Optimal energy scaling for a shear experiment in single-crystal plasticity with cross-hardening

TL;DR

This paper investigates the energy required for shear deformation in single-crystal plasticity with cross-hardening, revealing how the aspect ratio influences the energy scaling and identifying a critical aspect ratio for energy positivity.

Contribution

It introduces a non-convex variational model for shear in single-crystal plasticity with cross-hardening, analyzing energy dependence on aspect ratio and shear amount.

Findings

Existence of a critical aspect ratio separating zero and positive energy regimes.

Energy scaling bounds are established based on the prescribed shear amount.

Below the critical aspect ratio, shear can be imposed with zero energy.

Abstract

Consideration is given to a non-convex variational model for a shear experiment in the framework of single-crystal linearised plasticity with infinite cross-hardening. The rectangular shear sample is clamped at each end, and is subjected to a prescribed horizontal or diagonal shear, modelled by an appropriate hard Dirichlet condition. We ask: how much energy is required to impose such a shear, and how does it depend on the aspect ratio? Assuming that just two slip systems are active, we show that there is a critical aspect ratio, above which the energy is strictly positive, and below which it is zero. Furthermore, in the respective regimes determined by the aspect ratio, we prove energy scaling bounds, expressed in terms of the amount of prescribed shear.