Effective behavior of an interface propagating through a periodic elastic medium

TL;DR

This paper models the dynamics of interfaces in periodic elastic media, revealing a fractional Laplacian evolution, pinning-depinning behavior, and a square-root scaling law for velocity near depinning.

Contribution

It introduces a fractional Laplacian-based evolution equation for interfaces in heterogeneous media and demonstrates pinning-depinning phenomena with numerical scaling insights.

Findings

Interfaces follow a fractional Laplacian evolution.

Pinning and depinning behavior observed in periodic media.

Velocity scales as the square root of excess force near depinning.

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 show that a nearly flat interface is given by the graph of the function $g$ which evolves according to the equation $g_t (x) = -(-\Delta)^{1/2}g (x) + \varphi(x, g(x)) + F$. This equation also arises in the study of dislocations and fracture. We show in the periodic setting that such interfaces exhibit a stick-slip behavior associated with pinning and depinning. Further, we present some numerical evidence that the effective velocity of the phase boundary scales as the square-root of the excess macroscopic force above the depinning transition.