Fracture surfaces: Apparent roughness, relevant length scales, and fracture toughness

The Griffith criterion is rewritten to account for the propagation of a self-affine crack. It is shown that the stress-field singularity in the close vicinity of the crack tip involves an exponent -(${\mathit{d}}_{\mathit{F}}$-1)/2 instead of -1/2. Furthermore, the fracture toughness is shown to be related to the ratio of the maximum height ${\mathit{z}}_{\max }$ to the self-affine correlation length ξ.