A weakened form of Ilyushin's postulate and the structure of self-consistent Eulerian finite elastoplasticity

Abstract

From a general standpoint in terms of internal variables, we formulate a general theory of self-consistent Eulerian finite elastoplasticity based on the additive decomposition of the Eulerian strain rate, i.e., D= D e+ D p, as well as two consistency criteria. In this theory, the elastic behaviour is characterized by an exactly integrable elastic rate equation for D e with a general form of complementary elastic potential. It is assumed that the yield function depends in a general manner on the Kirchhoff stress and the internal variables. Moreover, the plastic rate equation for D p and the evolution equation for each internal variable are allowed to assume general forms relying on the just-mentioned variables and the stress rate. It is indicated that two consistency criteria, i.e., the self-consistency for the elastic rate equation and Prager's yielding stationarity, lead to the unique choice of objective rates, i.e., the logarithmic rate. The structure of the above theory is further studied and examined by virtue of a weakened form of Ilyushin's postulate. In a spinning frame defining the logarithmic rate, we introduce the notion of standard elastoplastic strain cycle, which starts at a point not on but inside a yield surface and incorporates only one infinitesimal plastic subpath. We show that this type of strain cycle is always possible. Then, by ruling out strain cycles starting at points on yield surfaces we propose a weakened form of Ilyushin's postulate, which says that the changing rate of the stress work done along every standard strain cycle should be non-negative, whenever the incorporated plastic subpath tends to vanish. By virtue of simple, rigorous procedures, we demonstrate that this weakened form of Ilyushin's postulate is adequate to ensure direct results concerning the normality rule and the convexity of the yield surface in the context of the foregoing Eulerian finite elastoplasticity theory. Specifically, with an exactly integrable elastic rate equation defining D e, we prove that, in the space of the Kirchhoff stresses, the difference ( D− D e) is just the gradient of the yield function multiplied by a plastic multiplier, and thus bears the very kinematical and physical feature of plastic strain rate. Furthermore, we prove that, in the space of the Kirchhoff stresses, the elastic domain bounded by each yield surface should be convex. The main results are derived in a self-contained manner within the context of an Eulerian theory of finite elastoplasticity, without involving issues concerning how to define intermediate stress-free states and plastic strains, etc.