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Informationen zum Autor Koichi Hashiguchi, Daiichi University, Japan, & Yuki Tamakawa, Tohuku University, Japan Koichi Hashiguchi is Professor, Daiichi University and Emeritus Professor of Kyushu University), Japan. He has taught applied mechanics for undergraduate and postgraduate students for 40 years and has supervised 34 Doctorates of applied mechanics. Current research in the field of plasticity includes the development of constitutive modelling of elastoplastic materials such as metals and soils which have been widely studied as elastoplastic materials for the last forty years. He has published circa 50 refereed journal papers on elastoplasticity since 2000. Yuki Tamakawa is Associate Professor, Dept. Civil and Environmental Eng., Tohoku University. He has taught applied mechanics for undergraduate and postgraduate students for 12 years, and his research interests include elastoplasticity, nonlinear mechanics, material and structural instability, and bifurcation. Klappentext Comprehensive introduction to finite elastoplasticity, addressing various analytical and numerical analyses & including state-of-the-art theoriesIntroduction to Finite Elastoplasticity presents introductory explanations that can be readily understood by readers with only a basic knowledge of elastoplasticity, showing physical backgrounds of concepts in detail and derivation processes of almost all equations. The authors address various analytical and numerical finite strain analyses, including new theories developed in recent years, and explain fundamentals including the push-forward and pull-back operations and the Lie derivatives of tensors.As a foundation to finite strain theory, the authors begin by addressing the advanced mathematical and physical properties of continuum mechanics. They progress to explain a finite elastoplastic constitutive model, discuss numerical issues on stress computation, implement the numerical algorithms for stress computation into large-deformation finite element analysis and illustrate several numerical examples of boundary-value problems. Programs for the stress computation of finite elastoplastic models explained in this book are included in an appendix, and the code can be downloaded from an accompanying website. Zusammenfassung This book provides an easy-to-understand introduction to finite elastoplasticity. It addresses various analytical and numerical finite strain analyses, including new theories developed in recent years, and explains fundamentals, including the push-forward and pull-back operations and the Lie derivatives of tensors. Inhaltsverzeichnis Preface xi Series Preface xv Introduction xvii 1 Mathematical Preliminaries 1 1.1 Basic Symbols and Conventions 1 1.2 Definition of Tensor 2 1.2.1 Objective Tensor 2 1.2.2 Quotient Law 4 1.3 Vector Analysis 5 1.3.1 Scalar Product 5 1.3.2 Vector Product 6 1.3.3 Scalar Triple Product 6 1.3.4 Vector Triple Product 7 1.3.5 Reciprocal Vectors 8 1.3.6 Tensor Product 9 1.4 Tensor Analysis 9 1.4.1 Properties of Second-Order Tensor 9 1.4.2 Tensor Components 10 1.4.3 Transposed Tensor 11 1.4.4 Inverse Tensor 12 1.4.5 Orthogonal Tensor 12 1.4.6 Tensor Decompositions 15 1.4.7 Axial Vector 17 1.4.8 Determinant 20 1.4.9 On Solutions of Simultaneous Equation 23 1.4.10 Scalar Triple Products with Invariants 24 1.4.11 Orthogonal Transformation of Scalar Triple Product 25 1.4.12 Pseudo Scalar, Vector and Tensor 26 1.5 Tensor Representations 27 1.5.1 Tensor Notations 27 1.5.2 Tensor Components and Transformation Rule 27 1.5.3 Notations of Tensor Operations 28 1.5.4 Operational Tensors 29 1.5.5 Isotropic Tensors 31 1.6 Eigenvalues and Eigenvectors 36
List of contents
Preface xi
Series Preface xv
Introduction xvii
1 Mathematical Preliminaries 1
1.1 Basic Symbols and Conventions 1
1.2 Definition of Tensor 2
1.2.1 Objective Tensor 2
1.2.2 Quotient Law 4
1.3 Vector Analysis 5
1.3.1 Scalar Product 5
1.3.2 Vector Product 6
1.3.3 Scalar Triple Product 6
1.3.4 Vector Triple Product 7
1.3.5 Reciprocal Vectors 8
1.3.6 Tensor Product 9
1.4 Tensor Analysis 9
1.4.1 Properties of Second-Order Tensor 9
1.4.2 Tensor Components 10
1.4.3 Transposed Tensor 11
1.4.4 Inverse Tensor 12
1.4.5 Orthogonal Tensor 12
1.4.6 Tensor Decompositions 15
1.4.7 Axial Vector 17
1.4.8 Determinant 20
1.4.9 On Solutions of Simultaneous Equation 23
1.4.10 Scalar Triple Products with Invariants 24
1.4.11 Orthogonal Transformation of Scalar Triple Product 25
1.4.12 Pseudo Scalar, Vector and Tensor 26
1.5 Tensor Representations 27
1.5.1 Tensor Notations 27
1.5.2 Tensor Components and Transformation Rule 27
1.5.3 Notations of Tensor Operations 28
1.5.4 Operational Tensors 29
1.5.5 Isotropic Tensors 31
1.6 Eigenvalues and Eigenvectors 36
1.6.1 Eigenvalues and Eigenvectors of Second-Order Tensors 36
1.6.2 Spectral Representation and Elementary Tensor Functions 40
1.6.3 Calculation of Eigenvalues and Eigenvectors 42
1.6.4 Eigenvalues and Vectors of Orthogonal Tensor 45
1.6.5 Eigenvalues and Vectors of Skew-Symmetric Tensor and Axial Vector 46
1.6.6 Cayley-Hamilton Theorem 47
1.7 Polar Decomposition 47
1.8 Isotropy 49
1.8.1 Isotropic Material 49
1.8.2 Representation Theorem of Isotropic Tensor-Valued Tensor Function 50
1.9 Differential Formulae 54
1.9.1 Partial Derivatives 54
1.9.2 Directional Derivatives 59
1.9.3 Taylor Expansion 62
1.9.4 Time Derivatives in Lagrangian and Eulerian Descriptions 63
1.9.5 Derivatives of Tensor Field 68
1.9.6 Gauss's Divergence Theorem 71
1.9.7 Material-Time Derivative of Volume Integration 73
1.10 Variations and Rates of Geometrical Elements 74
1.10.1 Variations of Line, Surface and Volume 75
1.10.2 Rates of Changes of Surface and Volume 76
1.11 Continuity and Smoothness Conditions 79
1.11.1 Continuity Condition 79
1.11.2 Smoothness Condition 80
1.12 Unconventional Elasto-Plasticity Models 81
2 General (Curvilinear) Coordinate System 85
2.1 Primary and Reciprocal Base Vectors 85
2.2 Metric Tensors 89
2.3 Representations of Vectors and Tensors 95
2.4 Physical Components of Vectors and Tensors 102
2.5 Covariant Derivative of Base Vectors with Christoffel Symbol 103
2.6 Covariant Derivatives of Scalars, Vectors and Tensors 107
2.7 Riemann-Christoffel Curvature Tensor 112
2.8 Relations of Convected and Cartesian Coordinate Descriptions 115
3 Description of Physical Quantities in Convected Coordinate System 117
3.1 Necessity for Description in Embedded Coordinate System 117
3.2 Embedded Base Vectors 118
3.3 Deformation Gradient Tensor 121
3.4 Pull-Back and Push-Forward Operations 123
4 Strain and Strain Rate Tensors 131
4.1 Deformation Tensors 131
4.2 Strain Tensors 136
4.2.1 Green and Almansi Strain Tensors 136
4.
