Disordering of a dynamic planar crack front in a model elastic medium of randomly variable toughness

Rice et al. (1994, J. Mech. Phys. Solids 42, 813–843) analyse the propagation of a planar crack with a nominally straight front in a model elastic solid with a single displacement component. Using the form of their results for a strictly linearized perturbation from a straight crack front which moves at uniform speed, we give the corresponding first-order expression for the deviation of a crack front from straightness as a direct integral expression in the deviation of the material toughness from uniformity in the crack plane. We then use this expression to analyse the autocorrelation of the crack front position when the toughness deviations are random. We find that the root mean square deviation in position diverges logarithmically with travel distance across the random toughness region, as do the variances of the propagation velocity and slope of the crack front. That is, according to strictly linearized analysis, perturbed about the solution for a uniformly moving crack front, the perturbations from straightness and from uniform propagation speed should grow without bound in the presence of random deviations in toughness. What is remarkable about this result is that, according to the same strictly linearized analysis, if the toughness is completely uniform over the remaining part of the fracture plane, after encounter with a region of nonuniform toughness, the moving crack front becomes asymptotically straight with increase of time. Nonlinearities, not considered here, must control how statistically disordered the crack front can ultimately become as it propagates through a region of random toughness variation. Also, because of the logarithmic nature of the growth, significant disorder can occur in response to small perturbations only when the crack moves over a great distance compared to the correlation length scale in the fracture toughness.