# Modelling of ultrasonic testing for cracks near a non-planar surface

[Licentiatavhandling]

Nondestructive testing using ultrasound has important applications in e.g. the nuclear power and aerospace industries, where it is used to inspect safety-critical parts for flaws. For reliable inspections a proper adaptation to the specific testing situation is crucial, and in this process a mathematical model of nondestructive ultrasonic testing is a valuable tool. In this thesis a two-dimensional mathematical model of ultrasonic nondestructive testing for cracks near a non-planar surface is developed, with applications in the testing of thick-walled pipes with diameter transitions, pipe connections, etc. In the first part of the thesis the 2D anti-plane wave scattering problem is considered. The solution method employed is based on reformulating the problem as two coupled boundary integral equations (BIE): a displacement BIE for the back surface displacement and a traction BIE for the crack opening displacement. In order to avoid numerical integration of singular integrals a regularization approach is employed for the displacement BIE. The integral equations are subsequently solved simultaneously using a combination of a series expansion of the crack-opening displacement and a boundary element discretization of the back surface. The transmitting ultrasonic contact probe is modelled by prescribing the traction beneath it. To model the action of the receiving probe an electromechanical reciprocity relation is used. An inverse temporal Fourier transform is applied to obtain the time traces, and a few numerical examples are given to illustrate the model and the influence of the back surface. In the second part of the thesis the solution method developed in the first part for anti-plane wave scattering is adapted and applied to the in-plane wave scattering problem, featuring coupled P and SV waves. The action of the transmitting ultrasonic contact probe is modelled also for this case by prescribing the traction beneath the probe, but in order to realistically model both P and SV probes two different tractions are considered. The model also takes into account the influence of a couplant applied between the probe and component. The action of the receiving probe is again modelled using reciprocity. Finally, a few numerical examples are given.

**Nyckelord: **Elastic waves, Nondestructive testing, Ultrasonics, Crack, BIEM, BEM.

Denna post skapades 2008-11-24.

CPL Pubid: 78663