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