This paper presents a stability analysis for fractal cracks. First, the Westergaard stress functions are proposed for semi-infinite and finite smooth cracks embedded in the stress fields associated with the corresponding self-affine fractal cracks. These new stress functions satisfy all the required boundary conditions and according to Wnuk and Yavari's embedded crack model they are used to derive the stress and displacement fields generated around a fractal crack. These results are then used in conjunction with the final stretch criterion to study the quasi-static stable crack extension, which in ductile materials precedes the global failure.

In this paper we first obtain the order of stress singularity for a dynamically propagating self-affine fractal crack. We then show that there is always an upper bound to roughness, i.e. a propagating fractal crack reaches a terminal roughness. We then study the phenomenon of reaching a terminal velocity. Assuming that propagation of a fractal crack is discrete, we predict its terminal velocity using an asymptotic energy balance argument. In particular, we show that the limiting crack speed is a material-dependent fraction of the corresponding Rayleigh wave speed.

The fractal crack model described here incorporates the essential
features of the fractal view of fracture, the basic concepts of
the LEFM model, the concepts contained within the
Barenblatt-Dugdale cohesive crack model and the quantized
(discrete or finite) fracture mechanics assumptions. The
well-known entities such as the stress intensity factor and the
Barenblatt cohesion modulus, which is a measure of material
toughness, have been re-defined to accommodate the fractal view of
fracture.