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# Large strain finite element method : a practical course / Antonio Munjiza, Esteban Rougier, Earl E. Knight.

Material type: BookPublisher: West Sussex : Wiley, 2015Description: xiv, 469 p. : ill. ; 24 cm.ISBN: 9781118405307 ; 1118405307.DDC classification: 620.11230151825
Contents:
PART ONE FUNDAMENTALS, 1 Introduction : 1.1 Assumption of Small Displacements -- 1.2 Assumption of Small Strains -- 1.3 Geometric Nonlinearity -- 1.4 Stretches -- 1.5 Some Examples of Large Displacement Large Strain Finite Element Formulation -- 1.6 The Scope and Layout of the Book -- 1.7 Summary -- 2 Matrices : 2.1 Matrices in General -- 2.2 Matrix Algebra -- 2.3 Special Types of Matrices -- 2.4 Determinant of a Square Matrix -- 2.5 Quadratic Form -- 2.6 Eigenvalues and Eigenvectors -- 2.7 Positive Definite Matrix -- 2.8 Gaussian Elimination -- 2.9 Inverse of a Square Matrix -- 2.10 Column Matrices -- 2.11 Summary -- 3 Some Explicit and Iterative Solvers : 3.1 The Central Difference Solver -- 3.2 Generalized Direction Methods -- 3.3 The Method of Conjugate Directions -- 3.4 Summary -- 4 Numerical Integration -- 4.1 Newton-Cotes Numerical Integration -- 4.2 Gaussian Numerical Integration -- 4.3 Gaussian Integration in 2D -- 4.4 Gaussian Integration in 3D -- 4.5 Summary -- 5 Work of Internal Forces on Virtual Displacements -- 5.1 The Principle of Virtual Work -- 5.2 Summary -- PART TWO PHYSICAL QUANTITIES, 6 Scalars : 6.1 Scalars in General -- 6.2 Scalar Functions -- 6.3 Scalar Graphs -- 6.4 Empirical Formulas -- 6.5 Fonts -- 6.6 Units -- 6.7 Base and Derived Scalar Variables -- 6.8 Summary -- 7 Vectors in 2D : 7.1 Vectors in General -- 7.2 Vector Notation -- 7.3 Matrix Representation of Vectors -- 7.4 Scalar Product -- 7.5 General Vector Base in 2D -- 7.6 Dual Base -- 7.7 Changing Vector Base -- 7.8 Self-duality of the Orthonormal Base -- 7.9 Combining Bases -- 7.10 Examples -- 7.11 Summary -- 8 Vectors in 3D : 8.1 Vectors in 3D -- 8.2 Vector Bases -- 8.3 Summary -- 9 Vectors in n-Dimensional Space : 9.1 Extension from 3D to 4-Dimensional Space -- 9.2 The Dual Base in 4D -- 9.3 Changing the Base in 4D -- 9.4 Generalization to n-Dimensional Space -- 9.5 Changing the Base in n-Dimensional Space -- 9.6 Summary -- 10 First Order Tensors : 10.1 The Slope Tensor -- 10.2 First Order Tensors in 2D -- 10.3 Using First Order Tensors -- 10.4 Using Different Vector Bases in 2D -- 10.5 Differential of a 2D Scalar Field as the First Order Tensor -- 10.6 First Order Tensors in 3D -- 10.7 Changing the Vector Base in 3D -- 10.8 First Order Tensor in 4D -- 10.9 First Order Tensor in n-Dimensions -- 10.10 Differential of a 3D Scalar Field as the First Order Tensor -- 10.11 Scalar Field in n-Dimensional Space -- 10.12 Summary -- 11 Second Order Tensors in 2D : 11.1 Stress Tensor in 2D -- 11.2 Second Order Tensor in 2D -- 11.3 Physical Meaning of Tensor Matrix in 2D -- 11.4 Changing the Base -- 11.5 Using Two Different Bases in 2D -- 11.6 Some Special Cases of Stress Tensor Matrices in 2D -- 11.7 The First Piola-Kirchhoff Stress Tensor Matrix -- 11.8 The Second Piola-Kirchhoff Stress Tensor Matrix -- 11.9 Summary -- 12 Second Order Tensors in 3D : 12.1 Stress Tensor in 3D -- 12.2 General Base for Surfaces -- 12.3 General Base for Forces -- 12.4 General Base for Forces and Surfaces -- 12.5 The Cauchy Stress Tensor Matrix in 3D -- 12.6 The First Piola-Kirchhoff Stress Tensor Matrix in 3D -- 12.7 The Second Piola-Kirchhoff Stress Tensor Matrix in 3D -- 12.8 Summary -- 13 Second Order Tensors in nD : 13.1 Second Order Tensor in n-Dimensions -- 13.2 Summary -- PART THREE DEFORMABILITY AND MATERIAL MODELING, 14 Kinematics of Deformation in 1D -- 14.1 Geometric Nonlinearity in General -- 14.2 Stretch -- 14.3 Material Element and Continuum Assumption -- 14.4 Strain -- 14.5 Stress -- 14.6 Summary -- 15 Kinematics of Deformation in 2D, 15.1 Isotropic Solids -- 15.2 Homogeneous Solids -- 15.3 Homogeneous and Isotropic Solids -- 15.4 Nonhomogeneous and Anisotropic Solids -- 15.5 Material Element Deformation -- 15.6 Cauchy Stress Matrix for the Solid Element -- 15.7 Coordinate Systems in 2D -- 15.8 The Solid- and the Material-Embedded Vector Bases -- 15.9 Kinematics of 2D Deformation -- 15.10 2D Equilibrium Using the Virtual Work of Internal Forces -- 15.11 Examples -- 15.12 Summary -- 16 Kinematics of Deformation in 3D : 16.1 The Cartesian Coordinate System in 3D -- 16.2 The Solid-Embedded Coordinate System -- 16.3 The Global and the Solid-Embedded Vector Bases -- 16.4 Deformation of the Solid -- 16.5 Generalized Material Element -- 16.6 Kinematic of Deformation in 3D -- 16.7 The Virtual Work of Internal Forces -- 16.8 Summary -- 17 The Unified Constitutive Approach in 2D : 17.1 Introduction -- 17.2 Material Axes -- 17.3 Micromechanical Aspects and Homogenization -- 17.4 Generalized Homogenization -- 17.5 The Material Package -- 17.6 Hyper-Elastic Constitutive Law -- 17.7 Hypo-Elastic Constitutive Law -- 17.8 A Unified Framework for Developing Anisotropic Material Models in 2D -- 17.9 Generalized Hyper-Elastic Material -- 17.10 Converting the Munjiza Stress Matrix to the Cauchy Stress Matrix -- 17.11 Developing Constitutive Laws -- 17.12 Generalized Hypo-Elastic Material -- 17.13 Unified Constitutive Approach for Strain Rate and Viscosity -- 17.14 Summary -- 18 The Unified Constitutive Approach in 3D : 18.1 Material Package Framework -- 18.2 Generalized Hyper-Elastic Material -- 18.3 Generalized Hypo-Elastic Material -- 18.4 Developing Material Models -- 18.5 Calculation of the Cauchy Stress Tensor Matrix -- 18.6 Summary -- PART FOUR THE FINITE ELEMENT METHOD IN 2D, 2D Finite Element: Deformation Kinematics Using the Homogeneous Deformation Triangle : 19.1 The Finite Element Mesh -- 19.2 The Homogeneous Deformation Finite Element -- 19.3 Summary -- 20 2D Finite Element: Deformation Kinematics Using Iso-Parametric Finite Elements : 20.1 The Finite Element Library -- 20.2 The Shape Functions -- 20.3 Nodal Positions -- 20.4 Positions of Material Points inside a Single Finite Element -- 20.5 The Solid-Embedded Vector Base -- 20.6 The Material-Embedded Vector Base -- 20.7 Some Examples of 2D Finite Elements -- 20.8 Summary -- 21 Integration of Nodal Forces over Volume of 2D Finite Elements : 21.1 The Principle of Virtual Work in the 2D Finite Element Method -- 21.2 Nodal Forces for the Homogeneous Deformation Triangle -- 21.3 Nodal Forces for the Six-Noded Triangle -- 21.4 Nodal Forces for the Four-Noded Quadrilateral -- 21.5 Summary -- 22 Reduced and Selective Integration of Nodal Forces over Volume of 2D Finite Elements : 22.1 Volumetric Locking -- 22.2 Reduced Integration -- 22.3 Selective Integration -- 22.4 Shear Locking -- 22.5 Summary -- PART FIVE THE FINITE ELEMENT METHOD IN 3D, 23 3D Deformation Kinematics Using the Homogeneous Deformation Tetrahedron Finite Element : 23.1 Introduction -- 23.2 The Homogeneous Deformation Four-Noded Tetrahedron Finite Element -- 23.3 Summary -- 24 3D Deformation Kinematics Using Iso-Parametric Finite Elements : 24.1 The Finite Element Library -- 24.2 The Shape Functions -- 24.3 Nodal Positions -- 24.4 Positions of Material Points inside a Single Finite Element -- 24.5 The Solid-Embedded Infinitesimal Vector Base -- 24.6 The Material-Embedded Infinitesimal Vector Base -- 24.7 Examples of Deformation Kinematics -- 24.8 Summary -- 25 Integration of Nodal Forces over Volume of 3D Finite Elements : 25.1 Nodal Forces Using Virtual Work -- 25.2 Four-Noded Tetrahedron Finite Element -- 25.3 Reduce Integration for Eight-Noded 3D Solid -- 25.4 Selective Stretch Sampling-Based Integration for the Eight-Noded Solid Finite Element -- 25.5 Summary -- 26 Integration of Nodal Forces over Boundaries of Finite Elements : 26.1 Stress at Element Boundaries -- 26.2 Integration of the Equivalent Nodal Forces over the Triangle Finite Element -- 26.3 Integration over the Boundary of the Composite Triangle -- 26.4 Integration over the Boundary of the Six-Noded Triangle -- 26.5 Integration of the Equivalent Internal Nodal Forces over the Tetrahedron Boundaries -- 26.6 Summary -- PART SIX THE FINITE ELEMENT METHOD IN 2.5D, 27 Deformation in 2.5D Using Membrane Finite Elements : 27.1 Solids in 2.5D -- 27.2 The Homogeneous Deformation Three-Noded Triangular Membrane Finite Element -- 27.3 Summary -- 28 Deformation in 2.5D Using Shell Finite Elements : 28.1 Introduction -- 28.2 The Six-Noded Triangular Shell Finite Element -- 28.3 The Solid-Embedded Coordinate System -- 28.4 Nodal Coordinates -- 28.5 The Coordinates of the Finite Element s Material Points -- 28.6 The Solid-Embedded Infinitesimal Vector Base -- 28.7 The Solid-Embedded Vector Base versus the Material-Embedded Vector Base -- 28.8 The Constitutive Law -- 28.9 Selective Stretch Sampling Based Integration of the Equivalent Nodal Forces -- 28.10 Multi-Layered Shell as an Assembly of Single Layer Shells -- 28.11 Improving the CPU Performance of the Shell Element -- 28.12 Summary
Summary: An introductory approach to the subject of large strains and large displacements in finite elements.
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Index : p. 461-469.

Includes bibliographical references.

PART ONE FUNDAMENTALS, 1 Introduction : 1.1 Assumption of Small Displacements -- 1.2 Assumption of Small Strains -- 1.3 Geometric Nonlinearity -- 1.4 Stretches -- 1.5 Some Examples of Large Displacement Large Strain Finite Element Formulation -- 1.6 The Scope and Layout of the Book -- 1.7 Summary -- 2 Matrices : 2.1 Matrices in General -- 2.2 Matrix Algebra -- 2.3 Special Types of Matrices -- 2.4 Determinant of a Square Matrix -- 2.5 Quadratic Form -- 2.6 Eigenvalues and Eigenvectors -- 2.7 Positive Definite Matrix -- 2.8 Gaussian Elimination -- 2.9 Inverse of a Square Matrix -- 2.10 Column Matrices -- 2.11 Summary -- 3 Some Explicit and Iterative Solvers : 3.1 The Central Difference Solver -- 3.2 Generalized Direction Methods -- 3.3 The Method of Conjugate Directions -- 3.4 Summary -- 4 Numerical Integration -- 4.1 Newton-Cotes Numerical Integration -- 4.2 Gaussian Numerical Integration -- 4.3 Gaussian Integration in 2D -- 4.4 Gaussian Integration in 3D -- 4.5 Summary -- 5 Work of Internal Forces on Virtual Displacements -- 5.1 The Principle of Virtual Work -- 5.2 Summary -- PART TWO PHYSICAL QUANTITIES, 6 Scalars : 6.1 Scalars in General -- 6.2 Scalar Functions -- 6.3 Scalar Graphs -- 6.4 Empirical Formulas -- 6.5 Fonts -- 6.6 Units -- 6.7 Base and Derived Scalar Variables -- 6.8 Summary -- 7 Vectors in 2D : 7.1 Vectors in General -- 7.2 Vector Notation -- 7.3 Matrix Representation of Vectors -- 7.4 Scalar Product -- 7.5 General Vector Base in 2D -- 7.6 Dual Base -- 7.7 Changing Vector Base -- 7.8 Self-duality of the Orthonormal Base -- 7.9 Combining Bases -- 7.10 Examples -- 7.11 Summary -- 8 Vectors in 3D : 8.1 Vectors in 3D -- 8.2 Vector Bases -- 8.3 Summary -- 9 Vectors in n-Dimensional Space : 9.1 Extension from 3D to 4-Dimensional Space -- 9.2 The Dual Base in 4D -- 9.3 Changing the Base in 4D -- 9.4 Generalization to n-Dimensional Space -- 9.5 Changing the Base in n-Dimensional Space -- 9.6 Summary -- 10 First Order Tensors : 10.1 The Slope Tensor -- 10.2 First Order Tensors in 2D -- 10.3 Using First Order Tensors -- 10.4 Using Different Vector Bases in 2D -- 10.5 Differential of a 2D Scalar Field as the First Order Tensor -- 10.6 First Order Tensors in 3D -- 10.7 Changing the Vector Base in 3D -- 10.8 First Order Tensor in 4D -- 10.9 First Order Tensor in n-Dimensions -- 10.10 Differential of a 3D Scalar Field as the First Order Tensor -- 10.11 Scalar Field in n-Dimensional Space -- 10.12 Summary -- 11 Second Order Tensors in 2D : 11.1 Stress Tensor in 2D -- 11.2 Second Order Tensor in 2D -- 11.3 Physical Meaning of Tensor Matrix in 2D -- 11.4 Changing the Base -- 11.5 Using Two Different Bases in 2D -- 11.6 Some Special Cases of Stress Tensor Matrices in 2D -- 11.7 The First Piola-Kirchhoff Stress Tensor Matrix -- 11.8 The Second Piola-Kirchhoff Stress Tensor Matrix -- 11.9 Summary -- 12 Second Order Tensors in 3D : 12.1 Stress Tensor in 3D -- 12.2 General Base for Surfaces -- 12.3 General Base for Forces -- 12.4 General Base for Forces and Surfaces -- 12.5 The Cauchy Stress Tensor Matrix in 3D -- 12.6 The First Piola-Kirchhoff Stress Tensor Matrix in 3D -- 12.7 The Second Piola-Kirchhoff Stress Tensor Matrix in 3D -- 12.8 Summary -- 13 Second Order Tensors in nD : 13.1 Second Order Tensor in n-Dimensions -- 13.2 Summary -- PART THREE DEFORMABILITY AND MATERIAL MODELING, 14 Kinematics of Deformation in 1D -- 14.1 Geometric Nonlinearity in General -- 14.2 Stretch -- 14.3 Material Element and Continuum Assumption -- 14.4 Strain -- 14.5 Stress -- 14.6 Summary -- 15 Kinematics of Deformation in 2D, 15.1 Isotropic Solids -- 15.2 Homogeneous Solids -- 15.3 Homogeneous and Isotropic Solids -- 15.4 Nonhomogeneous and Anisotropic Solids -- 15.5 Material Element Deformation -- 15.6 Cauchy Stress Matrix for the Solid Element -- 15.7 Coordinate Systems in 2D -- 15.8 The Solid- and the Material-Embedded Vector Bases -- 15.9 Kinematics of 2D Deformation -- 15.10 2D Equilibrium Using the Virtual Work of Internal Forces -- 15.11 Examples -- 15.12 Summary -- 16 Kinematics of Deformation in 3D : 16.1 The Cartesian Coordinate System in 3D -- 16.2 The Solid-Embedded Coordinate System -- 16.3 The Global and the Solid-Embedded Vector Bases -- 16.4 Deformation of the Solid -- 16.5 Generalized Material Element -- 16.6 Kinematic of Deformation in 3D -- 16.7 The Virtual Work of Internal Forces -- 16.8 Summary -- 17 The Unified Constitutive Approach in 2D : 17.1 Introduction -- 17.2 Material Axes -- 17.3 Micromechanical Aspects and Homogenization -- 17.4 Generalized Homogenization -- 17.5 The Material Package -- 17.6 Hyper-Elastic Constitutive Law -- 17.7 Hypo-Elastic Constitutive Law -- 17.8 A Unified Framework for Developing Anisotropic Material Models in 2D -- 17.9 Generalized Hyper-Elastic Material -- 17.10 Converting the Munjiza Stress Matrix to the Cauchy Stress Matrix -- 17.11 Developing Constitutive Laws -- 17.12 Generalized Hypo-Elastic Material -- 17.13 Unified Constitutive Approach for Strain Rate and Viscosity -- 17.14 Summary -- 18 The Unified Constitutive Approach in 3D : 18.1 Material Package Framework -- 18.2 Generalized Hyper-Elastic Material -- 18.3 Generalized Hypo-Elastic Material -- 18.4 Developing Material Models -- 18.5 Calculation of the Cauchy Stress Tensor Matrix -- 18.6 Summary -- PART FOUR THE FINITE ELEMENT METHOD IN 2D, 2D Finite Element: Deformation Kinematics Using the Homogeneous Deformation Triangle : 19.1 The Finite Element Mesh -- 19.2 The Homogeneous Deformation Finite Element -- 19.3 Summary -- 20 2D Finite Element: Deformation Kinematics Using Iso-Parametric Finite Elements : 20.1 The Finite Element Library -- 20.2 The Shape Functions -- 20.3 Nodal Positions -- 20.4 Positions of Material Points inside a Single Finite Element -- 20.5 The Solid-Embedded Vector Base -- 20.6 The Material-Embedded Vector Base -- 20.7 Some Examples of 2D Finite Elements -- 20.8 Summary -- 21 Integration of Nodal Forces over Volume of 2D Finite Elements : 21.1 The Principle of Virtual Work in the 2D Finite Element Method -- 21.2 Nodal Forces for the Homogeneous Deformation Triangle -- 21.3 Nodal Forces for the Six-Noded Triangle -- 21.4 Nodal Forces for the Four-Noded Quadrilateral -- 21.5 Summary -- 22 Reduced and Selective Integration of Nodal Forces over Volume of 2D Finite Elements : 22.1 Volumetric Locking -- 22.2 Reduced Integration -- 22.3 Selective Integration -- 22.4 Shear Locking -- 22.5 Summary -- PART FIVE THE FINITE ELEMENT METHOD IN 3D, 23 3D Deformation Kinematics Using the Homogeneous Deformation Tetrahedron Finite Element : 23.1 Introduction -- 23.2 The Homogeneous Deformation Four-Noded Tetrahedron Finite Element -- 23.3 Summary -- 24 3D Deformation Kinematics Using Iso-Parametric Finite Elements : 24.1 The Finite Element Library -- 24.2 The Shape Functions -- 24.3 Nodal Positions -- 24.4 Positions of Material Points inside a Single Finite Element -- 24.5 The Solid-Embedded Infinitesimal Vector Base -- 24.6 The Material-Embedded Infinitesimal Vector Base -- 24.7 Examples of Deformation Kinematics -- 24.8 Summary -- 25 Integration of Nodal Forces over Volume of 3D Finite Elements : 25.1 Nodal Forces Using Virtual Work -- 25.2 Four-Noded Tetrahedron Finite Element -- 25.3 Reduce Integration for Eight-Noded 3D Solid -- 25.4 Selective Stretch Sampling-Based Integration for the Eight-Noded Solid Finite Element -- 25.5 Summary -- 26 Integration of Nodal Forces over Boundaries of Finite Elements : 26.1 Stress at Element Boundaries -- 26.2 Integration of the Equivalent Nodal Forces over the Triangle Finite Element -- 26.3 Integration over the Boundary of the Composite Triangle -- 26.4 Integration over the Boundary of the Six-Noded Triangle -- 26.5 Integration of the Equivalent Internal Nodal Forces over the Tetrahedron Boundaries -- 26.6 Summary -- PART SIX THE FINITE ELEMENT METHOD IN 2.5D, 27 Deformation in 2.5D Using Membrane Finite Elements : 27.1 Solids in 2.5D -- 27.2 The Homogeneous Deformation Three-Noded Triangular Membrane Finite Element -- 27.3 Summary -- 28 Deformation in 2.5D Using Shell Finite Elements : 28.1 Introduction -- 28.2 The Six-Noded Triangular Shell Finite Element -- 28.3 The Solid-Embedded Coordinate System -- 28.4 Nodal Coordinates -- 28.5 The Coordinates of the Finite Element s Material Points -- 28.6 The Solid-Embedded Infinitesimal Vector Base -- 28.7 The Solid-Embedded Vector Base versus the Material-Embedded Vector Base -- 28.8 The Constitutive Law -- 28.9 Selective Stretch Sampling Based Integration of the Equivalent Nodal Forces -- 28.10 Multi-Layered Shell as an Assembly of Single Layer Shells -- 28.11 Improving the CPU Performance of the Shell Element -- 28.12 Summary

An introductory approach to the subject of large strains and large displacements in finite elements.

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