## Abstract

This paper describes some recent studies associated with the development of an instrument, the Crack Microgauge, at University College London, for the detection and measurement of surface cracks in metals by using a.c. electric currents in the metal surface. The design of the instrument has enhanced the accuracy of measurement of surface voltages in the neighbourhood of surface-breaking cracks to the point where it is now advantageous to consider the distribution of surface voltages by mathematical analysis. This enables better use to be made of the instrument since measurements can then be more accurately interpreted in terms of crack size and geometry. In contrast, earlier applications of the method relied on the calibration of the instruments against test blocks. The first generation of users of the Crack Microgauge interpreted their readings according to a simple one-dimensional normal crack model. This, however, is an oversimplification for cracks of finite aspect ratio and it leads to underestimates of centre-line crack depth typically of the order of 30-40<%> when the aspect ratio is about 3. The first major contribution of the theory was therefore to provide a model for surface current flow around cracks of finite aspect ratio. For this purpose a useful unfolding theorem was deduced in cases in which the current skin depth is small compared with crack dimensions. This allows the surface field to be unfolded into a plane distribution of potential satisfying Laplace's equation, and it enables us to obtain solutions for many different crack shapes by using standard mathematical methods. An account of these developments is given in $\xi$ 2 of this paper. The unfolding of the surface field gives the global distribution of surface current around a crack, that is, on the length scale of the crack dimensions. On the length scale of the skin depth, however, the solutions need modification in the neighbourhood of the surface and interior edges of the crack. In developing the unfolding theorem it was argued that these modifications, which we describe as corner solutions, could be neglected. The purpose of this paper is to construct these corner solutions to confirm that this is so. The mathematical theory described in $\xi$ 2 was developed for the most important case in which cracks develop in planes perpendicular to the surface of the metal. It is clear from the construction of that solution, however, that the unfolding may be applied with the same result to plane cracks inclined to the surface. Thus the global solution is not able to distinguish the inclination of the crack plane to the surface. We shall find, however, that the corner solutions are sensitive to this inclination, and they are given in $\S$ 3 for arbitrary angles of inclination. These studies suggest, therefore, that information on crack inclination could be deduced from experimental investigation of these corner solutions. Since the solutions are found to possess a phase of $\frac{1}{4} \pi$, it might be possible to extract this information by measuring perturbation signals in quadrature with the main field or perhaps by increasing the skin depth so that the scale of the solutions is enhanced. For a given specimen this could be done by reducing the frequency of the a.c. field, but, if the frequency is reduced significantly, the field distributes itself through the interior of the specimen on the scale of the crack dimensions and may no longer be described as a surface distribution. Further work to describe such field distributions when the skin depth is of the order of the crack depths is in progress.