Scaling invariance of crack surfaces

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Abstract
The morphology of fractured rock surfaces is studied in terms of their scaling invariance. Fresh brittle fractures of granite and gneiss were sampled with a mechanical laboratory profilometer, and (1 + 1)‐dimensional parallel profiles were added to build actual maps of the surfaces. A first step in the scaling invariance description is a self‐affine analysis using three independent methods. The root‐mean‐square and the maximum‐minimum difference of the height are shown to follow a power law with the sample length. The return probability and the Fourier spectrum are also computed. All these approaches converge to a unique self‐affine exponent: ζ = 0.80. Analysis over a broad statistical set provides a reproducibility error of ±0.05. No significant differences between the isotropic granite and the markedly anisotropic gneiss appear for the scaling exponents. An analysis of the profilometer shows that the two main drawbacks of the setup are not significant in these analyses. The systematic errors of the scaling analysis are estimated for the different methods. Isotropy of the scaling invariance within the mean fracture plane is shown either with the result obtained from different fracture orientations or with the two‐dimensional Fourier spectrum of the surface topography itself. The analysis is brought further into the multifractal framework. The structure functions are shown to have power law behavior, and their scaling exponent varies nonlinearly with the moment order. Finally, the corresponding conserved process belongs to a universal multifractal class with α = 1.5 for the Levy index and C1 = 0.3 for the fractal codimension of the mean singularities. The three indices (ζ, α and C1) completely characterize the scale invariance. The multifractal behavior is significant for physical properties which depend on high‐order moments like contact. According to this study and that of other groups, the self‐affine exponent ζ is constant over a large range of scales and for different fracture modes and various materials. This opens the possibility that there exists a form of universality in the cracking process. It appears that only the prefactor of the roughness is dependent on material and mode.

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