Correlation between the scale-dependent fractal dimension of fracture surfaces and the fracture toughness

Usually, a negative correlation between the fractal dimension and the fracture toughness of fracture surfaces is obtained experimentally with the slit-island method (SIM). We studied the application of the perimeter-area relation (the basis of the SLM) theoretically and with computer simulations. The conditions under which the fractal dimension obtained by SIM depends on a standard length are found, and we studied the dependence of the fractal dimension on that length. A technique is presented to estimate the fractal dimension of fracture surfaces more accurately. We predict self-similarity, in that the substructure of the fractal has the same dimension as the fractal. In a unified framework, the negative correlation found in experiments can be explained systematically. It is emphasized that before a correlation can be claimed to exist, it is essential to examine the real fractal dimension of fracture surfaces. Our investigation also uncovers additional characteristics of fracture surfaces that can be described by a parameter.