Zohdi / Wriggers An Introduction to Computational Micromechanics

1. Auflage 2008
ISBN: 978-3-540-32360-0
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 195 Seiten

Reihe: Engineering

In this, its second corrected printing, Zohdi and Wriggers' illuminating text presents a comprehensive introduction to the subject. The authors include in their scope basic homogenization theory, microstructural optimization and multifield analysis of heterogeneous materials. This volume is ideal for researchers and engineers, and can be used in a first-year course for graduate students with an interest in the computational micromechanical analysis of new materials.

Autoren/Hrsg.

Zohdi, Tarek I.

Wriggers, Peter

Inhaltsverzeichnis

1;Preface;6
2;Contents;7
3;Introduction;11
3.1;1.1 Basic Concepts in Micro–Macro Modeling;13
3.2;1.2 Historical Overview;14
3.3;1.3 Objectives of this Monograph;15
4;Some Basics of the Mechanics of Solid Continua;17
4.1;2.1 Kinematics of Deformations;18
4.2;2.1.1 Deformation of Line Elements;18
4.3;2.1.2 Infinitesimal Strain Measures;20
4.4;2.1.3 The Jacobian of the Deformation Gradient;20
4.5;2.2 Equilibrium/ Kinetics of Solid Continua;21
4.6;2.2.1 Postulates on Volume and Surface Quantities;22
4.7;2.2.2 Balance Law Formulations;23
4.8;2.3 Referential Descriptions of Balance Laws;24
4.9;2.4 The First Law of Thermodynamics/An Energy Balance;26
4.10;2.5 The Second Law of Thermodynamics/A Restriction;27
4.11;2.6 Linearly Elastic Constitutive Equations;28
4.12;2.6.1 The Infinitesimal Strain Case;30
4.13;2.6.2 Material Symmetry;31
4.14;2.6.3 Material Constant Interpretation;36
4.15;2.6.4 Consequences of Positive-Definiteness;37
4.16;2.7 Hyperelastic Finite Strain Material Laws;39
4.17;2.7.1 Basic Requirements for Finite Strain Laws;39
4.18;2.7.2 Determination of Material Constants;41
4.19;2.8 Moderate Strain Constitutive Relations;42
5;FundamentalWeak Formulations;46
5.1;3.1 DirectWeak Formulations;46
5.2;3.1.1 An Example;47
5.3;3.1.2 Some Restrictions;48
5.4;3.1.3 The Principle of Minimum Potential Energy;50
5.5;3.1.4 Complementary Weak Forms;51
6;Fundamental Micro–Macro Concepts;53
6.1;4.1 Testing Procedures;54
6.2;4.1.1 The Average Strain Theorem;55
6.3;4.1.2 The Average Stress Theorem;56
6.4;4.1.3 Satisfaction of HillÌs Energy Condition;57
6.5;4.2 The Hill-Reuss-Voigt Bounds;57
6.6;4.3 Observations;59
6.7;4.4 ClassicalMicro–Macro Mechanical Approximations;60
6.8;4.4.1 The Asymptotic Hashin-Shtrikman Bounds;60
6.9;4.4.2 The Concentration Tensor: Microfield Behavior;61
6.10;4.4.3 The Eshelby Result;62
6.11;4.4.4 Dilute Methods;63
6.12;4.4.5 The Mori-Tanaka Method;64
6.13;4.4.6 Further Methods;66
6.14;4.5 Micro-Geometrical (Manufacturing) Idealizations;67
6.15;4.5.1 Upper and Lower Variational Bounds;67
6.16;4.5.2 Proof of Energetic Ordering;68
6.17;4.5.3 Uses to Approximate the Effective Property;70
7;A Basic Finite Element Implementation;71
7.1;5.1 Finite Element Method Implementation;71
7.2;5.2 FEM Approximation;72
7.3;5.3 Global/local Transformations;74
7.4;5.4 Differential Properties of Shape Functions;75
7.5;5.5 Differentiation in the Referential Coordinates;78
7.6;5.6 Post Processing;82
7.7;5.7 Accuracy of the Finite Element Method;83
7.8;5.8 Local Adaptive Mesh Refinement;84
7.9;5.8.1 A-Posteriori Recovery Methods;85
7.10;5.8.2 A-Posteriori Residual Methods;85
7.11;5.9 Solution of Algebraic Equations;86
7.12;5.9.1 Krylov Searches and Minimum Principles;88
7.13;5.9.2 The Method of Steepest Descent;88
7.14;5.9.3 The Conjugate Gradient Method;90
7.15;5.9.4 Accelerating Computations;91
7.16;5.10 Remark on Penalty Methods;92
8;Computational/Statistical Testing Methods;93
8.1;6.1 A;93
8.2;Boundary;93
8.3;Value Formulation;93
8.4;6.2 Numerical Discretization;94
8.5;6.2.1 Topological Resolution;95
8.6;6.3 Elementally Averaged Quantities;97
8.7;6.4 Iterative Krylov Solvers/ Microstructural “Correctors”;99
8.8;6.5 Overall Testing Process: Numerical Examples;100
8.9;6.5.1 Successive Sample Enlargement;100
8.10;6.5.2 Multiple Sample Tests;102
8.11;6.6 A Minimum Principle Interpretation;104
8.12;6.6.1 A Proof Based on Partitioning Results;105
8.13;6.6.2 Relation to the Material Tests;105
8.14;6.7 Dependency on Volume Fraction;106
8.15;6.8 Increasing the Number of Samples;111
8.16;6.9 Increasing Sample Size;112
9;Various Extensions and Further Interpretations of Partitioning;114
9.1;7.1 Partitioning and Traction Test Cases;114
9.2;7.1.1 Isolating the Subsampling Error/Numerical Error Orthogonality;115
9.3;7.2 An Ergodic Interpretation of the Results;116
9.4;7.3 Statistical Shifting Theorems;117
9.5;7.4 Partitioning and Ensemble Averaging;119
9.6;7.4.1 Primal Partitioning;119
9.7;7.4.2 Complementary Partitioning;120
9.8;7.4.3 Homogenized Material Orderings;122
9.9;7.4.4 Embedded Orthogonal Monotonicities;122
9.10;7.5 Moment Bounds on Population Responses;124
9.11;7.5.1 First Order (Average) Bounds;124
9.12;7.5.2 Second Order (Standard Deviation) Bounds;124
9.13;7.5.3 Third Order (Skewness) Bounds;125
9.14;7.6 Remarks;126
10;Domain Decomposition Analogies and Extensions;128
10.1;8.1 Boundary Value Problem Formulations;128
10.2;8.2 Error in the “Broken” Problems;130
10.3;8.2.1 Multiscale Proximity Bounds;132
10.4;8.2.2 Uses of the Bounds for Domain Decomposition;136
10.5;8.3 The Connection to Material Testing;138
10.6;8.4 A “total” Orthogonal Sum;141
10.7;8.5 Iterative Extensions;143
10.8;8.5.1 Method I: Global/Local CG Iterations;143
10.9;8.5.2 Method II: Iterative Equilibration;144
11;Nonconvex–Nonderivative Genetic Material Design;151
11.1;9.1 Computational Material Design;152
11.2;9.2 Characteristic of Such Objectives;153
11.3;9.2.1 Nonconvexity;153
11.4;9.2.2 Size Effects;154
11.5;9.3 Introduction of Constraints;156
11.6;9.4 Fatigue-Type Constraints;158
11.7;9.4.1 Classical Fatigue Relations;158
11.8;9.4.2 Construction of a Constraint;159
11.9;9.4.3 Qualitative Behavior of the Fatigue Constraints;160
11.10;9.5 Nonconvex–Nonderivative Genetic Search;161
11.11;9.6 Numerical Examples;163
11.12;9.7 Scope of Use;166
12;Modeling Coupled Multifield Processes;168
12.1;10.1 Introduction;168
12.2;10.2 A Model Problem Involving Multifield Processes in Multiphase Solids;169
12.3;10.3 Constitutive Assumptions;170
12.4;10.3.1 An Energy Balance Including Growth;171
12.5;10.3.2 Mass Transfer and Reaction-Diffusion Models;173
12.6;10.4 Staggered MultifieldWeak Formulations;175
12.7;10.4.1 A Recursive Algorithm;175
12.8;10.4.2 Convergence and Contraction-Mapping Time Stepping Control;178
12.9;10.5 Numerical Experiments;180
12.10;10.6 Concluding Remarks;186
13;Closing Comments;188
14;References;189