Computational Continuum Mechanics.

Cover -- Title Page -- Copyright -- Contents -- Preface -- Chapter 1 Introduction -- 1.1 Matrices -- Definitions -- Determinant -- Inverse and Orthogonality -- Matrix Operations -- 1.2 Vectors -- Dot Product -- Cross Product -- Dyadic Product -- Projection -- 1.3 Summation Convention -- Unit Dyads -- 1.4 Cartesian Tensors -- Double Product or Double Contraction -- Invariants of the Second-Order Tensor -- Symmetric Tensors -- Higher-Order Tensors -- 1.5 Polar Decomposition Theorem -- Other Decompositions -- 1.6 D'Alembert's Principle -- Particle Mechanics -- Rigid-Body Kinematics -- Application of D'Alembert's Principle -- Continuum Forces -- 1.7 Virtual Work Principle -- Relationship with D'Alembert's Principle -- 1.8 Approximation Methods -- 1.9 Discrete Equations -- 1.10 Momentum, Work, and Energy -- Linear and Angular Momentum -- Work and Energy -- 1.11 Parameter Change and Coordinate Transformation -- Change of Parameters -- Coordinate Transformation -- Deformation and Strains -- Position Vector Gradients and Rigid Body Kinematics -- Problems -- Chapter 2 Kinematics -- 2.1 Motion Description -- Line Elements -- Rigid-Body Motion -- Floating Frame of Reference (FFR) -- Displacement Vector Gradients -- 2.2 Strain Components -- Geometric Interpretation of the Strains -- Eulerian Strain Tensor -- 2.3 Other Deformation Measures -- Right and Left Cauchy-Green Deformation Tensors -- Infinitesimal Strain Tensor -- 2.4 Decomposition of Displacement -- Homogeneous Motion -- Nonhomogeneous Motion -- 2.5 Velocity and Acceleration -- Eulerian Description -- Rate of Deformation and Spin Tensors -- Rate of Change of the Green-Lagrange Strain -- 2.6 Coordinate Transformation -- Strain Transformation -- Gradients and Strains -- Principal Strains -- Strain Invariants -- 2.7 Objectivity -- 2.8 Change of Volume and Area -- Volume -- Area.

2.9 Continuity Equation -- 2.10 Reynolds' Transport Theorem -- 2.11 Examples of Deformation -- Planar Displacement -- Extension and Stretch -- Shear Deformation -- 2.12 Geometry Concepts -- Problems -- Chapter 3 Forces and Stresses -- 3.1 Equilibrium of Forces -- 3.2 Transformation of Stresses -- 3.3 Equations of Equilibrium -- 3.4 Symmetry of the cauchy Stress Tensor -- Principal Stresses -- 3.5 Virtual Work of the Forces -- Tensor Double Product (Contraction) -- Volume Change -- Virtual Work -- Other Stress Measures -- First and Second Piola-Kirchhoff Stress Tensors -- Notation and Procedure -- Surface Forces -- Total and Updated Lagrangian Formulations -- Physical Interpretation -- 3.6 Deviatoric Stresses -- 3.7 Stress Objectivity -- Stress Rate -- Truesdell Stress Rate o -- Oldroyd and Convective Stress Rates o and o -- Green-Naghdi Stress Rate -- Jaumann Stress Rate -- 3.8 Energy Balance -- Problems -- Chapter 4 Constitutive Equations -- 4.1 Generalized Hooke's Law -- 4.2 Anisotropic Linearly Elastic Materials -- 4.3 Material Symmetry -- Reflection -- Rotations -- 4.4 Homogeneous Isotropic Material -- Poisson Effect and Locking -- Stress and Strain Invariants -- Plane-Stress and Plane-Strain Problems -- Finite Dimensional Model -- Generalized Elastic Forces -- Homogeneous Displacement -- 4.5 Principal Strain Invariants -- 4.6 Special Material Models for Large Deformations -- Compressible Neo-Hookean Material Models -- Incompressible Mooney-Rivlin Materials -- Objectivity -- 4.7 Linear Viscoelasticity -- One-Dimensional Model -- Other Viscoelastic Models -- Generalization -- Elastic Energy and Dissipation -- Another Form of the Viscoelastic Equations -- Three-Dimensional Linear Viscoelasticity -- 4.8 Nonlinear Viscoelasticity -- Another Model -- 4.9 A Simple Viscoelastic Model for Isotropic Materials -- 4.10 Fluid Constitutive Equations.

4.11 Navier-Stokes Equations -- Problems -- Chapter 5 Finite Element Formulation: Large-Deformation, Large-Rotation Problem -- Small- and Large-Deformation Problems -- Absolute Nodal Coordinate Formulation (ANCF) -- Organization -- 5.1 Displacement Field -- Separation of Variables -- Modes of Displacement -- Nodal Coordinates -- 5.2 Element Connectivity -- 5.3 Inertia and Elastic Forces -- Inertia Forces -- Elastic Forces -- 5.4 Equations of Motion -- Curved Geometry -- 5.5 Numerical Evaluation of the Elastic Forces -- Gaussian Quadrature -- Generalization -- 5.6 Finite Elements and Geometry -- General Continuum Mechanics Approach and Classical Theories -- Gradient Vectors -- Locking Problems -- Theory of Curves -- Theory of Surfaces -- Surface Curvature -- 5.7 Two-Dimensional Euler-Bernoulli Beam Element -- Kinematics of the Element -- Formulation of the Element Elastic Forces -- Special Case -- 5.8 Two-Dimensional Shear Deformable Beam Element -- Formulation of the Elastic Forces -- 5.9 Three-Dimensional Cable Element -- 5.10 Three-Dimensional Beam Element -- 5.11 Thin-Plate Element -- 5.12 Higher-Order Plate Element -- 5.13 Brick Element -- 5.14 Element Performance -- Patch Test -- Locking Problem -- Reduced Integration -- 5.15 Other Finite Element Formulations -- Isoparametric Finite Elements -- Use of Infinitesimal Rotation Coordinates -- Use of Finite Rotation Coordinates -- 5.16 Updated Lagrangian and Eulerian Formulations -- 5.17 Concluding Remarks -- ANCF Finite Elements -- Constrained Motion -- ANCF Reference Node -- Deformation Modes -- Problems -- Chapter 6 Finite Element Formulation: Small-Deformation, Large-Rotation Problem -- 6.1 Background -- Rigid-Body Motion -- Translations -- 6.2 Rotation and Angular Velocity -- Identities -- General Displacement -- Illustrative Example -- Euler Angles Singularity.

6.3 Floating Frame of Reference (FFR) -- 6.4 Intermediate Element Coordinate System -- 6.5 Connectivity and Reference Conditions -- Connectivity Conditions -- Reference Conditions -- Rigid-Body and Reference Motion -- 6.6 Kinematic Equations -- 6.7 Formulation of the Inertia Forces -- Body Inertia Shape Integrals -- 6.8 Elastic Forces -- 6.9 Equations of Motion -- 6.10 Coordinate Reduction -- 6.11 Integration of Finite Element and Multibody System Algorithms -- Linear Theory of Elastodynamics -- Nodal and Modal Coordinates -- Numerical Evaluation of the Inertia Shape Integrals -- Scaling of the Modal Coordinates -- Limitations of the FFR Formulation -- Problems -- Chapter 7 Computational Geometry and Finite Element Analysis -- 7.1 Geometry and Finite Element Method -- Bezier Geometry -- B-Spline Geometry -- NURBS Geometry -- 7.2 Ancf Geometry -- ANCF Element Geometry -- Control-Point Representation -- 7.3 Bezier Geometry -- 7.4 B-Spline Curve Representation -- Control Points and Degree of Continuity -- Illustrative Example -- Knot Insertion -- Comparison with FE Formulations -- 7.5 Conversion of B-Spline Geometry to ANCF Geometry -- 7.6 ANCF and B-Spline Surfaces -- B-Spline Surfaces -- ANCF Surfaces -- 7.7 Structural and Nonstructural Discontinuities -- B-Spline Model -- ANCF Model -- Problems -- Chapter 8 Plasticity Formulations -- 8.1 One-Dimensional Problem -- 8.2 Loading and Unloading Conditions -- 8.3 Solution of the Plasticity Equations -- Numerical Solution -- Plasticity Equations -- Trial Step -- The Return Mapping Algorithm -- 8.4 Generalization of the Plasticity Theory: Small Strains -- Associative Plasticity -- Numerical Solution of the Plasticity Equations -- Explicit Solution -- Implicit Solution -- 8.5 J2 Flow Theory with Isotropic/Kinematic Hardening -- Nonlinear Isotropic/Kinematic Hardening.

Return Mapping Algorithm for Nonlinear Isotropic/Kinematic Hardening -- Linear Kinematic/Isotropic Hardening -- 8.6 Nonlinear Formulation for Hyperelastic-Plastic Materials -- Multiplicative Decomposition -- Hyperelastic Potential -- Rate of Deformation Tensors -- Flow Rule and Hardening Law -- Numerical Solution -- Rate-Dependent Plasticity -- 8.7 Hyperelastic-Plastic J2 Flow Theory -- Problems -- References -- Index -- EULA.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.