Half Title -- Author Biography -- Title Page -- Copyright -- Dedication -- Contents -- List of Figures -- List of Tables -- Preface -- 1 General Problems in Solid Mechanics and Nonlinearity -- 1.1 Introduction -- 1.2 Small deformation solid mechanics problems -- 1.2.1 Strong form of equation: Indicial notation -- 1.2.1.1 Coordinates and displacements -- 1.2.1.2 Strain-displacement relations -- 1.2.1.3 Equilibrium equations: Balance of momentum -- 1.2.1.4 Boundary conditions -- 1.2.1.5 Initial conditions -- 1.2.1.6 Constitutive relations -- 1.2.2 Matrix notation -- 1.2.3 Two-dimensional problems -- 1.2.3.1 Plane stress and plane strain -- 1.2.3.2 Axisymmetric problems -- 1.3 Variational forms for nonlinear elasticity -- 1.4 Weak forms of governing equations -- 1.4.1 Weak form for equilibrium equation -- 1.5 Concluding remarks -- References -- 2 Galerkin Method of Approximation: Irreducible and Mixed Forms -- 2.1 Introduction -- 2.2 Finite element approximation: Galerkin method -- 2.2.1 Displacement approximation -- 2.2.2 Derivatives -- 2.2.3 Strain-displacement equations -- 2.2.4 Weak form -- 2.2.5 Irreducible displacement method -- 2.3 Numerical integration: Quadrature -- 2.3.1 Volume integrals -- 2.3.2 Surface integrals -- 2.4 Nonlinear transient and steady-state problems -- 2.4.1 Explicit Newmark method -- 2.4.2 Implicit Newmark method -- 2.4.3 Generalized midpoint implicit form -- 2.5 Boundary conditions: Nonlinear problems -- 2.5.1 Displacement condition -- 2.5.1.1 Nonlinear explicit problems -- 2.5.1.2 Nonlinear implicit problems -- 2.5.2 Traction condition -- 2.5.2.1 Pressure loading -- 2.5.3 Mixed displacement/traction condition -- 2.6 Mixed or irreducible forms -- 2.6.1 Deviatoric and mean stress and strain components -- 2.6.2 A three-field mixed method for general constitutive models -- 2.6.3 Local approximation of p and.

2.6.4 Continuous u-p approximation -- 2.7 Nonlinear quasi-harmonic field problems -- 2.8 Typical examples of transient nonlinear calculations -- 2.8.1 Transient heat conduction -- 2.8.2 Structural dynamics -- 2.8.3 Earthquake response of soil structures -- 2.9 Concluding remarks -- References -- 3 Solution of Nonlinear Algebraic Equations -- 3.1 Introduction -- 3.2 Iterative techniques -- 3.2.1 General remarks -- 3.2.2 Newton's method -- 3.2.3 Modified Newton's method -- 3.2.4 Incremental-secant or quasi-Newton methods -- 3.2.5 Line search procedures: Acceleration of convergence -- 3.2.6 ``Softening'' behavior and displacement control -- 3.2.7 Convergence criteria -- 3.3 General remarks: Incremental and rate methods -- References -- 4 Inelastic and Nonlinear Materials -- 4.1 Introduction -- 4.2 Tensor to matrix representation -- 4.3 Viscoelasticity: History dependence of deformation -- 4.3.1 Linear models for viscoelasticity -- 4.3.2 Isotropic models -- 4.3.2.1 Differential equation model -- 4.3.2.2 Integral equation model -- 4.3.2.3 Solution to integral equation with Prony series -- 4.3.3 Solution by analogies -- 4.4 Classical time-independent plasticity theory -- 4.4.1 Yield functions -- 4.4.2 Flow rule (normality principle) -- 4.4.3 Hardening/softening rules -- 4.4.3.1 Isotropic hardening -- 4.4.3.2 Kinematic hardening -- 4.4.4 Plastic stress-strain relations -- 4.5 Computation of stress increments -- 4.5.1 Explicit methods -- 4.5.2 Implicit methods: Return map algorithm -- 4.5.2.1 Return map algorithm -- 4.6 Isotropic plasticity models -- 4.6.1 Isotropic yield surfaces -- 4.6.2 J2 model with isotropic and kinematic hardening (Prandtl-Reuss equations) -- 4.6.2.1 Continuum rate form -- 4.6.2.2 Incremental return map form -- 4.6.3 Plane stress -- 4.7 Generalized plasticity -- 4.7.1 Nonassociative case: Frictional materials.

4.7.2 Associative case: J2 generalized plasticity -- 4.8 Some examples of plastic computation -- 4.8.1 Perforated plate: Plane stress solutions -- 4.8.2 Perforated plate: Plane strain solutions -- 4.8.3 Steel pressure vessel -- 4.9 Basic formulation of creep problems -- 4.9.1 Fully explicit solutions -- 4.9.1.1 ``Initial strain'' procedure: γ= 0 -- 4.9.1.2 Fully explicit process with modified stiffness: -- 4.10 Viscoplasticity: A generalization -- 4.10.1 General remarks -- 4.10.2 Implicit solution -- 4.10.3 Creep of metals -- 4.10.4 Soil mechanics applications -- 4.11 Some special problems of brittle materials -- 4.11.1 The no-tension material -- 4.11.1.1 An underground power station -- 4.11.1.2 Reinforced concrete -- 4.11.2 ``Laminar'' material and joint elements -- 4.12 Nonuniqueness and localization in elasto-plastic -- 4.13 Nonlinear quasi-harmonic field problems -- 4.14 Concluding remarks -- References -- 5 Geometrically Nonlinear Problems: Finite Deformation -- 5.1 Introduction -- 5.2 Governing equations -- 5.2.1 Kinematics and deformation -- 5.2.2 Stress and traction for reference and deformed states -- 5.2.2.1 Stress measures -- 5.2.2.2 Traction measures -- 5.2.3 Equilibrium equations -- 5.2.4 Boundary conditions -- 5.2.5 Initial conditions -- 5.2.6 Constitutive equations: Hyperelastic material -- 5.3 Variational description for finite deformation -- 5.3.1 Reference configuration formulation -- 5.3.1.1 Matrix form -- 5.3.1.2 Finite element approximation -- 5.3.1.3 Transient problems -- 5.3.2 First Piola-Kirchhoff formulation -- 5.3.3 Current configuration formulation -- 5.3.3.1 Finite element formulation -- 5.4 Two-dimensional forms -- 5.4.1 Plane strain -- 5.4.2 Plane stress -- 5.4.3 Axisymmetric with torsion -- 5.5 A three-field, mixed finite deformation formulation -- 5.5.1 Finite element equations: Matrix notation.

5.6 Forces dependent on deformation: Pressure loads -- 5.7 Concluding remarks -- References -- 6 Material Constitution for Finite Deformation -- 6.1 Introduction -- 6.2 Isotropic elasticity -- 6.2.1 Isotropic elasticity: Formulation in invariants -- 6.2.2 Isotropic elasticity: Formulation in modified invariants -- 6.2.3 Isotropic elasticity: Formulation in principal stretches -- 6.2.4 Plane stress applications -- 6.3 Isotropic viscoelasticity -- 6.4 Plasticity models -- 6.5 Incremental formulations -- 6.6 Rate constitutive models -- 6.7 Numerical examples -- 6.7.1 Necking of circular bar -- 6.7.2 Adaptive refinement and localization (slip-line) capture -- 6.7.2.1 Adaptive refinement based on energy norm error estimates -- 6.7.2.2 Alternate refinement using error indicators: Discontinuity capture -- 6.8 Concluding remarks -- References -- 7 Material Constitution Using Representative Volume Elements -- 7.1 Introduction -- 7.2 Coupling between scales -- 7.2.1 RVE with specified boundary displacements -- 7.2.1.1 Stress computation -- 7.2.1.2 Tangent computation -- 7.2.2 Kirchhoff and Cauchy stress forms -- 7.2.2.1 Stress computation -- 7.2.2.2 Tangent computation -- 7.2.3 Periodic boundary conditions -- 7.2.3.1 Stress computation -- 7.2.3.2 Tangent computation -- 7.2.4 Small strains -- 7.3 Quasi-harmonic problems -- 7.4 Numerical examples -- 7.4.1 Linear elastic properties -- 7.4.2 Uniformly loaded plate: Cylindrical bending -- 7.4.3 Moment-curvature: Elastic-plastic response -- 7.5 Concluding remarks -- References -- 8 Treatment of Constraints: Contact and Tied Interfaces -- 8.1 Introduction -- 8.2 Node-node contact: Hertzian contact -- 8.2.1 Geometric modeling -- 8.2.2 Contact models -- 8.2.2.1 Lagrange multiplier form -- 8.2.2.2 Perturbed Lagrangian -- 8.2.2.3 Penalty function form -- 8.2.2.4 Augmented Lagrangian form -- 8.3 Tied interfaces.

8.3.1 Surface-surface tied interface -- 8.4 Node-surface contact -- 8.4.1 Geometric modeling -- 8.4.1.1 Normal and tangent vector definitions -- 8.4.2 Contact modeling: Frictionless case -- 8.4.2.1 Contact residual -- 8.4.2.2 Contact tangent -- 8.4.3 Contact modeling: Frictional case -- 8.4.3.1 Residual and tangent -- 8.5 Surface-surface contact -- 8.5.1 Frictionless case -- 8.6 Numerical examples -- 8.6.1 Contact between two disks -- 8.6.2 Contact between a disk and a block -- 8.6.3 Frictional sliding of a flexible disk on a sloping block -- 8.6.4 Upsetting of a cylindrical billet -- 8.7 Concluding remarks -- References -- 9 Pseudo-Rigid and Rigid-Flexible Bodies -- 9.1 Introduction -- 9.2 Pseudo-rigid motions -- 9.3 Rigid motions -- 9.3.1 Equations of motion for a rigid body -- 9.3.2 Construction from a finite element model -- 9.3.3 Transient solutions -- 9.4 Connecting a rigid body to a flexible body -- 9.4.1 Lagrange multiplier constraints -- 9.5 Multibody coupling by joints -- 9.5.1 Translation constraints -- 9.5.2 Rotation constraints -- 9.5.3 Library of joints -- 9.6 Numerical examples -- 9.6.1 Rotating disk -- 9.6.2 Beam with attached mass -- 9.6.3 Biofidelic rear impact dummy -- 9.6.4 Sorting of randomly sized particles -- 9.7 Concluding remarks -- References -- 10 Background Mathematics and Linear Shell Theory -- 10.1 Introduction -- 10.2 Basic notation and differential calculus -- 10.2.1 Calculus in several variables -- 10.2.1.1 Distance in Reals n -- 10.2.1.2 Open and closed sets -- 10.2.2 Differential calculus: Frechet derivative -- 10.2.2.1 Real-valued functions -- 10.2.2.2 The Frechet derivative: General case -- 10.2.2.3 Properties of the Frechet derivative -- 10.2.3 Tangent spaces and tangent maps -- 10.2.4 Parameterizations, curvilinear coordinates, and the Jacobian transformation.

10.2.4.1 The reference configuration of a continuum body in Reals 3.