On the higher order asymptotic analysis of a non-uniformly propagating dynamic crack along an arbitrary path

Transient mixed-mode elastodynamic crack growth along arbitrary smoothly varying paths is considered. Asymptotically, the crack tip stress field is square root singular with the angular variation of the singular term depending weakly on the instantaneous values of the crack tip speed and on the mode-I and mode-II stress intensity factors. However, for a material particle at a small distance away from the moving crack tip, the local stress field will depend not only on the instantaneous values of the crack tip speed and stress intensity factors, but also on the past history of these time dependent quantities. In addition, for cracks propagating along curved paths the stress field is also expected to depend on the nature of the curved crack path. Here, a representation of the crack tip fields in the form of an expansion about the crack tip is obtained in powers of radial distance from the tip. The higher order coefficients of this expansion are found to depend on the time derivative of crack tip speed, the time derivatives of the two stress intensity factors as well as on the instantaneous value of the local curvature of the crack path. It is also demonstrated that even if cracks follow a curved path dictated by the criterion K~ = 0, the stress field may still retain higher order asymmetric components related to non-zero local curvature of the crack path.