A self-consistent Eulerian rate type model for finite deformation elastoplasticity with isotropic damage

Continuum models for coupled behaviour of elastoplasticity and isotropic damage at ®nite deformation are usually formulated by ®rst postulating the additive decomposition of the stretching tensor D into the elastic and the plastic part and then relating each part to an objective rate of the e€ective stress, etc. It is pointed out that, according to the existing models with several widely used objective stress rates, none of the rate equations intended for characterizing the damaged elastic response is exactly integrable to really deliver a damaged elastic relation between the e€ective stress and an elastic strain measure. The existing models are thus self-inconsistent in the sense of formulating the damaged elastic response. By consistently combining additive and multiplicative decomposition of the stretching D and the deformation gradient F and adopting the logarithmic stress rate, in this article, we propose a general Eulerian rate type model for ®nite deformation elastoplasticity coupled with isotropic damage. The new model is shown to be self-consistent in the sense that the incorporated rate equation for the damaged elastic response is exactly integrable to yield a damaged elastic relation between the e€ective Kirchho€ stress and the elastic logarithmic strain. The rate form of the new model in a rotating frame in which the foregoing logarithmic rate is de®ned, is derived and from it an integral form is obtained. The former is found to have the same structure as the counterpart of the small deformation theory and may be appropriate for numerical integration. The latter indicates, in a clear and direct manner, the e€ect of ®nite rotation and deformation history on the current stress and the hardening and damage behaviours. Further, it is pointed out that in the foregoing self-consistency sense of formulating the damaged elastic response, the suggested model is unique among all objective Eulerian rate type models of its kind with in®nitely many objective stress rates to be chosen. In particular, it is indicated that, within the context of the proposed theory, a natural combination of the two widely used decompositions concerning D and F can consistently and uniquely determine the elastic and the plastic parts in the two decompositions as well as all their related kinematical quantities, without recourse to any ad hoc assumption concerning a special form of the elastic part Fe in the decomposition F ˆ FeFp or a related relaxed intermediate con®guration. As an application, the proposed general model is applied to derive a self-consistent Eulerian rate type model for void growth and nucleation in metals experiencing ®nite elastic±plastic deformation by incorporating a modi®ed Gursonʹs yield function and an associated ¯ow rule, etc. Two issues involved in previous relevant literature are detected and raised for consideration. As a test problem, the ®nite simple shear response of the just-mentioned model is studied by means of numerical integration.