Exact transient full–field analysis of a finite crack subjected to dynamic anti–plane concentrated loadings in anisotropic materials

Abstract

In this study, the elastodynamic full–field response of a finite crack in an anisotropic material subjected to a dynamic anti–plane concentrated loading with Heaviside–function time dependence is investigated. A linear coordinate transformation is introduced to simplify the problem. The linear coordinate transformation reduces the anisotropic finite–crack problem to an equivalent isotropic problem. An alternative methodology, different from the conventional superposition method, is developed to construct the reflected and diffracted wave fields. The transient solutions are determined by superposition of two proposed fundamental solutions in the Laplace transform domain. The fundamental solutions to be used are the problems for applying exponentially distributed traction and displacement on the crack faces and along the crack–tip line in the Laplace transform domain, respectively. Exact analytical transient solutions for dynamic shear stresses, displacement and stress–intensity factor are obtained by using the Cagniard–de Hoop method of Laplace inversion and are expressed in explicitly compact formulations. The solutions have accounted for the contributions of all diffracted waves generated from two crack tips. Numerical results for the time history of shear stresses and stress–intensity factors during the transient process are calculated based on analytical solutions and are discussed in detail. The transient solutions of stresses have been shown to approach the corresponding static values after the first eight waves have passed the field point. The dynamic stress–intensity factor will reach a maximum value when the incident wave arrives at the crack tip, and remain constant before the first diffracted wave generated from the other crack tip arrives, and then will oscillate near the static value. A simple explicit expression of the dynamic overshoot for stress–intensity factors is derived as a function of the location for applied loadings, the crack length and material constants.