Estimates and Asymptotics for the stress concentration between closely spaced stiff $C^{1, \gamma}$ inclusions in linear elasticity

TL;DR

This paper derives optimal stress gradient estimates and asymptotic formulas for closely spaced $C^{1, \gamma}$ inclusions in linear elasticity, revealing how stress concentrates as inclusions approach each other.

Contribution

It extends stress concentration analysis to $C^{1, \gamma}$ inclusions, using $W^{1, p}$ estimates and Campanato's approach, improving upon previous $C^{2, \gamma}$ assumptions.

Findings

Gradient blow-up estimates depend on the distance between inclusions.

Asymptotic formulas describe stress behavior near blow-up points.

Optimal bounds for stress concentration are established.

Abstract

This paper is concerned with the stress concentration phenomenon in elastic composite materials when the inclusions are very closely spaced. We investigate the gradient blow-up estimates for the Lam\'{e} system of linear elasticity with partially infinite coefficients to show the dependence of the degree of stress enhancement on the distance between inclusions in a composite containing densely placed stiff inclusions. In this paper we assume that the inclusions to be of $C^{1, \gamma}$, weaker than the previous $C^{2, \gamma}$ assumption. To overcome this new difficulty, we make use of $W^{1, p}$ estimates for elliptic system with right hand side in divergence form, instead of a direct $W^{2, p}$ argument for $C^{2, \gamma}$ inclusion case, and combine with the Campanato's approach to establish the optimal gradient estimates, including upper and lower bounds. Moreover, an asymptotic formula of the gradient near the blow-up point is obtained for some symmetric $C^{1, \gamma}$ inclusions.