DEVELOPMENT AND IMPLEMENTATION OF ROBUST LARGE DEFORMATION AND CONTACT MECHANICS CAPABILITIES IN PROCESS MODELLING OF COMPOSITES

by AMIR OSOOLY

B.Sc., Sharif University of Technology, 1989

M.Sc., Sharif University of Technology, 1993

A thesis submitted in partial fulfilment of

the requirements for the degree of

Doctor of Philosophy

in

The Faculty of Graduate Studies

(Civil Engineering)

The University of British Columbia

(Vancouver)

April 2008

© Amir Osooly, 2008

Abstract Autoclave processing of large scale, one-piece structural parts made of carbon fiber- reinforced polymer composite materials is the key to decreasing manufacturing costs while at the same time increasing quality. Nonetheless, even in manufacturing simple flat parts, residual strains and stresses are unavoidable. For structural design purposes and to aid in the assembly procedures, it is desirable to have proven numerical tools that can be used to predict these residual geometrical and material properties in advance, thus avoid the costly experimental trial and error methods.

A 2-D finite element-based code, COMPRO, has previously been developed in-house for predicting autoclave process-induced deformations and residual stresses in composite parts undergoing an entire cure cycle. To simulate the tool-part interaction, an important source of residual deformations/stresses, COMPRO used a non-zero thickness elastic shear layer as its only interface option. Moreover, the code did not account for the large deformations and strains and the resulting nonlinear effects that can arise during the early stages of the cure cycle when the material is rather compliant.

In the present work, a contact surface employing a penalty method formulation is introduced at the tool-part interface. Its material-dependent parameters are a function of temperature, degree of cure, pressure and so forth. This makes the stick-slip condition plus separation between the part and the tool possible. The large displacements/rotations and large shear strains that develop at the early stages of the cure cycle when the resin has a very low elastic modulus provided the impetus to include a large strain/deformation option in COMPRO. A new “co-rotational stress formulation” was developed and found to provide a robust method for numerical treatment of very large deformation/strain problems involving anisotropic materials of interest here.

Several verification and validation examples are used to calibrate the contact interface parameters and to demonstrate the correctness of implementation and the accuracy of the proposed method. A number of comparisons are made with exact solutions, other

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methods, other experiments and the same models in other commercial codes. Finally, several interesting cases are examined to explore the results of COMPRO predictions with the added options.

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Table of Contents Abstract...... ii Table of Contents...... iv List of Tables ...... vii List of Figures...... ix List of Symbols...... xviii Abbreviations...... xxiv Acknowledgements...... xxv

Chapter 1 - INTRODUCTION ...... 1 1.1 Autoclave Processing Overview...... 2 1.2 Process Modeling...... 4 1.2.1 Liquid Moulding...... 5 1.2.2 Sheet-Forming...... 8 1.2.3 Autoclave Processing...... 10 1.3 Residual Stresses in the Autoclave Process...... 14 1.3.1 Tool-Part Interaction...... 15 1.3.2 Sliding Approach...... 17 1.4 Motivation and Research Objectives ...... 20

Chapter 2 - LARGE DEFORMATIONS...... 25 2.1 Background...... 26 2.1.1 Stress and Strain Measures ...... 26 2.1.2 Conjugate Stress and Strain Definition...... 29 2.1.3 Polar Decomposition of the Deformation Gradient Tensor...... 32 2.1.4 Objectivity of the Stress Rate ...... 33 2.1.5 Examples of Objective Stress Rates ...... 35 2.1.5.1 Truesdell Stress Rate...... 35 2.1.5.2 Jaumann Stress Rate ...... 37 2.1.5.3 Green-Naghdi Stress Rate...... 38 2.1.5.4 Logarithmic Stress Rate...... 39 2.1.6 General Form of Co-rotational Stress Rate...... 39 2.1.7 Role of Material Stiffness Tensor in Objective Stress Rates...... 41 2.2 Co-rotational Stress Formulation for Isotropic Materials...... 42 2.2.1 Linear Hyperelastic Materials...... 47 2.2.2 Non-linear Hyperelastic Materials...... 48 2.3 Co-rotational Stress Formulation for Anisotropic Materials ...... 49 2.3.1 Rotation of the Material Stiffness Tensor...... 50 2.3.2 Hypoelastic Constitutive Equation...... 52 2.4 Summary and Discussion...... 53

Chapter 3 - NUMERICAL IMPLEMENTATION AND VERIFICATION OF LARGE DEFORMATION FORMULATION...... 57

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3.1 Logarithmic Strain Rate...... 57 3.2 Co-rotational Constitutive Equation ...... 60 3.3 Determination of Total True Stress...... 61 3.4 Determination of the Element Tangent Stiffness Matrix...... 62 3.5 Determination of the Internal Forces ...... 67 3.6 Determination of the External Loads...... 68 3.7 Iteration Scheme ...... 70 3.8 Verification ...... 71 3.8.1 Linear Elastic Isotropic Materials...... 72 3.8.1.1 Simple Shear...... 73 3.8.1.2 Uniaxial Loading...... 78 3.8.1.3 Closed Loop Stress Path ...... 79 3.8.2 Non-linear Elastic Isotropic Materials...... 79 3.8.3 Linear Elastic Transversely Isotropic Materials ...... 80 3.8.3.1 Simple Shear...... 81 3.8.4 Thermal Loading...... 81 3.8.4.1 Isotropic Element...... 82 3.8.4.2 Bi-material Bar...... 82 3.8.5 Response of a Cantilever Beam Undergoing Large Rotations ...... 83 3.8.6 Applicable Range of Strains in Finite Element Codes...... 84 3.9 Summary and Discussion...... 86

Chapter 4 - CONTACT FORMULATION...... 111 4.1 Background...... 111 4.2 Development of Constitutive Equation...... 114 4.2.1 Normal Spring...... 116 4.2.2 Tangential Spring...... 118 4.2.3 Interfacial shear strength...... 122

4.2.3.1 Fluid Friction Phase (c

4.2.3.2 Bonding Phase ( c>c1 ) ...... 123 4.2.3.3 Friction at Interface...... 124 4.2.4 Master and Slave Surfaces...... 126 4.2.5 Higher Order Element...... 128 4.2.6 Stiffness Matrix...... 129 4.3 Verification ...... 132 4.4 Summary and Discussion...... 134

Chapter 5 - VALIDATION AND NUMERICAL CASE STUDIES...... 148 5.1 Simulation of Instrumented Tool Experiments...... 149 5.1.1 Experimental Results...... 151 5.1.2 COMPRO with Perfect Bonding Interface ...... 153 5.1.3 COMPRO with Shear Layer Interface...... 154 5.1.4 COMPRO with Contact Interface...... 156 5.2 Sensitivity Analyses on Instrumented Tool Experiments...... 159

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5.2.1 Low Initial Resin Modulus (4.71 kPa)...... 159 5.2.2 High Initial Resin Modulus (4.71 MPa) ...... 162 5.3 Prediction of Process Induced Distortions...... 163 5.3.1 Flat Parts...... 164 5.3.1.1 Models with Contact Interface...... 164 5.3.1.2 Models with Shear Layer Interface...... 166 5.3.2 Angle Parts...... 167 5.3.2.1 L-shaped Parts...... 167 5.3.2.2 C-shaped Parts...... 168 5.4 Numerical Case Studies...... 169 5.4.1 Semi Circular Parts ...... 170 5.4.2 Large Scale Flat Parts ...... 171 5.5 Summary and Discussion...... 172

Chapter 6 - SUMMARY, CONCLUSIONS AND FUTURE WORK ...... 219 6.1 Summary...... 219 6.2 Conclusions...... 221 6.3 Contributions...... 222 6.4 Recommendations and Future Work ...... 223

Bibliography ...... 225

Appendix A - Modified Hypoelastic Constitutive Equation...... 232

Appendix B - Bˆ Matrix ...... 234

Appendix C - Stiffness Reduction Factor Algorithm of Tangential Spring Used in Contact Interface...... 237

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List of Tables Table 3-1: Logarithm function of a symmetric matrix calculated using the decomposition (Jirasek and Bazant [2001]) method and the Taylor series expansion method in the case of uniaxial loading of an element with Poisson’s ratio equal to zero...... 88 Table 3-2: Different isotropic material properties used for verification examples..... 88 Table 3-3: Orthotropic material properties used for verification example...... 88 Table 3-4: Material properties of bars...... 88 Table 5-1: Computational runs for the instrumented tool set-up...... 176 Table 5-2: Files used in each project file of COMPRO runs (cases) for instrumented tool experiments...... 177 Table 5-3: Mesh properties used for each PATRAN files...... 177 Table 5-4: Material names used in each material files for instrumented tool runs... 178 Table 5-5: Properties of composite materials used in COMPRO runs...... 178 Table 5-6: Properties of other materials used in COMPRO runs...... 178 Table 5-7: Sensitivity analysis for case IT-14; varied parameters are the fluid friction

shear strength when the degree of cure is zero or c1 , the bonding shear strength when the material is 100% cured and the coefficient of friction...... 179 Table 5-8: Sensitivity analysis for case IT-12; varied parameters are the fluid friction

shear strength when the degree of cure is zero or c1 , the bonding shear strength when the material is 100% cured and the coefficient of friction...... 179 Table 5-9: Files used in each project files of COMPRO runs for warpage simulation of flat parts...... 180 Table 5-10: COMPRO sliding prediction for warpage when initial resin modulus is 1.0E5 (Pa) and experimental raw and adjusted results...... 181 Table 5-11: Material designation used in each material files for flat part runs...... 182 Table 5-12: Lay-up designation used in each Lay-up files for flat part runs...... 182 Table 5-13: COMPRO prediction for warpage when initial resin modulus is 1.0E5 (Pa). A) Sliding option; B) Shear layer with large deformation option; C) Shear layer without large deformation option; D) Closed-form solution (see Arafath [2007])...... 182

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Table 5-14: COMPRO prediction for warpage when initial resin modulus is as high as 1.0E7 (Pa). A) Shear layer with large deformation option; B) Shear layer without large deformation option...... 183 Table 5-15: Project file names for semi circle male and female tools and its same size flat part, and L-* and C-shaped parts...... 183 Table 5-16: Material file names...... 184 Table 5-17: Tool material properties...... 184 Table 5-18: Composite material properties...... 184 Table 5-19: Description of one- and Two-hold cure cycles(CYC9 and CYC8, respectively) used for L- and C- shape and long flat parts of 10.0 m length...... 185 Table 5-20: Part geometry, Tool’s material and number of holds in the cure circle for L- and C-shaped composite parts...... 185 Table 5-21: Lay-up designation used in each Lay-up files for Long scale flat part runs ...... 186 Table 5-22: Warpage results for L-shaped composite parts ...... 186 Table 5-23: Warpage results for C-shaped composite parts ...... 186 Table 5-24: Warpage results for 1200 mm flat part, 8-ply thickness...... 187 Table 5-25: Warpage results for 1200 mm semi-circle parts on convex and concave tools of 12.0 mm thickness. The line with bold values represents the results of the benchmark interfacial shear strength of 30 kPa...... 187

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List of Figures Figure1-1: A schematic showing the various elements in a vacuum-bagged composite part...... 23 Figure 1-2: Forming mechanisms for unidirectional composites (McEntee and O’Bradaigh [1998])...... 23 Figure 1-3: An example of COMPRO interface showing a composite part (representative of the spar of an aircraft) modelled with a shear layer. The shear layer and composite part elements (particularly at corners) undergo large distortions and strains...... 24 Figure 2-1: Non-linear structural response or force-displacement behaviour (left) nd while having nonlinear Nominal and 2 Piola-Kirchhoff stress (σ Ν and S ) versus Normal and Green strain relationships, or linear True (or Kirchhoff) stress (σ and τ ) versus Logarithmic strain relationship (right)...... 55 Figure 2-2: (a) A non-linear load-displacement curve obtained from experiments. (b) A linear Kirchhoff stress (τ ) versus Logarithmic strain assumption, which results in a non-linear True stress versus Logarithmic strain relationship. (c) A linear True stress (σ ) versus Logarithmic strain assumption, which results in a non-linear Kirchhoff stress versus Logarithmic strain relationship...... 55 Figure 2-3: Global shear stress for isotropic material in simple shear example based on equations: A) τˆ& = CD: ˆ and B) τ&ˆ = CU:ln(& ). µ is the Lame elastic constant...... 56 Figure 2-4: Fibres in off-axis composite materials rotate with respect to the global coordinate system in pure stretching (a) resulting in rotation of material coordinate system. Simple shear of an element (b) showing rigid body rotation (θ) of the element and rotation (-θ) of material coordinate system cancel each other resulting in zero net rotation for fibers...... 56 Figure 3-1: A unit square under rail (simple) shear...... 89 Figure 3-2: Normalized True normal and shear stresses for simple shear example involving an isotropic material. Comparison of: A) One-step solution, based on 22 terms of Taylor series of a logarithm of a matrix, B) COMPRO, C) Logarithmic stress rate based on papers by Xiao et al. [1997], Lin [2002, 2003] and Liu and Hong [1999], D) Eigenvalue-based (exact analytical) solution based on paper by Lin [2002, 2003]. µ is the Lame elastic constant...... 89 Figure 3-3: Simple shear example in ABAQUS for an isotropic material using a 2D plane strain element and the Jaumann stress rate. (a) All normalized

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normal and shear stresses vary in a sinusoidal manner with respect to normalized deformation, γ, (even principal stresses follow a sinusoidal shape). (b) All normalized x- and z-direction forces at nodes 3 and 4 show oscillation. µ is the Lame elastic constant...... 90 Figure 3-4: Simple shear example in LS-DYNA for an isotropic material using a constrained 2D element and the isotropic elastic material model (material type 1- MAT1) with Jaumann stress rate. (a) All normalized normal and shear stresses behave sinusoidally. (b) Normalized x- and z- direction forces at node 4 are shown to oscillate. µ is the Lame elastic constant...... 91 Figure 3-5: Simple shear example in LS-DYNA for an isotropic material using a non- isotropic elastic material model (material type 2- MAT2) and the Truesdell stress rate.(a) All normalized normal and shear stresses exhibit a hardening behaviour. (b) All normalized x- and z-direction forces at node 4 exhibit a hardening behaviour as well. µ is the Lame elastic constant...... 92 Figure 3-6: Simple shear in isotropic materials. Principal stresses increase monotonically. µ is the Lame elastic constant...... 93 Figure 3-7: A unit square under uniaxial loading...... 93 Figure 3-8: Normalized True and Kirchhoff stresses versus normalized displacement for an isotropic material with Poisson’s ratio of zero undergoing uniaxial loading. Comparison of exact solution (based on logarithm of a scalar value), one-step solution (hyperelastic solution Equation (2.96)) and COMPRO...... 94 Figure 3-9: Stress versus logarithmic strain curves for an uniaxial loading case showing both the Kirchhoff and True stresses as a function of the logarithmic strain when the Kirchhoff material stiffness, C Kirchhoff , is constant...... 94 Figure 3-10: Uniaxial loading of an isotropic material in LS-DYNA using an isotropic material model. µ is the Lame elastic constant...... 95 Figure 3-11: Uniaxial loading of an isotropic material in LS-DYNA using a non- isotropic material model. µ is the Lame elastic constant...... 95 Figure 3-12: Comparisons among COMPRO, ABAQUS and LS-DYNA for an isotropic material in a uniaxial loading case...... 96 Figure 3-13: A unit square under a closed deformation path, 1 to 4...... 96 Figure 3-14: (a) Kirchhoff and (b) True stress components due to a closed deformation path applied to a unit square. Figures compare the predictions of the Logarithmic stress rate method based on the analytical solution of Lin et

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al. [2003] and the current Co-rotational stress formulation implemented in COMPRO...... 97 Figure 3-15: Normalized True stress versus Logarithmic strain curve: Elastic modulus increases from its initial value to twice that value at 100% Nominal strain or about 69.3% Logarithmic strain...... 98 Figure 3-16: Normalized force versus displacement for a uniaxial loading case where the material follows the stress-strain curve above in Figure 3-15. This figure compares the predictions of ABAQUS, LS-DYNA and COMPRO and exact analytical solutions. ABAQUS results do not converge beyond the small strain range, while COMPRO and LS-DYNA predictions coincide...... 98 Figure 3-17: Simple shear of an orthotropic material. (a) Equivalent model in ABAQUS consists of an isotropic solid element and 2 vertical bars. (b) COMPRO model using lumped orthotropic material properties...... 99 Figure 3-18: Simple shear in orthotropic materials. Principal stresses increase monotonically while global stresses vary non- monotonically versus strain. µ is the Lame elastic constant...... 99 Figure 3-19: Simple shear of an orthotropic material. Force components predicted by COMPRO and the equivalent model in ABAQUS...... 100 Figure 3-20: Simple shear of an orthotropic material. x- and z-direction forces at node 3 and 4 in COMPRO run compared to x- and z-direction forces in node 3 (or 4) in ABAQUS run...... 101 Figure 3-21: The isotropic unit element with zero Poisson’s ratio used for verification of COMPRO large strain capability under thermal loading...... 101 Figure 3-22: Comparison of COMPRO predictions (circle and square legends) with analytical solution, Equation 3.88 (solid and dash lines) for thermal loading of the element showing in Figure 3-21...... 102 Figure 3-23: A bimaterial bar: Poisson’s ratio = 0, Elastic modulus = 1.0E+10 Pa for both materials while CTE = 0 for material 2 (top) and CTE = 0.01 for material 1 (bottom)...... 102 Figure 3-24: Comparison of COMPRO and ABAQUS predicted average normalized stresses and tip deformation versus temperature change for the bimaterial bar shown in Figure 3-23...... 103 Figure 3-25: A cantilever beam under pressure loading over half of its length with 400 (=100 x 4) elements...... 104 Figure 3-26: Pressure loading on the cantilever beam shown in Figure 3-25. Pressure increases linearly versus time...... 104

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Figure 3-27: Deformation of the bottom and top surfaces of the cantilever beam shown in Figure 3-25 and subjected to the pressure loading of Figure 3-26 as predicted by COMPRO at different times...... 105 Figure 3-28: Comparison of the ABAQUS and COMPRO predicted deformation profile of the bottom surface of the cantilever beam of Figure 3-25 subjected to the pressure loading shown in Figure 3-26. The results obtained using both 4- and 8-noded elements and compared...... 106 Figure 3-29: Centre line tip displacement (both in x- and z-directions) of the cantilever beam of Figure 3-25 subjected to pressure loading shown in Figure 3-26 using COMPRO with and without Large deformation option...... 107 Figure 3-30: True stresses at first integration points of the bottom row elements in the loaded portion of the cantilever beam of Figure 3-25 at time equal to 300 minutes. Comparisons are made between the predictions of COMPRO (4-and 8-noded elements) and ABAQUS (4- and 8-noded elements). Also shown are the predictions of COMPRO 4-noded elements without large deformation option...... 108 Figure 3-31: Heavily constrained unit length 1-, 4- and 16-elements cases...... 109 Figure 3-32: Normalized stresses, in 3rd integration points of right bottom corner element...... 110 Figure 4-1: Coulomb’s friction law and two regularized friction laws (Ling and Stolarski [1997])...... 137 Figure 4-2: Minimum energy and minimum feasible energy when there is an impenetrable surface...... 137 Figure 4-3: Penalty method: Replacing the impenetrable surface by a spring...... 138 Figure 4-4: Composite elements on top of tool elements at the interface...... 138 Figure 4-5: Normal spring element, contact laws...... 138 Figure 4-6: First ply on top of a tool. Resin thickness between the tool and the first fibre is about 20 µm...... 139 Figure 4-7: Tangential spring element contact laws...... 139 Figure 4-8: Schematic example of one tangential spring element. Tangential spring stiffness calculated at each time step base on the following

equationKTi= K T / SRF i ...... 140 Figure 4-9: Fluid friction shear strength at the interface as a function of degree of cure...... 140 Figure 4-10: Bonding shear strength at the interface when (a) there is no debonding, and (b) when there is one debonding at degree of cure of 0.78...... 141 Figure 4-11: Interfacial shear strength due to fluid friction and bonding effects...... 141

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Figure 4-12: Coefficient of friction between the resin and an aluminium tool...... 142 Figure 4-13: General behaviour of a contact element at the interface during a cure cycle while applying pressure and temperature...... 142 Figure 4-14: Coefficient of friction between FEP sheets...... 143 Figure 4-15: Slave node 2 passes one master segment (i.e., moves from A to B), while nodes 1 and 3 remain at the same master segments, A and B...... 143 Figure 4-16: Changing corners and curved sections to a smooth surface at boundaries...... 144

Figure 4-17: Virtual location of a tangential spring is between 41 and 4i at step i , and

between 41 and 4i+1 at step i +1...... 144 Figure 4-18: 2- and 3-noded contact elements for 4- and 8-noded master elements... 145 Figure 4-19: Forces in contact springs due to unit displacement at node 1 in the X- direction...... 145 Figure 4-20: A cantilever beam example used to verify the contact surface implementation in COMPRO...... 145 Figure 4-21: Normal and tangential spring properties for the cantilever example of Figure 4-20 with a contact element at the interface of the top and bottom beams (no sliding resistance at the interface)...... 146 Figure 4-22: Normal and tangential spring properties for the cantilever example of Figure 4-20 with a contact element at the interface of the top and bottom beams (perfect bonding at the interface)...... 146 Figure 4-23: Deformation of the top surface of a bi-metallic beam under uniform pressure loading for different contact conditions between the two beams. The solid lines are COMPRO’s results, while the thin lines with symbols are ABAQUS’s results...... 147 Figure 5-1: (a) Schematic of the instrumented tool set-up used by Twigg [2001]. (b) Location of strain gauges ...... 188 Figure 5-2: Mechanical strains in gauges A, A’, B, B’, C and C’ for autoclave pressure of 586 kPa. (a) Release agent at the interface (b) FEP at the interface. Experimental results are from Twigg [2001]...... 189 Figure 5-3: Mechanical strains in gauges A, A’, B, B’, C and C’ for autoclave pressure of 103 kPa. (a) Release agent at the interface (b) FEP at the interface. Experimental results are from Twigg [2001]...... 190 Figure 5-4: IT-IP1 is a fine mesh with no shear layer...... 191 Figure 5-5: IT-IP2 is a fine mesh with shear layer...... 191 Figure 5-6: IT-IP3 is a coarse mesh with no shear layer...... 191

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Figure 5-7: Cases IT-1 and IT-2: Mechanical strain versus time for low autoclave pressure (103 kPa). COMPRO results at strain gauges A, B and C with perfect bonding at the interface and a high initial resin modulus (47.1 MPa) are shown for both cases of with and without large deformation options. Experimental mechanical strain histories are also shown at locations A, B, B’, C and C’ for the case of release agent at the interface...... 192 Figure 5-8: Case IT-3: Mechanical strain versus time for low autoclave pressure (103 kPa). COMPRO results at locations A, B and C with perfect bonding at the interface and a low initial resin modulus (0.471 MPa) are shown for the case of no large deformation option. Experimental mechanical strain histories are also shown at locations A, B, B’, C and C’ for the case of release agent at the interface...... 193 Figure 5-9: Case IT-5: Mechanical strain versus time for low autoclave pressure (103 kPa). COMPRO results at locations A, B and C with shear layer at the interface are shown here. Experimental mechanical strain histories are also shown at locations A, B, B’, C and C’ for the case of release agent at the interface...... 194 Figure 5-10: Case IT-7: Mechanical strain versus time for high autoclave pressure (586 kPa). COMPRO results at locations A, B and C with shear layer at the interface are shown here. Experimental mechanical strain histories are also shown at locations A, A’, B, B’, C and C’ for the case of release agent at the interface...... 194 Figure 5-11: Case IT-9: Mechanical strain versus time for low autoclave pressure (103 kPa) and FEP layers at the interface. Experimental mechanical strain histories at locations A, A’, B, B’, C and C’; and COMPRO results at locations A, B and C (correspond to the case where sliding option is employed at the interface) are shown here...... 195 Figure 5-12: Case IT-11: Mechanical strain versus time for high autoclave pressure (586 kPa) and FEP layers at interface. Experimental mechanical strain histories at locations A, A’, B, B’, C and C’; and COMPRO results at locations A, B and C (correspond to the case where sliding option is employed at the interface) are shown here...... 196 Figure 5-13: Case IT-10: Mechanical strain versus time for low autoclave pressure (103 kPa) and release agent at the interface. Experimental mechanical strain histories at locations A, A’, B, B’, C and C’; and COMPRO results at locations A, B and C (correspond to the case where sliding option is employed at the interface) are shown here...... 197 Figure 5-14: Case IT-12: Mechanical strain versus time for high autoclave pressure (586 kPa) and release agent at the interface. Experimental mechanical strain histories at locations A, A’, B, B’, C and C’; and COMPRO results

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at locations A, B and C (correspond to the case where sliding option is employed at the interface) are shown here...... 198 Figure 5-15: Cases IT-13 and 14: COMPRO runs for different initial resin moduli. Mechanical strains at locations C, B and A are shown when autoclave pressure is high (586 kPa) and there is release agent at the interface. . 199 Figure 5-16: Debonding starts at the tip of the flat part and grows linearly. It reaches the centre line in approximately 50 minutes...... 200 Figure 5-17: IT-12, 16, 17 & 18 compares mechanical strain results of different COMPRO runs including all combination of 4- and 8-noded elements with coarse and fine meshes. Conditions are high autoclave pressure (586 kPa) and release agent at the interface...... 201 Figure 5-18: IT-14-S-1 and 2 are cases of sensitivity analysis when degree of cure at which the fluid friction mechanism stops is varied. Conditions are high autoclave pressure (586 kPa), release agent at the interface and low

initial resin modulus (4.71 kPa). IT-14: c1 = 0.65, IT-14-S-1: c1 = 0.50

and IT-14-S-2: c1 = 0.75 ...... 202 Figure 5-19: IT-14-S-3 and 4 are cases of sensitivity analysis when the maximum fluid friction shear strength is varied. Conditions are high autoclave pressure (586 kPa), release agent at the interface and low initial resin modulus m m (4.71 kPa). IT-14: τ ff = 30.0 kPa, IT-14-S-3: τ ff =15.0 kPa and IT-14- m S-4: τ ff = 50.0 kPa...... 203 Figure 5-20: IT-14-S-5 and 6 are cases of sensitivity analysis when the maximum bonding shear strength (at fully cured) is varied. Conditions are high pressure (586 kPa), release agent at the interface and low initial resin

modulus (4.71 kPa). IT-14: τ b = 5.0 MPa, IT-14-S-5: τ b = 3.0 MPa and

IT-14-S-6: τ b = 7.0 MPa...... 204 Figure 5-21: IT-14-S-7 and 8 are cases of sensitivity analysis when the coefficient of friction is varied. Conditions are high pressure (586 kPa), release agent at the interface and low initial resin modulus (4.71 kPa). IT-14: µ = 0.28, IT-14-S-7: µ = 0.20 and IT-14-S-8: µ = 0.40 ...... 205 Figure 5-22: IT-14-S-9 and 10 are cases of sensitivity analysis when the fluid friction shear strength at zero degree of cure is varied. Conditions are high pressure (586 kPa), release agent at the interface and low initial resin 0 0 modulus (4.71 kPa). IT-14: τ ff = 0.0 kPa, IT-14-S-9: τ ff =10.0 kPa 0 and IT-14-S-10: τ ff = 20.0 kPa...... 206 Figure 5-23: IT-12-S-1 and 2 are cases of sensitivity analysis when the degree of cure at which the fluid friction mechanism stops is varied. Conditions are high

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pressure (586 kPa), release agent at the interface and high initial resin

modulus (4.71 MPa). IT-12: c1 = 0.65 , IT-12-S-1: c1 = 0.75 and IT-12-

S-2: c1 = 0.50 ...... 207 Figure 5-24: IT-12-S-3 and 4 are cases of sensitivity analysis when the maximum fluid friction shear strength is varied. Conditions are high pressure (586 kPa), release agent at the interface and high initial resin modulus (4.71 MPa). m m IT-12: τ ff = 30.0 kPa, IT-12-S-3: τ ff =15.0 kPa and IT-12-S-4: m τ ff = 50.0 kPa...... 208 Figure 5-25: IT-12-S-5 and 6 are cases of sensitivity analysis when the fluid friction shear strength at zero degree of cure is varied. Conditions are high pressure (586 kPa), release agent at the interface and high initial resin 0 0 modulus (4.71 MPa). IT-12: τ ff = 0.0 kPa, IT-12-S-5: τ ff =10.0 kPa 0 and IT-12-S-6: τ ff = 20.0 kPa...... 209 Figure 5-26: Flat part model: Composite part on top of an aluminium tool. Tool has been modelled with an extra 5 mm length in order to support any expansion of the composite part...... 210 Figure 5-27: Warpage prediction of COMPRO are compared with adjusted experimental measurements of warpage for a flat composite part on an aluminium tool with 3 different lengths (300, 600 and 1200 mm) and 3 different thicknesses (4-, 8- and 16-ply). (a) High autoclave pressure of 586 kPa and release agent at interface. (b) Low autoclave pressure of 103 kPa and release agent at interface...... 211 Figure 5-28: Warpage prediction of COMPRO are compared with adjusted experimental measurements of warpage for a flat composite part on an aluminium tool with 3 different lengths (300, 600 and 1200 mm) and 3 different thicknesses (4-, 8- and 16-ply). (a) High autoclave pressure of 586 kPa and FEP sheets at interface. (b) Low autoclave pressure of 103 kPa and FEP sheets at interface...... 212 Figure 5-29: Flat part model: A shear layer introduced between the composite part and the aluminium tool...... 213 Figure 5-30: Total spring-in in an L-shaped part consists of spring-in due to corner

effect and spring-in due to flange warpage, θtcw= θθ+ ...... 213 Figure 5-31: An L-shaped composite part on top of an aluminum tool modeled in COMPRO. The shear strain is determined at the first integration point of the first element (number 93) in the straight segment of the composite part right after the curved segment ends. The element has 2 layers and each layer has 4 integration points...... 214

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Figure 5-32: Autoclave temperature and degree of cure versus time for element number 93 shown in Figure 5-31. The shear strain at the first integration point of this element reaches minimum and maximum values of -11 and 96%, respectively. The shear strain then drops to a small value as shear modulus of the composite part starts increasing...... 215 Figure 5-33: Total spring-in in a C-shaped part consists of different spring-ins due to corner effect, flange warpage and web warpage,

θt= θθ c++ w−− flange θ w web ...... 216 Figure 5-34: A flat composite part on an aluminium tool (as modelled in COMPRO) cured in an autoclave processing...... 216 Figure 5-35: Deformed shape of a half-circle 8-ply thickness composite part on 12.0 mm thick convex and concave tools after tool removal. Original outer and inner surface of convex and concave tools, respectively are shown as well...... 217 Figure 5-36: Deformed shape of virtual cases of perfect bonding at the interface for different lengths of 300, 600 and 1200 mm 4-ply composite parts on an aluminium tool after tool removal. Project files are FP-1, FP-4 and FP-7...... 218 Figure 5-37: Warpage of 10 m length composite flat part (half-length shown here): A) 32-ply thick; B) 64-ply thick...... 218 Figure B-1: Simple shear example in COMPRO. Determining strains based on: A) logarithmic function; B) Bˆ matrix with regular ∆=um0.01 and ˆ smaller time step of ∆=um 0.0001 ; C) BModified matrix...... 236

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List of Symbols A Current cross sectional area

A0 Original cross sectional area A Fibre direction vector at original configuration A′ Fibre direction vector at current configuration A′′ Fibre direction vector at material configuration after stretching B Strain-displacement matrix in global coordinate system Bˆ Strain-displacement matrix in material coordinate system C Right Cauchy-green deformation matrix; Material stiffness tensor C ◊ Truesdell material stiffness tensor or matrix C J Jaumann material stiffness tensor or matrix C GN Green-Naghdi material stiffness tensor or matrix

CD Material stiffness tensor D matrix

CG Green material stiffness tensor or matrix

CL Logarithmic material stiffness tensor or matrix

CC Rotated Logarithmic material stiffness tensor or matrix c Degree of cure; Cosines of an angle; Adhesion between two surfaces c1 Degree of cure at the end of fluid friction phenomenon

CG Green material stiffness value

CL Logarithmic material stiffness value

CN Nominal material stiffness value D Rate of deformation matrix in global coordinate system Dˆ Rate of deformation matrix in material coordinate system D Rate of deformation in vector form in global coordinate system E Material elastic modulus

-xviii-

E0 Material initial elastic modulus

Er Resin modulus E Green-Lagrangian strain matrix or vector f Normal force at contact interface F Deformation Gradient measure

FN Force in normal spring

FT Force in tangential spring F Deformation Gradient matrix g Modification number; Gravitational constant G Shear modulus G Modification matrix h Thickness or distance H Thickness

H1 Thickness Η A matrix i Counter; iteration number I First invariant II Second invariant I Identity matrix J Determinant of deformation gradient matrix K Structural stiffnesses

KN Stiffness of normal spring

KT Stiffness of tangential spring K Structural stiffness matrix K S Structural stiffness matrix of contact element l Length L Current length

-xix-

L0 Original length l Velocity gradient matrix m Mass n Counter N Shape function N Shape function vector p Pressure p0 Tolerance value, Initial pressure equal to 586 kPa p1 , p2 Tolerance values Q An intermediate matrix R Rigid body rotation matrix

RC Rotation matrix for material stiffness matrix due to stretching

RN Rotation matrix for strain vector

RS Rotation matrix for stress vector r& Nodes velocity vector s Sinus of an angle S 2nd Piola-Kirchhoff stress measure; Displacement S 2nd Piola-Kirchhoff stress matrix or vector SRF Stiffness reduction factor t Time; Thickness T Temperature Tfac The ratio of existing shear stress to the interfacial shear strength u Deformation value u Deformation vector U Right stretching matrix V Left stretching matrix w& Specific power (work rate per unit original volume) W Anti-symmetric spin tensor

-xx-

x,y,z Current coordinate system

X, Y, Z Original coordinate system

α Coefficient of thermal expansion vector δ Displacement; Variation

δ A Allowable gap within which tensile stress is permitted

δC Distance of outer limit of 1-ply to the first fiber ε Logarithmic strain measure

εG Green strain measure

ε N Nominal strain measure ε Logarithmic strain matrix in global coordinate system εˆ Logarithmic strain matrix in material coordinate system εˆ& Logarithmic strain rate matrix in material coordinate system εˆ& Logarithmic strain rate vector in material coordinate system εˆ Logarithmic strain vector in material coordinate system

εˆTC Thermal plus cure shrinkage strain tensor in material coordinate system

εˆTH Thermal strain vector in material coordinate system

εˆCS Cure shrinkage strain vector in material coordinate system

ε&ˆσ Mechanical stress rate tensor in material coordinate system

Φ& Virtual power γ Shear strain; Displacement κ Spring stiffnesses λ Eigenvalue µ Lame elastic constant; Coefficient of friction ν Poisson’s ratio

Πmin Minimum potential energy

-xxi-

c Πmin Minimum feasible potential energy θ Angle between fibers in their original and rotated configurations measured in material coordinate system; Angle θ ' Angle between fibers in their original and rotated configurations measured in global coordinate system

ρ0 Original material density ρ Current material density σ Cauchy (True) stress measure; Stress

σ N Nominal stress measure σ Cauchy (True) stress matrix or vector in global coordinate system σˆ Cauchy (True) stress matrix or vector in material coordinate system σ&ˆ Cauchy (True) stress rate matrix or vector in material coordinate system

O σ Arbitrary objective stress rate in global coordinate system

O σLog Logarithmic objective stress rate in global coordinate system

σ ◊ Truesdell stress matrix σ J Jaumann stress matrix σGN Green-Naghdi stress matrix ζ Natural coordinate system τ Shear stress

τ A Interfacial shear strength

τb Interfacial shear strength due to chemical bonding of the resin to the tool

τ f Interfacial shear strength due to friction

0 m τ ff,τ ff Initial and maximum interfacial fluid friction shear strengths

-xxii-

τ Kirchhoff stress matrix or vector in global coordinate system τˆ Kirchhoff stress matrix in material coordinate system τ&ˆ Rate of Kirchhoff stress matrix in material coordinate system ϒ Residual force vector Ω Angular velocity skew-symmetric tensor ΩLog Logarithmic angular velocity skew-symmetric tensor Ψ Force value Ψ Force vector

Ψint Internal load vector

Ψext External load vector

Ψ Pseudo− ext Pseudo external load vector : Double contraction operator in tensor multiplications

-xxiii-

Abbreviations 2PK Second Piola-Kirchhoff CFRP Carbon fibre reinforced polymer CFRTP Continuous fibre-reinforced thermoplastic sheet press forming CFS Co-rotational stress formulation CHILE Cure-hardening, instantaneously-linear elastic CTE Coefficient of thermal expansion FEP Fluorinated ethylene-propylene LCM Liquid composite moulding LD Large deformation LPT Laminated plate theory RA Release agent RFI Resin film infusion RTM Resin transfer moulding SCRIMP Seemann composites resin infusion molding process SMC Sheet-moulding compound SRF Stiffness reduction factor SRIM Structural reaction injection moulding VBM Vacuum bag moulding

-xxiv-

Acknowledgements I would like to thank my supervisor Dr. Reza Vaziri, for his exceptional guidance, help and knowledge throughout the course of my research work. I would also like to thank my co-supervisors Dr. Anoush Poursartip and Dr. Goran Fernlund for many helpful discussion and feedbacks. I sincerely thank all members of the Composites Group, whom I value for both their friendship and assistance.

I am grateful to Mr. Robert Courdji for his friendship and help. I also thank Mr. Mehdi Haghshenas, Mr. Graham Twigg, Dr. Nima Zobeiry, Dr. Ahamed Arafath and especially Dr. Ali Shahkarami for their valuable time, help, recommendations and friendship.

Many thanks to my parents, sister and brother, who encouraged me throughout this study. Most of all, I thank my lovely wife, Ramesh, who sacrificed many things to support me over these years.

-xxv- Chapter 1 – Introduction

CHAPTER 1 ‐ INTRODUCTION

New materials with better qualities and performance compared to traditional metal alloys are now more readily available in the market at higher raw material costs. Lightweight, high stiffness and high strength fibre-reinforced polymer composites are becoming the dominant non-structural and structural materials used in the aerospace industry. Recently, due to the widespread use of these materials in aircraft, sporting goods, wind turbine blades and military applications, raw material costs are decreasing. With lower manufacturing costs, these materials can be competitive in relation to traditional materials.

The manufacture of composite materials into large, complex and one-piece parts with fewer fastening requirements and machining needs is the key to low manufacturing costs. Although new manufacturing techniques and better understanding of the materials behaviour have increased the quality of the parts, it is still difficult to achieve dimensional fidelity. The main problem with composite manufacturing is that the final shape of the parts differs from their as-designed geometry, and consequently can be out of the tolerance range imposed by the tight standards within such industries as aerospace. Therefore, going through traditional shimming procedures in part assemblies, and even the worse case scenario of remanufacturing composite parts, are the major sources of increased composite manufacturing costs. As a result, any

-1- Chapter 1 – Introduction

inexpensive method capable of reducing or predicting the undesired deformation can save time and money.

Recently, much effort has gone into the development of processing models to simulate the manufacturing process. These computer-based models are available for sheet moulding, resin transfer moulding, autoclave processing, and so forth. Use of these numerical models provides designers with a good prediction of the final shape of a part, as well as insight into ways in which the entire process can be optimized.

1.1 AUTOCLAVE PROCESSING OVERVIEW

A large number of techniques, such as electron beam curing and resin transfer moulding, is available to process and build a composite part. The most common processing method used to manufacture large parts is autoclave processing. The products of this method are used in high-performance structural applications such as those employed in the aerospace industry.

The autoclave processing technique is a simple but precise method of manufacturing advanced composite materials. Prepregs, which consist of thin sheets of (usually high- modulus) fibres impregnated with partially-cured resin, are available in roll form. At this stage, the resin is in a semi-solid state at room temperature. To prevent more curing before processing time, the prepreg should be stored at a relatively low temperature. Prepregs are cut in various orientations and stacked on top of a tool to form the desired thickness, lay-up and shape. Sometimes different materials and parts, such as paper or metal honeycomb cores, pre-cured layers, adhesives and stiffeners, are included in the final part.

Using a chemical release agent on the tool surface, or placing at least one thin polymer film between the tool and the part, prevents the formation of a strong bond between the part and the tool (particularly at the end of processing). Nonetheless, such procedures

-2- Chapter 1 – Introduction cannot totally prevent tool-part interaction during curing. The part is covered with breather to let air and generated gases out and might be covered with bleeder to let extra resin flow out. The part and layers of cloth are sealed inside a vacuum bag. A tool can be part of a vacuum bag, as shown in Figure 1-1, or it can be placed inside a vacuum bag along with the composite part and layers of cloth. Vacuum is applied to the sealed bag, causing compaction of the laminate stack. The entire assembly is then placed in an autoclave, and subjected to a temperature and pressure cycle termed the cure cycle. Resin is flammable, and, although it is sealed inside the vacuum bag, nitrogen gas is often used to increase autoclave pressure for added safety.

While properties of fibres and pre-cured/non-curing parts remain the same throughout the cure cycle (except for temperature change), the thermoset resin undergoes a significant chemical and thus mechanical change. The advancement of the curing reaction in the resin is quantified by the parameter known as the degree of cure. Degree of cure is determined by the fraction of the total heat that has been released. A fully cured condition, which means that all possible bondings have occurred, corresponds to a degree of cure reaching a value of one asymptotically.

Kaw [1997] defines the curing process as follows. At the beginning of the cure cycle, when temperature increases, resin becomes a liquid monomer. As temperature increases further, cure advances, small molecules attach to each other to form chains and resin viscosity and stiffness increase. Through half of the cure, chains begin to connect to each other, resin stiffness increases further and the resin becomes gel-like. As cure proceeds, cross-linking between chains occurs, polymerization advances and resin becomes solid with a much higher stiffness value than it had initially. Usually complete cure will occur, with a final degree of cure of about 0.9.

The most important goal of autoclave processing is to produce a part with good quality and minimum dimensional distortions and residual stresses. To achieve this goal, first the cure cycle must be optimized and then the tool must be compensated for undesired part deformation. The traditional approach involves trial-and-error, which is time-

-3- Chapter 1 – Introduction

consuming, expensive, and may still require the use of expensive shimming. Also, there will be no insight into other part parameters such as the degree of cure and residual stress distribution along the part.

Another choice is to use an intelligent process control technique (Holl and Rehfield [1992]) and place various instruments such as thermocouples and strain gauges in the part and tool to control the real-time on-going autoclave process cycle. Although measured information is useful, this approach has many deficiencies. Deformations cannot be predicted and an expert system is required to decipher the information gathered. However, system rules are material-dependent and difficult to modify.

The final technique mentioned here includes a computational model to mimic the processing cycle and develop varying material behaviours during the entire cure cycle, even prior to performance of a real autoclave process.

1.2 PROCESS MODELING

Today, numerical models are used frequently for various types of processing of composite structures such as resin transfer moulding and autoclave processing. These models vary in complexity from a simple model to capture just one event such as cure kinetics to a complicated model for capturing the whole processing event.

Recently, many computational models with various degrees of complexity have been developed for various types of process modelling. For example, an integrated process model was developed for a liquid composite moulding by Long et al. [1998]. A flow model to simulate the vacuum bag moulding process was proposed by Han et al. [2000]. A 2-D finite element process model was developed by Johnston [1996] at the University of British Columbia for autoclave processing. This model can predict temperature, residual stress and degree of cure of the part at the element or integration point level.

-4- Chapter 1 – Introduction

Both tool and part can be modelled and virtually subjected to a simulated autoclave process cycle. It is possible and economical to perform as many runs as necessary to achieve an optimal process cycle and tool shape. By performing sensitivity analyses, parameters that play significant roles in the response (e.g., final part shape, residual stresses) can be identified. Process modelling has the strong advantage of providing insight into the problem. It can predict all process-induced residual stresses and deformations. Also, it can yield the values of all existing parameters in the model, such as degree of cure and material stiffness, at any given time and location.

Knowing some of the parameters, such as residual stresses, is critical. For example, residual stresses change a part’s reserve capacity and affect its fatigue behaviour. Also, residual deformation indicates whether the part: (1) is within its acceptable tolerance limit, (2) should go through a force-to-fit or shim-to-fit process, or (3) is not in the acceptable range and should be compensated for.

This review includes a brief look at various processing models and their capabilities. A more extensive treatment of existing autoclave processing models is included to assess their relative advantages and disadvantages. Because better prediction of the process- induced residual stresses and deformations are the main concern in this study, emphasis of the review is placed on the methodology of modelling the tool-part interaction and its effects on the process-induced stresses and deformations.

Generally, composite processes and process models can be placed into one of two categories: open moulding processes and closed moulding processes.

1.2.1 Liquid Moulding

Liquid composite moulding (LCM) is a closed moulding process. It includes a large group of processes that involve the injection of resin into a fibre-reinforced preform. Resin transfer moulding (RTM) and structural reaction injection moulding (SRIM) are two examples of this group. The RTM process entails the injection of resin through a

-5- Chapter 1 – Introduction

fibre-reinforced network placed in a closed mould. A successful process results in a perfect impregnation of the fibre-reinforced network, a complete filling of the mould, an absence of voids and dry spots, and the shortest cycle time possible. Many 2-, 2½- and 3-dimensional numerical computer-based flow models have been developed to optimize this process.

Reboredo and Rojas [1988] and Chan and Hwang [1991] developed one-dimensional models to simulate the mould-filling process in a rectangular mould while Young [1994] developed a 3-dimensional non-isothermal model for the mould filling process.

In the case of LCM, due to imperfect cutting of fibre preform there is a possibility of gap between the preform fibre network and the mould edge. This high-permeability path causes disruption when filling the mould cavity. In existing codes, the permeability parameter is usually changed locally, which is unsatisfactory. Hammami et al. [1998] developed two equations for the edge effect in LCM and implemented them into a finite element code called RTMFLOT. They achieved a good agreement between experimental results and their models, but in some cases, as flow through thickness became non-negligible, the results were inaccurate.

Long et al. [1998] developed an integrated process model by combining a kinematic fabric drape model with a control/finite element flow simulation. They linked these two codes by a semi-empirical reinforcement permeability model, which predicts the preform flow characteristics using the fibre architecture. The flow simulation segment predicts the degree of cure and heat transfer. Moreover, it allows one to design the locations of injection gates and vents to obtain a non-defective part and an optimal filling time. The preform process simulation segment predicts fibre orientation, the volume fraction and wrinkles. Long et al. [1998] were the first group to combine these two aspects of modelling.

Blest et al. [1999] proposed a curing model for the autoclave resin film infusion (RFI) process, which is an alternative to the autoclave processing technique using prepregs.

-6- Chapter 1 – Introduction

The lay-up of the RFI process includes dry fibre plies interspersed with resin. By putting this lay-up in a vacuum bag and applying temperature, an exothermic chemical reaction is initiated while the applied pressure pushes the resin to infuse through the dry fibres. They used a 1-D flow model with Newtonian and constant viscosity resin. Their model is able to establish the temperature and degree of cure distributions during the process cycle.

Loos and MacRae [1996] developed an analytical 2-D finite element/control volume model for non-isothermal infiltration of resin into a preformed fibre network. They could predict the transient temperature and degree of cure distributions, resin viscosity, compaction of preform fibre network and resin movement through the fibre network.

Han et al. [2000] suggested that for environmental protection, the vacuum bag moulding (VBM) process and its variations (e.g., SCRIMP) could replace traditional open moulding processes. They concluded that because it is suitable for fabricating complicated large components – which cannot be designed by trial and error – SCRIMP requires a modelling tool to design the process. They proposed hybrid 2½-D and 3-D flow models to simulate SCRIMP processes.

Han’s model is able to determine resin flow fronts, filling time, pressure, temperature, and conversion distributions. One important difference between the Han’s model and the usual RTM process models is that fibre compressibility and flow in the thickness of the fibre preform are considered in the former. The main advantage of Han’s model compared to other 3-D flow models (e.g., the Ni et al. [1998] model) is its great computational efficiency.

In the RTM process model, the calculated domain is a 3-D geometry. However, if the resin flow is neglected in the thickness direction, it is termed as a 2½-D flow model. In the VBM process model, grooves, channels and high permeability media are often used to reduce the flow resistance in the longitudinal and transverse directions. Therefore,

-7- Chapter 1 – Introduction

thickness flow must be considered between high permeability layers or channels and the fibre-reinforced network.

Saouab et al. [2001] developed a 2-D model to simulate resin flow and designed their experimental device to allow the characterization of any reinforcement thickness. In their 2-D finite difference model, they simulated flow in the horizontal plane of the network. They assumed that their fluid is Newtonian, incompressible and non-reactive. Also, they employed a numerical finite element code called RTM3D, which was developed for industrial applications by Renault S.A. This code is able to model a composite part with non-isotropic fibre stacking.

1.2.2 Sheet‐Forming

The sheet-forming process is another method of making complex, curved composite parts. If a thermoset resin is used, the method is known as the hot-draping process, which is a cost-effective alternative to the prepreg process. Thermoplastic forming such as diaphragm forming and press forming provide a one-step moulding process involving heating, forming and cooling. Difficulties in this approach include fibre buckling plus washing, warping and void formation.

de Luca et al. [1998] developed a numerical simulation tool to model the continuous fibre-reinforced thermoplastic sheet press forming (CFRTP) process. They modelled both unidirectional materials and woven fabrics by shell elements. The commercial metal stamping software, PAM-STAMP, was modified by adding temperature-flow modelling and new material models. Other improvements to the code include representation of the deformation modes in thermo-viscoelastic fibre-reinforced sheets and identification of material failure and forming limits. They solved the governing equations as a dynamic problem using explicit integration. This kind of approach is suitable for very non-linear material and geometry particularly in the presence of non- linear boundary conditions (e.g., contact surfaces). They used shell elements for each ply and viscous-frictional contact surfaces between the shells to model the inter-ply

-8- Chapter 1 – Introduction

slip. Fibres and matrix are assumed to be elastic and thermo-viscous, respectively. The model is able to capture ply buckling caused by fibre locking at corners. The friction model for the contact interfaces is a function of temperature, contact normal pressure, relative fibre orientation and sliding velocity between neighbouring plies. Heat transfer is modelled as well. de Luca et al. [1998] concluded that slow press forming would decrease the intra-ply and inter-ply viscous forces and thereby result in less wrinkling.

McEntee and O’Bradaigh [1998] introduced a finite element model for sheet forming of unidirectional continuous fibre-reinforced composite laminates. This model includes large deformation, contact surface between tool and part, and inter-ply slip. They improved an existing 2-D plane strain model by adding contact interfaces between tool and part and between plies. They assumed an ideal fibre-reinforced fluid body, which has inextensibility in the fibre direction and material incompressibility. As a plane strain model, it can deal with single-curvature forming.

Although McEntee and Obradaigh’s model can model inter-ply and intra-ply shear (slip), it cannot model resin percolation and transverse flow in the out-of-plane direction (plane strain). These four mechanisms are depicted in Figure 1-2. Regarding inter-ply slip, they state that, when a multi-ply laminate is stacked, a resin-rich layer of about 10 µm thickness (for thermoplastic plies of 0.125 mm thickness) forms between all plies. They model this layer by frictional contact elements, which join meshes of each individual ply. There is no delamination, separation or penetration between plies. Tools are assumed to be rigid. Also, a penalty method is used for contact elements, and large deformation is allowed.

Lin and Weng [1999] developed a model for a sheet-moulding compound (SMC) during compression moulding to determine the flow model that best fits the experimental results of short fibre-reinforced thermosets. They used a 2-D finite element approach to model the composite part at room temperature in order to avoid curing. They assumed that the composite material is rigid-viscoelastic where the elastic response is negligible. They also assumed that the friction between the part and the tool

-9- Chapter 1 – Introduction

is a function of shear viscosity of the composite part. Lin and Weng [1999] concluded that the experimental results agree better with results from the varying anisotropic model than those of the constant anisotropic and isotropic models.

1.2.3 Autoclave Processing

Autoclave processing using prepregs is of interest in this study. In this process, layers of impregnated fibres in partially-cured resin in tape shape are stacked on a tool in desired orientations to make the composite part. The entire tool-part assembly is then placed inside a vacuum bag, which in turn is placed in an autoclave. During the process, heat and pressure are applied to the vacuum bag to initiate the exothermal chemical reaction and squeeze the air and excess resin out of the laminate.

There is a relatively large number of numerical tools available for the autoclave process simulation. One of the original works is that of Loos and Springer [1983], who developed a 1-D model to determine the cure and temperature distribution in a flat composite plate. They assumed a non-deformable porous system, and calculated the flow variables in normal and parallel directions to the plate surface independently.

Dave [1990] and Gutowski et al. [1987] developed two similar numerical tools to model 3-D flow. They assumed that flow in all directions is coupled and that consolidation occurs in the thickness direction. Telikicherla et al. [1994] developed a 2- D time-dependent heat-conducting finite difference model with heat generation capability.

Kim and Hahn [1989] developed an elasticity-based approach based on Boggeti and Gillespie’s [1992] formulation. They combined a 1-D model with the laminate plate theory (LPT) to capture the residual stresses and deformations in each time-step. Resin cure shrinkage and cure-dependent resin modulus were also included in the model.

-10- Chapter 1 – Introduction

White and Hahn [1992] developed an inclusive viscoelastic 1-D model combined with laminated plate theory. The viscoelastic behaviour was a function of time and degree of cure, and capable of stress development.

Tredoux and Ven der Westhuizen [1995] developed a 2-D finite element code (FCURE) that can model heat transfer and the Newtonian resin flow jointly in an autoclave process. This code has sub-models of resin cure development, fibre permeability and void growth suppression. They solved both the resin flow and the heat transfer equations separately, and linked them together through the resin viscosity. This code is also capable of modelling the filament-winding process. Moreover, it can model both the tool and the part at the same time.

Teplinsky and Gutman [1996] coupled a 1-D anisotropic cure simulation of the autoclave curing process to an incremental LPT model. The result of this combination became an elastic 2-D model to capture stress development and other mechanical properties during the process for any arbitrary cross-section. Prediction of this model was reliable for thin laminates. This model could also determine the degree of cure and temperature at any arbitrary section of the composite part.

Wiersma et al. [1998] used a 3-D finite element model of forming process, DIEKA, which was developed at the University of Twente to predict the spring-forward in continuous fibre/polymer L-shaped parts. They assumed that the fibres are elastic (spring) and the resin is viscoelastic (spring and dash-pot). They used a 1-D model for the heat transfer analysis and modelled both the tool and the unidirectional composite part, with a perfect bonding interface condition between them. They only calculated the residual stresses and deformation during the cooling time. They assumed that at the end of the hold, the composite part was fully cured and free of residual stresses. They had made numerous assumptions and simplifications that at the end their model prediction for spring-forward was very poor.

-11- Chapter 1 – Introduction

Blest et al. [1999] developed a 2-D finite difference scheme on a moving grid to model the autoclave processing of thermoset composite laminates. They used an uncoupled heat transfer model and a resin flow model. Blest assumed that parameters such as specific heat, density and thermal conductivity are not a function of temperature or degree of cure. Permeability is assumed to be isotropic and not a function of degree of cure. No mention is made of process-induced stresses or modelling of the tool.

Zhu et al. [2001] developed a 3-D thermo-chemical, viscoelastic finite element model for simulation of the heat transfer, curing, residual stresses and deformations of a composite part during the autoclave processing. They modelled both the tool and the part at the same time, and assumed perfect bonding between them. Because this model considers resin chemical shrinkage, thermal mismatch between constituents and non- uniform curing as well, its prediction of residual stresses and deformation is highly consistent with experimental results. Anisotropic linear-viscoelastic materials are modelled by considering their properties as functions of temperature and degree of cure at each time step. They were amongst a few researchers who determined the residual stresses during the whole cure cycle and not only in the cool-down phase. Their results showed that a significant amount of residual stress is built up before cool-down. Their model revealed the effect of the part thickness on the spring-forward since it was relatively a comprehensive model. However, Zhu et al.’s [2001] model did not account for the tool-part contact interface and the non-linear effects due to large deformations.

Li et al. [2002] used the commercial finite element code, ABAQUS, to model thick filament-wound tubes during the cure cycle prior to cool down. They investigated residual stresses, delamination, interlaminar cracks, distortion and fiber wrinkling in a composite part using a quasi 3-D linear elastic model. The simulation included two uncoupled thermo-chemical and stress-strain models. Li et al. [2002] assumed the resin modulus to be zero before the gel point, and a low number until the modulus development point at a degree of cure equal to 0.45, and fully-relaxed elastic modulus afterward. They indicated that their metal tool affects the development of residual

-12- Chapter 1 – Introduction stresses in the composite part, therefore, a frictional elasto-plastic contact with separation and stick/slide conditions between the tool and the composite part was also used. The friction coefficient was assumed to be 0.5. They concluded that a preheated tool, in which its thermal expansions have already occurred, reduces the residual stresses because it reduces the tool-part interaction due to mismatch in thermal expansion of the tool and the part. They indicated that “the chemical cure shrinkage of the resin is the key to determine the residual stresses of the composite”.

A multi-physics 2-D finite element model, COMPRO, has been developed at the University of British Columbia to analyze autoclave processing of composite structures of intermediate size and complexity by Johnston [1996] and Hubert [1996]. The model caters for a number of important processing parameters and the development of residual stresses and deformations. This model also accounts for the effects of tool-part interaction, which have been neglected by most investigators. However, currently this tool-part interaction is simulated by a non-physical thin shear layer. Twigg [2001], after an extensive experimental study, recommended replacing this shear layer by a more realistic contact surface in COMPRO.

A cure-hardening, instantaneously-linear elastic (CHILE) constitutive law – which is essentially a pseudo-viscoelastic behaviour (see Zobeiry [2006]) – was considered for the resin. Micromechanical models are used to determine mechanical properties of a composite ply, including thermal expansion, chemical cure shrinkage and elasticity moduli. Adaptive time step is incorporated to reduce the run time and maintain the accuracy of results. Heat transfer, resin cure and resin flow models are integrated with the stress-deformation model to capture all identified sources of process-induced residual stresses and deformations during the whole autoclave process. Plies can have various materials and orientations. The COMPRO model is of interest here and forms the basis of the new developments in the current research study.

Fernlund et al. [2003] used a method to predict the deformed shape of a 3-D structure using 2-D process analysis of different cross sections from COMPRO. Although this

-13- Chapter 1 – Introduction method is a complement to the 2-D COMPRO code, it has its own limits such as not applicable to structures with non-symmetric lay-ups.

1.3 RESIDUAL STRESSES IN THE AUTOCLAVE PROCESS

The goal of most earlier-developed process models was to capture the temperature and degree of cure profiles in the composite part. Blest et al. [1999] suggested that by knowing temperature and degree of cure profiles, one can infer whether or not a high- quality composite part was produced. But the recently-developed process models are capable of capturing the residual stresses and deformations as well.

There are many sources of process-induced residual stresses:

● Anisotropy of thermal strains

● Non-homogeneous resin curing

● Variation of fibre volume fraction in thickness direction

● Non-homogeneous heat distribution during cure cycle

● Autoclave pressure and temperature

● Uneven flow

● Thermal strain mismatch between the tool and the part

● Tool-part interface condition

The last two sources are collectively called the tool-part interaction which is one of the major contributors to the process-induced residual stresses and deformations. A physically based and mechanistic representation of the tool-part interaction will be the main focus of this study.

-14- Chapter 1 – Introduction

1.3.1 Tool‐Part Interaction

As the mismatch between the part and the tool coefficient of thermal expansions (CTEs) increases, tool-part interaction becomes more significant. Experimental studies have shown that part geometry, autoclave pressure, tool surface conditions, cure cycle, and tool and part materials affect the tool-part interaction. The CTE for a carbon fibre reinforced polymer (CFRP) part in the fibre direction is almost zero. Therefore, as the CTE of tools (non-composites) are non-zero, the residual stresses in the part can increase considerably due to the tool-part interaction.

Autoclave processing tools are mostly composite, aluminium, steel and invar. The CTEs of composite tools are generally quite low and close to the CTE of the curing part. Composite tools can also be tailored, but their service life is shorter than that of metal tools. Aluminium and steel tools are relatively inexpensive. However, the trend is toward tool materials with greater dimensional stability (i.e., with CTE values that are negligible). An example is invar, despite its relatively high cost.

Twigg [2000] experimentally quantified the tool-part interaction during autoclave processing. Strain gauges were placed on a flat thin aluminium tool to measure the strains in two perpendicular directions. A stack of prepreg was then placed on the opposite side of the tool. Tool and part were enclosed in a vacuum bag and subjected to only a temperature (no pressure) cycle in an autoclave. The strains in the tool were measured for two sets of tool-part interface conditions. In the first set of experiments, the part was placed on the tool without any release agent applied to the tool surface. In the second set of experiments, two layers of Mylar release film were placed on the tool surface before placing the part on the tool. For the second set of experiments with a Mylar film interface, the measured strains in the tool during whole cure cycle were approximately the same as those due to free thermal expansion of the tool. This implies that there was almost no mechanical interaction between the tool and the part. For the first set of experiments without release agent, the result showed that there was no interaction during the heat-up until the point of gelation. After gelation, and especially

-15- Chapter 1 – Introduction during cool-down, there were considerable strains in the tool. This indicated that, for the untreated interface, stress transfer between the tool and the part was present and that the tool-part interaction was significant.

In the case of other models where the tool-part interaction is not considered, the only explanation for differences between the model predictions and the experimental results is the exclusion of the tool-part interaction. For instance, Wiersma et al. [1998] who modelled the cool-down portion of the cure cycle stated that the tool-part interaction has a significant effect on deformation and residual stresses. They added that the tool- part interaction, which was not modelled in their model, would be included in the later version of their model.

Amongst all the autoclave process models, only those of Li et al. [2002] and Johnston [1996] considered a specific model for the tool-part interaction. Because Li et al. [2002] employed a commercial finite-element code, their model did not consider the evolving resin properties during the cure cycle. Also, it was difficult to input data such as mesh geometry and fibre direction of the elements in their commercial code. Doing so is a time-consuming task. Therefore, the only usable model is that of Johnston [1996], which is designed specifically for autoclave processing and has all the required features of modeling and meshing the composite parts.

Out of all the numerical models developed to simulate the autoclave processing, only two models, COMPRO and the 3-D model developed at the University of Illinois, incorporated all the well known sources of residual stresses identified in the literature while The University of Illinois model does not have the feature to simulate the tool- part interaction. In COMPRO, the tool-part interaction is currently simulated by incorporating a non-realistic thin shear layer.

Residual deformation can be a function of a part’s geometry, material properties, interface conditions, and autoclave processing parameters such as pressure. Twigg [2001] has investigated tool-part interaction by comparing numerical results from the

-16- Chapter 1 – Introduction

autoclave process model COMPRO with experimental data. He stated that all efforts to incorporate tool-part interface behaviour into the process model have shown that tool- part interaction has an enormous effect on the level of residual deformations. He concluded that a better model for tool-part interface must be used instead of a thin shear layer at the interface. The latter is mechanistically flawed and does not adequately represent the real behaviour at the interface.

It seems obvious that introducing a contact surface approach in COMPRO can improve its ability to predict better residual stresses and deformations due to tool-part interaction.

1.3.2 Sliding Approach

Based on the inter-ply slippage model of Ridgard [1993], Flanagan [1997] suggested the first distinct theory for tool-part interaction. He assumed that sliding occurs between the tool and the part, and between the first ply and the remainder of the composite part, when the shear stress exceeds a critical value at these two interfaces. He performed a parametric study on eight-ply unidirectional and cross-ply laminates. The conclusions from Flanagan’s experiments are as follows:

i. Almost a linear increase of part warpage due to increasing part length.

ii. A significant decrease in warpage by increasing ply count.

iii. A higher warpage by increasing the autoclave pressure.

A large variation in warpage was observed between using an FEP (fluorinated ethylene- propylene) release film and a Freekote release agent (RA). He also observed warpage for the parts fabricated on CFRP tooling despite the similar values of CTE for the tool and the part. It should be noted that RA at interface decreases the adhesion or bonding between the tool and the part significantly, while FEP sheets create multi-interfaces, which completely eliminate adhesion or bonding between tool and FEP sheet.

-17- Chapter 1 – Introduction

Flanagan [1997] developed a two-phase theory based on his experiments. First, he assumed that stresses are developed at an early stage of the cure cycle, while there is interfacial and inter-ply sliding of low modulus resin. He then considered an elastic analysis for post-gelation developments. He thereby determined the temperature- dependent friction coefficients from experimental results. He then incorporated friction coefficients into a contact surface model in ABAQUS. Sliding was permitted only at the tool-part interface and between the first and second plies.

The results of the first ABAQUS model generated an in-plane strain distribution through the thickness of the part. A second elastic finite element model of the part with cured material properties was created. Computed strains from the first ABAQUS model were inputs to the second model as initial thermal strains. The deformed shape of the part was then determined.

Even though the absolute value of the predicted warpage results were not in agreement with the experimental results, the predicted results agreed with the experimentally observed trend for variation of warpage with part length. Twigg [2001] suggested that the inability of the pure friction model to distinguish between tools of different CTEs was a shortcoming of Flanagan’s approach.

A more realistic and acceptable approach for modelling the tool-part interaction was presented by Johnston et al. [2001] in their 2-D finite element code, COMPRO. They introduced an orthotropic thin low-shear modulus layer between the tool and the part. This layer has a high out-of-plane elastic modulus to prevent penetration of the part into the tool (resulting undesired no-lifting capabilities as well) and a very low in-plane elastic modulus to allow the tooling effect on the part to transfer only through the shear modulus. Poisson’s ratios and CTEs of the shear layer are assumed to be zero.

By calibrating the shear modulus of this shear layer for various tools and part geometries, the model can predict the experimentally observed process-induced deformation accurately. One of the main disadvantages of this approach is that the

-18- Chapter 1 – Introduction

shear layer properties depend on the part geometry (see Twigg [2001] and Twigg, et al [2004]). For example, by changing the part length, the deformation prediction is inaccurate and fails to follow expected trend, if the shear modulus of the shear layer is kept constant. Johnston et al. [2001] showed that the predicted results are highly dependent on shear layer properties, with the predicted deformations of a given panel spanning several orders of magnitude.

Twigg [2001] performed a comprehensive experimental and analytical procedure to study the tool-part interaction of a flat unidirectional composite part on a solid tool. He used COMPRO with the shear layer feature to model the experimentally observed deformations and identified another shortcoming of the shear layer approach. In some cases, even with the highest possible shear modulus for the shear layer (equal to the tool’s shear modulus) – which is similar to a perfect bonding condition at interface – the maximum predicted deformation was less than the experimental result. Twigg [2001] showed that using a lower initial resin modulus in the same example could overcome this problem.

It should be noted that warpage is caused by rotation at each cross section and its distribution along the length. Rotation at each cross section is caused by changes of strain through the thickness. The bigger the strain variation through thickness, the larger the rotation is. Then numbers and locations of these rotations along part length dictate the amount of warpage and deformed shape of the part. Perfect bonding moves the location of maximum strain variation through thickness toward the tip of the part and causes almost uniform strain distribution through thickness for some length of the part. Therefore the part remains almost flat for most of the length and a sudden rotation occurs close to the tip of the part.

From the instrumented tool experimental results, Twigg [2001] found that the slipping initiates at the tip of the part and moves towards the centre. According to his experimental results, the critical sliding strength is a function of time/degree of cure with the maximum value of 30 to 50 kPa at a degree of cure of 0.65. Similarly, the

-19- Chapter 1 – Introduction

sticking (bonding) shear strength is also a function of time/degree of cure and its value is always greater than or equal to the sliding stress value. The maximum sticking shear strength was estimated to be around 4.0 MPa at resin final degree of cure of about 0.9.

Numerically predicting the experimentally observed warpage using COMPRO with a shear layer interface, Twigg [2001] concluded that the current elastic shear layer approach is not sufficient to capture the stick-slide behaviour of the interface and that the shear layer should be replaced by a more realistic contact surface.

1.4 MOTIVATION AND RESEARCH OBJECTIVES

The major commercial airplane manufacturers are building the next generation of aircraft (e.g. Boeing 787 and Airbus A350 XWB) using large integrated autoclave processed composite parts. These large integrated composite structures have many advantages but are challenging to produce. One of the many issues that are involved as the scale of the structure increases is to maintain dimensional control. Poor dimensional control of composite parts during autoclave processing manifests itself in huge assembly costs and reduced quality. Therefore it is desirable to have reliable numerical tools to predict the residual geometrical and material properties of these structures to guide their manufacturing.

UBC process model family, COMPRO, used heavily in the Boeing 787 program, has provided significant insight into what is needed for the next round of improvements. COMPRO uses non-physical/non-zero thickness contact interface (shear layer) which does not allow for stick-slip and gap mechanisms. It assumes that deformations and strains are small, while composite parts (particularly at corner regions) may undergo large rotations/displacements and distortions/strains during cure as shown in Figure 1-3. Also, it lacks the ability to handle non-linear boundary conditions and non-linear geometry and currently does not include a realistic constitutive model necessary for the stick-slip and contact-gap mechanisms at the tool/part interface. All of the above

-20- Chapter 1 – Introduction

shortcomings provide the motivatation for us to develop the next round of improvements in COMPRO.

The objective of the current study is to develop the constitutive formulation for a contact interface that represents the physics of the tool–part interaction during processing of composite structures. A proper implementation of contact interface requires updated geometry and possibly large strain capabilities in the finite element model. Therefore, considerable effort will be devoted to the numerical enhancements of COMPRO that are required to accommodate contact surfaces. These include implementation of nonlinear geometry (both large deformation and large strain), and nonlinear solvers (e.g., Newton-Raphson iterative scheme) among others. The layout of this thesis is as follows:

Chapter 1. Introduction. Chapter 1 presents an introduction to this research and an outline of the goals.

Chapter 2. Large Deformations. Chapter 2 presents a proper definition of a hypoelastic constitutive equation for the case of large displacement, large rotation and large strain. The co-rotational stress method is developed for orthotropic materials.

Chapter 3. Numerical Implementation and Verification of Large Deformation Formulation. This chapter presents the numerical implementation of the proposed co- rotational stress formulation method including the form of the stiffness matrix and internal forces into COMPRO. Benchmark examples are presented to verify the implementation.

Chapter 4. Contact Formulation. This chapter starts by a review of the literature on contact surface modelling, particularly in composite materials. Formulation of a penalty method for the contact interface and its corresponding algorithm are then presented. A numerical example is then carried out to verify the contact surface implementation.

-21- Chapter 1 – Introduction

Chapter 5. Validation and Numerical Case Studies. In this chapter Twigg’s [2001] instrumented tool experiments are used to calibrate the contact interface constitutive parameters in COMPRO. COMPRO predictions are then validated with experimental results of Twigg for flat parts with various lengths and thicknesses. Sensitivity analyses are also performed to highlight the importance of each contact parameter. Analysis of several curved and large scale parts using COMPRO with a contact surface option are introduced here. At the end of this chapter, predictions of the as-manufactured geometrical and material properties of these parts are presented.

Chapter 6. Summary, Conclusions and Future Work. This final chapter presents summary of the thesis along with major conclusions derived from this research. Suggestions and recommendations are also made for future research.

-22- Chapter 1 – Introduction

bagging material breather release ply prepreg sealant tape

tool/mould vacuum port

Figure1-1: A schematic showing the various elements in a vacuum-bagged composite part.

Intraply shear Interply shear

Resin percolation Transverse squeeze flow

Figure 1-2: Forming mechanisms for unidirectional composites (McEntee and O’Bradaigh [1998]).

-23- Chapter 1 – Introduction

Figure 1-3: An example of COMPRO interface showing a composite part (representative of the spar of an aircraft) modelled with a shear layer. The shear layer and composite part elements (particularly at corners) undergo large distortions and strains.

-24- Chapter 2 –Large deformations

CHAPTER 2 ‐ LARGE DEFORMATIONS

Autoclave process modelling of composite parts generally involves consideration of large strains in the presence of a significant mismatch of the coefficients of thermal expansion between the composite part and the tool.

Occurrence of large deformations and strains in the finite element analysis of a composite part stems from various sources. Incorporation of a thin shear layer at the interface between the part and the tool can result in unrealistically large shear strain values in the elements within this layer. In most cases, shear strain in a thin shear layer exceeds 100%. This implies that the small strain assumption is not suitable for runs with a shear layer, and that a large strain assumption is needed.

Eliminating this shear layer and simulating the tool-part interface through contact algorithms available in the literature may also lead to large displacements in flat segments and large rotations in curved interfaces. In addition, the extremely low modulus of the resin in the initial stages of the curing process results in a behaviour that resembles a viscous fluid more than a solid. Although employing assumptions such as high resistance to compressibility (through adopting a Poisson’s ratio close to 0.5) is a good remedy for preventing unstable deformation modes, doing so rarely reduces the occurrence of large shear deformations and strains. Furthermore, when the material is in its initial stages of cure and is therefore extremely compliant in the resin dominated

-25- Chapter 2 –Large deformations

direction; it is conceivable that it experiences large strains particularly in the presence of applied autoclave pressure.

2.1 BACKGROUND

Consideration of large deformations (which includes large strains, large displacements and large rotations) is a basic requirement of composite process modelling. In order to improve the predictions of the finite element code COMPRO (see Johnston [1996]), large-deformation formulations were developed and implemented in this study. This chapter highlights some background information on this topic and describes the methodology adopted to achieve this goal. Some basic definitions of stress and strain are reviewed here, followed by a summary of existing formulations on large deformations and strains.

2.1.1 Stress and Strain Measures

Most materials known to exhibit linear elasticity preserve such behaviour within a limited range of strain values, beyond which they show significant non-linearity. In fact, it is well known that under such large strains, these materials exhibit a softening behaviour under tension and show signs of hardening under compression (see Belytschko et al. [2000]). This implies that under special circumstances realistic modelling of even linear elastic material requires consideration of large deformations and strains.

Generally, the mechanical behaviour of a discretized structure or a skeletal structure consisting of a system of springs can be modelled by relating the load and the deformation vectors through the structural stiffness matrix, as shown in Equation (2.1). Such behaviour is considered linear if the structural stiffness matrix is constant.

ψ = Ku (2.1)

-26- Chapter 2 –Large deformations

where K is the structural stiffness matrix, and u and ψ are nodal displacements and force vectors, respectively.

In parallel, the above relation governing the behaviour at a structural level can be redefined at the material level by relating a stress-strain pair through a unique material stiffness tensor shown in Equation (2.2).

σ = C : ε (2.2) where C is material stiffness tensor, and ε and σ are strain and stress matrices, respectively.

Many definitions exist for stresses and strains, and are related to each other through a relationship similar to the one defined above. For simplicity of discussion, such relationships are reviewed here for the 1-D case (uniaxial bar), which can be directly expanded into full 3-D cases.

In general, Nominal stresses and strains can be defined through the following equations:

σ NNN= C ε (2.3) Ψ σ N = (2.4) A0 u ε N = (2.5) L0

where CN is the Nominal material stiffness, ε N and σ N are the Nominal strain and stress measures, L0 and A0 are original length and cross-sectional area of a bar, and u and Ψ are deformation and force values applied at the end of the bar.

By substituting Equations (2.3), (2.4) and (2.5) in Equation (2.1), the relationship between the material and structural stiffnesses can be determined as:

-27- Chapter 2 –Large deformations

KL0 CN = (2.6) A0 It should be noted that a constant value of the structural stiffness (i.e., linear elastic materials) results in a constant Nominal material stiffness that implies linear behaviour in the Nominal stress and strain domain. Similarly, a varying structural stiffness (non- linear elastic materials) directly yields a varying Nominal material stiffness.

Another measure of strain and stress referred to as the Second Piola-Kirchhoff (2PK) stress and Green strain is defined below:

SC= GGε (2.7) Ψ 1 S = (2.8) AF0 1 ε = (1)F 2 − (2.9) G 2

where CG is the Green material stiffness, and εG and S are the Green strain and 2PK stress measures.

The one-dimensional deformation gradient, F , can thus be calculated as shown below:

LuLu+ F ==0 =+1 (2.10) LL00 L 0 where L is the current length of the bar.

Therefore, the relationship between the Green material stiffness and structural stiffness can be readily determined:

2KL0 CG = (2.11) FF(1)+ A0

Equation (2.11) shows that the Green material stiffness, CG , is dependent on the current deformation value through the deformation gradient, F . As a result, while the

-28- Chapter 2 –Large deformations

structural stiffness value is constant, the relationship between 2PK stress and Green

strain is nonlinear (i.e., CG is a function of deformation).

At large strains, most materials display nonlinear behaviour at the structure level by being soft in tension and hard in compression. It is possible to capture this nonlinear response through a linear stress-strain relationship called Logarithmic strain and True (Cauchy) stress. The advantage of this approach is its simplicity in dealing with linear equations in the mathematical manipulations of finite element equations. These measures in 1-D can be written as:

σ = CLε (2.12) Ψ σ = (2.13) A ε = ln(1+ u ) (2.14)

where CL is the Logarithmic material stiffness, andε and σ are Logarithmic strain and True stress measures. A is the current cross-sectional area of the bar.

This concept is summarized in Figure 2-1, where the nonlinear softening force- deformation behaviour is captured by three stress and strain pair definitions. It can be seen that, while the material stiffnesses based on Green and Nominal strain measures show dependency on the deformation itself, the Logarithmic strain consideration along with its stress conjugate, True stress, results in a constant material stiffness.

2.1.2 Conjugate Stress and Strain Definition

Conjugate stresses and strains are defined such that their combination in a system results in an internal energy equal to the external work done. This definition can easily be extracted from the virtual specific power, w& (work rate per unit original volume), equation defined similarly by Hill [1970] and Bonet and Wood [2000] as shown below:

-29- Chapter 2 –Large deformations

w 1 & = σ : D (2.15) ρρ0 or

ρ wJ& ===0 σ :::D σ D τ D (2.16) ρ

where ρ0 and ρ are the original and current material density measures, J , σ and τ are the determinant of the deformation gradient matrix, True and Kirchhoff stress matrices (or vectors) in a global coordinate system, and D is the rate of deformation matrix in a global coordinate system and is defined as:

1 D =+()FF&&−−1 FTT F (2.17) 2 The pair of conjugate stress and strain can be determined in the original or current configuration, as expressed below:

ρ SE::::& ===0 σ D Jσ D τ D (2.18) ρ The above equation can be rewritten as follows:

SE:tr()& = J σD (2.19) where tr() denotes trace of matrix product. The stress and strain pairs, S and E , that satisfy the above equations are called conjugated tensors in the original configuration. For example, if E is assumed to be a Green-Lagrangian strain tensor given by:

1 E = ()FFT − I (2.20) 2 where the deformation gradient tensor, F , is defined as:

dx F ==+δ u (2.21) dX ij i, j

-30- Chapter 2 –Large deformations

where x , X and u are the current coordinate, original coordinate and deformation vectors. Determinant of a deformation gradient tensor referred to as the Jacobian, J , is defined as:

ρ F ==J 0 (2.22) ρ The first derivative of the Green-Lagrangian strain tensor with respect to time can be written as:

1 E&&=+=()FFTT FF & FDF T (2.23) 2 Therefore, the 2PK stress tensor can be determined by substituting Equation (2.23) into Equation (2.18):

J 1 S :ESFDF& == : (TT ) tr( SFDF ) =J tr( FSFD T ) (2.24) JJ From the above equation and Equation (2.19), it can be concluded that:

1 JJtr(σDFSFD )= tr(T ) (2.25) J Therefore,

1 σ = FSF T (2.26) J or

SF= J −1σF −T (2.27) Equations (2.26) and (2.27) provide the relationship between the 2PK and True stress tensors. It should be noted that 2PK stress and Green-Lagrangian strain are conjugates in the original configuration. More information on this topic can be found in Bonet and Wood [2000], Belytschko et al. [2000], Zienkiewicz and Taylor [1991] and Crisfield [1991, 1997].

-31- Chapter 2 –Large deformations

2.1.3 Polar Decomposition of the Deformation Gradient Tensor

The deformation gradient tensor, F , can be decomposed into the multiplication of two parts, referred to as the rigid body rotation tensor and the stretch tensor. Mathematically, this can be expressed as:

F =RU=VR (2.28) where R is the orthogonal rotation tensor, and U and V are the symmetric right and left stretch tensors, respectively. The tensor, R , is specifically needed for the mathematical calculations of the methodology developed in this study. Some of the mathematical properties of these tensors are listed below:

RRIT = (2.29)

UU==TT RVR (2.30)

VV==TT RUR (2.31) In order to determine the decomposed tensors, specifically the right stretch tensor, U , and the rigid body rotation tensor, R , the following equation is used:

CFFRURUUUU==TTT() ⋅==2 (2.32) where C is the right Cauchy-Green deformation tensor (matrix). In the case of a 3x3 positive definite matrix, the square root of a matrix can be determined using the method developed by Franca [1989]. In the case of a plane-strain problem, the root-finding operation is essentially performed on a 2x2 rather than a 3x3 matrix. The process of determining U from C is based on the determination of their eigenvalues and invariants. Considering that the eigenvalues of C are the square of those of U , they can be calculated as a function of the invariants of matrix C .

Upon determination of the eigenvalues of the matrix U , λ1 and λ2 , its invariants, IU and IIU can easily be calculated:

-32- Chapter 2 –Large deformations

22 IC =+λλ12 ⎫ ⎪ 2 IIC II ⎬ ⇒=+IC λ1 II =⇒=λλ22 λ 2 C 2 C 12 2 2 ⎪ λ1 λ1 ⎭

42 λλ11−+=IIICC0

II±−2 4 II λ 2 = CC C 1 2

⎧ I +−III2 4 ⎪ CC C 2 λ1 = IICC±−4 II C⎪ 2 λ1 =⇒⎨ (2.33) 2 ⎪ I −−III2 4 ⎪λ = CC C ⎩ 2 2

IU = λ12+ λ (2.34) IIIIUC==λλ12 Knowing the invariants of C and hence U , the right stretch tensor and its inverse can be computed using the Cayley-Hamilton theorem (see Franca [1989]):

2 UUI0− IIIUU+= (2.35) 1 UIC= ()IIU + (2.36) IU

−1 1 UIU=−()IU (2.37) IIU 2.1.4 Objectivity of the Stress Rate

Any conjugate pair of strain and stress tensors (e.g., 2PK stress and Green strain tensors) should have specific characteristics to be suitable for large deformation/strain consideration using the stress rate method. The most important of these characteristics is having objective rates of stress and strain tensors. This objectivity condition is also referred to as frame invariant, since stress and strain in the material coordinate system should not be affected by rigid body rotations.

-33- Chapter 2 –Large deformations

It must be noted that in the co-rotational stress method proposed later in this study, stresses are determined in the material coordinate system and would not be affected by rigid body rotations. Consequently, objectivity of the stress rate does not pose a problem in this method.

Based on the formulation of the 2PK stress and Green strain tensors, they are the only conjugate pairs that are objective along with their rates. This is shown by substituting Equation (2.28) in Equation (2.20):

11 1 E =−=−=−()()()FFTT I URRUI UUI (2.38) 22 2 The first derivative of the Green strain with respect to time is then calculated as follows:

1 E&&=+()UU UU & (2.39) 2 Equations (2.38)and (2.39) indicates that the Green strain and its rate are not dependent on rigid body rotation, and therefore deemed to be objective tensors. The 2PK stress rate is derived from multiplication of the Green strain rate tensor and the symmetric 4th order material stiffness tensor, determined in its original configuration. Objectivity of the Green strain tensor and the frame-invariant symmetry of the material stiffness matrix result in objectivity of the 2PK stress tensor. This is referred to as a rate constitutive equation, and is given by:

S& = CEG : & (2.40)

where CG is the Green material stiffness tensor. Because 2PK stress is a summation of its stress rates, it holds the same objectivity property.

SS= ∑ &∆=tt∑(:) CEG & ∆ (2.41) In contrast to the 2PK stress rate tensor, the True stress rate is not objective (as will be discussed in Section 2.1.5). The fact that material behaviour can be best modelled with

-34- Chapter 2 –Large deformations

True stress and its conjugate Logarithmic strain, and their rates, has enabled other scientists to develop equivalent objective True stress rates, which will be discussed in the following sections.

Various commercial finite element codes exercise different objective stress rates for the large deformation consideration. The following section discusses some of the common objective stress rates, along with their advantages and disadvantages.

2.1.5 Examples of Objective Stress Rates

Rate constitutive equation that relates the stress rate to strain rate through a material stiffness tensor is used in many theories, including plasticity and viscoelasticity (see Hughes and Winget, [1980]). Such a relationship is the basis of “hypoelasticity” in materials. Equation (2.40) shows this kind of relation between the 2PK stress rate and the Green strain rate. However, the same relationship between True stress rate and Logarithmic strain rate cannot be established. Therefore:

σ& ≠ CUL :ln(& ) (2.42) where CL is the Logarithmic material stiffness tensor. To overcome this deficiency, alternative objective stress rates have been derived from the True stress tensor, some of which are discussed in the following sections.

2.1.5.1 Truesdell Stress Rate As discussed earlier, the 2PK stress rate is an objective tensor. From the existing relationship between 2PK and True stress tensors (Equation (2.26)), a relationship between their rates can also be established:

dd111TTT d σ& ==++()()()FSF FSF F SF dt J dt J J dt (2.43) 11dd F ()SFTT+ FS ( F ) Jdt J dt or

-35- Chapter 2 –Large deformations

Jtr(D )TTT 1−1 1 σ& =−FSF + F& () F F SF + FSF& + JJ2 J (2.44) 1 FSF()−1 FTT F& J or

TT1 σ& =−tr(D )σ +lσσ +lFSF + & (2.45) J or

T ◊ σ& = −+++tr(D )σ lσσl σ (2.46) where σ ◊ is referred to as the Truesdell stress rate, defined as:

1 σ ◊ = FSF& T (2.47) J , l is the velocity gradient tensor, given by:

lFF= & −1 (2.48) and time derivative of J (see Bonet and Wood [2000]) is:

d (J ) = Jtr(D) (2.49) dt By definition, the Truesdell stress rate is the multiplication (double contraction) of a 4th order material stiffness tensor (C ◊ ) and a rate of deformation tensor ( D ), as follows:

σ ◊◊= CD: (2.50) Because the above material stiffness tensor, C ◊ , is not known, it is generally assumed to be the same as the Logarithmic material stiffness tensor, CL . Such an approximation introduces significant errors as the strains become large (see simple shear example in the next chapter for confirmation). Incorporating the above assumption into the general equation of Truesdell stress rate (Equation (2.45)) leads to:

T σ& ≅−tr(D )σ +lσσ +lCD + L : (2.51)

-36- Chapter 2 –Large deformations

It should be noted that other objective stress rate definitions can be derived from the Truesdell stress rate defined above with further simplifications, such as the Jaumann stress rate discussed in the following section.

2.1.5.2 Jaumann Stress Rate The Jaumann stress rate can be obtained from the Truesdell stress rate by assuming small strains, while the rotations can be large. The Jaumann stress rate is a powerful stress rate measure used in finite element computations.

Several additional variables must be defined before deriving Jaumann stress rate equation. These definitions can be found in many finite element books such as Belytschko et al. [2000] and Bonet and Wood [2000]. It is possible to subdivide the velocity gradient tensor, l , into a symmetric part referred to as the rate of deformation, D , and a skew-symmetric part called the spin tensor, W . In mathematical form, this can be written as:

lW= + D (2.52) From the definition of the skew-symmetric spin tensor, one can write:

−=WWT (2.53) Employing another assumption of small strains ( D → 0 ), Equation (2.52) can be simplified as:

l ≅ W (2.54) Equation (2.51) can now be re-written as:

TJ σ& ≅+0Wσσ +WCDW +L : =σσ −W +σ (2.55) Further assuming that the Jaumann and Logarithmic material stiffness tensors are the same, the following equation can be established:

JJ σ =≅CDCD::L (2.56)

-37- Chapter 2 –Large deformations

The above assumptions (small strains and the equivalence of Jaumann and Logarithmic material stiffness tensors) are made to obtain the Jaumann stress rate from Equation (2.45), which leads to reasonable predictions in the small strains regime. However, Jaumann stress rate and analytical solution begin to deviate as the strains become larger, limiting the applicability of this stress rate tensor.

2.1.5.3 Green‐Naghdi Stress Rate The Green-Naghdi stress rate is derived from the Jaumann stress rate by redefining the spin tensor and substituting Equations (2.48) and (2.28) into Equation (2.52):

1 WlDRRRUUUUR=− =&&&TT +()−−11 − (2.57) 2 Ignoring the terms in the bracket in Equation (2.57), the spin tensor becomes:

WRR≅ & T (2.58) Substituting the above Equation in Equation (2.55) relationship between the True stress rate and the Green-Naghdi stress rate can be written as:

TTGNTTGN σ& ≅−+=−+RR&&σσRR σ RR &&σσRR C: D (2.59) As in the cases of the objective rates discussed earlier, it is assumed that the Green- Naghdi material stiffness tensor is equal to the Logarithmic material stiffness tensor.

TT σ& ≅−+RR&&σσRR CL : D (2.60) The Green-Naghdi stress rate yields good results for small strains, but because of its assumptions it gives wrong results for large strains.

In the last three stress rates, different assumptions have been made, but assuming the same material stiffness matrices as logarithmic material stiffness matrix is the main cause of results differs from exact solution when strains become large. Later in the next chapter, this fact will be shown by comparing results of simple shear example.

-38- Chapter 2 –Large deformations

2.1.5.4 Logarithmic Stress Rate The objective Logarithmic stress rate is the only measure that provides an exact solution in the case of isotropic materials. A detailed discussion of this stress rate (see below) can be found in Xiao et al. [1997], Liu and Hong [1999] Lin [2002, 2003] and Lin et al. [2003].

OLogLog σσσΩΩσLog =+& − (2.61) ΩLog is a skew-symmetric matrix, and can be found in the above-mentioned references. Because the material stiffness tensor is known for the Logarithmic stress rate (and is the

same as the Logarithmic material stiffness tensor, CL ), the exact solution can be derived for isotropic materials.

O σLog= CD L : (2.62) Furthermore, the True stress rate can be determined as:

O Log Log Log Log σσ& =−Log σΩΩσ + =CD L : −σΩ + Ω σ (2.63) The objective stress rate from this approach can be directly obtained from co-rotational stress formulation method for isotropic materials (coming up later in Equation (2.100)), and is discussed in Section 2.2.2. Therefore, the Logarithmic stress rate method is a subclass of the general method of co-rotational stress formulation, and as will be shown later it is only valid for isotropic materials.

2.1.6 General Form of Co‐rotational Stress Rate

The co-rotational stress rate is a general form and given by:

O σσ=−& 2( Ησ )Sym (2.64) where the subscript “sym” denotes the symmetric part of the term in the brackets and Η is an arbitrary matrix.

-39- Chapter 2 –Large deformations

It should be noted that the “co-rotational stress rate” is different from the “co-rotational stress formulation” proposed in this study. The co-rotational stress rate is a general objective stress rate definition that can reduce to all the stress rates mentioned before in special cases (see Liu and Hong [1999]) as shown in Equation (2.65).

⎧ 1 ⎪Hl=− tr( D )Ι for Truesdell stress rate ⎪ 2 ⎪ ⎪ ⎨HW= for Jaumann stress rate (2.65) ⎪ ⎪HRR= & T for Green - Naghdi stress rate ⎪ Log ⎩⎪H = Ω for Logarithmic stress rate Therefore, an infinite number of objective stress rates exist that can be derived from the True stress rate. These stress rates are generally divided into two categories:

● Non-spinning objective stress rates such as the Truesdell stress rate

● Spinning objective stress rates such as the Jaumann and Green-Naghdi stress rates

By definition, each objective stress rate is associated with its own unique material stiffness tensor that is different from the others. A common assumption is the equality of the material stiffness tensors amongst all objective stress rate definitions and use of the Logarithmic material stiffness tensor in all cases. This assumption is valid for the small strains range, but leads to significant error once the strains become larger. Belytschko et al. [2000] and Lin [2002, 2003] discussed these differences for several stress rate definitions. This can also be seen for the Jaumann and the Truesdell stress rates discussed later in simple shear example in the next chapter.

There are other examples of stress rates that fall within the general formulation of Equation (2.64). These include the work by Xia and Ellyin [1993] and Gadala and Wang [2000].

-40- Chapter 2 –Large deformations

It should be noted that the pairs of stress and strain along with their material stiffness tensor should reflect a unique material behaviour. Having the same material stiffness tensor for different pairs of stress and strain measures results in different material behaviours, contradicting the uniqueness rule stated above. However, this error is insignificant in the small strain range, leading to the overall agreement of various stress rate definitions. This is discussed further in the following section.

2.1.7 Role of Material Stiffness Tensor in Objective Stress Rates

As mentioned above, the Logarithmic material stiffness tensor is used in all the objective stress rate definitions. However, these approaches break down in the large strain range, making these objective stress rates inappropriate for such applications.

All objective stress rates are unique in their definitions, therefore, for two arbitrary objective rates, it can be written as:

OO σσij≠ (2.66) Replacing these stress rates by the product of their material stiffness tensor and the rate of deformation tensor results in:

CDCij::≠ D⇒≠ C ij C (2.67)

This implies that the material stiffness tensors must be different for different objective stress rates. Consequently, one can deduce that the material stiffness tensors of all objective stress rates (with the exception of one) cannot be replaced by the Logarithmic material stiffness tensor.

-41- Chapter 2 –Large deformations

2.2 CO‐ROTATIONAL STRESS FORMULATION FOR ISOTROPIC

MATERIALS

The motivation behind this work is to establish a novel methodology to capture the large-strain behaviour of “hypoelastic materials.” In contrast to the objective stress rate method commonly used for large-deformation considerations, this approach does not require the objectivity of stresses and their rates. This is the case because the stresses are always determined in the material coordinates, and rigid body rotations are applied manually.

The method proposed herein to overcome the lack of objectivity of the rate of the True stress tensor is to embed a coordinate system in the material/element, a system that rotates with the material/element. This method is based on determining the change of stress due to stretching of the material in the principal material coordinate system (material configuration), adding the change of stress to the original stress in the material configuration and then rotating the total stress to its current configuration given by hypoelastic equations (see Belytschko et al. [2000]):

tt+∆σˆ& = C : tt +∆ εˆ& (2.68)

tt+∆σσˆˆ= t+∆ tt +∆ σ& ˆt (2.69)

tt+∆ tt +∆ tt +∆ ttT +∆ σ = 00R σˆ R (2.70) The symbol “ ” on stresses and strains denotes that they are defined in the material configuration. Lower and upper left subscripts define original or base and current or desired configurations, respectively.

Two distinct approaches exist for the coordinate definition in the “co-rotational stress formulation” method:

● A coordinate system is embedded in the element that rotates with it. As a result, all material points in the element will experience the same amount of rotation.

-42- Chapter 2 –Large deformations

● A coordinate system is embedded in each element integration point that rotates with the material at that point. As a result, each integration point has its own rotation.

The second approach is more accurate for large strains and rotations (see Belytschko et al. [2000]). For the remainder of this study, the second approach stated above is implied when the terminology “co-rotational stress formulation” (CSF) is used. Belytschko et al. [2000] describes the CSF method as a viable candidate for not only isotropic but also anisotropic materials, given that the rotation tensor is determined accurately. Because the coordinate system attached to each integration point is rotating with the material, the material stiffness tensor remains constant and unaffected by rigid body rotations. Therefore, the stress rate-based constitutive equation is:

σˆ& = CUL :ln(& ) (2.71) The CSF method is the only approach that yields exact results for both isotropic and anisotropic materials. All the other methods such as the objective stress rate method discussed previously (except the Logarithmic stress rate) provide an approximation for isotropic materials; and none are valid for anisotropic materials.

Aside from the fact that the rotated True stress, σˆ , and Logarithmic strain, εˆ , are conjugates in the current configuration, Hoger [1987] proved that the rotated Kirchhoff stress, τˆ , is the conjugate of the Logarithmic strain in the original configuration as well.

If the material stiffness tensor is constant in the Kirchhoff stress domain as shown in Figure 2-2b and not in the True stress domain, then it implies that the total True stress cannot be determined from Equations (2.72) and (2.73). Rather, it should be determined indirectly from Equations (2.74), (2.75) and (2.78).

tt+∆ σσσˆˆˆ≠ t+∆& t (2.72) and

-43- Chapter 2 –Large deformations

σˆ& ≠ CUL :ln(& ) (2.73) or

tt+∆ t 00τττˆˆˆ= +∆& t (2.74) and

τˆ& = CUL :ln(& ) (2.75) in which

tt+∆ tt +∆ tt +∆ 00τσˆˆ= J (2.76) and

tt+∆ tt +∆ 00J = F (2.77) Therefore, the total rotated True stress can be determined as:

tt+∆1 tt +∆ στˆˆ= tt+∆ 0 (2.78) 0 J Although most researchers support the validity of Equation (2.75), many finite element codes such as ABAQUS use Equation (2.71) in their underlying formulations. However, since strain values are assumed to be small in those commercial codes, the Jacobian, J , approaches unity, and the Kirchhoff and True stress tensors becomes identical. In the updated COMPRO, both Equations (2.71) and (2.75) have been implemented and the user has the option of choosing either one.

The above-mentioned equations imply that, for elastic materials that are linear in the Logarithmic strain domain, two different equations can be defined to relate the Logarithmic strain to True or Kirchhoff stresses:

σˆ = CUL :ln( ) (2.79)

τˆ = CUL :ln( ) (2.80)

-44- Chapter 2 –Large deformations

For consistency, the Kirchhoff stress Equation (2.80) is used in this study, and all other findings are based on this assumption. It is worth noting that the Kirchhoff and True stress tensors can be used interchangeably through Equation (2.76).

Hoger [1987] proved that the rotated Kirchhoff stress, τˆ , and the time derivative of the logarithm of the right stretch tensor are conjugates in the case of isotropic materials, i.e.:

τ ::ln()D = τˆ & U (2.81) Belytschko et al. [2000] proved that the rotated Kirchhoff stress, τˆ , and the rotated velocity strain, Dˆ , tensors are a conjugate pair. In this approach, the rotated Kirchhoff stress and rotated velocity strain tensors are defined as:

τˆ = RT τR (2.82)

Dˆ = RDRT (2.83) Substituting Equations (2.82) and (2.83) into the virtual power Equation (2.16) results in:

TTˆˆˆ TTT w& ==τ :(DRτˆˆˆRRDRR ):()tr( =τRRDR )tr() = RRτD (2.84) Therefore:

ˆˆ w& ==τ :tr():D τˆˆD =τ D (2.85) Using Equations (2.81) and (2.85), we can write:

τˆˆ::ln()Dˆ = τ & U (2.86) It can be concluded that there are two different constitutive equations associated with different material stiffness tensors, as follows:

ˆ τˆ& ==CDCDL::ln()& U (2.87) In order for the above equation to remain valid, one of the following two conditions must hold true:

-45- Chapter 2 –Large deformations

1. DUˆ = ln(& )

2. Dˆ ≠ ln(& U ) (2.88)

It is known that condition (1) is not true. To further reinforce this statement, a rail shear test example has been solved using two definitions for rotated Kirchhoff stress based on the left and right sides of Equation (2.86). In both approaches, it is assumed that material stiffness tensors are equivalent and equal to the Logarithmic material stiffness tensor. As the strains increase, the rotated velocity strain tensor, Dˆ , yields erroneous results, while the exact result is the one with hypoelastic yielding which is shown by the rate of the logarithm of the right stretch tensor. These results are depicted in Figure 2-3. This example will be discussed later in section 3.8.1.1.

Therefore, the second statement of Equation (2.88) should be valid. This equation and

Equation (2.87) imply that there are two different material stiffness tensors, CD and CL as shown below:

CCD ≠ L (2.89) Considering the rail shear example and recalling that in both cases the material stiffness tensors are set equal to the Logarithmic material stiffness tensor, the definition that provides the exact solution is the one based on the Logarithmic material stiffness tensor, as follows:

τˆ& = CUL :ln(& ) (2.90) This can also be proven by considering the case of isotropic hyperelastic materials. Hyperelasticity and hypoelasticity will be further discussed in details in the following sections.

-46- Chapter 2 –Large deformations

2.2.1 Linear Hyperelastic Materials

By definition, hyperelastic materials are path-independent, and their total stress and strain are computable if the original and final configurations are known. Generally, the rotated total stress tensor is defined as the multiplication of the material stiffness tensor (itself a function of total stress) by the rotated total strain tensor, as shown below.

τˆ = CL ():τεˆ (2.91) For linear elastic isotropic materials in the Kirchhoff stress and Logarithmic strain domain, these equations can be written as:

τˆ = CUL :ln( ) (2.92)

where CL is constant.

In other words, the constitutive Equation (2.92) defines the type of material that behaves linearly in relation to a predetermined domain of strain and stress. The time derivative of Equation (2.92) yields the hypoelastic constitutive equation given by:

τˆ& = CUL :ln(& ) (2.93) This equation is identical to Equation (2.90), which is used in the rail shear example.

One-step solution

For any hyperelastic constitutive behaviour, in general, the total stress tensor can be determined from the total strain tensor in just one step. The Kirchhoff stress tensor can be evaluated using Equations (2.92) and (2.82), as depicted below for the case of linear elastic isotropic material:

TT τ ==RτˆRRCUR[:ln()]L (2.94) Because rigid body rotations do not affect the isotropic material stiffness tensor, Equation (2.94) can be re-written as:

-47- Chapter 2 –Large deformations

T τ = CRURL :[ ln( ) ] (2.95) The rotation tensor can be moved inside the logarithmic function of a tensor, based on the proof presented by Lu [1998]. Therefore:

T τ ===CRURCVCLLL:ln( ) :ln( ) :ε (2.96) where the total strain can be defined as:

ε = ln(V ) (2.97) Equation (2.96) is the one step solution for the Kirchhoff stress tensor from the knowledge of the total strain tensor in the case of linear isotropic materials.

2.2.2 Non‐linear Hyperelastic Materials

A similar approach is applicable to the non-linear elastic isotropic materials with respect to a predefined domain of strain and stress. The only difference in this case is that upon performing time differentiation on the total rotated stress equation, a hypoelastic constitutive equation will be obtained, where the secant material stiffness tensor is replaced by the tangent material stiffness tensor. In other words, Equation

(2.92) is written in terms of the secant material stiffness tensor, CL, Secant :

τˆ = CUL, Secant :ln( ) (2.98) The time derivative equation will be based on the tangent material stiffness tensor as shown below:

τˆ& = CUL, Tangent :ln(& ) (2.99) Finally, the total Kirchhoff stress tensor can be determined in just one step by knowing the total strain tensor. The appropriate equation can be derived from Equation (2.98) as follows:

T τ ==RτˆRCL, Secant :ln( V ) (2.100)

-48- Chapter 2 –Large deformations

The above equation holds true only for hyperelastic isotropic materials, and not for anisotropic materials. As mentioned earlier, the Logarithmic stress rate tensor can be determined from the above equation, and because Equation (2.96) is valid only for isotropic materials, the Logarithmic stress rate tensor is valid for isotropic materials as well. This is a limitation that does not exist in the CSF method. The logarithmic derivative of Kirchhoff stress is then given by:

O O Log τ = CVL, Tangent :(ln( )) (2.101) Xiao et al. [1997] has proved that:

O (ln(VD )) = (2.102) Therefore:

O Log τ = CDL, Tangent : (2.103) The remaining calculations are similar to those of the other objective stress rates.

2.3 CO‐ROTATIONAL STRESS FORMULATION FOR ANISOTROPIC

MATERIALS

It is rather intuitive that the material stiffness tensor of an anisotropic material in a global coordinate system changes with rigid body rotations. The material stiffness tensor remains constant once evaluated with respect to the material coordinate system, since this coordinate system rotates with rigid body rotation. However, the material stiffness tensor of an anisotropic material changes with the stretch tensor, U , even in the material coordinate system as depicted in Figure 2-4. This implies that the rotated Logarithmic strain and the rotated Kirchhoff stress tensors are not coaxial. The following two examples of uniaxial and simple shear loading demonstrate the rotation of anisotropic material stiffness tensor due to the applied deformation.

-49- Chapter 2 –Large deformations

Example 1: A uniaxially loaded bar with fibres at an inclined angle (off-axis). By increasing the load, the angle between the fibres and the X-direction decreases.

Example 2: Simple shear of a square element with fibres parallel to X-direction. The direction of the fibres remains constant while increasing the horizontal displacement. Despite the presence of a rigid body rotation, for an observer attached to a global coordinate system, no net rotation of the material coordinate system occurs. In this case the rigid body rotation in the clockwise direction is negated by an equal counter- clockwise rotation due to stretching in the Z-direction.

Belytschko et al. [2000] states that the co-rotational stress formulation is valid for anisotropic materials if and only if the rigid body rotation tensor is exactly determined. He also states that the rotation tensor, derived from the decomposition of deformation gradient tensor, is not exact. Belytschko et al. [2000] used the rotated velocity strain ˆ tensor, D , and the material stiffness tensor, CL , and not the material stiffness tensor,

CD . Therefore, as discussed in Section 2.2, this assumption would not yield the correct solution even with the exact rigid body rotation tensor, R .

In this study, we establish that the CSF method is an exact and valid method to capture material behaviour in True/Kirchhoff stress and the Logarithmic strain domain for both isotropic and anisotropic materials. It should be noted that the material stiffness tensor of an anisotropic material changes with the stretch tensor in the material coordinate system.

2.3.1 Rotation of the Material Stiffness Tensor

As mentioned earlier, stretching causes rotation in the anisotropic material stiffness tensor. In the case of the two-dimensional finite element code COMPRO, rotation of stiffness tensor is only considered in the X-Z plane, (i.e., around Y-axis, see Johnson [1996]), while in a general case, rotation of the material stiffness tensor should be

-50- Chapter 2 –Large deformations

permitted around all these axes. In the following the material stiffness tensor rotation due to finite stretching is determined exactly.

Assume that the fibre direction in the original and the current configurations are given by vectors A and A′ defined below. The angle between these two vectors can be determined in 2-D, as follows:

T A = [a12 a ] (2.104)

⎡⎤a1′ A′ ==⎢⎥FA (2.105) ⎣⎦a′2 AA′ θ′ = ArcCos() (2.106) A A′

where θ′ is rotation due to both stretching and rigid body rotation. Therefore, the total rotation matrix (including rigid body rotation and rotation due to stretching) can be determined as:

⎡Cos(')θ − Sin (')θ ⎤ R′ = ⎢ ⎥ (2.107) ⎣ Sin(')θθ Cos (')⎦ The rotation of fibres, θ , due to stretching is only visible to an observer attached to the material coordinate system. Therefore, the rigid body rotation should be cancelled as given by:

−1 T RC ==RR′ RR′ (2.108) A simplified method to directly determine the rotation, θ , only due to stretching is:

⎡⎤a1′′ −−11 A′′==⎢⎥RA ′ =()() UFFAUA = (2.109) ⎣⎦a′′2 AA′′ θ = ArcCos() (2.110) A A′′

-51- Chapter 2 –Large deformations

⎡cs− ⎤ RC = ⎢ ⎥ (2.111) ⎣sc⎦ in which:

cCos= ()θ (2.112) and

sSin= ()θ (2.113) Therefore, the anisotropic material stiffness tensor in the current configuration defined th in the material coordinate system can be easily determined. Because CL is a 4 order tensor, its rotation is given by:

TT CRRCRRCCCCC= L (2.114)

In the more conventional form, i.e., expressing the strain and stress as 6x1 vectors, CC ,

nd CL and RC transform into 2 order 6x6 matrices and the above equation can be written as:

T CRCRCCC= L (2.115) It should be noted again that in the co-rotational stress formulation approach developed for anisotropic materials, rigid body rotations do not affect material stiffness tensor in the material coordinate system, while stretching does affect the material stiffness tensor, as shown in Equation (2.114).

2.3.2 Hypoelastic Constitutive Equation

For the case of a hypoelastic material, the constitutive equation involving the rotated Kirchhoff stress in the material coordinate system and allowing for the rotation of the anisotropic stiffness tensor due to finite stretching can be written as:

τˆ& = CUC :ln(& ) (2.116)

-52- Chapter 2 –Large deformations

The above hypoelastic constitutive equation, which replaces the Equation (2.90) for isotropic materials, is the only one that is suitable for composite materials during cure, which their orthotropic materials evolve in time.

2.4 SUMMARY AND DISCUSSION

Modelling of structures undergoing large deformations need to address stress objectivity and rotation of anisotropic material stiffness tensor. Researchers mostly have addressed only the first issue by considering different objective stress rates. Our findings here suggest that “Co-rotational Stress Formulation” is the only method within which one can address both issues.

Many objective stress rate definitions are conjugates of the rate of deformation tensor D , and can satisfy the virtual work rate equation (or alternatively, virtual power equation). With the exception of one (Logarithmic stress rate definition), all of these stress rate definitions have unknown material stiffness tensors that are consequently assumed to be equal to the Logarithmic material stiffness tensor. As a result these stress rates are applicable to a limited range of strains, and produce erroneous predictions at finite strains. The Logarithmic stress rate is the only objective stress rate that takes the advantage of having the actual isotropic material stiffness tensor.

It appears that none of the existing objective stress rate definitions are applicable to anisotropic materials.

For linear elastic isotropic materials, the final solution can be obtained directly in one step using the hyperelastic constitutive relation, which utilizes the definition of logarithmic function of the strain tensor in its formulation.

The Rotated True/Kirchhoff stresses and the Logarithmic strain are computed in the material coordinate system, and therefore are inherently objective. Each of these pairs of strain and stress tensors is related through the actual material stiffness tensor; and

-53- Chapter 2 –Large deformations

thus they are theoretically applicable to anisotropic materials for any range of strain values.

The anisotropic material stiffness tensor which rotates due to stretching can be precisely determined at any configuration based on its rotation matrix, as shown in Equation (2.115). Since the anisotropic material stiffness tensor rotates due to stretching, therefore, strains and stresses would end up with different principal axes (i.e., cease to remain coaxial). The proposed method of “Co-rotational Stress Formulation” easily copes with non-coaxial stresses and strains.

The hypoelastic material model is the only constitutive equation suitable for curing materials of interest in this study. The stiffness of these materials increases during the curing process, while the existing stresses remain unaffected by this change.

-54- Chapter 2 –Large deformations

σ or τ Force Stress σN S

Displacement Strain

Figure 2-1: Non-linear structural response or force-displacement behaviour (left) while nd having nonlinear Nominal and 2 Piola-Kirchhoff stress (σ Ν and S ) versus Normal and Green strain relationships, or linear True (or Kirchhoff) stress (σ and τ ) versus Logarithmic strain relationship (right).

ψ

u τ τ σ Stress σ Stress (a)

ε = ln(u) ε = ln(u) (b) (c)

Figure 2-2: (a) A non-linear load-displacement curve obtained from experiments. (b) A linear Kirchhoff stress (τ ) versus Logarithmic strain assumption, which results in a non-linear True stress versus Logarithmic strain relationship. (c) A linear True stress (σ ) versus Logarithmic strain assumption, which results in a non-linear Kirchhoff stress versus Logarithmic strain relationship.

-55- Chapter 2 –Large deformations

2 ˆ τ&ˆ =CD: γ γ

/µ) γ/2

xz θ σ ( L0 =1 1 Z stress L0 =1 τˆ& =CU:ln(& ) X

Normalized global shear 0 0 15304560

γ/L0 (m/m)

Figure 2-3: Global shear stress for isotropic material in simple shear example based on equations: A) τˆ& = CD: ˆ and B) τˆ& = CU:ln(& ). µ is the Lame elastic constant.

γ γ γ/2

p p θ tθ t+∆tθ

(a) Z (b) X

Figure 2-4: Fibres in off-axis composite materials rotate with respect to the global coordinate system in pure stretching (a) resulting in rotation of material coordinate system. Simple shear of an element (b) showing rigid body rotation (θ) of the element and rotation (-θ) of material coordinate system cancel each other resulting in zero net rotation for fibers.

-56- Chapter 3 – Numerical implementation and verification of large deformation formulation

CHAPTER 3 ‐ NUMERICAL IMPLEMENTATION AND

VERIFICATION OF LARGE DEFORMATION

FORMULATION

This chapter is devoted to the numerical implementation of the constitutive formulation which is presented in chapter 2 and is based on corotational stress formulation method. These implementations are made in COMPRO. Following the presentation of the algorithms, the chapter concludes with benchmark examples for verification purposes.

3.1 LOGARITHMIC STRAIN RATE

Logarithmic strain and Kirchhoff/True stress is the conjugate pair of interest. Therefore, determining the Logarithmic strain rate, which is the time derivative of the logarithmic function of a symmetric matrix, U , as defined below is needed:

tt+∆ t tt+∆ ln(UU )− ln( ) ε&ˆ =≅ln(& U ) (3.1) ∆t Based on right side of Equation (3.1), in order to determine the Logarithmic strain rate, one should be able to determine the logarithmic function of a symmetric matrix. There exist two different methods of the decomposition and the Taylor series expansion for this purpose.

-57- Chapter 3 – Numerical implementation and verification of large deformation formulation Decomposition Method

This method is based on finding the root of the stretch tensor, U , as shown in the following equation:

i 11− ln(UUU )=− (ii ) (3.2) 2 in which i is a positive integers (i =1, 2, 3, …). The decomposition method has been used by many researchers, e.g., Jirasek and Bazant [2001] or Bazant et al. [2000]. Higher values of i results in better accuracy at the expense of increasing the numerical computations. Given that the algorithm for determining the square root of a symmetric matrix is available (introduced in Section 2.1.3); only those values of i in Equations (3.2) that are equal to two to the power of a natural number are of interest. This increases the efficiency of the whole computations, as in each iteration, values from the previous iterations can be readily used, therefore:

in==2n 1, 2, 4, 8, ... = 0, 1, 2, ... (3.3)

Taylor Series Expansion Method

This method, which to the author’s knowledge has not been used by other researchers, is an extension of the Taylor series of the logarithm of a scalar value to that of the logarithm of a symmetric matrix. The Taylor series expansion of the logarithm of a number is:

⎧ xx−−11 135 1 x − 1 ⎫ ln(xx )=+++ 2⎨⎬ ( ) ( ) ( ) ... if > 0 (3.4) ⎩ xx++13 1 5 x + 1 ⎭ Similarly, the Taylor series expansion of the logarithm of a positive definite matrix is:

⎧ −−111 −3 ⎫ ln(U )=−++−+ 2⎨⎬ ( U IU )( I )⎣⎦⎡⎤ ( U IU )( I ) + ... (3.5) ⎩ 3 ⎭ If we now define:

-58- Chapter 3 – Numerical implementation and verification of large deformation formulation

QUIUI=−()() +−1 (3.6) then the logarithm function of a matrix can be rewritten as:

11 ln(UQQQ )= 2(+++35 ...) (3.7) 35 The required number of terms that should be calculated is determined by the conditional convergence equation below. The convergence condition is determined by comparing the norm of the nth term of Equation (3.7) to a percentage ratio of the norm of summation of all previous terms, including the nth term. In COMPRO, the threshold

p0 is assumed to be equal to 0.1%.

Q21i−

21i − ≤ p (3.8) n 21i− 0 ∑ Q i=1 21i − A comparison is performed between these two methods for the case of a bar with zero Poisson’s ratio under axial loading. In this case, the only non-zero strain value is in the longitudinal direction, and the deformation gradient matrix, which is a diagonal matrix, is given by:

⎡100+ εˆ11 ⎤ ⎢ ⎥ FRUIU===⎢ 010⎥ (3.9) ⎣⎢ 001⎦⎥ A diagonal deformation gradient matrix has been used on purpose, since we can find its exact logarithmic function by only finding the logarithm of diagonal terms. Results of the exact solution, the decomposition and the Taylor series expansion methods are summarized in Table 3-1. It can be seen that the Taylor series expansion method is more accurate than the decomposition method. Also, it seems that as the strain increases, the error in the decomposition method grows faster than that of the Taylor series expansion method. Even in a general case, where all six components of strains

-59- Chapter 3 – Numerical implementation and verification of large deformation formulation are non-zero, the Taylor series method gives much more accurate results than the decomposition method (results are not presented here).

3.2 CO‐ROTATIONAL CONSTITUTIVE EQUATION

As stated in chapter 2, the proposed co-rotational stress formulation along with its hypoelastic constitutive equation is applicable to all isotropic and anisotropic materials. CSF is an accurate method that can also cope with varying anisotropic material stiffness tensor.

The hypoelastic constitutive equation of interest, applicable to large deformations and strains for anisotropic materials, was formulated previously in Equation (2.116). In the scheme established in this study, which is based on the approach employed in the finite element code COMPRO, all material properties are updated at each time step. An iterative approach is added, in which variables such as displacements, stresses, and rotation matrices are updated at each iteration within a single time step. Rewriting the previously-developed constitutive relationships for such iterative scheme results in:

tt+∆ tt +∆ tt +∆ τˆ& = CUC ln(& t ) (3.10) Similarly, for the material stiffness tensor we can write:

tt+∆ tt +∆ t ttT +∆ CRCRCCCC= tt (3.11)

where CC and RC are 6x6 rotated logarithmic material stiffness and rotation (due to stretching) matrices, respectively. The rotation angle needed to construct the rotation matrix, RC , is determined in Equation (2.110).

Using hypoelastic constitutive model for elastic materials causes a problem when the material stiffness tensor changes (e.g., due to temperature changes). This problem arises since hypoelastic constitutive model cannot account these changes for already stored stresses. Although this deficiency in COMPRO is minimal, however it can be

-60- Chapter 3 – Numerical implementation and verification of large deformation formulation compensated for by adding an extra term to Equation (3.10), as illustrated in detail in Appendix A.

3.3 DETERMINATION OF TOTAL TRUE STRESS

The total rotated Kirchhoff stress is determined according to the following equations:

tt+∆ττˆˆ= t+∆ tt +∆ τ& ˆt (3.12) At any time step, t, and each iteration, i, incremental changes in the stress are added to the original stress tensor at the material coordinate system, as shown in Equation (3.13). In order to avoid numerical convergence problems such as oscillating deformations and stresses, the material coordinate system is assumed to be unchanged during these iterations. Therefore, rigid body rotations are not applied to the stresses at an iteration level. The equation below is used to calculate the total rotated Kirchhoff stresses in the material coordinate system.

tiτττˆˆˆ= ti−1 +∆& it (3.13) The total rotated stress tensor obtained at the end of the numerical iteration, tt−∆ , is carried to the next time step, t, and serves as an initial value of stress tensor for the new set of iterations performed at time step, t.

tttττˆˆ0 = −∆ (3.14) Then total rotated True stress can be determined as:

1 tt+∆ˆˆ tt +∆ στ= tt+∆ (3.15) 0 J Finally the total True stress tensor can be determined as follows:

tt+∆σ = tt +∆R tt +∆σˆ ttT +∆ R (3.16)

It should be noted that most researchers assume a linear relationship between the Kirchhoff stress and the Logarithmic strain tensors. In the LS-DYNA and ABAQUS

-61- Chapter 3 – Numerical implementation and verification of large deformation formulation commercial finite element codes, it is assumed that the True stress and the Logarithmic strain tensors have a linear relationship, as shown in Figure 2-2c. Both assumptions are implemented in COMPRO based on Equations (2.71) and (2.75).

3.4 DETERMINATION OF THE ELEMENT TANGENT STIFFNESS MATRIX

The virtual work done in a specific time period on a physical system or, alternatively, the virtual power (equal to the rate of virtual work) by external and internal sources, should equal zero. Internal virtual power can be determined in both material and global coordinate systems, as formulated below:

δδΦ&&t=Φ int +Φ δ & ext =0 (3.17)

where Φ& t , Φ& int and Φ& ext are the total, internal, and external virtual power terms, respectively. The internal virtual power can be calculated for both cases of the True or Kirchhoff stresses in material or global coordinate systems based on constant material stiffness in their domain, as follows:

⎧ TT δδΦ=−& int σε&dv =− σεˆ δˆ& dv ⎪ ∫∫ ⎨ (3.18)a,b 11TT ⎪δδδΦ=−& int ττεε&dv =− ˆ ˆ& dv ⎩ JJ∫∫ and

δδΦ=−ττTTεεdV =−ˆ δˆ& dV (3.19) & int ∫ & ∫ in which J is the determinant of deformation gradient matrix, v and V are current and original volume related to each other as:

vJV= (3.20) The external virtual power in the global coordinate system is formulated as:

T δΦ=& extΨ extδ r& (3.21)

-62- Chapter 3 – Numerical implementation and verification of large deformation formulation where Ψext is the external force vector and r& is the nodal velocity vector. Equating the internal and external power equations (Equations (3.18), (3.19) and (3.21)) in both the material and the global coordinate system results in:

Ψ TTδδr ==σεdv σεˆ T δˆ& dv (3.22) ext &&∫ ∫ and

Ψ TTδδr ==τεdV τεˆ T δˆ& dV (3.23) ext &&∫ ∫ The first time derivative of Equation (3.22) is:

ddTT d T ()(Ψextδδr&&==σεdv )( σεˆ δˆ& dv ) (3.24) dt dt∫∫ dt or

Ψ& TTδδr +=Ψ r σε T δdv + σε T δ dv ext&&&& ext ∫∫& && (3.25) =+∫∫σε&ˆˆTTδδ&&&ˆˆdv σε dv where Ψ& ext is referred to as the rate of the external force vector.

To further expand the above equations, the relationship between the strain rate and rate of deformation vectors is defined through a matrix, B , as follows:

ε&&= Br (3.26) In a given time step, the B matrix is assumed to be independent of r as is the case in updated Lagrangian scheme, therefore, the time derivative of the B matrix, as well as its variations, should equal zero:

δ BB0= & = (3.27) Because the second time derivative of deformation is zero ( &&r0= ) for static problems of interest here, it can be concluded from the time derivative of Equation (3.26) that:

&&ε = 0 (3.28)

-63- Chapter 3 – Numerical implementation and verification of large deformation formulation This relation remains valid for rotated strain and velocity vectors. Therefore, the second term on each side of Equation (3.25) is zero, and this equation can be rewritten as:

Ψ& TTδδr = σεdv ext &&∫ & (3.29)a,b Ψ& TTδδr = σεˆ& ˆ&dv ext & ∫ Similarly, the above equation can be rewritten for the Kirchhoff stress as follows:

TT1 T Ψ& extδδr&&==τε&&dV τε δ & dv ∫∫J (3.30)a,b TT1 T Ψ& extδδr& ==τε&&ˆˆ&&ˆˆdV τε δ dv ∫∫J Determination of B Matrix

In general, it is customary to define the B matrix to relate the velocity vector to the rate of deformation vector in a global coordinate system as shown below, which is inconsistent with the previous definition of Equation (3.26):

DBr= & (3.31) Because of interest here is the calculation of strain in a local coordinate system, a new matrix, Bˆ , is defined to relate the velocity vector in the global coordinate system to the strain vector in the local coordinate system, as expressed below. The ultimate goal is to find the exact terms for the Bˆ matrix.

εˆ& = Brˆ & (3.32) In order to achieve that goal, we can define the rotated velocity strain in vector form as:

ˆ −−11 DR= εε⋅= DR ⋅⋅ Br& (3.33)

where Rε is the rigid body rotation matrix applied to the strain vector, defined as:

-64- Chapter 3 – Numerical implementation and verification of large deformation formulation

⎡ cs22− cs⎤ ⎢ ⎥ 22 (3.34) Rε = ⎢ sc cs⎥ ⎢ 22⎥ ⎣22cs−− cs c s ⎦ In the method proposed by Hoger [1986], the True stress and the Logarithmic strain tensors are of interest. These strain and stress tensors are related to each other through the Logarithmic material stiffness matrix, as shown below. Because the exact material stiffness matrix is used here, there is no approximation in this equation.

τˆ& ==CCCεˆ& CUln(& ) (3.35) Hoger [1987] determined the derivative of the Logarithmic strain tensor to be a function of the rate of deformation tensor, Dˆ , and eigenvalues of the right stretch tensor, U . For a 2D plane strain case (as implemented in COMPRO), one of the eigenvalues ( λ2 ,) would always be equal to one.

λ2 = λy =1.0 (3.36)

If the other two eigenvalues are equal (λ13= λ ), then:

T ε&ˆ ===ln(& UDRDR ) ˆ (3.37)

Otherwise, if λ13≠ λ , then:

⎡ DgD110(1)+ 13 ⎤ ˆ& T ⎢ ⎥ ε ==ln(& UR )⎢ 0 1 0 ⎥ R (3.38) ⎣⎢(1+ gD )13 0 D 33 ⎦⎥ Rewriting the above equations in vector form results in:

⎡⎤⎡⎤⎡⎤DD1110 0 11 −−11⎢⎥⎢⎥⎢⎥ − 1 εˆ& ==RRDD01 0 = RGD (3.39) εε⎢⎥⎢⎥⎢⎥33 33 ε ⎣⎦⎣⎦⎣⎦⎢⎥⎢⎥⎢⎥2(1++⋅gD )13 0 0 1 g 2 D 13 where

-65- Chapter 3 – Numerical implementation and verification of large deformation formulation

2λ13λ λ1 g = 22⋅−ln( ) 1 (3.40) λλ13− λ 3 and

⎡10 0⎤ ⎢ ⎥ G = ⎢01 0⎥ (3.41) ⎣⎢001+ g⎦⎥ From Equations (3.39) and (3.33), it can be concluded that:

−1 ˆ εˆ& ==RGBrBrε && (3.42) where

ˆ −1 BRGB= ε (3.43) Therefore, Bˆ can be rewritten as:

22 ⎡⎤cs(1+ gcsN )⎡ 1,XX 0 N 2, 0 ...⎤ ⎢⎥⎢ ⎥ ˆ 22 (3.44) B =−+⎢⎥sc(1 gcs )⎢ 0 N1,ZZ 0 N 2, ...⎥ ⎢⎥22⎢ ⎥ ⎣⎦−+−2cs 2 cs (1 g )( c s )⎣ N1,ZX N 1, N 2, Z N 2, X ...⎦ Detailed explanations concerning Bˆ can be found in Appendix B.

Now that Bˆ is determined, we can go back to calculate the element tangent stiffness matrix which relates the rate of force and the velocity vector as follows:

Ψ& ext = Kr& (3.45) By substituting Equations (3.35), (3.37) or (3.38) and (3.45) in Equation (3.30)b, the tangent stiffness matrix in the Kirchhoff stress domain can be determined:

TT11ˆˆ T ()Kr&&δδ r== ( CCCεε&&ˆˆ )dv (CBr & )() δ Br & dv (3.46) ∫∫JJ

TTT1 ˆˆ rKr&&δδ= rBCBr &C &dv (3.47) J ∫

-66- Chapter 3 – Numerical implementation and verification of large deformation formulation

TT1 ˆˆ rK&&{}0− BCBC dv δ r= (3.48) J ∫ Because δ r& is an arbitrary vector and r& cannot be zero, the term inside the brackets in Equation (3.48) should be zero, resulting in the rotated tangential element stiffness matrix given by:

1 KBCB= ˆˆT dv (3.49) J ∫ C The above equation for rate-independent material stiffness in the True stress and the Logarithmic strain domain can be rewritten as:

KBCB= ˆˆT dv (3.50) ∫ C For more information on determining the tangent stiffness matrix, the reader is referred to Volume one of Crisfield [1991] and McMeeking and Rice [1975].

3.5 DETERMINATION OF THE INTERNAL FORCES

The equation of the virtual power of a physical system was developed earlier (Equation (3.18)b). The internal force vector can then be derived from that equation:

TT11 Tˆˆ 1 T Ψintδδr&&&==ττˆˆε&ˆdv δ()Br dv = τ ˆ B δ r dv (3.51) JJ∫∫ J ∫ or

TT1 ˆ ()0Ψint − τˆ Brdv δ & = (3.52) J ∫ Because the variation of velocity expressed as δ r& is an arbitrary vector, then:

1 Ψ TT= τˆ Bˆdv (3.53) int J ∫ or

-67- Chapter 3 – Numerical implementation and verification of large deformation formulation 11 Ψ ==BBRˆˆTTτˆdv−1τ dv (3.54) int JJ∫∫σ where

−1 τˆ = Rσ τ (3.55)

In the above equations, Rσ is the rigid body rotation matrix for stresses expressed in vector form. In mathematical form, Rσ is written as:

⎡cs22−2 cs⎤ ⎢ ⎥ 22 (3.56) Rσ = ⎢sc2 cs⎥ ⎢ 22⎥ ⎣cs−− cs c s ⎦ Comparing Equations (3.34) and (3.56), it can be concluded that:

−T RRσ = ε (3.57) Therefore, the internal force can be obtained from:

11 Ψ ==()RGBR−−11TTTτ dv ( R −− RGB 1 )τ dv (3.58) int JJ∫∫εσ σε 1 Ψ = ()GB T τ dv (3.59) int J ∫ As mentioned earlier, the above relations are developed assuming linear behaviour in the Kirchhoff stress and Logarithmic strain domain. In the case of similar behaviour in the True stress and Logarithmic strain domain, this relation takes in the following form:

Ψ ===(RGBR−−11 )TTσ dv ... ( GB ) σ dv (3.60) int ∫∫εσ

3.6 DETERMINATION OF THE EXTERNAL LOADS

In finite element simulation of the autoclave composite processing, the only source of external loading is the applied pressure. Due to the specific nature of the pressure loads, they are considered to be “follower” loads. As a result, the mechanical external forces,

-68- Chapter 3 – Numerical implementation and verification of large deformation formulation

Ψσ , applied at each node of the elements must be recomputed at the beginning of each iteration to consider the changes in the elements caused by rigid body rotations and changes in the element dimensions due to processing deformations.

Ψ = Npds (3.61) ext ∫ In which p, s and N are autoclave pressure, current surface of structure and shape function vector, respectively.

Temperature variations in both curing and non-curing materials result in volumetric changes. Cure shrinkage is another source of volumetric change in curing materials. For the development presented in this thesis, it is assumed that the coefficient of thermal expansion (CTE) and the volumetric cure shrinkage factor are defined in the Logarithmic strain domain. Total thermal and cure shrinkage strain vector in material coordinate system is determined as follows:

εεεˆˆˆTC= TH+=∆+ CSα T ε ˆ CS (3.62)

where εˆTH , εˆCS and εˆTC are thermal, cure shrinkage and total thermal and cure shrinkage strain vectors, respectively. α is a vector includes material CTEs in principle directions.

T α = [CTE123 CTE CTE 000] (3.63)

Total rotated mechanical strain matrix, εˆσ , is determined by deducting the total thermal and cure shrinkage strain matrix from the total strain matrix in material coordinate system as follows:

εεεˆˆˆσ = − TC (3.64) Equation (3.22) can be rewritten as:

TT−1 Ψextδδr& + τεˆ ˆ&dv = 0 (3.65) J ∫ By substituting in and simplifying the above equation, we can get:

-69- Chapter 3 – Numerical implementation and verification of large deformation formulation

TT1 ˆ ΨextδδrC&&− ()()0C εˆσ Br⋅=dv (3.66) J ∫

TT1 ˆ {(NCpds )− (C (εεˆˆ−=TC ))Br dv }δ & 0 (3.67) ∫∫J Since δ r& is an arbitrary vector therefore the term in the brackets in Equation (3.67) must vanish:

11 NBCBrBCpds−+ˆˆTT dv ˆεˆ dv =0 (3.68) ∫∫JJCC ∫TC Again by substituting in and simplifying the above equation, we can get:

1 NKrBCpds− +=ˆ T εˆ dv 0 (3.69) ∫∫J C TC or

1 Kr=+ Npds Bˆ T C εˆ dv (3.70) ∫∫J C TC The thermal expansion and cure shrinkage effects of materials are taken into consideration by adding an equivalent mechanical load vector as shown in Equation (3.71) to the external load vector. Therefore the modified external load vector,

Ψ Pseudo− ext , is defined as:

1 Ψ =+NBCpdsˆ T εˆ dv (3.71) Psudo− ext∫∫J C TC

3.7 ITERATION SCHEME

To ensure that the computed stresses and deformations converge to their mathematically exact values for a given mesh density, iterative schemes are commonly employed in finite element methods to deal with issues such as large strains, updated geometry, nonlinear materials and contact. At each time step (e.g., in COMPRO), it is assumed that the material properties are constant, and thus the iterations only account for changes in deformations, stresses and external loads within a single time step.

-70- Chapter 3 – Numerical implementation and verification of large deformation formulation Two distinct criteria for convergence exist: the convergence of deformation and that of the residual forces. In mathematical form, these criteria can be formulated as follows:

nn++11 n + 1 ΨΨext−ϒ int ResidualForce_ Norm= =≤ p (3.72) nn++111 ΨΨext ext

nn+1rr− Deformation_ Norm= ≤ p (3.73) 1r 2

where p1 and p2 are the convergence threshold values. These values are usually

defined by the user. As an example, the default value used for p1 in the commercial finite element code ABAQUS is equal to 0.5%. The default values used in COMPRO are as follows:

p1 = 0.5% (3.74) p2 = 0.1% The residual force vector is determined as the difference between the external and internal force vectors, expressed as:

ϒ = ΨΨext− int (3.75) In order to increase the efficiency of the analysis, an additional constraint is commonly applied to the iterative scheme based on the convergence rate (speed). This is usually done to terminate the iterations in cases where the convergence rate (speed) is very

slow, or the maximum number of iterations exceeds a certain threshold, p3 , i.e., np≥ 3 where p3 is taken to be 40 in COMPRO.

3.8 VERIFICATION

To verify the correct implementation of the formulation developed in this chapter for the overall approach proposed to deal with the issue of large deformations/strains, solutions to some basic benchmark problems are exercised in this section. These

-71- Chapter 3 – Numerical implementation and verification of large deformation formulation examples include studies of a single-element undergoing rail shear (or simple shear), uniaxial loading, closed loop stress path and the simulation of bi-material bars and beams.

The study of simple shearing of isotropic linear elastic materials has been of great interest to researchers in this field. This is so because the solution to this problem is readily applicable to many different objective rates. The exact analytical solution of a single-element undergoing a simple shear is fully developed and presented by Lin [2002, 2003]. Furthermore, all the points in the element undergoing simple shear loading experience the same strain and stress values (i.e., the strain and stress fields are homogeneous).

The uniaxial loading of linear elastic isotropic materials is another suitable example for verification purposes, since the analytical results can be directly determined by using the Logarithmic definition of strain. The uniaxial tensile loading example will also be applied to non-linear elastic isotropic materials. It should be noted that COMPRO uses a cure hardening instantaneously linear elastic (CHILE) constitutive model that implies constant material properties at each time step. Therefore, in this example the CHILE constitutive model in COMPRO for evolving the elastic modulus in the presence of large deformation/strain will be tested as well.

Both examples will be reconsidered for linear elastic transversely isotropic materials (unidirectional composite material), so that the material stiffness rotation due to the stretching can also be tested and verified.

3.8.1 Linear Elastic Isotropic Materials

The simple case of a linear elastic isotropic material is considered in this section for all the examples.

-72- Chapter 3 – Numerical implementation and verification of large deformation formulation

3.8.1.1 Simple Shear The example of the simple shear of a 2-D plane-strain unit square is depicted in Figure 3-1. The main characteristics of the simple shear example are summarized below:

● Strains and stresses are homogeneous throughout the volume of the element.

● All the points in the element undergo the same rigid body rotations.

● Global strains and stresses start from pure shear at the beginning and approach pure normal at very large deformations.

● Hypoelastic yielding for shear stress is visible. Global shear stress approaches zero. From a thermodynamic standpoint, the principal normal stresses increase monotonically and asymptotically. Also, the slope of the principal stresses versus deformation asymptotically approaches a zero limit.

● At any instant, the in-plane area (volume) of the element is constant; therefore, there is a specific plane or angle on Mohr’s circle, where a state of pure shear exists.

The deformation field resulting from simple shear loading can be described by the following equations, in both X and Z directions:

⎧x =+XZγ ⎨ (3.76)a, b ⎩zZ= where X and Z are the original coordinates, and x and z are the current coordinates. γ is the deformation applied to the moving nodes (top of the element in Figure 3-1).

The deformation gradient matrix and its determinant can be written as follows:

⎡⎤∂∂xx ⎢⎥∂∂XZ⎡1 γ ⎤ F ==⎢⎥⎢ ⎥ (3.77) ⎢⎥∂∂zz⎣01⎦ ⎣⎦⎢⎥∂∂XZ

-73- Chapter 3 – Numerical implementation and verification of large deformation formulation and

J ==F 1.0 (3.78)

Rigid body rotations and stretch tensors can be defined by the following equations, Liu and Hong [1999]:

⎡ cs⎤ R = ⎢ ⎥ (3.79) ⎣−sc⎦ ⎡cs⎤ ⎢ ⎥ U = 1+ s2 (3.80) ⎢s ⎥ ⎣⎢ c ⎦⎥ where the rigid body rotation angle is calculated from:

γ θ = Arctan() (3.81) 2 Various methods are used to obtain the solution to the simple shear problem as depicted in Figure 3-2. These methods are described below.

• One step solution

The total True strain in a global coordinate system can be determined using Equations(2.97), (3.79) and (3.80), resulting in:

T ⎡cssc+−+γ γ ⎤ ε ==ln[]VFR ln⎣⎦⎡⎤ = ln ⎢ ⎥ (3.82) ⎣ sc⎦ Therefore:

⎡1+ s2 ⎤ s ε = ln ⎢ ⎥ (3.83) ⎢ c ⎥ ⎣⎢ sc⎦⎥

-74- Chapter 3 – Numerical implementation and verification of large deformation formulation The logarithm of this matrix at any deformation, γ , is thus determined in one step, using the first 20 terms (for more accuracy) of the Taylor series expansion of the logarithmic function of a matrix obtained before (Equation (3.7)).

• Logarithmic stress rate

The only “exact” objective stress rate, known as the Logarithmic stress rate, is provided by Xiao et al. [1997], Lin [2002, 2003] and Liu and Hong [1999]. This measure is added to Figure 3-2 for comparative purposes.

• Eigenvalue-based (or exact) solution

Lin [2002, 2003] used eigenvalues to decompose the deformation gradient matrix into a diagonal one, pre- and post-multiplied by the matrix of eigenvectors. He then applied the logarithmic function to each term of the diagonal matrix and determined the True stress tensor. These results are also added to Figure 3-2.

• “Co-rotational stress formulation” method

The method proposed in this thesis is the Co-rotational Stress Formulation (CSF), which is implemented in the finite element code COMPRO. Values of the True stress tensor are determined from COMPRO and added to Figure 3-2, as well.

COMPRO employs a load-controlled approach and yields a coarse force-deformation curve at large deformations involving a softening behaviour. Nonetheless, the computed deformations are exact for the input loads. In order to overcome this problem and generate a continuous load- deformation curve, some minor changes have been made in COMPRO for this specific example to enable a deformation-controlled analysis. Because iteration is not possible in specifically-modified COMPRO, time steps are artificially chosen to be very small to avoid numerical errors.

-75- Chapter 3 – Numerical implementation and verification of large deformation formulation Results from all the above-mentioned methods agree very well. This reinforces the premise that the CSF method yields the exact solution, as mentioned earlier. It is important to emphasize that the proposed CSF method is accurate. Theoretically, its applicable range is unlimited. However, the applicable range of strains and deformations in COMPRO is numerically limited, even though it is much larger than many other commercial codes (e.g., ABAQUS and LS-DYNA). For comparison, the simple shear example is modeled using ABAQUS and LS-DYNA commercial codes, and the results are compared with those obtained using COMPRO. The results are presented in Figure 3-3 to Figure 3-5.

The ABAQUS finite element code considers non-linear geometry and large deformations employing the Jaumann stress rate algorithm (see section 4.4.1 of ABAQUS Theory Manual [2003] and ABAQUS Standard User’s Manual [2003] version 6.4). The restrictions that exist in this approach are listed below:

1. The material must be isotropic.

2. The constitutive equation must be hyperelastic with large strain or hypoelastic with small strain assumptions.

Clearly it is possible to go beyond the bounds of these limitations. However, there will be no guarantee concerning the accuracy of the results obtained from ABAQUS in such a case. The response predictions using ABAQUS are shown in Figure 3-3a, b. The oscillations observed in the normalized stress and force responses indicate the inability of ABAQUS to capture the true behaviour of the system at large strains. The same 2-D solid element is used in ABAQUS (version 6.4). It is further notable that even at early stages of the loading (i.e., when the strains are small), the sign of the normal stresses are opposite of the exact solution, further reinforcing the error in the solution and Jaumann stress rate method employed in ABAQUS.

The same example has been modelled in the commercial explicit finite element code, LS-DYNA (see LS-DYNA Manual [1997]). This code employs a hypoelastic

-76- Chapter 3 – Numerical implementation and verification of large deformation formulation constitutive equation, and uses the Jaumann and Truesdell stress rate formulations for isotropic and anisotropic materials, respectively. In order to demonstrate the capabilities of LS-DYNA, an isotropic material is modelled using both the isotropic and non-isotropic material models with parameters that theoretically should provide the same results.

The results of the normalized stresses and forces versus γ are depicted in Figure 3-4a, b. Predictions of LS-DYNA using a brick element (constraint in the out-of-plane direction) and the Jaumann stress rate appear to be oscillatory, as was the case for ABAQUS. Contrary to the predictions of ABAQUS, the signs of the normal stresses match those of the exact solution at the early stage of loading.

Predictions of LS-DYNA using the Truesdell stress rate show a hardening behaviour compared to that of the Jaumann stress rate predictions. Clearly the Truesdell stress rate predictions deviate from the expected hypoelastic-yielding behaviour. These results are summarized in Figure 3-5 and for normalized stresses and forces, respectively.

Furthermore, the normal stress component, σZ , has the wrong sign at the early stage of loading.

The behaviour of shear stress σ XZ in the global coordinate system in the simple shear example (shown in Figure 3-6) is of special interest, since it increases in the early stages of loading and then asymptotically approaches zero afterwards. The principal

stresses, σ1 and σ2 , have different signs and equal absolute values (see Figure 3-6). With increasing deformation, γ , the absolute values of principal stresses increase monotonically with a decreasing slope, which is an indication of hypoelastic yielding. Although the slope of the curves of principal stresses as a function of γ decreases, it remains positive at all times. As a result, the thermodynamic and energy conservation laws are not violated. It means that as deformation increases, which translates to positive external energy, all principal strains increase as well, which translates to positive internal energy (see Hill [1970, 1978]).

-77- Chapter 3 – Numerical implementation and verification of large deformation formulation

3.8.1.2 Uniaxial Loading To simulate a uniaxial loading case, a 2-D plane-strain unit square element is loaded in the X-direction, as shown in Figure 3-7. An additional assumption in this example is that of the Poisson’s ratio is considered to be zero. The material properties chosen for this analysis are summarized in Table 3- 2. Predictions of the behaviour from COMPRO and the exact one-step hyperelastic solution of Equation (2.96) are identical, and match the exact solution (based on logarithm of a scalar value since this example is a very simple 1-D case). All three methods assume a constant material stiffness matrix in both the Kirchhoff and the True stress domains and results are available for both cases. These predictions are shown in Figure 3-8.

In this example, with the Kirchhoff material stiffness constant, the Kirchhoff stress varies linearly with Logarithmic strain, as shown in Figure 3-9. Deriving the True stress from the Kirchhoff stress based on Equation (2.76) results in a curved relation between the True stress and the Logarithmic strain, as depicted in Figure 3-9. The True stress response shows a softening behaviour in tension and a hardening behaviour in compression versus the Logarithmic strain.

As seen in Figure 3-9, the slope of the True stress becomes negative at large tensile strain values, which is caused by the unrealistic zero Poisson’s ratio assumption. This assumption results in the maximum change in the volume and the largest value of deformation gradient determinant, J . Consequently, the True stress computed by dividing the Kirchhoff stress by J will be unrealistically small as strain increases. Although this example of a zero Poisson’s ratio is not practically valid, it is chosen due to the inherent ease of determining the exact solution by finding the logarithm of a scalar value.

The same example is modelled using LS-DYNA for both isotropic and anisotropic material models. The results are summarized in Figure 3-10 and Figure 3-11, respectively. While the anisotropic material model predicts an erroneous hardening

-78- Chapter 3 – Numerical implementation and verification of large deformation formulation behaviour, the isotropic material model shows a softening behaviour for the normalized axial stress (which is equivalent to the normalized force) versus the Nominal strain.

The same example is modelled using ABAQUS and COMPRO finite element codes. In COMPRO, the options of using constant Kirchhoff and True material stiffness matrices are exercised. All the results from LS-DYNA, ABAQUS and COMPRO are depicted in Figure 3-12. The prediction of LS-DYNA isotropic material model and the ABAQUS and COMPRO with constant True material stiffness matrix show perfect agreement.

3.8.1.3 Closed Loop Stress Path To perform this test, the deformation is applied to a unit 2-D plane-strain element in four phases, as shown in Figure 3-13. While all the initial stresses and strains are zero, they should ideally revert back to zero at the end of phase four, since the final deformation is also zero. Material properties used to set up this example are summarized in Table 3-2 (see Material #3). For this example, the CFS algorithm is implemented in MATLAB (see MATLAB & Simulink [2001] version 6.1). Both cases of constant Kirchhoff and True material stiffness matrices are considered and compared with the Logarithmic stress rate method by Lin et al. [2003]. Results (summarized in Figure 3-14a, b) show perfect agreement regardless of the stress measure employed, indicating that the CSF approach yield the exact solution.

3.8.2 Non‐linear Elastic Isotropic Materials

Non-linearity refers to the material stiffness matrix in the constitutive equation not being constant. As composite materials cure, material stiffness changes without affecting the already-existing strains and stresses. All changes in material stiffness, except those from temperature effects, are assumed to be irreversible.

For this case, only one example of uniaxial loading is considered. The same 2D plane strain unit element under axial loading is modelled using COMPRO, ABAQUS and LS- DYNA finite element codes. The one-step exact solution for a nonlinear material is

-79- Chapter 3 – Numerical implementation and verification of large deformation formulation based on Equation (2.100). Having a material with zero Poisson’s ratio results in a 1D state of strain, which simplifies the calculation of Equation (2.100). For simplicity, the tangential elastic modulus is defined as a function of deformation that goes from an original value to twice the initial modulus at 100% Nominal strain (εN = 1). That is:

ε EE/10 = +=εN e (3.84) Based on Equation (3.84) and tangent stiffness definition, the True stress can be determined as:

dσ E = (3.85) dε resulting in

ε dEdEedσε==0 ε (3.86) and

ε σ = Ee0 (1)− (3.87) The stress-strain relationship expressed in Equation (3.87) is depicted in Figure 3-15.

For this analysis, Material #3 in Table 3-2 with a zero Poisson’s ratio is used. All the predictions from the three different codes, plus that of the exact one-step solution (Equation (3.87)), are depicted in Figure 3-16. A perfect agreement is observed between the predictions of LS-DYNA, COMPRO and the exact one-step solution. Analysis using ABAQUS yielded similar results, although it did not continue beyond 20% Nominal strain because the large deformation option is not allowed with a user- defined non-hyperelastic constitutive equation.

3.8.3 Linear Elastic Transversely Isotropic Materials

This subcategory of non-isotropic materials is the focus of this section, since composite materials are usually transversely isotropic. For simplicity, a unit element is modelled

-80- Chapter 3 – Numerical implementation and verification of large deformation formulation in COMPRO with lumped material properties, as summarized in Table 3-3. The element and the original principal material directions of material properties are shown in Figure 3-17B.

3.8.3.1 Simple Shear For this example, COMPRO and ABAQUS finite element codes are considered. It was found that ABAQUS could not cope with the rotation of the anisotropic material stiffness matrix caused by excessive stretching. Therefore, an equivalent model consisting of two bars and one 2D plane-strain solid element with isotropic material properties, as shown in Figure 3-17a, was constructed. The bars act as reinforcements and therefore result in an orthotropic behaviour with the strong direction aligned with the direction one. The properties of bars and solid elements are summarized in Table 3-4, and as Material #6 in Table 3-2. Because this equivalent pseudo-2-D model is different from an actual 2-D model, it is expected that the results from this equivalent model in ABAQUS differ from those of COMPRO.

Normalized principal stresses in orthotropic materials (similar to isotropic materials) vary monotonically versus strain as depicted in Figure 3-18. Plots of the Normalized forces versus γ are shown in Figure 3-19. Predictions are relatively close to each other, showing similar trends. ABAQUS predicts the same values of forces in X and Z directions in nodes 3, while COMPRO predicts significantly different values of forces at these two nodes as shown in Figure 3-20. Therefore, forces shown in Figure 3-19 for the COMPRO run are the average of forces in nodes 3 and 4 in each direction. It is interesting that the forces at node 3 change directions. This performance can be seen in the 2-D solid element model due to the rigid body rotation of the element, which needs counteracting forces in the opposite direction of rotation (or movement) at node 3.

3.8.4 Thermal Loading

Temperature changes can generate deformations and stresses in undetermined structures. The most important source of loads in process modelling using COMPRO is

-81- Chapter 3 – Numerical implementation and verification of large deformation formulation thermal loading. Verification of large deformation implementation in COMPRO is crucial, and is continued in the following two examples. It should be noted that the coefficient of thermal expansion (and degree of cure) is assumed to be known in the Logarithmic strain domain.

3.8.4.1 Isotropic Element A single isotropic square element, as shown in Figure 3-21, is modelled in COMPRO to investigate thermal effects. It is assumed that the Poisson’s ratio is zero. This element is constrained in the Z -direction, while it is unconstrained in the X -direction. Material properties used for this example are summarized in Table 3-2 (see Material #4). The CTE is assigned a large value of 0.01/°C. As a result, a temperature change of +100°C can generate 100% thermal Logarithmic strain (equivalent to 173% Nominal strain) in the unconstrained direction. Results from COMPRO and the exact solution are summarized in Figure 3-22 for ±100°C change in temperature. The normalized True stress in the constrained Z -direction changes linearly from +1.0 to -1.0. The exact solution can be derived as:

σε/()()E = =∆CTE T (3.88)

3.8.4.2 Bi‐material Bar A bi-material bar, as shown in Figure 3-23, is modelled with both COMPRO and ABAQUS using twenty 2-D plane strain elements in the X -direction and four elements through the thickness. Material properties used in the analysis are provided in Table 3-2 (Material #3 and #4). Both materials have the same elastic modulus and a zero Poisson’s ratio. While the top material has a CTE of zero, the bottom one has a CTE of 0.01/°C.

Figure 3-24 shows the deformation of the tip of the bar in the X -direction, along with normalized True stresses versus temperature change. The analysis in ABAQUS terminates at about 120°C due to convergence problems, while that in COMPRO reaches the target temperature change of 200°C.

-82- Chapter 3 – Numerical implementation and verification of large deformation formulation Monitoring the X -direction stresses in two adjacent elements of different materials located at the tip of the bar (elements 78 and 79 in Figure 3-24) shows equal tensile (positive) stresses for both materials in ABAQUS run, which are physically not possible. As mentioned earlier, the predictions of ABAQUS lose credibility after the strains become large.

3.8.5 Response of a Cantilever Beam Undergoing Large Rotations

To further generalize the verification of the formulation developed in this study, a single cantilever beam under external pressure over half of its length is modelled using COMPRO and ABAQUS. This model, as shown in Figure 3-25, is comprised of one hundred 2-D plane strain elements along its 20 m length and four elements through its 0.8 m thickness. Mechanical properties of the beam material are presented in Table 3-2 as Material #3, with a Poisson’s ratio of zero. The external pressure applied to the beam increased linearly versus time as shown in Figure 3-26.

It should be noted that any results from ABAQUS at large strains, if present, may not be reliable. The dimensions of this model are chosen such that the convergence problems arising from the occurrence of large strains in the commercial finite element code, ABAQUS, are possibly avoided. Nonetheless, ABAQUS failed to complete the analysis for the entire time span, resulting in premature termination of the simulation at strain values of about 20%. The analysis carried out in COMPRO was stable for the entire time.

The termination time of the analysis was set at 1500 minutes, determined such that no interpenetration occurs in the elements during the deformation of the beam (beam does not reach and pass through supports). The deformed profiles of the top and bottom surfaces of the beam as predicted by COMPRO at certain time instances are depicted in Figure 3-27. Comparison of the beam’s deformed profile of the bottom surface predicted by COMPRO and ABAQUS are shown in Figure 3-28. These simulations are carried out using both 4- and 8-noded element options in ABAQUS and COMPRO. As

-83- Chapter 3 – Numerical implementation and verification of large deformation formulation can be seen for this figure, the predictions of ABAQUS using 4- and 8-noded elements are very close. This is the case because the 4-noded formulation in ABAQUS takes advantage of an extra shape function that improves the accuracy of the results. In COMPRO, however, the model using the 8-noded element shows a more flexible response that is more accurate than the 4-noded element, and is closer to the predictions of ABAQUS.

Further analyses were performed on the same model using COMPRO, without considering the large deformation option. Figure 3-29 shows a comparison of the beam tip displacement (at the middle surface) in X - and Z - directions for the runs with and without the consideration of large deformation option. In the case where large deformation option is ignored, no displacement in the X direction is predicted, while the displacement in the Z direction appears to increase linearly (in contrast to the fully non-linear behaviour observed in the tip displacements when the large deformation option is considered).

Values of stress in the global coordinate system are plotted in Figure 3-30 at 300 minutes for the elements located at the bottom of the loaded portion of the beam, considering 4- and 8-noded element options of ABAQUS and COMPRO, as well as the 4-noded COMPRO model with no large-deformation effects. Good agreement exists amongst the results shown, except for the 4-noded COMPRO predictions with no large- deformation consideration. It is rather intuitive that the 4-noded element option of COMPRO with no large deformation option is not a suitable candidate for this category of problems, where very large rotations are presented.

3.8.6 Applicable Range of Strains in Finite Element Codes

Consideration of large deformations in a finite element code can create artificial sensitivity to element distortion. This is the case because shear strain is the source of distortion in an element in a finite element model. Finite element codes with large deformation options become highly responsive to the amount of shear strains in the

-84- Chapter 3 – Numerical implementation and verification of large deformation formulation elements. To further explain this point, a 2-D plane-strain square of unit side-length that is heavily constrained (as shown in Figure 3-31) is considered under a pre-defined temperature change. This square is meshed uniformly using 1, 4 and 16 elements, and modelled using COMPRO. Material properties used for this analysis are provided in Table 3-2 as Material #4, which has an elastic modulus of 10 GPa, a Poisson’s ratio of zero and a CTE of 0.01/°C. The applied temperature is set to increase linearly with time at a slope of 1 °C/min for 1000 minutes.

Predictions of normalized stresses at the integration point marked by the letter “O” are plotted versus the displacement of the top right corner node of the model, as shown in Figure 3-32 for the 1-, 4- and 16-element models. Normalizing the stress values relative

to the shear modulus, G , implies that the normalised shear stress component σxz is indeed no different than the shear strain γXZ , since γXZ = σ XZ / G . It can be concluded from the above analysis that increasing the number of elements in the model leads to drastically capturing higher strain values at much lower loadings.

Performing the above simulations, the 1-element model showed stability throughout the analysis, even when taken far beyond the applicable engineering limits (temperature change of 1000°C, a displacement of ten million units and 1000% shear strain). The 4- element model, however, was interrupted earlier due to instability at 99°C, showing a displacement of 1.2 units and a shear strain of 250%. The behaviour resulting from reaching a higher shear strain value at a lower temperature as a function of mesh size can be clearly observed when moving to the 16-element model. Shear strain of about 200% is observed at a 50°C temperature change, along with a displacement of 0.5 units at the top right corner node. It can thus be concluded that conservatively, the applicable range of the CSF method, based on an average mesh size, is for the maximum shear strain values of 100%.

-85- Chapter 3 – Numerical implementation and verification of large deformation formulation

3.9 SUMMARY AND DISCUSSION

Many finite element commercial codes provide options to consider non-linear geometry and large deformation in the analysis of physical systems. However, most of them are bound by severe limitations in their range of applicability.

The commercial code ABAQUS, for instance, uses only hyperelastic isotropic materials for the large deformation option. As a result, the hypoelastic material model and its constitutive equation cannot be implemented in ABAQUS through a “User Material Subroutine.” ABAQUS employs a Jaumann stress rate, which is an approximation of the Truesdell stress rate given a small strain assumption. On the other hand, the simplifying assumption in the material stiffness matrix definition of the Truesdell stress rate approach itself results in a non-exact method. The errors caused by these two assumptions in the Jaumann stress rate approach appear to cancel each other in the strain range below 20%. While ABAQUS cannot cope with anisotropic materials, its predictions improve when shear stresses are small.

LS-DYNA is a robust commercial code that can accommodate a hypoelastic constitutive model. LS-DYNA employs the Jaumann stress rate for isotropic materials and the Truesdell stress rate for anisotropic materials. Results from LS-DYNA are more accurate for isotropic materials when shear strain values are small. For instance, in the uniaxial loading example, LS-DYNA provides predictions that agree well with the exact solution for isotropic materials. With the increase in the shear strain values, the predictions of LS-DYNA become more erroneous. For anisotropic materials, LS- DYNA results are inaccurate regardless of the level of shear strain in the system.

In both the above mentioned codes, strain level should remain small. Otherwise, numerical problems – such as oscillation of stresses and forces in the Jaumann stress rate, and hardening behaviour in the Truesdell stress rate – would arise. These commercial codes are also unable to accurately model anisotropic materials undergoing large deformations.

-86- Chapter 3 – Numerical implementation and verification of large deformation formulation COMPRO with the newly implemented “Co-rotational Stress Formulation” has all the requirements for isotropic and anisotropic materials to successfully deal with the large deformations/strains cases. The new COMPRO copes with all issues arising from large deformations such as the rotation of the material stiffness matrix due to stretching in anisotropic materials.

COMPRO can practically be used to analyse systems undergoing large deformation/strain up to about 100% shear strain values for both isotropic and anisotropic materials.

One of the basic assumptions made in objective stress rates is the assumption of small strains and deformations. However, this assumption may not be valid for the range of displacements that may occur under circumstances involving specialized materials and/or structures. As a result, the prediction of the system’s behaviour deviates from reality and, in such cases, significant inaccuracies may occur in the simulations.

-87- Chapter 3 – Numerical implementation and verification of large deformation formulation

Table 3-1: Logarithm function of a symmetric matrix calculated using the decomposition (Jirasek and Bazant [2001]) method and the Taylor series expansion method in the case of uniaxial loading of an element with Poisson’s ratio equal to zero.

ln(U) Taylor series expansion method (# of terms used) Decomposition method (Equation (3.2)) Analytical Strain No.=1 Error No.=2 Error No.=3 Error I = 1 Error I = 2 Error I = 4 Error Solution -90% -1.6364 -28.9% -2.0015 -13.1% -2.1482 -6.7% -4.9500 115.0% -2.8460 23.6% -2.4319 5.6% -2.3024 -80% -1.3333 -17.% -1.5309 -4.9% -1.5835 -1.6% -2.4000 49.1% -1.7889 11.2% -1.6532 2.7% -1.6094 -70% -1.0769 -10.6% -1.1810 -1.9% -1.1991 -0.4% -1.5167 26.0% -1.2780 6.1% -1.2222 1.5% -1.2040 -60% -0.8571 -6.5% -0.9096 -0.7% -0.9154 -0.1% -1.0500 14.6% -0.9487 3.5% -0.9243 0.9% -0.9163 -50% -0.6667 -3.8% -0.6914 -0.2% -0.6930 0.0% -0.7500 8.2% -0.7071 2.0% -0.6966 0.5% -0.6931 -40% -0.5000 -2.1% -0.5104 -0.1% -0.5108 0.0% -0.5333 4.4% -0.5164 1.1% -0.5122 0.3% -0.5108 -30% -0.3529 -1.1% -0.3566 0.0% -0.3567 0.0% -0.3643 2.1% -0.3586 0.5% -0.3571 0.1% -0.3567 -20% -0.2222 -0.4% -0.2231 0.0% -0.2231 0.0% -0.2250 0.9% -0.2236 0.2% -0.2233 0.1% -0.2231 -10% -0.1053 -0.1% -0.1054 0.0% -0.1054 0.0% -0.1056 0.2% -0.1054 0.0% -0.1054 0.0% -0.1054 10% 0.0952 -0.1% 0.0953 0.0% 0.0953 0.0% 0.0955 0.2% 0.0953 0.0% 0.0953 0.0% 0.0953 20% 0.1818 -0.3% 0.1823 0.0% 0.1823 0.0% 0.1833 0.5% 0.1826 0.2% 0.1824 0.1% 0.1823 30% 0.2609 -0.6% 0.2623 0.0% 0.2624 0.0% 0.2654 1.1% 0.2631 0.3% 0.2626 0.1% 0.2624 40% 0.3333 -1.0% 0.3364 0.0% 0.3365 0.0% 0.3429 1.9% 0.3381 0.5% 0.3369 0.1% 0.3365 50% 0.4000 -1.4% 0.4053 0.0% 0.4055 0.0% 0.4167 2.8% 0.4082 0.7% 0.4062 0.2% 0.4055 60% 0.4615 -1.8% 0.4697 -0.1% 0.4700 0.0% 0.4875 3.7% 0.4743 0.9% 0.4711 0.2% 0.4700 70% 0.5185 -2.3% 0.5301 -0.1% 0.5306 0.0% 0.5559 4.8% 0.5369 1.2% 0.5322 0.3% 0.5306 80% 0.5714 -2.8% 0.5870 -0.1% 0.5877 0.0% 0.6222 5.9% 0.5963 1.4% 0.5899 0.4% 0.5878 90% 0.6207 -3.3% 0.6406 -0.2% 0.6418 0.0% 0.6868 7.0% 0.6529 1.7% 0.6446 0.4% 0.6419 100% 0.6667 -3.8% 0.6914 -0.2% 0.6930 0.0% 0.7500 8.2% 0.7071 2.0% 0.6966 0.5% 0.6931 150% 0.8571 -6.5% 0.9096 -0.7% 0.9154 -0.1% 1.0500 14.6% 0.9487 3.5% 0.9243 0.9% 0.9163 200% 1.0000 -9.0% 1.0833 -1.4% 1.0958 -0.3% 1.3333 21.4% 1.1547 5.1% 1.1125 1.3% 1.0986 300% 1.2000 -13.4% 1.3440 -3.1% 1.3751 -0.8% 1.8750 35.3% 1.5000 8.2% 1.4142 2.0% 1.3863 400% 1.3333 -17.2% 1.5309 -4.9% 1.5835 -1.6% 2.4000 49.1% 1.7889 11.2% 1.6532 2.7% 1.6094

Table 3-2: Different isotropic material properties used for verification examples.

Material #1 Material #3 Material #4 Material #6 Elastic modulus = E 10 MPa 10 GPa 10 GPa 10 kPa Poisson’s ratio = υ 0 0 0 0 CTE 0 0 0.01 m/(°C·m) 0

Table 3-3: Orthotropic material properties used for verification example.

E1 E2 = E3 υ12 = υ13 = υ23 100 GPa 10 kPa 0.0

Table 3-4: Material properties of bars. Axial stiffness = E Poisson’s ratio 50 GPa 0.5

-88- Chapter 3 – Numerical implementation and verification of large deformation formulation

γ γ γ/2 θ L0 =1

Z L0 =1 X

Figure 3-1: A unit square under rail (simple) shear.

8 γ γ Analytical Solution γ/2 θ µ One-Step Solution L0 =1 )/ )/ Logarithmic Stress Rate Z Xz σ 6 , COMPRO X z L0 =1 σ -

, σ σ X σ , -σ 2 X, - Z σ X Z 4

1 σXZ

2 0 σXZ 024 Normalized stress ( Normalized 0 0 10203040 γ/L (m/m) 0

Figure 3-2: Normalized True normal and shear stresses for simple shear example involving an isotropic material. Comparison of: A) One-step solution, based on 22 terms of Taylor series of a logarithm of a matrix, B) COMPRO, C) Logarithmic stress rate based on papers by Xiao et al. [1997], Lin [2002, 2003] and Liu and Hong [1999], D) Eigenvalue-based (exact analytical) solution based on paper by Lin [2002, 2003]. µ is the Lame elastic constant.

-89- Chapter 3 – Numerical implementation and verification of large deformation formulation

2.2 )

µ σ / σ Z 1 σ 1.1

σ XZ 0 σ X -1.1 σ 2 Normalized stress ( Normalized -2.2 0 2.5 5 7.5 10

(a) γ γ γ/2 4 3 θ L0 =1 5.4 1 2 ) ψ Z3 /EA Z L0 =1 ψ 2.7 X ψ X3 0

ψ -2.7 X4 ψ

Normalized force ( Z4 -5.4 02.557.510

(b) γ/L0 (m/m)

Figure 3-3: Simple shear example in ABAQUS for an isotropic material using a 2D plane strain element and the Jaumann stress rate. (a) All normalized normal and shear stresses vary in a sinusoidal manner with respect to normalized deformation, γ, (even principal stresses follow a sinusoidal shape). (b) All normalized x- and z-direction forces at nodes 3 and 4 show oscillation. µ is the Lame elastic constant.

-90- Chapter 3 – Numerical implementation and verification of large deformation formulation

2.2

) σ X µ /

σ 1.1 σ XZ 0

-1.1 σ Z

Normalized stress ( Normalized -2.2 02.557.510 (a) γ γ γ/2 4 3 θ L0 =1 5.2 1 2 ) ψ X4 /EA Z L0 =1 ψ 0 X

-5.2 ψ Z4

Normalized force ( -10.4 02.557.510 (b) γ/L0 (m/m)

Figure 3-4: Simple shear example in LS-DYNA for an isotropic material using a constrained 2D element and the isotropic elastic material model (material type 1- MAT1) with Jaumann stress rate. (a) All normalized normal and shear stresses behave sinusoidally. (b) Normalized x- and z-direction forces at node 4 are shown to oscillate. µ is the Lame elastic constant.

-91- Chapter 3 – Numerical implementation and verification of large deformation formulation

1000 ) µ / σ 750 σ X σ XZ 500

250

Normalized stress ( Normalized σ Z 0 02.557.510 (a)

5000

) γ γ γ/2 4000 /EA 4 3 θ ψ L0 =1 1 2 3000 ψ X4 Z L0 =1 2000 X

1000 ψ

Normalized force ( Z4 0 02.557.510 (b) γ/L (m/m) 0

Figure 3-5: Simple shear example in LS-DYNA for an isotropic material using a non- isotropic elastic material model (material type 2- MAT2) and the Truesdell stress rate.(a) All normalized normal and shear stresses exhibit a hardening behaviour. (b) All normalized x- and z-direction forces at node 4 exhibit a hardening behaviour as well. µ is the Lame elastic constant.

-92- Chapter 3 – Numerical implementation and verification of large deformation formulation

8

/µ) 6 σ γ γ γ/2 θ L0 =1 4 σ1, -σ2

σx, -σz Z L0 =1 2

Normalized stress ( X

σxz 0 0 10203040 γ/L0 (m/m)

Figure 3-6: Simple shear in isotropic materials. Principal stresses increase monotonically. µ is the Lame elastic constant.

p Z L0=1 u X L =1 0

Figure 3-7: A unit square under uniaxial loading.

-93- Chapter 3 – Numerical implementation and verification of large deformation formulation

1.5 Analytical Solution )/E

X One-Step Solution (Equation (2.96)) τ τ X ,

X COMPRO σ σ 0.5 X

-0.5 0.5 1.5 2.5 3.5 -0.5 Z p

X u

Normalized stress ( stress Normalized -1.5 u/L (m/m) 0

Figure 3-8: Normalized True and Kirchhoff stresses versus normalized displacement for an isotropic material with Poisson’s ratio of zero undergoing uniaxial loading. Comparison of exact solution (based on logarithm of a scalar value), one-step solution (hyperelastic solution Equation (2.96)) and COMPRO.

2 τ X Kirchhoff )/E C = Constant X σ

τ X ,

X 0 σ -2 -1 0 1 2

-2 p

-4 u Z

Normalized stressNormalized ( -6 X ε =ln((L +u)/L ) 0 0

Figure 3-9: Stress versus logarithmic strain curves for an uniaxial loading case showing both the Kirchhoff and True stresses as a function of the logarithmic strain when the Kirchhoff material stiffness, C Kirchhoff , is constant.

-94- Chapter 3 – Numerical implementation and verification of large deformation formulation )

µ σ / X X 4 σ

Z L =1 2 0 p u X L0=1 Normalized stress ( Normalized 0 0369 u/L0 (m/m)

Figure 3-10: Uniaxial loading of an isotropic material in LS-DYNA using an isotropic material model. µ is the Lame elastic constant.

1000 ) µ / X σ 750 Z L0=1 p u X L0=1 500 σ X

250 Normalized stressNormalized (

0 0369

u/L0 (m/m)

Figure 3-11: Uniaxial loading of an isotropic material in LS-DYNA using a non- isotropic material model. µ is the Lame elastic constant.

-95- Chapter 3 – Numerical implementation and verification of large deformation formulation

LS-DYNA, non-isotropic material model

ABAQUS and LS-DYNA, /E) COMPRO, X Isotropic material model True σ 1.5 C = Constant

Z L0=1 p

L =1 u 0 X 0 -2 0 2 4 6 8 10 Normalized stress ( Normalized COMPRO, CKirchhoff = Constant -1.5

u/L0 (m/m)

Figure 3-12: Comparisons among COMPRO, ABAQUS and LS-DYNA for an isotropic material in a uniaxial loading case.

Z

2 S 1

um u 0 4 3

X

Figure 3-13: A unit square under a closed deformation path, 1 to 4.

-96- Chapter 3 – Numerical implementation and verification of large deformation formulation

0.8 /E)

τ Z 2 S 1 u τ 0 3 0.6 z 4

τx 0.4 X

0.2 τy τ 0 xz

Logarithmic Stress Rate COMPRO Algorithm -0.2 Normalized Kirchhoffstress components ( 01234 (a) /E)

σ 0.4 σz

σx 0.2

σy σxz 0

Logarithmic Stress Rate COMPRO Algorithm

Normalized True stress components ( -0.2 01234 Phase (b)

Figure 3-14: (a) Kirchhoff and (b) True stress components due to a closed deformation path applied to a unit square. Figures compare the predictions of the Logarithmic stress rate method based on the analytical solution of Lin et al. [2003] and the current Co-rotational stress formulation implemented in COMPRO.

-97- Chapter 3 – Numerical implementation and verification of large deformation formulation

) 1.0 0 /E X 2E0 /E0 σ 1.0

0.5

E /E 0.0 0 0 Normalized stress ( Normalized 1.0 0 0.2 0.4 0.6 ε (m/m)

Figure 3-15: Normalized True stress versus Logarithmic strain curve: Elastic modulus increases from its initial value to twice that value at 100% Nominal strain or about 69.3% Logarithmic strain.

)

0 1.0 A 0

/E p ψ Z 0.5 u COMPRO X LS-DYNA ABAQUS+CHILE Model Analytical Solution

Normalized force ( 0.0 0.0 0.3 0.5 0.8 1.0 u/L (m/m) 0

Figure 3-16: Normalized force versus displacement for a uniaxial loading case where the material follows the stress-strain curve above in Figure 3-15. This figure compares the predictions of ABAQUS, LS-DYNA and COMPRO and exact analytical solutions. ABAQUS results do not converge beyond the small strain range, while COMPRO and LS-DYNA predictions coincide.

-98- Chapter 3 – Numerical implementation and verification of large deformation formulation

γ γ γ γ Z 1 L0=1 Bars X 2

L0=1

(a) (b)

Figure 3-17: Simple shear of an orthotropic material. (a) Equivalent model in ABAQUS consists of an isotropic solid element and 2 vertical bars. (b) COMPRO model using lumped orthotropic material properties.

7 ) µ

/ γ γ γ/2 X 5 σ θ L0 =1

3 Z σ1 L0 =1 σx X 1 σ Normalized stress ( Normalized σz 2 σxz -1 010203040 γ/L (m/m) 0

Figure 3-18: Simple shear in orthotropic materials. Principal stresses increase monotonically while global stresses vary non- monotonically versus strain. µ is the Lame elastic constant.

-99- Chapter 3 – Numerical implementation and verification of large deformation formulation

0.3 ψz COMPRO ψx COMPRO ) /E

ψ 0.2 ψx ABAQUS

0.1 ψz ABAQUS Normalized force (

02.557.510

-0.1

0.01 ) γ γ γ/2 /E ψz COMPRO ψ 4 3 L0 =1 θ 2 1 ψz ABAQUS 0.005 Z L =1 ψx COMPRO 0 X

ABAQUS

Normalized force ( ψ 0 x 0 0.05 0.1 0.15 0.2 γ/L (m/m) 0

Figure 3-19: Simple shear of an orthotropic material. Force components predicted by COMPRO and the equivalent model in ABAQUS.

-100- Chapter 3 – Numerical implementation and verification of large deformation formulation

0.2 ψ ABAQUS ψx 4 COMPRO x γ γ γ/2 4 3 θ ) L0 =1

/E 2 1 ψ COMPRO 0.1 ψz 4 Z L0 =1 ψz ABAQUS X

0.0

Normalized force ( ψz 3 COMPRO ψx 3 COMPRO -0.1 0 2.5 5 7.5 10 γ/L (m/m) 0

Figure 3-20: Simple shear of an orthotropic material. x- and z-direction forces at node 3 and 4 in COMPRO run compared to x- and z-direction forces in node 3 (or 4) in ABAQUS run.

u

L0=1 Z u X L =1 0

Figure 3-21: The isotropic unit element with zero Poisson’s ratio used for verification of COMPRO large strain capability under thermal loading.

-101- Chapter 3 – Numerical implementation and verification of large deformation formulation

1 1.8 /E) z σ 1.2 0.5 σz u 0.6

0 0 (mm) 0

-0.6 u/L -0.5 Z u -1.2 X u

Normalized True stress ( -1 -1.8 -100 -50 0 50 100 ∆T (°C)

Figure 3-22: Comparison of COMPRO predictions (circle and square legends) with analytical solution, Equation 3.88 (solid and dash lines) for thermal loading of the element showing in Figure 3-21.

Material 2 h = 0.8 mm Material 1

L = 20 mm 0

Figure 3-23: A bimaterial bar: Poisson’s ratio = 0, Elastic modulus = 1.0E+10 Pa for both materials while CTE = 0 for material 2 (top) and CTE = 0.01 for material 1 (bottom).

-102- Chapter 3 – Numerical implementation and verification of large deformation formulation

1.25

/E) σ

σ x 79 ABAQUS COMPRO 0.75 78 σx

Element 79 Material 2 0.25 σxz

Normalized True stress ( stress True Normalized σ ABAQUS z -0.25 0 50 100 150 200

1.25 5 ) u 0 /E) O σ 4 R P M O 0.75 C σx Element 78 3 Material 1 ABAQUS

2 0.25 US AQ COMPRO AB σz 1 σ Normalized True stress ( stress True Normalized xz σx -0.25 0 (u/L displacement Tip Normalized 0 50 100 150 200 ∆T(°C)

Figure 3-24: Comparison of COMPRO and ABAQUS predicted average normalized stresses and tip deformation versus temperature change for the bimaterial bar shown in Figure 3-23.

-103- Chapter 3 – Numerical implementation and verification of large deformation formulation

p Material 3 h =0.8 m

L/2 =10 m L/2 = 10 m

Figure 3-25: A cantilever beam under pressure loading over half of its length with 400 (=100 x 4) elements.

30

20

20 kPa/min (MPa)

p 10

0 0 500 1000 1500 t (min)

Figure 3-26: Pressure loading on the cantilever beam shown in Figure 3-25. Pressure increases linearly versus time.

-104- Chapter 3 – Numerical implementation and verification of large deformation formulation

p t=1500 min 10

L/2 L/2

t=1000 min 5 Z

X t=0 min 0 -15 -10 -5 0 5 10 15 20 t=600 min -5

Deformation (m) Deformation t=500 min t=50 min -10

t=100 min -15 t=200 min t=300 min

-20 Length (m)

Figure 3-27: Deformation of the bottom and top surfaces of the cantilever beam shown in Figure 3-25 and subjected to the pressure loading of Figure 3-26 as predicted by COMPRO at different times.

-105- Chapter 3 – Numerical implementation and verification of large deformation formulation

t=1500 min 10 p

t=1000 min 5 L/2 L/2

t=600 min t=0 min 0 -15 -10 -5 0 5 10 15 20

-5 t=50 min

t=500 min Deformation (m) Deformation

-10 t=100 min t=200 min

COMPRO 4-Noded -15 COMPRO 8-Noded t=300 min ABAQUS 4-Noded ABAQUS 8-Noded -20 Length (m)

Figure 3-28: Comparison of the ABAQUS and COMPRO predicted deformation profile of the bottom surface of the cantilever beam of Figure 3-25 subjected to the pressure loading shown in Figure 3-26. The results obtained using both 4- and 8-noded elements and compared.

-106- Chapter 3 – Numerical implementation and verification of large deformation formulation

5 Z dir. with no

(m) large deformation -5

-15 Z dir.

-25 X dir. X dir. with no Displacement -35 large deformation 0 500 1000 1500 t (min)

Figure 3-29: Centre line tip displacement (both in x- and z-directions) of the cantilever beam of Figure 3-25 subjected to pressure loading shown in Figure 3-26 using COMPRO with and without Large deformation option.

-107- Chapter 3 – Numerical implementation and verification of large deformation formulation

600 σ XZ

σ XZ -400 σ Z (MPa)

σ COMPRO 4-noded -1400 COMPRO 8-noded σ X σ X ABAQUS 4-noded ABAQUS 8-noded COMPRO 4-noded, no LD option -2400 0510 Length (m)

Figure 3-30: True stresses at first integration points of the bottom row elements in the loaded portion of the cantilever beam of Figure 3-25 at time equal to 300 minutes. Comparisons are made between the predictions of COMPRO (4- and 8-noded elements) and ABAQUS (4- and 8-noded elements). Also shown are the predictions of COMPRO 4-noded elements without large deformation option.

-108- Chapter 3 – Numerical implementation and verification of large deformation formulation

u x x

L0 x o

u

L0 x x x o

u

L0

x x x o

L 0

Figure 3-31: Heavily constrained unit length 1-, 4- and 16-elements cases.

-109- Chapter 3 – Numerical implementation and verification of large deformation formulation

10 σz τxz,γxz 0 2 u x x 0

-10 x o -2 -4 σx -20 σy 048 0246810 millions /G) 10 σ

5 τxz,γxz 0 σy u σ -5 z x x x o σx -10 Normalized True stress ( 00.40.81.2

10

5 τxz,γxz 0 σy u σz -5

x x σ x o x -10 0 0.2 0.4 0.6 Normalized deformation (u/L ) 0

Figure 3-32: Normalized stresses, in 3rd integration points of right bottom corner element.

-110- Chapter 4 – Contact formulation

CHAPTER 4 ‐ CONTACT FORMULATION

The need for a contact interface feature in processing modelling of composites was discussed in sections 1.3. This chapter is devoted to the development of appropriate constitutive relations that represent contact between the composite part and the tool during processing. In other words, a frictional contact surface is introduced as the replacement for the shear layer interface currently used in COMPRO to model tool-part interaction. This is then followed by the numerical implementation of the contact in the process modelling code, COMPRO.

4.1 BACKGROUND

Due to its inherent non-linear behaviour, contact is one of the most difficult problems in mechanics to simulate. Leonardo Da Vinci and Coulomb were among the first scientists to work on contact problems. Non-linearities in mechanics are classified into three categories: material, geometry and boundary conditions. Contact falls under the boundary conditions category.

The numerical treatment of contact problems includes the formulation of the geometry, the interface constitutive laws, and the development of algorithms. The interface laws include tangential and normal stress-strain (force-deformation) relationships. Owing to the vast application of contact problems, those relationships have often been combined

-111- Chapter 4 – Contact formulation with large elastic or inelastic deformations. Two distinct categories of contact methods are penetration and non-penetration methods. The Penalty method and the Lagrange Multiplier method are the two numerical approaches employed to solve these categories of problems, respectively.

As Tworzydlo et al. [1998] stated “the phenomena of contact and the friction of solid bodies are among the most complex and difficult to model of all mechanical events”. They included the complex structure of engineering surfaces, the severe elasto-plastic deformation, damage, heat generation and atomic-range interactions as their reasons for that conclusion. Moreover, they stated that in order to achieve better insight, an asperity-based model should be used. That model is based on micro-scale analysis of deformation and the relative sliding of surface asperities. Based on the chosen level of model complexity of the asperity, visco-elasto-plastic, hyperelastic or brittle materials can be used, damage evolution at the surface can be modelled, and effects of lubrication and surface contamination can be.

Zavarise et al. [1998] suggested that solving the contact problem is related to a special modification of the unconstrained functional to include inequality constraints. In the case of the penalty method, a wrong choice of its parameter could lead to poor conditioning in the global stiffness matrix or a non-acceptable penetration. However, its implementation into an existing code is simple. The Lagrange multiplier method forces the constraint to be satisfied, but increases the number of unknowns. Also, continuous change of the total number of equations occurs when a non-linear contact problem is modelled. Moreover, the diagonal terms of the global stiffness matrix due to contact constraints can be zero. Both methods are characterized by two opposite states: when the gap is open and contact constraints are inactive, and vice versa. Therefore, their algorithm should account for both states. Now knowing some advantages and disadvantages of penalty and Lagrange multiplier methods, Wriggers [2002] explained that if the mesh density increases, the penalty method becomes more stable and numerical ill conditioning is less likely to occur.

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Ling and Stolarski [1997] suggested that the Coulomb friction law could be used to solve various contact problems, but its effectiveness is still arguable. A general form of the Coulomb friction law is given by:

τ A = c + µσ (4.1)

where τ A , c , µ and σ are the interfacial shear strength, adhesion between two contact surfaces, coefficient of Coulomb friction and normal stress at the interface.

They proposed a number of regularized friction laws to make the Coulomb friction law smoother as depicted in Figure 4-1 in order to avoid the numerical complexities involved in stick-slip conditions.

By analysing the stress distribution under a wheel in a wheel-soil interaction, Hiroma et al. [1997] suggested that there is adhesion between the wheel and the soil in addition to the friction. Tworzydlo et al. [1998] explained that adhesion forces could be very large for highly-polished uncontaminated surfaces. Due to this condition, virtual welding can occur between two surfaces. Adhesion forces and its effects are negligible for contaminated surfaces. For a rough engineering surface under common working conditions, adhesion effects can still be significant. Those effects should be taken into account in the case of rational models.

Wriggers [1996] states that the key in applying the frictional model, which is similar to an elasto-plastic formulation, is to separate the tangential deformation into an elastic stick section and a plastic slip section. Wriggers [1996] suggested that “for the general case of contact including large deformations, arbitrary sliding of a node over the entire contact area has to be allowed”. This issue can be captured by the node-to-segment approach.

Ling and Stolarski [1997] stated that, for the majority of contact problems, contact areas are unknown prior to the analysis. The contact problems are thus nonlinear since the boundary conditions continuously change during the analysis and must be

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determined at each time step. They mentioned that if both contact bodies deform then considerable changes in the contact area including relative sliding and separation can occur.

Wriggers and Scherf [1998] suggested that a high gradient of stress can take place at the contact interface, which in turn can lead the modeller to refine the mesh in those areas and thereby perform a more accurate analysis. They recommended that, because numerical models of contact problems give approximate solutions, a method to control the error is needed, especially in the presence of friction. They introduced an adaptive technique to find an optimal mesh, one that is very fine at the interface and coarse in other areas. They noted that both bodies in contact can have arbitrary constitutive laws, which do not affect the contact interface algorithm.

Wriggers et al. [2001] stated that in a contact surface discretized by low-order finite elements (e.g., 4-noded 2-D solid elements), surface normals change suddenly at common nodes. This may lead to convergence problems. They asserted that “a smooth deformed surface with no slope discontinuities between segments is obtainable by having a C1-continuous interpolation at the master surface”.

Wriggers [2002] explained that there are three approaches to model contact between slave and master bodies: slave node to master node, slave node to master segment, and slave node to a defined smooth surface on the master body.

4.2 DEVELOPMENT OF CONSTITUTIVE EQUATION

Consider the system shown in Figure 4-2 which has a point mass m, connected to a spring with stiffness K. Due to a rigid surface or an impenetrable surface, the maximum possible deformation of mass m is limited to the original distance, h, between the mass and the rigid surface. This kind of contact surface is called a unilateral contact surface. Minimum potential energy for this system without considering the impenetrable surface provides the maximum deformation for the mass and is given by:

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1 Π=−+()umguKu2 (4.2) min 2

δΠ=−+min ()umguKuumgKuuδδ =−+= ( ) δ 0 (4.3) mg u = (4.4) max K where u is the displacement of mass. The above energy equation is only valid if the maximum deformation is less than h, otherwise a contact occurs and the above equation becomes invalid. When the contact occurs (see Figure 4-2), an extra term should be added to the above energy equation. Definition of this extra term depends on the method used for contact formulation. For instance, in the Lagrange multiplier method, this extra term is derived from the following constraint equation:

cu()= h−≥ u 0 (4.5) Contact⇒= c() u 0, f ≤ 0 ⎪⎧ ⎨ ⇒⋅fcu() = 0 (4.6) Gap⇒> c() u 0, f = 0 ⎩⎪ where f is the contact force. Now, the potential energy equation becomes:

1 Π=−++−(,uf ) mgu Ku2 fh ( u ) (4.7) min 2 This extra term increases the number of unknowns in the system of equations from one unknown, u , to two unknowns, u and f .

In the penalty method, the impenetrable surface is replaced with a spring as shown in Figure 4-3. The stiffness of this spring, κ , (in compression) must be as high as numerically possible to minimize penetration, while the stiffness should ideally be zero in tension (if there is no adhesion). The extra term for the above energy equation is given by:

1 κ()uh− 2 (4.8) 2 resulting in the energy equation:

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11 Π=−++−()umguKuuh22κ ( ) (4.9) min 22 and its minimization:

δΠ=−++−=min ()umgKuuhu (κδ ( )) 0 (4.10)

−mg++−= Kuκ uκ h 0 (4.11) Because of the impenetrable surface, there are two extra terms in Equation (4.11). The first term,κ ⋅u , is similar to the term Ku⋅ of the main element (spring). The second term, −⋅κ h , derives from the fact that part of the deformation of the mass was filling the gap h. If the gap is filled in the next increment, this term no longer exists and should be manually deducted from the force vector at each time step. In the finite element implementation of the contact elements (springs), the same algorithm will be used. Since κ is a known value, the number of unknowns is unchanged and equal to one.

In the above example, we explained the behaviour of a frictionless contact surface, called a 1-D contact surface, where the contact occurs only in the normal direction. In a 2-D or 3-D contact surface, in addition to the contact in the normal direction, contact in the tangential direction should be modelled. In the curing process of a composite structure, stick and slip conditions with the combination of friction is the ideal model for the tangential behaviour. Tangential springs will be used to model this behaviour. One major advantage of the contact surface is its zero interface thickness, which means all springs have zero initial lengths.

4.2.1 Normal Spring

One contact element, which relates one slave node to two (in 4-noded element option) or three (in 8-noded element option) master nodes, consists of two sets of springs, one parallel and one normal to the interface. A normal spring is introduced at the interface between the part and the tool to prevent penetration of the two surfaces in contact with each other. Normal springs should have the maximum stiffness possible to minimize

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the penetration. Also, having the same stiffness for the normal spring elements as for other elements at the interface reduces the risk of encountering numerical problems. The stiffness of the normal spring can be determined from the equation below based on the assumption of having the same deformation, δ , due to a force, F, applied to a spring or a resin with thickness of H:

⎧ FH ⎪ ⎪ ELr (1) δ = ⎨ (4.12) ⎪ F ⎪ ⎩ KLN (1) If we assume that the element height, H, in the composite region is about 0.2 mm, as shown in Figure 4-4 and the out-of-plane dimension is taken to be unity, the above equation can be rewritten as:

E KE=≅r 5000 (/Nm3 ) (4.13) NrH Since the resin modulus evolves as the resin cures, the stiffness of the normal spring is varying too. The high autoclave pressure, p, can also cause penetration of the two contact surfaces. Numerical experiments for a variety of example problems have shown that to prevent penetration due to autoclave pressure, the stiffness of the normal spring should satisfy the following condition:

45 3 KMaxpxN ≥ (10 , 5 10 ) ( Nm / ) (4.14) The constitutive equation of a normal spring is depicted in Figure 4-5. In compression

the stiffness of the spring is K N , but in tension there are two zones where stiffness changes according to the following equations:

0 ≤<δδ ⇒KK = ⎪⎧ AN ⎨ (4.15) δδ≥⇒=KK/104 ⎩⎪ AN

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whereδ A is the allowable gap within which tensile stress is permitted due to adhesion between the composite part (resin) and the tool. Once there are FEP sheets (see section 1.3.2) at the interface, no tensile stress is allowed since no adhesion between FEP sheet

and tool could exist, thus, δ A = 0 . In the case of release agent at the interface, δ A becomes a function of resin adhesion to the tool. For simplicity, we assume that:

δ if c≤ c ⎪⎧ C 1 δ A = ⎨ (4.16) δ /2 if c> c ⎩⎪ C 1

where δC is the thickness of the top resin layer of a ply which is assumed to be around

20 µm (Figure 4-6). c and c1 are degree of cure and specific degree of cure at which the fluid friction event ends, respectively. δ A should be zero when cc> 1 . However, to prevent numerical instability it was set to an arbitrary small value δ AC= δ / 2 .

A normal spring has two purposes: one is to prevent penetration, and the other is to determine the normal contact force. Normal contact force is firstly needed to determine the existence of a gap or a contact and secondly needed to be used in the calculation of the interfacial shear strength. A normal spring is always perpendicular to the master surface, and rotates with the rotation of the master surface.

4.2.2 Tangential Spring

As mentioned before, a contact element, which relates one slave node to two or three master nodes, consists of two springs, one parallel and one normal to the interface. The tangential spring, which is parallel to the interface, is introduced at the part-tool interface to capture the interaction between the curing resin and the tool. The stiffness of the tangential spring depends on the composite material properties and the interface conditions (e.g., release agent or FEP sheets, see section 1.3.2).

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The tangential spring element has similar behaviour in both directions. Its initial stiffness can be determined by replacing the resin-rich region with a spring parallel to the surface based on the equilibrium of shear forces at the interface as shown in Figure 4-6:

⎧GLRγ (1) FL==τ (1) ⎨ ⇒ GRTγδ = K (4.17) ⎩KLT (1)δ

where GR is the resin shear modulus, KT is the tangential spring stiffness and δ is the tangential displacement. For an isotropic resin layer of thickness h, we can write:

E δ GKγ ==R δ (4.18) RT2(1+υ ) h EE K =≅RR (4.19) T 2(1++υ )hx 2(1 0.5)(20 10−6 )

KETR≅ 17000 (4.20) The same as the case for normal spring, since resin modulus evolves as the resin cures, the stiffness of the tangential spring is varying too.

If the existing shear stress at the interface surpasses the shear strength, the tangential spring stiffness, KT , will be reduced by a factor, SRF >1. In this case, increasing the SRF factor eliminates stress over-shooting and thereby prevents numerical problems.

If the existing shear stress at the interface is less than the interfacial shear strength, by decreasing this factor, we eliminate any possible over-slipping situations. The factor, SRF , is calculated based on an algorithm summarized in Appendix C. The tangential spring behaviour is schematically depicted in Figure 4-7.

To demonstrate the behaviour of the tangential spring element, a seven-step loading example is schematically shown in Figure 4-8 for one slave node. It should be noted

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that interfacial shear strength, τ A , is a varying function at each time step at each slave node, which will be discussed later.

At the beginning of step 1, the interfacial shear strength was determined, τ A1, while the tangential spring stiffness was initialized to KT . At the end of step 1, the existing shear stress was below the interfacial shear strength, τ A1, therefore, for the next step (step 2) the same tangential spring stiffness, KT , was used.

At the end of step 2, the interfacial shear strength was determined to be unchanged and

equal to τ A1. Now, the existing shear stress is above the interfacial shear strength of

τ A1, therefore, for the next step (step 3) two procedures should be performed: (1) The excess shear stress should be released by releasing the excess forces at the corresponding nodes of the spring of that slave node and (2) the tangential spring

stiffness, KT , should be reduced by SRF . Therefore, in step 3, first the excess shear stress was released and then one more run was performed based on the determined tangential spring stiffness of KKSRFTT2 = / . KT 2 is smaller than KT since the existing shear stress is more than the interfacial shear strength.

At the end of step 3, the interfacial shear strength was determined as τ A2 , which is larger than τ A1. The existing shear stress is still above the interfacial shear strength of

τ A2 . Therefore, for the next step (step 4) the same two procedures should be performed. First the excess shear stress was released and then one more run was

performed based on the determined tangential spring stiffness of KKSRFTT3 = / . It should be noted that SRF was calculated again and is different from the last one, therefore, KT 3 is smaller than KT 2 since the existing shear stress is more than the interfacial shear strength.

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At the end of step 4, the interfacial shear strength was determined to be τ A3 , which is larger than τ A2 . The existing shear stress is below the interfacial shear strength of

τ A3 , therefore, for the next step (step 5) just one procedure was performed. One run is performed based on the newly determined tangential spring stiffness of

KKSRFTT4 = / . KT 4 is larger than KT 3 since the existing shear stress is less than the interfacial shear strength.

At the end of step 5, the interfacial shear strength was determined to be unchanged as

τ A3 . The existing shear stress is still below the interfacial shear strength of τ A3 , therefore, for the next step (step 6) just one run was performed based on the determined tangential spring stiffness of KKSRFTT5 = / , which is larger than KT 4 .

At the end of step 6, the interfacial shear strength was determined to be unchanged as

τ A3 . The existing shear stress is still below the interfacial shear strength of τ A3 , therefore, for the next step (step 7) just one run was performed based on the determined tangential spring stiffness of KT , which is larger than KT 5 .

The various degrees of spring stiffness and the interfacial shear strength can be expressed mathematically as:

KKTT>>≤<2345 K T K T K T (4.21) and

τ A123<ττAA< (4.22) The above algorithm, along with the algorithm for the factor SRF , is only suitable for curing composite materials. In other cases, the SRF algorithm might not be applicable and should be switched off.

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4.2.3 Interfacial shear strength

The interfacial shear strength is a complex function of many variables such as normal pressure at the interface, the fluid friction shear strength, bonding shear strength and the coefficient of Coulomb friction. However, the interfacial shear strength can be said to be a function of chemical bonding between the tool and the resin and a function of mechanical shear strength (Coulomb friction) between surfaces. Determining this value at each descretized node of contact slave surface is crucial.

The interfacial shear strength can be determined using the experimental results by Twigg [2001]. Although the chemical bonding between the resin and the tool varies smoothly as the resin cures, for simplicity we assume that the chemical bonding has different behaviour in the two different fluid friction and bonding phases.

4.2.3.1 Fluid Friction Phase ( c

At a low degree of cure, (say c

0 suggested a linear behaviour for the chemical interfacial shear strength from τ ff = 0 at

m zero degree of cure toτ ff = 30 kPa at c1 = 0.65 , as shown in Figure 4-9. τ ff is the fluid friction shear strength.

During the fluid friction phase, if the resin separates from the tool or slides on it and becomes attached to it again, the amount of the chemical interfacial shear strength will not change. This means that after sliding occurs, the amount of shear stress needed to cause it to slide again would not change. When the resin slides on the tool, a very thin layer of the resin remains attached to the tool and only resin polymer molecules and

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chains slide on top of each other much like very low viscosity fluids. Friction between resin chains generates the fluid friction shear resistance, which is less than the sticking force between a thin resin layer and the tool. This resistance is proportional to the resin degree of cure, and a weak function of autoclave pressure, p . Twigg developed the following equation:

p 0.2 τ ff ∝ ()c (4.23) p0

where p0 is equal to 586 kPa.

After the degree of cure reaches c1 , the fluid friction behaviour of the resin ceases and bonding initiates. It is important to note that c1 depends on the resin and cure cycle.

The degree of cure c1 occurs when the resin viscosity increases and the resin enters into a more solid-like phase.

4.2.3.2 Bonding Phase ( c>c1 )

At this stage, the resin behaves more like a solid. As it cures, chemical bonding occurs between the resin and the tool. That bonding is much stronger than the fluid frictional bonding. Twigg [2001] suggested that shear strength of 3.0 to 4.0 MPa at a degree of cure of 0.9 is required at the interface to break the bond. He also suggested a linear behaviour between degrees of cure of c1 and 0.9. Linearly extrapolating this curve to degree of cure of 1.0, shear strength of 4.0 to 5.0 MPa will be obtained, as depicted in Figure 4-10A. Bonding shear strength can be written as:

τ b ∝ c (4.24) The fluid friction behaviour probably exists at this stage, but as the resin cures its effect becomes negligible in comparison to bonding shear strength. It is assumed that the

shear strength due to fluid friction no longer exists after the degree of cure of c1 is reached. It is obvious that if a material behaviour changes as it cures, the change must

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be gradual rather than sudden. For the sake of simplicity, it is assumed that when bonding behaviour begins, fluid friction behaviour ceases.

At the bonding stage when c>c1 , if the resin separates from or slides on the tool, all existing bonding breaks. For the resin to attach to the tool again, it should undergo further curing to rebuild chemical bonding to the tool. For instance, if there is no sliding between the degree of cure of c1 and the ideally degree of cure of 1.0 then we need 4MPa of shear stress at the interface to break the whole bonding. Now assume we have sliding at a degree of cure of 0.78, which needs 1.5 MPa of shear stress to break the bonding at interface (based on a linear behaviour as shown in Figure 4-10B). At this point, chemical bonding between the resin and the tool starts occurring and increasing at the interface again as the resin cures further (ideally to a degree of cure of 1.0). While assuming there is no more sliding, a shear stress of approximately 2.5 MPa is needed to break the rebuilt chemical bonding at the interface. Nonetheless, the summation of all peaks (two peaks in this case) must be the same as in the first case (where no sliding occurred) of 4.0 MPa (1.5+ 2.5= 4.0 ). This behaviour is a superposition of bonding shear strength, which is a linear function of degree of cure.

The chemical interfacial shear strength at the interface from fluid friction to bonding behaviours for the entire range of degree of cure is shown in Figure 4-11.

4.2.3.3 Friction at Interface Friction between the composite part and the tool, or the FEP sheet, is not only a function of the interface condition but also a function of the resin degree of cure, temperature and viscosity. The frictional behaviour of the resin should be investigated at both fluid friction and bonding phases separately for both interface conditions of release agent and FEP sheets.

4.2.3.3.1 Release Agent at Interface At the fluid friction phase, we assume that Coulomb friction between the part and the tool does not exist since the resin behaves like a fluid. In other words, friction is

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considered inside the fluid friction phenomenon and not as a separate event in this phase. Also, Twigg [2001] explained that the fluid friction shear strength is a weak function of interface pressure and thus in his fluid friction formulation, the Coulomb friction is indirectly included.

At the bonding phase, if debonding occurs, then the Coulomb friction becomes an additional factor at the interface. Now, both the Coulomb friction and the bonding phenomena exist simultaneously. Twigg [2001] suggested a constant coefficient of

Coulomb friction, µ , for the entire range of the cure cycle afterc1 , as depicted in Figure 4-12. Therefore the Coulomb friction shear strength can be written as:

0 cc< ⎪⎧ 1 τ f = ⎨ (4.25) µ p cc≥ ⎩⎪ 1

where p , µ and τ f are the pressure normal to the interface, the coefficient of Coulomb friction and Coulomb friction shear strength, respectively.

The total interfacial shear strength after the degree of cure of c1 should never be less than the Coulomb friction, τ f . Now the interfacial shear strength for the entire cure cycle can be expressed as:

⎧τ cc< ⎪ ff 1 τ A = ⎨ (4.26) Max(,ττbf ) c≥ c1 ⎩⎪ The interface behaviour at a slave node is schematically depicted in Figure 4-13. All the above behaviours and graphs apply to an interface with release agent. Effects of having FEP sheets at the interface require additional considerations, and are the subject of the next subsection.

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4.2.3.3.2 FEP Sheets at Interface FEP sheets between the part and the tool make the interface behaviour more complex. There will be one interface between the part and the FEP sheet, and other interfaces between FEP sheets themselves (i.e., multiple interfaces exist). The final interface will be between the last FEP sheet and the tool surface. Fortunately, the only phenomenon occurring at the FEP-to-FEP and FEP-to-tool interfaces is friction. Since all of these extra interfaces are parallel to each other, only one coefficient of Coulomb friction, which is the minimum of all coefficients of Coulomb friction between various FEP sheets and between the tool and the last FEP sheet, is of concern. Twigg [2001] suggested a constant coefficient of friction ofµ = 0.18, as shown in Figure 4-14 for 2 FEP layers.

τ f = µ p (4.27)

The Coulomb frictional shear strength, τ f , exists over the entire cure cycle. Since the coefficient of friction between FEP sheets themselves and between FEP sheet and the tool is much less than the coefficient of friction between the FEP and the part after debonding, the latter will be of no interest. Therefore, the interfacial shear strength can be determined as:

⎧Min(,)ττ c< c ⎪ ff f 1 τ A = ⎨ (4.28) Min(,ττ ) c≥ c ⎩⎪ bf 1 4.2.4 Master and Slave Surfaces

Usually each contact consists of a pair of surfaces, termed master and slave surfaces as explained before, there are three approaches to model contact between slave and master bodies: slave node to master node, slave node to master segment, and slave node to a defined smooth surface on the master body.

Here the chosen algorithm for contact formulation is a slave node to a master segment approach. The composite side of the contact is the slave surface and the tool side is the

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master surface. The tool is less deformable while the composite part deforms more and moves around the corners during processing. Therefore, in the node-to-segment approach, it is more logical to choose the tool as the master region and the composite part as the slave region. In addition, a master surface should preferably be larger than the slave surface. In our approach, all slave nodes should be located at one boundary, while the master surface can consist of one or two boundaries.

In this approach, each element in the master region at interface is one master segment. In order to prevent numerical instability due to slope discontinuity between segments (elements) for curved parts, it is strongly recommended that an 8-noded plane strain element option in COMPRO be employed. This new feature is introduced into COMPRO for several other reasons as well, but the reason just mentioned is the most important.

Each slave node is located on top of only one assigned master segment/element. Projection of the slave node is usually on the same master segment. It is quite possible that several slave nodes have the same master segment, but there is only one assigned master element for each slave node. If a slave node is not located on top of a master segment, then that slave node has passed all master segments and there is no assigned master segment for it.

A slave node passes one master segment/element when it passes the average of two normals to the surfaces of two adjacent elements at the common node, as shown in Figure 4-15. For example, nodes 1 and 3 in Figure 4-15 are on segments A and B, respectively in configurations i and i +1. Node 2 is located on top of segment A at configuration i and on top of segment B at configuration i +1. Node 2 passes the average of normals to segments A and B at the common node 4 at configuration i +1. It is important to note that, although node 2 at configuration i is located on top of segment A, its projection is not located on segment A.

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4.2.5 Higher Order Element

The 8-noded elements are implemented in COMPRO for various benefits such as more rapid convergence, accurate results, and a smooth segment-to-segment slope at the master surface of the interface. In the current COMPRO pre-processor, the tool-part assembly is discretized using the 4-noded solid elements. However, these elements can be converted to 8-noded elements by switching on the 8-noded element option. In so doing, extra nodes will be added to the mesh, elements and the boundaries automatically.

There is one important consequence at curved boundaries: all 8-noded elements will have a circular side instead of a piece-wise linear side. For example, nodes in a 90° corner of a 4-noded mesh on the tool side of the interface are numbered 1, 2, 3, 4, 5, 6 and 7, as shown in Figure 4-16. If the 8-noded element option is switched on, then nodes 8, 9, 10, 11, 12 and 13 will be added to the mesh as middle nodes of 8-noded elements. Only those sides of elements that are located at the interface are shown. A circle through nodes 2, 3 and 4 is then automatically traced. The middle node 9 is shifted to its new location on the circle as node 9’. This algorithm is repeated for nodes 3, 4 and 5, and node 10 is shifted to its new location 10’. For the last circle, which passes through nodes 4, 5 and 6, both nodes 11 and 12 are shifted to new locations 11’ and 12’.

It is obvious that if the corner is a quarter circle from the beginning, the curved interface becomes a quarter circle as well; therefore, slopes of adjacent elements at the common nodes are continuous. If the curved part is not a circle but rather a smooth curve, then the final curve at the interface will consist of small circular arcs connected to each other. This leads to discontinuous slopes at common element nodes. But this discontinuity is not severe and will not cause numerical and convergence problems.

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Triangular Element

One of the benefits of having a contact surface in a finite element code is that different mesh densities can exist on both sides of the contact surface. This will allow us to have a coarser mesh for a large tool at areas where the mesh size is not important. Since this option does not exist in COMPRO, triangular solid elements are introduced into COMPRO to compensate for this shortcoming. Now, we can go from a small-sized element at the interface to a large-sized element at the far side of the tool using triangular elements. To avoid using numerically-sensitive, distorted rectangular elements, a triangular element can also be used in places where we have ply drop-off and thickness change. It can also be used at the noodle area of an I-stringer. Hence, 3- and 6-noded triangular elements are implemented in COMRPO.

4.2.6 Stiffness Matrix

Each slave node on the composite side of the interface is located on top of one master segment. A contact element, consisting of a normal spring and a tangential spring, is assigned to each slave node. This contact element connects the slave node to two or three nodes of the master element. There is a probability that one or more slave nodes might pass all master segments during a computational run, and thus will no longer be on any master segment. In such a case, there are no contact elements for those slave nodes until they are back on a master segment.

A dummy node called “projected node” is a slave node projection on the master segment. A normal spring is virtually located between the slave and projected node. Therefore, any gap or penetration is measured based on the distance between the slave and the projected nodes in the current configuration.

A tangential spring is virtually located between a projected node at the current configuration and the projected node at the original configuration as depicted in Figure

4-17. For instance, at step i +1, the tangential spring is located between 41 and 4i+1 .

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Therefore, any sliding or sticking is measured based on the distance between the same projected nodes at two different configurations. As an incremental algorithm, change in the interfacial shear strength is always computed.

There are 2- and 3-noded contact elements for 4- and 8-noded master elements, respectively, as shown in Figure 4-18. The stiffness matrix of one contact element can be determined as follows.

The shape functions for a 2-noded contact element are:

N1 =1−ζ (4.29) N2 =1+ζ By moving node 1 in the X-direction by 1 unit, as shown in Figure 4-19, the terms of the first column of the stiffness matrix of the contact element can be determined. Unit displacement of node 1 in the X-direction can be broken down into normal and tangential displacements (at the interface), equal to s and c , respectively. Therefore, projected node 4 moves to 4’ in the amount of Ns1 ⋅ and Nc1 ⋅ . Thus, forces in the springs can be determined as:

ΨTT= KNc1 (4.30) and

Ψ NN= KNs1 (4.31)

where FT and FN are forces in normal and tangential springs.

Spring forces at slave node 3 can be resolved in a global coordinate system as:

22 Ψ XTN=−ΨΨcsKNcKNsNT − =− T1111 − N =− (4.32) and

ΨZTN=−ΨΨscKNcsKNscNT + =− T1113 + N =− (4.33) where

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22 TKcKs1 =+TN and TKKcs3 = ()TN− (4.34) Now, the contact element stiffness terms at node 3 can be determined as:

KNT51=Ψ X /1=− 1 1 (4.35) KNT61==−ΨZ /1 1 3 Moreover, the contact element stiffness terms (equal to the values of forces) at nodes 1 and 2 due to the forces at node 4’ can be determined as:

2 KN11=− 1Ψ X /1 = NT 1 1 KN=−Ψ /1 = NT2 21 1Z 1 3 (4.36) KN31=− 2Ψ X /1 = NNT 1 2 1

KN41=− 2ΨZ /1 = NNT 1 2 3 Similarly, other columns of the contact element stiffness matrix can be determined. The stiffness matrix of a 2-noded contact element is given by:

22 ⎡⎤N11⋅ T N 13 ⋅ T NNT 121 ⋅⋅ NNT 123 ⋅⋅ −⋅ NT 11 −⋅ NT 13 ⎢⎥22 ⎢⎥NT13 NT 12 NNT 123 NNT 122−− NT 13 NT 12 ⎢NNT NNT N22 T N T−− NT NT ⎥ S 121 12 21 23 21 23 (4.37) K = ⎢ 22 ⎥ ⎢NNT123 NNT 122 N 23 T N 22 T−− NT 23 NT 22⎥ ⎢ −−NT NT − NT − NT T T ⎥ ⎢ 11 13 21 2 3 1 3 ⎥ ⎣⎦⎢ −−NT13 NT 12 − NT 23 − NT 22 T 3 T 2 ⎥ where

22 TKsKc2 =+TN (4.38) The shape function for a 3-noded contact element is:

N1 = (1)/2ζ − ζ

N2 =+(1)/2ζζ (4.39) 2 N3 =−(1ζ ) Similarly, the stiffness matrix of a 3-noded contact element can be determined by:

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22 ⎡⎤N11 T N 13 T NNT 121 NNT 123 NNT 131 NNT 133−− NT 11 NT 13 ⎢⎥22 ⎢⎥N13 T N 12 T NNT 123 NNT 122 NNT 133 NNT 132−− NT 13 NT 12 ⎢⎥22 NNT1 21 NNT 1 23 N 2 T 1 N 2 T 3 NNT 2 31 NNT 2 33−− NT 21 NT 23 ⎢⎥22 S ⎢⎥NNT1 23 NNT 1 22 N 2 T 3 N 2 T 2 NNT 2 33 NNT 2 32−− NT 23 NT 22 K = (4.40) ⎢⎥NNT NNT NNT NNTNTNTNTNT22−− ⎢⎥131 133 231 233 3 1 3 3 31 33 22 ⎢⎥NNT1 33 NNT 1 32 NNT 2 33 NNT 2 32 NT 3 3 NT 3 2−− NT 33 NT 32 ⎢⎥−−−−−−NT NT NT NT NT NT T T ⎢⎥11 13 21 2 3 31 33 1 3 ⎣⎦⎢⎥−−−−−−NT13 NT 12 NT 23 NT 22 NT 33 NT 32 T 3 T 2 The stiffness matrix of a contact element, K S , of each slave node are determined and added to the stiffness matrix of the structure at each time step. It is assumed that the contact element stiffness matrix remains constant during iteration to prevent numerical instability.

4.3 VERIFICATION

To verify the implementation of the contact element in COMPRO, a bi-material cantilever beam is modelled both in COMPRO and ABAQUS. A close agreement of the results would ensure the correct implementation of the formulation in COMPRO. In order to make a meaningful comparison possible, the ABAQUS contact surface should have the same features that exist in COMPRO. Therefore, spring elements are used as the contact surface in the ABAQUS model, which is equivalent to the contact surface model in COMRPO.

Both top and bottom parts of a bi-material cantilever beam are made of steel with an elastic modulus of 200 GPa and a Poisson’s ratio of 0.3. Both parts are 5 mm long and 1 mm thick. The parts are modelled with five 2-D solid elements in the length direction and only one element is used in the thickness direction as shown in Figure 4-20. A total pressure of 700 kPa is applied at the top of the beam in 20 equal steps.

The bi-material cantilever beam is modelled both in COMPRO and ABAQUS for 4 different interface conditions.

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1. Contact element at interface

A normal spring has zero stiffness in tension and a very high stiffness of K N = 10.0 GPa/m in compression until the compressive force reaches 10 GN/m, and is zero thereafter. A tangential spring has a symmetrical behaviour with a high stiffness of

KT =1.0 GPa/m until the force in the spring reaches ±100 N/m. After that point, the tangential spring has no resistance. These properties are depicted in Figure 4-21.

2. Perfect bonding interface

In this case a normal spring has a symmetric behaviour with a high stiffness of

K N =10.0 GPa/m, with no limit on the force. Also, a tangential spring has a symmetric behaviour with a high stiffness of KT = 1.0 GPa/m with no force limit. All these properties are shown in Figure 4-22.

3. No contact element at the interface

In this case the normal and tangential springs have zero stiffnesses.

4. Only normal spring and no tangential spring at the interface

In this case the normal spring has a symmetrical behaviour with a high stiffness of

K N =10.0 GPa/m with no force limit, while the tangential spring has zero stiffness.

In the case 3 of no contact element, the top part of the beam virtually passes through the bottom part. Maximum deformation occurs in this case because the bottom part of the beam has no resistance. Deformation profiles of the top surface of the beam for all four cases are depicted in Figure 4-23. In all cases, the results from COMPRO and ABAQUS match perfectly. This simple example shows the validity of the implementation of contact surface in COMPRO. Some practical processing problems will be given in the next chapter.

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4.4 SUMMARY AND DISCUSSION

Although an elastic shear layer was previously used in COMPRO to simulate the tool- part interaction observed in the experiments, it provided a much better model than the perfect bonding condition used by other researchers. A contact interface with penalty method and a slave node to a master segment approach is chosen here to simulate the tool-part interaction because it is suitable for implementation in existing codes.

To avoid discontinuity at the interface, a higher-order 8-noded 2-D solid element was introduced into COMPRO. Triangular elements were implemented in COMPRO to allow the possibility of having a fine mesh at the interface while having a coarser mesh away from the interface.

The contact surface implemented in COMPRO enables us to model the complex tool- part interaction mechanisms, such as stick-slip condition, fluid friction and adhesion, observed in the experiments. It also enables us to model gaps. The benefits of the implemented contact surface are summarized below:

• Because there is no need for an elastic shear layer with a certain thickness, a zero interface thickness for a contact surface is possible.

• When using a thin shear layer, the element size in the planar direction should also be small to maintain a reasonable element aspect ratio. This usually increases the mesh density in the tool and the part resulting in inefficient computational runs and large output files, while employing a contact surface avoids all these problems.

• In shear layer modelling, the Poisson’s ratio, elastic moduli and coefficient of thermal expansions that are assigned to the material of the shear layer are unknown but important parameters. These parameters do not have a physical value or meaning, but can alter the results significantly. They are dummy

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quantities that should be manipulated by the user to decrease their effects on the runs and results. In contact surface modelling, all these parameters are eliminated.

• A significant of penetration in shear layer modelling is possible due to possible element inversion (negative jacobians caused by excessive distortions of compliant elements used in the shear layer). In the contact surface approach, only minimal penetration is allowed.

• Stick-slip modelling is possible.

• Separation and gap between two sides of a contact (usually composite part and tool) is possible. This event is important at corners, where separation can occur.

• Contact surface (springs) parameters are not geometry-dependent and depend only on the constitutive behaviour of the interface.

The normal spring stiffness in a contact element must be as high as numerically possible. It is not a material property that should be determined by experimentation.

The tangential spring stiffness in a contact element can be determined based on the resin modulus and the thickness of the resin-rich layer between the tool and the first ply of the laminate. Therefore, it evolves with the degree of cure.

There are several other contact parameters that should be determined from experiments. These parameters are the chemical interfacial shear strength (in both fluid friction and bonding phases) and coefficient of friction, which vary during the cure cycle.

The interfacial shear strength is a function of pressure, the fluid friction shear strength, bonding shear strength and the coefficient of friction. It varies as the resin cures, but for simplicity we assume that it has two different behaviour at the fluid friction and bonding phases separated by the specific degree of cure of c1 .

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The fluid friction shear strength is a weak function of pressure and independent of sliding. The bonding shear strength is a function of degree of cure and deviation of the degree of cure from the last sliding or separation that occurred after the specific degree

of cure, c1 .

Having FEP sheets between the part and the tool creates parallel sub-interfaces, which include Coulomb friction shear strength in determining the interfacial shear strength during the whole cure cycle.

The degree of cure c1 represents the point in the cure cycle where an appreciable modulus of the resin has developed.

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Coulomb friction (Pa) τ Two different regularized friction laws

δ (mm)

Figure 4-1: Coulomb’s friction law and two regularized friction laws (Ling and Stolarski [1997]).

K

C Πmin Πmin Π m u h mg/K

u

Figure 4-2: Minimum energy and minimum feasible energy when there is an impenetrable surface.

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K

m u h

κ

Figure 4-3: Penalty method: Replacing the impenetrable surface by a spring.

L

Composite elements H Interface at interface

Tool elements at interface H1

Figure 4-4: Composite elements on top of tool elements at the interface.

σ K /104 1 N δ δA KN 1 δC

Figure 4-5: Normal spring element, contact laws.

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L Resin 1-ply = 0.2 mm h h = 20 µm τ δ Interface

Figure 4-6: First ply on top of a tool. Resin thickness between the tool and the first fibre is about 20 µm.

τ 1 KT /SRF τA

KT 1 δ -τA KT /SRF 1

Figure 4-7: Tangential spring element contact laws.

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τ 2 τ 3 4 KT τA2 KT2 KT3 τA1 1

KT δ A) B) δ

τ τ τA3 τA3 KT4 5 KT5 6

KT 7

δ δ C) D)

Figure 4-8: Schematic example of one tangential spring element. Tangential spring stiffness calculated at each time step base on the following

equation KKSRFTi= T/ i .

m τff = 30.0 (kPa) ff τ

c1 = 0.65 c

Figure 4-9: Fluid friction shear strength at the interface as a function of degree of cure.

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τb = 4.0 τb = 2.5 (MPa) (MPa) b b τb = 1.5 τ τ

τb = 0.030 τb = 0.030

c1 = 0.65 1.0 c c1 = 0.65 0.78 1.0 c (a) (b)

Figure 4-10: Bonding shear strength at the interface when (a) there is no debonding, and (b) when there is one debonding at degree of cure of 0.78.

4.0 (MPa) A τ

0.030

c = 0.65 1.0 c 1

Figure 4-11: Interfacial shear strength due to fluid friction and bonding effects.

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µ

0.28

c1 = 0.65 1.0 c

Figure 4-12: Coefficient of friction between the resin and an aluminium tool.

τ T, p 4.0 MPa

Interfacial shear

strength, τA

Existing Shear stress at interface, τ

40.0 kPa Temperature 180.0 °C 30.0 kPa 25.0 kPa

Pressure 106 kPa

Fluid Friction Bonding phase time phase c = c 1

Figure 4-13: General behaviour of a contact element at the interface during a cure cycle while applying pressure and temperature.

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µ

0.18

c1 = 0.65 1.0 c

Figure 4-14: Coefficient of friction between FEP sheets.

2

4

1 2 3 1 2 3 4 4 A A B B C C

Step i Step i+1

Figure 4-15: Slave node 2 passes one master segment (i.e., moves from A to B), while nodes 1 and 3 remain at the same master segments, A and B.

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9’ 1 8 2 9 3 10’ 10 4 11’ 11

5 12’ 12

C 6

13

7

Figure 4-16: Changing corners and curved sections to a smooth surface at boundaries.

1 1 41 1 4i+1 41 4i 2 41 2 2 3 3 3

Step 1 Step i Step i+1

Figure 4-17: Virtual location of a tangential spring is between 41 and 4i at step i , and

between 41 and 4i+1 at step i +1 .

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4 3 4 2 3 Ө Ө 2 Z 1 1 5 B X A

Figure 4-18: 2- and 3-noded contact elements for 4- and 8-noded master elements.

2 3 Ψ 4 4’ N Z c s Ψ Ө N1·c T 1 X δ=1 N1·s

Figure 4-19: Forces in contact springs due to unit displacement at node 1 in the X- direction.

2 mm

5 mm

Figure 4-20: A cantilever beam example used to verify the contact surface implementation in COMPRO.

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5.E+09 100

KT (N/m) (N/m) 0.E+00 T N

Ψ -10 -5 0 5 10 Ψ 0 1 -1.E-06 0.E+00 1.E-06 -5.E+09 1 unit Length, unit Length, unit Length, unit Length, KN Force in normal spring per -1.E+10 Force spring in tangential per -100 Relative normal displacement (m) Relative tangential displacement (m)

Figure 4-21: Normal and tangential spring properties for the cantilever example of Figure 4-20 with a contact element at the interface of the top and bottom beams (no sliding resistance at the interface).

1.E+10 100

5.E+09 KN KT (N/m) (N/m) N T Ψ 1 Ψ 1 0.E+00 0 -10 -5 0 5 10 -1.E-06 0.E+00 1.E-06

-5.E+09 unit Length, Length, unit unit Length, Length, unit Force in normalspring per

-1.E+10 springForce per in tangential -100 Relative normal displacement (m) Relative tangential displacement (m)

Figure 4-22: Normal and tangential spring properties for the cantilever example of Figure 4-20 with a contact element at the interface of the top and bottom beams (perfect bonding at the interface).

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X-Coordinate (mm) 012345 0.0 Per fect Bonding B oth Norm al a -0.5 nd T ange On ntia ly N l Sp p orm ring al s Spri -1.0 ng

N o S pr Deformation (µm) Deformation -1.5 in gs

-2.0

Figure 4-23: Deformation of the top surface of a bi-metallic beam under uniform pressure loading for different contact conditions between the two beams. The solid lines are COMPRO’s results, while the thin lines with symbols are ABAQUS’s results.

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CHAPTER 5 ‐ VALI D ATI ON AND NUMERICAL CASE

STUDIES

Autoclave processing of composite parts on tools generates stresses and mechanical strains both in the part and the tool. Some of those stresses and strains result in residual deformations. Although these deformations have undesirable consequences for dimensional fidelity of various components, the residual stresses can be of equal importance for the performance of the composite part in service. Fatigue resistance and ultimate strength can be significantly affected by the residual stresses. If a modified COMPRO with large deformation/strain and contact surface options can accurately predict both deformations and residual stresses during the entire cure cycle, then it will be a very useful numerical tool for the industry.

In this chapter, first the parameters for the contact surface developed in the previous chapter will be calibrated using the instrumented tool experiment carried out by Twigg [2001]. Then results of COMPRO with shear layer and perfect bonding interfaces are compared with those of COMPRO with contact interface to investigate the improvement of the results due to this new feature. Second, the calibrated contact surface is used to predict the experimentally observed warpage of flat unidirectional composite parts of different lengths and thicknesses. Sensitivity analysis is performed on cases with both low and high initial resin moduli to study the importance of each

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contact parameters. Also, one example is performed using COMPRO with a contact interface for both 4- and 8-noded elements to verify sufficiency of mesh density and convergence. Third, COMPRO predictions of L- and C-shaped parts spring-in will be compared with experimental results performed by Albert [2001]. Finally, a few other cases will be modelled in COMPRO to explore COMPRO’s capability.

5.1 SIMULATION OF INSTRUMENTED TOOL EXPERIMENTS

Twigg [2001] carried out two sets of experimental studies at the University of British Columbia to investigate the tool-part interaction phenomena in composite processing. The first set, instrumented tool experiments consisting of a relatively thick flat composite part (T-800H/3900-2) on a thin aluminium tool with different pressure and interface conditions, was designed to measure the mechanical strains in the tool and part during cure. The second set of experiments was designed to measure the warpage of a flat composite part on a thick aluminium tool. Each experiment used different part thicknesses and lengths, autoclave pressures and interface conditions.

In the instrumented tool experiments, 8 strain gauges were placed on a thin aluminium tool. 6 strain gauges were placed in the longitudinal direction of the tool as shown in Figure 5-1. This instrumented tool was placed on top of an identically-sized 16-layer carbon fibre reinforced polymer (CFRP) composite, as shown in the same figure. The total strains at these 6 strain gauge locations were recorded during the entire cure cycle. Since the strain gauges were calibrated for free thermal expansion of the tool plate, the mechanical strains could be easily computed. The recorded mechanical strains for different autoclave pressures and interface conditions are shown in Figure 5-2 and Figure 5-3. These conditions are combinations of low and high autoclave pressures (103 and 586 kPa) with two different interface conditions (release agent (RA) and two plies of FEP sheets). These two interface conditions will be referred to as RA- and FEP-interface, respectively. The reader is referred to Twigg [2001] for more details of the experimental procedure. All strains are measured in pairs with respect to the

-149- Chapter 5 – Validation and numerical case studies centreline of the part, such as A’ and A, which located at the left and right ends of the tool (see Figure 5-1). Experimental results showed that each pair of strain gauges recorded almost identical values except the pair A and A’ for the FEP-interface. The mechanical strains at these strain gauges are similar, but they deviate from each other at the beginning of the cure cycle after about 50 minutes. In a highly nonlinear phenomenon such as contact large differences in reading of pairs of strain gauges are expected to occur.

As explained in the previous chapter, the contact surface has been developed such that different mechanisms during curing of a thermoset polymer under various interface conditions can be modelled precisely. The contact surface parameters are determined based on the data obtained from Twigg’s instrumented tool experiments.

For example, Twigg found that debonding of cured material (c ≅ 0.90 ) occurs at approximately 4.0 MPa for the RA-interface. For the FEP-interface, FEP layers slide on top of each other soon after the shear stress at the interface(s) surpasses the Coulomb friction shear strength (which is usually less than the maximum bonding shear strength). Therefore, debonding rarely occurs if there is more than one FEP layer at the interface.

At the beginning of the cure cycle, the fluid friction phenomenon is active at the ply closest to the tool, where the tool-part interaction generates maximum shear stresses. This fluid friction mechanism occurs only in the resin-rich region of the first ply. This mechanism depends on material properties and is different from the Coulomb friction, which is a function of the two materials in contact, their surface roughness and most importantly, the amount of pressure. The fluid friction mechanism (for both interface conditions of RA and FEP layers) is a weak function of pressure. Twigg assumed that the interfacial shear strength grows linearly from zero for not-cured materials to a value of approximately 30.0 kPa for materials at degree of cure of c1 ≅ 0.65 .

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The FEP-interface adds the Coulomb friction mechanism to the interface for the entire cure cycle. Twigg measured a coefficient of (Coulomb) friction of µ ≅ 0.18 for two layers of FEP. It should be reminded that for the RA-interface, the Coulomb friction mechanism is active only when the fluid friction phenomenon is over (with a coefficient of Coulomb friction of µ ≅ 0.28 ).

A set of COMPRO runs, listed in Table 5-1, is designed firstly to validate our proposed contact surface model, secondly to examine various existing options in COMPRO when the contact surface option is used and thirdly to calibrate the contact surface parameters and related composite material properties. All the COMPRO files used for boundary conditions, program control, finite element mesh, cure cycle, part lay-up and material properties for each project (run) are listed in Table 5-2. The instrumented tool experiment was modelled in COMPRO using both fine and coarse finite element meshes. A fine mesh with a shear layer and another mesh without a shear layer are shown in Figure 5-4 and Figure 5-5, respectively. A coarse mesh with no shear layer is shown in Figure 5-6. In all cases, there is only one element through the thickness of the aluminium tool and 2 or 4 elements through the thickness of the composite part. Different mesh geometries (files) are summarized in Table 5-3. The names of the materials used in COMPRO runs for each material file are summarized in Table 5-4. Properties of materials used in this set of runs are summarized in Table 5-5 and Table 5-6.

After running all cases in COMPRO, mechanical strains are plotted versus time at strain gauge locations A, B and C. These results are compared with experimental results in the next subsection.

5.1.1 Experimental Results

From the early stage of the cure cycle to approximately 70% of heat-up section, the amount of mechanical strains and stresses are negligible, and temperature dips have very small effects on mechanical strains. At this stage, the aluminium tool expands and

-151- Chapter 5 – Validation and numerical case studies the resin is a low-viscosity fluid. The composite part does not experience the effects of the aluminium tool expansion because the resin monomers close to the interface slide on top of each other with little resistance. The fluid friction phenomenon is at its early stage, with low shear stress resistance.

In the last 30% of the heat-up section, when the resin degree of cure is higher, additional tool expansion causes higher stresses and mechanical strains in the composite part. Later, in the hold section of the cure cycle, resin behaves more like an elastic solid or a very high-viscosity fluid material. Therefore, all temperature dips, which cause a change in tool length, create stresses or mechanical strains in the composite part.

Later on at cool-down, the resin behaves like a perfectly elastic material. Decreasing the temperature, which translates into contraction of the tool while composite part length does not change (due to almost zero CTE in unidirectional composites), causes tensile or positive stress in the tool. These increasing normal stresses (mechanical strains) result from increasing shear stresses at the interface. This situation continues until the shear stress at the interface surpasses the interfacial shear strength. Based on the interface condition, various scenarios can occur. For the RA-interface, the interfacial shear strength at any time is the bonding shear strength at the corresponding degree of cure. At this point, debonding occurs and the interfacial shear strength drops to a point determined based on the Coulomb friction between the composite part and the tool. This behaviour is observable in Figure 5-2a and Figure 5-3a. For the FEP- interface, the interfacial shear strength is determined by the minimum bonding shear strength between the resin and the FEP layer, and the Coulomb friction shear strength between two FEP plies. The latter seems smaller and becomes the shear strength of the interface. Therefore, for FEP-interface condition, there is no debonding, and as soon as the shear stress at the interface reaches the shear strength, sliding occurs. During sliding, the shear stress at the interface remains constant even though the temperature is decreasing. This behaviour can also be seen in Figure 5-2b and Figure 5-3b.

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At the end of the cure cycle, when the pressure is removed, all mechanical strains approach zero for the FEP-interface, but for the RA-interface there is still some residual shear stresses. Not having absolute zero stress for FEP sheets at the interface after pressure removal is probably due to two reasons:

1. Calibration of large size strain gauges and

3. Existence of local friction between FEP layers.

5.1.2 COMPRO with Perfect Bonding Interface

A set of virtual experiments is modelled in COMPRO with perfect bonding between the composite part and the aluminium tool (this is considered to be an extreme case as far as the build-up of residual stresses is concerned). The COMPRO runs are carried out with and without the large deformation (LD) option. Numerical results from COMPRO are compared with each other and with the experimental results. The COMPRO project files for these examples are labelled IT-1 to IT-4, as listed in Table 5-1.

Since COMPRO is a 2-D plane strain finite element code and the experimental cases are closer to plane stress (out-of-plane dimension of the part and the tool are very small), it is assumed that the aluminium tool has a zero CTE in the out-of-plane direction.

The first two COMPRO runs, IT-1 and IT-2, are without and with the LD option, respectively and for an autoclave pressure of 103 kPa. Figure 5-7 shows the comparison between the COMPRO predictions and the experimental results. As shown in the figure, the numerical results are very different from the experimental results (predicted mechanical strains are at least one order of magnitude higher than the experimental results). This is mainly due to the fact that the full bonding condition cannot capture the mechanisms such as fluid friction and debonding at the interface. Although COMPRO with the LD option yields slightly smaller mechanical strains, it is still very different

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from the experimental results. For these runs, a value of 47.1 MPa was assumed for the initial resin modulus.

The next two runs, IT-3 and IT-4, are similar to the previous two runs except for the initial resin modulus. In these cases, the initial resin modulus was assumed to be relatively low value of 0.471 MPa. The COMPRO run with LD option (IT-4) did not converge due to numerical problems (high values of shear strains). The results from COMPRO run without LD option (IT-3) are compared with experimental results as shown in Figure 5-8. The results are similar to the previous run (IT-1, higher initial resin modulus) except that the magnitude of mechanical strains is a little lower.

The above results show that the full bonding condition is not a realistic interface condition to model the experimentally observed behaviour. In the next subsection, a thin shear layer is used to simulate the experimentally observed interface condition.

5.1.3 COMPRO with Shear Layer Interface

In this study, the tool-part interface is simulated by introducing a thin shear layer between the tool and the part. The thickness of the shear layer should be as small as possible and is governed by the aspect ratio of the element (to avoid numerical problems). The shear layer is also assumed to be either isotropic or orthotropic linear elastic material.

The through-thickness elastic modulus of the shear layer is assumed to be high (usually equal to the modulus of the tool) to prevent penetrations, while the other two elastic moduli are assumed to be as low as possible to minimize unnecessary shear layer effects. The shear modulus of the shear layer is the parameter of interest to be calibrated to capture the same experimental warpages for different tool-part interface conditions, part geometries and materials. The Poisson’s ratios of the shear layer in all directions are assumed to be zero for simplicity. The out-of-plane and the through- thickness CTEs of the shear layer are also assumed to be zero. The CTE of shear layer

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along the interface direction is assumed to be equal to the summation of the tool CTE and the effect of the tool’s Poisson’s ratio in the out-of-plane direction, as given by Equation (5.1) for an aluminium tool:

SL AL CTE11=+=+= CTE(1υµ ) 23.6(1 0.327) 31.32 ( / ° C ) (5.1) The COMPRO runs from IT-5 to IT-8 are with the shear layer interface for both low and high autoclave pressure cases (103 and 586 kPa). For these cases, the initial resin modulus is assumed to be 0.471 MPa. A shear layer thickness of 0.4 mm was assumed to keep the aspect ratio within the allowable limit (i.e., <5) as shown in Figure 5-5.

The COMPRO runs, IT-6 and IT-8, with the LD option did not converge due to numerical errors (very high values of shear strains). The results from COMPRO runs without the LD option, IT-5 and IT-7, are compared with the experimental results in Figure 5-9 and Figure 5-10 for low and high autoclave pressures, respectively. Even though the order of the magnitudes of the mechanical strains is the same, the COMPRO results are four times higher than the experimental results. Other differences of the shear layer results from the experimental results are as follows:

1. The effect of temperature dips on mechanical strains in the heat-up and hold sections of the cure cycle is not captured properly.

2. The experimentally observed mechanisms such as debonding and sliding are not captured.

3. The shear layer behaviour is independent of pressure loading.

By changing the shear modulus of the shear layer, the level of mechanical strains at the hold stage can be matched. However, the debonding and the sliding mechanism at the cool-down regime cannot be captured. Strains and deformations at other locations in the composite part are probably incorrect. Even in different geometry and complex shapes, it will be impossible to obtain correct results for all variables simultaneously.

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Therefore, although the shear layer is a better model for the interface than perfect bonding, it does not capture the physical reality well.

5.1.4 COMPRO with Contact Interface

A contact surface is the most recent modification in COMPRO, and the best option for modelling the experimentally observed interface behaviour. In general, a contact surface is characterized by non-linear behaviour, and each point along the interface can have different behaviour regardless of neighbouring points. Stick, slip and gap are available options in our contact surface model. The LD option must always be used when a contact interface is employed.

The COMPRO runs IT-9 to IT-12 are with contact surface for different autoclave pressures and interface conditions. The COMPRO run IT-9 is for autoclave pressure of 103 kPa and FEP-interface. The contact surface parameters for the FEP-interface are as

m m follows: µ = 0.18, τ ff = 30 kPa and τb = 5.0 MPa.

The mechanical strains at strain gauge locations C, B and A are compared with experimental results as shown in Figure 5-11. The predicted strains at locations C and B compare very well with the experimental results. However, there is a difference between the predicted and the experimental results at location A. There is also a marked discrepancy between the experimental results at locations A and A’. With the exception of location A, the predicted results seem to capture the experimental behaviour well.

The observed discrepancy can be analytically proven using the following assumptions. It is appropriate to assume that the coefficient of friction at any given time should be the same along the interface. Keeping this assumption in mind and looking at the end of the cure cycle, where sliding between FEP layers is occurring, the value of the coefficient of friction at location A and A’ (in experimental results) is several times larger than the value of the coefficient of friction at location C and B. Therefore, all or part of the experimental mechanical strains histories at A’ and A must be shifted

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downward. This action places the experimental mechanical strains at locations A and A’ on top of each other and the same as the numerical results.

The COMPRO run IT-11 is for an autoclave pressure of 586 kPa and FEP interface. The predicted results for mechanical strains are compared with the experimental results as shown in Figure 5-12. The numerical and experimental results at location C’ are very similar, while the experimental results at C are slightly different. The experimental strains at location C and C’ are fairly similar up to 170 minutes into the cure cycle (this point marked as D1 in the figure). At this point, there is some noise in the readings and the strains in the two locations deviate. After point D1 the curves at C and C’ appear to be spaced apart by a constant shift. Numerical and experimental results at B are very close, while the experimental results at B’ are shifted slightly upward over the entire cure cycle. At location A, the numerical curve falls between the two experimental curves at A and A’. The same discrepancy in experimental results at locations A and A’ exist here as it was for the previous case.

The COMPRO runs IT-10 and IT-12 are for low (103 kPa) and high (586 kPa) autoclave pressures with RA-interface. The contact surface parameters for the RA- interface are similar to the FEP-interface except the coefficient of friction, which is µ = 0.28. The predicted results for mechanical strains are compared with experimental results as shown in Figure 5-13 and Figure 5-14 for low and high autoclave pressures, respectively. For the low pressure case, the experimental results at location A’ are not available. The predicted results for the high pressure case compare rather well with the experimental results.

For the low pressure case, the predicted results capture the behaviour well, but there is a slight shift in the strain values. It is expected that the mechanical strains in the high pressure case is larger than those in the low pressure case. However, the experimental results show the opposite trend. This may be due to some error in strain gauge readings. The predicted results show the right trend and the discrepancy between the predicted and experimental results may be due to this error in strain gauge readings.

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Overall, it can be seen that the developed contact surface is capable of capturing the experimentally observed tool-part interface behaviour throughout the entire cure cycle.

Effect of Initial Resin Modulus

In this study, the effects of the initial resin modulus on the mechanical strain at the interface are investigated. The cases considered here are for high autoclave pressure (586 kPa) and RA-interface. The COMPRO runs IT-13 and IT-14 are for the initial resin moduli of 471.0 kPa and 4.71 kPa. The predicted results from both runs are compared with experimental results in Figure 5-15. From the figure, it can be seen that by lowering the initial resin modulus, mechanical strains become smaller at the hold section of the cure cycle, especially at strain gauges close to the centre of the part. However, the effect of the temperature dip and the mechanical strains after debonding are not affected by the initial resin modulus. The mechanical strains are the same at debonding since the evolved resin modulus at this stage are the same in both cases.

Debonding Front

One of the interesting phenomena is debonding of a composite part from the tool during cool-down. The advancement of debonding is shown in Figure 5-16. The debonding starts at the tip at about 260 minutes of the cure cycle and proceeds linearly toward the centre of the part in 50 minutes. After debonding of the entire part from the tool, there is slight bonding between them due to molecular attraction.

Convergence

Comparing results of different runs (cases IT-12, IT-16, IT-17 and IT-18) for high autoclave pressure and RA-interface, as depicted in Figure 5-17, shows a very good match amongst them. These COMPRO runs included 4- and 8-noded element options with fine and coarse mesh density. Only at debonding event, the 4-noded coarse mesh (case IT-17) shows an expected lower debonding strain.

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5.2 SENSITIVITY ANALYSES ON INSTRUMENTED TOOL EXPERIMENTS

In this study, a set of sensitivity analyses is performed using the first set of experimental results to verify the contact surface parameters developed in the previous chapter and to clarify the importance of each parameter.

In COMPRO with a bonded interface, the input value for the resin initial modulus changes the results (mechanical strains) dramatically. Although not presented in this thesis, changes in the resin initial modulus also have a major effect on results when using a shear layer. Therefore, sensitivity analyses are performed for two different initial resin moduli of 4.71 kPa and 4.71 MPa.

The case considered here is for an autoclave pressure of 586 kPa and RA-interface. The contact surface parameters for the RA-interface are:

0 mm τ ff==0.0kPa ,ττ ff 30.0 kPa , b == 5.0 MPa ,c1 0.65 and µ = 0.28 The above values, which are referred to as the “baseline values”, are obtained from the experimental results done by Twigg [2001]. The sensitivity of each of these parameters is investigated here.

5.2.1 Low Initial Resin Modulus (4.71 kPa)

The COMPRO runs for the sensitivity analysis are listed in Table 5-7. Sensitivity of each parameter is tested by keeping the rest of the parameters constant at the baseline values. All results in this section are compared with predicted mechanical strains obtained from the case IT-14.

Effect of c1

Figure 5-18 shows the effect of c1 on the mechanical strains (cases IT-14-S-1 and 2).

As shown in the figure, changing c1 has small effects on mechanical strains except at

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debonding. As explained in the previous chapter, in order to produce a more general equation, the maximum bonding shear strength is defined at a degree of cure of c =1.00 instead of the actual final degree of cure, which is usually around c ≅ 0.90 .

Thus a slight change in c1 changes the value of maximum bonding shear strength, τb , at a degree of cure of 0.9. To solve this problem, if c1 is varied, the user should also change the maximum bonding shear strength in order to have the same desired bonding shear strength at its final degree of cure, which is less than one. Therefore, it can be concluded that changing c1 has little effect on the mechanical strains at the interface during whole cure cycle.

m Effect of τ ff

m Figure 5-19 shows the influence of the maximum fluid friction shear strength, τ ff , on the mechanical strains at the interface (cases IT-14-S-3 and 4). As shown in the figure,

m τ ff has an appreciable effect on mechanical strains in the heat-up and consequently on the hold section of the cure cycle. At the end of the heat-up, differences between the mechanical strains reach their maximum (22 and 28%), then remain constant during hold and cool-down sections. After debonding, strains become identical, as they are now purely a function of Coulomb friction.

It is important to note that after debonding, the composite part and the tool become two separate structures in contact with each other, and the only interaction is through Coulomb friction at the interface. At this stage, the elastic aluminium tool, which has strain gauges embedded on top of it, has only frictional shear stresses as its external loads. Therefore, all previous interactions from the composite part (which due to its inelastic behaviour has history dependency) and remaining residual stresses in the composite part lose its effects on strain gauges readings after debonding. In other words, while the composite part has its residual stress, strain gauges cannot show them after debonding.

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Effect of τb

Figure 5-20 shows the effect of the maximum bonding shear strength, τb , on the mechanical strain at the interface (cases IT-14-S-5 and 6). As shown in the figure, this variable only changes the time of the occurrence of debonding during cool-down. Although its effect is high (for this section of a mechanical strain curve), it occurs at a time when almost all residual stresses in the composite part have already built up and composite part is almost elastic. Therefore, the maximum bonding shear strength is an insignificant parameter.

Effect of µ

Figure 5-21 shows the influence of the coefficient of Coulomb friction, µ , on the mechanical strains (cases IT-14-S-7 and 8). This parameter affects the timing and amount of mechanical strains at debonding at locations B and C; and the constant value of the residual mechanical strains after debonding. After debonding at the tip (location A), the amount of force transferred toward the centre is based on whatever is being released, which is the debonding force minus the frictional force. By increasing or decreasing the coefficient of friction, friction forces increase or decrease and the magnitude of forces transferred toward the centre diminishes or increases. In such a case, debonding at central locations occur later or sooner, which leads to higher or lower values of mechanical strains at those locations. In any case, since debonding occurs when the composite part is almost elastic, the coefficient of Coulomb friction parameter has less of practical importance.

0 Effect of τ ff

Figure 5-22 shows the effect of the mechanical strains at the interface on the initial

0 fluid friction shear strength (at zero degree of cure), τ ff (cases IT-14-S-9 and 10). This parameter affects the mechanical strains in the heat-up section of a cure cycle

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significantly. However, their values at the end of the fluid friction mechanism, which is very important to mechanical strains, are the same. This parameter appears to be insignificant, but more research on the subject is needed.

When there is a release agent at the interface, in the case of a low initial resin modulus

m it can be concluded that the maximum fluid friction shear strength, τ ff , is the most important contact surface parameter affecting the residual stresses in the composite part, while other parameters merely change the mechanical strain profiles temporarily.

5.2.2 High Initial Resin Modulus (4.71 MPa)

The COMPRO runs for the sensitivity analyses are listed in Table 5-8. Because the effects of varying the maximum bonding shear strength and the coefficient of friction are obvious, they are excluded from this set of case sensitivity analyses. Just as before, for the remaining contact parameters, sensitivity of each parameter is tested by varying that parameter while keeping the rest of the parameters constant at the baseline values (i.e., first row of Table 5-8). All results in this section are compared with predicted mechanical strains obtained from the case IT-12.

Effect of c1

Figure 5-23 shows the effect of c1 on the mechanical strains (cases IT-12-S-1 and 2). In contrast to the case of low initial resin modulus (Figure 5-18), the effect of c1 on mechanical strains is significant. The maximum shear stress that can be transferred through the interface is the minimum of the interface shear strength of (in this case) the fluid friction shear strength and composite part shear stress capability. Composite part shear stress capability is a function of resin modulus. Therefore, when the initial resin modulus is low, the shear stress capability decreases and vice versa. Increasing the resin initial elastic modulus increases the effect of c1 on mechanical strains significantly.

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m Effect of τ ff

m Figure 5-24 shows the effect of the maximum fluid friction shear strength, τ ff , on the mechanical strains at the interface (cases IT-12-S-3 and 4). As for the case of low initial resin modulus, this parameter has a large effect on the mechanical strains during the heat-up section of the cure cycle. At the end of the heat-up, differences between mechanical strains reach their maximum and remain constant during the hold and cool- down sections.

It is clear from a comparison of Figure 5-19 and Figure 5-24 that the maximum fluid friction shear strength has a larger effect on residual mechanical strains as initial resin modulus increases.

0 Effect of τ ff

Figure 5-25 shows the effect of the initial fluid friction shear strength (at zero degree of

0 cure), τ ff , on the mechanical strains at the interface (cases IT-12-S-5 and 6). The effect is insignificant as for the case of low initial resin modulus.

In summary, for low and high initial resin moduli, when there is a release agent at the

m interface the maximum fluid friction shear strength, τ ff , is the most significant contact surface parameter and c1 is the second-most important one. Other parameters merely change the mechanical strain profiles.

5.3 PREDICTION OF PROCESS INDUCED DISTORTIONS

Having established all the contact surface parameters for a given combination of tool, part and interface materials allows us to use the COMPRO code to predict the residual deformation profile of different part geometries (sizes).

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5.3.1 Flat Parts

The experimental results of a flat composite part on an aluminium tool of Twigg [2001] are simulated in COMPRO using both contact and shear layer interfaces. These models are mainly made for comparison of contact and shear layer interfaces results with each other and with experimental results.

5.3.1.1 Models with Contact Interface In this section, the calibrated contact surface parameters for FEP-interface and RA- interface will be used to predict the experimental results of Twigg [2001]. Twigg’s experiments consist of any combination of 3 lengths (300, 600 and 1200 mm), 3 thicknesses (4-, 8- and 16-ply), 2 interface conditions of 2 sheets of FEP and release agent, and autoclave pressures of 103 and 586 kPa.

The finite element model of the part and the tool and the boundary conditions applied on the model are shown in Figure 5-26. Due to symmetry, only half of the problem is modelled.

The contact surface parameters extracted from the instrumented tool experiments are valid here. These parameters are the same as before, except for the degree of cure at which the fluid friction behaviour of the resin ceases. This degree of cure is changed from 0.65 to 0.83 due to the difference in the cure cycle used in this case. It is at this degree of cure in which the resin starts to behave elastically.

Twigg found from his comprehensive experiments on flat unidirectional parts that the maximum part warpage varies with part thickness and length according to the following equation:

l 3 warpage ∝ (5.2) t 2

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Since not all the experimental results exactly follow the average trend observed by Twigg, first the experimental results are adjusted according to Equation (5.2). The COMPRO predictions are then compared with the adjusted experimental results.

Material properties are the same as for the instrumented tool experiment, and are summarized in Table 5-5. COMPRO files for all runs are listed in Table 5-9. Materials and material file names used for the flat part simulations are summarized in Table 5-11, while lay-up designations used in each lay-up files are listed in Table 5-12.

Results from COMPRO with the contact surface option are summarized in Table 5-10 for 8-noded element runs. Warpage predictions follow the experimentally observed trend well. For instance, in case FP-15, the autoclave pressure is 103 kPa and there is a release agent at the interface. The part is 600 mm long and 8 plies in thickness. Warpage from the adjusted experimental results is -1.11 mm, while COMPRO predicts -1.14 mm. Thus the match is good. Four 3-D bar charts as shown in Figure 5-27 and Figure 5-28 compare COMPRO predictions and the adjusted experimental results.

However, there are some differences between COMPRO predictions and the adjusted experimental results, some of which seems to be due to the discrepancies in experiments. For example, in the case of the longest part (1200 mm) and 4-ply thickness, Twigg measured 42.04 mm of warpage. In contrast, Petrescue [2005] measured 24.0 mm of warpage in a separate experiment. This discrepancy exists for very thin and long parts, since a small change in experimental conditions can change the results dramatically. Also, there is another set of discrepancies amongst the shortest and thickest parts. In these parts, warpages are very small and sometimes their measurements are inaccurate. Otherwise, results of COMPRO agree well with the experiments, and the average difference for all cases is about 28%. Even though some times the absolute values of warpages are not in good agreement, the important point is that the predictions using the contact surface agree with the experimental trends (i.e., they satisfy the scaling law, Equation (5.2)) as shown in Figure 5-27 and Figure 5-28.

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5.3.1.2 Models with Shear Layer Interface Another set of runs is designed with the previously used shear layer as an interface condition to enable us to better understand the benefits of having the new sliding surface at the interface. Project files and their corresponding COMPRO files are summarized at the bottom of Table 5-9. The shear layer model in COMPRO is depicted in Figure 5-29. In both cases the computations are run with and without the LD option. The results are summarized in Table 5-13, along with results of a closed- form solution performed by Arafath [2007], which is a simplified plane stress version of COMPRO employing a shear layer without the LD option.

A shear modulus of 7.5 kPa is chosen for shear layer material such that experimental warpage of a 600 mm part with 8-ply thickness will be the same as the warpage prediction of a COMRO shear layer with the LD option. Results of all other runs with the same shear layer material are then obtained.

As expected, the closed-form solution and the COMPRO shear layer without the LD option have very similar results. While these results are close to the COMPRO shear layer with the LD option for short length parts, they become very different for the longest parts. For a 300 mm part with 4-ply thickness, the experimental warpage is 200% of that predicted by COMPRO shear layer without the LD option, while for a 1200 mm part with 8-ply thickness this ratio becomes 40%. These results suggest that even for a simple flat part case, the predictions of COMPRO with a shear layer interface are inconsistent.

Three additional cases have been modelled in COMPRO with a shear layer interface, using a higher initial resin modulus of 10 MPa. The part’s length is 1200 mm, with three different thicknesses of 4-, 8- and 16-ply. The results of these runs are summarized in Table 5-14. No trend is evident. Nor is there a difference between COMPRO results with and without LD options when thickness changes. These examples demonstrate that a shear layer at the interface is not an acceptable option for modelling and simulating the real behaviour of a curing composite part on a tool.

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5.3.2 Angle Parts

Verification and validation of a new constitutive model for the contact interface between the tool and the part and its implementation in COMPRO using relatively simple and small geometry composite parts have been achieved and demonstrated in previous sections. Proceeding to curved-geometry parts is the next step toward establishing and extending the applicable range of the proposed model to more complex cases.

Warpage is defined as an undesired deformation due to the curing process of a flat composite part while spring-in is the change of the angle at the corners of a composite part (after tool removal). Separation of warpage and spring-in in curved parts helps one better understand the components of residual deformation. Two sets of examples are studied here: L-shaped and C-shaped parts.

5.3.2.1 L‐shaped Parts Albert [2001] manufactured and tested several L-shaped composite parts on an aluminium tool to investigate separation of spring-in at corners and warpage at flanges. She suggested that the total spring-in is a combination of these two factors, as shown in Figure 5-30. Several of her experiments are modelled in COMPRO using a contact

m interface with τ ff = 30.0 kPa at a degree of cure of c1 = 0.83. Project file names and COMPRO input file names are summarized in Table 5-15. Material file names and properties are summarized in Table 5-16 to Table 5-18. Part geometry, tool’s material and number of hold in the cure cycle are summarized in Table 5-20. Two 1- and 2-hold cure cycles, as shown in Table 5-19 , are used to cure the composite parts. In order to have a plane stress condition for this set of examples, all thermal and cure shrinkage strains in the out-of-plane direction are assumed to be zero, except in the quasi- symmetrical case.

Total spring-in and their components are presented in Table 5-22. By comparing these results with the experimental results for cases involving both 1- and 2-hold cure cycles,

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we observe an excellent match. Only in the case of quasi-symmetrical lay-up the results are relatively different.

Large strain values:

The amount of the total strain (based on dividing the total stress in global direction by its corresponding shear modulus) is determined at the first integration point of the first element in the straight segment of the composite part right after the curved segment ends. This element (number 93) is at the interface with the aluminum tool as depicted in Figure 5-31. The shear strain at this integration point reaches minimum and maximum values of -11 and 96%, respectively, as shown in Figure 5-32. The shear modulus builds up rapidly as curing advances. At this stage the shear strain drops to a very small value and remains small until the end of the cure cycle.

This high value of shear strain occurs while the resin modulus has not yet developed. Due to the tool expansion and resin cure shrinkage, the element in the straight segment of the composite part goes around the corner and the square elements adjacent to that corner undergo high distortions. This behaviour shows the need for large deformation option in COMPRO as well.

5.3.2.2 C‐shaped Parts Albert [2001] also manufactured and tested several C-shaped composite parts on an aluminium tool to investigate the same idea of separation of spring-in into corner components, webs and flange warpage components. Total spring-in as depicted in Figure 5-33 can be expressed as:

θt= θθ c++ w−− flange θ w web (5.3) The total rotation of the flange is equal to:

θv=+θθ t h =+ θθ c w−− flange + θ w web + θ h (5.4)

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where θt , θv , θh , θc ,θw− flange and θwweb− are total, vertical and horizontal rotations of flange and web, respectively and rotations due to corner effect, due to warpage at flange and due to warpage at web, respectively.

Some of Albert’s experiments are modelled in COMPRO using a contact interface with

m τ ff = 30.0 kPa at a degree of cure of c1 = 0.83. Project file names and COMPRO input file names are summarized in Table 5-15. Material file names and properties are summarized in Table 5-16 to Table 5-18. Part geometry, tool’s material and number of hold in the cure cycle are summarized in Table 5-20. Two 1- and 2-hold cure cycles, as shown in Table 5-19, are used to cure the composite parts. In order to have a plane stress condition for this set of examples, all thermal and cure shrinkage strains in the out-of-plane direction are assigned a zero value.

C-shaped parts are characterized by relatively complex behaviour during the cure process. Their final deformed shapes support this statement. Distinctly different from L-shaped parts, flanges remain straight with no deformation, while webs undergo a large amount of warpage (see Albert and Fernlund [2002]).

Total spring-ins and their components are presented in Table 5-23. Predicted results and the experimental results are slightly different from each other in both cases of 1- and 2- hold cure cycles. The numerical results are less than experimental results only for the case of a 16-ply composite part on a steel tool. It is expected that for steel tools with smaller coefficients of thermal expansion, total warpage should become smaller than that of aluminium tools. Although this trend is seen in experimental and numerical results, several exceptions do exist.

5.4 NUMERICAL CASE STUDIES

Here we present numerical exercises where the focus is the prediction of distortion in realistic size composite structures for which there are no experimental results available.

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The objective is to test the capability of the numerical tool that has been developed in this study.

5.4.1 Semi Circular Parts

A flat part 1200 mm in length and 8-ply thick is modelled similar to case FP-18 in the previous section, but with tool thickness of 12.0 mm instead of 6.35 mm. This project file is called FP-18C, and its mesh is shown in Figure 5- 34. COMPRO input-file names for these projects are summarized in Table 5-15. The composite material is T- 800H/3900-2 with initial resin modulus of 100 kPa; and other properties summarized in Table 5-18, while the tool material properties are summarized in Table 5-17. The contact interface parameters are the same as in the case FP-18, with the maximum fluid friction shear strength of 30.0 kPa at a degree of cure of 0.83. The cure cycle is the same as for the FP-18 project.

In addition, two half-circle parts are modelled on convex and concave tools separately, with a 1200 mm perimeter. Tool thickness is 12.0 mm, and the interface radius is 382.0 mm for both cases. The COMPRO input file names, material file names, and composite and tool properties are summarized in Table 5-15 to Table 5-18. These cases are being re-run for the maximum fluid friction shear strengths of 0.0, 10.0, 20.0, 30.0 and 60.0 kPa. All other contact interface parameters are the same as for the flat part modelled previously.

Warpage results for the equivalent flat part geometries (two different tool thicknesses) are summarized in Table 5-24. Deformation results for half-circle cases are determined based on the distance of the tip from its original location, summarized in Table 5-25. Deformed shapes are depicted in Figure 5-35. These results are broken down into two components of warpage and spring-in based on these two linear equations:

Convex tool: wwwtw= + θ (5.5)

Concave tool: wwwtw= −+θ (5.6)

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where wt , ww and wθ are total tip displacement, displacement due to warpage and displacement due to corner effect, respectively.

The flat part results suggest that a thicker tool increases the amount of warpage. The half-circle part results suggest that the above two linear equations are not valid. When the interfacial shear strength is zero, there is still warpage due to the tool-part interaction through normal forces at the interface. In a convex tool, by increasing the interfacial shear strength, the total deformation at first increases but then decreases. This indicates the behaviour at the interface is similar to the one for perfect bonding condition at the interface, which always displays asymptotic deformation.

To better understand this behaviour, the results of perfect bonding for three different lengths but identical thickness composite parts on an aluminium tool are discussed here. In the previous cases of FP-1, FP-4 and FP-7, the interface is changed to perfect bonding condition. As it can be seen in Figure 5-36, total downward warpages at the tip remain the same for all three models with different lengths. This behaviour is similar to the shear lag boundary layer behaviour. These cases prove that perfect bonding does not necessarily yield maximum warpage. Also, we can see from these cases that in half- circle cases, by increasing the interfacial shear strength, we approach virtual perfect bonding behaviour.

5.4.2 Large Scale Flat Parts

Todays, large one-piece composite structures are being employed in primary structures of aircraft. For example, the Boeing 787 aircraft has a 30 m one-piece wing, with a cross-section of up to several metres. In this section, two ten-metre flat parts with thicknesses of 32 and 64 plies are modelled while assuming release agent condition for the interface. A 2-hold cure cycle, as shown in Table 5-19, is used to cure the composite parts. Project file names are summarized in Table 5-15. Material names and properties are also summarized in Table 5-16 to Table 5-18, while lay-up designations used in each lay-up files are listed in Table 5-21. As in the case of L- and C-shaped

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Deformed shapes of these long flat parts are depicted in Figure 5-37. We observe a similar behaviour as we saw in the perfect bonding case (a perfect bonding interface between the tool and the flat part). It seems that if flat parts are sufficiently long then shear lag boundary layers (development lengths) are much shorter than the part’s length. Therefore, similar to the perfect bonding case, composite parts tend to remain virtually flat for much of their length from the center line of the part after tool removal. However, they eventually bend downward at the tip.

A thicker part increases the shear lag length, but the total warpage doesn’t necessarily increase. For example, in the 32- and 64-ply thickness cases, the total warpages are almost equal. It can be stated that for a length shorter than the shear lag length of the thinner part, the warpage of the thinner part is much greater than that of the thicker part. However, the warpage of the thinner part remains constant when the parts’ length surpasses the shear lag length of the thinner part. Warpage of the thicker part still increases as its length increases. When the length of the thicker part is greater than its own shear lag length, the thick part’s warpage remains constant. In this example, it seems that firstly the five meter length of both thin and thick parts is much greater than the shear lag length of both and secondly the warpages are the same.

Based on COMPRO’s results, the shear lag lengths for 32- and 64-ply CFRP composite parts on an aluminium tool are calculated to be approximately 1700 and 2400 mm, respectively.

5.5 SUMMARY AND DISCUSSION

Twigg [2001] showed that for the same combination of tool and part materials to obtain reasonable prediction of the experimental results when using COMPRO with a shear layer at the interface, the modulus of the shear layer must change for different

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geometries and interface conditions. This is clearly undesirable as for a given material set (tool and part) the model parameters should not be dependent on the geometry that is being modelled.

Modelling of the strain histories in the instrumented tool experiments

The original version of COMPRO with a shear layer was shown to be incapable of capturing magnitude and the time history of the mechanical shear strains at all times during a cure cycle. Thus, all other internal variables derived from these strains such as stresses and internal forces are expected to be erroneous.

Although the shear layer interface is an improvement over perfect bonding, it is not reliable for modelling the interface behaviour when attempting to capture the part-tool interaction. The introduction of a contact surface at the interface between the part and the tool is a more viable solution for autoclave processing of a composite part. It has been shown here that COMPRO with the contact surface option developed in this study yields the correct mechanical strains at all times during a cure cycle.

Fluid friction, bonding and Coulomb friction are mechanisms occurring at the interface of part and tool during curing of composite parts in an autoclave process. Most residual stresses in a composite part are built up during the heat-up section of a cure cycle, particularly during the final stages of heat-up. They normally remain constant during a cure cycle hold.

The most important contact surface parameter is the maximum fluid friction shear strength, which is the dominant factor in the heat-up portion of a cure cycle. The second most important contact surface parameter is the degree of cure at which fluid friction shear stress reaches its maximum (that is, when the fluid friction phase terminates).

The initial fluid friction shear strength at a zero degree of cure has some effects on mechanical strains during the heat-up section, but it does not affect the mechanical strains or final residual stresses significantly. This parameter must be researched

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When there are FEP layers at the interface, the coefficient of Coulomb friction between FEP plies becomes an important contact surface parameter. The coefficient of friction can reduce the interfacial shear strength during the heat-up stage, since the interfacial shear strength is the minimum of the fluid friction shear strength of the resin-rich layer and the friction shear strength between FEP plies.

Modelling of warpage in flat part experiments

Numerical results of COMPRO with a contact interface agree well with experimental results except for very thick or very long flat parts. This discrepancy occurs when: (1) Parts are long and thin, and therefore warpage is large. Due to the strongly non-linear nature of the problem, a small change in experimental conditions changes the warpage amount significantly. (2) Parts are short and thick, and therefore warpage is small. Measurements can be difficult and errors can be comparable to the average warpage.

Only a different cure cycle can change the time when the fluid friction phase terminates. This is the case because we made the fluid friction a function of the degree of cure, whereas it is really a function of viscosity, which in turn is a function of temperature and the degree of cure. Since viscosity is an increasing function of the degree of cure and a decreasing function of temperature, a certain viscosity at which the fluid friction phase ends can be reached at a different degree of cure when the cure cycle is varied.

The contact surface option appears to be a representative mechanism for the tool-part interface. Part deformations can be predicted for different geometries, with the same materials and interface conditions if contact parameters are determined once. Here we showed that the contact surface parameters calibrated using instrumented tool

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Warpage in flat parts increases with part length. Warpage stops increasing after the length surpasses a specific length termed “shear lag length.” This behaviour is similar to cases with perfect bonding condition at the interface. Shear lag length increases as part thickness increases.

Although corner spring-in and warpage are different and distinct phenomena, they interact with each other – particularly on curved parts. Parts with flat sections and corners such as L-shaped parts have a total spring-in involving linear addition of corner spring-ins and spring-in due to warpage in the flat section.

Curved parts such as half-circle parts have a total spring-in comprised of warpage and spring-in of curved sections; these components cannot be added linearly to obtain the total spring-in because they interact with each other.

In curved parts with the assumption of zero fluid friction shear strength at the interface, the tool-part interaction exists and causes warpage due to the existence of normal surface forces.

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Table 5-1: Computational runs for the instrumented tool set-up.

COMPRO’s options

† (Pa) 0

Interface r Interface Element parameter E Large Large Others Others Pressure (kPa) (kPa) Pressure Project file (run) deformation deformation

IT-1 No Perfect bonding - 4-noded 103 4.71E7 IT-2 Yes Perfect bonding - 4-noded 103 4.71E7 IT-3 No Perfect bonding - 4-noded 103 4.71E5 IT-4 Yes Perfect bonding - 4-noded 103 4.71E5 IT-5 No Shear layer G=1.0E5 (Pa) 4-noded 103 4.71E5 IT-6 Yes Shear layer G=1.0E5 (Pa) 4-noded 103 4.71E5 IT-7 No Shear layer G=1.0E5 (Pa) 4-noded 586 4.71E5 IT-8 Yes Shear layer G=1.0E5 (Pa) 4-noded 586 4.71E5 IT-9 Yes Contact FEP sheets 4-noded 103 4.71E6 IT-10 Yes Contact Release agent 4-noded 103 4.71E6 IT-11 Yes Contact FEP sheets 4-noded 586 4.71E6 IT-12 Yes Contact Release agent 4-noded 586 4.71E6 IT-13 Yes Contact Release agent 4-noded 586 4.71E5 IT-14 Yes Contact Release agent 4-noded 586 4.71E3 IT-15 Yes Contact Release agent 4-noded 586 4.71E7 IT-16 Yes Contact Release agent 8-noded 586 4.71E6 Contact IT-17 Yes Release agent 4-noded 586 4.71E6 Contact mesh IT-18 Yes Release agent 8-noded Coarse 586 4.71E6

† Initial resin modulus

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Table 5-2: Files used in each project file of COMPRO runs (cases) for instrumented tool experiments.

Cure Project Control† Boundary condition Lay-up+ Material PATRAN ‡ cycle* ₪ IT-1 CTL1 BCI1 CYC1 MAT1 PAT1 IT-2 CTL1 BCI1 CYC1 MAT1 PAT1 IT-3 CTL1 BCI1 CYC1 MAT2 PAT1 IT-4 CTL1 BCI1 CYC1 MAT2 PAT1 IT-5 CTL1 BCI2 CYC1 MAT3 PAT2 IT-6 CTL1 BCI2 CYC1 MAT3 PAT2 IT-7 CTL2 BCI2 CYC2 MAT3 PAT2 IT-8 CTL2 BCI3 CYC2 MAT3 PAT2 IT-9 CTL2 BCI4 CYC3 MAT4 PAT1 LAY1 IT-10 CTL2 BCI5 CYC1 MAT4 PAT1 IT-11 CTL2 BCI6 CYC4 MAT4 PAT1 IT-12 CTL2 BCI7 CYC2 MAT4 PAT1 IT-13 CTL2 BCI7 CYC2 MAT5 PAT1 IT-14 CTL2 BCI7 CYC2 MAT6 PAT1 IT-15 CTL2 BCI7 CYC2 MAT1 PAT1 IT-16 CTL2 BCI7 CYC2 MAT4 PAT3 IT-17 CTL2 BCI7 CYC2 MAT4 PAT3 IT-18 CTL2 BCI7 CYC2 MAT4 PAT3 † Refer to COMPRO file which controls minimum and maximum time steps, number of time steps, frequency of writing the outputs etc. ‡ Refer to COMPRO file which contains stress and thermal boundary conditions. * Temperature profile is shown in the same figure of the results. Pressure is constant and its value is mentioned in the label of the same figure as well. + Refer to 16 unidirectional ply laminate.  Refer to Table 5-4. ₪ Refer to COMPRO file which contains mesh and nodes in each boundary (see Table 5-3).

Table 5-3: Mesh properties used for each PATRAN files.

Thin aluminium tool Composite part Shear layer

PATRAN file (mm) (mm) (mm) (mm) (mm) (mm) # of # of elem. # of # of elem. # of # of elem. Elem. size Elem. size Elem. size PAT1 300 x 1 1.0 x 0.7624 300 x 4 1.0 x 0.8 - - PAT2 300 x 1 1.0 x 0.7624 300 x 4 1.0 x 0.8 300 x 1 1.0 x 4.0 PAT3 150 x 1 1.0 x 0.7624 150 x 4 2.0 x 1.6 - -

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Table 5-4: Material names used in each material files for instrumented tool runs.

Material file Thin aluminium Tool† Shear layer† Composite part‡ MAT1 Tool2 - [0]16 T-800H/3900-2 MAT2 Tool2 - [0]16 T-800H/3900-2 -E5 MAT3 Tool2 SL1 [0]16 T-800H/3900-2 -E5 MAT4 Tool2 - [0]16 T-800H/3900-2 -E6 MAT5 Tool2 - [0]16 T-800H/3900-2 -E5 MAT6 Tool2 - [0]16 T-800H/3900-2 -E3 MAT7 Tool1 - [0]16 T-800H/3900-2 † refer to Table 5-6. ‡ refer to Table 5-5.

Table 5-5: Properties of composite materials used in COMPRO runs.

Fiber properties Resin properties Composite properties

3

Material† 0 13 23 ∞ (µ/°C) (µ/°C) =CTE (GPa) υ (MPa) 1

υ υ υ (GPa) 2 (GPa) (GPa) 1 3 0 ∞ (µ/°C) (µ/°C) 13 r r E E G E E CTE CTE

T-800H/3900-2 47.1 T-800H/3900-2 -E6 4.71 T-800H/3900-2 -E5 0.471 4.71 4.71 29.5 T-800H/3900-2 0.20 0.25 0.50 0.37 -E3 210.0 17.24 27.60 0.00471 0.0374 0.0374 T-800H/3900-2 -1E5 0.1 T-800H/3900-2 -1E7 10.0 †Resin modulus development, kinetics and Poisson’s ratio models in COMPRO are 2, 6 and 2, respectively.

Table 5-6: Properties of other materials used in COMPRO runs.

G E E CTE CTE CTE Material 13 E (GPa) 2 3 1 2 3 υ (kPa) 1 (GPa) (GPa) (µ/°C) (µ/°C) (µ/°C) Tool1 0.327 69.0 69.0 69.0 23.6 23.6 23.6 Tool2 0.327 69.0 69.0 69.0 23.6 0.0 23.6 SL1 0.0 100.0 1.0E-4 1.0E-4 10.0 31.32 0.0 0.0 SL2 0.0 7.5 15.0E-6 69.0 69.0 23.6 0.0 0.0

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Table 5-7: Sensitivity analysis for case IT-14; varied parameters are the fluid friction

shear strength when the degree of cure is zero or c1 , the bonding shear strength when the material is 100% cured and the coefficient of friction.

0 m Project file τ ff (kPa) c1 τ ff (kPa) τ b (MPa) µ IT-14 0.0 0.65 30.0 5.0 0.28 IT-14-S-1 0.0 0.50 30.0 5.0 0.28 IT-14-S-2 0.0 0.75 30.0 5.0 0.28 IT-14-S-3 0.0 0.65 15.0 5.0 0.28 IT-14-S-4 0.0 0.65 50.0 5.0 0.28 IT-14-S-5 0.0 0.65 30.0 3.0 0.28 IT-14-S-6 0.0 0.65 30.0 7.0 0.28 IT-14-S-7 0.0 0.65 30.0 5.0 0.20 IT-14-S-8 0.0 0.65 30.0 5.0 0.40 IT-14-S-9 10.0 0.65 30.0 5.0 0.28 IT-14-S-10 20.0 0.65 30.0 5.0 0.28

Table 5-8: Sensitivity analysis for case IT-12; varied parameters are the fluid friction

shear strength when the degree of cure is zero or c1 , the bonding shear strength when the material is 100% cured and the coefficient of friction.

0 m Project file τ ff (kPa) c1 τ ff (kPa) τ b (MPa) µ IT-12 0.0 0.65 30.0 5.0 0.28 IT-12-S-1 0.0 0.50 30.0 5.0 0.28 IT-12-S-2 0.0 0.75 30.0 5.0 0.28 IT-12-S-3 0.0 0.65 15.0 5.0 0.28 IT-12-S-4 0.0 0.65 50.0 5.0 0.28 IT-12-S-5 10.0 0.65 30.0 5.0 0.28 IT-12-S-6 20.0 0.65 30.0 5.0 0.28

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Table 5-9: Files used in each project files of COMPRO runs for warpage simulation of flat parts.

Boundary Cure Project (run) Control† Lay-up+ Material PATRAN₪ condition‡ cycle*

FP-1, FP-21 LAY2 PAT4 FP-2, FP-22 LAY3 PAT5 FP-3, FP-23 LAY4 PAT6 FP-4, FP-24 LAY2 PAT7 FP-5, FP-25 CTL3 BCI8 CYC5 LAY3 MAT8 PAT8 FP-6, FP-26 LAY4 PAT9 FP-7, FP-27 LAY2 PAT10 FP-8, FP-28 LAY3 PAT11 FP-9, FP-29 LAY4 PAT12 FP-11, FP-31 LAY2 PAT4 FP-12, FP-32 LAY3 PAT5 FP-13, FP-33 LAY4 PAT6 FP-14, FP-34 LAY2 PAT7 FP-15, FP-35 CTL3 BCI8 CYC6 LAY3 MAT8 PAT8 FP-16, FP-36 LAY4 PAT9 FP-17, FP-37 LAY2 PAT10 FP-18, FP-38 LAY3 PAT11 FP-19, FP-39 LAY4 PAT12 FP-S-1 LAY2 PAT13 FP-S-2 LAY3 PAT14 FP-S-3 LAY4 PAT15 FP-S-4 LAY2 PAT16 FP-S-5 CTL3 BCI9 CYC5 LAY3 MAT9 PAT17 FP-S-6 LAY4 PAT18 FP-S-7 LAY2 PAT19 FP-S-8 LAY3 PAT20 FP-S-9 LAY4 PAT21 FP-S-17 LAY2 PAT19 FP-S-18 CTL3 BCI9 CYC5 LAY3 MAT10 PAT20 FP-S-19 LAY4 PAT21 † Refer to COMPRO file which controls minimum and maximum time steps, number of time steps, frequency of writing the outputs etc. ‡ Refer to COMPRO file which contains stress and thermal boundary conditions. * Temperature profile is shown in the same figure of the results. Pressure is constant and its value is mentioned in the label of the same figure as well. + Refer to Table 5-12.  Refer to Table 5-11. ₪ Refer to COMPRO file which contains mesh and nodes in each boundary.

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Table 5-10: COMPRO sliding prediction for warpage when initial resin modulus is 1.0E5 (Pa) and experimental raw and adjusted results.

COMPRO Autoclave Part Twigg’s Twigg’s COMPRO # of Percentage project file pressure & length experimental raw adjusted prediction plies difference (run) interface (mm) data (mm) results (mm) (mm) FP-1 4 -0.78 -0.64 -0.88 38 FP-2 300 8 -0.25 -0.16 -0.22 40 FP-3 16 -0.08 -0.04 -0.05 24 FP-4 4 -4.09 -5.10 -4.99 -2

FP-5 & 600 8 -1.23 -1.27 -1.47 15

FP-6 kPa 586 16 -0.11 -0.32 -0.40 26

FP-7 Release Agent 4 -42.04† -40.77 -24.04 -41 FP-8 1200 8 -7.71 -10.19 -6.42 -37 FP-9 16 -1.58 -2.55 -2.45 -4 FP-11 4 -0.67 -0.55 -0.73 -31 FP-12 300 8 -0.26 -0.14 -0.18 -26 FP-13 16 -0.07 -0.03 -0.03 0 FP-14 4 -2.03 -4.44 -3.82 14

FP-15 & 600 8 -0.97 -1.11 -1.14 -3

FP-16 kPa 103 16 -0.31 -0.28 -0.30 -9

FP-17 Release Agent 4 -15.72 -35.49 -17.50 51 FP-18 1200 8 -2.67 -8.87 -8.60 36 FP-19 16 -1.56 -2.22 -1.93 13 FP-21 4 -0.43 -0.50 -0.71 40 FP-22 300 8 -0.15 -0.13 -0.20 59 FP-23 16 -0.16 -0.03 -0.05 66 FP-24 4 -3.94 -4.02 -3.88 -3

FP-25 & 600 8 -0.86 -1.00 -1.30 30

FP-26 kPa 586 16 -0.28 -0.25 -0.38 53 FP-27 4 -39.6 -32.2 -19.92 -38 Two FEP Layers Layers Two FEP FP-28 1200 8 -8.97 -8.04 -5.67 -30 FP-29 16 -1.31 -2.01 -2.39 19 FP-31 4 -0.70 -0.47 -0.55 -18 FP-32 300 8 -0.13 -0.12 -0.16 -37 FP-33 16 -0.04 -0.03 -0.04 -39 FP-34 4 -3.84 -3.72 -2.66 29

FP-35 & 600 8 -0.81 -0.93 -1.01 -8

FP-36 kPa 103 16 -0.12 -0.23 -0.30 -30 FP-37 4 -29.1 -29.8 -9.13 69 Two FEP Layers Layers Two FEP FP-38 1200 8 -7.26 -7.45 -4.62 38 FP-39 16 -1.18 -1.86 -1.88 -1 † Petrescue’s [2005] experiments resulted in 24.00 mm warpage.

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Table 5-11: Material designation used in each material files for flat part runs.

Material file Aluminium tool † Shear layer† Composite part‡ MAT8 Tool2 - T-800H/3900-2 -1E5 MAT9 Tool2 SL2 T-800H/3900-2 -1E5 MAT10 Tool2 SL2 T-800H/3900-2 -1E7 † Refer to Table 5-6. ‡ Refer to Table 5-5.

Table 5-12: Lay-up designation used in each Lay-up files for flat part runs

Lay-up file Plies†

LAY2 [0]4

LAY3 [0]8

LAY4 [0]16 † Each ply has 0.2 mm thickness.

Table 5-13: COMPRO prediction for warpage when initial resin modulus is 1.0E5 (Pa). A) Sliding option; B) Shear layer with large deformation option; C) Shear layer without large deformation option; D) Closed-form solution (see Arafath [2007]).

COMPRO COMPRO, COMPRO, Twigg’s Closed- Part project file shear layer shear layer COMPRO # of experimental form length (run) for with LD without contact plies raw data solution (mm) shear layer option LD option (mm)‡ (mm) (mm) cases† (mm) (mm) 4 FP-S-1 -0.31 -0.34 -0.78 -0.88 -0.44 300 8 FP-S-2 -0.10 -0.10 -0.25 -0.22 -0.12 16 FP-S-3 -0.03 -0.03 -0.08 -0.05 -0.03 4 FP-S-4 -2.86 -4.29 -4.09 -4.99 -5.30 600 8 FP-S-5 -1.21 -1.55 -1.23 -1.47 -1.73 16 FP-S-6 -0.38 -0.45 -0.11 -0.40 -0.48 4 FP-S-7 -8.34 -31.86 -42.04 -24.04 -44.83 1200 8 FP-S-8 -6.62 -17.30 -7.71 -6.42 -20.39 16 FP-S-9 -3.17 -6.23 -1.58 -2.45 -6.66 † Project files are the same for shear layer cases with and without LD option. ‡ Results are imported from Table 5-10.

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Table 5-14: COMPRO prediction for warpage when initial resin modulus is as high as 1.0E7 (Pa). A) Shear layer with large deformation option; B) Shear layer without large deformation option.

COMPRO Twigg’s COMPRO, COMPRO, project file Part # of experimental shear layer shear layer (run) for length plies raw data with LD without LD shear layer (mm) (mm) option (mm) option (mm) cases FP-S-17 4 -42.04 -0.452 -0.453 FP-S-18 1200 8 -7.71 -0.501 -0.501 FP-S-19 16 -1.58 -0.436 -0.436

Table 5-15: Project file names for semi circle male and female tools and its same size flat part, and L-* and C-shaped parts.

Project Boundary Control† Cure cycle* Lay-up+ Material PATRAN₪ (run) condition‡ FP-18C1 CTL4 BCI8 CYC6 PAT11 FP-18C CTL5 BCI9 CYC6 MAT8 PAT22 LAY3 Circle-M CTL5 BCI9 CYC6 PAT23 Circle-F CTL5 BCI9 CYC6 PAT24 FP-LF1 CTL5 BCI9 CYC8 LAY11 PAT30 MAT8 FP-LF2 CTL5 BCI9 CYC8 LAY12 PAT31 LS1 CYC8 LAY8 PAT25 LS2 CYC8 LAY8 PAT26 MAT12 LS3 CYC9 LAY9 PAT27 CTL6 BCI10 LS4 CYC9 LAY8 PAT25 LS5 CYC8 LAY8 PAT25 MAT13 LS6 CYC9 LAY8 PAT25 CS1 CYC9 LAY8 PAT28 CS2 CYC8 LAY8 MAT12 PAT28 CS3 CYC8 LAY10 PAT29 CTL7 BCI10 CS4 CYC9 LAY8 PAT28 CS5 CYC8 LAY8 MAT13 PAT28 CS6 CYC8 LAY10 PAT29 † Refer to COMPRO file which controls minimum and maximum time steps, number of time steps, frequency of writing the outputs etc. ‡ Refer to COMPRO file which contains stress and thermal boundary conditions. * Cure cycle 8 and 9 are summarized in Table 5-19. Temperature profile of cure cycle 6 is shown in the same figure of the results. Pressure is constant and its value is mentioned in the label of the same figure or table as well. + Refer to Table 5-20 and Table 5-21.  Refer to Table 5-16. ₪ Refer to COMPRO file which contains mesh and nodes in each boundary.

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Table 5-16: Material file names.

Material file Tool material† Composite material‡ MAT8 TOOL2 T-800H/3900-2 -1E5 MAT12 TOOL3 T-800H/3900-2 -1E5u MAT13 TOOL4 T-800H/3900-2 -1E5u † Refer to Table 5-18. ‡ Refer to Table 5-17.

Table 5-17: Tool material properties.

E CTE CTE CTE Material 1 2 3 υ (GPa) (µ/°C) (µ/°C) (µ/°C) TOOL2 0.327 69.0 23.6 0.0 23.6 TOOL3 0.327 6900.0 23.6 0.0 23.6 TOOL4 0.299 200.0 12.5 0.0 12.5

Table 5-18: Composite material properties.

Fiber properties Resin properties Composite properties

0 ∞ Material E1 E3 G13 Er E CTE1 CTE2 CTE3 0 ∞ r υ13, υ23 υ , υ (GPa) (MPa) (µ/°C) T-800H/3900-2 -1E5 210 17.24 27.6 0.2, 0.25 0.5, 0.37 0.1 4710 .0374 29.5 29.5 T-800H/3900-2 -1E5u 210 17.24 27.6 0.2, 0.25 0.5, 0.37 0.1 4710 .0374 54.0 54.0

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Table 5-19: Description of one- and Two-hold cure cycles(CYC9 and CYC8, respectively) used for L- and C- shape and long flat parts of 10.0 m length.

1-hold cure cycle (CYC9) Apply a vacuum of -101 kPa Pressurize to 586 kPa (release vacuum when autoclave pressure reaches 103 kPa) When pressure reaches 448 kPa, heat to 180˚C at a rate of 2.2˚C/min Hold for 140 min (after lagging thermocouple reaches 174˚C) Cool down to room temperature Depressurize when the lagging temperature reaches 82˚C 2-hold cure cycle (CYC8) Apply a vacuum of -101 kPa Pressurize to 586 kPa (release vacuum when autoclave pressure reaches 103 kPa) When pressure reaches 448 kPa, heat to 135˚C at a rate of 2.2˚C/min Hold for 160 min (after lagging thermocouple reaches 130˚C) Heat to 180˚C at a rate of 2.2˚C/min Hold for 120min (after lagging temperature reaches 174˚C) Cool down to room temperature Depressurize when the lagging temperature reaches 82˚C

Table 5-20: Part geometry, Tool’s material and number of holds in the cure circle for L- and C-shaped composite parts.

Flange Project length Lay-up† Tool’s material # of holds file (mm)

LS1 57 [0]8 LAY8 Aluminium, TOOL2 2 LS2 89

LS3 57 [0,45,-45,90]2S LAY9 Aluminium (TOOL2) 1 LS4 Aluminium (TOOL2) 1

LS5 57 [0]8 LAY8 Steel (TOOL4) 2 LS6 Steel (TOOL4) 1 CS1 1 [0] LAY8 Aluminium (TOOL2) CS2 57 8 2

CS3 [0]16 LAY10 Aluminium (TOOL2) 2 CS4 1 [0]8 LAY8 Steel (TOOL4) CS5 57 2

CS6 [0]16 LAY10 Steel (TOOL4) 2 † Each ply has 0.2 mm thickness.

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Table 5-21: Lay-up designation used in each Lay-up files for Long scale flat part runs

Lay-up file Plies†

LAY11 [0]32

LAY12 [0]64 † Each ply has 0.2 mm thickness.

Table 5-22: Warpage results for L-shaped composite parts

Experiment spring-in (°) COMPRO spring-in (°) Experiment COMPRO Project total spring-in total spring-in file ∆ӨWarpage ∆ӨCorner ∆ӨCorner ∆ӨWarpage ∆Өt (°) ∆Өt (°)

LS1 0.80 0.96 0.93 0.89 1.76 1.83 LS5 0.63 0.74 0.93 0.56 1.37 1.50 LS2 1.04 1.10 0.93 1.47 2.14 2.40 LS3 0.07 1.34 1.03 0.12 1.41 1.16 LS4 0.19 0.84 1.05 0.01 1.03 1.05 LS6 0.04 0.88 0.97 0.08 1.02 1.05

Table 5-23: Warpage results for C-shaped composite parts

Experiment spring-in (°) COMPRO spring-in (°) Experiment COMPRO Project total spring-in total spring-in file ∆ӨWarpage ∆ӨCorner ∆ӨCorner ∆ӨWarpage ∆Өt (°) ∆Өt (°) CS1 0.18 0.94 0.93 0.14 1.12 1.07 CS4 0.18 0.86 0.87 0.22 1.04 1.09 0.92 0.97 1.89 CS2 0.85 0.75 1.60 0.79 0.85 1.64 0.55 0.81 1.36 CS5 0.67 0.55 1.22 0.88 0.75 1.63 0.28 0.83 1.11 CS3 0.92 0.26 1.18 0.25 0.95 1.20 0.38 0.77 1.15 CS6 0.78 0.35 1.13 0.37 0.88 1.25

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Table 5-24: Warpage results for 1200 mm flat part, 8-ply thickness.

Project file Tool thickness (mm) Warpage (mm) FP-18C 12.0 11.53 FP-18C1 6.35 8.60

Table 5-25: Warpage results for 1200 mm semi-circle parts on convex and concave tools of 12.0 mm thickness. The line with bold values represents the results of the benchmark interfacial shear strength of 30 kPa.

m Deformation (mm) Deformation components (mm) τ ff

(kPa) Convex tool Concave tool Warpage Corner effect 60.0 6.83 -6.32 6.58 0.25 30.0 8.08 -2.82 5.45 2.63 20.0 9.17 -1.27 5.22 3.95 10.0 3.71 1.23 1.24 2.47 0.0 1.26 4.68 -1.71 2.97

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Strain Gauges at FEP or Release Longitudinal Dir. Agent Interface

FEP or Release 0.76 mm Thin Aluminum Agent Interface Instrumented Tool A B 16 Ply (3.2 mm) 2 Sheets of 0.0254 mm C’ C CFRP Laminate FEP Release Film B’ A’

16 Ply (3.2 mm) CFRP Laminate 6.35 mm Aluminum Baseplate (a)

Line of Symmetry

102mm 102 51 51 102 102 mm

51mm

A’ B’ C’ C B A 51mm

305 mm 305 mm

(b)

Figure 5-1: (a) Schematic of the instrumented tool set-up used by Twigg [2001]. (b) Location of strain gauges

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1500 180 Temperature 1200 150 C’ 900 Line of Symmetry C 120 C) °

600 B’ 90 A’B’ C’ C B A B A’ 300 60

A Temperature ( Mechanical strain (µm/m) strain Mechanical 0 30

-300 0 0 50 100 150 200 250 300 350 (a)

1500 180

Temperature 1200 150

900 120 C) °

600 C 90 C’ B’

300 60 Temperature (

Mechanical strain (µm/m) strain Mechanical B A A’ 0 30

-300 0 0 50 100 150 200 250 300 350 Time (min) (b)

Figure 5-2: Mechanical strains in gauges A, A’, B, B’, C and C’ for autoclave pressure of 586 kPa. (a) Release agent at the interface (b) FEP at the interface. Experimental results are from Twigg [2001].

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1500 180 Temperature 1200 150

900 120 C) °

600 C 90

300 C’ 60 Temperature ( Mechanical strain (µm/m) strain Mechanical A 0 30 B B’ -300 0 0 50 100 150 200 250 300 350 (a)

1500 180

Temperature 1200 150

900 120 Line of Symmetry C) °

600 90 A’B’ C’ C B A

300 60 Temperature ( Mechanical strain (µm/m) strain Mechanical B B’ A 0 A’ 30 C C’ -300 0 0 50 100 150 200 250 300 350 Time (min) (b)

Figure 5-3: Mechanical strains in gauges A, A’, B, B’, C and C’ for autoclave pressure of 103 kPa. (a) Release agent at the interface (b) FEP at the interface. Experimental results are from Twigg [2001].

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Line of symmetry 300 mm

0.76 mm

3.2 mm

Figure 5-4: IT-IP1 is a fine mesh with no shear layer.

Line of symmetry 300 mm 0.76 mm 0.4 mm 3.2 mm

Figure 5-5: IT-IP2 is a fine mesh with shear layer.

Line of symmetry 300 mm

3.2 mm

Figure 5-6: IT-IP3 is a coarse mesh with no shear layer.

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1000 180 Temperature Exp. at A 0 Line of Symmetry 51 102 102 mm Exp. at B, B’, C & C’ 120 m/m) C) µ -1000 51mm ° C B A 51mm

-2000 305 mm 60 Temperature ( -3000 Mechanical strain ( COMPRO at A, B & C; with LD option COMPRO at A, B & C; without LD option -4000 0 0 50 100 150 200 250 300 350 Time (min)

Figure 5-7: Cases IT-1 and IT-2: Mechanical strain versus time for low autoclave pressure (103 kPa). COMPRO results at strain gauges A, B and C with perfect bonding at the interface and a high initial resin modulus (47.1 MPa) are shown for both cases of with and without large deformation options. Experimental mechanical strain histories are also shown at locations A, B, B’, C and C’ for the case of release agent at the interface.

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1000 180 Temperature Exp. at A 0

Exp. at B, B’, C & C’ 120 m/m)

-1000 C) µ °

-2000 Line of Symmetry 51 102 102 mm 60 51mm COMPRO at A, B & C; -3000 without LD option Temperature ( C B A 51mm Mechanical strain ( strain Mechanical 305 mm -4000 0 0 50 100 150 200 250 300 350 Time (min)

Figure 5-8: Case IT-3: Mechanical strain versus time for low autoclave pressure (103 kPa). COMPRO results at locations A, B and C with perfect bonding at the interface and a low initial resin modulus (0.471 MPa) are shown for the case of no large deformation option. Experimental mechanical strain histories are also shown at locations A, B, B’, C and C’ for the case of release agent at the interface.

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1200 180 Temperature Line of Symmetry 900 51 102 102 mm 150 51mm C B A 51mm C) m/m) 600 120 ° µ 305 mm 300 90 Exp. at A, B, B’, C & C’

0 60 Temperature ( -300 COMPRO at C, B & A 30 Mechanical strain ( -600 0 0 50 100 150 200 250 300 350 Time (min)

Figure 5-9: Case IT-5: Mechanical strain versus time for low autoclave pressure (103 kPa). COMPRO results at locations A, B and C with shear layer at the interface are shown here. Experimental mechanical strain histories are also shown at locations A, B, B’, C and C’ for the case of release agent at the interface.

180 Exp. at C & C’ 1200 Temperature Line of Symmetry 51 102 102 mm 150 900 51mm m/m) µ

C B A 51mm 120 C) 600 ° 305 mm 90 300 Exp. at A & A’ Exp. at B & B’ 0 60 Temperature ( -300 30

Mechanical strain( COMPRO at C, B & A

-600 0 0 50 100 150 200 250 300 350 Time (min)

Figure 5-10: Case IT-7: Mechanical strain versus time for high autoclave pressure (586 kPa). COMPRO results at locations A, B and C with shear layer at the interface are shown here. Experimental mechanical strain histories are also shown at locations A, A’, B, B’, C and C’ for the case of release agent at the interface.

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400 180 Temperature Line of Symmetry

m/m) 51 102 102 mm µ

200 COMPRO at C 120 C) 51mm ° C B A 51mm 305 mm 0 60 Temperature ( Mechanical strain ( Exp. at C’ Exp. at C -200 0

400 m/m) µ 200 Exp. at B

0

COMPRO B Mechanical strain ( Exp. at B’ -200

400

Exp. at A m/m) µ 200

0

COMPRO at A Exp. at A’ Mechanical strain ( -200 0 50 100 150 200 250 300 350 Time (min)

Figure 5-11: Case IT-9: Mechanical strain versus time for low autoclave pressure (103 kPa) and FEP layers at the interface. Experimental mechanical strain histories at locations A, A’, B, B’, C and C’; and COMPRO results at locations A, B and C (correspond to the case where sliding option is employed at the interface) are shown here.

-195- Chapter 5 – Validation and numerical case studies

180 Exp. at C

600 Temperature m/m) µ

120 C) °

Exp. at C’ 200 D1 COMPRO at C 60 Temperature ( Mechanical strain ( -200 0

Line of Symmetry Exp. at B’

m/m) 600 51 102 102 mm

µ Exp. at B 51mm C B A 51mm 200 305 mm COMPRO B

Mechanical strain ( -200 m/m)

µ 600

Exp. at A COMPRO at A 200

Mechanical strain ( Exp. at A’ -200 0 50 100 150 200 250 300 350 Time (min)

Figure 5-12: Case IT-11: Mechanical strain versus time for high autoclave pressure (586 kPa) and FEP layers at interface. Experimental mechanical strain histories at locations A, A’, B, B’, C and C’; and COMPRO results at locations A, B and C (correspond to the case where sliding option is employed at the interface) are shown here.

-196- Chapter 5 – Validation and numerical case studies

180 Temperature 900

m/m) Line of Symmetry µ

51 102 102 mm 120 C) ° 500 51mm Exp. at C’ C B A 51mm 305 mm 60 100 Temperature (

Mechanical strain ( COMPRO at C Exp. at C -300 0

900 Exp. at B’ m/m) µ

500

COMPRO B 100

Exp. at B Mechanical strain ( -300

m/m) 900 µ

500 Exp. at A

100 Mechanical strain ( COMPRO at A -300 0 50 100 150 200 250 300 350 Time (min)

Figure 5-13: Case IT-10: Mechanical strain versus time for low autoclave pressure (103 kPa) and release agent at the interface. Experimental mechanical strain histories at locations A, A’, B, B’, C and C’; and COMPRO results at locations A, B and C (correspond to the case where sliding option is employed at the interface) are shown here.

-197- Chapter 5 – Validation and numerical case studies

180

1000 Temperature m/m) Exp. at C’ µ

120 C) °

600 COMPRO at C

Exp. at C 60 200 Temperature ( Mechanical strain ( -200 0

Line of Symmetry 1000 51 102 102 mm m/m)

µ Exp. at B’ 51mm 600 C B A 51mm 305 mm Exp. at B

200 COMPRO B Mechanical strain ( -200

1000 m/m) µ Exp. at A’ 600 Exp. at A

200

Mechanical strain ( COMPRO at A -200 0 50 100 150 200 250 300 350 Time (min)

Figure 5-14: Case IT-12: Mechanical strain versus time for high autoclave pressure (586 kPa) and release agent at the interface. Experimental mechanical strain histories at locations A, A’, B, B’, C and C’; and COMPRO results at locations A, B and C (correspond to the case where sliding option is employed at the interface) are shown here.

-198- Chapter 5 – Validation and numerical case studies

180

1000 Temperature m/m)

µ 120 C) ° 600

Exp. at C COMPRO; IT-14 60 200 COMPRO; IT-13 Temperature ( Mechanical strain ( strain Mechanical

-200 0

1000 m/m) µ Exp. at B

600

COMPRO; IT-14 200 COMPRO; IT-13 Mechanical strain (

-200

1000 Line of Symmetry m/m) 51 102 102 mm µ 51mm 600 C B A 51mm Exp. at A

305 mm

200 Mechanical strain (

-200 0 50 100 150 200 250 300 350 Time (min)

Figure 5-15: Cases IT-13 and 14: COMPRO runs for different initial resin moduli. Mechanical strains at locations C, B and A are shown when autoclave pressure is high (586 kPa) and there is release agent at the interface.

-199- Chapter 5 – Validation and numerical case studies

320 Center line Location of gauge B 300 Location of gauge A Location of 280 gauge C Time (min) Time

Tip 260 0 50 100 150 200 250 300

X-coordinate (mm)

Figure 5-16: Debonding starts at the tip of the flat part and grows linearly. It reaches the centre line in approximately 50 minutes.

-200- Chapter 5 – Validation and numerical case studies

180 C Other 1000 Temperature COMPRO; IT-17 COMPRO runs

m/m) 4-noded, coarse mesh µ 120 C)

Line of Symmetry ° 600 51 102 102 mm 51mm C B A 51mm Exp. at C 60 200 305 mm Temperature ( Mechanical strain (

-200 0

Other B Line of symmetry COMPRO runs 1000 300 mm m/m)

µ Exp. at B 0.76 mm

600 3.2 mm

200 COMPRO; IT-17 4-noded, coarse mesh -200

A COMPRO; IT-17 1000 300 mm 4-noded, coarse mesh m/m) strain Mechanical ( µ 0.76 mm

600 3.2 mm Other 3 COMPRO runs

200

Mechanical strain ( strain Mechanical Exp. at A -200 0 50 100 150 200 250 300 350 Time (min)

Figure 5-17: IT-12, 16, 17 & 18 compares mechanical strain results of different COMPRO runs including all combination of 4- and 8-noded elements with coarse and fine meshes. Conditions are high autoclave pressure (586 kPa) and release agent at the interface.

-201- Chapter 5 – Validation and numerical case studies

180

C COMPRO; IT-14 1000 Temperature IT-14-S-1

m/m) IT-14-S-2

µ 120

600

Exp. at C 60 200 Temperature (°C)

-200 0

B 1000 Line of Symmetry Exp. at B m/m) ( Mechanical strain

µ 51 102 102 mm COMPRO; IT-14 IT-14-S-1 600 51mm IT-14-S-2 C B A 51mm

200 305 mm Mechanical strain ( Mechanical strain -200

A 1000 m/m)

µ Exp. at A 600 COMPRO; IT-14 IT-14-S-1 IT-14-S-2 200

Mechanical strain ( Mechanical strain -200 0 50 100 150 200 250 300 350

Time (min)

Figure 5-18: IT-14-S-1 and 2 are cases of sensitivity analysis when degree of cure at which the fluid friction mechanism stops is varied. Conditions are high autoclave pressure (586 kPa), release agent at the interface and low initial

resin modulus (4.71 kPa). IT-14: c1 = 0.65 , IT-14-S-1: c1 = 0.50 and IT-

14-S-2: c1 = 0.75 .

-202- Chapter 5 – Validation and numerical case studies

180 C 1000 Temperature

m/m) 120 µ

600

COMPRO; IT-14, IT-14-S-3 Exp. at C 60 200 Temperature (°C) Temperature

COMPRO; IT-14-S-4 -200 0 B 1000

m/m) ( strain Mechanical Line of Symmetry µ 51 102 102 mm Exp. at B 51mm 600 C B A 51mm

305 mm 200

-200 A 1000 m/m) Mechanical strain ( µ

Exp. at A 600

200 Mechanical strain ( strain Mechanical

-200 0 50 100 150 200 250 300 350 Time (min)

Figure 5-19: IT-14-S-3 and 4 are cases of sensitivity analysis when the maximum fluid friction shear strength is varied. Conditions are high autoclave pressure (586 kPa), release agent at the interface and low initial resin modulus (4.71 m m kPa). IT-14: τ ff = 30.0 kPa, IT-14-S-3: τ ff =15.0 kPa and IT-14-S-4: m τ ff = 50.0 kPa.

-203- Chapter 5 – Validation and numerical case studies

1400 180 C Temperature COMPRO; IT-14 1000 IT-14-S-5

m/m) IT-14-S-6 120 µ

600

Exp. at C 60 200 Temperature (°C)

-200 0 1400 B COMPRO; IT-14 m/m)1000 ( strain Mechanical Line of Symmetry µ 51 102 102 mm IT-14-S-5 IT-14-S-6 Exp. at B 51mm 600 C B A 51mm

305 mm 200

-200 1400 A

m/m)1000 Mechanical strain ( µ

Exp. at A 600 COMPRO; IT-14 IT-14-S-5 IT-14-S-6 200 Mechanical strain (

-200 0 50 100 150 200 250 300 350 Time (min)

Figure 5-20: IT-14-S-5 and 6 are cases of sensitivity analysis when the maximum bonding shear strength (at fully cured) is varied. Conditions are high pressure (586 kPa), release agent at the interface and low initial resin

modulus (4.71 kPa). IT-14: τ b = 5.0 MPa, IT-14-S-5: τ b = 3.0 MPa and

IT-14-S-6: τ b = 7.0 MPa.

-204- Chapter 5 – Validation and numerical case studies

1400 180 C Temperature 1000 COMPRO; IT-14

m/m) 120

µ IT-14-S-7 IT-14-S-8 600

Exp. at C 60 200 Temperature (°C)

-200 0 1400 B Line of Symmetry 102 mm m/m)1000 strain ( Mechanical 51 102 µ 51mm C B A 51mm 600 COMPRO; IT-14 305 mm IT-14-S-7 IT-14-S-8

200

Exp. at B -200 1400

A Exp. at A m/m) Mechanical strain (

µ 1000

600 COMPRO; IT-14 IT-14-S-7 IT-14-S-8 200 Mechanical strain (

-200 0 50 100 150 200 250 300 350 Time (min)

Figure 5-21: IT-14-S-7 and 8 are cases of sensitivity analysis when the coefficient of friction is varied. Conditions are high pressure (586 kPa), release agent at the interface and low initial resin modulus (4.71 kPa). IT-14: µ = 0.28, IT-14-S-7: µ = 0.20 and IT-14-S-8: µ = 0.40 .

-205- Chapter 5 – Validation and numerical case studies

180 C 1000 m/m)

µ 120

600

Exp. at C 60 COMPRO; IT-14, IT-14-S-9, IT-14-S-10 200 Temperature (°C)

-200 0 B 1000 Line of Symmetry 102 mm m/m) ( strain Mechanical 51 102 µ 51mm Exp. at B 600 C B A 51mm 305 mm

200

-200 A 1000 m/m) ( strain Mechanical µ

Exp. at A 600

200 Mechanical strain ( strain Mechanical

-200 0 50 100 150 200 250 300 350 Time (min)

Figure 5-22: IT-14-S-9 and 10 are cases of sensitivity analysis when the fluid friction shear strength at zero degree of cure is varied. Conditions are high pressure (586 kPa), release agent at the interface and low initial resin 0 0 modulus (4.71 kPa). IT-14: τ ff = 0.0 kPa, IT-14-S-9: τ ff =10.0 kPa and 0 IT-14-S-10: τ ff = 20.0 kPa.

-206- Chapter 5 – Validation and numerical case studies

180 C 1000 COMPRO; IT-12

m/m) IT-12-S-1

µ 120 IT-12-S-2 600

Exp. at C 60 200 Temperature (°C)

-200 0

B 1000 Line of Symmetry 51 102 102 mm m/m)( strain Mechanical

µ Exp. at B 51mm 600 C B A 51mm

305 mm

200

-200

A 1000 m/m) ( strain Mechanical µ Exp. at A 600

200 Mechanical strain ( Mechanical -200 0 50 100 150 200 250 300 350 Time (min)

Figure 5-23: IT-12-S-1 and 2 are cases of sensitivity analysis when the degree of cure at which the fluid friction mechanism stops is varied. Conditions are high pressure (586 kPa), release agent at the interface and high initial resin

modulus (4.71 MPa). IT-12: c1 = 0.65 , IT-12-S-1: c1 = 0.75 and IT-12-S-

2: c1 = 0.50 .

-207- Chapter 5 – Validation and numerical case studies

180 C 1000 m/m)

µ 120

600

Exp. at C COMPRO; IT-12, IT-12-S-3, IT-12-S-4 60 200 Temperature (°C)

-200 0

B Line of Symmetry 1000 51 102 102 mm

m/m) ( strain Mechanical Exp. at B µ 51mm C B A 600 51mm 305 mm

200

-200 A 1000 m/m) Mechanical strain ( µ Exp. at A 600

200 Mechanical strain ( strain Mechanical -200 0 50 100 150 200 250 300 350 Time (min)

Figure 5-24: IT-12-S-3 and 4 are cases of sensitivity analysis when the maximum fluid friction shear strength is varied. Conditions are high pressure (586 kPa), release agent at the interface and high initial resin modulus (4.71 MPa). m m IT-12: τ ff = 30.0 kPa, IT-12-S-3: τ ff =15.0 kPa and IT-12-S-4: m τ ff = 50.0 kPa.

-208- Chapter 5 – Validation and numerical case studies

180 C 1000 m/m)

µ 120

600

Exp. at C COMPRO; IT-12, IT-12-S-5, IT-12-S-6 60 200 Temperature (°C)

-200 0 B 1000 Line of Symmetry 51 102 102 mm

m/m) ( strain Mechanical Exp. at B µ 51mm 600 C B A 51mm

305 mm

200

-200 A 1000 m/m) ( strain Mechanical µ Exp. at A 600

200 Mechanical strain ( -200 0 50 100 150 200 250 300 350 Time (min)

Figure 5-25: IT-12-S-5 and 6 are cases of sensitivity analysis when the fluid friction shear strength at zero degree of cure is varied. Conditions are high pressure (586 kPa), release agent at the interface and high initial resin 0 0 modulus (4.71 MPa). IT-12: τ ff = 0.0 kPa, IT-12-S-5: τ ff =10.0 kPa and 0 IT-12-S-6: τ ff = 20.0 kPa.

-209- Chapter 5 – Validation and numerical case studies

Line of Symmetry

4 x 0.2, 0.4 or 0.8 mm

6.35 mm

150, 300 or 600 x 1 mm 5 mm

Figure 5-26: Flat part model: Composite part on top of an aluminium tool. Tool has been modelled with an extra 5 mm length in order to support any expansion of the composite part.

-210- Chapter 5 – Validation and numerical case studies

-45 COMPRO (a) -35 Adjusted -25 mm experiments -15

-5

y 120 -pl 6 0 m 4 00 m m ly 300 m -p m 8 120 m ly 6 0 m -p 00 m 16 mm 300 mm

-45 (b) COMPRO -35 Adjusted mm -25 experiments -15

-5

y 120 pl 6 0 m 4- 00 m 30 mm ly 0 m -p 120 m 8 0 m ly 600 m -p mm 16 300 mm

Figure 5-27: Warpage prediction of COMPRO are compared with adjusted experimental measurements of warpage for a flat composite part on an aluminium tool with 3 different lengths (300, 600 and 1200 mm) and 3 different thicknesses (4-, 8- and 16-ply). (a) High autoclave pressure of 586 kPa and release agent at interface. (b) Low autoclave pressure of 103 kPa and release agent at interface.

-211- Chapter 5 – Validation and numerical case studies

-45

(a) COMPRO -35

Adjusted -25 mm experiments -15

-5

1 200 y 60 mm -pl 0 m 4 300 m y mm pl 120 8- 0 m y 600 m pl mm 6- 300 1 mm

-45

(b) -35 mm Adjusted COMPRO -25 experiments -15

-5

12 y 00 pl 60 mm 4- 0 m 300 m ly mm -p 120 8 0 m ly 600 m -p mm 6 300 1 mm

Figure 5-28: Warpage prediction of COMPRO are compared with adjusted experimental measurements of warpage for a flat composite part on an aluminium tool with 3 different lengths (300, 600 and 1200 mm) and 3 different thicknesses (4-, 8- and 16-ply). (a) High autoclave pressure of 586 kPa and FEP sheets at interface. (b) Low autoclave pressure of 103 kPa and FEP sheets at interface.

-212- Chapter 5 – Validation and numerical case studies

Line of Symmetry

4 x 0.2, 0.4 or 0.8 mm

0.4 mm

6.35 mm

150, 300 or 600 x 1 mm

Figure 5-29: Flat part model: A shear layer introduced between the composite part and the aluminium tool.

Өw

Өc

Өt

Figure 5-30: Total spring-in in an L-shaped part consists of spring-in due to corner

effect and spring-in due to flange warpage, θtcw= θθ+ .

-213- Chapter 5 – Validation and numerical case studies

} Composite part } Aluminum tool

x x x x x x Element 93

o x

Figure 5-31: An L-shaped composite part on top of an aluminum tool modeled in COMPRO. The shear strain is determined at the first integration point of the first element (number 93) in the straight segment of the composite part right after the curved segment ends. The element has 2 layers and each layer has 4 integration points.

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1.0 240 Degree of cure

0.8 Autoclave temperature 180 C) C) 0.6 °

) and degree of cure (c) 0.4 120 xz /G xz τ 0.2 ( Temperature 60 0.0 0 50 100 150 200 250 300 350 τxz/Gxz Shear strain ( strain Shear -0.2 0 Time (min)

Figure 5-32: Autoclave temperature and degree of cure versus time for element number 93 shown in Figure 5-31. The shear strain at the first integration point of this element reaches minimum and maximum values of -11 and 96%, respectively. The shear strain then drops to a small value as shear modulus of the composite part starts increasing.

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Ө h Өw-web

90°-Өc

Өw-flange

Өv

Figure 5-33: Total spring-in in a C-shaped part consists of different spring-ins due to

corner effect, flange warpage and web warpage, θt= θθ c++ w−− flange θ w web .

Line of Symmetry

8 x 0.2 = 1.6 mm

6.35 or 12.0 mm

600 x 1 mm 5 mm

Figure 5-34: A flat composite part on an aluminium tool (as modelled in COMPRO) cured in an autoclave processing.

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400

Tool side of Composite side of convex mold convex mold 300

m τ ff = 20, 30, 60, 10 & 0 (kPa) 20 200

10 100 Original interfaces

0 0 375 380 385

-100 Z-coordinate (mm) -10

-200 -20

m τ ff = 0, 10, 20, 30 & 60 (kPa) -300 Composite side of Tool side of concave mold concave mold

-400 0 50 100 150 200 250 300 350 400 X-coordinate (mm)

Figure 5-35: Deformed shape of a half-circle 8-ply thickness composite part on 12.0 mm thick convex and concave tools after tool removal. Original outer and inner surface of convex and concave tools, respectively are shown as well.

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3 300, 600 & 1200 mm parts

-1 Deformation (mm)

-5 0 200 400 600

Length (mm)

Figure 5-36: Deformed shape of virtual cases of perfect bonding at the interface for different lengths of 300, 600 and 1200 mm 4-ply composite parts on an aluminium tool after tool removal. Project files are FP-1, FP-4 and FP-7.

X-coordinate (mm)

0 10002000300040005000 0

32 Plies

64 Plies -20

-40 Deformation (mm)

-60

Figure 5-37: Warpage of 10 m length composite flat part (half-length shown here): A) 32-ply thick; B) 64-ply thick.

-218- Chapter 6 – Summary, conclusions and future work

CHAPTER 6 ‐ SUMMARY, CONCLUSIONS AND FUTURE

WORK

6.1 SUMMARY

A composite part with a low initial elastic modulus (~ 100.0 kPa) for the resin at the beginning of an autoclave cure cycle yields a very soft material/element. An autoclave pressure of 100.0 to 600.0 kPa and major mismatches between part and tool properties cause large shear strains in elements on the composite side and a high probability of sliding at the tool-part interface. This research project is aimed at improving the prediction of the end results of autoclave processing of structural composite parts within the in-house developed code, COMPRO. To achieve this goal the following two important features have been added:

1. An appropriate hypoelastic constitutive model that accounts for continuously changing elastic modulus of the material in the presence of large deformations/strains; and

2. An appropriate contact interface between a composite part and a tool.

The proposed “co-rotational stress formulation” method has been shown to have all the requirements for dealing with large deformations/strains in isotropic and anisotropic

-219- Chapter 6 – Summary, conclusions and future work

materials in a consistent and accurate fashion. Co-rotational Kirchhoff (or True) stress and Logarithmic strain in material coordinate system are shown to be objective and the only conjugate pairs of interest. Other commercial codes, such as ABAQUS and LS- DYNA, utilize objective stress rates such as Jaumann and Truesdell, which are valid only for small strain range (a value less than a few percentages) in hypoelastic materials and isotropic materials. Moreover, some of them are limited to hyperelastic material models. Our proposed “co-rotational stress formulation” method is theoretically valid for any amount of strain and rotation. Although a few percentages of strain is relatively large compared to a yield strain of 0.2% for steel, it is small compared to a possible 100% shear strain in composite materials/elements at the beginning of the cure cycle when the material is very compliant and fluid-like.

The contact interface rids us of having to use a shear layer, which has a non-zero thickness and unknown parameters with an imposed non-physical mechanism. Though contact interface capability exists in commercial codes, our proposed contact interface is uniquely designed for a composite part and its specific interface. The possibility of a stick-slip condition at the interface with a gap exists, while “fluid friction,” “bonding- debonding” mechanisms and the static Coulomb friction option between the part and the tool or FEP sheets are modelled. One important feature of our contact algorithm is to avoid force oscillation in contact elements by skipping the updating of some of the contact parameters in each iteration. This feature renders the proposed contact surface very stable and robust. A contact interface with an appropriate constitutive law is a viable method to capture the tool-part interaction.

Several well-established examples are used to verify the implementation of the proposed constitutive equation and contact surface in COMPRO. Experiments involving instrumented tools are used to assign and calibrate contact interface parameters. Other experiments are used to validate the large deformations/strains and contact interface implementations.

-220- Chapter 6 – Summary, conclusions and future work

6.2 CONCLUSIONS

Co-rotational stress formulation

The co-rotational stress formulation method developed here:

• is the only accurate and robust technique for both isotropic and non-isotropic materials undergoing large deformations/strains.

• can use a hypoelastic constitutive equation with non-linear geometry (large deformations/strains).

• is theoretically valid for all strain ranges, but practically it is valid only for shear strains less than 100% due to numerical implementation issues.

• is valid for any rotations.

• can deal with rotation of an anisotropic material stiffness matrix due to stretching. This causes a non-co-axial stress and strain behaviour.

Contact interface

• Contact element (spring) stiffness is not geometry-dependent.

• Two contact parameters need to be calibrated using experimentation. These parameters are the chemical interfacial shear strength and coefficient of friction, both of which vary during the cure cycle.

• The chemical interfacial shear strength is a function of pressure, degree of cure and change of degree of cure from prior sliding or separation.

• Most residual stresses in a composite part are built up during the heat-up stage of a cure cycle, particularly at the upper level of a heat-up segment. Most of

-221- Chapter 6 – Summary, conclusions and future work

these residual stresses during cool-down and after tool removal are released in the form of undesired residual deformations.

• The most important contact surface parameters are maximum fluid friction shear strength and the degree of cure at which it occurs. Other contact surface parameters are less important in determining the total residual stresses and deformations, since most of them are active after the heat-up stage of a cure cycle when the composite part is almost elastic.

Numerical results from COMPRO with the above added features agree with experimental results, except for those cases in which: (i) Warpage is very small and within the errors of the measurement technique. (ii) Warpage is very large and a small change in experimental conditions can apparently cause very different results (e.g., warpage of 42 and 24 mm obtained for the same thin long flat part performed by G. Twigg and L. Petrescue, respectively). Interestingly, numerical models follow this trend in experiments as a small change in contact parameters causes significant changes in predictions in similar cases.

Contact surface parameters determined through experimentation are applicable to other geometries and models for the same set of materials. Thus, these parameters are constant for all geometries as they should be for any robust numerical model. Because a cure cycle can change the time that a fluid friction phenomenon ends, the specific degree of cure, c1 , should be modified for different cure cycles.

6.3 CONTRIBUTIONS

The uniqueness of this research lies in its introduction of a new general hypoelastic constitutive equation (co-rotational stress formulation method) that theoretically has no limits on kinds of material and strain range. The CSF general approach covers hard-to- deal-with non-isotropic materials, and their non-co-axial stress and strain behaviour.

-222- Chapter 6 – Summary, conclusions and future work

Rotation of material stiffness matrix due to stretching, which creates new axes for strains in material coordinate system (source of non-co-axial stress and strain) can be dealt with properly.

The CSF method practically can cope with as much as 100% shear strain in materials, while other commercial codes fail to yield meaningful and converged results at strains well below this limit. Although cured materials are elastic in a very small range of strain, curing materials can experience high strains easily while they are still developing elastic modulus.

Another strength of this research is its stable approach to a contact interface specifically designed for curing composite parts. Contact interfaces are difficult problems in mechanics. This stability problem becomes even more difficult to work with when one side of an interface cures and its stiffness changes by at least four orders of magnitude. By adding this feature, the deformed shape of a cured part can be predicted in advance, while residual stresses that are critical for structural design can be determined.

6.4 RECOMMENDATIONS AND FUTURE WORK

More research is needed in the following areas in order to take the advancements accomplished here to a higher level.

Most of this research focused on a 2-D approach to the constitutive model and contact surface at the interface given that the developments were made in COMPRO, which is a 2-D finite element code. More work should be done in order to extend the applicability of these research results to a 3-D model. For instance:

• Although in the large deformation/strain approach most equations are valid for 3-D cases, some – such as determining the rotation of the material axis due to rotation and stretching in 3-D – are totally different.

-223- Chapter 6 – Summary, conclusions and future work

• In a 2-D contact surface, the direction of sliding is known, while in a 3-D contact surface this angle is unknown and should be determined with respect to a projected local axis x-y on a plane projection of a contact surface.

• Having consistent equations forced us to use the Bˆ matrix instead of the logarithmic function of a strain matrix to determine strains. Although theoretically there is no assumption in determining the Bˆ matrix, strains greater than 100% determined by the Bˆ matrix begin to diverge from the exact results determined by the logarithmic function. Therefore, an opportunity exists here to explore this behaviour.

• More research is needed to precisely calibrate and determine contact surface parameters for the same material and also for different materials. An example is the determination of the initial fluid friction shear strength at a zero degree of cure.

-224- Biblography

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-231- Appendix A – Modified hypoelastic constitutive equation

APPENDIX A ‐ MODIFIED HYPOELASTIC CONSTITUTIVE EQUATION As was explained earlier, the hypoelastic constitutive equation is valid except when changes in the elastic modulus will affect the total built-up stresses in the materials. This deficiency in the hypoelastic constitutive equation can be compensated for by adding an extra term to Equation (3.10). This extra term can be determined by Equations A.1 and A.3 for two different kinds of materials that can face this problem.

1. Hyperelastic materials such as aluminium when its material stiffness is varying due to reversible phenomenon of temperature changes.

Hypoelastic constitutive equation of this material can be modified by adding the second term of equation A.1 to our constitutive Equation (3.10).

dddT()τεˆ dd()C εˆ ()ˆ ()C o τˆ& ==CC =C +εˆˆˆ =+C ε& C ε A.1 dt dtCCC dt dT dt in which:

o d()C C = C T& A.2 C dT 2. Curing materials when temperature changes decrease their material stiffness.

-232- Appendix A – Modified hypoelastic constitutive equation

In curing material decreasing temperature does not change the degree of cure. Material stiffness only increases due to effect of decreasing temperature. Therefore, a hypoelastic equation can cope with decreasing temperature in curing materials.

While increasing temperature increases the degree of cure which can lead to increase in material stiffness it directly decrease material stiffness. Therefore, increasing temperature has a mix effect on curing material stiffness. However, if material stiffness increases, then the hypoelastic equation can cope with increasing temperature; on the other hand, if material stiffness is decreasing, a second term should be added to hypoelastic equation as shown below:

∂()C dT oo τεˆˆ& ===C ˆˆElasticC ε Elastic CC−1τ A.3 ∂Tdt CCC in which the elastic logarithmic strain in material configuration is:

Elastic −1 εˆ = CC τˆ A.4 Therefore, the final constitutive equation will become:

o tt+∆ tt +∆ tt +∆ tt +∆ tt +∆ −1 t τˆˆ& =⋅CUCCCCCln(& t ) + τ A.5

o It is worth mentioning that C C is the changes in material stiffness matrix due to temperature changes and for curing material it should be assumed zero when

o determinant of C C is positive.

-233- Appendix B – Bˆ matrix

APPENDIX B ‐ Bˆ MATRIX The relationship between the strain vector and displacements is determined through a matrix referred to as the B matrix. In the large deformation formulation developed in this appendix, Bˆ is used for the same purpose. This matrix has a wide range of applicability in finite element codes, such as in determining strains (Equation (3.42)), stiffness matrix (Equation (3.49)) and thermal and cure shrinkage loads (Equation (3.71)). As mentioned earlier, the Bˆ matrix is determined using an exact solution proposed by Hoger [1986, 1987]. In this project, two approaches were adopted in order to determine this matrix:

−−11ˆ ε&ˆ ==RGDRGBrBrNN&& = B.1 as a result

ˆ −1 BRGB= N B.2 Or alternatively,

−−11ˆ ε&ˆ ==GRNN D GR Br&& = Br B.3 therefore

ˆ −1 BGRB= N B.4 The definition used in this research is the one derived from Equation (3.39), as shown in Equation B.1 above.

-234- Appendix B – Bˆ matrix

Although it is straight forward to use the logarithmic function of a matrix (Equation (3.1)) to determine the exact strain, for consistency purposes Bˆ matrix should be used for strain calculations because it is also used to determine loads and the stiffness matrix. It is rather interesting to note that neither of the above definitions for the Bˆ matrix provides the exact results in the simple shear example at very high strain values. An ˆ average of these two definitions, which is referred to in this thesis as BM odified , provides a better approximation by improving normal stress values.

To further expand this discussion, the same simple shear example was considered under a displacement-controlled loading in COMPRO (especially modified for this purpose) ˆ ˆ using the three approaches of the logarithmic function, B matrix and BM odified matrix, to calculate the components of the strain vector. By reducing the size of the time steps, the displacement at each time step becomes small enough that the iterations can be avoided.

The resulting displacement in each time step was 0.01 m, while an additional case with an incremental displacement of 0.0001 m was also considered for the Bˆ matrix approach. Results from all four sets of simulations are depicted in Figure B-1. Good agreement exists between the results of all approaches, up to a γ of one, after which the predictions begin to deviate. This implies that the choice of approach employed would not affect the results within the range considered in this study, i.e., below a 100% logarithmic shear strain.

-235- Appendix B – Bˆ matrix

8 1 )

/µ σ ,ln σ x ( & ˆ 0.5 σx , BModified

) 6 µ /

σ γ γ γ/2 0 θ 00.51L =1 γ/L0 (m/m) 0 4 Z X L0 =1 ˆ σx , B ˆ Normalized stress ( stress Normalized 2 σ , B ˆ xz Modified σxz , B σ ,ln 0 xz & 010203040

γ/L0 (m/m)

Figure B-1: Simple shear example in COMPRO. Determining strains based on: A) logarithmic function; B) Bˆ matrix with regular ∆=um0.01 and ˆ smaller time step of ∆=um0.0001 ; C) BModified matrix.

-236- Appendix C – Stiffness reduction factor algorithm of tangential spring used in contact interface

APPENDIX C ‐ STIFFNESS REDUCTION FACTOR ALGORITHM OF TANGENTIAL SPRING USED IN CONTACT INTERFACE 1. Initialize stiffness reduction factor to one at time step one, SRF =1.0

2. Run stress analysis.

3. Loop over each slave node.

4. Determine the ratio of existing shear stress to the interfacial shear strength, τ Tfac = . τ A

5. If Tfac >1.01 GOTO 6 otherwise GOTO 10.

6. If Tfac >1.09 GOTO 7 otherwise GOTO 8

7. Then SRF= 3( SRF )( Tfac )

8. Then SRF= 2 SRF

9. If SRF >1020 then SRF =1020

10. If Tfac < 0.995 and there is a temperature ramp then GOTO 11 otherwise GOTO 13.

-237- Appendix C – Stiffness reduction factor algorithm of tangential spring used in contact interface 11. Decrease SRF= 0.5( SRF )( Tfac )

12. If SRF <1.0 then SRF =1.0

13. Keep difference of SRFs of two adjacent slave nodes (i and i −1) not larger than 4 orders:

4 4 14. If SRFii>10 SRF −1 then SRFii=10 SRF −1

−4 −4 15. If SRFii<10 SRF −1 then SRFii=10 SRF −1

16. Reduce/increase tangential spring stiffness by its SRF (i.e., KKSRFTT= / )

17. End Loop

-238-