Asymptotic fields at the tip of a cohesive crack

Recently, the authors (Xiao and Karihaloo, J Mech Mater Struct 1:881-910, 2006) obtained universal asymptotic expansions at a cohesive crack tip, analogous to the Williams (ASME J Appl Mech 24:109-114, 1957) expansions at a traction-free crack tip for any normal cohesion-separation law (i.e. softening law) that can be expressed in a special polynomial. This special form ensures that the radial and angular variations of the asymptotic fields are separable as in the Williams expansions. The coefficients of the expansions of course depend nonlinearly on the softening law and the boundary conditions. They demonstrated that many commonly-used cohesion-separation laws, e.g., rectangular, linear, bilinear and exponential, can indeed be expressed very accurately in this special form. They also obtained universal asymptotic expansions when the cohesive crack faces are subjected to Coulomb friction. The special polynomial involves fractional powers which seem rather contrived. In this paper, we will show that the asymptotic expansions can be obtained in a separable form even when the cohesion-separation law is in a special polynomial form involving only integer powers.