Disentangling Scaling Properties in Anisotropic Fracture

Structure functions of rough fracture surfaces in isotropic materials exhibit complicated scaling properties due to the broken isotropy in the fracture plane generated by a preferred propagation direction. Decomposing the structure functions into the even order irreducible representations of the SO(2) symmetry group indexed by ($m=0,2,4,\dots$) results in a lucid and quickly convergent description. The scaling exponent of the isotropic sector ($m=0$) dominates at small length scales. One can reconstruct the anisotropic structure functions using only the isotropic and the first nonvanishing anisotropic sector ($m=2$) [or at most the next one ($m=4$)]. The scaling exponent of the isotropic sector should be observed in a proposed, yet unperformed, experiment.

Figure 1

Sketch of a hypothetical fracture experiment arranged to allow a crack to develop in an isotropic fashion, i.e., with all angles $\theta$ being statistically equivalent. On the left, the full three dimensional experiment is shown, where the tensile axis is along $z$ and a circular cavity is in the $xy$ plane. On the right, a magnified version of the circular cavity in the $xy$ plane is shown.

Figure 2

The raw fracture surface of the aluminum alloy obtained in Ref. 4.

Figure 3

Contour plot of the second order structure function of the surface shown in Fig. 2.

Figure 4

Log-log plots of the amplitudes $a_0(\ell )$, $2|a_2(\ell )|$, and $2|a_4(\ell )|$ vs $\ell$ for the aluminum alloy.

Figure 5

The experimental $S_2(\ell ,\theta )$ for the aluminum alloy (circles) and the representation Eq. (5) (line), for $\ell =5$, 15, and $25\mu \mathrm{m}$.

Figure 6

The experimental $S_2(\ell ,\theta )$ for the aluminum alloy (circles) and the SO(2) expansion up to the third even order irreducible representation (line), for $\ell =35\mu \mathrm{m}$.

Figure 7

Log-log plots of the amplitudes $a_0(\ell )$ and $2|a_2(\ell )|$ for the artificial rock.

Figure 8

Comparison of the experimental $S_2(\ell ,\theta )$ (circles) and the SO(2) expansion up to the third even irreducible representation (line) for the artificial rock with $\ell =2\mathrm{mm}$.