Sharp interface limit of a phase-field model of crystal grains.

We analyze a two-dimensional phase field model designed to describe the dynamics of crystalline grains. The phenomenological free energy is a functional of two order parameters. The first one reflects the orientational order, while the second reflects the predominantally local orientation of the crystal. We consider the gradient flow of this free energy. Solutions can be interpreted as ensembles of grains (in which the orientation is constant in space) separated by grain boundaries. We study the dynamics of the boundaries as well as the rotation of the grains. In the limit of an infinitely sharp interface, the normal velocity of the boundary is proportional to both its curvature and its energy. We obtain explicit formulas for the interfacial energy and mobility, and study their behavior in the limit of a small misorientation. We calculate the rate of rotation of a grain in the sharp interface limit, and find that it depends sensitively on the choice of the model.

Rate-and-state theory of plastic deformation near a circular hole.

We show that a simple rate-and-state theory accounts for most features of both time-independent and time-dependent plasticity in a spatially inhomogeneous situation, specifically, a circular hole in a large stressed plate. Those features include linear viscoelastic flow at small applied stresses, strain hardening at larger stresses, and a dynamic transition to viscoplasticity at a yield stress. In the static limit, this theory predicts the existence of a plastic zone near the hole for some but not all ranges of parameters. The rate-and-state theory also predicts dynamic failure modes that we believe may be relevant to fracture mechanics.