Thermomechanics of Elastoplastic Deformation

Contents:

Introduction

Chapter 1: Kinematic and dynamic equations of motion of the deformed solid • The theory of deformation • The law of motion of the continuum • Kinematics of the infinitesimal particle of the material space • Description of the deformation of a material particle • Speed characteristics of continuum motion • Differentiation of various types of tensors with respect to time • Non-holonomic strain measures • Euler's approach to the description of the kinematic characteristics • The theory of stress • Interactions between particles of continuum, the stress tensor • Stress tensors used in the non-linear theory • Stress distribution along the edges of the cube • The principal axes and principal values of the symmetric stress tensor • The system of invariants associated with the expansion in octahedral basis • The system of invariants associated with the areas of maximum tangential stresses • Dynamics of motion of continuous medium • Equations of motion of the continuous medium in the local form • Equations of motion in variational forms • The continuity equation • The theorem of the change of kinetic energy • Description of the elementary work of the internal forces and stress power

Chapter 2: Fundamentals of thermomechanics of equilibrium processes? Equilibrium thermomechanical processes • Conditions of thermomechanical equilibrium of the medium • Principle of thermomechanical definability • The principle of material equality of adiabatic processes, the reaction of the medium to uniform motion • Differentiation of the true stress tensor in time • The general postulate of isotropy for equilibrium processes • The laws of thermomechanics • The law of variation of internal energy • The law of entropy change in equilibrium processes • Thermomechanical potentials, restrictions on dissipation production • Conditions of stability of equilibrium states • Description of the properties of anisotropic materials • Symmetry groups of anisotropic materials • Invariant tensor bases and systems of algebraic invariants

Chapter 3: Thermomechanics of reversible processes, partial postulate of isotropy for anisotropic materials? Thermomechanics of reversible deformation. Finite strains in anisotropic media • Construction of thermomechanical models of reversible processes • Tensor-linear relationship between stress and finite strains • Stresses in a transversely isotropic material • Hooke's law and elastic eigenstates • Properties of linearly elastic materials. The canonical representation of elasticity tensors in the six-dimensional space • The concept of elastic eigenstates of materials • Elastic eigenstates of the isotropic material • Elastic eigenstates of anisotropic materials • Determination of the type of initial elastic anisotropy of the material • Partial Il?yushin postulate and its generalization • Formulation of the partial Il?yushin isotropy postulate • Formulation of the partial postulate in the case of finite strains of initially isotropic materials • The conditions of applicability of the partial postulate of isotropy at finite strains • Generalization of the partial postulate for initially anisotropic media • Nonlinear relations of thermoelasticity and the partial postulate of isotropy • Model of anisotropic thermoelasticity in the framework of the partial postulate of isotropy • Quasi-linear relations for isotropic materials • Defining relations that take into account the possibility of deviation of the properties of the material from the partial isotropy postulate • Methods of processing experiments based on the finite deformation of a solid cylinder • Description of the stress-strain state of the cylinders when loaded by axial force and torque • The asymptotic solution of the problem of torsion of a solid cylinder • The program of experiments to determine the constants in defining relations (10.55) • Investigation of nonlinear effects in torsion of a cylinder

Chapter 4: Thermomechanics of irreversible equilibrium deformation • Elastoplastic deformation of initially isotropic materials • Basic relations that determine the behaviour of the body in equilibrium irreversible processes • Terms of reversible and irreversible deformation, reversibility surface • Equilibrium deformation of the initially isotropic material • Thermomechanics of equilibrium processes in the strain space • Variant of the thermomechanical model of equilibrium irreversible process • Description of irreversible deformation in the stress space • Specification of the form of defining relations • Analysis of deformation anisotropy • Residual stresses in a thick-wall cylinder • Irreversible equilibrium processes in anisotropic materials • Construction of thermomechanical basis in anisotropic materials • The variant of the flow theory • Variant of the deformation theory of plasticity • Changes of the properties of anisotropic materials under plastic deformation • Description of deformation anisotropy • Changing the orientation of the principal axes of anisotropy in finite homogeneous deformation • Changing the orientation of the axes of anisotropy of biaxial deformation • Changing the orientation of the axes of anisotropy in simple shear

Chapter 5: Variational formulation of thermomechanical problems, non-uniqueness and stability of finite deformation processes? Statement of thermomechanical problems in the reference configuration • Variational forms of the equations of motion and heat conductivity • Formulation of the coupled thermoelasticity problem in the reference configuration • Formulation of thermomechanical problem of finite deformation of anisotropic elastoplastic solids • Stability in finite deformation • Stability and uniqueness of finite deformation processes • Construction of defining relations in the neighbourhood of the inflection point of the deformation path • Uniqueness and stability in the homogeneous subcritical state • The non-uniqueness and instability with respect to perturbations in a given direction
Appendix A. Six-dimensional vector images of symmetric tensors
Appendix B. Six-dimensional images of the fourth-rank tensor
Appendix C. Isotropic tensors of the second and fourth rank
Appendix D. Generating elements of symmetry groups in the six-dimensional space
Literature
Index