Thermodynamics of Solids

Abstract

Material properties are represented by constitutive equations, which complete the set of balance equations for momentum and energy. According to experimental observations, material behavior may be idealized to be rate independent or rate dependent. Moreover, rate dependent or rate independent hysteresis effects may occur: This implies four different categories of constitutive models, namely the theories of elasticity, viscoelasticity, plasticity and viscoplasticity. In case of elasticity the stress depends only on the present strain. For inelastic materials the stress is related to the present strain and its past history. This relationship can be represented explicitly as a functional of the strain process. An implicit representation of a functional relation is obtained from the formulation of a system of ordinary differential equations for a set of additional internal variables. These evolution equations specify the rate of change of internal variables depending on their present values and the strain (or stress) input. Different constitutive models imply different mathematical characteristics of the evolution equations. These concern the existence of equilibrium solutions and their stability properties. Some principal ideas for a modelling of material properties are outlined for the purely mechanical case and then generalized to thermodynamics. In the thermodynamic context the existence of rate independent hysteresis effects, encountered in plasticity and viscoplasticity, leads to a basic problem: It is impossible to approximate reversible processes through slow deformation and temperature histories. In the theory of viscoplasticity the evolution equations have equilibrium solutions, which depend on the process history and represent equilibrium hysteresis behavior.