In high-contrast composite materials, the stress concentration is a common phenomenon when inclusions are close to touch. It always causes damage initiation. This problem was proposed mathematically by Ivo Babuska, concerning the system of linear elasticity, modeled by a class of second order elliptic systems of divergence form, with discontinuous coefficients. I will first review some of our results on upper bound estimates by developing an iteration technique with respect to the energy integral to overcome the difficulty from the lack of maximal principle for elliptic systems, then will present two very recent results of myself on lower bound estimates and asymptotics to show that the blow-up rates of the gradients are actually optimal and the convexity of inclusions plays more important role than the curvature in such blow-up analysis.
