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