Nonlinear Solid Mechanics : Theoretical Formulations and Finite Element Solution Methods

1 Introduction; 1.1 Motivation and objectives; 1.2 Outline of the main topics; 1.3 Further studies recommendations; 1.4 Summary of main notations; 2 Boundary value problem in linear and nonlinear elasticity; 2.1 Boundary value problem in elasticity with small displacement gradients; 2.1.1 Domain and boundary conditions; 2.1.2 Strong form of boundary value problem in 1D elasticity; 2.1.3 Weak form of boundary value problem in 1D elasticity and the principle of virtual work; 2.1.4 Variational formulation of boundary value problem in 1D elasticity and principle of minimum potential energy; 2.2 Finite element solution of boundary value problems in 1D linear and nonlinear elasticity; 2.2.1 Qualitative methods of functional analysis for solution existence and uniqueness; 2.2.2 Approximate solution construction by Galerkin, Ritz and finite element methods; 2.2.3 Approximation error and convergence of finite element method; 2.2.4 Solving a system of linear algebraic equations by Gauss elimination method; 2.2.5 Solving a system of nonlinear algebraic equations by incremental analysis; 2.2.6 Solving a system of nonlinear algebraic equations by Newtons iterative method; 2.3 Implementation of finite element method in ID boundary value problems; 2.3.1 Local or elementary description; 2.3.2 Consistence of finite element approximation; 2.3.3 Equivalent nodal external load vector; 2.3.4 Higher order finite elements; 2.3.5 Role of numerical integration; 2.3.6 Finite element assembly procedure; 2.4 Boundary value problems in 2D and 3D elasticity; 2.4.1 Tensor, index and matrix notations; 2.4.2 Strong form of a boundary value problem in 2D and 3D elasticity; 2.4.3 Weak form of boundary value problem in 2D and 3D elasticity; 2.5 Detailed aspects of the finite element method; 2.5.1 Isoparametric finite elements; 2.5.2 Order of numerical integration; 2.5.3 The patch test; 2.5.4 Hu-Washizu (mixed) variational principle and method of incompatible modes; 2.5.5 Hu-Washizu (mixed)variational principle and assumed strain method for quasi-incompressible behavior; 3 Inelastic behavior at small strains; 3.1 Boundary value problem in thermomechanics; 3.1.1 Rigid conductor and heat equation; 3.1.2 Numerical solution by time-integration scheme for heat transfer problem; 3.1.3 Thermo-mechanical coupling in elasticity; 3.1.4 Thermodynamics potentials in elasticity; 3.1.5 Thermodynamics of inelastic behavior: constitutive models with internal variables; 3.1.6 Internal variables in viscoelasticity; 3.1.7 Internal variables in viscoplasticity; 3.2 1D models of perfect plasticity and plasticity with hardening; 3.2.1 1D perfect plasticity; 3.2.2 1D plasticity with isotropic hardening; 3.2.3 Boundary value problem for 1D plasticity; 3.3 3D plasticity; 3.3.1 Standard format of 3D plasticity model: Prandtl-Reuss equations; 3.3.2 J2 plasticity model with von Mises plasticity criterion; 3.3.3 Implicit backward Euler scheme and operator split for von Mises plasticity; 3.3.4 Finite element numerical implementation in 3D plasticity; 3.4 Refined models of 3D plasticity; 3.4.1 Nonlinear isotropic hardening; 3.4.2 Kinematic hardening; 3.4.3 Plasticity model dependent on rate of deformation or viscoplasticity; 3.4.4 Multi-surface plasticity criterion; 3.4.5 Plasticity model with nonlinear elastic response; 3.5 Damage models; 3.5.1 1D damage model; 3.5.2 3D damage model; 3.5.3 Refinements of 3D damage model; 3.5.4 Isotropic damage model of Kachanov; 3.5.5 Numerical examples: damage model combining isotropic and multisurface criteria; 3.6 Coupled plasticity-damage model; 3.6.1 Theoretical formulation of 3D coupled model; 3.6.2 Time integration of stress for coupled plasticitydamagemodel; 3.6.3 Direct stress interpolation for coupled plasticitydamagemodel; 4 Large displacements and deformations; 4.1 Kinematics of large displacements; 4.1.1 Motion in large displacements; 4.1.2 Deformation gradient; 4.1.3 Large deformation measures; 4.2 Equilibrium equations