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In high-contrast elastic composites, it is vitally important to investigatethe stress concentration from an engineering point of view. The purpose of thispaper is to show that the blowup rate of the stress depends not only on theshape of the inclusions, but also on the given boundary data, when hardinclusions are close to matrix boundary. First, when the boundary of inclusionis partially relatively parallel to that of matrix, we establish the gradientestimates for Lam\'{e} systems with partially infinite coefficients and findthat they are bounded for some boundary data $\varphi$ while some $\varphi$will increase the blow-up rate. In order to identify such novel blowupphenomenon, we further consider the general $m$-convex inclusion cases anduncover the dependence of blow-up rate on the inclusion's convexity $m$ and theboundary data's order of growth $k$ in all dimensions. In particular, thesharpness of these blow-up rates is also presented for some prescribed boundarydata.

Boundary blow-up analysis of gradient estimates for Lam\'{e} systems in the presence of $M$-convex hard inclusions