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

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 Lamé systems of linear elasticity, may exhibit the singularities with respect to the distance ε \varepsilon between them. In this paper, we establish the asymptotic formulas of the stress concentration for core-shell geometry with C 1 , α C^{1,\alpha } boundaries in all dimensions by precisely capturing all the blow-up factor matrices, as the distance ε \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.