Crystal elasto-plasticity on the Poincaré half-plane

Abstract

We explore the nonlinear variational modeling of two-dimensional (2D) crystal plasticity based on strain energies which are invariant under the full symmetry group of 2D lattices. We use a natural parameterization of strain space via the upper complex Poincaré half-plane. This transparently displays the constraints imposed by lattice symmetry on the energy landscape. Quasi-static energy minimization naturally induces bursty plastic flow and shape change in the crystal due to the underlying coordinated basin-hopping local strain activity. This is mediated by the nucleation, interaction, and annihilation of lattice defects occurring with no need for auxiliary hypotheses. Numerical simulations highlight the marked effect of symmetry on all these processes. The kinematical atlas induced by symmetry on strain space elucidates how the arrangement of the energy extremals and the possible bifurcations of the strain-jump paths affect the plastification mechanisms and defect-pattern complexity in the lattice.