1.9.5 Derivatives of Tensor Field1.9.6 Gauss's Divergence Theorem; 1.9.7 Material-Time Derivative of Volume Integration; 1.10 Variations and Rates of Geometrical Elements; 1.10.1 Variations of Line, Surface and Volume; 1.10.2 Rates of Changes of Surface and Volume; 1.11 Continuity and Smoothness Conditions; 1.11.1 Continuity Condition; 1.11.2 Smoothness Condition; 2 General (Curvilinear) Coordinate System; 2.1 Primary and Reciprocal Base Vectors; 2.2 Metric Tensors; 2.3 Representations of Vectors and Tensors; 2.4 Physical Components of Vectors and Tensors.

1.4.6 Tensor Decompositions1.4.7 Axial Vector; 1.4.8 Determinant; 1.4.9 On Solutions of Simultaneous Equation; 1.4.10 Scalar Triple Products with Invariants; 1.4.11 Orthogonal Transformation of Scalar Triple Product; 1.4.12 Pseudo Scalar, Vector and Tensor; 1.5 Tensor Representations; 1.5.1 Tensor Notations; 1.5.2 Tensor Components and Transformation Rule; 1.5.3 Notations of Tensor Operations; 1.5.4 Operational Tensors; 1.5.5 Isotropic Tensors; 1.6 Eigenvalues and Eigenvectors; 1.6.1 Eigenvalues and Eigenvectors of Second-Order Tensors.

1.6.2 Spectral Representation and Elementary Tensor Functions1.6.3 Calculation of Eigenvalues and Eigenvectors; 1.6.4 Eigenvalues and Vectors of Orthogonal Tensor; 1.6.5 Eigenvalues and Vectors of Skew-Symmetric Tensor and Axial Vector; 1.6.6 Cayley-Hamilton Theorem; 1.7 Polar Decomposition; 1.8 Isotropy; 1.8.1 Isotropic Material; 1.8.2 Representation Theorem of Isotropic Tensor-Valued Tensor Function; 1.9 Differential Formulae; 1.9.1 Partial Derivatives; 1.9.2 Directional Derivatives; 1.9.3 Taylor Expansion; 1.9.4 Time Derivatives in Lagrangian and Eulerian Descriptions.

2.5 Covariant Derivative of Base Vectors with Christoffel Symbol2.6 Covariant Derivatives of Scalars, Vectors and Tensors; 2.7 Riemann-Christoffel Curvature Tensor; 2.8 Relations of Convected and Cartesian Coordinate Descriptions; 3 Description of Physical Quantities in Convected Coordinate System; 3.1 Necessity for Description in Embedded Coordinate System; 3.2 Embedded Base Vectors; 3.3 Deformation Gradient Tensor; 3.4 Pull-Back and Push-Forward Operations; 4 Strain and Strain Rate Tensors; 4.1 Deformation Tensors; 4.2 Strain Tensors; 4.2.1 Green and Almansi Strain Tensors.

Comprehensive introduction to finite elastoplasticity, addressing various analytical and numerical analyses & including state-of-the-art theories Introduction to Finite Elastoplasticitypresents 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

INTRODUCTION TOFINITE STRAIN THEORYFOR CONTINUUMELASTO-PLASTICITY; Contents; Preface; Series Preface; Introduction; 1 Mathematical Preliminaries; 1.1 Basic Symbols and Conventions; 1.2 Definition of Tensor; 1.2.1 Objective Tensor; 1.2.2 Quotient Law; 1.3 Vector Analysis; 1.3.1 Scalar Product; 1.3.2 Vector Product; 1.3.3 Scalar Triple Product; 1.3.4 Vector Triple Product; 1.3.5 Reciprocal Vectors; 1.3.6 Tensor Product; 1.4 Tensor Analysis; 1.4.1 Properties of Second-Order Tensor; 1.4.2 Tensor Components; 1.4.3 Transposed Tensor; 1.4.4 Inverse Tensor; 1.4.5 Orthogonal Tensor.