Energy Estimates, Relaxation, and Existence for Strain-Gradient Plasticity with Cross-Hardening

We consider a variational formulation of gradient elasto-plasticity subject to a class of single-slip side conditions, and show how the nonconvexity effects induced by such conditions can be not only resolved mathematically, but also tested physically. We first show that, for a large class of plastic deformations, a given single-slip condition (specification of Burgers’ vectors and slip planes) can be relaxed by introducing a microstructure through a two-stage process of mollification and lamination. This yields a relaxed side condition which only prescribes certain slip planes, and allows for arbitrary slip directions in these planes. The relaxed model should be a useful tool for simulating macroscopic plastic behavior without the need to resolve arbitrarily fine spatial scales. After deriving the relaxed model, we discuss a partial result on the existence of minimizers. Finally, we apply the relaxed model to a specific physical system, in order to be able to compare the analytical results with experiments. In particular, a rectangular shear sample in which only two slip planes are active is clamped at each end, and is subjected to a prescribed horizontal shear, which requires a certain amount of energy. We show that above some critical aspect ratio the energy is strictly positive and below that aspect ratio it is zero. Moreover, in the respective regimes determined by the aspect ratio, we prove energy scaling bounds, expressed in terms of the amount of prescribed shear, and we show that the scalings as well as the critical aspect ratio change radically if the single-slip condition or the strain gradient penalization is neglected.