On the boundary conditions for ultrasonic transmission by partially closed cracks

The probability of detecting crack-like defects using ultrasonic techniques can be severly reduced if the crack is closed by a static background pressure. In this paper, we model the contacting faces of a partially closed crack by an array of circular spot-welds randomly distributed over an infinite plane. We give an exact derivation of the reflection and transmission coefficients for a plane elastic wave at such a boundary in terms of the mean interfacial stresses. The latter are estimated in the limit when the contact radius is much smaller than the wavelength and the contacts are sparsely distributed. This calculation is then related to a distributed spring model of the interface. The latter replaces the real interface by an effective homogeneous linear boundary condition which relates the crack opening displacement to the boundary stresses by effective stiffnesses. These unknown parameters are chosen to ensure that the model condition predicts the exact values of the mean interfacial stresses and the reflection and transmission coefficients in the limit already described. Our results are consistent with and complement thoses of Baik and Thompson who introduced the distributed spring model in this and a number of other contexts. Our analysis provides a systematic assessment of the range of validity of the model and suggests ways in which the present estimates may be improved