Lower bounds of gradient's blow-up for the Lamé system with partially infinite coefficients

Abstract

In composite material, the stress may be arbitrarily large in the narrow region between two close-to-touching hard inclusions. The stress is represented by the gradient of a solution to the Lamé system of linear elasticity. The aim of this paper is to establish lower bounds of the gradients of solutions of the Lamé system with partially infinite coefficients as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero. Combining it with the pointwise upper bounds obtained in our previous work, the optimality of the blow-up rate of gradients is proved for inclusions with arbitrary shape in dimensions two and three. The key to show this is that we find a blow-up factor, a linear functional of the boundary data, to determine whether the blow-up will occur or not.

33 pages; submitted

References in corpus (2)

• Quantitative characterization of stress concentration in the presence of closely spaced hard inclusions in two-dimensional linear elasticity
• Optimal boundary gradient estimates for Lamé systems with partially infinite coefficients

Cited by in corpus (3)

• Boundary blow-up analysis of gradient estimates for Lamé systems in the presence of $M$-convex hard inclusions
• Blow-up analysis of hydrodynamic forces exerted on two adjacent $M$-convex particles
• Estimates and Asymptotics for the stress concentration between closely spaced stiff $C^{1,\gamma }$ inclusions in linear elasticity