An exact method for cracked elastic strips under concentrated loads — time-harmonic response

Abstract

An exact method is presented for the determination of the near-tip stress field arising from the scattering of SH waves by a long crack in a strip-like elastic body. The waves are generated by a concentrated anti-plane shear force acting on each face of the crack. Time-harmonic variation of the external loading is assumed. The problem has two characteristic lengths, i.e. the strip width, and the distance between the point of application of the concentrated forces and the crack tip. It is well-known that the second characteristic length introduces a serious difficulty in the mathematical analysis of the problem: a non-standard Wiener-Hopf (W-H) equation arises, one that contains a forcing term with unbounded behavior at infinity in the transform plane. In addition, the presence of the strip's finite width results in a complicated W-H kernel. Nevertheless, a procedure is described here which circumvents the aforementioned difficulties and holds hope for solving more complicated problems (e.g. the plane-stress/strain version of the present problem) having similar features. The method is based on integral transform analysis, an exact kernel factorization and usage of certain theorems of analytic function theory. Numerical results for the stress-intensity-factor dependence upon the ratio of characteristic lengths and the external load frequency are presented.