An Introduction to Computational Micromechanics (eBook)

Preface 6
Contents 7
Introduction 11
1.1 Basic Concepts in Micro–Macro Modeling 13
1.2 Historical Overview 14
1.3 Objectives of this Monograph 15
Some Basics of the Mechanics of Solid Continua 17
2.1 Kinematics of Deformations 18
2.1.1 Deformation of Line Elements 18
2.1.2 Infinitesimal Strain Measures 20
2.1.3 The Jacobian of the Deformation Gradient 20
2.2 Equilibrium/ Kinetics of Solid Continua 21
2.2.1 Postulates on Volume and Surface Quantities 22
2.2.2 Balance Law Formulations 23
2.3 Referential Descriptions of Balance Laws 24
2.4 The First Law of Thermodynamics/An Energy Balance 26
2.5 The Second Law of Thermodynamics/A Restriction 27
2.6 Linearly Elastic Constitutive Equations 28
2.6.1 The Infinitesimal Strain Case 30
2.6.2 Material Symmetry 31
2.6.3 Material Constant Interpretation 36
2.6.4 Consequences of Positive-Definiteness 37
2.7 Hyperelastic Finite Strain Material Laws 39
2.7.1 Basic Requirements for Finite Strain Laws 39
2.7.2 Determination of Material Constants 41
2.8 Moderate Strain Constitutive Relations 42
FundamentalWeak Formulations 46
3.1 DirectWeak Formulations 46
3.1.1 An Example 47
3.1.2 Some Restrictions 48
3.1.3 The Principle of Minimum Potential Energy 50
3.1.4 Complementary Weak Forms 51
Fundamental Micro–Macro Concepts 53
4.1 Testing Procedures 54
4.1.1 The Average Strain Theorem 55
4.1.2 The Average Stress Theorem 56
4.1.3 Satisfaction of HillÌs Energy Condition 57
4.2 The Hill-Reuss-Voigt Bounds 57
4.3 Observations 59
4.4 ClassicalMicro–Macro Mechanical Approximations 60
4.4.1 The Asymptotic Hashin-Shtrikman Bounds 60
4.4.2 The Concentration Tensor: Microfield Behavior 61
4.4.3 The Eshelby Result 62
4.4.4 Dilute Methods 63
4.4.5 The Mori-Tanaka Method 64
4.4.6 Further Methods 66
4.5 Micro-Geometrical (Manufacturing) Idealizations 67
4.5.1 Upper and Lower Variational Bounds 67
4.5.2 Proof of Energetic Ordering 68
4.5.3 Uses to Approximate the Effective Property 70
A Basic Finite Element Implementation 71
5.1 Finite Element Method Implementation 71
5.2 FEM Approximation 72
5.3 Global/local Transformations 74
5.4 Differential Properties of Shape Functions 75
5.5 Differentiation in the Referential Coordinates 78
5.6 Post Processing 82
5.7 Accuracy of the Finite Element Method 83
5.8 Local Adaptive Mesh Refinement 84
5.8.1 A-Posteriori Recovery Methods 85
5.8.2 A-Posteriori Residual Methods 85
5.9 Solution of Algebraic Equations 86
5.9.1 Krylov Searches and Minimum Principles 88
5.9.2 The Method of Steepest Descent 88
5.9.3 The Conjugate Gradient Method 90
5.9.4 Accelerating Computations 91
5.10 Remark on Penalty Methods 92
Computational/Statistical Testing Methods 93
6.1 A 93
Boundary 93
Value Formulation 93
6.2 Numerical Discretization 94
6.2.1 Topological Resolution 95
6.3 Elementally Averaged Quantities 97
6.4 Iterative Krylov Solvers/ Microstructural “Correctors” 99
6.5 Overall Testing Process: Numerical Examples 100
6.5.1 Successive Sample Enlargement 100
6.5.2 Multiple Sample Tests 102
6.6 A Minimum Principle Interpretation 104
6.6.1 A Proof Based on Partitioning Results 105
6.6.2 Relation to the Material Tests 105
6.7 Dependency on Volume Fraction 106
6.8 Increasing the Number of Samples 111
6.9 Increasing Sample Size 112
Various Extensions and Further Interpretations of Partitioning 114
7.1 Partitioning and Traction Test Cases 114
7.1.1 Isolating the Subsampling Error/Numerical Error Orthogonality 115
7.2 An Ergodic Interpretation of the Results 116
7.3 Statistical Shifting Theorems 117
7.4 Partitioning and Ensemble Averaging 119
7.4.1 Primal Partitioning 119
7.4.2 Complementary Partitioning 120
7.4.3 Homogenized Material Orderings 122
7.4.4 Embedded Orthogonal Monotonicities 122
7.5 Moment Bounds on Population Responses 124
7.5.1 First Order (Average) Bounds 124
7.5.2 Second Order (Standard Deviation) Bounds 124
7.5.3 Third Order (Skewness) Bounds 125
7.6 Remarks 126
Domain Decomposition Analogies and Extensions 128
8.1 Boundary Value Problem Formulations 128
8.2 Error in the “Broken” Problems 130
8.2.1 Multiscale Proximity Bounds 132
8.2.2 Uses of the Bounds for Domain Decomposition 136
8.3 The Connection to Material Testing 138
8.4 A “total” Orthogonal Sum 141
8.5 Iterative Extensions 143
8.5.1 Method I: Global/Local CG Iterations 143
8.5.2 Method II: Iterative Equilibration 144
Nonconvex–Nonderivative Genetic Material Design 151
9.1 Computational Material Design 152
9.2 Characteristic of Such Objectives 153
9.2.1 Nonconvexity 153
9.2.2 Size Effects 154
9.3 Introduction of Constraints 156
9.4 Fatigue-Type Constraints 158
9.4.1 Classical Fatigue Relations 158
9.4.2 Construction of a Constraint 159
9.4.3 Qualitative Behavior of the Fatigue Constraints 160
9.5 Nonconvex–Nonderivative Genetic Search 161
9.6 Numerical Examples 163
9.7 Scope of Use 166
Modeling Coupled Multifield Processes 168
10.1 Introduction 168
10.2 A Model Problem Involving Multifield Processes in Multiphase Solids 169
10.3 Constitutive Assumptions 170
10.3.1 An Energy Balance Including Growth 171
10.3.2 Mass Transfer and Reaction-Diffusion Models 173
10.4 Staggered MultifieldWeak Formulations 175
10.4.1 A Recursive Algorithm 175
10.4.2 Convergence and Contraction-Mapping Time Stepping Control 178
10.5 Numerical Experiments 180
10.6 Concluding Remarks 186
Closing Comments 188
References 189