Computational Continuum Mechanics

**An updated and expanded edition of the popular guide to basic continuum mechanics and computational techniques** This updated third edition of the popular reference covers state-of-the-art computational techniques for basic continuum mechanics modeling of both small and large deformations. Approaches to developing complex models are described in detail, and numerous examples are presented demonstrating how computational algorithms can be developed using basic continuum mechanics approaches. The integration of geometry and analysis for the study of the motion and behaviors of materials under varying conditions is an increasingly popular approach in continuum mechanics, and absolute nodal coordinate formulation (ANCF) is rapidly emerging as the best way to achieve that integration. At the same time, simulation software is undergoing significant changes which will lead to the seamless fusion of CAD, finite element, and multibody system computer codes in one computational environment. __Computational Continuum Mechanics____, Third Edition__ is the only book to provide in-depth coverage of the formulations required to achieve this integration. * Provides detailed coverage of the absolute nodal coordinate formulation (ANCF), a popular new approach to the integration of geometry and analysis * Provides detailed coverage of the floating frame of reference (FFR) formulation, a popular well-established approach for solving small deformation problems * Supplies numerous examples of how complex models have been developed to solve an array of real-world problems * Covers modeling of both small and large deformations in detail * Demonstrates how to develop computational algorithms using basic continuum mechanics approaches __Computational Continuum Mechanics____, Third Edition__ is designed to function equally well as a text for advanced undergraduates and first-year graduate students and as a working reference for researchers, practicing 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. Computational Continuum Mechanics, 3rd Edition 4 Contents 6 Preface 10 1 Introduction 14 1.1 Matrices 15 1.2 Vectors 19 1.3 Summation Convention 24 1.4 Cartesian Tensors 25 1.5 Polar Decomposition Theorem 34 1.6 D’Alembert’s Principle 36 1.7 Virtual Work Principle 42 1.8 Approximation Methods 45 1.9 Discrete Equations 47 1.10 Momentum, Work, and Energy 50 1.11 Parameter Change and Coordinate Transformation 52 Problems 57 2 Kinematics 60 2.1 Motion Description 61 2.2 Strain Components 68 2.3 Other Deformation Measures 73 2.4 Decomposition of Displacement 75 2.5 Velocity and Acceleration 77 2.6 Coordinate Transformation 81 2.7 Objectivity 87 2.8 Change of Volume and Area 90 2.9 Continuity Equation 94 2.10 Reynolds’ Transport Theorem 95 2.11 Examples of Deformation 97 2.12 Geometry Concepts 105 Problems 107 3 Forces and Stresses 110 3.1 Equilibrium of Forces 110 3.2 Transformation of Stresses 113 3.3 Equations of Equilibrium 113 3.4 Symmetry of the Cauchy Stress Tensor 115 3.5 Virtual Work of the Forces 116 3.6 Deviatoric Stresses 126 3.7 Stress Objectivity 128 3.8 Energy Balance 132 Problems 133 4 Constitutive Equations 136 4.1 Generalized Hooke’s Law 137 4.2 Anisotropic Linearly Elastic Materials 139 4.3 Material Symmetry 140 4.4 Homogeneous Isotropic Material 142 4.5 Principal Strain Invariants 149 4.6 Special Material Models for Large Deformations 150 4.7 Linear Viscoelasticity 154 4.8 Nonlinear Viscoelasticity 168 4.9 A Simple Viscoelastic Model for Isotropic Materials 174 4.10 Fluid Constitutive Equations 175 4.11 Navier–Stokes Equations 177 Problems 177 5 Finite Element Formulation: Large-Deformation, Large-Rotation Problem 180 5.1 Displacement Field 182 5.2 Element Connectivity 189 5.3 Inertia and Elastic Forces 191 5.4 Equations of Motion 193 5.5 Numerical Evaluation of The Elastic Forces 201 5.6 Finite Elements and Geometry 206 5.7 Two-Dimensional Euler–Bernoulli Beam Element 212 5.8 Two-Dimensional Shear Deformable Beam Element 216 5.9 Three-Dimensional Cable Element 218 5.10 Three-Dimensional Beam Element 219 5.11 Thin-Plate Element 221 5.12 Higher-Order Plate Element 223 5.13 Brick Element 224 5.14 Element Performance 225 5.15 Other Finite Element Formulations 229 5.16 Updated Lagrangian and Eulerian Formulations 231 5.17 Concluding Remarks 234 Problems 236 6 Finite Element Formulation: Small-Deformation, Large-Rotation Problem 238 6.1 Background 239 6.2 Rotation and Angular Velocity 242 6.3 Floating Frame of Reference (FFR) 247 6.4 Intermediate Element Coordinate System 249 6.5 Connectivity and Reference Conditions 251 6.6 Kinematic Equations 256 6.7 Formulation of The Inertia Forces 258 6.8 Elastic Forces 261 6.9 Equations of Motion 263 6.10 Coordinate Reduction 264 6.11 Integration of Finite Element and Multibody System Algorithms 266 Problems 271 7 Computational Geometry and Finite Element Analysis 274 7.1 Geometry and Finite Element Method 275 7.2 ANCF Geometry 277 7.3 Bezier Geometry 279 7.4 B-Spline Curve Representation 280 7.5 Conversion of B-Spline Geometry to ANCF Geometry 284 7.6 ANCF and B-Spline Surfaces 286 7.7 Structural and Nonstructural Discontinuities 288 Problems 290 8 Plasticity Formulations 292 8.1 One-Dimensional Problem 294 8.2 Loading and Unloading Conditions 295 8.3 Solution of the Plasticity Equations 296 8.4 Generalization of The Plasticity Theory: Small Strains 304 8.5 J2 Flow Theory with Isotropic/Kinematic Hardening 311 8.6 Nonlinear Formulation for Hyperelastic–Plastic Materials 325 8.7 Hyperelastic–Plastic J2 Flow Theory 335 Problems 339 References 342 Index 352 This work 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 for large displacement analysis. An updated and expanded edition of the popular guide to basic continuum mechanics and computational techniques This updated third edition of the popular reference covers state-of-the-art computational techniques for basic continuum mechanics modeling of both small and large deformations. Approaches to developing complex models are described in detail, and numerous examples are presented demonstrating how computational algorithms can be developed using basic continuum mechanics approaches. The integration of geometry and analysis for the study of the motion and behaviors of materials under varying conditions is an increasingly popular approach in continuum mechanics, and absolute nodal coordinate formulation (ANCF) is rapidly emerging as the best way to achieve that integration. At the same time, simulation software is undergoing significant changes which will lead to the seamless fusion of CAD, finite element, and multibody system computer codes in one computational environment. Computational Continuum Mechanics, Third Edition is the only bookto provide in-depth coverage of the formulations required to achieve this integration. Provides detailed coverage of the absolute nodal coordinate formulation (ANCF), a popular new approach to the integration of geometry and analysis Provides detailed coverage of the floating frame of reference (FFR) formulation, a popular well-established approach for solving small deformation problems Supplies numerous examples of how complex models have been developed to solve an array of real-world problems Covers modeling of both small and large deformations in detail Demonstrates how to develop computational algorithms using basic continuum mechanics approaches Computational Continuum Mechanics, Third Edition is designed to function equally well as a text for advanced undergraduates and first-year graduate students and as a working reference for researchers, practicing 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