Singularities of the stress concentration in the presence of $C^{1,\alpha}$-inclusions with core-shell geometry

TL;DR

This paper derives precise asymptotic formulas for stress concentration in high-contrast composites with core-shell inclusions having $C^{1,eta}$ boundaries, revealing how stress singularities behave as the core and shell boundaries approach each other.

Contribution

It provides the first comprehensive asymptotic analysis of stress concentration in $C^{1,eta}$ core-shell geometries, including explicit blow-up factor matrices and optimal gradient estimates.

Findings

Asymptotic formulas for stress blow-up as the core and shell boundaries approach.

Explicit characterization of blow-up factor matrices in all dimensions.

Optimal gradient estimates derived from the asymptotic analysis.

Abstract

In high-contrast composites, if an inclusion is in close proximity to the matrix boundary, then the stress, which is represented by the gradient of a solution to the Lam\'{e} systems of linear elasticity, may exhibits the singularities with respect to the distance $\varepsilon$ between them. In this paper, we establish the asymptotic formulas of the stress concentration for core-shell geometry with $C^{1,\alpha}$ boundaries in all dimensions by precisely capturing all the blow-up factor matrices, as the distance $\varepsilon$ between interfacial boundaries of a core and a surrounding shell goes to zero. Further, a direct application of these blow-up factor matrices gives the optimal gradient estimates.