Objective stress rates, cyclic deformation paths, and residual stress accumulation

As one of the basic constitutive ingredients in finite elastoplasticity, Truesdell's hypoelastic equation of grade zero has been widely used to present an objective Eulerian rate formulation of the elastic behaviour. Although it has been known that a stress rate employed in this basic rate equation would actually result in path-dependence behaviour inconsistent with the notion of hyperelasticity, sometimes it has been believed that the definition of the objective stress rate in this basic rate equation might be insubstantial, on account of the fact that the "elastic deformation" may be small for metals and alloys, etc. Towards clarifying this issue, here we present a unified, direct study on the path-dependence properties of Truesdell equation for all possible deformation paths. Towards this goal, explicit closed-form solutions for the stress response for any given deformation path are derived for classical stress rates including Zaremba-Jaumann rate, Green-Naghdi rate, Oldroyd rate, Cotter-Rivlin rate, and Truesdell rate, etc. From these solutions in terms of explicit path integrations, it may become straightforward to understand how the stress response relies on deformation paths. In particular, cyclic deformation paths composed of any number of deformation cycles are taken into consideration. Explicit exact expressions in simple form are established between the residual stress and the cycle number. These results clearly suggest that the monotone accumulation phenomena of residual stresses with increasing cycle number, found in previous numerical examples, may actually be general facts for all possible cyclic deformation paths. These phenomena imply that, even for a cyclic deformation path undergoing small deformation, the final residual stress would reach such a noticeable level that in any event it could not be regarded to be negligible. Examples are given in the cases of the simple shearing and smooth cyclic plane deformation paths. (c) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.