It was shown that a complete description of the statistical scaling properties of fracture surfaces in heterogeneous materials call for the use of the two-dimensional (2D) height-height correlation function /1/. This function was observed to exhibit anisotropic scaling properties similar to the Family-Viseck scaling predicted in interface growth models, characterized by three critical exponents ζ=0.75, β =0.6 and Z= ζ/β=1.2 independent to some extent of the considered material, the loading condition, and the crack growth velocity.
All the investigated fracture surfaces were shown to be self-affine with an exponent ζ~0.4 ±0.04, remarkably lower than the ‘‘universal’’ value ζ~0.75 reported in the previous section and usually measured for many materials /3/. This low value is similar to that found for sandstone samples of similar microstructure. Furthermore, the 2D height-height correlation function is found to exhibit Family-Viseck scaling as for the materials investigated in the previous section (silica glass, aluminium alloy, mortar, wood) but with a different set of critical exponents ζ=0.4, β=0.5 and Z=ζ/β=0.8 /4/. This set of exponents is practically independent on the porosity within the range investigated as well as on the bead diameterand of the crack growth velocity. This suggests the existence of a second universality class for failure problems.
In contrast, the fracture surfaces of sandstone were found to be self-affine with a roughness exponent significantly smaller, around ζ~0.4-0.5 /2/. To shed light on the origin of this anomalously low roughness exponent, we investigated the scaling properties of an artificial material comparable to sandstone but with a microstructure that can be tuned experimentally. Packing made of sintered glass beads have been selected for that purpose because, like sandstone, they are made of cemented grains. However, contrary to natural rocks, the characteristic grain size and cohesion can be adjusted by modifying the bead diameter (from 50 µm to 128µm) and the porosity (from 3% to 26%).
Top: Log-log representation of the 1D height-height correlation function Δh(Δz) for surface roughness profiles measured along the crack front direction z in sintered glass packing with porosities Φ=3% (o), Φ=7% (), Φ=15% (),Φ=25% () and ,Φ=26% (+).Straight lines are power-law fits with exponents ζ~0.4. Bottom, inset: 2D height-height correlation function Δh with Δz for various values of Δx. Main: The Family-Viseck data collapse was obtained using ζ=0.4, β =0.5, Z=0.8.