Asymptotic Analysis of a Coupled System of Nonlocal Equations with Oscillatory Coefficients

TL;DR

This paper analyzes the asymptotic behavior of solutions to strongly coupled nonlocal integral equations with oscillatory coefficients, revealing an effective local elliptic system characterized by a tensor similar to those in classical elasticity.

Contribution

It provides a rigorous derivation of the local limit system for a coupled nonlocal system with oscillatory coefficients, motivated by a peridynamic model of heterogeneous media.

Findings

Effective local elliptic system derived in the vanishing nonlocality limit.

The effective tensor shares properties with classical elasticity tensors.

The analysis applies to systems modeling deformation of heterogeneous media.

Abstract

In this paper we study the asymptotic behavior of solutions to systems of strongly coupled integral equations with oscillatory coefficients. The system of equations is motivated by a peridynamic model of the deformation of heterogeneous media that additionally accounts for short-range forces. We consider the vanishing nonlocality limit on the same length scale as the heterogeneity and show that the system's effective behavior is characterized by a coupled system of local equations that are elliptic in the sense of Legendre-Hadamard. This effective system is characterized by a fourth-order tensor that shares properties with Cauchy elasticity tensors that appear in the classical equilibrium equations for linearized elasticity.