Nonlinear Solid 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 in large displacements; 4.2.1 Strong form of equilibrium equations; 4.2.2 Weak form of equilibrium equations; 4.3 Linear elastic behavior in large displacements: Saint-Venant- Kirchhoff material model; 4.3.1 Weak form of Saint-Venant-Kirchhoff 3D elasticity model and its consistent linearization; 4.4 Numerical implementation of finite element method in large displacements elasticity; 4.4.1 1D boundary value problem: elastic bar in large displacements; 4.4.2 2D plane elastic membrane in large displacements; 4.5 Spatial description of elasticity in large displacements; 4.5.1 Finite element approximation of spatial description of elasticity in large displacements; 4.6 Mixed variational formulation in large displacements and discrete approximations; 4.6.1 Mixed Hu-Washizu variational principle in large displacements and method of incompatible modes; 4.6.2 Mixed Hu-Washizu variational principle in large displacements and assurned strain methods for quasi-incompressible behavior; 4.7 Constitutive models for large strains; 4.7.1 Invariance restrictioris on elastic response; 4.7.2 Constitutive laws for large deformations in terms of principal stretches; 4.8 Plasticity and viscoplasticity for large deformations; 4.8.1 Multiplicative decomposition of deformation gradient; 4.8.2 Perfect plasticity for large deformations; 4.8.3 Isotropic and kinematic hardening in large deformation plasticity; 4.8.4 Spatial description of large deformation plasticity; 4.8.5 Numerical implementation of large deformation plasticity; 5 Changing boundary conditions: contact problems; 5.1 Unilateral 1D contact problem; 5.1.1 Strong form of ID elasticity in presence of unilateral contact constraint; 5.1.2 Wcak form of unilatcral 1D contact problcm and its finite element solution; 5.2 Contact problems in 2D and 3D; 5.2.1 Contact between two deformable bodies in 2D case; 5.2.2 Mortar element method for contact; 5.2.3 Numerical examples of contact problems; 5.2.4 Refinement of contact model; 6 Dynamics and time-integration schemes; 6.1 Initial boundary value problem; 6.1.1 Strong form of elastodynamics; 6.1.2 Weak form of equations of motion; 6.1.3 Finite element approximation for mass matrix; 6.2 Time-integration schemes; 6.2.1 Central difference (explicit) scheme; 6.2.2 Trapezoidal rule or average acceleration (implicit) scheme; 6.2.3 Mid-point (implicit) scheme and its modifications for energy conservation and energy dissipation; 6.3 Mid-point (implicit) scheme for finite deformation plasticity; 6.4 Contact problem and time-int.egration schemes; 6.4.1 Mid-point (implicit) scheme for contact problem in dynamics; 6.4.2 Central difference (explicit) scheme arid impact problem; 7 Thermodynamics and solution methods for coupled problems; 7.1 Thermodynamics of reversible processes; 7.1.1 Thermodynamical coupling in ID elasticity; 7.1.2 Thermodynamics coupling in 3D elasticity and constitutive relations; 7.2 Initial-boundary value problem in thermoelasticity and operator split solution method; 7.2.1 Weak form of initial-boundary value problem in 3D elasticity and its discrete approximation; 7.2.2 Operator split solution method for 3D thermoelasticity; 7.2.3 Numerical examples in thermoelasticity; 7.3 Thermodynamics of irreversible processus; 7.3.1 Thermodynamics coupling for 1D plasticity; 7.3.2 Thermodynamics coupling in 3D plasticity; 7.3.3 Operator split solution method for 3D thermoplasticity; 7.3.4 Numerical example: thermodynamics coupling in 3D plasticity; 7.4 Thermomechanical coupling in contact; 8 Geometric and material instabilities; 8.1 Geometric instabilities; 8.1.1 Buckling, nonlinear instability and detection criteria; 8.1.2 Solution methods for boundary value problem in presence of instabilities; 8.2 Material instabilities; 8.2.1 Detection criteria for material instabilities; 8.2.2 Illustration of finite element mesh lack of objectivity for localization problems; 8.3 Localization limiters; 8.3.1 List of localization limiters; 8.3.2 Localization limiter based on mesh-dependent softening rriodulus - 1D case; 8.3.3 Localization limiter based on viscoplastic regularization - 1D case; 8.3.4 Localization limiter based on displacement or deformation discontinuity - 1D case; 8.4 Localization limiter in plasticity for massive structure; 8.4.1 Theoretical formulation of limiter with displacement discontinuity - 2D/3D case; 8.4.2 Numerical implementation within framework of incompatible mode method; 8.4.3 Numerical examples for localization problems; 8.5 Localization problem in large strain plasticity; 9 Multi-scale modelling of inelastic behavior; 9.1 Scale coupling for inelastic behavior in quasi-static problems; 9.1.1 Weak coupling: nonlinear homogenization; 9.1.2 Strong coupling micro-macro; 9.2 Microstructure representation; 9.2.1 Microstructure representation by structured mesh with isoparametric finite elements; 9.2.2 Microstructure representation by structured mesh with incompatible mode elements; 9.2.3 Microstructure representation with uncertain geometry and probabilistic interpretation of size effect for dominant failure mechanism; 9.3 Conclusions and remarks on current research works; References; Index.