Orientational phenomena in a plastic flow of a two-dimensional square crystal

Within the framework of irreversible thermodynamics, a nonlinear relaxation equation for the 4-rank alignment tensor is formulated for a two-dimensional square crystal subjected to the shear flow. An evolution equation governing the dynamics of the orientation of the crystal, analogous to the Ericksen–Leslie balance equation used in the theory of liquid crystals, is derived from the relaxation equation and analyzed for a Couette flow geometry. It is found that two solutions of the evolution equation are possible: (i) the crystal aligns at a specific angle or (ii) rotates in the shear plane (tumbling). The flow alignment angle is predicted to depend on the value of the shear rate. Furthermore, a generalized Fokker–Planck equation for the probability distribution function is formulated within the mean-field approximation. It is used to derive a scalar counterpart, expressed in terms of microscopic quantities, of the phenomenological evolution equation. The stationary solution is also investigated, based on the kinetic equation.