Slow Patterns in Multilayer Dislocation Evolution with Dynamic Boundary Conditions

TL;DR

This paper investigates slow dislocation patterns in multilayer systems with dynamic boundary conditions, revealing how bulk and interface dynamics interact and evolve over specific time scales, with implications for understanding material behavior.

Contribution

It introduces a multiscale parabolic model with dynamic boundary conditions, analyzing the coupled bulk-interface evolution and deriving explicit transition profiles and their governing ODE system.

Findings

Dynamic solutions approach explicit transition profiles over time.

The ODE system matches that of fractional Allen-Cahn with static bulk.

New corrector functions stabilize coupled bulk-interface dynamics.

Abstract

In this paper, we study the slow patterns of multilayer dislocation dynamics modeled by a multiscale parabolic equation in the half-plane coupled with a dynamic boundary condition on the interface. We focus on the influence of bulk dynamics with various relaxation time scales, on the slow motion pattern on the interface governed by an ODE system. Starting from a superposition of N stationary transition layers, at a specific time scale for the interface dynamics, we prove that the dynamic solution approaches the superposition of N explicit transition profiles whose centers solve the ODE system with a repulsive force. Notably, this ODE system is identical to the one obtained in the slow motion patterns of the one-dimensional fractional Allen-Cahn equation, where the elastic bulk is assumed to be static. Due to the fully coupled bulk and interface dynamics, new corrector functions with delicate estimates are constructed to stabilize the bulk dynamics and characterize the limiting behavior of the dynamic solution throughout the entire half-plane.