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| Foreword | |
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| Preface | |
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| Acknowledgements | |
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| Introduction | |
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| Motivation and objectives | |
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| Outline of the main topics | |
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| Further studies recommendations | |
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| Summary of main notations | |
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| Boundary value problem in linear and nonlinear elasticity | |
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| Boundary value problem in elasticity with small displacement gradients | |
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| Domain and boundary conditions | |
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| Strong form of boundary value problem in 1D elasticity | |
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| Weak from of boundary value problem in 1D elasticity and the principle of virtual work | |
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| Variational formulation of boundary value problem in 1D elasticity and principle of minimum potential energy | |
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| Finite element solution of boundary value problems in 1D linear and nonlinear elasticity | |
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| Qualitative methods of functional analysis for solution existence and uniqueness | |
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| Approximate solution construction by Galerkin, Ritz and finite element methods | |
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| Approximation error and convergence of finite element method | |
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| Solving a system of linear algebraic equations by Gauss elimination method | |
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| Solving a system of nonlinear algebraic equations by incremental analysis | |
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| Solving a system of nonlinear algebraic equations by Newton's iterative method | |
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| Implementation of finite element method in 1D boundary value problems | |
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| Local or elementary description | |
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| Consistence of finite element approximation | |
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| Equivalent nodal external load vector | |
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| Higher order finite elements | |
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| Role of numerical integration | |
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| Finite element assembly procedure | |
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| Boundary value problems in 2D and 3D elasticity | |
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| Tensor, index and matrix notations | |
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| Strong from of a boundary value problem in 2D and 3D elasticity | |
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| Weak form of boundary value problem in 2D and 3D elasticity | |
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| Detailed aspects of the finite element method | |
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| Isoparametric finite elements | |
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| Order of numerical integration | |
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| The patch test | |
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| Hu-Washizu (mixed) variational principle and method of incompatible modes | |
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| Hu-Washizu (mixed) variational principle and assumed strain method for quasi-incompressible behavior | |
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| Inelastic behavior at small strains | |
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| Boundary value problem in thermomechanics | |
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| Rigid conductor and heat equation | |
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| Numerical solution by time-integration scheme for heat transfer problem | |
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| Thermomechanical coupling in elasticity | |
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| Thermodynamics potentials in elasticity | |
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| Thermodynamics of inelastic behavior: constitutive models with internal variables | |
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| Internal variables in viscoelasticity | |
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| Internal variables in viscoelasticity | |
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| 1D models of perfect plasticity and plasticity with hardening | |
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| 1D perfect plasticity | |
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| 1D plasticity with isotropic hardening | |
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| Boundary value problem for 1D plasticity | |
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| 3D plasticity | |
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| Standard format of 3D plasticity model: Prandtl-Reuss equations | |
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| J2 plasticity model with von Mises plasticity criterion | |
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| Implicit backward Euler scheme and operator split for von Mises plasticity | |
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| Finite element numerical implementation in 3D plasticity | |
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| Refined models of 3D plasticity | |
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| Nonlinear isotropic hardening | |
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| Kinematic hardening | |
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| Plasticity model dependent on rate of deformation or viscoplasticity | |
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| Multi-surface plasticity criterion | |
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| Plasticity model with nonlinear elastic response | |
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| Damage models | |
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| 1D damage model | |
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| 3D damage model | |
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| Refinements of 3D damage model | |
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| Isotropic damage model of Kachanov | |
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| Numerical examples: damage model combining isotropic and multisurface criteria | |
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| Coupled plasticity-damage model | |
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| Theoretical formulation of 3D coupled model | |
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| Time integration of stress for coupled plasticity-damage model | |
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| Direct stress interpolation for coupled plasticity-damage model | |
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| Large displacements and deformations | |
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| Kinematics of large displacements | |
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| Motion in large displacements | |
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| Deformation gradient | |
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| Large deformation measures | |
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| Equilibrium equations in large displacements | |
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| Strong form of equilibrium equations | |
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| Weak form of equilibrium equations | |
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| Linear elastic behavior in large displacements: Saint-Venant-Kirchhoff material model | |
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| Weak form of Saint-Venant-Kirchhoff 3D elasticity model and its consistent linearization | |
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| Numerical implementation of finite element method in large displacements elasticity | |
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| 1D boundary value problem: elastic bar in large displacements | |
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| 2D plane elastic membrane in large displacements | |
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| Spatial description of elasticity in large displacements | |
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| Finite element approximation of spatial description of elasticity in large displacements | |
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| Mixed variational formulation in large displacements and discrete approximations | |
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| Mixed Hu-Washizu variational principle in large displacements and method of incompatible modes | |
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| Mixed Hu-Washizu variational principle in large displacements and assumed strain methods for quasi-incompressible behavior | |
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| Constitutive models for large strains | |
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| Invariance restrictions on elastic response | |
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| Constitutive laws for large deformations in terms of principal stretches | |
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| Plasticity and viscoplasticity for large deformations | |
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| Multiplicative decomposition of deformation gradient | |
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| Perfect plasticity for large deformations | |
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| Isotropic and kinematic hardening in large deformation plasticity | |
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| Spatial description of large deformation plasticity | |
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| Numerical implementation of large deformation plasticity | |
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| Changing boundary conditions: contact problems | |
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| Unilateral 1D contact problem | |
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| Strong form of 1D elasticity in presence of unilateral contact constraint | |
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| Weak form of unilateral 1D contact problem and its finite element solution | |
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| Contact problems in 2D and 3D | |
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| Contact between two deformable bodies in 2D case | |
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| Mortar element method for contact | |
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| Numerical examples of contact problems | |
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| Refinement of contact model | |
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| Dynamics and time-integration schemes | |
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| Initial boundary value problem | |
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| Strong form of elastodynamics | |
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| Weak form of equations of motion | |
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| Finite element approximation for mass matrix | |
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| Time-integration schemes | |
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| Central difference (explicit) scheme | |
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| Trapezoidal rule or average acceleration (implicit) scheme | |
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| Mid-point (implicit) scheme and its modifications for energy conservation and energy dissipation | |
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| Mid-point (implicit) scheme for finite deformation plasticity | |
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| Contact problem and time-integration schemes | |
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| Mid-point (implicit) scheme for contact problem in dynamics | |
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| Central difference (explicit) scheme and impact problem | |
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| Thermodynamics and solution methods for coupled problems | |
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| Thermodynamics of reversible processes | |
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| Thermodynamical coupling in 1D elasticity | |
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| Thermodynamics coupling in 3D elasticity and constitutive relations | |
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| Initial-boundary value problem in thermoelasticity and operator split solution method | |
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| Weak form of initial-boundary value problem in 3D elasticity and its discrete approximation | |
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| Operator split solution method for 3D thermoelasticity | |
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| Numerical examples in thermoelasticity | |
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| Thermodynamics of irreversible processes | |
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| Thermodynamics coupling for 1D plasticity | |
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| Thermodynamics coupling in 3D plasticity | |
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| Operator split solution method for 3D thermoplasticity | |
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| Numerical example: thermodynamics coupling in 3D plasticity | |
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| Thermomechanical coupling in contact | |
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| Geometric and material instabilities | |
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| Geometric instabilities | |
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| Buckling, nonlinear instability and detection criteria | |
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| Solution methods for boundary value problem in presence of instabilities | |
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| Material instabilities | |
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| Detection criteria for material instabilities | |
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| Illustration of finite element mesh lack of objectivity for localization problems | |
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| Localization limiters | |
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| List of localization limiters | |
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| Localization limiter based on mesh-dependent softening modulus - 1D case | |
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| Localization limiter based on viscoplastic regularization - 1D case | |
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| Localization limiter based on displacement or deformation discontinuity - 1D case | |
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| Localization limiter in plasticity for massive structure | |
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| Theoretical formulation of limiter with displacement discontinuity - 2D/3D case | |
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| Numerical implementation within framework of incompatible mode method | |
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| Numerical examples for localization problems | |
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| Localization problem in large strain plasticity | |
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| Multi-scale modelling of inelastic behavior | |
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| Scale coupling for inelastic behavior in quasi-static problems | |
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| Weak coupling: nonlinear homogenization | |
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| Strong coupling micro-macro | |
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| Microstructure representation | |
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| Microstructure representation by structured mesh with isoparametric finite elements | |
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| Microstructure representation by structured mesh with incompatible mode elements | |
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| Microstructure representation with uncertain geometry and probabilistic interpretation of size effect for dominant failure mechanism | |
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| Conclusions and remarks on current research works | |
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| References | |
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| Index | |