- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:04
Interfaces and Free Boundaries
Interfaces Free Bound.
IFB
1463-9963
1463-9971
Partial differential equations
Numerical analysis
Fluid mechanics
Biology and other natural sciences
10.4171/IFB
http://www.ems-ph.org/doi/10.4171/IFB
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
18
2016
1
Effective behavior of an interface propagating through a periodic elastic medium
Patrick
Dondl
Universität Freiburg, FREIBURG, GERMANY
Kaushik
Bhattacharya
California Institute of Technology, PASADENA, UNITED STATES
Fractional Laplacian, interfaces, heterogeneous media, effective properties, homogenization, periodic media, free boundary problems, pinning
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
Partial differential equations
91
113
10.4171/IFB/358
http://www.ems-ph.org/doi/10.4171/IFB/358