Boundary Blow-Up Analysis of Gradient Estimates for Lamé Systems in the Presence of $m$-Convex Hard Inclusions
Haigang Li, Zhiwen Zhao · SIAM Journal on Mathematical Analysis · 2020
In high-contrast elastic composites, it is vitally important to investigate the stress concentration from an engineering point of view. The purpose of this paper is to show that the blow-up rate of the stress depends not only on the shape of the inclusions, but also on the given boundary data, when hard inclusions are close to the matrix boundary. First, when the boundary of inclusion is partially relatively parallel to that of the matrix, we establish the gradient estimates for Lamé systems with partially infinite coefficients and find that they are bounded for some boundary data $\varphi$ while some $\varphi$ will increase the blow-up rate. In order to identify such a novel blow-up phenomenon, we further consider the general $m$-convex inclusion cases and uncover the dependence of the blow-up rate on the inclusion's convexity $m$ and the boundary data's order of growth $k$ in all dimensions. In particular, the sharpness of these blow-up rates is also presented for some prescribed boundary data.