On estimating stress intensity factors and modulus of cohesion for fractal cracks

Existing solutions for the singular stress field in the vicinity of a fractal crack tip have been adapted for a somewhat modified problem. Since the integration along the fractal curve is prohibitive and does not lend itself to the presently available mathematical treatments, a simplified one has replaced the original problem. The latter involves a smooth crack embedded in a singular stress field, for which the order of singularity is adjusted to match exactly the one obtained from the analyses pertaining to the fractal crack. Of course, this is only an approximation, and we may only hope that it leads toward correct results, at least in a cursory sense. The advantage of such an approach becomes obvious when one inspects the final closed-form solutions for (a) the stress intensity factor in mode I fractal fracture, and (b) cohesion modulus, which results from the cohesive zone model and serves as a measure of the material resistance to crack propagation. As expected for the fractal geometry employed here, our results are strongly dependent on the fractal dimension D (or roughness exponent H).