materials. However, some assumptions such as constant Poisson’s ratio (similar to (Zienkiewicz et al. 1968; Jurkiewiez et al. 1999)), iso-thermal, and linear spring moduli were made to simplify the incremental formulation.  In terms of computational efficiency, both the finite difference schemes used to solve differential equations in DF of viscoelasticity and the internal variable approach for IF of viscoelasticity, 60  overcome storage problem discussed earlier and are considered efficient approaches compared to direct integration schemes. Moreover, both approaches are easy-to-implement in available finite element codes. This is due to the fact that they can be implemented at the material integration point level. However, the DF that uses a FD scheme does not need to involve the assumptions that most internal variable approaches developed based on IF of viscoelasticity have made (see Section 3.2.1).  Recognizing that the DF approach is a simpler and easier to implement approach for structural modelling of viscoelastic composites, here we will expand the 1-D differential form to the 3-D multi-axial stress state and develop a FE formulation consistent with the one presented in (Zobeiry 2006). It should be noted that in all the above models that have been developed based on the DF, the material is assumed to be isotropic or transversely isotropic (Zobeiry et al. 2006), while we relax these assumptions and consider the material to be generally orthotropic in the present work.   3.3 Development of Viscoelastic Constitutive Equations for Orthotropic Materials  3.3.1 Formulation In a general 3-D state of stress, the constitutive equation relating the stress ij  and the strain kl  for a linear elastic material can be written as follows:  ij ijkl klC    (3.8) 61  where ijklC are components of the material stiffness tensor. Using matrix notation, the stiffness tensor can also be written in a 6 × 6 matrix form, denoted by the symbol C  with the double underscore indicating matrix quantities and single underscore denoting vectors.  For an isotropic material, the stiffness matrix can be expressed in terms of two independent constants (moduli),G  and K , representing the shear and bulk behaviour of the material, respectively.  4 2 23 3 34 23 3430 0 00 0 00 0 00 0. 0G K K G K GG K K GG KCGSym GG                (3.9) Therefore, the stress-strain relationship can also be written as follows:   211 22 33 11 22 333222 11 33 11 22 333233 11 22 11 22 333121323(2 )(2 )(2 )2 02 02 0G K                                                       (3.10) In other words, the total stress vector for an isotropic material can be decomposed into two vectors:  G K     (3.11) where the vectors G  and K ,  represent the shear and bulk components of the total stress, respectively.  62   For viscoelastic behaviour of isotropic materials, researchers have usually employed separation of the viscoelastic behaviour into two distinct behaviours,  shear and bulk  (e.g. (Brinson and Knauss 1992; Ellyin et al. 2007; Muliana and Khan 2008; Tran et al. 2011; Webber 1969; Wiersma et al. 1998; Taylor et al. 1970; Adolf and Martin 1996; Haj-Ali and Muliana 2004)). Here, we extend this approach to orthotropic viscoelastic materials by separating the viscoelastic behaviour of such materials into distinct behaviours and employing the DF of viscoelasticity to describe each behaviour.  In order to describe the behaviour of an elastic, orthotropic material, nine constants are needed and the stress-strain relationship can be expressed in terms of these nine constants (stiffness matrix components) as follows:  11 22 33 44 55 66 12 13 2311 22 33 44 55 66 12 13 23C C C C C C C C CC C C C C C C C C                  (3.12)  where the stress tensor   has been decomposed into nine tensors each corresponding to a  stiffness matrix component or a material property, P :  11 22 2391...C C C pp         (3.13) In which the indices p  associate each stress or strain tensor with the corresponding stiffness matrix component, (e.g. 11C ). In expanded form and using matrix notation, the above equations can be written as: 63   1111 22 22 333312 1344 12 55 66 2313 2322 3312 11 130 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 2 0 0 0 2 0 0 02 0 0 0 0 0 0 0 20 0 0 2 0 0 0 2 00 0 0 00 0 00 0 0C C CC C CC C                                                                 23 3311 220 0 00 0 0 00 0 0 0C                   (3.14)  For simplicity, the above equation may also be written in vector form as:  11 1122 2233 3311 22 3312132344 55 6612130 00 00 00 0 00 0 00 0 000 000 00 02 00 20 0C C CC C C                                                                                                    2322 3311 3311 2212 13 2300020000 0 00 0 00 0 0C C C                                                               (3.15) Now, we assume that the viscoelastic behaviour of an orthotropic material can be expressed using nine sets of differential equations. For each component of the stiffness matrix (propertyP ), 64  similar to the isotropic or transversely isotropic case in (Zobeiry 2006), a differential equation is employed to relate the associated stresses and strains as follows:     11 22 2311 ; , , ....,NuP P P ii P iP P C C C     (3.16)  In deriving the above equations, for each component of the stiffness matrix (e.g. 11C ) a generalized Maxwell model has been employed. For each element i  of the generalized Maxwell model a governing differential equation can be written as follows:        11 22 231 ; , , ....,iP P Pi iP iP P C C C      (3.17) In the above equation, the spring stiffness and time constant of Maxwell element i  associated with component P  of the stiffness matrix are denoted by iP  and  P i , respectively. The unrelaxed values of each component of stiffness matrix, denoted by superscript u  in Equation (3.16),  is obtained by summing over all Maxwell elements (see Equation (3.17)). Therefore, the unrelaxed value of component P  is obtained as:  11 22 231; , , ....,Nu iiP P P C C C    (3.18) where N  is the number of Maxwell elements.  The overall governing differential equation is obtained by summing over all nine stress vectors in Equation (3.16):     11NuP P iP P i P iP     (3.19) 65  The first term on the right side of the above equation can be replaced by uC   ( uC  is the unrelaxed stiffness matrix of the material), therefore the overall governing differential equation can be written as:     11NuP iP i P iC    (3.20) It should be noted that the above equations are very similar to those obtained for transversely isotropic composites by Zobeiry (Zobeiry 2006).   According to Equation (3.15), different stress and strain vectors (6×1) are associated with each component of stiffness matrix in the above equations. By neglecting the zero components in these vectors, they can be simplified as:          1112223313445523662211112233 3312 1113332322;222CCCCCCCCC                            (3.21)        11111222221233331344 131255132323662311122223311331213 223323;CCC CCC CCC CCC CCCC C                         (3.22) 66  Once these vectors have been determined, the total stress vector can be obtained as follows:                    11 12 1312 22 2313 33 2344556611 11 1122 22 2233 33 33121323C C CC C CC C CCCC                       (3.23)  3.3.2 Numerical Implementation Having obtained the governing differential equations for viscoelastic behaviour of an orthotropic material in a format consistent with those obtained for a transversely isotropic material by Zobeiry (Zobeiry 2006), the same solution technique can be used to solve the differential equations. The solution technique has been presented in (Zobeiry 2006) and it is repeated here for completeness.  Rewriting Equation (3.17) at time steps n   and  1n   for element i  of the generalized Maxwell model, including the free strain Pf (e.g. strain generated due to cure shrinkage or thermal expansion) leads to:         1 11 111( )1( )n nn niP P Pf Pni iP in nn niP P Pf Pni iP iPP               (3.24) Using the Crank–Nicolson method for solving Equation (3.17) , we can write: 67          1 12n n n nP P P Pi i i it           (3.25) Substituting Equation (3.24) in Equation (3.25):         1 1 111 11 1 ( ) ( )2 2 2n n n n n niP P P P Pf Pfn ni iP Pi it t t P                              (3.26) To simplify the above equation, the Crank–Nicolson method is employed for strain:  1 11 1( )2( )2n n n nP P P Pn n n nPf Pf Pf Pftt               (3.27) Substituting Equation (3.27) in Equation (3.26) and rearranging, the stress vector in Maxwell element “ i ”, associated with stiffness matrix component P ,  at time increment ( 1n ) can be calculated as:       1 1 111112( ) ( )111122nin n n n n nP iP P P P Pf Pfi innP iP itPtt                           (3.28) The total stress vector at time increment ( 1n ) can be calculated by summing over all Maxwell elements ( 1, 2, 3,...,i N ) and stiffness matrix components P , as follows:    11 11 22 231; , , ....,N nnP iP iP C C C     (3.29)  A finite element formulation has been developed in (Zobeiry 2006) for transversely isotropic materials. The same formulation can be employed with some minor changes due to the same format of final equations. The formulation is given in Appendix D. For more details the reader is 68  referred to (Zobeiry 2006). Based on this formulation and the equations presented in the previous section, a user material subroutine (UMAT) has been developed for orthotropic materials in this thesis and linked with the commercial computer code (ABAQUS®). This UMAT serves as a tool to define the material properties and constitutive relations which are required for viscoelastic analysis.  3.4 Verification Examples In this section, numerical examples are presented for verification purposes. Solutions to four illustrative examples are given to demonstrate the capabilities of the ABAQUS® implementation of the DF formulations. Numerical results using the “DF code” are compared with available analytical solutions as well as numerical results using the ABAQUS® built-in viscoelastic model which is based on the IF of viscoelasticity. Some issues such as accuracy and convergence are briefly discussed. In the first two examples, the material is isotropic, while in the third the material is transversely isotropic. In the fourth and final example, an orthotropic material is considered to demonstrate the capability of the DF code in modelling such materials. The isotropic and transversely isotropic runs use special cases of the generally orthotropic UMAT.  3.4.1 Isotropic Material under Shear Stress In order to compare the accuracy of the two approaches (i.e. IF and DF) based on their 1-D forms, the DF and the IF of viscoelasticity are employed for predicting the stress in a pure shear loading test (1-D problem). Two cases with two different types of isotropic viscoelastic materials are considered for the analysis. For Case I, a simple isotropic material is considered with a single 69  relaxation time of 0.9899s   and the relaxation shear modulus of the material, G , is expressed as:   333.7 3037.1 exp / 0.9899G t    (3.30) A constant Poisson’s ratio of 0.3 is assumed. The material behaviour is given in Section 3.1.1 of the ABAQUS® Benchmark Manual (ABAQUS Inc. 2010) .   For Case II, the 3501-6 epoxy resin, with the experimentally measured properties from (Kim and White 1996) is employed. In (Kim and White 1996) a constant Poisson’s ratio of 0.35 was assumed and the relaxation modulus was described by the Prony series parameters listed in Table 2-3.   To examine the effect of different loading functions on the predicted stress at a specific time (/ 2t  sec (Case I) and t  109 min (Case II)), three different strain-time functions are prescribed in each case. The stress in the DF is calculated using Equation (3.28), while in the IF the shear stress increment,  is calculated using the following equation (see Section 4.8.1 of the ABAQUS® Theory Manual (ABAQUS Inc. 2010)):  12iNui iuiGG w G              (3.31) where 1n n      is the total strain increment and 1n ni i i      is the viscous strain increment in the ith Maxwell element. Note that, / 2i i   (where i  is the engineering shear strain). The viscous strain increment in the ith Maxwell element is estimated as follows: 70   1 exp exp 1 1 expn ni i ii i it t tt t                                                     (3.32) In other words, the total strain n and the viscous strain ni  at the beginning of the increment are employed to calculate the viscous strain in the ith Maxwell element at the end of the increment, knowing its relaxation time, i , and the time increment, 1n nt t t    (see Appendix E  for the derivation of Equation (3.32)).   Figure 3-3 Comparison between DF and IF solutions for the shear stress in a 1-D problem (pure shear loading) under 3 different applied strain-time functions: (i), (ii) and (iii). The viscoelastic behaviour of the material was characterized by (a) Case I, single element Prony series given in (ABAQUS Inc. 2010), and (b) Case II, multiple Prony series given in (Kim and White 1996). The predicted stress (shear stress) at t = π/2 sec and 109 min, for cases I and II, respectively is shown as a function of the time step Δt used in the numerical computation.  ( )i( )ii( )iiit (a) (b)σ(GPa)σ(GPa)T71  The predicted stresses at t T where / 2T  sec and 910T  min for Case I and Case II, respectively using both approaches (IF and DF) are compared in Figure 3-3. A convergence study is performed employing different time increments for each prescribed load function. For load functions (i) and (ii), the predicted stress using IF is not dependent on the time increment being used and match with the exact solution obtained by the application of the correspondence principle. This is independent from the material behaviour and has been observed in both cases (Figure 3-3 (a) and (b)). It should be noted that in the IF formulation employed for this example, a linear variation of strain within each time step has been assumed (see Equation (E.8)) and therefore, the results of the IF should be insensitive to the time increment size. However, in the DF approach, this assumption was not made. Using DF, the predicted shear stress depends on the time increment being used. By using smaller time increments in the load function (i) and (ii), the accuracy of the DF prediction for shear stress increases at the end of t T  and the result approaches the IF result which matches with the exact solution.  Applying a non-linear strain-time loading function (i.e. load function (iii)), the IF predictions become sensitive to the time step size being used in both cases. It should be noted that the example shown here is just a mathematical one and both methods give accurate results for relatively large time steps employed in practical cases. This 1-D example indicates that the accuracy of the DF is comparable to the IF of viscoelasticity for non-linear strain-time loading function. In the next examples, the accuracy of DF and IF for more realistic 3-D solids is further examined.  72  3.4.2 Isotropic Material under Uniaxial Stress In this example, an isotropic viscoelastic solid rod with dimensions of 10×1×0.5 mm is examined in a creep test with an applied longitudinal stress of 100 psi (0.689 MPa), as shown in Figure 3-4 (a). Note that the longitudinal direction in this example is the direction 2. The rod is modeled with a single solid element and the relaxation shear modulus expressed by Equation (3.30). The bulk modulus is assumed to be constant and equal to 100,000 psi. This example is taken from Section 3.1.1 of the ABAQUS® Benchmark Manual (ABAQUS Inc. 2010), in which the exact analytical solution for longitudinal strain is also given as:    /1022 0.1 1 0.9 te     (3.33) The same example has also been used by Zobeiry (Zobeiry 2006) for verification purposes. Here, the same example is analysed using the orthotropic UMAT.  In order to compare the DF and IF results in this example, automatic time stepping as well as fixed time steps, are employed. All analyses were performed by the ABAQUS® software. The automatic time stepping algorithm being used is the one described in ABAQUS® software (ABAQUS Inc. 2010). According to (ABAQUS Inc. 2010), this algorithm is based on the number of iterations required to find a converged solution. By choosing automatic time incrementation, the computation begins with an initial increment size. The size of subsequent time increments are adjusted based on the convergence rate. For this example, an initial increment size of 10-5 times the total time period (50 sec) was selected.    73   Figure 3-4 Isotropic rod under a constant uniaxial stress (creep condition). (a) Comparison of the longitudinal strain, as predicted numerically (with both DF and IF), and analytically using ABAQUS® automatic time stepping algorithm. (b) Comparison of the longitudinal strain predictions at t = 30 sec using fixed time steps (strains are normalized by the exact value,   ).  The resulting longitudinal strains are compared in Figure 3-4(a). It can be seen that even though both results (DF and IF) are very close to the analytical solution, the IF requires more time steps (14 versus 9 time steps) to satisfy the convergence criterion.  To examine the dependency of solution on the time step size, the analysis was also performed with fixed time steps, similar to the previous example. The longitudinal strains at t  30 sec after the initial load application are computed with both approaches using various fixed time steps. These strains are plotted as a function of t  in Figure 3-4(b). The results imply that for this example, the DF approach is able to predict the strain with greater accuracy compared to IF using the same increment of time.  123100 psi(a) (b)74  3.4.3 Transversely Isotropic Material under Thermal Loading As an application of the current development, a thermo-viscoelastic analysis of a transversely isotropic rod (Figure 3-5(a)) during a temperature change is performed. The dimensions of the rod are 10×1×0.5 mm, and it is completely confined and subjected to an increase of temperature, equal to 10 oC. The analysis has been performed in (Zobeiry et al. 2014) using a single solid element with the material properties listed in Table 3-1. Here, the analysis is also performed with the orthotropic UMAT using the stiffness matrix components given in Table 3-2 (derived from properties listed in Table 3-1).  Table 3-1 Viscoelastic properties of the transversely isotropic material given in (Zobeiry 2006) .          η (MPa) 1270.0 1260.0 1.0K2 (MPa) 64.0 18.0 1.0G12 (MPa) 40.0 0.7 1.0G23(MPa) 27.0 0.7 2.0l (MPa) 64.0 22.0 1.5α1 (με/°C) 0.010 0.010 -α2 (με/°C) 0.001 0.001 -Property Glassy Rubbery τk (s)75   Table 3-2 Viscoelastic properties of the transversely isotropic material employed by Zobeiry (Zobeiry 2006).   The analytical solution for this problem has been given in (Zobeiry 2006). The principal stresses 11  and 22  are calculated as:            /1.511 11 22 33/1.522 11 2 22 33126.44 0.842.56 0.92 4.2t tt te eK e e                    (3.34) The principal stresses from the DF and the analytical solution are compared in Figure 3-5. According to this figure, the finite element predictions using DF and the analytical solution, for both principal stresses, agree very well. It should be noted that the DF results using the orthotropic UMAT (current work) and the transversely isotropic UMAT (Zobeiry 2006; Zobeiry et al. 2014) are identical.   C11 (MPa) 1270.0 1260.0 1.0C22 (MPa) 91.0 18.7 1.0C33 (MPa) 91.0 18.7 1.0C44 (MPa) 40.0 0.7 1.0C55 (MPa) 40.0 0.7 1.0C66 (MPa) 27.0 0.7 2.0C12 (MPa) 64.0 22.0 1.5C13 (MPa) 64.0 22.0 1.5C23(MPa) 37.0 17.3 1.0α1 (με/°C) 0.010 0.010 -α2 (με/°C) 0.001 0.001 -α3 (με/°C) 0.001 0.001 -Property Glassy Rubbery τk (s)76   Figure 3-5 (a) Transversely isotropic rod under temperature change. Comparison of numerical predictions (using DF) and analytical solution for (b) longitudinal stress, and (c) transverse stress. A constant time step size of 1 sec is used.  3.4.4 Orthotropic Material under Uniaxial Stress This example is taken from (Poon and Ahmad 1998) in which an orthotropic rod is subjected to an applied uniaxial stress in the 1-direction prescribed as:    , 0 10001000tt t     (3.35) The viscoelastic properties of the orthotropic material are given in Table 3-3.      123t (sec)10( )oT C10t (sec)10( )oT C10(b)(a)(c)77  Table 3-3 Viscoelastic properties of the orthotropic material taken from (Poon and Ahmad 1998).   The variation of non-zero strain components are plotted in Figure 3-6.  In this figure, the analytical solutions have been obtained using Laplace transform and shown with solid lines. The software package Mathematica® has been employed for this purpose. The numerical predictions, depicted with symbols, show very good agreement with analytical exact results in all three directions. This validates the capabilities of current approach in modelling orthotropic viscoelastic materials.    C113.857 0.319 0.100C223.705 0.303 0.090C33 3.582 0.291 0.080C44 2.000 0.160 0.100C55 1.800 0.150 0.090C661.600 0.140 0.080C12 1.674 0.139 0.100C13 1.659 0.137 0.100C23 1.614 0.291 0.100Property Cijg (GPa) Cijr (GPa) τk (s)78   Figure 3-6 An orthotropic rod subjected to an applied uniaxial stress in the longitudinal direction, and free of stress in other directions (Poon and Ahmad 1998). Comparison of numerical predictions (using DF) and analytical solution for strains in the longitudinal and transverse directions.  3.5 Reverse Micromechanics and Application to Process Modelling Evaluation of phase average stresses and strains is an important task in many applications of composite materials. For instance, in design of composite parts it is crucial to determine the stresses and strains in each constituent (e.g. resin). Zobeiry (Zobeiry 2006) proposed a procedure to determine the stress and strain of each phase (at micro-scale) from the stress and strain in the composite structure (at macro-scale). The procedure was called “Reverse Micromechanics” in (Zobeiry 2006). Knowing the stress and strain field at the macro-level, this procedure can be employed in the presented multi-scale approach as a post-processing step to determine the phase average stresses and strains at each integration point of the macro-structure. This would enable us to obtain the resin pressure from the resin stress tensor and combine the presented multi-scale t (sec)1( )t10-3123( )t10-379  approach with the recently developed integrated flow-stress framework (Haghshenas 2012) for process modelling of composite structures. This is the subject of our future research.  3.6 Summary and Conclusions As a last step in multi-scale modelling of complex viscoelastic composite structures, an efficient modelling strategy is required. For this purpose, the differential approach developed by Zobeiry (Zobeiry et al. 2006; Zobeiry 2006; Zobeiry et al. 2014) for transversely isotropic materials was extended to model the response of generally orthotropic viscoelastic materials. The presented approach enables the simulation the viscoelastic behaviour of orthotropic composite structures that consist of at least one viscoelastic phase.  A user material subroutine (UMAT) for orthotropic viscoelastic materials, based on the UMAT developed previously by Zobeiry (Zobeiry 2006)  for transversely isotropic materials, was coded in FORTRAN to illustrate how this approach can be implemented in a general purpose finite element code. In this work, ABAQUS® was employed as a commercial finite element code within which this subroutine was implemented. The developed subroutine enables analysis of complex structures at the macro-scale using the input parameters obtained from the output results of the micromechanical analysis at the lower-scale. The lower-scale analysis can be performed either analytically (e.g. analytical closed-form micromechanics equations) or numerically (e.g. computational homogenization technique) as described in Chapter 2 to provide these input parameters.   80  The constant Poisson’s ratio assumption employed in the other 3-D DF approaches (e.g. (Chazal and Pitti 2009)) was not made in the current approach. Therefore, the presented DF formulation can be employed to connect the micro-scale analysis to macro-scale analysis of orthotropic structures without restrictive assumptions. Moreover, it can be incorporated in other available frameworks (e.g. integrated framework for process modelling (Haghshenas 2012)) to increase the versatility of these frameworks and their ability to tackle orthotropic problems.   Although some advantages of the DF approach over the IF approach were presented in a previous work for 1-D problems (Zobeiry et al. 2006), extension for 3-D multi-axial modelling of orthotropic materials, time step size effect and convergence of results were presented and discussed in this Chapter. Numerical examples demonstrated the capability and accuracy of the presented approach in viscoelastic analysis of composites. 81  Chapter 4: Micro-Mechanical Modelling of Strand-Based Composites  Using the principles of classical micromechanics, analytical equations are developed in this chapter to estimate the effective orthotropic properties of a unit cell of strand-based composites according to their constituent phase properties and their microstructural features such as resin thickness, void content and strand geometrical characteristics. Although a special type of strand-based wood composite product, Parallel Strand Lumber, is considered here as an illustrative example, the methodology can be used for other wood composites consisting of high volume fraction of wood strands.  The predictive accuracy of the derived analytical equations is investigated through comparisons with numerical results. Applications of these equations in a linear viscoelastic analysis are also discussed. The analytical micromechanics models developed here provide an efficient means of computing effective properties of a unit cell of strand-based composites. These models can then be used within a multi-scale modelling framework (see Chapter 5) to simulate the macroscopic behaviour of structures made of such materials.  4.1 Background With the advent of numerous engineered construction materials and new processes used to manufacture them, it is desirable to have tools that can reliably predict the behaviour of structures made with such materials. The motivation for this chapter comes from structural wood composite industry. Strand-based wood composites are a new category of building materials that are widely being used in the construction industry, especially in North America. This type of 82  material which belongs to a class of wood-based composites made of wood strands (Forest Products Laboratory 1999; Sugimori and Lam 1999) is becoming popular in other parts of the world due to its reasonable cost compared to conventional construction materials, environmental friendliness and tailorable properties.  Strand-based wood composites consist of discontinuous strands. Strands are chips of wood cut from trees which are bonded together with a thermoset resin (e.g. Phenol Formaldehyde) (Forest Products Laboratory 1999). Due to the cutting method widely used in the wood composite industry, strands are close to being rectangular in shape (Nishimura et al. 2004). The mechanical properties of strands are direction dependent. Wood strands are usually stiffer in the longitudinal direction than in the transverse direction. The directions are defined according to the annual ring structure of the tree as illustrated in Figure 4-1. Radial and tangential directions, denoted by R and T, refer to directions normal and tangent to the annual rings, respectively.    Figure 4-1 Geometrical characteristics of wood strands.  231L (1)R (2)T (3)Tree Cross-SectionLRT83  Unlike conventional fibre-reinforced composites (e.g. carbon fibre reinforced plastics), strand-based wood composites (e.g. Parallel Strand Lumber or PSL) consist of high volume fractions of fully orthotropic fibres (wood strands) with a complex cellular microstructure. In these composites, resin content is usually less than 5% by weight. A typical sample of a PSL, which is used as beams and columns in wood frame buildings, is shown in Figure 4-2. Complex manufacturing process of strand-based wood composite materials induces voids in their microstructure. The void size and void content depend on the processing parameters and can affect the properties of these materials (Clouston and Liu 2006; Sugimori and Lam 1999).    Figure 4-2 (a) Small sample of PSL (Parallel Strand Lumber) beam and (b) PSL cross-section  Macroscopic behaviour of composite structures, including both their time-independent and time-dependent elastic response, depends on the microstructure description of the material at different scales. Based on homogenization techniques, several multi-scale models have been developed to take into account the effect of microstructure on macroscopic behaviour of engineering structures (Gereke et al. 2012; Kanoute et al. 2009; Stuerzenbecher et al. 2010). In these models, the 20 mm(a) (b)84  effective properties of a unit cell of the material at the lower scales are determined and used in the higher scales.  Benabou and Duchanois (Benabou and Duchanois 2007) and Stuerzenbecher et al. (Stuerzenbecher et al. 2010) used analytical continuum micromechanics to estimate the properties of each layer of strand-based boards. Once the properties of each layer were determined in the first step, lamination theory was used in the second step to predict the elastic moduli of strand-based boards. Gereke et al. (Gereke et al. 2012) proposed a two-step numerical multi-scale framework for predicting the elastic properties of strand-based wood composites. In the first step, the effective elastic properties of a material unit cell were estimated using a computational homogenization technique. Knowing the effective properties of the unit cell, stochastic finite element structural analysis of a beam (assembly of several unit cells) under three-point bending was performed in the second step. As noted in (Gereke et al. 2012), the key step was the first one in which the properties of the material unit cell were estimated based on the constituents’ properties and geometrical features. In (Gereke et al. 2012), strands were assumed to be fully covered with resin and perfectly bonded. In other words, the presence of voids in the microstructure was ignored.   Dai et al. (Dai et al. 2007) showed that resin distribution has a key effect on the final properties of oriented strand boards. Thus, in the present chapter, we improve the previously developed unit cell model (first step of the multi-scale approach in (Gereke et al. 2012)) by introducing voids in the resin phase (Section 4.2.2). This enables us to estimate the effective properties of a unit cell of partially covered strands with resin. Moreover, computational homogenization method used in 85  (Gereke et al. 2012) to estimate the unit cell properties is replaced here by an analytical (closed-form) approach in order to further improve the computational efficiency of the multi-scale framework. Analytical expressions for orthotropic, elastic, discontinuous, rectangular shaped fibre composites are thus developed in this chapter. This is first done for the case of full resin coverage (Section 4.2.1.1) and then for partial coverage (Section 4.2.1.2). The resulting analytical expressions will help in better understanding the load transfer mechanism in strand-based composite materials as well as understanding the effect of voids on engineering elastic constants.  Another essential benefit of developing an analytical approach is its application in predicting the time-dependent (viscoelastic) properties. Computational viscoelastic homogenization techniques are indeed usually computationally intensive thus hindering their application in certain industries such as the wood composite industry. Therefore, with the aid of the correspondence principle analytical expressions are also used to estimate the viscoelastic properties of a material unit cell, namely the time evolution of its elastic constants (Young’s and shear moduli as well as Poisson’s ratios). The results are discussed in Section 4.3.3.  4.2 Elastic analysis 4.2.1 Analytical Approach Throughout the literature, different analytical models have been proposed to predict the properties of various types of wood composites. Facca et al. (Facca et al. 2006) used different analytical micromechanics models to predict Young’s modulus of different natural short fibre composites including wood fibre plastics. Halpin-Tsai equations were found to predict the 86  stiffness of natural short fibre composites more accurately than other classical micromechanical schemes such as Rule of Mixtures and Narin’s Shear Lag model (Nairn 2004). According to (Facca et al. 2006), increasing the fibre volume fraction decreases the accuracy of all micromechanical models in predicting the stiffness of composites.   Although strand-based composites consist of high volume fractions of strands, many researchers (Fan and Enjily 2009; Mundy and Bonfield 1998; Shaler and Blankenhorn 1990) used simple analytical micromechanics equations, such as Rule of Mixtures and Halpin-Tsai (Halpin and Kardos 1976) equations, developed for conventional composites in order to predict the flexural modulus of oriented flakeboards. Halpin and Kardos (Halpin and Kardos 1976) employed Halpin-Tsai equations (see Chapter 2) for rectangular shaped short fibre composites.   Table 4-1 Parameter   used in calculating the elastic moduli of rectangular shaped short fibre composites.   For rectangular shaped short fibre composites, the parameter   depends on the fibre dimensionsL , T and R  (see Figure 4-1). Table 4-1 summarizes the expressions used in calculating  for an anisotropic fibre. In this chapter, the accuracy of the analytical models developed for rectangular shaped strand-based wood composites (considered in this thesis) are compared with Halpin-Tsai predictions as well as those from finite element simulations (Section 4.3.3). The proposed E 1 E 2 E 32 (L/R)Parameter1 1 1PropertyG 12 G 13 G 23RT    LR   TR   87  analytical approach for predicting the effective properties of highly filled strand-based wood composites at the meso-scale are described below.   Unlike conventional fibre reinforced composites, in strand-based wood composites fibres (strands) are discontinuous. As the first assumption, strands are assumed to be rectangular prisms covered with a thin layer of resin with constant thickness throughout the material. Additionally, we assume that the resin is fully cured. Depending on the amount of resin and manufacturing process, strands are either partially or fully covered with a thin layer of the resin in different wood composite products. Using this assumption a unit cell of the material can be identified as shown in Figure 4-3. This figure simply shows a wood strand surrounded by 6 blocks of resin in a selected unit cell of the material.    Figure 4-3 A unit cell of a strand-based composite. For clarity, the resin blocks are shown detached from the strand.  TLRResinWoodResinResin32 1r1t288  In Section 4.2.1.1, analytical equations for strands fully covered with resin are derived. Afterwards, it is shown how these equations could be used for partially covered strands with minor modifications. It should be mentioned that the idealized unit cell depicted in Figure 4-3, is discretized and used in the numerical approach (Section 4.2.2) as well.  Throughout the rest of the thesis, in order to distinguish the equations derived for strand-based composites from the conventional short fibre composites equations, the term “strands” is used instead of “fibres” to describe the rectangular shaped fibres.  4.2.1.1 Full Resin Coverage A micromechanical scheme is first developed for estimating the Young’s moduli of a material unit cell as defined in Figure 4-3. This scheme is based on a combination of Voigt and Reuss models (Tucker and Liang 1999) and enforcement of isotress and isostrain conditions between the unit cell blocks. For each of the Young’s moduli, two successive steps are applied as illustrated in Figure 4-4.   89   Figure 4-4 Micromechanical model used for normal properties. (a) Isostress condition between a strand and two resin blocks in front and back and (b) Isostrain condition between the homogenized new strand and the four resin blocks surrounding it.  For loading applied along the strand direction, the first step consists of assuming that an isostress condition prevails between the strand and the two resin blocks in front and back of it. As a consequence, the homogenized Young’s modulus in the longitudinal direction of the three-block system in Figure 4-4 (a) is given by the Reuss model:  1 11-1 s sns s rV VE E E  (4.1) It should be noted that wood strand is a fully orthotropic material whereas resin is considered as isotropic. Therefore, a subscript is associated with each of the three directions except for the resin modulus. The strand volume fraction in the three-block system is defined using the strand length ( L ) and the resin thickness (rt ) between the two strands as follows:  srLV L t  (4.2) (b)ResinNew Strand32 1(a)90  In Equation (4.1) the homogenized Young’s modulus of the three-block system is denoted by 1nsE and we refer to it as the longitudinal Young’s modulus of the new strand.  In the second step, it is assumed that this new strand undergoes a uniform deformation with the four resin blocks surrounding it. In other words, an isostrain condition is assumed to prevail. This leads to the following expression (Voigt model) for the longitudinal Young’s modulus of the material unit cell:   1 1 1c ns ns s rE V E V E   (4.3) where nsV  denotes the volume fraction of the new strand, which in terms of the total volume is given by:     ns r rTRV T t R t   (4.4) By substituting Equations (4.1), (4.2) and (4.4) in Equation (4.3), the longitudinal modulus of the material unit cell is obtained as follows:     111s r r rc rr r s r r rE E L t tTRE ET t R t E t E L L t              (4.5)  The transverse Young’s moduli in the other two directions are derived in a similar way. Since the resin layer is very thin, its thickness may be considered to be negligible compared to the strand dimensions (1 2 3/ 1; with 1, 2, 3 such that , ,r it L i L L L R L T    ). Therefore, high order terms (greater than the 2nd order) of /r it L   are neglected in the expressions. Thus, the Young’s 91  moduli of the unit cell, when normalized by the Young’s modulus of the resin, may be simplified as follows:  1 1 1 11iii j kiiici j krEEE                  (4.6) where icE  is the unit cell modulus in the “i ” direction and i  is the ratio of strand modulus in the “ i ” direction to the resin modulus:  1,2,3isirE iE   (4.7) Also, i  denotes the ratio of strand length in the “ i ” direction ( iL ) to resin thickness ( rt ):   , 1, 2, 3iirL it    (4.8) Similarly, j  and k  are the relative strand length to resin thickness ratio in the other two directions.   A similar two-step approach with successive application of isostress and isostrain assumptions as depicted in Figure 4-5 leads to the normalized shear properties of the unit cell in different planes as follows:  1 11ij ijiji j kiji j kijcr ij ijGGG                    (4.9) 92  Here, ij  is defined as the ratio of the shear moduli of the strand to resin in the corresponding plane:  , 1,2,3ijsijrG i jG   (4.10)   Figure 4-5 Micromechanical model used for shear properties. (a) Isostress condition between embedded strand and resin surrounding it. (b) Isostrain condition between the homogenized new strand and two resin blocks in front and back.  If the isostrain condition is first applied between the strand and the two resin blocks, followed by application of isostress conditions, other estimates can be obtained for effective properties of the material unit cell. The unit cell Young's moduli in this case are given by:  1 11i iii j kicrii ii j kEEE                   (4.11) and the shear moduli are given by: New StrandWood StrandResin 23 nsResin 23 ns 23 r 23 r32 1(b)(a)93    1 111 1iji jijijcr ikiji j kjGGG                  (4.12) In Section 4.2.3, estimates of effective normal and shear properties obtained with Equations (4.6) and (4.9) will be compared to those of Equations (4.11) and (4.12).  For Poisson’s ratios 12  and 13 , a two-step approach as depicted in Figure 4-6 is proposed. Assuming first an isostrain condition in the longitudinal direction between the strand and the four resin blocks that cover it (see Figure 4-6 (b)), the longitudinal and transverse displacements of the five-block system due to a global applied stress 1  may be written as follows:  1 1111 2 32 12 131 2 32 312s rs rs s ss s ss s ss rrr r r rr r ru L LE Eu RE E Eu tE E E                    (4.13)  94   Figure 4-6 Stress-strain analysis in a material unit cell for determining     and    . (a) A unit cell under normal between the resin and the strand inside; and (c) Isostress condition between the front and rear resin blocks and the homogenized new strand in the middle.stress; (b) Isostrain condition in the longitudinal direction and isostress condition in the transverse directions   Application of load along the strand direction induces stresses due to the mismatch between the strand and resin properties in the other directions. As shown in Figure 4-6 (b), while isostrain condition is assumed in the longitudinal direction between the strand and all resin blocks (except the front and rear blocks), isostress conditions are assumed to prevail between strand and resin in the transverse directions: 2 2s r   (resin top and bottom blocks) and  3 3s r   (resin left and right blocks). +b)=c)2F3FTLRResinWoodResinResinr1t21s3s2s2r3s1r2s3r1r(a)32 1((c)95  Due to equilibrium of forces in the transverse directions, transverse stresses in the resin blocks may be expressed using transverse stresses in the strand and geometrical features:   2 23 3r srr srTtRt   (4.14) Knowledge of the stress state in the resin blocks surrounding the strand allows calculation of the displacements in the transverse directions of the five-block system in the following way.   Considering compatibility conditions between side resin blocks and the three-block system (in vertical direction for 2u  and in horizontal direction for 3u ) sandwiched between them ( Figure 4-6 (b)), the face and core displacements should be equal, therefore:        2 23 3face coreface coreu uu u (4.15) where face displacements in  2 faceu  corresponds to the displacements of left and right side resin blocks and  3 faceu  corresponds to the displacements of the top and bottom side resin blocks. They are expressed as follows:        31 222 313sr rr r rfacer r rs rrr r rfacer r ru R tE E Eu T tE E E                   (4.16) The core consists of two materials, strand in the centre and resin top and bottom blocks for  2 coreu , and strand in the centre and resin left and right blocks for   3 coreu . Therefore their transverse displacements are composed of two additive contributions: 96     2 2 23 3 3s rcores rcoreu u uu u u   (4.17) Substituting Equation (4.14) in Equation (4.16) and in Equation (4.17), with Equation (4.13) to express  2ru  and 2su , and then using the compatibility conditions (Equation (4.15)), allows computing the transverse displacements of the five-block systems ( 2nsu  and 3nsu ) as functions of normal stress and geometrical parameters:            2 2 2 13 3 3 1, , ,, , ,ns s rcore facens s rcore faceu u u f R T tu u u g R T t     (4.18)  Finally, Poisson’s ratios of the five-block system (strand plus 4 resin blocks) are calculated by using the following expression:    21213131////ns rnsns rnsu R tu Lu T tu L   (4.19) These expressions describe the Poisson’s ratios for the new strand system (coated strand). By neglecting terms with high order (greater than the 2nd order) of resin thickness and resin Poisson’s ratios, these expressions may be simplified further. The simplified version of 13ns  is given below:  97      2 22 3 3 23 2 3 2 3 122 2 2 22 3 2 3 2 3 2 3 2 3 3 2 3 2 3 13132 2 2 212 3 2 3 3 2 2 3 2 2 3 2 3 2 3 2 32 2 2 22 3 2 3 2 3 3 232 21 2 3 2 3 3 2 2 3 3 2-122 2s r ssnsrr sDD                                                                              2 3 2 33   (4.20) In the above equation, parameters i  and i  are the same as those defined in Equations (4.7) and (4.8), respectively.  By interchanging the subscripts 2 and 3 in Equation (4.20), similar equation is derived for 12ns . Once the two Poisson’s ratios of the new strand (12ns and 13ns ) have been determined, the Poisson’s ratios of the unit cell are estimated using isostress condition depicted in Figure 4-6 (c). For simplicity the final expressions are not provided here but the corresponding results will be shown in Section 4.3.3.  98   Figure 4-7 Stress-strain analysis in a material unit cell for determining    . (a) A unit cell under transverse stress; (b) Isostress condition between top and bottom resin blocks and the strand inside; and (c) Isostrain condition between the side resin blocks and the homogenized new strand in the middle.  In order to calculate the third Poisson’s ratio (23c ), a two-step approach as depicted in Figure 4-7 is adopted. Due to the small dimension of the strand in the radial (2) direction, isostrain condition was followed by an isostress condition for estimating of 12c and 13c . Along the strand direction, stress is transferred mostly through the strand and the two resin blocks in front and rear. However, perpendicular to the strand direction, stress is transferred through both the strand and the resin blocks surrounding it. Therefore, to obtain23c , isostrain condition between the new strand and the resin around it (Figure 4-7 (c)) seems to be more relevant after an isostress condition in the new strand system as defined in Figure 4-7 (b). By neglecting terms with high = +1F2F3F2ns2ns2ns 2r 2ns2r2r32 1(b)(a)(c)TLRResinWoodResinResinr1t22299  orders (greater than 2) of resin thickness and resin Poisson’s ratio, the overall unit cell Poisson’s ratio is finally obtained as follows:           2 2 3 23s 13s 21s 2 1 2 1 32 1 2 2 2 3221 232 213 23 23 2 3 32 1 1 1222322 2 1 2 3 2 22 23 321 3 1 1 3 1 3 1 213 31 21 23+ + -1+=111r s r sr r r s s r s s rr r r rs s s s s scv v v v v v vv v v v v v v v vv v vDDv v v v v v v v                                              2 1 3 12 23 32 23r r rsv v v      (4.21)  4.2.1.2 Partial Resin Coverage Partial coverage is analysed using interface parameters (Hashin 1990; Nairn 2007). Interface parameters are defined as:  ntttD u (4.22)  nnnnD u (4.23) where nt  and nn  are tractions tangential and normal to the interface area, tu  and nu  are the interface displacement jumps described by (Hashin 1990). According to (Hashin 1990), infinite values of tD   and nD  imply zero displacement jumps, namely perfect interface conditions. At the other extreme, zero values correspond to debonded interface. Any finite positive values of interface parameters define an imperfect interface due to the presence of an interphase, for instance.  100  For fully covered strands with a thin layer of resin and an imperfect interface in the sense previously recalled due to (Hashin 1990), tD   and nD  may be defined in terms of strand and resin properties (see (Nairn 2007)):  1 1 1rt r stD G G     (4.24)  1 1 1rn r stD E E     (4.25) It may be noticed from Equations (4.24) and (4.25) that the interface becomes perfect as soon as resin properties (rG and rE ) approach strand properties ( sG and sE ).  The interface parameters of a partially covered wood composite may now be deduced by introducing the concept of resin area coverage of strands. For this purpose, Figure 4-8 shows two strands partially bonded together by a thin layer of resin with thickness rt . The behaviour of this system in shear and normal loadings are analysed. In Figure 4-8 (a), the tangential displacement jump is depicted. The latter may be written as a function of the shear properties of the strand       (sG ) and the resin ( rG ), and the resin thickness ( rt ) as follows:  srt r r s r rr su t t tG G         (4.26) where r   and s  are the resin and strand shear strains due to shear stresses in the resin and strand, r  and s , respectively.   Assuming all the shear force from one strand is transferred uniformly to the other strand through the bond leads to: 101   ssr st rr sAAu tG G            (4.27)  By defining Resin Area Coverage (aR ) as:  rasAR A (4.28)   Figure 4-8 Incorporating interface properties for modelling the effect of partial resin coverage. (a) Shear stress; and (b) Normal stress.  Equation (4.27) may be rewritten in terms of resin area coverage as:  1 1 st s ra r s tu tR G G D      (4.29) R(a) (b)102  Therefore the tangential interface parameter may be derived in terms of resin thickness, resin area coverage and the shear moduli of resin and strand:  1 1 1rt a r stD R G G     (4.30) By comparing Equation (4.30) with Equation (4.24) for fully bonded strands, the behaviour of partially bonded strands in shear may be modeled as the behaviour of fully bonded strands with equivalent shear property defined as:   r a rEqG R G  (4.31) With a similar approach, normal interface parameter nD  is derived in terms of resin thickness and the Young’s moduli of the resin and strand using Equations (4.32) to (4.35).     2 2n s r r s ru L t L t      (4.32) Assuming full transfer of load through the interface, the normal stress in the resin phase may be estimated as:  rr nsAA      (4.33) Substituting Equation (4.33) in Equation (4.32) leads to:  r rn na r st tu R E E     (4.34) Finally the normal interface parameter nD  is given by:  1 1 1rn a r stD R E E     (4.35) 103  Comparing Equation (4.35) with Equation (4.25) for fully bonded strands, the behaviour of partially covered strands in tension may be modeled using the concept of resin equivalent Young’s modulus.   r a rEqE R E  (4.36) Equation (4.31) and (4.36) are subsequently used in analytical approach for estimating the effective properties of a unit cell of wood composites with partially covered strands. The accuracy of this approach is evaluated in the next section using numerically generated reference data.  4.2.2 Numerical Approach In this section, assuming periodic microstructure for the strand-based wood composite, an appropriate unit cell of the material is identified and its homogenized properties are determined. For modeling purposes, the material is idealized as a periodic assembly of rectangular strands with the same size and covered by a thin layer of resin with constant thickness (Figure 4-9(b)). Figure 4-9(c) shows the corresponding unit cell for strands that are fully covered with resin. This unit cell (Figure 4-9(c)) is then discretized using 8-node linear brick elements, C3D8 in the finite element software, ABAQUS® (ABAQUS Inc. 2010) (see Figure 4-9(d)).   To estimate the elastic properties, the stiffness tensor of the unit cell is computed using six elementary loadings (three uniaxial and three simple shear loadings) and periodic boundary conditions as explained in Appendix B. In order to compare the analytical and numerical results, the strand thickness ( R ), width (T ) and length ( L ) are chosen to be 5 mm, 13 mm and 600 mm, 104  respectively. These values are taken from the experimental study conducted by Arwade et al. (Arwade et al. 2009).    Figure 4-9 Identifying the material unit cell  (a) PSL meso-structure; (b) idealized meso-structure; (c) material unit cell; (d) discretized unit cell.  It should be emphasized that in earlier works (Gereke et al. 2010; Gereke et al. 2011; Gereke et al. 2012) a similar approach was pursued for modelling the effective elastic properties of the material unit cell. There, it was assumed that PSL was free of voids and that there was perfect bonding between strands and resin. However, voids are present in the microstructure of most strand-based composite products such as PSL and OSB and should be taken into account in our numerical approach.   Here we extend the approach by considering strands that are partially covered with resin. The size of these voids varies throughout wood composite products (Clouston and Liu 2006; Dai and Steiner 1994; Sugimori and Lam 1999) and therefore, two types of voids are considered. In Section 4.2.3, the effect of these two types of voids on elastic properties will be briefly discussed.  (a)ResinWood Strand(a)(b) (c) (d)105  In order to investigate the effect of strands’ resin coverage or voids, two scenarios are considered. In the first scenario, the thickness of the resin (rt ) that bonds two adjacent strands is assumed to be constant. In other words, by increasing the amount of resin, only the resin area coverage (aR ) defined in Equation (4.37) increases:     / 1a C r r rRTLR R R t T t L t       (4.37) where CR  is the resin content by volume. Figure 4-10 depicts the linear relationship used between resin area coverage (aR ) and resin content by volume ( CR ) according to the first scenario.    Figure 4-10 Resin area coverage as a linear function of resin content for partial resin coverage, scenario 1 (constant resin thickness).  In the second scenario, which is more realistic, the resin thickness is increased as more resin is applied (Figure 4-11). This is due to the possibility of resin droplets overlapping each other in some local points. Dai et al. (Dai et al. 2007) studied the resin distribution in oriented strand 106  boards. According to this study, resin area coverage increases with resin content nonlinearly. Using statistical analysis, Dai et al. (Dai et al. 2007) derived a relationship between resin area coverage (aR ) and resin content by weight. This relationship can be rewritten on the basis of volume. The resulting relation (Equation (4.38)) is used for the second scenario and referred to as Dai’s model, Figure 4-11.      1 p 2 1 1exs CC s Car rRRR t MC R R       (4.38)  In the above, MC  is the wood strand moisture content (here we assume it to be 12%), s  and r  are densities of the strand and resin, respectively.    Figure 4-11 Resin area coverage as a function of resin content according to Dai’s model (Dai et al. 2007) for partial resin coverage, scenario 2 (variable resin thickness).  107  In order to model partial coverage (i.e. for aR <100%) according to both scenarios (constant resin thickness and Dai’s model), resin elements are selected randomly and replaced by “void” elements (elastic properties significantly lower than those of the resin) with a constraint of periodicity at the boundary of the unit cell. Specific Python® scripts have been developed for this purpose. Ten realizations are performed for each case considered in both scenarios. For each realization, the stiffness tensor of the unit cell is classically computed using six elementary loadings and periodic boundary conditions. The arithmetic average and standard deviation of resulting engineering constants over the realizations are then calculated. Each point on the numerical curves will thus represent the ensemble average over ten realizations.   An alternative methodology to the employment of void elements is also envisioned to account for partial coverage. It entails using modified, degraded, elastic properties for the whole resin phase as it was described for the analytical approach in Section 4.2.1.2. In this method, the effective properties of partially bonded strands are modeled using the unit cell of fully covered strand but with degraded resin properties.  4.2.3 Comparison of Analytical and Numerical Predictions The elastic properties of resin and wood strand used in both numerical and analytical approaches are listed in Table 4-2. Resin is assumed to be isotropic and elastic in this section. The elastic properties of Phenol Formaldehyde (PF) resin, which is widely used in wood composite industry, are taken from the literature (Shaler and Blankenhorn 1990).  The orthotropic elastic properties of strands are those of Pine wood taken from wood handbook (Forest Products Laboratory 1999).   108  Table 4-2 Constituents’ elastic properties of wood and resin.   Figure 4-12 shows the longitudinal modulus 1E  estimates for both fully and partially covered unit cells using numerical finite element method. For partially covered strands, three cases are studied. In the first two cases, resin thicknesses are held constant (scenario 1, Section 4.2.2). Resin thicknesses of 0.28 mm and 0.08 mm are considered. The corresponding results are indicated by filled and hollow squares, respectively. The third case corresponds to the second scenario introduced in Section 4.2.2 i.e. resin thickness increasing with the resin content according to Dai’s model (Equation (4.38)).  MaterialE 1 E 2 E 3Wood 13 1.393 0.856Resin0.467 0.456 0.4887.6 2.92 0.30.991 0.909 0.162G 12 G 13 G 23Young's Modulus (GPa ) Shear Modulus (GPa ) Poisson’s ratioν12 ν13 ν23109   Figure 4-12 Numerical results for longitudinal Young’s modulus under partial resin coverage (constant and variable resin thicknesses) using void elements versus full coverage.  It is recalled that each point on the curves corresponds to the arithmetic average over ten realizations of the unit cell with randomly distributed void elements. Standard deviation is indicated and shows that the effective longitudinal Young’s modulus does not vary significantly with the voids spatial distribution except for partially covered strand with the constant resin thickness of 0.28 mm and the resin content below 2%.  It should be noted that the resin area coverage in this case is below 20% (see Figure 4-10). For such very low resin area coverage, the relevance of finite element results obtained using void elements becomes questionable. The same observation has been made for other properties.   110   Figure 4-13 Numerical results for partial resin coverage (Dai’s model) using void elements showing void type effect on longitudinal modulus at different resin contents. Voids Type I have a volume range of 0.1–1mm3 while the volume of Voids type II varies in the range of 1–10mm3 depending on the resin content of the unit cell. The solid curve shows the resin area coverage variation with resin content.  In order to investigate the sensitivity of results with void sizes, numerical simulations were also performed with two different types of voids. Whatever the mechanical property, the effect of void content was found to be greater than the void size. As an illustration, Figure 4-13 shows the numerical results for longitudinal modulus with two different types of voids. In simulations with Void Type I, single resin elements were selected randomly and replaced by void elements, while in those with Void Type II, single resin elements with all their neighbouring elements were considered as void elements. Therefore, in simulations with Void Type II, voids have larger sizes at a given resin content.  Due to the small effect of void type, results are shown only for unit cells with smaller voids (Voids Type I) throughout the rest of the chapter. 111   Figure 4-14 Effect of resin content on the longitudinal Young’s modulus considering two scenarios (constant and variable thicknesses) for partial coverage: (a) Using void elements. (b) Using equivalent resin properties. Finite element results are shown with symbols. Solid lines refer to estimates using equations (4.6) and (4.36). Dashed lines refer to estimates using equations (4.11) and (4.36).  Using both scenarios (constant and variable resin thicknesses) for modelling partial resin coverage, the effects of resin content on the effective elastic properties are presented in Figure 4-14 to Figure 4-16. Analytical estimates are systematically compared to numerical results. Estimates obtained with Equations (4.6) and (4.9) are shown with solid lines while results obtained from Equations (4.11) and (4.12) are presented with dashed lines. In both cases, Equations (4.31) and (4.36) are considered to account for the modification of resin properties due to partial coverage. For more clarity, only the results with resin thickness of 0.28 mm are shown for constant resin thickness.    (a) (b)112   Figure 4-15 Comparisons between analytical and numerical results assuming constant resin thickness and using equivalent resin properties (equations (4.31) and (4.36)) for partial coverage. (a) Transverse Young’s Moduli; and (b) Shear moduli. Finite element results are shown with symbols. Solid lines refer to estimates using a two-step approach with successive isostress and isostrain conditions equations (4.6) and (4.9). Dashed lines refer to estimates using a two-step approach with successive isostrain and isostress conditions (equations (4.11) and (4.12)).  Figure 4-14 shows the comparisons for the longitudinal Young’s modulus. In Figure 4-14 (a), finite element simulations have been performed using void elements while in Figure 4-14 (b) they have been performed using the model of fully covered strand with equivalent resin properties (Equation (4.36)). Only results with resin area coverage higher than 20% are reported in Figure 4-14 (a) because of the low accuracy of the numerical solution (with void elements) below this limit. A very good agreement is observed between analytical estimates by Equation (4.6) (and Equation (4.36)) and numerical results for both constant and variable resin thicknesses. Moreover, Figure 4-14 (a) and Figure 4-14 (b) show that finite element results using void elements or equivalent resin properties are very close. This observation was also made for other effective properties and has been presented in (Malekmohammadi et al. 2013a). These (b)(a)113  similarities between both types of finite element predictions prove the validity of equivalent resin properties, based on the concept of interface parameters, to account for partial coverage.   Figure 4-16 Comparisons between analytical and numerical results assuming variable resin thickness (Dai’s model) and using equivalent resin properties (equations (4.31) and (4.36)) for partial coverage. (a) Transverse Young’s moduli. (b) Shear moduli. Symbols, solid and dashed lines have the same significances as those in Figure 4-15.  Thus, in order to provide reliable numerical reference data to check the validity of the analytical estimates of transverse Young’s moduli and shear properties throughout a wide range of resin contents (including low resin area coverage), only the finite element results obtained with equivalent resin properties are presented in Figure 4-15 (for constant resin thickness) and in Figure 4-16 (for variable resin thickness). Comparison between the analytical and numerical results shows that Equations (4.6) and (4.9) provide very good estimates except for 2E  and 3E . For transverse moduli in directions 2 and 3, the accuracy of analytical estimates diminishes with increasing resin content (see e.g. Figure 4-15 (a)). This is due to small size of strands in these directions. Simple shear-lag analysis shows (a) (b)114  that the stress is not fully developed in these directions and initial isostress assumption made in deriving Equation (4.6) is not perfectly valid. However, it should be noted that the error between the analytical estimates and the numerical results (used as reference) remains below 5% throughout the studied range of resin content (maximum error of 4.4% for the estimate of 2E  by Equation (4.6). The resin content of industrially manufactured strand-based wood composite products is well below the maximum value of this range.  Due to small variation of Poisson’s ratios with resin content, these results are not presented in this secton. However, estimates of Poisson’s ratios will be used and evaluated in the viscoelastic context in the next section.  4.3 Viscoelastic Analysis In the previous sections, both constituents of strand-based composite were assumed to remain linear elastic. However, there are many studies throughout the literature showing the time-dependent behaviour of both PF resin and wood, especially at different moisture contents. Here we only consider the resin phase as a linear-viscoelastic material and investigate the accuracy of our analytical approach in modeling the stress relaxation behaviour of strand-based wood composites.  4.3.1 Analytical Approach The viscoelastic properties of the material unit cell (Figure 4-3) are estimated using the correspondence principle employed by most researchers, e.g. see (Christensen 1979; Matzenmiller and Gerlach 2004; Zobeiry 2006). Using the correspondence principle, the linear 115  viscoelastic heterogeneous problem in the real time domain is first transformed to a virtual linear elastic problem in Laplace space. The latter is then solved using linear micromechanical schemes. Finally, the effective viscoelastic properties are obtained using numerical inversion to time domain. This classical approach is applied here using linear estimates developed in Section 4.2.1.1.  4.3.2 Numerical Approach  In order to validate our analytical approach, the viscoelastic behaviour of the material unit cell is modelled using finite element method with ABAQUS® software. The geometrical and mechanical characteristics of the strand are the same as for elastic analysis. Full resin coverage is considered here. The resin content is 7.3% by volume corresponding to resin thickness of 0.28 mm. The resin phase shear and bulk moduli (G and K , respectively) are defined by Prony series expansions (see Section 2.3.2).   The Prony series for the shear and bulk moduli are distributed over twelve relaxation times ranging over twelve decades. Shear weight factors kg  and associated relaxation times k  are taken from the literature (Naik et al. 2008). Elastic properties given in Table 4-2 are used for the instantaneous, unrelaxed, shear and bulk modulus values. Since the bulk modulus of the resin is difficult to measure during relaxation, identical distributions are prescribed for G  and K  (i.e.,k kg k ), meaning that a constant Poisson’s ratio of 0.3 is assumed. The resulting parameters defining the Prony series are given in Table 4-3.   116   Table 4-3 Resin viscoelastic properties (Prony series parameters for the shear and bulk moduli)   It should be noted that the viscoelastic behaviour of the PF resin used in wood composite industry may be different from the data given in Table 4-3. However, using the same Prony series for both numerical and analytical approaches is sufficient to evaluate analytical estimates by comparison with numerical reference data.   Relaxation loading paths (10000 seconds) for each of the six elementary loadings are here prescribed to the unit cell thanks to kinematical periodic conditions. This allows computing the effective stiffness tensor at each time step in order to finally deduce the time evolution of Young’s and shear moduli as well as of Poisson’s ratios.  No gkkkτk (s )1 0.01262 0.01262 1.01E–092 0.02552 0.02552 1.01E–083 0.06252 0.06252 1.01E–074 0.07686 0.07686 1.01E–065 0.12858 0.12858 1.01E–056 0.14718 0.14718 1.01E–047 0.18163 0.18163 1.01E–038 0.17148 0.17148 1.01E–029 0.08336 0.08336 1.01E–0110 0.03488 0.03488 1.01E+0011 0.01204 0.01204 1.01E+0112 0.00964 0.00964 1.01E+02117  4.3.3 Comparison of Analytical and Numerical Predictions Unit cell relaxation Young’s and shear moduli are shown in Figure 4-17 and Figure 4-18. Numerical results are shown with symbols while analytical results are plotted with lines. As shown by the elastic analysis, estimates by Equations (4.6) and (4.9) are in better agreement with numerical reference data than Equations (4.11) and (4.12). Therefore, in the viscoelastic context, only the estimates resulting from Equations (4.6) and (4.9) are used for comparison with numerical results. Results obtained from Halpin-Tsai estimates are also reported. The maximum error of 4.4% occurs in predicting 2E  in the unrelaxed state which could be attributed to comparable strand dimension in transverse direction to resin thickness. As the resin relaxes with time, the mismatch between the resin modulus and the strand modulus in transverse direction diminishes. Lower property mismatch results in a more uniform stress field throughout the unit cell, and therefore reduces the effects of geometrical features. Although Halpin-Tsai equations give reasonably good estimates over a wide range of time for Young’s moduli, only advanced analytical estimates give accurate results throughout the whole relaxation time for all moduli (both Young’s and shear moduli).  118   Figure 4-17 Comparison between analytical (lines) and numerical (symbols) longitudinal relaxation moduli for full resin coverage, using resin viscoelastic properties. Resin modulus    and numerical reference transverse relaxation moduli (   and   ) are given for comparison. Halpin-Tsai prediction for    (dotted dashed line) overlaps with current estimate (Equation (4.6)).  Estimates of Poisson’s ratios are compared with numerical results in Figure 4-19. According to this figure, current micromechanical approach is able to estimate accurately all three Poisson’s ratios of fully orthotropic strand-based composite during the entire relaxation period.   119   Figure 4-18 Comparisons between analytical (lines) and numerical (symbols) relaxation moduli for full resin coverage, using resin viscoelastic properties. (a) Transverse Young’s moduli; and (b) Shear moduli. Resin relaxation shear modulus    is added for comparison. Halpin-Tsai predictions are presented with dotted and dashed lines.   Figure 4-19 Comparisons between analytical predictions (solid lines) and numerical results (symbols) for full resin coverage, using resin viscoelastic properties; Poisson’s ratios.  (a) (b)120  4.4 Summary and Conclusions In this chapter, micromechanical equations are developed based on classical micromechanics principles. In a previous investigation (Gereke et al. 2012), the elastic behaviour of a PSL beam at both macro and micro scales was studied using the finite element method. Here we have introduced analytical equations to replace the time consuming and computationally intensive simulations required at the microscale. The analytical estimates for a material unit cell may be used for orthotropic strands that are either fully or partially covered by the resin. Partial coverage is taken into account through equivalent resin properties arising from the concept of interface parameters (Hashin 1990). Partial coverage modelling (assuming constant or variable resin thicknesses) constitutes a specific contribution of this chapter.    In the case of full coverage, it is shown that the linear estimates may be used directly to obtain viscoelastic properties with the aid of the correspondence principle. The benefit of this work is in its simplicity and versatility for estimating the microscopic properties of different strand-based composites.   The analytical predictions for effective elastic properties of composites with orthotropic strands that are partially covered with resin are in good agreement with reference finite element simulations for different resin volume contents. In the viscoelastic case, the estimates are also shown to be very close to numerical results with correlations that, in most cases, are better than classical estimates obtained from the Halpin-Tsai model.   121  The analytical micromechanics equations developed in this chapter are incorporated in the multi-scale modelling framework developed for strand-based wood composites in the next chapter.   122  Chapter 5: Multi-Scale Modelling of Strand-Based Composites  In this chapter, a multi-scale modelling framework developed for predicting the mechanical properties of strand-based wood composites is presented. This framework is based on closed-form analytical models at three different resolution levels; micro-, meso- and macro-mechanical. A preprocessing step is performed to provide the input data for the three main modelling steps in this framework. Finite element based mechanical analyses are employed to validate the analytical models developed for the first two steps. The predictive capability of the entire framework is validated using a set of experimental data reported in the literature. Although the methodology presented is general, it is specifically applied here to predict an important structural property (Modulus of Elasticity, MOE) of a special strand-based wood composite product, namely the Oriented Strand Board (OSB). The MOE predictions of OSB panels show very good agreement with the available experimental data, thus instilling confidence in the practical utility of this easy-to-use and efficient analytical modelling tool for predicting the properties of wood composites employed in structural members.  Applications of this framework in optimization and reliability analysis of strand-based composites are also discussed. Finally, the application of the numerical version of the framework in creep modelling of PSL beams is demonstrated using a simple case study.  5.1  Development of the Multi-scale Modelling Framework  Figure 5-1 illustrates the multi-scale analysis framework in its general form for predicting the MOE of OSB panels. In a pre-processing step, parameters such as strand density, resin thickness 123  and resin area coverage are back-calculated from panel characteristics (e.g. overall density, species, resin content, etc.) based on the following assumptions:   Panel can be divided into multiple layers (depending on the required accuracy)  Each layer consists of  strands, resin, wax, fines, and voids   Strand and resin volume fractions are constant within a layer but may vary from one layer to another, according to the Vertical Density Profile (VDP)  Moisture, fines and wax content are constant throughout the whole panel  Uncompacted wood strands have the same physical and mechanical properties as those of the neat wood (e.g. given in Wood Handbook (Forest Products Laboratory 1999))  Fines have the same density as strands within each specific layer  Resin and wax are incompressible while strands and fines are compressible  Strands are compressed in their radial direction (2X   direction in Figure 5-2 (c))  Resin thickness is uniform around a single strand in each layer  Fines do not make any contribution to load transfer. Their presence only contributes to the density of the wood composite product.   124   Figure 5-1 Steps involved in the multi-scale analysis framework showing the input and output parameters for each step   Figure 5-2 Unit cell of softwood: a) schematic representation, b) cell wall parameters, c) discretized unit cell  5.2 Preprocessing Step Initially, and in order to provide inputs for other steps, a preprocessing step needs to be performed. In this step, the following parameters are determined for “each layer” first: S2t3S1tCMLt(a)  (b) (c)RtTtRlTlα1X3X 2125   Uncompacted strand density,  s UC .Density of an uncompacted wood strand is calculated using the following equation given in the Wood Handbook (Forest Products Laboratory 1999):      1000 1s UC SG MC    (5.1) where the specific gravity ( SG ) values for several neat strands are provided in the Wood Handbook (Forest Products Laboratory 1999) and the moisture content (MC ) of the strand is assumed to be equal to the moisture content of the panel for a specific layer of interest.  Resin volume fraction, rV . Assuming that the resin density, r , is constant throughout the panel, the resin volume fraction in each layer is estimated based on the panel resin content by weight, rW , and the layer density, layer , taken from panel vertical density profile (see Figure 5-1).   layerr rrV W      (5.2)  Wax volume fraction, wV . Assuming a constant value for wax density, the wax volume fraction in each layer is estimated based on the panel wax content by weight, wW , and wax density, w , as follows:  layerw wwV W      (5.3)  Void volume fraction, vV . The void volume fraction in each layer, is assumed to be proportional to the density of the layer, layer . In other words, voids shrink at the same 126  rate as the panel densifies during the manufacturing of the panel. The core layer in the center of the panel is chosen as a reference layer (zero compaction) with known void volume fraction,  v coreV , and density, core , for this purpose. Therefore, as the layer density changes through the panel thickness, void volume fraction in each layer varies accordingly:   corev v corelayerV V       (5.4)  Strand volume fraction, sV , and  fines volume fraction, fiV . Assuming that fines have the same density as the strand (i.e. s ), in a given layer, their volume fraction is estimated as:   1fifi v w rsWV V V VW        (5.5) where fiW  and sW  are the fines and strand weight fractions, respectively. Therefore, the strand volume fraction, sV , can be written as:   1s v w fi rV V V V V     (5.6)  Compacted strand density, s . Having determined the strand volume fraction, sV , the strand density in each layer (compacted strand) is determined using the following equation:  layers ssWV       (5.7)  Compaction ratio, rC , is the ratio which is a measure of strand compaction in each layer and is defined as: 127    srs UCC   (5.8)  Strand thickness,  R . Strands are compressed in the direction normal to the panel (2X   direction in Figure 5-2(c)). Therefore, strand size varies through the thickness of the panel accordingly:  UCrRR C  (5.9)  Resin thickness, rt . Assuming the uniform coverage of strands by the resin, the resin thickness is constant in all three directions. Therefore, Equation (5.6) may also be written in terms of the strand thickness, width, length ( R , T , L ) and the resin thickness, rt , as:  ( )( )( )s r r rRTLV R t T t L t     (5.10) Assuming that the resin thickness is negligible compared to the strand length (rt L ), the above equation may be simplified and solved for resin thickness, rt :     2 2 22 4 (1 )2s s s s srsT R V T R V V V Ttt V       (5.11)  Resin area coverage, aR . This parameter is defined as the ratio of actual area covered with resin (partial strand coverage) to the maximum available area to be covered by resin if there were only resin and strands in the panel (full strand coverage):  max( )( )( )( )r r ra rr rA T t R tR VA T t R t RT       (5.12) In deriving the above equation, it is assumed that fines act as voids and there is no bonding between the fines and the strands or between the fines and the resin. 128   5.3 Micro-mechanical Step Mechanical properties of strands in each layer (compacted strands) can be estimated from their constituents’ properties and microstructural features such as micro-cellular geometrical parameters. Researchers have used both numerical (Qing and Mishnaevsky 2009; Gereke et al. 2011; Guo and Gibson 1999) and analytical methods (Gibson and Ashby 1999; Hofstetter et al. 2005) for this purpose. The numerical approach and the analytical approach employed in this framework are presented in Section 5.3.1 and Section 5.3.2, respectively. The validity of the analytical approach in predicting the elastic moduli of both normal (uncompacted) and compacted strands are investigated in Section 5.3.3.   5.3.1 Numerical Approach Knowing the specific gravity ( SG ) of a given wood strand and the density of the solid cell wall material, cw , at moisture content of 12 %, the micro-cellular geometrical parameters (, , , ,R T R Tl l t t  ) of earlywood and latewood (see Figure 5-2(a) for what these parameters signify) are computed using the following assumptions:  • Wood has a uniform, periodic cellular microstructure from which a unit cell of the material can be identified. • The wood cellular microstructure is hexagonal and remains hexagonal (Figure 5-2) during compaction. See (Gereke et al. 2011; Qing and Mishnaevsky 2009) for more details. 129   The cell wall is modelled as an elastic laminate consisting of four layers: S1, S2, S3 and a compound middle lamella, CML (Figure 5-2 (b)). The connection between the cell walls is realized with the CML, which homogenises the properties of the middle lamella and the primary wall in the model. The cell wall is a fibre-reinforced composite with cellulose fibres (micro-fibrils) embedded in the natural polymer lignin. The orientation of the micro-fibrils influences the mechanical properties of the cell wall. The micro-fibril angle (MFA) varies in the dominating S2-layer between 10° and 30° depending on the species, the age and the source of the cell, earlywood or latewood (Dinwoodie 1975; Kollmann and Cote Jr 1968). The MFA is incorporated as the inclination of the local 1X  material orientation of each secondary layer (S1, S2, S3).  The microstructure of wood strands is idealized as a regular honeycomb structure consisting of hexagonal cells (Figure 5-2). A three-dimensional representation of the microstructure is realized in an FE model of the unit cell. The dimensions of the air-filled core (lumen), Rl  and Tl  (see Figure 5-2 (a)), and the thicknesses of the cell wall layers, 1St , 2St , 3St  and CMLt  (see Figure 5-2(b)), represent the cellular geometry of the wood in the model. The angle   defines the inclination of the radial cell wall (see Figure 5-2(a)).   Once, the ratio of earlywood to latewood is estimated, the property of a single strand is computed through a computational homogenization technique, wherein periodic boundary conditions are applied. This procedure was described in (Gereke et al. 2012; Michel et al. 1999). In wood 130  composites where resin penetration into the wood cellular structure (lumen) is possible, an interface layer (wood with penetrated resin into the lumen) may also be considered. The properties of this interface layer can be estimated by assuming that the lumen is partially filled with resin.   It should be noted that due to the last assumption (hexagonal cellular microstructure), the accuracy of this model diminishes at higher compaction levels. As the compaction level increases, the cellular structure of the strands may not be represented accurately by hexagonal cells.  5.3.2 Analytical Approach Gibson and Ashby (Gibson and Ashby 1999) proposed the following equations to predict the properties of different wood species based on their density:  2s cwsL LcwE C E       (5.13) As an alternative to the numerical method, the above equation may be employed to predict the strand’s Young’s modulus in the longitudinal direction, sLE, knowing the strand density, s , solid cell wall density, cw , and cell wall axial Young’s modulus, cwLE. The coefficient 2C  is a parameter which has been found to be close to unity for most wood species (see (Gibson and Ashby 1999)). Other moduli are calculated based on the ratio of the strands’ Young’s modulus in the longitudinal direction to their value for a given species which can be found in the Wood 131  Handbook (Forest Products Laboratory 1999). In the current framework, these ratios are assumed to remain constant during compaction.  5.3.3 Results  5.3.3.1 Validation and Comparison between Experimental Data and Predictions  In this section, predictions of both numerical and analytical approaches for the elastic moduli of normal wood strands are compared with reference data given in the Wood Handbook (Forest Products Laboratory 1999). Figure 5-3 shows comparisons between the model predictions and experimental values for the longitudinal Young’s modulus of both softwood and hardwood species. For these uncompacted strands, the results show fairly good agreement between the predicted strands properties (both numerically and analytically) and experimental data from the Wood Handbook (Forest Products Laboratory 1999). 132   Figure 5-3 Comparisons between the model predictions (both analytical (Gibson and Ashby 1999) and numerical (Gereke et al. 2011)) and experimental data (Forest Products Laboratory 1999) for longitudinal modulus of different wood species: (a) softwoods and (b) hardwoods  Comparisons for other elastic moduli are shown in Figure 5-4. For clarity, only results for selected softwood species (pine and spruce) are presented. Although there are good agreements between the model predictions and experimental data for longitudinal Young’s moduli, the numerical and analytical models are not able to predict other moduli with the same level of accuracy for all wood species. This could be attributed to inaccurate representation of cell wall 133  geometry. As stated by Qing and Mishnaevsk (Qing and Mishnaevsky 2009), the longitudinal Young’s modulus of wood is less dependent on cell geometry.   Figure 5-4 Comparisons between the model predictions (both analytical (Gibson and Ashby 1999) and numerical (Gereke et al. 2011)) and experimental data  (Forest Products Laboratory 1999) for Young’s modulus in (a) radial direction,  (b) tangential direction and shear modulus in (c) LT plane, and (d) RT plane for different softwoods  5.3.3.2 Comparisons between Numerical and Analytical Predictions For compacted strand properties at low compaction levels, analytical equations based on Equation (5.13) proposed by Gibson and Ashby (Gibson and Ashby 1999) are used. To show the 134  validity of the analytical approach, a wood strand with cellular properties of a uniform, regular hexagonal microstructure has been chosen. Elastic moduli of this strand, estimated by both numerical and analytical approaches are compared.  Figure 5-5 (a) shows the very good agreement between predictions of the analytical equations proposed by Gibson and Ashby (Gibson and Ashby 1999) and the numerical estimates for the Young’s moduli. It should be noted that the transverse moduli in the 2 and 3 directions are different in the numerical approach due to the selected cellular microstructure. The numerical results are consistent with the ones given by Guo and Gibson (Guo and Gibson 1999). As also mentioned by Guo and Gibson (Guo and Gibson 1999), the analytical approach only considers bending of the cell walls and neglects their axial and shears deformations during loading. Therefore, the material is treated similarly under loading in radial and tangential directions and only one set of values are given for the two transverse directions which lies between the numerical values.  As shown in Figure 5-5 (b), the analytical approach underestimates the transverse shear modulus during compaction. Underestimating transverse shear modulus for regular hexagonal honeycombs with different geometrical parameters (Rl , Tl , Rt and Tt  ) has also been noted in (Gibson and Ashby 1999) for experimentally measured values.  For other shear moduli the analytical predictions lie within the numerical estimates.  135   Figure 5-5 Elastic moduli of a hypothetical wood strand with regular hexagonal cellular microstructure ( L cwE = 35 GPa, cw = 1500 kg/m3,  = 30°) under various level of compaction. (a) Young’s moduli (b) Shear moduli  The results shown in Figure 5-5 are for a wood strand with simplified microstructure and isotropic solid cell wall properties. In reality, wood strands have very complex cellular microstructure consisting of multi-layers of anisotropic cell wall material (see (Gibson and 136  Ashby 1999; Qing and Mishnaevsky 2009; Hofstetter et al. 2005) ). For these strands, Equation (5.13) still provides a good estimate for the longitudinal modulus as shown by Gibson and Ashby (Gibson and Ashby 1999). For other moduli of different wood species, various coefficients are introduced. These coefficients have been determined empirically for different wood species (e.g. see (Gibson and Ashby 1999)) and have been used here (see Table 5-1).  Table 5-1 Longitudinal Young’s modulus of wood species in Figure 5-4. Other elastic moduli may be calculated using the coefficients representing their value with respect to the longitudinal Young’s modulus (Forest Products Laboratory 1999) Species   EL  (GPa) ET/EL ER/EL GLR/EL GLT/EL GRT/EL Pine Loblolly 13.53 0.078 0.113 0.082 0.081 0.013  Lodgepole 10.12 0.068 0.102 0.049 0.046 0.005  Longleaf 15.07 0.055 0.102 0.071 0.06 0.012  Pond 13.31 0.041 0.071 0.05 0.045 0.009  Ponderosa 9.79 0.083 0.122 0.138 0.115 0.017  Red 12.32 0.044 0.088 0.096 0.081 0.011  Slash 15.07 0.045 0.074 0.055 0.053 0.01  Sugar 9.02 0.087 0.131 0.124 0.113 0.019 Spruce Sitka 10.89 0.043 0.078 0.064 0.061 0.003   Engelmann 9.79 0.059 0.128 0.124 0.12 0.01  In the numerical approach presented in Section 5.3.1, the hexagonal cellular microstructure of wood is assumed to be preserved during compaction. Although this may seem to be an unrealistic assumption at high compaction levels (e.g. on the face of OSB panels), it will suffice to establish a framework for predicting MOE properties of OSB panels (see Section 5.6.2) due to the fact that the strand’s longitudinal modulus makes the highest contribution to the final MOE than other moduli. 137  5.4 Meso-mechanical Step In the meso-mechanical step, a unit cell of the material at the meso-scale (Figure 5-6) is identified. It is assumed that the material has a periodic microstructure at this scale and the properties of the constituent compacted strands are obtained from the previous step based on the compaction level (Figure 5-7).    Figure 5-6 (a) Cross-section of the material, and (b) Material unit cell at the meso-scale  Resin area coverage, strand and resin thicknesses are obtained from the preprocessing step. Resin mechanical properties are modified using the resin area coverage to incorporate the effect of voids in the analytical approach (see Section 4.2.1.2). In the numerical approach, voids may be taken into account either by incorporating void elements or using equivalent resin properties (similar to the analytical approach, see (Malekmohammadi et al. 2013b; Malekmohammadi et al. 2013a)).  Having determined the required parameters for a unit cell of the material at the meso-scale, the unit cell effective properties (corresponding to strands that are fully aligned with the panel Unit Cell(b)(a)Void138  longitudinal direction) are calculated. In order to consider the effect of strands with different sizes, resin area coverage, or resin thickness, the effective properties of several unit cells need to be calculated and used in the next step of analysis (macro-mechanical step).     Figure 5-7 Schematic representation of the various input and output quantities in the meso-mechanical step  The numerical approach employs the methodology previously presented in Chapter 4 for the unit cell and, for brevity, its description is not repeated in this chapter.  The same unit cell as the one employed in the numerical approach is used here. However, using the principles of classical micro-mechanics, the engineering constants for this unit cell are estimated. The closed-form analytical equations employed for estimating the effective properties of the material unit cell at the meso-scale are those given in Chapter 4. It should be noted that these equations were derived for fully covered strands. For partially resin-covered strands, the Resin Properties E, νCompacted  Strand Mechanical  Properties(EL, ER, ET, GRT, …. νLR,… )Equivalent Resin Properties (Er,)eq,(Gr,)eqResin Area Coverage, RaResin Thickness, trCompacted Strand Thickness, tsPreprocessingMicro-mechanical Step* A partially covered strand with resin at the base orientation of 0 degree.Void Content(Analytically / Numerically)(Numerically)Effective properties of a Unit Cell (0o)*139  equations are modified as described in Chapter 4 by introducing the concept of equivalent resin properties (see Section 4.2.1.2).  5.5 Macro-mechanical Step  5.5.1 Strand Orientation Distribution (in-plane and through-thickness) Strands are not fully aligned in most strand-based wood composite products (Shaler and Blankenhorn 1990; Fan and Enjily 2009; Forest Products Laboratory 1999). Having determined the properties of the material unit cell at the meso-scale (a strand covered fully or partially with resin), the strand orientation also needs to be incorporated in the computational scheme. For this purpose, the panel is divided into several thin layers and idealized by combining many unit cells with the same resin thickness as shown in Figure 5-8. Each layer consists of several unit cells having the same density and same wood species (different wood species may also be considered in the later versions of the code), and distributed with a specific distribution function for the strand orientation.   140   Figure 5-8 Dividing the panel into multiple layers (a) real meso-structure, and (b) idealized meso-structure. Each layer has a different density as denoted by colors  Local and global coordinates are defined in a way that the local coordinate system {1, 2, 3} of the strands  aligned with the panel longitudinal direction, coincides with the panel global coordinate system {x, y, z} (see Figure 5-9). The strand orientation distribution may vary throughout the thickness of the strand-based wood composite product. It is assumed that strands in each layer have been compacted with the same compaction ratio.   yz(a)(b)One Layer141   Figure 5-9 Incorporation of strand orientation in the macro-mechanical step of the framework (a) top view of the real panel, (b) top view of the idealized panel, and (c) cross-section (side view) of the idealized panel  The effective stiffness matrix of each layer in the global panel coordinate system is estimated through averaging of the stiffness matrix of unit cells orientated at different angles using:   1 jnjjC w C       (5.14) where   jC  is the transformed stiffness matrix of a unit cell with orientation of j  and is calculated as:         11 20jC T C T   (5.15) The transformation matrices,  1T  and   2T  are defined as follows: 2 2 2 22 2 2 21 22 2 2 20 0 0 2 0 0 00 0 0 2 0 0 00 0 1 0 0 0 0 0 1 0 0 0,0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 2 2 0 0 0m n mn m n mnn m mn n m mnT Tm n m nn m n mmn nm m n mn nm m n                                       (5.16) Effective properties of a Unit C llθiOrientationAngleFrequencyθiSTDV, MeanE1, E2, E3, G12,….v12,… E1, E2, E3, G12,….v12,… 1,2, 3, 12,.12,  Layer effective properties (E1, E2, E3, G12,….v12,… )xzTransformation*by θiFrom Step 2ρyVDP(a) (b) (c)31zbt/2t/2142   where    cossinmn  (5.17) Also, jw is the volume fraction of the jth strand in a single layer and 11njjw . The effective properties of each off-axis unit cell are estimated through the transformation matrix. A similar approach has been proposed in the literature for estimating effective properties of braided (Ayranci and Carey 2008), short fibre (Laspalas et al. 2008) and woven (Sankar and Marrey 1997; Shrotriya and Sottos 2005; Vandeurzen et al. 1996) composites.   For panels with mixed species, a species distribution function in addition to orientation distribution function may be defined to incorporate the effect of different species. A panel may be considered as a multi-layer laminate with known layer effective engineering constants according to the panel vertical density profile (see Figure 5-9). The effective engineering constants of each layer are calculated from the effective stiffness matrix.  5.5.2 Flexural Modulus Calculation Based on the ASTM standard (ASTM 2006), the MOE of a wood-based panel is determined as:  348PLMOE I  (5.18) where P  is the applied load, L  is the panel length and   is the total mid-span deflection. For a panel with width, b , and thickness, t , the above equation can be written as:  334L PMOE bt       (5.19) 143   In order to estimate the MOE, the total deflection needs to be calculated accurately. The total deflection  due to load P , has two main contributors: deflection due to bending, b , and deflection due to shear deformation, s :  b s      (5.20) The second term on the right-hand side of Equation (5.19) can be written as (see e.g. (Gibson and Ashby 1999) ):     348 4eff effL LP EI GA   (5.21) where  effEI  and  effGA are the effective bending rigidity and effective shear rigidity, calculated by integrating the Young’s and shear moduli through the beam cross-section , respectively, as follows:   /22/2( )tefftEI E y by dy   (5.22)   /2/2( )tefftGA G y bdy   (5.23) In the above equations, y  is the out-of-plane (through-thickness) direction of the panel as shown in Figure 5-9.   In deriving Equation (5.21), first-order shear deformation laminated plate theory (FSDT) has been used. In this theory, the transverse planes do not necessarily remain normal to the plate mid-surface during deformation. Therefore, shear strains are non-zero and shear deformation is 144  considered. However, in this theory, as a first order approximation, transverse shear strain is uniform through the thickness of the laminate. This assumption is unrealistic and will generate errors (Reddy 2003). In order to compensate for this assumption, a shear correction factor,  , is introduced (Birman and Bert 2002) (see Equation (5.21)).   For a general laminate, the shear correction factor,  , depends on layers’ properties and their configuration. This factor is determined as the ratio of the transverse shear strain energy in the first-order shear deformation laminated plate theory, fU , to the actual transverse shear strain energy, aU (see (Reddy 2003)):  faUU   (5.24)  The two shear strain energies are calculated based on the transverse shear stress as follows:  22xyfU AG   (5.25)  22xyaAU dAG       (5.26) where A  is the cross-sectional area and xy  is the average shear stress (transverse shear stress in FSDT, which is constant) throughout the panel, and is calculated based on the applied load P, width, b , and thickness,  t , of the panel:  2xyPbt   (5.27)  145  In order to determine the transverse shear stress, a simple analytical approach (based on laminated beam theory) provided in (Birman and Bert 2002) is employed:    11/2=yxxytQy Q ydy D       (5.28) Here, 11Q is the transformed layer stiffness in the panel global coordinate system and D  is the laminate bending stiffness defined as:  /211/2 /2ytt tD Q ydy dy         (5.29) In Equation (5.28), xQ  is the transverse shear stress resultant and is calculated as:  /2/2( ) 2tx xytPQ y dy b   (5.30) Once the shear correction factor,  , is determined using Equation (5.24), the MOE is predicted from combining Equation (5.19) and Equation (5.21).  5.6 Validation  5.6.1 Experimental Data from Literature Chen et al. (Chen et al. 2010) conducted an extensive and detailed experimental study on Oriented Strand Board (OSB) properties. In their study, the influence of panel density on MOE and Modulus of Rupture (MOR) of OSB made from aspen strands, in both parallel and perpendicular directions, was investigated. Panels manufactured for this purpose were from commercial aspen strands with average dimensions of 108mm × 14.25mm × 0.69mm and face-146  to-core weight ratio of 1.5. The core strands were oriented perpendicular to the face strands. Strand mats were hand-formed and a hot press with a platen temperature of 205°C was used to press them. Manufactured panels consist of strands, fines, resin, and slack wax. The composition of the panel on the weight percentage basis is given in Table 5-2.   Panels with an overall density ranging from 449 to 705 3kg/m were manufactured and tested for several panel properties such as MOE and MOR, in both parallel and perpendicular directions. Here, the experimentally measured values for MOE in both directions are used to validate the multi-scale modelling predictions. Panels with five different vertical panel density profiles (Figure 5-10) were examined for this purpose. The figure in (Chen et al. 2010) was digitized to extract the point densities and reproduce the vertical density profiles used here.      Table 5-2 Panel composition as given in (Chen et al. 2010). Note that the values are the overall value and vary through the layers depending on the layer compaction ratio Constituent Weight Fraction (%)                          Strand 85.8 Fines 10.0 Resin 3.0 Slack Wax 1.2  147    Figure 5-10 Vertical panel density profiles for five panels with different overall densities. Data are extracted from (Chen et al. 2010)  For bending test, panels with the same size of 711mm × 711mm × 11.1 mm were tested as shown in Figure 5-11.  148   Figure 5-11 Three-point-bending test on the panels. Strands in face layers are (a) parallel, and (b) perpendicular to the beam longitudinal axis  5.6.2 MOE Predictions and Comparisons with Experimental Data The experimentally measured values for MOE by Chen et al. (Chen et al. 2010) are compared with the multi-scale framework predictions for the five panels described in the previous section. A moisture content of 7% was assumed to be constant throughout all panels. No data on orientation distribution of strands were reported in (Chen et al. 2010). However, based on some experimental studies on commercial panels and experimental results reported for strand distribution of OSB panels, a normal orientation distribution with a mean value of 0° and standard deviation of 30° was employed as input data for the macro-mechanical step. These (b)(a)149  values resulted in MOE predictions which match with experimental data both in the parallel and perpendicular directions. Parallel refers to the case where strands in the face layer are parallel to the beam longitudinal axis. It is assumed that strands in the face and core layers are placed in the forming mat following a single methodology. Therefore, a single distribution function is sufficient for both face and core layers.   Figure 5-12 Comparison between current predictions (both analytical and numerical) and experimental results (filled triangles). Predictions using the numerical approach (at micro- and meso-scale) and the analytical approach (at all three scales) are shown with hollow squares and filled circles, respectively  The current predictions are shown in Figure 5-12 for five different panels. Very good agreement was found between the results of the multi-scale modelling predictions (using both numerical and analytical approaches) and the reported experimental data. The capability of the modelling framework in estimating the MOE of OSB panels having a wide range of overall panel densities 150  justifies the validity of approximations and assumptions made in the analytical approaches at all three levels: micro-, meso- and macro-scale.  It should be emphasized that in this chapter, the validity of the current multi-scale modelling methodology has only been examined for OSB panels. Validating it for other wood composite products requires an extensive set of experimental data for those products which may be of interest in future studies.  5.7 Applications The presented multi-scale framework has several practical applications. In this section, some of these applications are presented.  5.7.1 Optimization of OSB Panels There are many parameters which affect the vertical (through-thickness) density profile of the manufactured OSB panels. Vertical density profile along with some parameters such as mechanical properties of resin and strands, strand orientation distribution, fines content, moisture content, etc., all contribute to the structural behaviour (e.g. MOE) of OSB panels.   Using the analytical version of the multi-scale framework, the optimized vertical density profile (VDP) of the OSB panels can be obtained to maximize the MOE for a desired overall panel density. Having an analytical tool which can be implemented in an easy to use environment such as Microsoft EXCEL® enables OSB manufacturers to optimize the vertical density profile of their products easily. For this purpose, the following steps are followed: 151    Mathematical relations between elastic properties of each layer, including Young’s modulus, shear modulus and layer densities of the panel are assumed (Figure 5-13). These relations can be obtained using the analytical equations presented in the micro-mechanical and meso-mechanical steps.   Figure 5-13 Schematic of Young’s and shear modulus vs. layer density of OSB panels obtained from analytical equations.   Using the above relations and assuming an arbitrary VDP (a polynomial function), distributions of Young’s and shear moduli along the thickness of the panels are obtained.  These distributions are used to obtain the MOE of the OSB panels based on the formulation of the three-point-bending test to calculate the displacement, δ, as in Equation (5.21) and in the multi-scale modelling framework.  A computer code was developed to find the coefficients of the assumed polynomial function representing the density profile of the panel such that the MOE obtained from the multi-scale framework is maximized or the overall panel density is minimized for a given panel MOE.  E ( )G ( )152  Figure 5-14 depicts an as-manufactured VDP versus an optimized VDP obtained using the OSB optimization code developed in a spreadsheet (EXCEL) environment.    Figure 5-14 Vertical Density Profile (VDP) of an OSB panel. Panel with the optimized density profile has an overall density that is 12% lower than the as-manufactured one while maintaining the same MOE of 8.05 GPa  It should be noted that the optimized density profile can be used as a guideline in the manufacturing process to reduce the overall cost of the production. For example, by reducing the pressing time to obtain panels with the optimized density profile, the overall cost can be reduced.  5.7.2 Reliability Analysis of PSL Beams The multi-scale modelling framework developed involves several variables as input with certain degree of uncertainty that can be assigned to each variable. Therefore, the final predictions of the model (e.g MOE) should have an associated level of confidence (or degree of uncertainty) which can be determined through reliability analysis. 0200400600800100012000 2 4 6 8 10Thickness (mm)(kg/m3)as manufacturedoptimized153   Through integrating the analytical equations employed in the analytical multi-scale modelling framework with an available reliability tool (RELAN, (Foschi 1988)), a probabilistic analysis can be performed to investigate the uncertainty in the structural response of a PSL beam under a 3-point loading condition (Malekmohammadi et al. 2013c). A schematic of the procedure is depicted below.   Figure 5-15 Schematic of the procedure used in reliability analysis of a PSL beam  A PSL beam with square cross-section and fixed dimensions (65mm×65mm×1200mm) under 3-point bending load is chosen for the analysis. The resin employed to bond the strands in the beam is assumed to be elastic and with isotropic properties of phenol formaldehyde (PF) (see Chapter 4). Strands (from pine (Longleaf)) are assumed to be elastic and distributed according to a normal orientation distribution function with a mean value of 0° and standard deviation of 30°. Two cases are considered to perform the reliability analysis. In the first case, all variables except strand modulus are assumed to be constant. In order to consider the effect of variation of strand properties on predicted MOE, a lognormal distribution with the mean value and coefficient of FrequencyValue1x 2xxnF(c)MOE10cici+1cmci-1F(ci)f(c)MOEcif(ci)R ndom VariablesPr bability DensityMOE Cumulative DistributionFunction (CDF)MOE Probability DensityFunction (PDF)Inputs for RELAN Outputs of RELAN Post processing154  variation of 15GPa and 10%, respectively, are assigned to strand modulus. In the second case, the strand’s properties are kept constant while resin modulus and strand size ( L , R and T ) are assumed to vary by 20%. The resulting probability density functions of MOE in both cases are shown in Figure 5-16. Comparison of the probability distribution in Case I and Case II indicates that the strand modulus affects the structural response more significantly than resin modulus and strand size while the other parameters are assumed to be fixed.   Figure 5-16 MOE Probability density function of a PSL beam (a) Case I (b) Case II  It should be noted that an accurate analysis, requires a comprehensive characterization of important variables and parameters. Performing some case studies similar to those presented here would guide the experimentalist in designing an efficient testing program to characterize these parameters.  5.7.3 Creep Modelling of PSL Beams There are many studies throughout the literature showing the time-dependent behaviour of both PF resin and wood, especially at different moisture contents. One of the main applications of 0.000.020.040.060.080.100.120.140.16f (c)MOE (GPa)0.000.010.020.030.040.05f (c)MOE (GPa)(a) (b)155  multi-scale modelling framework is predicting the response of strand-based wood composite structures.   A numerical stochastic multi-scale modelling approach has been presented in (Gereke et al. 2012) for elastic properties of PSL beams. Having developed a differential approach for orthotropic viscoelastic materials in Chapter 3, the numerical stochastic multi-scale modelling approach has been extended for predicting the time-dependent behaviour of PSL beams subjected to a constant load.  The PSL beam is assumed to have an idealized meso-structure similar to OSB panels but with different strand size and full coverage of strands with resin. In order to predict the creep of a PSL beam, the same unit cell employed for meso-mechanical simulations (see Figure 5-6) is used. It should be mentioned both constituting phases (wood and resin) in PSL exhibit viscoelastic behaviour. However, for simplicity reasons, only the resin phase (phenol formaldehyde) is considered to be viscoelastic and its behaviour is described with the same viscoelastic parameters used in Chapter 4. The three-dimensional orthotropic material constants for wood are average values for different pine species taken from (Forest Products Laboratory 1999) representing the actual PSL products (see Chapter 4). No strand compaction is considered here for simplicity. However, compacted strand properties obtained from the micro-mechanical step can be used to consider the compaction of strands.  In order to predict the creep response of the beams, the components of the unit cell effective relaxation matrix, ijC , are required. These can be estimated either analytically using the 156  correspondence principle or numerically as described in Chapter 4. Here, the numerical procedure presented in Chapter 4 was employed to derive components of the unit cell relaxation matrix. These components are required to characterize the viscoelastic behaviour of the material on the meso-scale. Variation of these components during time is shown in Figure 5-17.  157   Figure 5-17 Variation of the effective relaxation matrix components ijC versus time obtained from numerical analysis of PSL unit cell. Symbols denote numerical results and lines are the associated fitted curves.  (a)(b)(c)158  In Figure 5-17, numerical results are shown with symbols while the fitted curves are plotted with lines. The differential approach for modelling the macroscopic behaviour of viscoelastic structures requires the components of relaxation matrix of the orthotropic material be defined by Prony series expansions following the formulation described in Chapter 3. The unrelaxed and relaxed values and the Prony series parameters associated with each component are obtained from a curve fitting procedure on the numerical results. These parameters are given in Table 5-3 and Table 5-4 for reference.  Table 5-3 Unrelaxed and relaxed values of components of the relaxation matrix for a PSL unit cell.         C1113.475 12.596C221.947 1.515C33 1.481 0.989C44 1.068 0.760C55 1.030 0.790C660.177 0.162C12 1.125 0.898C13 0.870 0.658C23 0.595 0.488Component Ciju (GPa) Cijr (GPa)159  Table 5-4 Prony series parameters obtained from curve fitting of the numerical results   Having characterized the viscoelastic behaviour of the material unit cell numerically (meso-mechanical step), the creep behaviour of the beam can now be determined from FE analysis of a beam consisting of several strands distributed within it (macro-mechanical step). In other words, results of the meso-mechanical step (viscoelastic properties of the meso-mechanical unit cell) are used as input parameters for the macro-mechanical step (PSL beam under a specific load). The differential approach proposed for analysis of viscoelastic orthotropic materials in Chapter 3 has been employed in the macro-mechanical simulations of the PSL beam under a specific load.  In the macro-mechanical step, PSL beams under three-point bending are analysed. Each beam consists of 24 unit cells (strands with resin) arranged in a regular structure. The beam is assumed to have a square cross-section of 39 mm × 39 mm and a span of 380 mm between the supports as described in (Arwade et al. 2009). A constant distributed load with the total magnitude of 1.5 kN is applied at the centre of the beam on the top surface. The average displacement at the m  centre nodes at the bottom surface of the beam is defined as follows: i w11w22w33w44w55w66w12w13w23τi (s)1 0.009 0.006 0.012 0.003 0.009 0.003 0.004 0.008 0.002 1.01E-092 0.018 0.012 0.024 0.007 0.017 0.007 0.008 0.017 0.003 1.01E-083 0.045 0.030 0.058 0.017 0.043 0.016 0.019 0.042 0.008 1.01E-074 0.056 0.037 0.072 0.022 0.053 0.022 0.024 0.052 0.012 1.01E-065 0.093 0.063 0.120 0.038 0.089 0.036 0.041 0.087 0.020 1.01E-056 0.110 0.078 0.139 0.052 0.106 0.050 0.055 0.104 0.033 1.01E-047 0.138 0.100 0.172 0.071 0.133 0.067 0.073 0.131 0.047 1.01E-038 0.154 0.137 0.171 0.131 0.152 0.120 0.125 0.152 0.112 1.01E-029 0.127 0.163 0.104 0.202 0.134 0.180 0.185 0.137 0.207 1.01E-0110 0.124 0.186 0.070 0.232 0.133 0.240 0.230 0.136 0.273 1.01E+0011 0.059 0.090 0.030 0.109 0.064 0.121 0.113 0.064 0.135 1.01E+0112 0.065 0.098 0.030 0.115 0.069 0.137 0.123 0.069 0.148 1.01E+02160   11 miim    (5.31) and recorded during time while the applied loads is maintained. The average displacement,  , is used to construct deflection –time curves.  To evaluate the effect of strand orientation, two cases are considered. In the first case, all strands are assumed to be aligned with the beam length. In the second case, random strand orientation is considered within each beam and is introduced by defining the material orientation within each unit cell making up the beam accordingly. For PSL beam consisting of random strands, strands are randomly distributed according to the orientation distribution described in (Gereke et al. 2012) which is also given in figure below:   Figure 5-18 Strand orientation in the PSL beam consisting of random strands according to (Gereke et al. 2012)  161  A convergence study showed that a mesh size of 100 × 12 × 16 provides converged values for the node displacement on the bottom face of the beam in both cases. The results for the two PSL beams considered are presented in Figure 5-19. The deformed shape of the beam and the total displacement contour for the second case are also shown on the top right as well.   Figure 5-19 Creep curve for PSL beams consisting of aligned strands and random strands.   Note that the average deflection has been normalized by the initial elastic deflection of the beam consisting of aligned strands (0o strands) in Figure 5-19. As it is expected, the PSL beam with aligned strands experiences lower deflections during time due to more contribution of elastic stiff strands. Figure 5-19 demonstrates the capability of the multi-scale framework in predicting viscoelastic behaviour of strand-based composite structures considering several variables including constituents’ properties, microstructural features and geometrical parameters.  162  5.8 Summary and Conclusions A mechanistic framework was presented in this chapter based on micro-structural features of strand-based wood composites considering such factors as: wood species and their combination; strand dimensions; strand orientation; compaction and density profile; resin type, content, and distribution; wax, fines and void contents; face-to-core weight ratio.  The analytical equations derived from the fundamentals of micro-mechanics were employed in this framework. This would enable the optimization of the panel properties based on microstructural features with fewer experiments. Moreover, the analytical equations were incorporated in a reliability analysis software RELAN® to enable stochastic modelling of quantities of interest.  Numerical simulations as well as experimental data were used to validate the accuracy of the developed analytical framework. Very good agreements were found between the current model predictions and the experimental data. Although the capability of this framework is only presented for the MOE of OSB panels, a similar approach can be used for other strand-based wood composite products such as PSL, OSL, etc. However, relevant experimental data will be needed to validate the presented framework for other wood composite products.  In the context of this framework, a simple case study showed how an orthotropic viscoelastic model would enable us to predict the creep behaviour of a PSL beam. The same methodology can be employed in predicting the viscoelastic response of more complex strand-based wood composite structures. 163  Chapter 6: Conclusions and Future Work  6.1 Summary An efficient multi-scale modelling approach for predicting the viscoelastic response of composite structures was presented in this thesis. This modelling approach consists of two main steps for composites with fairly simple microstructures (e.g. UD continuous fibre composites). The first step involves predicting the effective stiffness matrix of a representative volume element of the material at the micro-scale. Estimates can be obtained either using a numerical approach or an analytical approach if previously validated analytical micromechanics equations are available. Having determined the relaxation functions from each component of the effective stiffness matrix, in the second step structural analysis (macro-scale simulation) is performed based on the differential form of viscoelasticity.  Efficiency and accuracy of computations in both steps are very important and were considered in this thesis. Some analytical micromechanics models are already available for the analyses required in the first step. There are certain assumptions involved in these models which make their applications limited to composites with certain idealized microstructures. For composites with more complex microstructures (e.g. strand-based wood composites) the micromechanics models have been less explored.  With the power of modern computers, an alternative approach is using a numerical method (e.g. FE based) to model the microstructure of composites with all its details. However, setting up and using these models in practice is rather complex and may decrease the efficiency of the whole 164  modelling strategy. There are certain advantages in using analytical micromechanics equations for different composites and therefore their application for the micro-scale modelling was explored in this work. Composites with different microstructures were considered. In particular, two classes of composites were examined in more detail: advanced fibre reinforced composites employed in aerospace structures and strand-based wood composites which are increasingly being used in construction of low to mid-rise buildings. Computational homogenization technique using the finite element method was employed as a tool to investigate the accuracy and the validity range of analytical micromechanics equations. Additionally, capabilities of the computational homogenization technique in modelling composites with different microstructures were also demonstrated.  Computational efficiency in the second step, involving macro-scale structural analysis, is of great importance for large composite structures with complex geometries and loading conditions. Through some examples, the efficiency of viscoelastic computations in the second step was studied by comparing results obtained using the differential form of viscoelasticity (adapted in this study) with those obtained from the more conventional integral form of viscoelasticity.  Python scripts and a user material subroutine (UMAT) were developed and implemented successfully in a commercial general-purpose finite element code (ABAQUS®) for numerical simulations. Using these computer codes, the developed multi-scale modelling approach can be employed in design and analysis of viscoelastic composites with generally orthotropic properties. As special cases, the same routines can also be used to conduct viscoelastic analyses of isotropic or transversely isotropic composites. 165   Due to the importance of a specific structural property, namely, modulus of elasticity or MOE for manufacturers of strand-based wood composite products, an analytical multi-scale modelling framework for strand-based wood composite panels was developed and implemented in an easy-to-use environment (Microsoft Excel®) to predict the MOE of Oriented Strand Board (OSB) panels. The framework was validated using experimental data taken from the literature. Furthermore, applications of this multi-scale modelling framework were explored and some case studies were presented to illustrate its capabilities for industrial use.  6.2 Conclusions The following conclusions can be drawn from the presented work:    Although several multi-scale models have been proposed in the literature, their implementation into commercial general-purpose codes is quite challenging and they have remained unused by industry. Only recently, this aspect has been addressed by researchers due to the increasing need from industry to solve practical engineering problems associated with hierarchical nature of composites. Efficiency and the structural complexity of most multi-scale “in-house” codes were identified as the main barriers in this area.   For industrial applications, the analytical approach can be considered to be efficient for predicting the properties of the unidirectional (UD) continuous fibre composites at micro-scale. Very good agreements were found between the numerical solutions (assuming hexagonal packing of fibres) and analytical predictions obtained from the combination of Hashin’s CCA model and Christensen and Lo’s GSC model for elastic properties of UD 166  continuous fibre composites over a wide range of fibre volume fractions. Furthermore, both the numerical and analytical predictions were found to agree very well with the experimental data reported in the literature for elastic properties of UD continuous fibre composites.   Analytical micromechanics models for thermo-elastic properties of composites may be used to obtain their time-dependent properties with the aid of the correspondence principle. The benefit of this approach is in its simplicity, efficiency and versatility for estimating the microscopic properties of viscoelastic composites.   Modification of classical solid micromechanics to predict prepreg storage moduli when the resin phase is in a fluid state is a promising approach which can be incorporated in process modelling of thermoset matrix composites. The presented approach here is not only simple to use, but physically based. This is an important step in the development and implementation of integrated process models that link the fluid and solid behaviour of the resin during composites manufacturing, as in the recent work by Haghshenas (Haghshenas 2012).   Expressing constitutive relations in a differential form and employing generalized Maxwell representation of viscoelastic behaviour of the material would enable us to analyse the viscoelastic response of composite structures efficiently. Moreover, this approach leads to an easy-to-implement numerical algorithm which can be used in any finite element code structure for orthotropic composites. Therefore, it can be employed to solve complex viscoelastic problems in different industries.  167   Expressing the variation of the time-dependent effective stiffness matrix components (at lower scales) in terms of exponential functions of time (i.e. Prony series) provides inputs, in a suitable format, for analysis of viscoelastic composites at higher scales (e.g. macro-scale).   Analytical micromechanics equations developed for strand-based composites can be employed to replace the time consuming and computationally intensive simulations required to obtain the effective properties (elastic or viscoelastic) of the material unit cell at lower scales. The analytical predictions may be used for orthotropic strands that are either fully or partially covered by the resin.   Taking an analytical multi-scale modelling approach (as presented here) for the analysis of strand-based wood composite structures has valuable applications for wood composite industry due to the analytical equations employed in its development. The application of this framework in reliability analysis of PSL beams and optimization of OSB panels were demonstrated.   Using the analytical multi-scale modelling framework developed for wood composites, the optimized vertical density profile of the OSB panels can be obtained to maximize the Modulus of Elasticity (MOE) for a desired overall panel density. Having an analytical tool which can be implemented in an easy-to -use environment such as Microsoft EXCEL®, enables OSB manufacturers to optimize the vertical density profile of their products easily.  168   The multi-scale mechanistic modelling framework developed in this work is a significant advance over the largely empirical models that have been used in industry for estimating the mechanical properties of wood composites and furthermore reduces the reliance on testing for evaluating these properties. Therefore, it helps accelerate the insertion of new wood composite products into structural applications.   6.3 Contributions The following contributions were made in this work:   Solid micromechanical models were modified by incorporating the fibre bed interaction with resin for the purpose of predicting the shear modulus of prepregs continuously during the full cure history of thermoset composites.   A numerical algorithm was developed for viscoelastic modelling of orthotropic materials based on the differential form of viscoelasticity. This algorithm can be used as an alternative to other numerical algorithms developed based on the integral form of viscoelasticity to analyse the time-dependent response of orthotropic materials efficiently.   The numerical algorithm developed for viscoelastic modelling of orthotropic materials was programmed in a user material subroutine (UMAT) and implemented in a commercial general-purpose finite element code for structural analysis of viscoelastic orthotropic composites.   Using the principles of classical micromechanics, analytical closed-form equations were developed in this work to estimate the effective orthotropic properties of strand-based 169  composites (consisting of high volume fraction of strands) according to their constituent properties and microstructural features such as resin thickness and strand geometrical characteristics.    For modelling the behaviour of strand-based composites containing voids, the concept of interface parameters were employed and an efficient modelling strategy for partially resin-covered strands were developed.   A multi-scale modelling framework was developed and employed in predicting an important structural property of strand-based wood composites (i.e. MOE) which is of interest to wood composite industry. Micro-structural features of strand-based wood composites and factors such as wood species and their combination; strand dimension; strand orientation; compaction and density profile; resin type, content, and its distribution; wax, fines and void contents; were taken into account. Since this framework is grounded on rigorous mechanics at different scales as opposed to empirical assumptions, it is a significant advance over the largely empirical models that have been used in industry. The predictive capabilities of this framework will help manufactures to optimize their current products or develop new wood composite structural products.   6.4 Future Areas of Research Many interesting areas for further research can be proposed based on this work. A few of these research areas are outlined below:   The developments accomplished in this thesis (e.g. wavy unit cell model, modified solid micromechanics, UMAT for viscoelastic orthotropic composites) can be extended and 170  employed in current process models or the development of next generation of process models (e.g. (Haghshenas 2012)). Therefore, integration of the viscoelastic multi-scale modelling approach into the next generation of process models can be a subject for further research. These multi-scale process models should consider the viscoelastic behaviour and multi-scale nature of composites with different microstructures and constituents (e.g. short fibre thermoplastic composites) during their manufacturing process.   The unit cells considered in this work for the numerical analysis at the micro-scale are idealized ones within which the interaction of fibres has been ignored. A unit cell in which the interaction of fibres and buckling of fibres in compression are included would enable us to predict some local effects such as wrinkling. Additionally, it would enable us to analyse the viscoelastic behaviour of woven and braided composites.   In the multi-scale modelling framework developed for strand-based wood composites, variation of moisture content and temperature has been ignored. Therefore, one future research area could be to incorporate these effects and enhance the capability of this framework in modelling structures under different environmental conditions.   In order to validate the multi-scale model for orthotropic composite structures with different applications, material viscoelastic characterization as well as creep tests on some simple structures (e.g. beams) are necessary. An extensive experimental study in this area could be beneficial in validating and revisiting the assumptions made in this work if necessary.   171   With the aid of the numerical multi-scale modelling approach it is now possible to consider the stress at a point in a wood composite panel during a Concentrated Static Load (CSL) test and use the reverse micromechanics to calculate the stress in the individual constituents; resin and strand.  This provides a level of detail that would allow damage modelling where one could apply different failure criteria to the resin and strand.   The application of the multi-scale modelling approach presented here can be extended to predict the response of honeycomb structures by using the developed models at different scales, namely, at micro-scale (cell wall material), meso-scale (honeycomb cell)  and macro-scale (honeycomb beam). Engineering honeycombs can potentially be made with composite cell walls (Gibson 2012). 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Texas A&M University, USA.    189  Appendices  Appendix A  Multi Scale Modelling Approach and Computational Homogenization  A.1 Brief Review of Multi-scale Modelling Approach for Composite Materials Developing micromechanical models for new composite materials can be quite challenging and their application may be limited to a specific composite or for a specific purpose. With the power of modern computers, a feasible approach to analysis of the response of a structure made from these materials may seem to be modelling the whole structure numerically, as a heterogeneous material with all of its constituent phases and complex morphology. However, extraction of results and usage of these models in practice involves more complexity (Ladeveze 2004; Kanouté et al. 2009).  To avoid these challenges, researchers have examined multi-scale approaches through coupling macro-scale and micro-scale analysis using homogenization theory (Sanchez-Palencia et al. 1980) and applied it to linear elastic composites (Guedes and Kikuchi 1990; Fish and Wagiman 1993; Ghosh et al. 1995) and later to composites with non-linear behaviour (Fish et al. 1997; Fish and Yu 2001; Terada and Kikuchi 2001; Feyel 2003). Computational homogenization, e.g. by numerical finite element method, is a powerful technique in finding the effective properties of composite materials (Guedes and Kikuchi 1990) and is key in most multi-scale modelling approaches. This method is versatile and may be used for composites with: (i) complex morphological characteristics, (ii) different contrasts between constituents’ properties, (iii) high volume fraction of fibres, or (iv) composite materials with generally orthotropic properties (e.g. 190  strand-based wood composites). Computational homogenization is also utilized in this thesis as a numerical approach for determining the effective properties at lower scales.  Several multi-scale methods have emerged in the past two decades based on the computational homogenization technique. In most of these methods, the goal is to employ the accuracy of micromechanical models with the efficiency of macro-mechanical models by coupling the two models (Kanouté et al. 2009). However, the way to couple these models efficiently in a multi-scale framework is still questionable and strongly depends on the final application. Moreover, being able to move forward from the lower level to higher level properties, is not always our primary concern. For instance, in structural failure (damage) analysis, it is desirable to move in the reverse direction, i.e. from higher level properties to lower level properties. In these problems, it is necessary to un-smear or perform reverse micromechanics to obtain average stresses or strains in each phase of interest for purposes of failure (damage) initiation calculations.  Recently, some researchers have reviewed several multi-scale approaches and discussed the challenges involved (Kanouté et al. 2009; Geers et al. 2010; Nguyen et al. 2011; Böhm 2014). In order to reduce the computational time in most of these approaches, a periodic microstructure is assumed throughout the whole structure. Therefore, a repeating unit cell of the material (sometimes also called Representative Volume Element (RVE), see (Pindera et al. 2009)) can be identified.   Geers and coworkers (Geers et al. 2010) emphasized that in most well-established multi-scale schemes, e.g. (Feyel and Chaboche 2000; Ghosh et al. 1995; Yuan and Fish 2008), the 191  macroscopic deformation gradient is assumed to be constant over the RVE length scale. This assumption is valid when the characteristic microscopic length scale (e.g. fibre diameter) is much smaller than the characteristic macroscopic length scale of the structure (e.g. width of a beam). This is known as the principle of separation of scales (Nguyen et al. 2011). Separation of scales implies that the RVE size should be very small so that macroscopic stress and strain fields are uniform throughout the RVE. Therefore, many multi-scale models are not appropriate for problems involving localization such as damage where large macroscopic solution gradients exist. In the following, some of advanced multi-scale modelling approaches and their developments are briefly reviewed.  Feyel and coworkers (e.g. (Feyel and Chaboche 2000; Feyel 2003)) introduced a two-scale modelling scheme in which, for a given macroscopic strain at each integration point,  the stress and strain fields within a repeating unit cell of the material are computed numerically. Finite element is then used to compute the macroscopic stress tensor through homogenization. This approach is known as FE2 approach because computations are performed simultaneously at two scales and two sets of mesh are required (i.e. one at micro-scale (unit cell) and the other at macro-scale (structure)). As noted by the authors, for large complex structures, parallel computing is necessary using this approach.  More complex approaches were proposed by Fish, Ghosh and their colleagues based on mathematical homogenization theory. Using asymptotic expansions of displacements and superposing microscopic displacement fields on macroscopic fields, Fish and Wagiman (Fish and Wagiman 1993) analysed the locally non-periodic linear elastic composites. In this approach, 192  the problem is decomposed into two separate macro and micro problems and the structure is divided into substructures consisting of a main periodic region and locally non-periodic regions. In the non-periodic regions (or regions susceptible to edge effects), the displacement field at any point is obtained by superposing the microscopic and macroscopic displacement fields. Appropriate boundary conditions are enforced to ensure the compatibility between the periodic and non-periodic regions. This approach was later extended for solving non-linear problems (Fish et al. 1997) and for damage modelling (Fish et al. 1999; Fish and Yu 2001).  Ghosh and coworkers (Ghosh et al. 1995) also employed asymptotic homogenization to analyse heterogeneous structures with arbitrary microstructures by coupling Voronoi cell finite element method (Ghosh and Mallett 1994) with homogenization method. To overcome the limitation of the previous models (e.g. (Ghosh et al. 1995; Fish and Yu 2001)) in critical regions (e.g. free edges, interfaces, macro cracks), Ghosh and coworkers (Ghosh et al. 2001) proposed an adaptive multi-level methodology using mesh enrichment technique. Using this technique, the macro-scale mesh is refined into regions with high gradients of macroscopic field variables. Voronoi cell finite element method was used to analyse the micro-structure while conventional finite element method was employed for the macro-structural analysis. The adaptive multi-level methodology has been applied to modelling the evolution of damage in composites (Ghosh et al. 2007).  To overcome the localization problem and edge effects, Ladeveze (Ladeveze et al. 2001; Ladeveze 2004) proposed a different multi-scale computational approach where the structure is decomposed into an assembly of substructures and interfaces. The substructure may contain one 193  or several unit cells. An interface between two neighbouring substructures transfers both the displacement field and traction field. The application of this approach to contact problems has been demonstrated in (Ladeveze et al. 2002).  Although several multi-scale finite element methods have been well established for solving nonlinear problems, they have been quite under-utilized in solving practical engineering problems in industry (e.g. process modelling of thermoset composites) because of the difficulties in their implementations in commercial software (Yuan and Fish 2008; Tchalla et al. 2013). The main barrier is attributed to the complex structure (e.g. multiple loadings using periodic boundary conditions and manipulation of inputs and outputs) of the in-house “academic” codes developed based on different multi-scale finite element schemes (Yuan and Fish 2008). Only recently, some researchers have tried to demonstrate that utilizing existing capabilities of commercial codes (e.g ABAQUS®) may facilitate the application of these multi-scale schemes in solving practical engineering problems (Yuan and Fish 2008; Tchalla et al. 2013). Moreover, Python scripting is shown to be a powerful tool for bridging the scales in the analysis.  To overcome the storage problem and CPU time, Tachalla et al. (Tchalla et al. 2013) proposed a two-scale 2-D finite element toolbox, for the analysis of heterogeneous structures. They implemented this approach for both linear and non-linear problems in ABAQUS® as an integrated toolbox using a Python script and user-defined FORTRAN subroutines. The computational approach involves micro and macro-scale simulations and is fairly similar to the one being presented in this thesis for the 3-D elastic analysis. First, at the micro-scale a chosen RVE of the heterogeneous material is subjected to appropriate periodic boundary conditions 194  representing the macroscopic strain tensor obtained from solving the macro-scale problem. The micro-scale problem is then solved and the stress field is obtained. Knowing the stress field, the volume averaging of stress is performed to calculate the effective constitutive matrix which is used at the macro-scale. This matrix is employed in solving the macro-scale problem and updating the macro-scale strains and stresses. It should be noted that for linear problems, the constitutive matrix need to be computed only once due to applicability of principle of superposition. For non-linear problems, a standard iterative technique has been used by Tchalla et al. (Tchalla et al. 2013) to find the solution of the macro-scale problem. At each load increment applied in the macro-scale problem, the tangent modulus is computed at each integration point. This is carried out by solving the micro-scale problem through applying periodic boundary conditions representing the macro-scale strain increment.  A.2 Computational Homogenization Technique Owing to advances in computational power, computational homogenization has been, for example, widely employed for short fibre composites in the past decade (e.g. see (Harper et al. 2012; Kari et al. 2007; Hine et al. 2002)). In elasticity for instance, the effective stiffness tensor may be determined by solving six elementary boundary value problems (corresponding to three uniaxial extensions and three simple shear loadings), e.g. see (Kanit et al. 2003; Kanit et al. 2006). There are certain challenges involved in this technique as discussed in the literature (e.g. (Nguyen et al. 2011)). In the following, one of these challenges, the choice of boundary conditions is discussed.  195  Apparent stiffness tensor of a given volume of the material depends on the type of boundary conditions applied. Many researchers have proved that imposing homogeneous strain boundary conditions results in apparent properties close to upper bound while imposing homogeneous stress boundary conditions generate estimates close to lower bound results (Huet 1990; Kanit et al. 2003).   Periodic boundary conditions are ideally suited boundary conditions for both periodic and random media (Geers et al. 2010; Kanit et al. 2003; Xia et al. 2003). Whatever the size of the volume, apparent elastic properties obtained with periodic boundary conditions are closer to the effective properties than with other boundary conditions.   In this thesis, computational homogenization based on full-field microstructural simulations by finite elements, also serves as a tool to provide reference solutions and evaluate the relevance of different analytical micromechanics models. As described earlier, once the accuracy of these analytical models has been determined, they can be implemented in a multi-scale model.  A.3 Stiffness Matrix Computation The effective stiffness C  of a composite with linear elastic constituents is defined by:  :ı C   (A.1) For a periodic material, ı  and İ  are respectively the volume average of the stress and strain tensors in the unit cell. To determine the effective stiffness matrix of a unit cell of the material, six elementary loadings (three uniaxial extensions and three simple shear loadings) corresponding to pre-specified forms of the average strain tensor İ  have to be successively 196  applied to the unit cell. In each case, the local problem is solved by the finite element method. The volume average of the resulting stress field in the unit cell, namely the knowledge of ı , provides a specific column of the stiffness tensor (6×6 matrix representation), see e.g. (Michel et al. 1999) for more details. The anisotropic engineering coefficients that appear in the formulations of C  may thus be calculated.   A.4 Periodic Boundary Condition (PBC) Kinematical periodic boundary conditions are prescribed to impose each of the six elementary loadings. Periodicity of displacement on the boundaries is required in representing the behaviour of a material with periodic structure under different loading conditions. Applying periodic boundary conditions instead of homogeneous displacement or traction boundary conditions prevents any over-constrained conditions in the numerical model and results in more accurate predictions of effective properties (Xia et al. 2003).   197   Figure A-1 Directions and global coordinate systems at different resolution levels; (a) Structure (macro-scale) and (b) Repeating unit cell (micro-scale) representing the periodic micro-structure of the material. For more details of the unit cell and the assumed periodic micro-structure of different composites see Chapter 2 and Chapter 4.  For a given strain tensor   (corresponding to a prescribed elementary loading), the displacement field for any point 1 2 3( , , )x x xx  belonging to the boundary ∂Ω of the unit cell Ω must satisfy the following condition (see (Michel et al. 1999)):     P ,    u x İ x u x x  (A.2) The fluctuation of displacement, Pu , is periodic, which means that Pu  takes the same values at two homologous points i+ iΩ x  and i iΩ x  on opposite faces of Ω ( Figure A-1(b)). This finally leads to:     i i i- i-     u x İ x u x İ x  (A.3) ZXYL1L3L2zxyUnit cell ΩLoad, Pl1l3l2x3+x1+x2+x3-x2-x1-(a) (b)XStructure198  The correct periodic boundary conditions must ensure that the traction vector       t x ı x n x, with n  being the outer unit vector normal to the boundary ∂Ω, is anti-periodic. In other words, it is equal in magnitude but opposite in sign for i+ iΩ x and i iΩ x  on opposite faces of Ω. The periodic boundary conditions (Equation (A.2)) ensure the following average relationship:     1 dVV   İ İ x İ x  (A.4) where  İ x is the strain field in the unit cell  and  V  is the volume of the unit cell   with the dimensions 1l , 2l   and 3l . The brackets,  , denotes the average on the unit cell volume.  A.5 Implementation in ABAQUS® using Python Scripting The computational approach has been implemented in ABAQUS® through Python scripting. The main benefit of using Python scripting is the ability to parameterize the model, automate repetitive tasks, and read and extract data from ABAQUS® database. Python scripts were used to create geometries, generate lists of nodes and elements, and define material properties in an appropriate format as required to be employed in the ABAQUS® environment. All nodes were fully defined in the “global” coordinate system. Once the geometry of the unit cell and the meshes are defined, boundary conditions should be applied.   Implementation of periodic boundary conditions is an important task in solving the unit-cell problem. This can be accomplished in ABAQUS® through applying a certain displacement (or traction) on a pre-defined node in a single step and coupling opposite nodes on the surface of the unit cell through establishing equations similar to Equation (A.3), relating their displacements in 199  an input file. To prevent rigid body motion, the displacements of the central node of the model is constrained in all 3 main directions.  A Python script creates input files for six simulations (three uniaxial extensions and three simple shear loadings) with corresponding periodic boundary conditions (PBC). Executing the six input files, accessing the stress and strain values at each integration point and averaging them, calculating the stiffness matrix and extracting engineering constants are all performed sequentially using a set of Python scripts which are coupled together through a main Python script (see Figure A-2).      200    Figure A-2 Schematic overview of the implementation of numerical approach for predicting the unit cell effective properties in ABAQUS®. Python scripts denoted by “.py” extension. MainPython Script(unit_cell.py)6 Input Files (*.inp)Abaqus/Standard6 Output Files (*.odb)6 Report Files (*.rpt)Results(*.txt)Create Geometry(shape.py)Generate Input Files(Inp_write.py)Compiling Functions(compile.py)Report(report.py)Initialize(initial.py)201   In this work, Pythons scripts have been widely employed to perform the finite element simulations with ABAQUS® using 8-node linear brick elements, C3D8. Idealized unit cells of different composite materials have been identified and discretized for this purpose (see Chapter 2 and 4). The goal is to present a methodology for multi-scale modelling of composites and the computational homogenization is employed as a tool here to obtain the homogenized properties of the material required for the analysis at higher scales. For further works involving composites with complex microstructures, the methodology would remain valid (see (Terada et al. 2000; Kanit et al. 2003; Kanit et al. 2006)), provided a correct choice of RVE.               202  Appendix B  Micro-Mechanical Elastic Analysis of Circular Fibre Composites   The application of most analytical micromechanics equations have been well investigated by several researchers using numerical and experimental techniques. However, comprehensive studies using correct numerical approach or extensive experimental data for all effective properties over a wide range of microstructural parameters are quite rare. Performing such a comprehensive study is one of the main goals of this appendix.  In Section B.1 different analytical models for predicting the effective properties of solid composites are reviewed. Among different analytical for circular fibre composites reviewed in Section B.1, three models are quite well-known and have shown to give very good predictions of the effective engineering moduli of circular fibre composites. Therefore, they are described here in detail and their application in finding the effective elastic properties of circular long and short fibre composites are studied carefully by considering a wide range of parameters such as fibre volume fraction and fibre aspect ratios.   Specifically, we investigate the validity of the CCA and GSC models for continuous, long fibre composites and the Halpin-Tsai model for short, discontinuous fibre composites. The accuracy of these models are evaluated using numerical reference solutions obtained from the finite element analysis performed based on the computational homogenization technique (see Appendix A).   In order to verify the numerical approach developed here, its results for different engineering constants are compared against available experimental and numerical data reported in the 203  literature for circular fibre composites with different microstructures. The verification serves two purposes: (1) that the numerical tool can reliably be used to generate reference solutions for evaluating the accuracy of analytical micromechanics models, and (2) that it can be a component of a multi-scale framework for modelling composites with complex microstructures.  B.1 Introduction Knowledge of the effective properties of composites is of great interest for engineers. Throughout history, several micromechanical models have been proposed to predict the effective mechanical and thermal properties of composite materials. Various Researchers (Aboudi 1991; Christensen 1990; Hashin 1983; Pindera et al. 2009) have reviewed and analysed different aspects of these models. Among different micromechanical models, some of the best known ones which have been the basis for other models are presented in Table 2-1. Most analytical models are based on finding stress or strain concentration tensors of a single ellipsoidal inclusion embedded in another medium (matrix). In order to find these tensors, different authors have chosen different approaches or approximations.   Voigt and Reuss were among the first researchers who proposed the simplest models for predicting the elastic moduli of heterogeneous systems based on isostrain and isostress assumptions in 1889 and 1929, respectively (Aboudi 1991). These simple models can also be treated as bounds for effective properties of composites.  Using elasticity solutions Hashin introduced the Composite Spheres Assemblage (CSA) model (Hashin 1962) and found a closed-form solution for the effective bulk modulus and bounds for 204  the shear modulus of spherical composite systems. The microstructure considered in this model consists of spheres with some special characteristics. The size of all particles is not the same and they are distributed in a way that a shell of matrix can be associated to each fibre. Although the sizes of particles are variable, the volume fraction of each sphere in a fictitious spherical region to which it belongs to the same as the volume fraction of other spheres in their corresponding regions. Due to these assumptions, CSA model could not properly represent composites with high volume fraction of spheres with identical diameters. Two years later, Hashin and Rosen (Hashin and Rosen 1964)  proposed their famous Composite Cylinders Assemblage model (CCA) for continuous long fibre composites. They found closed-form solutions for four material properties (   ,    ,     and    ) and bounds for     of transversely isotropic composites using the principles of minimum potential and complementary energy (Hashin 1972).  For an ellipsoidal inclusion embedded in an infinite matrix, Eshelby showed that the strain field inside the inclusion is uniform (Eshelby 1957). He solved the inclusion problem and found the stress field inside and outside of an ellipsoidal inclusion embedded in an infinite matrix. Based on Eshelby’s solution to the inclusion problem, the effective stiffness of a composite with dilute concentration of inclusions can be determined. This has been the basis of many proposed methods and models for non-dilute composites in later years.  One of the proposed methods to determine the effective properties for non-dilute composites is the Self-Consistent Scheme. Hill (Hill 1965) and Budiansky (Budiansky 1965) used Eshelby solution for the strain field inside a single ellipsoidal inclusion to predict the overall elastic moduli of spherical particle composites and later extended it to continuous, unidirectional fibre 205  composites. In this method, the fibre/particle is embedded in a matrix whose properties are the unknown effective properties of the composite. The strain field in the inclusion is uniform and is expressed as a function of effective composite properties. This method was applied to short fibre composites by Laws and Mclaughlin (Laws and Mclaughlin 1979). The self-consistent method gives unrealistic and erroneous results for rigid inclusions and cavities (Christensen 1979) and as stated by Budiansky (Budiansky 1965), this method is not suitable for composites with high mismatch between phase properties. In a more realistic version, known as the Generalized Self-Consistent (GSC) method, the inclusion/fibre is surrounded by a matrix shell which is embedded in an effective medium of unknown properties. Hermans and Kerner (Kerner 1956; Hermans 1967) presented exact solutions for bulk and shear properties of spherical and cylindrical fibre composites using this approach. However, they both made mistakes in their work and the presented expressions for shear modulus are incorrect. Later using the GSC, Christensen and Lo (Christensen and Lo 1979) found exact solution for effective transverse shear modulus of spherical and cylindrical composite systems by satisfying all continuity conditions. They demonstrated that their results lie between the proposed bounds by Hashin and Rosen (Hashin and Rosen 1964) for transverse modulus     (Note: Hashin and Rosen found exact solution for four effective elastic properties but bounds for the fifth one).  Although the general version of the self-consistent method better represents the microstructure of composite materials than its original version and it addresses the problems such as satisfying interface conditions in the original version, researchers have only been able to apply it to spherical and cylindrical geometries.   206  As explained in (Hashin 1983), results of the GSC method for spherical and cylindrical fibre composites are similar to exact solutions of the CSA and CCA models for longitudinal properties. This is due to the geometrical similarities in defining fibre volume fraction in both models. However, there is a difference between the GSC and Hashin’s CSA and CCA models that is worth mentioning. Hashin’s models involve different sizes of particles/fibres such that their combination fully fills the composite media, while in the GSC there is no restriction on size distribution of particles. In the GSC all spheres/fibres (and the matrix surrounding them) except one, are represented by the equivalent homogeneous material. Therefore, although these two models use the same approach for defining the volume fraction, they are inherently two different models with similar results for most properties (see (Christensen and Lo 1979)).  In Section B.2.1 the micromechanical equations of both Hashin’s CCA model and Christensen’s GSC  model are described in more detail for calculating the mechanical properties of transversely isotropic long fibre composites. It should be mentioned that Hermans’ GSC and Christensen’s GSC equations give exactly the same results as CCA for longitudinal properties. However, the only closed-form solutions for transverse properties may be found with the GSC model proposed by Christensen and Lo (Christensen and Lo 1979). Therefore, it is proposed to use the CCA and GSC together for predicting all properties of continuous long fibre composites in this thesis (see Section B.2.1).  Halpin and Tsai rearranged the GSC equations in terms of engineering constants and proposed analytical expressions for the moduli of transversely isotropic short fibre composites (Halpin and Kardos 1976). They showed that the moduli of composites with transversely isotropic phases 207  could be written in a general form using a parameter called the geometry factor. Different methods have been proposed to determine this parameter. Among those are some mathematical expressions depending on the reinforcement type as well as curve fitting using experimental data. Halpin-Tsai equations are described in Section B.2.2 and employed for predicting the properties of short fibre composites.  Mori and Tanaka used the solution of Eshelby problem to find the effective stiffness matrix of polycrystalline materials (Mori and Tanaka 1973). This approach has been applied to composite materials by many researchers in the 1980s (Benveniste 1987; Tandon and Weng 1984; Weng 1984). By simply using a direct approach, Benveniste reformulated the Mori-Tanaka scheme and showed how it can be applied to two phase composites with anisotropic elastic constituents (Benveniste 1987). The main assumption in any model based on Mori-Tanaka scheme is that the inclusion (fibre) sees the average strain of the matrix which is different from the applied global strain (Tucker and Liang 1999). Although self-consistent models are similar to Mori-Tanaka in terms of embedding nature of the inclusion in an infinite matrix, the assumptions are quite different. A direct approach is used in deriving stiffness matrix in Mori-Tanaka while the self-consistent method is an implicit one.  Based on the periodic array of rectangular fibres, Aboudi (Aboudi 1991) proposed the Method of Cells for continuous fibre composites. He analysed a unit cell of the material by dividing it into four sub-cells and applying homogeneous boundary conditions. The displacement in each sub-cell is then represented by some linear functions. By applying continuity conditions for displacement and tractions, explicit expressions were derived for representative cell effective 208  properties. As it was noted by some researchers (Pindera et al. 2009; Sun and Vaidya 1996; Xia et al. 2003), applying homogeneous boundary conditions gives incorrect results for shear modulus due to over-constrained assumption of deformed boundaries remaining plane. In fact, this problem has also existed in most finite element unit cell modeling of composite materials for a long time. As it is explained in Appendix A and also in Appendix B (Section B.3), only periodic boundary conditions which satisfy periodicity of both stress and strain give correct results for effective properties when considering a unit cell of  a periodic composite and hence we will apply them in our numerical model presented in this thesis.   In addition to above analytical models, researchers have proposed bounds for elastic properties of composites based on variational principles in elasticity for heterogeneous materials. Hashin and Shtrikman derived bounds for the effective moduli of isotropic composites consisting of isotropic phases with arbitrary phase geometries (Hashin and Shtrikman 1963). They used principles of minimum potential energy and minimum complementary energy to bound the strain energy and the moduli of heterogeneous material. The derived bounds depend only on individual phase elastic moduli and their volume fractions. It should be noted that for spherical particulate composites, Hashin-Shtrikman bounds for bulk modulus give results equivalent to Hashin’s CSA model (Hashin 1962). Walpole extended the application of Hashin-Shtrikman bounds to composites with anisotropic constituents (Walpole 1966(a); Walpole 1966(b); Walpole 1969). Using this bounding approach, researchers (Eduljee et al. 1994; Willis 1977; Wu and McCullough 1977) derived estimates for the elastic moduli of aligned ellipsoidal and cylindrical short fibre composites with transversely isotropic properties. According to (Eduljee et al. 1994), 209  inclusion shape and geometry introduce significant complexities to models proposed for predicting the elastic moduli of discontinuous fibre composites.  Since the early 1960’s, researchers have examined the accuracy of micromechanical models and bounds using both experiments and numerical models. Most notably, (Tsai 1964; Whitney and Riley 1966; Adams and Doner 1967; Adams and Doner 1967) evaluated the accuracy of micromechanical models at different volume fractions of fibres by comparing prediction results with experimental data on Glass/Epoxy and Boron/Epoxy composites. According to their results, very good agreements were found for longitudinal properties. However, significant scatter and some discrepancies were found in transverse directions. These discrepancies were mostly attributed to errors in experimental measurements and assumed geometrical models.  One of the most comprehensive experimental works in this area was later conducted by Brintrup (Brintrup 1975) for Glass/Polyester composites at a wide range of volume fractions. Although very good agreement between analytical model predictions and experimental data for longitudinal Young’s modulus was reported, there are discrepancies between analytical closed-form solutions and experimental data for transverse Young’s modulus. In Figure B-1, some of the experimental data reported in the literature for transverse Young’s modulus of different composites at different volume fraction of fibres are shown. The solid line represents the analytical closed-form solutions. Note that only those experimental works are considered in this figure where the constituents’ properties are reported by the authors. These properties are also given in Appendix C for convenience.  210   Figure B-1 Comparisons between the experimental data reported in the literature and the corresponding analytical predictions (Christensen and Lo 1979) for transverse Young’s modulus of different composites at different volume fraction of fibres. The lines represent the analytical closed-form solutions for Glass/Epoxy, Boron/Epoxy and Glass/Polyester composites using the constituents’ properties given in Appendix C.  The discrepancy for Glass/Polyester composites has been addressed in (Hull 1980). Although there is a very good match between analytical micromechanics (both isostrain and the CCA model) and experimental data for longitudinal modulus (not shown here), the situation for modulus in the transverse direction is less clear as mentioned by Hull. This is mainly due to scatter in experimental data in this direction and the inelastic deformation of the matrix. It should be emphasized that these measurements were conducted by applying relatively large stresses for certain period of time. During loading, plastic deformation or creep may occur. This can lead to degradation of resin modulus in this direction. Experiments with more creep resistant materials 211  show better agreements between the experimental data and the analytical predictions in this figure.  Due to experimental challenges and advances in computational powers, researchers have evaluated the accuracy of micromechanical models using numerical simulations beside experiments as well. Adams and coworkers (Adams 1987; Adams and Doner 1967; Adams and Doner 1967; Adams and Crane 1984) was among the first who used numerical techniques (finite difference and finite element) to simulate the behaviour of fibre reinforced composites at the micro-scale. In later years researchers employed this technique and evaluated the accuracy of micromechanical models for a wide range of composites materials (Aboudi 1983; Hashin 1972; Facca et al. 2006; Tucker and Liang 1999; Weng 1984; Halpin and Kardos 1976; Sun and Vaidya 1996).   In a comprehensive study, Noor and Shah (Noor and Shah 1993) compared the predictions of several micromechanical models with 3D finite element results and experimental data of unidirectional fibre composites. Comparisons were shown for composites with a range of volume fractions from 10% to 80%. According to this study, spatial arrangement of fibres affects transverse properties (2E , 23G  and 23 ) more significantly than longitudinal properties. Among micromechanical models examined in this study, Mori-Tanaka predictions were believed to be the closest to experimental data as well as finite element simulations with hexagonal packing of fibres. This conclusion has also been confirmed later by Tucker and Liang (Tucker and Liang 1999) for short fibre composites for fibre volume fractions up to 30%.  212  Compared to mechanical properties of composites, the thermal properties of composites such as coefficients of thermal expansion (CTE) and thermal conductivities have not been analysed in the literature as extensively. Few researchers have proposed analytical approaches for thermal properties of composites and multi-phase materials (Levin 1967; Rosen and Hashin 1970; Schapery 1968). Although some other analytical approaches have also been proposed, reviewing the available theoretical predictions for CTE by Bowles and Tompkins (Bowles and Tompkins 1989) shows that most analytical predictions except those described by Rosen and Hashin (Rosen and Hashin 1970) which is an extension of Levin results, did not account for Poisson’s ratio effects in the transverse direction accurately. Rosen and Hashin’s analytical approach is presented in Section B.2.3.   As shown by Bowles and Tompkins, the analytical solutions presented by Rosen and Hashin (Rosen and Hashin 1970) which can be employed for predicting the effective coefficients of thermal expansion of any two-phase material regardless of its microstructural geometry, give the closest match with experimental data as well as numerical results. Therefore, in this thesis, the analytical expressions described in (Rosen and Hashin 1970) are employed for predicting the effective coefficients of thermal expansion of composites.  Using the elastic-viscoelastic correspondence principle and Laplace transform (see Section 2.3.1), researchers have extended the application of these micromechanical models for predicting the linear viscoelastic behaviour of composites (Abadi 2009; Brinson and Lin 1998; Hashin 1972; Pichler and Lackner 2009; Wang and Weng 1992; Yancey and Pindera 1990).  However, there are challenges and limitations involved and the accuracy of these models should be 213  examined carefully. To overcome some of these challenges, some researchers have used incorrect assumptions such as constant Poisson’s ratio with time (Hilton 2003). In the following sections two main aspects in using micromechanical models for micro-scale analysis of viscoelastic composites with applications in aerospace and automotive industries are addressed. The macro-scale analysis of structures made from these composites is the subject of next chapter (Chapter 3).  B.2 Analytical Approach B.2.1 Continuous Long Fibre Composites Based on a continuum approach, Hashin and Rosen (Hashin and Rosen 1964) proposed the Composite Cylinders Assemblage (CCA) model to estimate the effective properties of unidirectional cylindrical fibre composites. A simple version of this model is presented in (Hashin 1972). Using this approach, closed-form expressions were derived for four effective elastic moduli and bounds for the fifth elastic modulus of transversely isotropic fibre composites. A closed-form solution for the fifth elastic modulus was later derived by Christensen and Lo (Christensen and Lo 1979). The CCA direct approach can be presented as follows:   CCA Model Let us assume that the volume of a composite material can be filled with an assemblage of cylindrical fibres in a surrounding shell of resin as shown in Figure B-2. This requires having a wide range of fibre diameters in the system. Now, if in each fibre-resin cylinder, the ratio of the diameter of fibre ( b ) to the diameter of the surrounding resin ( a ) is assumed to be constant, it 214  can be shown that this ratio represents the volume fraction of the fibre (fV) in the whole system as:  2ffcA bV A a       (B.1)   Figure B-2 The Composite Spherical/Cylindrical Assemblage Model.  The effective transverse bulk modulus of the above system is derived by finding the solution to a boundary value problem with the following homogeneous displacement boundary conditions.     12 23 30u Su S xu S x  (B.2) ba213Matrix (resin)Long cylindrical fibreor Spherical reinforcement215  where the displacement field created by the applied uniform strain (  ) is denoted by “ u ”, and “S ” is the surface of a single composite cylinder.  Note that 1 refers to the fibre direction while 2 and 3 are directions perpendicular to the fibre direction. The components of position vector ( x ) are also denoted by (1 2 3, ,x x x ) accordingly. The above conditions may also be written in cylindrical coordinates. Considering symmetry, assuming plane strain conditions, and applying displacement and traction continuity at the fibre-resin interface described as:        f rf rrr rru b u bb b   (B.3) where superscripts f , r   refer to fibre and resin and subscripts r  denotes radial direction (in transverse direction of fibre), as shown in Figure B-2. The effective plane strain bulk modulus (23cK ) can be obtained as follows (Hashin 1972):   23 2rrrcaK   (B.4)   23 2323 23 2311fc rff r r rVK KVK K K G    (B.5) It should be noted that applying the following homogeneous tractions boundary conditions also leads to Equation (B.5).     12 23 30T ST S nT S n  (B.6) In Equation (B.5), the fibre and the resin plane strain bulk modulus, 23 fK  and 23rK , are respectively calculated as: 216        23 223 23 3 12 111/ 4 / 4 /f f f f f fK G E E    (B.7)   23 22 1 2rr r rEK     (B.8) where r and f are the resin and fibre Poisson’s ratios. Note that the resin plane strain bulk modulus, denoted by23rK , is different than the resin bulk modulus rK . The plane strain bulk modulus of the material, which is denoted by 23K in general, is derived based on the plane strain condition (11 0  ) as:  2322 332 2rP PK     (B.9) In the above equation, P  is the applied pressure on the material and 22  and 33  are the normal strains in transverse directions (2 and 3). Using the stress-strain relationship it can be shown that:  22 2323 2C CK   (B.10) For an isotropic material (e.g. resin), Equation (B.10) leads to Equation (B.8) through substituting the stiffness matrix components, 22C  and 33C , in terms of elastic Young’s modulus and Poisson’s ratio. Using Hill’s universal relations (see (Hill 1964)) which is written as:       2121 123 23 2323 234 1 111 1f r ffc f f r ff r cf rVVE E V E VK K KK K                      (B.11) 217      1212 1223 23 2323 231111 1ff r fc f f r fc f rf rVVV VK K KK K                      (B.12) and substituting the elasticity solution for 23cK  given in Equation (B.5) into the above equations, the exact solutions are obtained for longitudinal Young’s modulus (1cE ) and major Poisson’s ratio of the composite (12c ). This will lead to the following expressions for the effective properties of the unidirectional, continuous, long fibre reinforced composite:       2121 123 234 1-1-1- 1f r f fc f f r ff ff r rV VE E V E VV VK K G              (B.13)       1223 2312 13 1223 231 11-1-1- 1f r f fr fc c f f r ff ff r rV VK KV VV VK K G                      (B.14) A similar approach can be applied to derive closed-form solutions for longitudinal shear modulus by applying the following boundary conditions on the surface of a composite cylinder:     1 12 22 12 13 0u S xu S xu S  (B.15) and calculating the effective longitudinal shear modulus by:  1212 0122cG   (B.16) 218  where 12  is the average shear stress in the composite and 12 is the applied uniform shear strain. Details of finding the solution is given by Hashin (Hashin 1972) and not repeated here. However, the final solution obtained by Hashin (Hashin 1972) is used in this thesis. Therefore, it is given below for reference:        1212 13121 11 1r f f fc c rr f f fG V G VG G GG V G V            (B.17) The aforementioned approach fails to yield the exact solutions for transverse shear modulus (Hashin 1979). The reason for this is that displacement and traction boundary conditions cannot be satisfied simultaneously at the fibre-resin interface under pure transverse shear. In fact, for specified shearing displacement boundary conditions, an upper bound, while for specified shearing traction boundary conditions a lower bound can be obtained using variational methods.  The upper bound for transverse shear modulus can be written as:          23 223 ( ) 23 21 3 11 3 1f r f f f rc rf f f f rV V V VG GV V V V          (B.18) Parameters  , r  , f  and  are defined as:  23 2323 232323 23-, ,1 1,2 2f fr f rr rf ffr rfrr fr r f fG GG GG GG GKKK G K G         (B.19) The lower bound for transverse shear modulus can be expressed as: 219       23 ( )231 212fc rf r rr r rf rVG GV K GG K GG G    (B.20) Using (B.18) and (B.20) we can obtain estimates for transverse modulus and Poisson’s ratio:  23 23 ( )2 ( )23 23 ( )4 c ccc cK GE K mG  (B.21)   23 23 ( )23 ( )23 23 ( )c ccc cK mGK mG   (B.22) where m  is defined as:  12231241 cccEmK  (B.23)  Although the CCA model gives bounds for transverse properties of transversely isotropic long fibre composites, these bounds are very close and diverge from each other by increasing the mismatch of phase properties. In order to find an exact solution for effective transverse shear modulus of unidirectional long fibre composites, Christensen and Lo 1979) (Christensen and Lo 1979) employed the GSC model (Hermans 1967).   GSC Model The GSC model, also known as three phase model, consists of an inclusion (fibre), a matrix phase around it and an infinite homogeneous phase region having the effective properties of the composites surrounding these both phases (i.e. fibre and matrix). Using this model Christensen and Lo found the exact solution for effective transverse shear modulus of spherical and cylindrical composite systems by satisfying all continuity conditions. To obtain the exact 220  solutions, they used the equilibrium equations and applied the continuity of the stress and displacement at the two interfaces, i.e. the interface between the fibre and the matrix, and the one between the matrix and the infinite homogeneous region. Additionally, they used the Eshelby’s results for the strain energy of a homogeneous medium containing an inclusion (see (Eshelby 1957)) and set the total strain energy of the three phase model equal to the strain energy of its equivalent homogenous medium. They showed that the exact solution of 23cG  can finally be obtained by solving the following quadratic equation:  223c 23cr rG GA B C = 0G G             (B.24) where A , B  and C  are some mathematical expressions given below:    2 23 2323 23 23 2333 1 11 1f ff f fr rf f f fr r f r f f r f rr r r rG GA V VG GG G G GV VG G G G                                                                 (B.25)      223 2323 23 23 23323 23 23 36 1 11 1 1 21 1f ff f fr rf f f fr f r f f r fr r r rf f fr f f r f fr r rG GB V VG GG G G GV VG G G GG G GV VG G G                                                                              (B.26)  221    2 23 2323 23 23 23 33 1 11 1 (( ) )f ff f fr rf f f fr f f r f fr r r rG GC V VG GG G G GV VG G G G                                      (B.27) where  233 43 4f fr r      (B.28) A mistake was initially made in deriving the above equations which later was corrected in (Christensen 1990).  It should be noted that exact solutions for transverse Young’s modulus and transverse Poisson’s ratio can be obtained using the same equations as Equation (B.21), and Equation (B.22) knowing the exact 23cG . Although the CCA model and Christensen model are different, they also have similarities as described in Section B.1. In this thesis, the solution of Christensen’s GSC model is considered as the closed-form solution for the effective transverse shear modulus of unidirectional long fibre composites.  It is worth mentioning that one application of the above analytical expressions is in determining the elastic properties of transversely isotropic fibres (e.g. carbon fibres). Carbon fibres are known to be transversely isotropic materials and measuring their transverse properties directly is a difficult task due to the small diameter of carbon fibres (~0.01 mm) (Hashin 1979). Therefore the micromechanical equations presented here can be employed for this purpose in addition to estimating the effective properties of fibre composite materials. 222   Using some available experimental data and numerical reference solutions, the validity of the above analytical closed-form solutions in predicting the effective properties of continuous long fibre composites is investigated in Section B.4.1.  B.2.2 Short Fibre Composites Halpin and Tsai rearranged Herman’s results (Hermans 1967) for effective moduli (plane strain bulk modulus, longitudinal and transverse shear moduli) of continuous long fibre composites with transversely isotropic phases and put them in a general simple analytical form and extended its application to composites with different fibre geometries and transversely isotropic phases (Halpin and Kardos 1976). The Halpin-Tsai equations in a general form can be written as:  11fc rfVP P V      (B.29) Here, cP  is the composite modulus, rP  is the resin (matrix). The   parameter is a geometrical parameter defined according to the loading direction. Parameter   is defined as a function of  ,  and constituent’s moduli:  1cfcfPPPP (B.30)  Halpin and Tsai tried to extend the application of their equations to estimate engineering constants, specifically 2E , 12G , 23G and 23  of different composites (including short fibre composites) using some approximation (see (Halpin and Kardos 1976)). They examined the 223  limiting values of  and concluded that this parameter should be a measure of fibre geometry. However, one major difficulty in using Halpin-Tsai Equations is still to determine a correct value for   (see (Jones 1999)) depending on the fibre geometry.  For unidirectional long fibre composites, estimates of  parameter were obtained by comparison with numerical solutions of Adams and Doner (Adams and Doner 1967) for square packing of fibres. Halpin and Kardos (Halpin and Kardos 1976) found very good agreements up to fibre volume fraction of 0.7 between numerical results and Halpin-Tsai predictions for 2E  and 12Gusing 2   and 1  , respectively.  For circular short fibre composites, Table B-1 summarizes expressions for calculating  and estimating the effective moduli as suggested by Halpin and Kardos (Halpin and Kardos 1976) and Tucker and Liang (Tucker and Liang 1999) which are employed in this thesis. In Table B-1, the fibre length and fibre diameter are denoted by L and D , respectively. The fibre longitudinal direction is indicated by suffix 1. Since aspect ratio of the fibre is considered in Halpin-Tsai equations, these expressions are still widely being used by researchers for predicting the properties of short fibre thermoplastic composites (Facca et al. 2006; Hine et al. 2002) as well as nano-composites (Fornes and Paul 2003; Jia et al. 2013).  However, some researchers also suggested employing shear lag analysis to obtain the longitudinal modulus of the aligned short-fibre composites (mainly due to higher accuracy and simplicity of shear lag models) and approximate other engineering constants by micromechanical 224  models proposed for continuous long fibres (Tucker and Liang 1999; Hine et al. 2002). Therefore, revisiting the validity of these equations for short fibre composites is necessary and in this thesis (Section B.4.2) Halpin-Tsai equations along with analytical solutions for UD continuous long fibre composites are examined over a wide range of volume fractions and fibre aspect ratios, using numerically generated reference data.  Table B-1 Fibre geometry parameter in the Halpin-Tsai equations for circular short fibre composites as suggested by Halpin and Kardos (Halpin and Kardos 1976) and Tucker and Liang (Tucker and Liang 1999).   B.2.3 Coefficient of Thermal Expansion (CTE) The effective coefficient of thermal expansion of composites can be calculated analytically in a boundary value problem involving a composite subjected to zero tractions and a constant temperature change on the boundary. This problem has been analysed elegantly by Levin (Levin 1967) for a two-phase material in 1967. Rosen and Hashin (Rosen and Hashin 1970) generalized Levin's method for multi-phase materials with anisotropic phases and then derived bounds for the coefficients of thermal expansion and specific heat capacities using energy methods. Since Levin’s paper is in Russian, Rosen and Hashin’s general approach for a two-phase composite is presented here. E 1 2 (L/D )E 2 2G 12 1G 23 1Property ζ225   Assuming a two phase body of volume V and bounding surface S , is subjected to a constant boundary temperature,   and a homogeneous traction boundary condition, iT , then we can write:   S     (B.31)    ; , 1, 2, 3i ij jT S n i j   (B.32) where ij  are the components of the stress tensor and jn  are the components of the unit vector normal to the boundary. The thermo-elastic problem defined by (B.31) and (B.32) can be solved by superposing the solution of two linear problems of     0iST S   (B.33) and    0i ij jST S n  (B.34) The solution procedure is quite long and the reader is referred to (Rosen and Hashin 1970) for the details. According to (Hashin 1979; Hashin 1972), the effective coefficient of thermal expansion for an anisotropic fibre reinforced composite cij  can be calculated as follows:     c f r cij ij ij ij klrs rsij rsijP S S        (B.35) where rsijS  is the compliance tensor and the tensor klrsP  is defined as:   f rklrs rsij rsij klijP S S I   (B.36) 226  In Equation (B.35), ij  and rsijS  are the volume average coefficient of thermal expansion and the volume average compliance tensor, respectively, given by:   1f rij ij f ij fV V       (B.37)   1f rrsij rsij f rsij fS S V S V    (B.38) Similar to the elastic properties, through measuring the coefficient of thermal expansion of the composite experimentally and knowing the elastic properties of the resin, fibre and the composite, thermal expansion coefficients of transversely isotropic fibres can be estimated using Equation (B.35) (Hashin 1979).  For composites with isotropic phases, Equation (B.35) can be simplified to obtain the longitudinal coefficient of thermal expansion, 11c as:     1211 11113 1 21 1ff r fccc f rf rVVE K KK K                  (B.39) and the transverse  coefficient of thermal expansion, 22 33c c  , as:     12 1233 22 2223 113 1 231 1 2ff r fc cc cc c f rf rVVK E K KK K                     (B.40) In the above equations fK  and rK  are the bulk modulus of the isotropic fibre and the resin, respectively. It should be noted that the effective elastic properties of the composite, denoted by the subscript c  in the above equations, are needed to calculate the effective coefficients of 227  thermal expansion. The effective elastic properties of the composite can be obtained from micromechanical models (e.g. CCA model) presented earlier.  The above equations are different than Schapery’s results (Schapery 1968) which also give good predictions for effective coefficients of thermal expansion.           13 2 12 12 1111 1 1 1f f f m m fcf f m fc c f f f m m f f f m f cE V E VE V E VV V V V                       (B.41) Note that the coefficient of thermal expansion 1c in (B.41) can also be derived using the strength of material approach presented in (Hyer and White 1998).   Schapery’s results are obtained through employing minimum complementary and potential energy principles of thermo-elasticity combined with approximating values of potential and complementary energies and are limited to composites with isotropic phases. However, some researchers (e.g. (Strife and Prewo 1979; Bogetti and Gillespie 1992; White and Kim 1998)), simply modified Equation (B.41) and applied it to transversely isotropic fibres by replacing f  with 1 f  and 2 f  which is not a mathematically correct approach (see (Bowles and Tompkins 1989)). Therefore, the effective coefficients of thermal expansion are estimated analytically in this thesis using the original approach proposed by Levin (Levin 1967), and described by Rosen and Hashin (Rosen and Hashin 1970) as in Equation (B.35). All of the elastic solutions can be extended to Laplace transformed solutions for linear viscoelastic materials by means of the correspondence principle (see Section 2.3.1). 228  B.3 Numerical Approach Although analytical models have proven to be capable of predicting the elastic properties of conventional fibre composites, there are some limitations associated with these models in estimating the effective properties of other composites which should be noted. For instance, studies on high energy composites by Banerjee and coworkers (e.g. (Banerjee and Adams 2003))  have shown that analytical micromechanics (CSA, Hashin-Shtrikman, Self-Consistent Scheme) are not able to predict the effective elastic properties of these composites. In fact, predictions were an order of magnitudes higher (Self-Consistent Scheme) or lower (CSA) than the experimental data. The large discrepancies between analytical predictions and experimental data were attributed to the significant mismatch between the properties of the binder ( E   0.7 MPa) and the filler ( E   15,300 MPa) and the high volume fraction (92%) of the filler in such composites. However, numerical predictions using computational homogenization technique provided reasonable estimates for the effective shear and bulk properties at different temperatures (Banerjee 2002).   Therefore, as an alternative to the analytical approach, a numerical approach for determining the effective elastic properties of composites is presented here. For this purpose, computational homogenization based on full-field microstructure simulations (described in Appendix A) is employed to obtain the effective properties of heterogeneous composite materials. Such simulations are moreover used as reference solutions to evaluate the relevance of analytical micromechanics models described earlier.   229  Assuming a periodic microstructure for the composite, an appropriate unit cell of the material will be identified in order to determine its homogenized properties. This section presents the identification, geometrical characteristics and the morphological content of the unit cells considered, and the procedure used to estimate their effective properties. Once the unit cell is identified, it is discretized and subjected to six elementary loading conditions using periodic boundary conditions. Three axial and three shear loadings are applied separately for this purpose. Periodic boundary conditions described earlier are implemented using Python scripts (Python Software Foundation 2013). The resulting stress and strain fields are used to calculate the average stress and strain and hence the effective stiffness matrix for the unit cell of the material (see Appendix A).  The model and the methodology for each unit cell is verified by assigning the same elastic properties to both resin and fibre phases as the input parameters and comparing the homogenized output results with the given input properties. For composite unit cells where similar numerical approaches have been employed in the literature to estimate their effective properties, the same inputs are used and the results of the two models have been compared. The constituents are assumed to be linear elastic in this section of the thesis. Therefore, the effective macroscopic behaviour is also linear elastic. B.3.1  Continuous Long Fibre Composites Unidirectional (Hexagonal and Square packing) For modeling purposes, UD continuous long fibre composites may be idealized as a periodic array of square or hexagonal fibres.  Figure B-3 shows the idealized cross-section of a typical 230  continuous long fibre reinforced composite. Note that perfect bonding between the fibre and resin is assumed here and only the cross-section of the material is shown here. Assuming periodicity and a certain fibre packing geometry in the entire composite material, two different repeating 3-D unit cells of the material can be identified as indicated by the dashed lines in Figure B-3. These two different 3-D unit cells represent the microstructure of UD continuous long fibre composite and are referred to as Square (SQR) or Hexagonal (HEX) unit cells in the rest of this thesis.  Figure B-3 Idealized cross-sections of unidirectional long fibre reinforced composites and their smallest repeating unit cells assuming: (a) square packing (SQR), and (b) hexagonal packing (HEX) of fibres.  The geometrical parameters, a  and d , for the unit cell representing the square packing of fibres in Figure B-3(a)  are related to fibre volume fraction, fV as follows:  24 fda V  (B.42) where, a  is the cross-sectional height (and width) of the unit cell and d  is the fibre diameter.  (a)YZ(b)daaYZabdaa231  The cross-sectional width a , height b , for hexagonal packing are related to the fibre volume fraction, fV as follows:  22 3 fdaV  (B.43)  3b a   (B.44) A constant length of unity is considered for both unit cells. It should be noted that in both packing geometries, a  is also the closest distance between the centres of the fibres. Therefore, the maximum volume fraction of fibres in both packing geometries occur when the fibres are touching (i.e 2a R ) (see (Hull 1980)). It can be shown that the maximum fibre volume fraction occurs at 78.5% and 90.7%, in square and hexagonal unit cells, respectively. Both unit cells are then discretized for the finite element analysis. The cross-sections of the discretized unit cells are depicted in Figure B-4. The unit cells have been discretized using 8-noded linear brick elements, C3D8 in the commercial finite element software, ABAQUS® (ABAQUS Inc. 2010).  232   Figure B-4 Unidirectional long fibre composite. Discretized unit cell of (a) Square (b) Hexagonal packing at fibre volume fraction of 0.6.  Mathematically, it can be shown that the hexagonal packing maintains the transverse isotropy which is a special feature of unidirectional composites, while square packing of fibres does not. Therefore, hexagonal unit cell is a more realistic representation of the UD long fibre composite microstructure (Li 1999; Li 2001). It should be noted that there is a distinction between hexagonal symmetry and transverse isotropy symmetry (Cowin and Mehrabadi 1995). Nonetheless, it can be shown that the two symmetries are characterized by exactly the same stiffness tensor. It was shown by Cowin and Mehrabadi (Cowin and Mehrabadi 1995) that the six-fold symmetry of hexagonal systems and the infinite-fold symmetry of transversely isotropic materials place the same restrictions on the form of the stiffness tensor. Thus, the two symmetries are treated the same in elasticity theory.  Although the hexagonal unit cell is a more realistic representation of the composite microstructure, comparing the results of the two unit cells will provide information on the (a) (b)YZYZ233  influence of the spatial distribution of the fibres. Additionally, the variability of a  and b  parameters for a constant fibre diameter allows for investigating the effect of different fibre volume fractions.  Unidirectional with Waviness (Hexagonal packing) Although fibres were assumed to be straight in unidirectional continuous long fibre composites, small waviness may be introduced during the manufacturing process (Garnich and Karami 2004). Therefore, it is crucial to include waviness in micro-level analysis of composites and examine its effect on structural level properties of the composites.   Assuming the long fibre composite consists of a periodic arrangement of wavy continuous fibres (Figure B-5(a)), a repeating unit cell of the material can be identified as shown in Figure B-5(b).    Figure B-5 Wavy unidirectional long fibre composite. (a) idealized periodic microstructure, and (b) repeating unit cell  A hexagonal packing of fibres may also be considered due to its more realistic features explained previously. A discretized unit cell of the wavy material as shown in Figure B-6 can be employed (a)XY(b)234  to estimate effective properties of wavy unidirectional long fibre composites with the same procedure used for long fibre composites with hexagonal packing.   Figure B-6 Wavy unidirectional long fibre composite. Discretized wavy unit cell assuming hexagonal packing of fibres.  To generate the wavy geometry, the cross-sectional profile of the unit cell is extruded in the third direction ( X ). A centre point with coordinates  0 0 0, ,X Y Z  is chosen for this purpose. The height of this centre point  0Y  is assumed to vary along the fibre length ( x ) according to the following function:  02( ) cosY x A x      (B.45) where A  is the amplitude and   is the wavelength of the fibre geometry. Therefore, for a given waviness ratio ( /A  ), an arbitrary length (  ) is required in addition to fibre volume fraction, fV , and fibre diameter, d  , for generating the wavy unidirectional fibre composite unit cell. XZYAmplitude(A)Wavelength (λ)XY235    In a different but simpler approach, a straight unit cell may be used through incorporation of material orientation along the unit cell length. The material orientation is incorporated by defining local coordinate systems as depicted in Figure B-7. The straight unit cell is made of a homogeneous orthotropic (or transversely isotropic) material having its effective properties obtained from the UD long fibre composite unit cell (Micro-scale Unit cell). The homogenized properties in the straight unit cell can also be obtained from another micromechanical approach (e.g analytical) once its validity has been verified (see Section B.4.1).  The material orientation varies along the unit cell length according to the wavy geometry defined in Equation (B.45) so that the longitudinal axis ( x ) of the material orientation is always tangent to the corresponding fibre direction of the wavy unit cell. A similar approach has previously been used by Karami and Garnich 2005 (Karami and Garnich 2005). This approach can be employed in a multi-scale modelling framework to enhance the modelling efficiency by decreasing the computational time. It also enables us to examine the validity of using analytical micromechanics approach with appropriate tensorial transformation matrix (e.g.  (Quek et al. 2003; Shokrieh and Mazloomi 2010; Shrotriya and Sottos 2005)) to estimate the effective properties of composites with undulating fibres.  236   Figure B-7 Wavy unidirectional long fibre composite. Discretized straight unit cell with incorporated material orientation assuming hexagonal packing of fibres. Local coordinates are defined to represent the material orientation inside the straight unit cell.  B.3.2 Short Fibre The unit cell used for square packing of long fibre composites is employed here for short fibre composites with some modifications. Here, the fibre is assumed to have a general elliptical cross-section with a specific geometry as depicted in Figure B-8. The resin thicknesses (rxt , ryt and rzt ) in the three main directions ( , ,X Y Z ) and the fibre length ( , ,L R T )are the variable parameters in this unit cell. XZYxyzxyzMicro-ScaleUnit CellMeso-ScaleUnit CellGlobal Coordinate SystemLocal Coordinate System237   Figure B-8 Schematic of short fibre composite unit cell.  Figure B-9 shows the discretized unit cell for a circular short fibre composite which is employed for examining the validity of analytical approach for short fibre composites. The fibre is assumed to have a circular cross-section, therefore ( R T ) and is embedded in a matrix (resin) with a constant resin thickness in all three directions, i.e. rx ry rzt t t  . YXZYZXZ(a)(c)(b)238   Figure B-9 Discretized unit cell of a short fibre composite.   B.3.3 Coefficient of thermal expansion (CTE) Numerical approach can also be employed to determine the effective thermal properties of composites. The same unit cell as the one used to determine the effective elastic properties can be used to determine the effective coefficients of thermal expansion (CTE) of different composites. However, CTE calculation needs new inputs and a new set of loading to be imposed. Therefore, a new input file is required. In this new input file, the constituents’ CTE values are given as inputs with their elastic properties and an arbitrary temperature change of 1 C  is applied in a single step along a traction load. The unit cell boundaries are still subjected to the same periodic boundary conditions. As explained earlier, the periodicity conditions force opposite faces to have identical deformed shapes during the load application in order to keep the integrity of the whole material.  XYZ239  A temperature change of 1 C  is applied to all nodes in the model to simplify the calculation of the outputs. Therefore, using the stress-strain relationship, the components of unit cell average strain and stress vectors are related by:     1 1  1, 2, 3C iE iTi S T i       (B.46) or,      1 1  1, 2, 3i i iCTE S i      (B.47) In the above equation, 1iS denotes the components of the first column in the volume average compliance matrix and 1  is the applied longitudinal stress. The effective CTE of the material unit cell can now be calculated by subtracting the average strain vector in the unit cell with no temperature change from the average strain vector of the unit cell subjected to a temperature change of 1 C as follows:     CTEi i i     (B.48) where i  is the volume average strain vector component in the unit cell subjected to traction only:  1 1 i iS    (B.49)  B.4 Validations and Comparisons B.4.1 Long Fibre Composites In this section, the numerical finite element predictions of the elastic properties of the unidirectional fibre composites are compared with available experimental data, analytical solutions and some other numerical results. The numerical approach is first validated using 240  experimental data for 3 different types of composites: AS4/3501-6, Boron/Aluminum and E-Glass/Epoxy. The numerical results are also compared with those of other similar numerical approaches reported in the literature. In order to examine the validity of the analytical approach presented in Section B.2.1, numerical results of unit cells with both square and hexagonal packing are compared with the CCA model predictions for longitudinal properties and the GSC model solutions of Christensen and Lo (Christensen and Lo 1979) for transverse properties.  Summary of results in the form of engineering constants for the three composites, are given in Table B-2, Table B-3 and Table B-4. In general, very good agreement is observed between the current numerical predictions and the numerical results reported in the literature for all three composite systems. For most engineering constants the discrepancy between the two numerical approaches is below 2%, except for 23  of E-Glass/Epoxy composite in square packing and 23  of Boron/Aluminum composite in both packing geometries. For Boron/Aluminum, no experimental data are available for23 . However, the current numerical predictions (HEX) are in better agreement with analytical solutions of Christensen and Lo (Christensen and Lo 1979). Also, for E-Glass/Epoxy the experimental results are between current numerical predictions using HEX or SQR unit cells and the current numerical predictions for all engineering constants using HEX unit cell are in better agreement with the analytical solutions. Note that in reality, the fibres in UD composites are randomly distributed. However, a numerical study by Huang et al. (Huang et al. 2008) has shown that the predictions of HEX unit cells are very close to those of a unit cell of random fibre arrays.  241  As noted in (Sun and Vaidya 1996), large discrepancies are observed in the transverse properties between the numerical (as well as analytical) and experimental data. This could be due to the wide range of scatter in the properties of the transversely isotropic AS4 fibre reported in the literature (Sun and Vaidya 1996).  According to Table B-2, Table B-3 and Table B-4, the analytical approach is capable of predicting all engineering constants of the three types of composites with very high accuracy (maximum error below 1.4% between the numerical results (HEX) and the analytical solutions).  Table B-2 Engineering constants for AS4/3501-6 composite (fibre volume fraction of 0.60). Experimental data and predictions of micromechanical models (Numerical and Analytical) are given for comparison. Constituents’ properties are given in Appendix C.        FE (SQR) FE (HEX) FE (SQR) FE (HEX)E1 (GPa) 139 142.6 142.6 142.1 142.3 142.9 (b)E2 (GP ) 9.85 9.6 9.2 9.6 9.2 9.2 (c)G12 (GPa) 5.25 6.0 5.9 6.0 5.8 5.8 (b)G23 (GPa) - 3.1 3.4 3.1 3.3 3.34 (c)ν120.3 0.25 0.25 0.253 0.253 0.25 (b)ν23- 0.35 0.38 0.349 0.377 0.38 (c)(a) Sun & Vaidya (1996)(b) Hashin (1972)(c) Christensen and Lo (1979)Property Experiment (a)Numerical (a) Numerical (present work)Analytical242  Table B-3 Engineering constants for Boron/Aluminum composite (fibre volume fraction of 0.47). Experimental data and predictions of micromechanical models (Numerical and Analytical) are given for comparison. Constituents’ properties are given in Appendix C.    Table B-4 Engineering constants for E-Glass/Epoxy composite (fV 0.54). Experimental data and predictions of micromechanical models (Numerical and Analytical) are given for comparison. Constituents’ properties are given in Appendix C.   The volume fraction effect may be examined using the numerical approach. To validate the numerical approach for studying the volume fraction effect, the numerical results are compared with those reported in (Garnich and Karami 2004). Using the same constituents’ properties as in AnalyticalFE (SQR) FE (HEX) FE (SQR) FE (HEX) (b) & (c)E1 (GPa) 216 215 215 214.6 214.4 215 (b)E2 (GPa) 140 144 136.5 144.8 133.5 134.2 (c)G12 (GPa) 52 57.2 54 54.3 54.0 54 (b)G23 (GP ) - 45.9 52.5 45.7 51.3 51.5 (c)ν120.29 0.19 0.19 0.195 0.196 0.195 (b)ν23- 0.29 0.34 0.249 0.305 0.304 (c)(a) Sun & Vaidya (1996)(b) Hashin (1972)(c) Christensen and Lo (1979)Numerical (a)Property Experiment (a)Numerical (present work)AnalyticalFE (SQR) FE (HEX) FE (SQR) FE (HEX) (b) & (c)E1 GPa) 41.5 41.6 41.6 41.5 41.4 41.6 (b)E2 GPa) 17.1 18.2 15.1 18.4 15.3 15.5 (c)G12 (GPa) 5.6 5.8 5.8 5.6 5.5 5.47 (b)G23 GP 6.1 4.2 5.3 4.3 5.4 5.47 (c)ν120.316 0.265 0.265 0.267 0.268 0.268 (b)ν230.391 0.551 0.430 0.311 0.420 0.416 (c) (a) Gusev et al. (2000)(b) Hashin (1972)(c) Christensen and Lo (1979)Property Experiment (a)Numerical (a) Numerical (present work)243  (Garnich and Karami 2004), the engineering constants of AS4/3501-6 composite at four different volume fractions have been compared in Table B-5. Results show very good agreement between the two sets of numerical results.  Using the same constituents’ properties, the accuracy of the analytical approach is examined for a wider range of volume fractions. The results are plotted in Figure B-10 . Note that the analytical solutions are denoted by lines (dashed line for exact solutions and solid lines for bounds), while the numerical results assuming hexagonal and square packing of fibres are shown with square and cross symbols, respectively. Comparisons show very good agreement between numerical predictions assuming hexagonal packing of the fibres (crosses) and analytical closed-form solutions for a wide range of volume fractions (fV  < 90%) studied.  The analytical results of the CCA model provide bounds for the effective elastic properties based on variational energy principles (Hashin 1972). For longitudinal properties (1E , 12G  and 12 ) these bounds coincide and provide exact solutions. The exact solutions for the effective transverse properties (2E , 23G  and 23 ) are obtained using the model proposed by Christensen and Lo (Christensen and Lo 1979), Equations (B.24)-(B.28). According to Figure B-10, the closed-form solutions for the effective transverse properties as well as the numerical results using hexagonal packing of fibres, lie well between the predicted bounds obtained from the CCA model. This has also been reported by Christensen and Lo (Christensen and Lo 1979) for the effective transverse shear modulus.  244  Table B-5 Numerical predictions of engineering constants for AS4/3501-6 composite at different volume fraction assuming hexagonal packing of fibres. Constituents’ properties are those of Garnich and Karami (Garnich and Karami 2004) and  given in Appendix C.  Numerical              Numerical               (Garnich & Karami  2004) (present work)E1 (GPa) 85.180 85.680 0.59%E2 (GPa) 8.213 8.255 0.51%E3 (GPa) 8.212 8.260 0.58%G12 (Gpa) 3.777 3.795 0.48%G13 (GPa) 3.768 3.801 0.88%G23 (GPa) 2.820 2.836 0.57%ν120.295 0.295 0.00%ν130.295 0.295 0.00%ν230.457 0.456 0.22%E1 (GPa) 105.400 105.977 0.55%E2 (GPa) 9.516 9.575 0.62%E3 (GPa) 9.512 9.579 0.70%G12 (Gpa) 4.778 4.820 0.88%G13 (GPa) 4.776 4.818 0.88%G23 (GPa) 3.298 3.321 0.70%ν120.285 0.285 0.00%ν130.285 0.285 0.00%ν230.443 0.442 0.23%E1 (GPa) 125.800 126.273 0.38%E2 (GPa) 11.200 11.256 0.50%E3 (GPa) 11.190 11.257 0.60%G12 (Gpa) 6.282 6.338 0.89%G13 (GPa) 6.276 6.323 0.75%G23 (GPa) 3.925 3.948 0.59%ν120.275 0.275 0.00%ν130.275 0.275 0.00%ν230.426 0.425 0.23%E1 (GPa) 145.700 146.022 0.22%E2 (GPa) 13.350 13.358 0.06%E3 (GPa) 13.340 13.349 0.07%G12 (Gpa) 8.649 8.803 1.78%G13 (GPa) 8.715 8.790 0.86%G23 (GPa) 4.712 4.743 0.66%ν120.266 0.266 0.00%ν130.266 0.266 0.00%ν230.405 0.407 0.49%0.60.70.4Error  (%)0.5PropertyVf245   Figure B-10 Unidirectional long fibre composite. Comparison between numerical and analytical predictions of effective elastic properties for AS4/3501-6 composite at different volume fractions. (a) Young’s moduli; (b) Shear moduli; (c) Poisson’s ratios. Constituents’ properties are those employed by (Garnich and Karami (Garnich and Karami 2004) and are given in Appendix C. (a)(b)(c)246   The numerical results for1E  and 12 show the negligible effect of packing geometry on these properties within the range of volume fractions used in this study. For 12G  both packing geometries give almost identical results up to volume fraction of 50%.  The fibre packing geometry affect the transverse properties (2E , 23G  and 23 ) more significantly at volume fractions above 0.3. In general, by increasing the volume fraction of fibre, the discrepancy between the results of two packing geometries becomes more pronounced.  To further analyse the effect of fibre packing geometry, the normalized transverse modulus is plotted in Figure B-11 for a fibre reinforced composite with relaxed resin properties. Results are presented using the constituents’ properties reported by White and Kim (White and Kim 1998). Although, a larger discrepancy is observed between the results from the two packing geometries (HEX and SQR), a good agreement is still observed between numerical results assuming hexagonal packing of fibres and the analytical solution. The results presented in this section, increase our confidence in using the analytical approach (presented in Section B.2.1) for micromechanical elastic modelling of unidirectional fibre composites. The predictive capability of this approach for viscoelastic properties of unidirectional fibre composites will be examined in Section 2.3.3. 247   Figure B-11 Unidirectional long fibre composite. Comparison between numerical and analytical predictions of effective transverse modulus for AS4/3501-6 composite using resin relaxed modulus. Constituent’s properties are those reported in (White and Kim 1998).  In order to examine the fibre waviness effect, the two unit cells described in Section B.3.1 are employed. Note that the second unit cell employs a straight unit cell with homogenized properties but with incorporated wavy material orientation along its length. The homogenized properties in the straight unit cell may be obtained from any micromechanics (numerical or analytical). Therefore, the equivalence of the two unit cells will enable us to efficiently analyse composite structures made of wavy fibres in a multi-scale modelling framework.  The effective elastic properties of the two unit cells have extensively been investigated at four different waviness ratios ( /A  ) and fibre volume fractions by Garnich and Karami (Garnich and Karami 2004; Karami and Garnich 2005). The results of the two unit cells at four different 248  waviness ratios for AS4/3501-6 composite at fibre volume fraction of 0.66 are summarized in Table B-6. At waviness ratio of zero, the results of the two models are identical and as the waviness increases the results start to deviate. Garnich and Karami (Garnich and Karami 2004) attributed the main reason for the deviations to the different numerical discretization used in the two FE models.   In Table B-7 the wavy unit cell (Figure B-6) results and its corresponding straight unit cell (Figure B-7) results are compared using the current numerical approach. The effective elastic properties obtained from the unidirectional unit cell with hexagonal packing (Figure B-4(b)) were used as the homogenized properties for the straight unit cell. Therefore, the small differences between the values in the two tables may be attributed to the different meshing pattern used in (Karami and Garnich 2005).  According to Table B-6 (also Table B-7) by increasing the waviness, the unit cell becomes orthotropic. The longitudinal Young’s modulus decreases while the transverse Young’s modulus in the second direction (in the waviness plane) increases as expected (but not as much as the longitudinal modulus reduces). The transverse Young’s modulus in the third direction (out of waviness plane) remains almost constant. If the average axial stress at each cross-section is plotted along the wavelength for both unit cell models, very similar stress distributions are obtained (see (Karami and Garnich 2005)). The equivalence of the local stress field implies that the straight unit cell may be used in meso-level simulations and in failure prediction of laminates with wavy fibres.  249  Table B-6 Numerical predictions of engineering constants for AS4/3501-6 composite with different waviness ratios assuming hexagonal packing of fibres and fibre volume fraction of 0.66 (Garnich and Karami 2004). Predictions using two different unit cells are compared. Constituents’ properties are given in Appendix C.   Karami & Garnich 2004 Karami & Garnich 2004Wavy Unit Cell Straight Unit CellE1 (GPa) 133.330 133.330 0.00%E2 (GPa) 9.125 9.125 0.00%E3 (GPa) 9.128 9.128 0.00%G12 (Gpa) 7.232 7.232 0.00%G13 (GPa) 7.255 7.255 0.00%G23 (GPa) 3.158 3.158 0.00%ν120.261 0.261 0.00%ν130.261 0.261 0.00%ν230.373 0.373 0.00%E1 (GPa) 127.500 127.050 0.35%E2 (GPa) 9.145 9.151 0.07%E3 (GPa) 9.128 9.130 0.02%G12 (Gpa) 7.160 7.222 0.87%G13 (GPa) 7.235 7.229 0.08%G23 (GPa) 3.141 3.144 0.10%ν120.255 0.256 0.39%ν130.265 0.264 0.38%ν230.373 0.373 0.00%E1 (GPa) 92.249 89.366 3.13%E2 (GPa) 9.359 9.358 0.01%E3 (GPa) 9.130 9.131 0.01%G12 (Gpa) 7.442 7.461 0.26%G13 (GPa) 7.400 7.336 0.86%G23 (GPa) 3.087 3.092 0.16%ν120.207 0.223 7.73%ν130.287 0.282 1.74%ν230.372 0.371 0.27%E1 (GPa) 61.690 57.770 6.35%E2 (GPa) 9.808 9.801 0.07%E3 (GPa) 9.132 9.134 0.02%G12 (Gpa) 7.684 7.924 3.12%G13 (GPa) 7.775 7.621 1.98%G23 (GPa) 2.963 2.976 0.44%ν120.160 0.191 19.38%ν130.306 0.299 2.29%ν230.371 0.367 1.08%0.040.0670Error  (%)0.0125PropertyWaviness Ratio ( A/λ )250  Table B-7 Numerical predictions of engineering constants for AS4/3501-6 composite at different waviness assuming hexagonal packing of fibres and fibre volume fraction of 0.66 (Present Work). Predictions using two different unit cells are compared. Constituents’ properties are given in Appendix C.       Present Work     Present WorkWavy Unit Cell Straight Unit CellE1 (GPa) 134.183 133.896 0.21%E2 (GPa) 9.149 9.152 0.04%E3 (GPa) 9.149 9.151 0.02%G12 (GPa) 7.312 7.317 0.07%G13 (GPa) 7.315 7.312 0.04%G23 (GPa) 3.166 3.166 0.01%ν120.261 0.261 0.00%ν130.261 0.261 0.00%ν230.373 0.373 0.00%E1 (GPa) 128.369 127.716 0.51%E2 (GPa) 9.176 9.174 0.02%E3 (GPa) 9.151 9.151 0.00%G12 (GPa) 7.338 7.342 0.05%G13 (GPa) 7.304 7.299 0.06%G23 (GPa) 3.172 3.172 0.01%ν120.255 0.261 2.35%ν130.264 0.261 1.14%ν230.373 0.373 0.00%E1 (GPa) 93.856 90.986 3.06%E2 (GPa) 9.389 9.379 0.11%E3 (GPa) 9.151 9.152 0.01%G12 (GPa) 7.534 7.569 0.47%G13 (GPa) 7.231 7.186 0.62%G23 (GPa) 3.225 3.222 0.09%ν120.220 0.261 18.64%ν130.283 0.261 7.77%ν230.371 0.373 0.54%E1 (GPa) 63.373 60.890 3.92%E2 (GPa) 9.794 9.780 0.14%E3 (GPa) 9.154 9.155 0.01%G12 (GPa) 7.909 7.984 0.95%G13 (GPa) 7.102 6.983 1.67%G23 (GPa) 3.326 3.315 0.34%ν120.187 0.261 39.57%ν130.300 0.261 13.00%ν230.368 0.373 1.36%0.067Waviness Ratio ( A/λ ) Property Error  (%)00.01250.04251  B.4.2 Short Fibre Composites In order to investigate the validity of Halpin-Tsai equations presented in Section B.2.2, the short fibre unit cell presented in Section B.3.2 is employed. The elastic properties of the resin and the fibre used for comparison between numerical and analytical approaches are listed in Table B-8. Constituents’ properties are typical of fibre reinforced thermoplastics which are employed by Tucker and Liang (Tucker and Liang 1999). Figure B-12 compares the finite element results and the analytical predictions of Halpin-Tsai equations for different Young’s and shear moduli at different fibre aspect ratios (i.e. /L D  = 1, 2, 4, 8, 16, 32, 64, 128) and a constant volume fraction of 0.2. In order to demonstrate the effect of fibre length on the elastic moduli further, the analytical results for UD continuous long fibre composite at the same volume fraction of fibre are also shown in Figure B-12.   The results for 1E  (numerical and analytical) and for 2E  (numerical) show that Young’s moduli of aligned short fibre composites approach their corresponding long fibre composites values as the fibre aspect ratio increases. For all moduli except longitudinal Young’s modulus, 1E , the effect of aspect ratio is negligible in both approaches. This is consistent with the results of Tucker and Liang (Tucker and Liang 1999). For longitudinal Young’s modulus, 1E , both approaches show an increase in the modulus with increasing the fibre aspect ratio while the transverse Young’s modulus,2E , almost remains constant. The Halpin-Tsai equations underestimate the Young’s modulus for all aspect ratios and their predictions agree better with the numerical finite element results in the transverse direction for 2E  in the considered range of fibre aspect ratios ( /L D  < 130). 252   According to Figure B-12 (b) shear moduli variation with the fibre aspect ratio is negligible. The Halpin-Tsai predictions for 12G are in excellent agreement with numerical FE results. These results suggest that 12G of an aligned circular short fibre composite is insensitive to the fibre aspect ratio and depend only on constituent phase properties and their volume fraction.   Table B-8 Constituents’ elastic properties for a typical thermoplastic composite reported by Tucker and Liang (1999)     Figure B-12 Short fibre composite. Comparison between numerical and analytical predictions of effective elastic properties for the composite at different fibre aspect ratios and at a constant volume fraction of 0.2. (a) Young’s moduli; (b) Shear moduli. Constituents’ properties are typical of fibre reinforced thermoplastics which are employed by Tucker and Liang (1999) and are given in Table 6.  E  (GPa) 30.00 1.00ν 0.20 0.38Property  Fibre Matrix(a) (b)253  In general, the Halpin-Tsai equations with suggested fibre geometry parameters in Table B-1 show fairly good agreement with finite element reference data except for longitudinal Young’s modulus, 1E . For longitudinal Young’s modulus, 1E  using the shear lag models (e.g. (Cox 1952)) may be a better choice than Halpin-Tsai equations as discussed in (Hine et al. 2002; Tucker and Liang 1999).  In order to investigate the validity of the Halpin-Tsai assumptions made for constant fibre geometry parameters in Table B-1, the effective properties of short fibre composites have been studied at a wide range of fibre volume fractions in Figure B-13. According to this figure, by increasing the fibre volume fraction, the effect of fibre aspect ratio on the effective properties of the composite becomes more significant. By increasing the fibre aspect ratio, the effective longitudinal Young’s and shear moduli increase (see Figure B-13(a) and (c)), while the effective transverse Young’s and shear moduli decrease (see Figure B-13(b) and (d)). This can only be realized from numerical results. However, the fibre aspect ratio effect for all effective moduli (except 1E ) is negligible at low volume fractions of fibres ( fV < 0.3) and Halpin-Tsai equations serve us well in this low volume fraction region.  254   Figure B-13 Short fibre composite. Comparison between numerical and analytical predictions of effective elastic properties for the composite at different fibre aspect ratios and fibre volume fractions. (a) Young’s moduli; (b) Shear moduli. Constituents’ properties are typical of fibre reinforced thermoplastic composites which are employed by Tucker and Liang (1999) and are given in Table B-8.      (b)(a)Increasing Increasing Increasing Increasing 255  Appendix C  Measured Elastic Properties of Different Composites  Measured elastic properties of some composites at different fibre volume fractions are listed below. The values have been reported in the given references. Constituents’ properties have also been given in these references. Some of these constituents’ properties were employed as inputs for micro-scale models in this thesis. Note that this is not an exhaustive list. Only references that report both the constituents’ properties and composite properties have been cited.  256  Table C-1 Constituents’ properties of different composites reported in the literature. Composite Type Vf Fibre Properties Resin Properties Composite Properties Ref. E1 (GPa) E2 (GPa) G12 (GPa) G23 (GPa) ν12 ν23 E (GPa) ν E1 (GPa) E2 (GPa) G12 (GPa) G23 (GPa) ν12 ν23 Glass/Epoxy 0.54 72.5       0.2   5.32 0.365 41.5 17.1 5.63 6.07 0.32 0.39 Gusev et al (2000) Glass/Epoxy 0.52 73       0.2   2.7 0.4   14.1         Benzarti et al (2000) Glass/Epoxy 0.76 73.14       0.22   3.45 0.35   20.01         Tsai (1964)   0.71 73.14       0.22   3.45 0.35   19.32           0.71 73.14       0.22   3.45 0.35   22.08           0.70 73.14       0.22   3.45 0.35   15.18           0.64 73.14       0.22   3.45 0.35   15.18           0.64 73.14       0.22   3.45 0.35   13.11           0.62 73.14       0.22   3.45 0.35   13.11           0.61 73.14       0.22   3.45 0.35   11.73         S Glass/Epoxy 0.39 82.80       0.20   3.45 0.35   8.63         Adams & Doner (1967)   0.46 82.80       1.20   3.45 0.35   8.97           0.57 82.80       2.20   3.45 0.35   13.25           0.68 82.80       3.20   3.45 0.35   21.94           0.73 82.80       4.20   3.45 0.35   25.94         Glass/Polyester 0.18 73.00       0.25   3.80 0.42   4.18         Brintrup (1975)   0.19 73.00       0.25   3.80 0.42   4.99           0.19 73.00       0.25   3.80 0.42   4.54           0.19 73.00       0.25   3.80 0.42   3.73           0.20 73.00       0.25   3.80 0.42   4.00           0.20 73.00       0.25   3.80 0.42   5.36           0.20 73.00       0.25   3.80 0.42   5.18           0.21 73.00       0.25   3.80 0.42   4.72           0.21 73.00       0.25   3.80 0.42   5.09         257  Composite Type Vf Fibre Properties Resin Properties Composite Properties Ref. E1 (GPa) E2 (GPa) G12 (GPa) G23 (GPa) ν12 ν23 E (GPa) ν E1 (GPa) E2 (GPa) G12 (GPa) G23 (GPa) ν12 ν23   0.21 73.00       0.25   3.80 0.42 19.55 4.43           0.22 73.00       0.25   3.80 0.42 20.04 4.13           0.23 73.00       0.25   3.80 0.42 21.82 5.52           0.24 73.00       0.25   3.80 0.42 22.38 5.58           0.25 73.00       0.25   3.80 0.42 22.87 5.04           0.26 73.00       0.25   3.80 0.42 23.23 4.22           0.29 73.00       0.25   3.80 0.42 24.05 4.57           0.29 73.00       0.25   3.80 0.42 25.14 4.80           0.31 73.00       0.25   3.80 0.42 25.71 5.81           0.33 73.00       0.25   3.80 0.42 27.06 5.63           0.33 73.00       0.25   3.80 0.42   6.14           0.35 73.00       0.25   3.80 0.42   5.17           0.38 73.00       0.25   3.80 0.42 31.78 5.57           0.39 73.00       0.25   3.80 0.42   6.26           0.40 73.00       0.25   3.80 0.42   5.80           0.40 73.00       0.25   3.80 0.42   6.34           0.41 73.00       0.25   3.80 0.42   5.81           0.41 73.00       0.25   3.80 0.42 33.59 6.92           0.42 73.00       0.25   3.80 0.42 32.96 6.63           0.42 73.00       0.25   3.80 0.42   6.44           0.43 73.00       0.25   3.80 0.42   6.63           0.44 73.00       0.25   3.80 0.42   6.78           0.45 73.00       0.25   3.80 0.42   6.77          Glass/Polyester 0.47 73.00       0.25   3.80 0.42   7.08           0.48 73.00       0.25   3.80 0.42   7.19           0.48 73.00       0.25   3.80 0.42   7.05           0.48 73.00       0.25   3.80 0.42   6.88           0.48 73.00       0.25   3.80 0.42   7.83           0.49 73.00       0.25   3.80 0.42   7.12           0.49 73.00       0.25   3.80 0.42   7.37         258  Composite Type Vf Fibre Properties Resin Properties Composite Properties Ref. E1 (GPa) E2 (GPa) G12 (GPa) G23 (GPa) ν12 ν23 E (GPa) ν E1 (GPa) E2 (GPa) G12 (GPa) G23 (GPa) ν12 ν23   0.50 73.00       0.25   3.80 0.42   7.07           0.50 73.00       0.25   3.80 0.42 38.63 6.83           0.51 73.00       0.25   3.80 0.42 40.29 7.63           0.51 73.00       0.25   3.80 0.42 39.02 7.27           0.51 73.00       0.25   3.80 0.42 38.64 8.18           0.51 73.00       0.25   3.80 0.42   7.11           0.52 73.00       0.25   3.80 0.42   6.96           0.52 73.00       0.25   3.80 0.42   7.93           0.53 73.00       0.25   3.80 0.42   8.01           0.55 73.00       0.25   3.80 0.42 43.07 8.81           0.56 73.00       0.25   3.80 0.42   9.32           0.56 73.00       0.25   3.80 0.42 42.87 8.95           0.57 73.00       0.25   3.80 0.42   9.32           0.58 73.00       0.25   3.80 0.42   9.64          Glass/Polyester 0.58 73.00       0.25   3.80 0.42   9.04           0.59 73.00       0.25   3.80 0.42 46.62 9.71           0.60 73.00       0.25   3.80 0.42   9.51           0.63 73.00       0.25   3.80 0.42 47.09 10.04           0.64 73.00       0.25   3.80 0.42 48.18 10.32           0.65 73.00       0.25   3.80 0.42   10.39           0.65 73.00       0.25   3.80 0.42   10.02           0.66 73.00       0.25   3.80 0.42   11.23           0.67 73.00       0.25   3.80 0.42   11.44         Carbon/Epoxy 0.56 220   25   0.15   3.3 0.37 125.00 9.10 5.00   0.34   Harris (1999) Carbon/Epoxy 0.67 232 15 24 5.02 0.279 0.49 5.28 0.354 157 10.8 7.1 3.63 0.3 0.49 Kriz (1979) 259  Composite Type Vf Fibre Properties Resin Properties Composite Properties Ref. E1 (GPa) E2 (GPa) G12 (GPa) G23 (GPa) ν12 ν23 E (GPa) ν E1 (GPa) E2 (GPa) G12 (GPa) G23 (GPa) ν12 ν23 AS4/3501-6 0.6 235 14 28   0.2 0.34 4.8 0.34 139 9.85 5.25   0.3   Sun & Vaidya (1996) AS4/3501-6 0.66 201 13.5 95 4.9 0.22 0.38 4.3 0.35 134 9.14 7.29 3.16 0.261   Garnich & Karami (2004) Boron/ Aluminum 0.47 379.3       0.1   68.3 0.3 216 140 52   0.29   Sun & Vaidya (1996) Boron/Epoxy 0.20 414.00       0.20   4.14 0.35 80.61           Whitney &  Riley (1966)   0.55 414.00       0.20   4.14 0.35 207.39             0.60 414.00       0.20   4.14 0.35 245.97 21.36           0.65 414.00       0.20   4.14 1.35 244.60 23.43           0.70 414.00       0.20   4.14 2.35 237.71 26.77 12.20 12.20       0.75 414.00       0.20   4.14 3.35   33.81 16.74 16.74     SIC/CAS 0.39 163       0.19   98 0.2 121.00 112.00 52.00 47.46 0.18   Daniel et  al (1992) SIC/CAS 0.39 163       0.19   98 0.2     51.70 31.50     Tsai et al (1990)    260   Appendix D  Finite Element Formulation Based on (Zobeiry 2006)  The general FE equation in the material can be written as:  TVB dV F   (D.1) In which V  is the volume of the material, F  is the external load vector:  Bu   (C.2) Equation (D.1) can be written in the incremental form as follows:  T n nVB dV F   (D.3) where the increment of stress vector for time step n , n , is obtained from:  1n n n      (D.4) Using Equation (3.29), the above equation can be written as:   9 11 1Nn n npi pip i        (D.5) Substituting (Equation (3.28), in the above equation n  is derived as:  11 11 12 2[ ( )] ( )1 11 12 2n nn n n npi piip pf pip i p in npi pit tPt t               (D.6) Similar to Equation (3.20), The above equation can be written as:  ( )n u n n pifT p iD A       (D.7) 261  where,   1[ ( )]112n n u n nip pf pfTp inpiP Dt           (D.8)  111 12 2112n nnpi pipi pinpit tAt   (D.9) Therefore, Equation (D.3), can be written as:  ( )T u n n T npifTp iV VB D B u dV B A dV F          (D.10) By rearranging the above equation as:  T u n n T n TupifTTp iV V VB D BdV u F B D dV B A dV             (D.11) The governing FE equation can be written in terms of displacement increment u :  fTK u F F F     (D.12)  where TK is the global stiffness matrix expressed as:   T uT TVK B D B dV  (D.13) and  T uf fTVF B D dV    (D.14)  Tpip iVF B A dV       (D.15)  262  In order to calculate stress and strains, the displacement field is estimated from the FE equation, first. Knowing the displacements, strains are obtained and subsequently stresses are calculated from Equation (3.28).                     263  Appendix E  Integral Form (IF) of Viscoelasticity in ABAQUS®  The integral form of viscoelasticity for an isotropic material under pure shear loading can be written as:       0t dt G t dd      (E.1)  Expressing relaxation function  G t in terms of Prony series expansion (see Equation (3.1)), the above equation may be written as follows:        10102 ( )2 ( ) 1iitt Niitt Niir u ru u rdt G G G w e dddt G G G w e dd                       (E.2) The above equation may be simplified by introducing the viscous strain terms  i t as:              101 0( )2 1 12 ; 1iitt NiittNu ii iuiu ruudG Gt G w e dG ddGt G t t t e dG d                                   (E.3) The above equation should be written in an incremental form so that it can be employed in a finite element analysis. Thus:  .              1 1 112Nn n u n n n nii iuiGt t G t t t t                      (E.4) 264  Note that the stress value at the end of each increment of time  1nt   can be obtained from its value at the beginning of that time increment  nt  and the increment of stress expressed as:   12Nu iiuiGG G              (E.5) The second term on the right side of Equation (E.5) needs to be calculated before using this equation. The expression given in the ABAQUS® Theory Manual for i   is derived as follows:  The viscous strain at the end of the time increment can be expanded as:          1 111 101 1n nn ni int tt tn ni i td dt e d e dd d                               (E.6) The term inside the first parenthesis may rewritten by adding and subtracting a term, i te  , as:            111101001 1 1111n nn ni i i innini innint tt tt tnitttnittttttd de e e d e dd ddddde e ddde dd                                                             1  (E.7) The third integral can be simplified by assuming a linear variation of strain with time (i.e. constant strain rate) during an increment of time (similar to (Zocher et al. 1997) ) as: 265        1 11 11 1n nn ni in nt tt tt tde d e dd t                         (E.8) Therefore, Equation (E.7) can be written as:  1 1 1i i it t tn n ni i ie e t e t                                      (E.9) Now, the viscous strain increment can be obtained knowing the total strain increment,  , the total strain , n , and the viscous strain , ni , at the beginning of the time increment as follows:  1 1 1 1i i it t tn n n ni i i i ie e t e t                                           (E.10) 

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