Effective behavior of an interface propagating through a periodic elastic medium

Abstract

We consider a moving interface that is coupled to an elliptic equation in a heterogeneous medium. The problem is motivated by the study of displacive solid-solid phase transformations. We argue that a nearly flat interface is given by the graph of the function $g$ which evolves according to the equation $g_t (x) = -(-\Delta)^{1/2}g (x) + \varphi(x, g(x)) + F$ where $-(-\Delta)^{1/2}g$ describes the elasticity of the interface, $\varphi(x, g(x))$ the heterogeneity of the media and $F$ the external force driving the interface. This equation also arises in the study of ferroelectric and ferromagnetic domain walls, dislocations, fracture, peeling of adhesive tape and various other physical phenomena. We show in the periodic setting that such interfaces exhibit a stick-slip behavior associated with pinning and depinning. Specifically, there is a critical force $F^\star$ below which the interface is trapped, and beyond which the interface propagates with a well-described effective velocity that depends on $F$. We present numerical evidence that the effective velocity ranges from $v \sim (-\log |F-F^\star|)^{-1}$ to $(F-F^\star)^\beta$ for some $0<\beta\le1$ for $F$ close to $F^\star$ depending on $\varphi$. We obtain $\beta = 1/2$ for the case of non-degenerate smooth obstacles. We further present numerical evidence that $F^\star$ can depend on direction and sense of propagation.