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

Abstract

In the region between two closely located stiff inclusions, the stress, which is the gradient of a solution to the Lamé 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 in 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 ‐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.