A Maximum Eigenvalue Approximation for Crack-Sizing Using Ultrasonic Arrays

Ultrasonic phased array systems are becoming increasingly popular as tools for the inspection of safety-critical structures with in the non-destructive testing industry. The datasets captured by these arrays can be used to image the internal microstructure of individual components, all owing the location and nature of any defects to be deduced. Unfortunately, many of the current imaging algorithms require an arbitrary threshold at which the defect measurements can be taken and this aspect of subjectivity can lead to varying characterisations of a flaw between different operators. This paper puts forward an objective approach based on the Kirchoff scattering model and the approximation of the resulting scattering matrices by Toeplitz matrices. A mathematical expression relating the crack size to the maximum eigenvalue of the associated scattering matrix is thus derived. The formula is analysed numerically to assess its sensitivity to the system parameters and it is shown that the method is most effective for sizing defects that are commensurate with the wavelength of the ultrasonic wave (or just smaller than. The method is applied to simulated FMC data arising from finite element calculations where the crack length to wavelength ratios range between 0.6 and 1.8. The recovered objective crack size exhibits an error of 12%.