The theory of elastic-plastic deformation at finite strain with induced anisotropy modeled as combined isotropic-kinematic hardening

Abstract

A T heory of elastic-plastic deformation with strain induced anisotropy based on finite-deformation-valid continuum mechanics is presented. On the foundation of nonlinear kinematics which provides strict uncoupling of elastic and plastic deformation rate terms according to their physical origins, it introduces a basis for the modified plastic rate of deformation D̂ p suggested by G.J. Creus, A.G. Groehs and E.T. Onat in a report entitled “Constitutive Equations for Finite Deformations of Elastic-Plastic Solids,” 1984, in which this variable was suggested in order to give an elegant mathematical structure to the theory. D̂ p is shown to express the resultant rate of deformation in the current configuration of the elastically-plastically deforming material which is envisaged to be generated by the pure plastic flow and the anisotropy-caused spin, both considered to be occurring in the unstressed state. From this basis an elastic-plastic theory is developed in the case where the strain-induced anisotropy takes the form of combined isotropic-kinematic hardening, although the concepts involved also apply to more general anisotropic characteristics. A general evolution equation is adopted for the back stress α, the kinematic-hardening shift of the yield surface, its rate of growth being expressed as a general form-invariant function of α and D̂ p , including a general term expressing the influence of the spin of α because it is embedded in the deforming material. By providing an expression for the total strain rate as the sum of the strain rate D̂ p and an elastic term, linear in the Jaumann derivative of Kirchhoff stress, it is shown that ( D̂ p dt) is the residual strain increment following a loading/unloading cycle imposed by a stress increment. By considering materials which obey the normality rule it is also shown that the instantaneous elastic-plastic moduli have the symmetries necessary for generating a rate potential function and hence can be incorporated into Hill's variational principle valid for solving problems involving finite deformation and convenient for finite-element exploitation.