Modelling crack-induced ultrasonic scattering in a thick-walled pipe
Doctoral thesis, 2017
Ultrasonic testing is used in several industries where there are high demands on safety, e.g. nuclear power and aerospace industries. In the nuclear industry and elsewhere there are many pipes that need to be tested. To this end this thesis considers ultrasonic wave scattering from a crack inside a thick-walled pipe. Several different crack types are considered: an infinite axial-radial crack, a finite axial-radial crack, and a radial-angular crack. To solve these problems a hypersingular integral equation method has been employed. The hypersingular integral equation is derived from an integral representation for the elastodynamic field that involves the Green's tensor of the pipe. The primary unknown in the integral equation is the crack opening displacement (COD). The Green's tensor of the pipe consists of two terms, one is the free space part, called the singular part, and an added part, called the regular part, to fulfil the stress free boundary conditions on the walls of the pipe. The regular part is derived in the first paper and is the same in all papers. The singular part for both the infinite and finite axial-radial cracks is a double Fourier representation in rectangular coordinates. The Green's tensor for the angular-radial crack is instead a Hankel transform representation. The hypersingular integral equation must be regularized and this is accomplished by expansions of the COD in Chebyshev functions, which have the correct square root behaviour along the crack edges. The integral equation also needs to be projected on the same set of Chebyshev functions, and this concludes the regularization. The COD can thereby be determined. The ultrasound in the pipe is excited by a probe on the outer wall of the pipe and a model for such a probe acting as transmitter is developed. The ultrasound from this probe act as the source term in the integral equation. As a receiver another probe is used and the action of this probe is modelled by a reciprocity argument where the output is the electric signal from the receiving probe. As a special and common case the same probe is acting as both transmitter and receiver, this is called pulse-echo testing. A few examples are given for the different crack types showing the influence of varying pipe diameter and wall thickness.
Hypersingular integral method