Nonlinear Solid Mechanics

This book offers a recipe for constructing the numerical models for representing the complex nonlinear behavior of structures and their components, represented as deformable solid bodies. Its appeal extends to those interested in linear problems of mechanics.

This book offers a recipe for constructing the numerical models for representing the complex nonlinear behavior of structures and their components, represented as deformable solid bodies. Its appeal extends to those interested in linear problems of mechanics.

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 Newton's 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