Natural Frequencies of a Penny-Shaped Crack With Spring Boundary Condition

Abstract

Natural frequencies of a penny-shaped crack are calculated for the three-dimensional elastic problem. The crack is imbedded in a homogeneous medium and on the crack surface the spring boundary conditions are assumed. Only the symmetric problem is considered and the complex frequencies are given as the SEM (singularity expansion method) poles of the symmetric part of the transition (T) matrix. The T matrix is calculated with a direct integral equation method leading to integral equations relating normal stress and displacement on the crack surface. The location of the poles in the complex frequency plane are compared with the scattering cross-section versus frequency and with Rayleigh surface waves.