Perturbation Eigensolutions of Elastic Structures With Cracks

Abstract

The purpose of this paper is to determine approximate eigensolutions of a class of cracked mechanical systems governed by the two-dimensional Helmholtz equation through a perturbation approach. Shen (1993) shows that exact eigenvalues λm 2 , and their corresponding crack-opening shapes ΔΨ m of such mechanical systems satisfy a Fredholm integral equation A (λm 2 )ΔΨ m = 0. Following the integral equation approach, the approximation in this paper consists of formulating the Rayleigh quotient of the Fredholm operator A (λ2 ) and estimating eigenvalues μ(λ2 ) of the operator A (λ2 ) through perturbation and stationarity of the Rayleigh quotient. The zeros of μ(λ2 ) then approximate eigenvalues λm 2 of the cracked systems. Finally, approximate λm 2 are calculated for two-dimensional elastic solids under antiplane-strain vibration with an oblique internal crack and a boundary crack.