Statistics of the deformation of the front of a tunnel-crack propagating in some inhomogeneous medium

We present a statistical analysis of some geometrical features of the front of a tensile tunnel-crack propagating quasistatically, according to some Paris-type law, in some elastic solid with spatially varying Paris constant. The work is based on an earlier formula of the authors, which provides the first-order change of the distribution of the mode I stress intensity factor along the front of a tunnel-crack, arising from some small but otherwise arbitrary in-plane perturbation of this front. The quantities studied include the power spectrum and the autocorrelation function of the deviation of the two parts of the front from reference straight lines, the autocorrelation function of the derivative of this deviation in the direction of the crack front, the mean squared fluctuation of the deviation, and its correlation distance. The various measures of the magnitude of the deviation of the front from straightness are all found to increase in time at a considerable rate, which means in some sense that the “wavyness” of the front continuously grows. However, the correlation distance of the deviation also increases, which mitigates the preceding conclusion, since it means in another sense that the crack front tends to “straighten back” in time. Also, comparisons are made with the cases of a semi-infinite crack propagating quasistatically or dynamically, using some results of Rice and coworkers for the latter case. The rate of growth of the various measures of the magnitude of the deviation from straightness is much larger for the tunnel-crack than for the semi-infinite one. This is because the finite width of the tunnel-crack induces a “destabilizing” effect of the straight configuration of the front for sinusoidal perturbations with large wavelengths, which is typical of such finite crack geometries.