Effective behavior of an interface propagating through a periodic elastic medium

  • Patrick W. Dondl

    Universität Freiburg, Germany
  • Kaushik Bhattacharya

    California Institute of Technology, Pasadena, USA


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 gg which evolves according to the equation gt(x)=(Δ)1/2g(x)+φ(x,g(x))+Fg_t (x) = -(-\Delta)^{1/2}g (x) + \varphi(x, g(x)) + F where (Δ)1/2g-(-\Delta)^{1/2}g describes the elasticity of the interface, φ(x,g(x))\varphi(x, g(x)) the heterogeneity of the media and FF 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 FF^\star below which the interface is trapped, and beyond which the interface propagates with a well-described effective velocity that depends on FF. We present numerical evidence that the effective velocity ranges from v(logFF)1v \sim (-\log |F-F^\star|)^{-1} to (FF)β(F-F^\star)^\beta for some 0<β10<\beta\le1 for FF close to FF^\star depending on φ\varphi. We obtain β=1/2\beta = 1/2 for the case of non-degenerate smooth obstacles. We further present numerical evidence that FF^\star can depend on direction and sense of propagation.

Cite this article

Patrick W. Dondl, Kaushik Bhattacharya, Effective behavior of an interface propagating through a periodic elastic medium. Interfaces Free Bound. 18 (2016), no. 1, pp. 91–113

DOI 10.4171/IFB/358