Lower bounds of gradient's blow-up for the Lam\'{e} system with partially infinite coefficients

TL;DR

This paper establishes lower bounds for the gradient blow-up in the Lamé system with partially infinite coefficients, demonstrating the optimality of the blow-up rate for inclusions of arbitrary shape in 2D and 3D.

Contribution

It introduces a method to determine the blow-up factor for the gradient of solutions, proving the optimality of blow-up rates in elasticity problems with close-to-touching inclusions.

Findings

Lower bounds match upper bounds, confirming optimal blow-up rates.

The blow-up factor determines whether gradient blow-up occurs.

Results apply to inclusions of arbitrary shape in 2D and 3D.

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 Lam\'{e} system of linear elasticity. The aim of this paper is to establish lower bounds of the gradients of solutions of the Lam\'{e} 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.