Effective behavior of an interface propagating through a periodic elastic medium

Patrick W. Dondl; Kaushik Bhattacharya

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arXiv · 1206.2578

Effective behavior of an interface propagating through a periodic elastic medium

Patrick W. Dondl·Kaushik Bhattacharya

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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 $g_t (x) = -(-Δ)^{1/2}g (x) + φ(x, g(x)) + F$. 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.

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Patrick W. Dondl, Kaushik Bhattacharya. 2012-06-12. Effective behavior of an interface propagating through a periodic elastic medium. https://arxiv.org/abs/1206.2578

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