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.
Zeitschrift Angewandte Mathematik und Physik
- Pub Date:
- October 2014
- Mathematics - Analysis of PDEs
- 24 pages, 17 figures, to appear in Zeitschrift fuer angewandte Mathematik und Physik (ZAMP). Comments on mollification in intro adjusted. Comment on lamination above Thm 3.1 corrected