Torsional Vibration of an Elastic Solid Containing a Penny-Shaped Crack

The axisymmetric wave equation is solved for the problem of torsional elastic waves impinging on a penny-shaped crack the periphery of which is assumed to be infinitely sharp. Using Hankel transforms, the problem is reduced to the solution of two simultaneous integral equations of the Fredholm type. The proposed method of solution permits an examination of the complete scattered-wave field at points both near to and far from the penny-shaped plane of discontinuity. In elastodynamics, however, it is the nearfield stress solution that is of chief interest. To this end, the singular nature of the local dynamic stress field is determined in elementary closed form, while the magnitude of this stress field, which can be adequately described by a singularity parameter k3, is calculated numerically. The important results are that (1) the stresses are singular of the order r1−12 as r1 → 0 at the diffracting edge of the crack and (2) k3 is found to be proportional to the material constants, the crack radius, and the wavelength or frequency of the incoming waves. A knowledge of this parameter k3 is essential to a clear understanding of the propagation of cracks through structural components undergoing torsional oscillations, since its value has been known to control the stability or instability behavior of cracks.