Computational Continuum Mechanics (Hardcover, 3)

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

7 COMPUTATIONAL GEOMETRY AND FINITE ELEMENT ANALYSIS 261

7.1 Geometry and Finite Element Method / 262

7.2 ANCF Geometry / 264

7.3 Bezier Geometry / 266

7.4 B-Spline Curve Representation / 267

7.5 Conversion of B-Spline Geometry to ANCF Geometry / 271

7.6 ANCF and B-Spline Surfaces / 273

7.7 Structural and Nonstructural Discontinuities / 275

8 PLASTICITY FORMULATIONS 279

8.1 One-Dimensional Problem / 281

8.2 Loading and Unloading Conditions / 282

8.3 Solution of the Plasticity Equations / 283

8.4 Generalization of The Plasticity Theory: Small Strains / 291

8.5 J2 Flow Theory with Isotropic/Kinematic Hardening / 298

8.6 Nonlinear Formulation for Hyperelastic–Plastic Materials / 312

8.7 Hyperelastic–Plastic J2 FLOW THEORY / 322

Problems / 326

REFERENCES 329

INDEX 339