A Series Solution for the Diffraction of SH Waves by a Vertical Edge Crack

Abstract

Herein, a substitute series solution, which is unavailable in literature, pertaining to the antiplane diffraction problem by a vertical edge crack, is derived via the region-matching technique combining with the method of images. The dynamic stress intensity factor can be straightforwardly extracted from the expression of the radial stress field, fulfilling the crack-tip singularity inherently. Computed results of the extremely near-, near-, and far-field solutions agree well with those of exact anaiyticai solution and available data. The proposed approach can be applied to cope with the diffraction problems by multiple cracks or branched cracks under SH-wave incidence. a single vertical edge crack. In the work of Tsaur [9], the explicit analytical expressions of near-and far-field displacements are given in terms of radial and angular Mathieu functions. Unfortunately, owing to the lack of asymptotic expressions of radial Mathieu functions for a very small argument, the evaluation of the DSIF is merely via an approximate formula adopted by Datta and Shah [10]. In other words, the DSIF cannot be extracted from the expression of stress field in a straightforward way. Furthermore, when one attempts to exploit the MSV to deal with the case of multiple (a finite number of) cracks [4] or kinked cracks [II], a sequence of coordinate transformations between local systems will be indispensable, that is to say, the usage of addition formulas for Mathieu functions [12] is necessary. However, such formulas implicate numerous complicated calculations of multiple summations since Mathieu functions are functions of functions. This, as a matter of fact, considerably increases the computational complexity. From the viewpoint of application, addition formulas of Bessel functions [13] are much easier than those of Mathieu functions. Thus, in the present work, the authors aim to derive an alternative series solution, based on the Bessel and trigonometric functions, to the classical diffraction problem of SH waves triggered by a single vertical edge crack. To the authors' knowledge, the proposed series solution is nonexistent elsewhere. The regionmatching technique and the method of images are utilized to achieve the goal. Making good use of proper eigenfunctions of the Helmholtz equation, the trick presented herein recasts the boundary-value problem under consideration into the coupled systems of linear algebraic equations. Expressions of the DSIF and far-field scattered displacement are derived also. Comparisons with the results of exact analytical solution [9] and those obtained by the singular integral equation [2] are given. For antiplane diffraction problems, the proposed approach is promising and can be applicable to some wave-crack interaction problems [4,11,14], if suitable Graf's addition formulas of Bessel functions are introduced.