  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 blowup 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 matrix boundary. First, when the boundary of inclusion
is partially relatively parallel to that of matrix, we establish the gradient
estimates for Lam\'{e} 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 novel blowup
phenomenon, we further consider the general $m$-convex inclusion cases and
uncover the dependence of 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.
