This book presents the nonlinear theory of continuum mechanics and demonstrates its use in developing nonlinear computer formulations for large displacement dynamic analysis. Basic concepts used in continuum mechanics are presented and used to develop nonlinear general finite element formulations that can be effectively used in large displacement analysis. The book considers two nonlinear finite element dynamic formulations: a general large deformation finite element formulation and a formulation that can efficiently solve small deformation problems that characterize very stiff structures. The…mehr
This book presents the nonlinear theory of continuum mechanics and demonstrates its use in developing nonlinear computer formulations for large displacement dynamic analysis. Basic concepts used in continuum mechanics are presented and used to develop nonlinear general finite element formulations that can be effectively used in large displacement analysis. The book considers two nonlinear finite element dynamic formulations: a general large deformation finite element formulation and a formulation that can efficiently solve small deformation problems that characterize very stiff structures. The book presents material clearly and systematically, assuming the reader has only basic knowledge in matrix and vector algebra and dynamics. The book is designed for use by advanced undergraduates and first-year graduate students. It is also a reference for researchers, practising engineers, and scientists working in computational mechanics, bio-mechanics, computational biology, multibody system dynamics, and other fields of science and engineering using the general continuum mechanics theory.
1. Introduction 2. Kinematics 3. Forces and stresses 4. Constitutive equations 5. Plasticity formulations 6. Finite element formulation: large deformation, large rotation problem 7. Finite element formulation: small deformation, large rotation problem.
PREFACE ix
1 INTRODUCTION 1
1.1 Matrices / 2
1.2 Vectors / 6
1.3 Summation Convention / 11
1.4 Cartesian Tensors / 12
1.5 Polar Decomposition Theorem / 21
1.6 D'Alembert's Principle / 23
1.7 Virtual Work Principle / 29
1.8 Approximation Methods / 32
1.9 Discrete Equations / 34
1.10 Momentum, Work, and Energy / 37
1.11 Parameter Change and Coordinate Transformation / 39
Problems / 43
2 KINEMATICS 47
2.1 Motion Description / 48
2.2 Strain Components / 55
2.3 Other Deformation Measures / 60
2.4 Decomposition of Displacement / 62
2.5 Velocity and Acceleration / 64
2.6 Coordinate Transformation / 68
2.7 Objectivity / 74
2.8 Change of Volume and Area / 77
2.9 Continuity Equation / 81
2.10 Reynolds' Transport Theorem / 82
2.11 Examples of Deformation / 84
2.12 Important Geometry Concepts / 92
Problems / 94
3 FORCES AND STRESSES 97
3.1 Equilibrium of Forces / 97
3.2 Transformation of Stresses / 100
3.3 Equations of Equilibrium / 100
3.4 Symmetry of the cauchy Stress Tensor / 102
3.5 Virtual Work of the Forces / 103
3.6 Deviatoric Stresses / 113
3.7 Stress Objectivity / 115
3.8 Energy Balance / 119
Problems / 120
4 CONSTITUTIVE EQUATIONS 123
4.1 Generalized Hooke's Law / 124
4.2 Anisotropic Linearly Elastic Materials / 126
4.3 Material Symmetry / 127
4.4 Homogeneous Isotropic Material / 129
4.5 Principal Strain Invariants / 136
4.6 Special Material Models for Large Deformations / 137
4.7 Linear Viscoelasticity / 141
4.8 Nonlinear Viscoelasticity / 155
4.9 A Simple Viscoelastic Model for Isotropic Materials / 161
4.10 Fluid Constitutive Equations / 162
4.11 Navier-Stokes Equations / 164
Problems / 164
5 FINITE ELEMENT FORMULATION: LARGE-DEFORMATION, LARGE-ROTATION PROBLEM 167
5.1 Displacement Field / 169
5.2 Element Connectivity / 176
5.3 Inertia and Elastic Forces / 178
5.4 Equations of Motion / 180
5.5 Numerical Evaluation of The Elastic Forces / 188
5.6 Finite Elements and Geometry / 193
5.7 Two-Dimensional Euler-Bernoulli Beam Element / 199
5.8 Two-Dimensional Shear Deformable Beam Element / 203
5.9 Three-Dimensional Cable Element / 205
5.10 Three-Dimensional Beam Element / 206
5.11 Thin-Plate Element / 208
5.12 Higher-Order Plate Element / 210
5.13 Brick Element / 211
5.14 Element Performance / 212
5.15 Other Finite Element Formulations / 216
5.16 Updated Lagrangian and Eulerian Formulations / 218
5.17 Concluding Remarks / 221
Problems / 223
6 FINITE ELEMENT FORMULATION: SMALL-DEFORMATION, LARGE-ROTATION PROBLEM 225
6.1 Background / 226
6.2 Rotation and Angular Velocity / 229
6.3 Floating Frame of Reference (FFR) / 234
6.4 Intermediate Element Coordinate System / 236
6.5 Connectivity and Reference Conditions / 238
6.6 Kinematic Equations / 243
6.7 Formulation of The Inertia Forces / 245
6.8 Elastic Forces / 248
6.9 Equations of Motion / 250
6.10 Coordinate Reduction / 251
6.11 Integration of Finite Element and Multibody System Algorithms / 253
1. Introduction 2. Kinematics 3. Forces and stresses 4. Constitutive equations 5. Plasticity formulations 6. Finite element formulation: large deformation, large rotation problem 7. Finite element formulation: small deformation, large rotation problem.
PREFACE ix
1 INTRODUCTION 1
1.1 Matrices / 2
1.2 Vectors / 6
1.3 Summation Convention / 11
1.4 Cartesian Tensors / 12
1.5 Polar Decomposition Theorem / 21
1.6 D'Alembert's Principle / 23
1.7 Virtual Work Principle / 29
1.8 Approximation Methods / 32
1.9 Discrete Equations / 34
1.10 Momentum, Work, and Energy / 37
1.11 Parameter Change and Coordinate Transformation / 39
Problems / 43
2 KINEMATICS 47
2.1 Motion Description / 48
2.2 Strain Components / 55
2.3 Other Deformation Measures / 60
2.4 Decomposition of Displacement / 62
2.5 Velocity and Acceleration / 64
2.6 Coordinate Transformation / 68
2.7 Objectivity / 74
2.8 Change of Volume and Area / 77
2.9 Continuity Equation / 81
2.10 Reynolds' Transport Theorem / 82
2.11 Examples of Deformation / 84
2.12 Important Geometry Concepts / 92
Problems / 94
3 FORCES AND STRESSES 97
3.1 Equilibrium of Forces / 97
3.2 Transformation of Stresses / 100
3.3 Equations of Equilibrium / 100
3.4 Symmetry of the cauchy Stress Tensor / 102
3.5 Virtual Work of the Forces / 103
3.6 Deviatoric Stresses / 113
3.7 Stress Objectivity / 115
3.8 Energy Balance / 119
Problems / 120
4 CONSTITUTIVE EQUATIONS 123
4.1 Generalized Hooke's Law / 124
4.2 Anisotropic Linearly Elastic Materials / 126
4.3 Material Symmetry / 127
4.4 Homogeneous Isotropic Material / 129
4.5 Principal Strain Invariants / 136
4.6 Special Material Models for Large Deformations / 137
4.7 Linear Viscoelasticity / 141
4.8 Nonlinear Viscoelasticity / 155
4.9 A Simple Viscoelastic Model for Isotropic Materials / 161
4.10 Fluid Constitutive Equations / 162
4.11 Navier-Stokes Equations / 164
Problems / 164
5 FINITE ELEMENT FORMULATION: LARGE-DEFORMATION, LARGE-ROTATION PROBLEM 167
5.1 Displacement Field / 169
5.2 Element Connectivity / 176
5.3 Inertia and Elastic Forces / 178
5.4 Equations of Motion / 180
5.5 Numerical Evaluation of The Elastic Forces / 188
5.6 Finite Elements and Geometry / 193
5.7 Two-Dimensional Euler-Bernoulli Beam Element / 199
5.8 Two-Dimensional Shear Deformable Beam Element / 203
5.9 Three-Dimensional Cable Element / 205
5.10 Three-Dimensional Beam Element / 206
5.11 Thin-Plate Element / 208
5.12 Higher-Order Plate Element / 210
5.13 Brick Element / 211
5.14 Element Performance / 212
5.15 Other Finite Element Formulations / 216
5.16 Updated Lagrangian and Eulerian Formulations / 218
5.17 Concluding Remarks / 221
Problems / 223
6 FINITE ELEMENT FORMULATION: SMALL-DEFORMATION, LARGE-ROTATION PROBLEM 225
6.1 Background / 226
6.2 Rotation and Angular Velocity / 229
6.3 Floating Frame of Reference (FFR) / 234
6.4 Intermediate Element Coordinate System / 236
6.5 Connectivity and Reference Conditions / 238
6.6 Kinematic Equations / 243
6.7 Formulation of The Inertia Forces / 245
6.8 Elastic Forces / 248
6.9 Equations of Motion / 250
6.10 Coordinate Reduction / 251
6.11 Integration of Finite Element and Multibody System Algorithms / 253
Problems / 258
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