An analytical/numerical approach for cracked elastic strips under concentrated loads — transient response

Abstract

An analytical/numerical approach 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 suddenly on each face of the crack. 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. In particular, a non-standard Wiener-Hopf (W-H) equation arises, that contains a forcing term with unbounded behaviour at infinity in the transform plane. In addition, the presence of the strip's finite width results in a complicated W-H kernel introducing, therefore, further difficulties. 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. Our method is based on integral transform analysis, an exact kernel factorization, usage of certain theorems of analytic function theory, and numerical Laplace-transform inversion. Numerical results for the stress-intensity-factor dependence upon the ratio of characteristic lengths are presented.