On Formal Structure of Constitutive Equations for Materials Exhibiting Shape Memory Effects

Abstract

A derivation of constitutive equations in a general three-dimensional setting is described, based on an additive decomposition of the rate of deformation tensor. The rate of deformation tensor is assumed to consist of an elastic part, a thermoelastic part, a plastic part, a part due to shape memory transformation, and a part due to phase transformation. The thermoelastic part due to thermoelastic coupling accounts for the influence of temperature near phase transformation, while the plastic part is taken in the form of classical J 2 flow theory of plasticity with combined isotropic and kinematic hardening, where the back stress represents a tensor of orientational microstresses. It is assumed that the phase transformation part depends on the first and the second invariant of the tensor of crystallographic distortion, on the deviatoric part of the stress tensor, and on a special evolution parameter describing the rate of forming of a new phase. The elastic part of the rate of deformation tensor is connected with the objective rate of Cauchy stress through the tensor of elastic compliance. As a result, a general form of derived constitutive equations exhibits a similar structure as constitutive relations in finite deformation plasticity.