Eigenvalue asymptotics for an elastic strip and an elastic plate with a crack

TL;DR

This paper derives an asymptotic formula describing how embedded eigenvalues approach spectral thresholds in elastic structures with small cracks, providing insights into the spectral behavior of such systems.

Contribution

It introduces a new asymptotic analysis for the eigenvalues of elasticity operators with cracks, focusing on the zero Poisson's ratio case.

Findings

Derived asymptotic formulas for eigenvalues near spectral thresholds

Analyzed the effect of small cracks on the spectral properties of elastic operators

Provided mathematical insights into crack-induced spectral shifts in elastic plates

Abstract

We consider the elasticity operator with zero Poisson's ratio on an infinite strip and an infinite plate with a horizontal crack. We prove an asymptotic formula for the distance of the embedded eigenvalues to some spectral threshold of the operator as the crack becomes small.