On postulates of plasticity in classical elastoplasticity

Abstract

It is commonly known that Drucker’s postulate of plasticity in stress space is applicable to hardening materials only, but not to softening materials, and, on the other hand, that Ilyushin’s postulate of plasticity in strain space is applicable to both softening and hardening materials. Accordingly, it is usually thought that the latter would be less restrictive and hence more general than the former. In this work, we introduce the notions of standard elastoplastic stress and strain cycles, each starting at a point inside the yield surface and incorporating only one infinitesimal plastic strain increment. We show that these standard cycles always exist for a yield function depending continuously on stress and plastic strain history. By means of these standard cycles, we propose respective weakened forms of Drucker’s and Ilyushin’s postulates. These two weakened forms are less restrictive in two respects. One is that, unlike Drucker’s and Ilyushin’s postulates, they are not concerned with stress and strain cycles starting at points on the yield surface. The second is that they are formulated in terms of the rates of change of the net stress work and the stress work when the plastic strain increment incorporated tends to vanish. We demonstrate that either of the proposed weakened forms is adequate to result in the normality rule for the plastic strain rate and the convexity of the yield surface, as does either Drucker’s or Ilyushin’s postulate. These suggest that, in a less restrictive sense and hence in a broader scope, the two proposed weakened forms not only apply to both hardening and softening materials, but also ensure the same consequences.