Landau-Type Theory of Planar Crystal Plasticity

We show that nonlinear continuum elasticity can be effective in modeling plastic flows in crystals if it is viewed as a Landau theory with an infinite number of equivalent energy wells whose configuration is dictated by the symmetry group GL(2,Z). Quasistatic loading can be then handled by athermal dynamics, while lattice-based discretization can play the role of regularization. As a proof of principle, we study dislocation nucleation in a homogeneously sheared 2D crystal and show that the global tensorial invariance of the elastic energy foments the development of complexity in the configuration of collectively nucleating defects. A crucial role in this process is played by the unstable higher symmetry crystallographic phases, typically thought to be unrelated to plastic flow.