The crack surface anomalous scaling and its connection with the size-scale effects

The size-scale effects is one of the most important research topics in solid mechanics. Several theories have been proposed in order to describe the scaling of mechanical properties in fracture mechanics of quasi-brittle materials such as concrete, rock, wood and a broad class of fibrous or particulate composites. In the last two decades they were investigated by means of several techniques, including renormalisation group theory, intermediate asymptotics, dimensional analysis, statistics of extremes among the others. One of the most successful approaches is the fractal one. It is based on the assumption of a fractal-like damage localization at the mesostructural level and on the linking of mechanical properties to the fractal dimensions of the damage domains. In particular, the fractal dimension of fracture surface can be linked to the scaling properties of toughness. On the other side, recent experimental researches have shown that fracture surfaces present an anisotropic propagation in the longitudinal and transverse directions. To describe such anisotropy, it does not appear sufficient to characterize the fracture surface by a single fractal dimension, but the anomalous scaling (Morel et al., Physical Review E 58, 6999-7005 [1998]) should be introduced. This approach has proved to be very effective in describing the R-curve behaviour (Morel et al., International Journal of Fracture 114, 307-325 [2002]). Dealing with the size-scaling effects, a scaling law for both fracture toughness and tensile strength has been recently proposed. In this work, we point out some inconsistencies of the proposed approach, suggesting a more consistent way to derive the scaling laws and a correction on the scaling exponent at the larger scales. The phenomenon of scaling in notched and un-notched structures is summarized in a unified framework and the anomalous scaling is applied to the case of unnotched specimens, showing how it captures correctly only the convexity of the scaling law in a bilogarithmic plane and not the real asymptotes, thus indicating that the anomalous scaling can not be considered as a satisfactory explanation to the size-scale effects.