Perturbation Eigensolutions of Elastic Structures With Cracks

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 λm2, and their corresponding crack-opening shapes ΔΨm of such mechanical systems satisfy a Fredholm integral equation A(λm2)ΔΨm = 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 λm2 of the cracked systems. Finally, approximate λm2 are calculated for two-dimensional elastic solids under antiplane-strain vibration with an oblique internal crack and a boundary crack.