The convexity of inclusions and gradient's concentration for Lamé systems with partially infinite coefficients

Abstract

It is interesting to study the stress concentration between two adjacent stiff inclusions in composite materials, which can be modeled by the Lamé system with partially infinite coefficients. To overcome the difficulty from the lack of maximum principle for elliptic systems, we use the energy method and an iteration technique to study the gradient estimates of the solution. We first find a novel phenomenon that the gradient will not blow up any more once these two adjacent inclusions fail to be locally relatively strictly convex, namely, the top and bottom boundaries of the narrow region are partially "flat". This is contrary to our expectation. In order to further explore the blow-up mechanism of the gradient, 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 stress and the order of the relative convexity of the subdomains in all dimensions.

arXiv comment61 pages

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• Stress concentration for two nearly touching circular holes
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Cited by in corpus (4)

• Boundary blow-up analysis of gradient estimates for Lamé systems in the presence of $M$-convex hard inclusions
• Stress concentration factors for the Lamé system arising from composites
• Blow-up analysis of hydrodynamic forces exerted on two adjacent $M$-convex particles
• Asymptotic analysis of the stress concentration between two adjacent stiff inclusions in all dimensions