Fracture Surfaces as Multiscaling Graphs

Fracture paths in quasi-two-dimensional (2D) media (e.g., thin layers of materials or paper) are analyzed as self-affine graphs $h(x)$ of height $h$ as a function of length $x$. We show that these are multiscaling, in the sense that $n$th order moments of the height fluctuations across any distance $\ell$ scale with a characteristic exponent that depends nonlinearly on the order of the moment. Having demonstrated this, one rules out a widely held conjecture that fracture in 2D belongs to the universality class of directed polymers in random media. In fact, 2D fracture does not belong to any of the known kinetic roughening models. The presence of multiscaling offers a stringent test for any theoretical model; we show that a recently introduced model of quasistatic fracture passes this test.

Figure 1

A log-log plot of $S_3(\ell )$ as a function of $\ell$. The linear fit corresponds to a typical scaling range of about 1.5 orders of magnitude, with a slope of ${\zeta }_3\simeq 1.65$.

Figure 2

The spectrum ${\zeta }_n$ as a function of the moment order $n$ for rupture lines in paper. The function is fitted to the form ${\zeta }_n=nH-n^2\lambda$, and the parameters $H$ and $\lambda$ are given. The errors in the estimation of these parameters reflect both the variance between different samples and the fit quality. The linear plot $n{\zeta }_1$ is added to stress the nonlinear nature of ${\zeta }_n$.

Figure 3

An example of the convergence of the integral in Eq. (5) for $\ell ={10}^{2.25}$ and $n=8$.

Figure 4

The natural logarithm of $P\mathbf{(}\Delta h(\ell )\mathbf{)}{\ell }^H$ as a function of $\Delta h(\ell )/{\ell }^H$ for $H=0.64$. The legend gives the scale $\ell$ for each plot.

Figure 5

The spectrum ${\zeta }_n$ as a function of the moment order $n$ for rupture lines in the model of Refs. 16, 17. The function is fitted to the form ${\zeta }_n=nH-n^2\lambda$, and the parameters $H$ and $\lambda$ are given. The errors in the estimation of these parameters reflect both the variance between different realizations and fit quality. The linear plot $n{\zeta }_1$ is added for comparison.