Computational Continuum Mechanics Book

Preface     ix
Introduction     1
Matrices     2
Vectors     6
Summation Convention     12
Cartesian Tensors     13
Polar Decomposition Theorem     25
D'Alembert's Principle     27
Virtual Work Principle     34
Approximation Methods     37
Discrete Equations     40
Momentum, Work, and Energy     43
Parameter Change and Coordinate Transformation     45
Problems     48
Kinematics     51
Motion Description     52
Strain Components     60
Other Deformation Measures     67
Decomposition of Displacement     69
Velocity and Acceleration     71
Coordinate Transformation     75
Objectivity     82
Change of Volume and Area     85
Continuity Equation     89
Reynolds' Transport Theorem     90
Examples of Deformation     92
Problems     100
Forces and Stresses     103
Equilibrium of Forces     103
Transformation of Stresses     106
Equations of Equilibrium     107
Symmetry of the Cauchy Stress Tensor     109
Virtual Work of the Forces     111
Deviatoric Stresses     120
Stress Objectivity     123
Energy Balance     127
Problems     129
Constitutive Equations     131
Generalized Hooke's Law     132
Anisotropic Linearly Elastic Materials     134
Material Symmetry     135
Homogeneous Isotropic Material     137
Principal Strain Invariants     144
Special Material Models for Large Deformations     146
Linear Viscoelasticity     150
Nonlinear Viscoelasticity     164
A Simple Viscoelastic Model for Isotropic Materials     171
Fluid Constitutive Equations     173
Navier-Stokes Equations     174
Problems     175
Plasticity Formulations     177
One-Dimensional Problem     179
Loading and Unloading Conditions     180
Solution of the Plasticity Equations     181
Generalization of the Plasticity Theory: Small Strains     190
J[subscript 2] Flow Theory with Isotropic/Kinematic Hardening     197
Nonlinear Formulation for Hyperelastic-Plastic Materials      214
Hyperelastic-Plastic J[subscript 2] Flow Theory     225
Problems     230
Finite Element Formulation: Large-Deformation, Large-Rotation Problem     231
Displacement Field     233
Element Connectivity     240
Inertia and Elastic Forces     243
Equations of Motion     246
Numerical Evaluation of the Elastic Forces     250
Finite Elements and Geometry     256
Two-Dimensional Euler-Bernoulli Beam Element     263
Two-Dimensional Shear Deformable Beam Element     267
Three-Dimensional Cable Element     269
Three-Dimensional Beam Element     270
Thin-Plate Element     272
Higher-Order Plate Element     274
Element Performance     275
Other Finite Element Formulations     280
Updated Lagrangian and Eulerian Formulations     282
Problems     284
Finite Element Formulation: Small-Deformation, Large-Rotation Problem     286
Background     287
Rotation and Angular Velocity     291
Floating Frame of Reference     296
Intermediate Element Coordinate System     297
Connectivity and Reference Conditions      300
Kinematic Equations     306
Formulation of the Inertia Forces     307
Elastic Forces     311
Equations of Motion     313
Coordinate Reduction     314
Integration of Finite Element and Multibody System Algorithms     317
Problems     319
References     321
Index     327