Self-affine roughness of a crack front in heterogeneous media

The long-range elastic model, which is believed to describe the evolution of a self-affine rough crack front, is analyzed to linear and nonlinear orders. It is shown that the nonlinear terms, while important in changing the front dynamics, do not change the scaling exponent which characterizes the roughness of the front. The scaling exponent thus predicted by the model is much smaller than the one observed experimentally. The inevitable conclusion is that the gap between the results of experiments and the model that is supposed to describe them is too large and some new physics has to be invoked for another model.

Figure 1

(Color online) Demonstration of data collapse for the linear model, plotting ${\log }_{10}\mathbf{\left(}w(\ell ,L)L^{-\zeta }\mathbf{\right)}$ vs ${\log }_{10}(\ell ∕L)$ with $L=2^n$, $n=12$, 13, and 14, using $\zeta =0.362$. The results are obtained by averaging over 100 realizations.

Figure 2

(Color online) Demonstration of the failure of data collapse for the linear model, plotting ${\log }_{10}\mathbf{\left(}w(\ell ,L)L^{-\zeta }\mathbf{\right)}$ vs ${\log }_{10}(\ell ∕L)$ with $L=2^n$, $n=12$, 13, and 14, using $\zeta =0.4$. The results are obtained by averaging over 100 realizations.

Figure 3

(Color online) Comparison of the fronts obtained with the linear and the nonlinear model, where the initial condition and the quenched noise are all the same. Note the huge difference in scales between the abscissa and the ordinate.

Figure 4

(Color online) Demonstration of data collapse for the nonlinear model, plotting ${\log }_{10}\mathbf{\left(}w(\ell ,L)L^{-\zeta }\mathbf{\right)}$ vs ${\log }_{10}(\ell ∕L)$ with $L=2^n$, $n=12$, 13, and 14, using $\zeta =0.362$. The results are obtained by averaging over 100 realizations.