 | Roy S. Computational modeling of polymer composites: a study of creep and environmental effects / S.Roy, J.N.Reddy. - Boca Raton: CRC/Taylor & Francis, 2014. - xvi, 282 p., [1] l. col. ill.: ill. - (Computational mechanics and applied analysis). - Bibliogr. at the end of the chapters. - Ind.: p.275-282. - ISBN 978-1-4665-8649-9
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Preface ........................................................ xv
1 General Introduction and Equations of Solid Mechanics ...... 1
1.1 Introduction ............................................... 1
1.2 Vectors and Tensors ........................................ 2
1.2.1 Definitions ......................................... 2
1.2.2 Components of Vectors and Tensors ................... 2
1.2.3 Summation Convention ................................ 4
1.2.4 The Del Operator .................................... 5
1.2.5 Transformations of Components ....................... 6
1.3 Equations of Solid Mechanics ............................... 9
1.3.1 Introduction ........................................ 9
1.3.2 Kinematics ......................................... 10
1.3.3 Compatibility Equations ............................ 13
1.3.4 Stress Vector and Stress Tensor .................... 14
1.3.5 Equations of Motion ................................ 16
1.3.6 Constitutive Relations: Hooke's Law ................ 17
1.3.7 Linear Viscoelasticity ............................. 21
1.4 Energy Principles of Solid Mechanics ...................... 25
1.4.1 Introduction and Concept of Work Done .............. 25
1.4.2 The Principle of Virtual Displacements ............. 26
1.4.3 The Principle of Minimum Total Potential Energy .... 29
1.5 Summary ................................................... 31
References ................................................ 31
2 A Review of the Finite Element Method ..................... 33
2.1 Introduction .............................................. 33
2.2 Linear Plane Elasticity Problems .......................... 34
2.1 Governing Equations ....................................... 34
2.2.2 Finite Element Approximation ....................... 36
2.2.3 Virtual Work Statement ............................. 37
2.2.4 Finite Element Model ............................... 38
2.3 Finite Element Models of Nonlinear Continua ............... 40
2.3.1 Introduction ....................................... 40
2.3.2 Strain and Stress Measures ......................... 41
2.3.3 Principle of Virtual Displacements ................. 43
2.3.4 Total Lagrangian Formulation ....................... 44
2.3.5 Updated Lagrangian Formulation ..................... 45
2.3.6 2-D Finite Element Models .......................... 46
2.3.6.1 Total Lagrangian formulation .............. 46
2.3.6.2 Updated Lagrangian formulation ............ 47
2.4 Numerical Integration ..................................... 48
2.4.1 Preliminary Comments ............................... 48
2.4.2 Coordinate Transformations ......................... 48
2.4.3 Integration over a Master Rectangular Element ...... 52
2.4.4 Integration over a Master Triangular Element ....... 53
2.4.5 Numerical Integration over Three-Dimensional
Elements ........................................... 55
2.5 Two-Dimensional Finite Elements ........................... 56
2.5.1 Properties of Approximation Functions .............. 56
2.5.2 Linear Triangular Element .......................... 56
2.5.3 Linear Rectangular Element ......................... 58
2.6 Three-Dimensional Finite Elements ......................... 61
2.6.1 Hexahedral (Brick) Elements ........................ 61
2.6.2 Tetrahedral Elements ............................... 61
2.6.3 Prism Elements ..................................... 63
2.7 Summary ................................................... 65
References ................................................ 65
3 Finite Element Models of Linear Viscoelastic Materials .... 67
3.1 Introduction .............................................. 67
3.2 Linear Viscoelastic Formulation ........................... 67
3.2.1 Introduction ....................................... 67
3.2.2 Uniaxial Stress State .............................. 69
3.2.3 Multiaxial Stress State ............................ 71
3.2.4 Three-Dimensional Viscoelastic Constitutive
Relations .......................................... 76
3.3 Finite Element Analysis ................................... 79
3.3.1 Finite Element Model ............................... 79
3.3.2 Example Problems ................................... 80
3.3.2.1 Creep and recovery of axisymmetric
viscoelastic rod .......................... 80
3.3.2.2 Response of an axisymmetric viscoelastic
rod to cyclic load ........................ 83
3.3.2.3 Response of a three-dimensional
viscoleastic rod to cyclic load ........... 85
3.3.2.4 Delayed failure in a linear viscoelastic
material .................................. 85
3.4 Summary ................................................... 93
References ................................................ 93
4 Finite Element Analysis of Diffusion in Polymer and
Polymer Matrix Composites ................................. 95
4.1 Introduction .............................................. 95
4.1.1 Preliminary Comments ............................... 95
4.1.2 Diffusion in a Polymer ............................. 97
4.2 Modeling of Moisture Diffusion ............................ 99
4.2.1 Governing Equations ................................ 99
4.2.2 Finite Element Formulation ........................ 100
4.2.3 Solution of Nonlinear Equations ................... 102
4.2.3.1 Direct iteration scheme .................. 102
4.2.3.2 Newton's iteration scheme ................ 103
4.2.4 Axisymmetric Diffusion Problems ................... 103
4.2.4.1 Preliminary comments ..................... 103
4.2.4.2 Finite element model ..................... 104
4.2.5 Numerical Examples ................................. 105
4.2.5.1 One-dimensional linear Fickian
diffusion in a polymer film .............. 105
4.2.5.2 Two-dimensional Fickian diffusion in an
orthotropic material ..................... 107
4.2.5.3 Hygrothermal stresses in two-
dimensional Fickian diffusion in an
orthotropic material ..................... 111
4.3 Diffusion with Time-Varying Diffusivity .................. 113
4.3.1 Introduction ...................................... 113
4.3.2 Governing Equations ............................... 113
4.3.3 Analytical Solution ............................... 114
4.3.4 Variational (Weak) Form ........................... 116
4.3.5 Finite Element Model .............................. 116
4.3.6 A Numerical Example ............................... 117
4.4 Summary .................................................. 119
References ............................................... 119
5 Finite Element Models of Nonlinear Viscoelastic
Materials ................................................ 123
5.1 Introduction ............................................. 123
5.2 Uniaxial Stress State .................................... 124
5.3 Multiaxial Stress State .................................. 127
5.4 Constitutive Relations for Three-Dimensional
Viscoelasticity .......................................... 133
5.5 Finite Element Model ..................................... 136
5.6 Example Problems of Nonlinear Viscoelasticity ............ 137
5.6.1 Analysis of Adhesive Coupons ...................... 137
5.6.2 Isothermal Creep and Recovery in an Epoxy
Adhesive .......................................... 140
5.6.3 Analysis of a Model Joint ......................... 144
5.6.4 Analysis of a Composite Single Lap Joint .......... 150
5.6.5 Nonlinear Isochronous Creep in an Axisymmetric
Rod ............................................... 153
5.7 Delayed Failure .......................................... 156
5.7.1 Uniaxial Formulation .............................. 156
5.7.2 Multiaxial Formulation ............................ 157
5.7.3 Example: A Butt Joint ............................. 159
5.8 Summary .................................................. 162
References ............................................... 162
6 Finite Element Analysis of Nonlinear Diffusion in
Polymers ................................................. 167
6.1 Introduction to Nonlinear Fickian Diffusion .............. 167
6.2 Background on Nonlinear Diffusion Analysis ............... 168
6.3 Newton-Raphson Technique for Solving Nonlinear
Diffusion Problem ........................................ 169
6.4 Iterative Solution Procedure ............................. 170
6.5 Examples of Nonlinear Diffusion Problems ................. 171
6.5.1 Diffusion in a Semi-Infinite Media ................ 171
6.5.2 Gas Transport in Uniaxially Stretched
Polystyrene ....................................... 173
6.5.3 Analysis of a Butt Joint Including Moisture
Diffusion ......................................... 176
6.6 Summary .................................................. 186
References ............................................... 186
7 Non-Fickian Solvent Diffusion in a Solid with Large
Dilatation ............................................... 189
7.1 Introduction ............................................. 189
7.2 Governing Equations ...................................... 190
7.3 Swelling (Dilatation) due to Solvent Ingress in an
Orthotropic Solid ........................................ 191
7.3.1 Governing Equations ............................... 191
7.3.2 Finite Element Formulation ........................ 193
7.3.3 Time-Integration using θ-Family of Approximation .. 195
7.3.4 The Newton Iteration Scheme ....................... 195
7.3.5 Numerical Example: Diffusion in a One-
Dimensional Bar with Large Dilatation ............. 196
7.3.6 Effective Diffusivity and Diffusivity Correction
Factor ............................................ 200
7.3.7 Calculation of Shear Stresses ..................... 202
7.4 Summary .................................................. 204
References ............................................... 204
8 A Coupled Hygrothermal Cohesive Layer Model for
Simulating Debond Growth in Bimaterial Interfaces ........ 205
8.1 Preliminary Comments ..................................... 205
8.2 Introduction ............................................. 205
8.3 Cohesive Layer Model Development ......................... 207
8.4 Derivation of Consistent Diffusivities ................... 212
8.5 Cohesive Layer Diffusion Boundary Conditions ............. 212
8.6 Cohesive Work of Separation .............................. 213
8.7 Numerical Implementation ................................. 214
8.8 Finite Element Model Verification ........................ 215
8.8.1 Comparison with Analytical Solution for a DCB
Specimen .......................................... 215
8.8.2 Modification of DCB Solution (Modified Williams'
Model) ............................................ 219
8.9 Comparison Between Analytical Solution and Finite
Element Results .......................................... 222
8.10 Simulation of Debond Growth due to Bond Degradation:
Wedge Test Simulation .................................... 228
8.11 Summary .................................................. 234
References ............................................... 234
9 A Viscoelastic Cohesive Layer Model for Prediction of
Interlaminar Shear Strength of Carbon/Epoxy Composites ... 237
9.1 Introduction ............................................. 237
9.2 Background ............................................... 238
9.3 Finite Element Modeling .................................. 238
9.4 A Multi-Scale Viscoelastic Cohesive Layer Formulation
Including Damage Evolution ............................... 239
9.4.1 Governing Equations ............................... 239
9.4.2 Damage Evolution Law .............................. 241
9.4.3 Determination of Principal Stretch ................ 242
9.4.4 Damage Initiation Criterion ....................... 243
9.5 Hydrolysis of Epoxy Resins in a Polymer Composite ........ 244
9.5.1 Introduction ...................................... 244
9.5.2 Mechanism-Based Modeling of Degradation Due to
Hygrothermal Aging in Polymer Composites .......... 244
9.5.3 Calculation of Moisture Degradation Parameter r ... 245
9.5.4 Derivation of Internal State Variable for
Moisture Induced Degradation ...................... 245
9.5.5 Modeling of Strength Degradation due to
Hygrothermal Effects .............................. 247
9.5.6 Delamination Failure at the Interface between
Adjacent Lamina in a Unidirectional Carbon/Epoxy
Laminate .......................................... 248
9.6 Results and Discussion ................................... 248
9.6.1 Finite Element Simulation of Short Beam Shear
Experiments ....................................... 248
9.6.2 A Sensitivity Study of the Effect of
Displacement Rate on Cohesive Law ................. 250
9.6.3 Verification of Model Prediction with Test Data
for Interlaminar Shear Strength ................... 252
9.7 Summary .................................................. 254
References ............................................... 255
10 A Multi-Scale Viscoelastic Cohesive Layer Model for
Predicting Delamination in High Temperature Polymer
Composites ............................................... 257
10.1 Introduction ............................................. 257
10.2 Double Cantilever Beam (DCB) Experiment .................. 260
10.1 Specimen Preparation and DCB Specimen Geometry ........... 260
10.2.2 Experimental Method ............................... 261
10.3 Viscoelastic Cohesive Layer Model ........................ 261
10.3.1 Preliminary Comments .............................. 261
10.3.2 Damage Evolution Law for the Micromechanical RVE .. 262
10.4 Extraction of Cohesive Law from Experimental Data
Through J-Integral ....................................... 263
10.5 Evaluation of Damage Evolution Law ....................... 265
10.6 Numerical Results ........................................ 268
10.7 Summary .................................................. 271
References ............................................... 272
Index ......................................................... 275
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