The Convexity of Inclusions and Gradient’s Concentration for Lamé Systems with Partially Infinite Coefficients (Analysis in Theory and Applications, 2024)
It is interesting to study the stress concentration between two adjacent stiffinclusions in composite materials, which can be modeled by the Lamé system with partially infinite coefficients. To overcome the difficulty from the lack of maximumprinciple for elliptic systems, we use the energy method and an iteration technique tostudy the gradient estimates of the solution. We first find a novel phenomenon thatthe gradient will not blow up any more once these two adjacent inclusions fail to belocally relatively strictly convex, namely, the top and bottom boundaries of the narrowregion are partially “flat”. In order to further explore the blow-up mechanism of thegradient, 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 stressand the order of the relative convexity of the subdomains in all dimensions.
Keywords: Gradient estimates; convex inclusions; blowup analysis; linear elasticity; composite materials.