The convexity of inclusions and gradient's concentration for Lam\'e systems with partially infinite coefficients

TL;DR

This paper investigates how the shape of inclusions affects stress concentration in composite materials modeled by Lamé systems with infinite coefficients, revealing that non-convex shapes prevent gradient blow-up.

Contribution

It introduces a novel phenomenon showing that non-convex inclusions do not cause gradient blow-up, and establishes a relationship between convexity order and stress concentration rates.

Findings

Gradient does not blow up if inclusions are not locally convex.

The blow-up rate of stress relates to the order of convexity of inclusions.

Gradient estimates are obtained using energy and iteration methods.

Abstract

It is interesting to study the stress concentration between two adjacent stiff inclusions in composite materials, which can be modeled by the Lam\'e system with partially infinite coefficients. To overcome the difficulty from the lack of maximum principle for elliptic systems, we use the energy method and an iteration technique to study the gradient estimates of the solution. We first find a novel phenomenon that the gradient will not blow up any more once these two adjacent inclusions fail to be locally relatively strictly convex, namely, the top and bottom boundaries of the narrow region are partially "flat". This is contrary to our expectation. In order to further explore the blow-up mechanism of the gradient, we next investigate two adjacent inclusions with relative convexity of order m and finally reveal an underlying relationship between the blow-up rate of the stress and the order of the relative convexity of the subdomains in all dimensions.