# Crystal elasto-plasticity on the Poincar\'e half-plane

## TL;DR

This paper models 2D crystal plasticity using a nonlinear variational approach on the Poincaré half-plane, revealing how lattice symmetry influences defect dynamics and plastic flow without auxiliary hypotheses.

## Contribution

It introduces a symmetry-invariant strain energy framework on the Poincaré half-plane for 2D crystals, linking lattice symmetry to defect interactions and plasticity mechanisms.

## Key findings

- Symmetry constrains the energy landscape and defect behavior.
- Plastic flow occurs via basin-hopping strain activity.
- Numerical simulations show symmetry's impact on defect patterns.

## Abstract

We explore the nonlinear variational modelling 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 Poincar\'e 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.