Pinning of Interfaces by Localized Dry Friction

TL;DR

This paper models interface evolution in a heterogeneous environment with localized dry friction obstacles, demonstrating how interfaces become pinned and exhibit hysteresis until a critical force causes depinning.

Contribution

It introduces a new mathematical framework for interface pinning with dry friction obstacles, proving existence and equivalence of viscosity and weak solutions.

Findings

Interfaces are pinned by obstacles until a critical force is exceeded.

A comparison principle for the model is established.

Viscosity solutions are shown to be equivalent to weak solutions.

Abstract

We consider a model for the evolution of an interface in a heterogeneous environment governed by a parabolic equation. The heterogeneity is introduced as obstacles exerting a localized dry friction. Our main result establishes the emergence of a rate-independent hysteresis for suitable randomly distributed obstacles, i.e., interfaces are pinned by the obstacles until a certain critical applied driving force is exceeded. The treatment of such a model in the context of pinning and depinning requires a comparison principle. We prove this property and hence the existence of viscosity solutions. Moreover, under reasonable assumptions, we show that viscosity solutions are equivalent to weak solutions.