Finite Difference Methods Applied to Ultrasonic Non-Destructive Testing Problems

The scattering and reflection of ultrasonic pulses is the basis for ultrasonic non-destructive testing techniques. However problems are encountered in the interpretation of experimental results in many systems especially those which have target dimensions of the order of a wavelength, where there are no analytical solutions. It is shown that the lack of an adequate analytical theory can be overcome by the use of numerical models based on finite difference approximations of the basic elastic equations of motion. Finite difference methods, which have previously had a successful history in seismology, are introduced to study non-destructive testing problems and provide a complete description of the interactions of elastic waves, including mode-conversion as an intrinsic part of the formulations. A review is given of the types of problem, that are of interest to non-destructive testers and which finite difference methods are best suited to solve. Specific examples of the technique are shown applied to pulsed Rayleigh, compressional and shear wave scattering by such features as open slots. The descriptions given by the numerical models are confirmed by laboratory experiments on both aluminium and steel blocks. The prediction of a new family of mode-conversion techniques for crack detection and sizing, suggested by the numerical models, is confirmed by experimental results. INTRODUCTION Ultrasonic methods of. non-destructive testing have been developed considerably in recent years, both in terms of the range of systems that are examined and the importance placed on the results that are obtained. The importance of pre-service and in-service inspection is going to increase in the future as the consequences of the failure of a system, such as a nuclear plant or an aircraft, increase in magnitude. However although considerable progress has been made in improving ultrasonic testing techniques, many problems remain, especi~ ally in the interpretation of experimental results. These problems are in part due to the lack of adequate theories for the elastic wave propagation and scattering processes involved. If ultrasonic non-destructive testing methods are to become increasingly reliable and quantitative it is necessary to develop a well-founded theory for such problems as the interaction of pulsed waves with small targets, particularly those with dimensions of the order of a wavelength. [1] The lack of adequate analytical theories for such problems has caused the attention of some workers to turn to consider numerical methods. It is found that numerical methods, which give full-wave descriptions of the interaction of elastic waves with structures that are of relevance to engineers, can be found in the seismological literature. [2] The activities within the research group at The City University fall into three related areas. Firstly there is the development of practical ultrasonic non-destructive testing techniques, in particular using short pulses and spectroscopy. Secondly, there is the development of instrumenta310 tion required for broad-band and spectroscopic studies, together ~lith a transducer development programme which includes using visualisation techniques. Thirdly, there ~re the studies of wave propagation and scattering using numerical models supported by experimental measurements. This paper is based on the work in the third area of study and considers finite difference methods and the application of such techniques to problems which form the basis for ultrasonic nondestructive testing. BASIC EQUATIONS AND BOUNDARY CONDITIONS The basic elastic equations for wave propagation and scattering have been presented by many authors, including Graff {3], so only a brief summary is included in this paper. For a general heterogeneous, linear, isotropic perfectly elastic medium, an elastic wave can be described by the equation[4] (A + v)v(v.u) + v v2u + VA(v.u) + Y~ X (V X U) + 2{V~.V)U where u is the displacement, p is the density, t is the time and A and ~ are the Lame parameters of the medium. ( 1 ) With a restriction to two spatial dimensions and homogeneity, the basic equations 1~hich describe the displacements in the system are