Asymptotic analysis of the stress concentration between two adjacent stiff inclusions in all dimensions

TL;DR

This paper provides a detailed asymptotic analysis of stress concentration between two closely located stiff inclusions in any dimension, revealing how stress blows up as the inclusions nearly touch.

Contribution

It offers a sharp asymptotic description of stress concentration, including blow-up factors, for all dimensions and shapes, extending previous results to more general geometries.

Findings

Derived asymptotic formulas for stress concentration as inclusions approach contact

Captured blow-up factor matrices involving integrals of solutions

Presented an example with curvilinear squares for numerical applications

Abstract

In the region between two closely located stiff inclusions, the stress, which is the gradient of a solution to the Lam\'{e} system with partially infinite coefficients, may become arbitrarily large as the distance between interfacial boundaries of inclusions tends to zero. The primary aim of this paper is to give a sharp description in terms of the asymptotic behavior of the stress concentration, as the distance between interfacial boundaries of inclusions goes to zero. For that purpose we capture all the blow-up factor matrices, whose elements comprise of some certain integrals of the solutions to the case when two inclusions are touching. Then we are able to establish the asymptotic formulas of the stress concentration in the presence of two close-to-touching $m$-convex inclusions in all dimensions. Furthermore, an example of curvilinear squares with rounded-off angles is also presented for future application in numerical computations and simulations.