Strong convergence of the solutions of the linear elasticity and uniformity of asymptotic expansions in the presence of small inclusions

TL;DR

This paper proves the strong convergence of solutions to the linear elasticity equations with extreme elastic constants and demonstrates the uniformity of asymptotic expansions despite small inclusions.

Contribution

It establishes convergence results for the Lamé system with extreme parameters and shows the uniformity of asymptotic expansions in the presence of small inclusions.

Findings

Solutions converge in $H^1$-norm as elastic constants tend to extremes.

Asymptotic expansions are uniform with respect to Lamé parameters.

Results apply to inclusions with extreme elastic properties.

Abstract

We consider the Lam\'e system of linear elasticity when the inclusion has the extreme elastic constants. We show that the solutions to the Lam\'e system converge in appropriate $H^1$-norms when the shear modulus tends to infinity (the other modulus, the compressional modulus is fixed), and when the bulk modulus and the shear modulus tend to zero. Using this result, we show that the asymptotic expansion of the displacement vector in the presence of small inclusion is uniform with respect to Lam\'e parameters.