Fractal structure and fractal dimension determination at nanometer scale

Three-dimensional fractures of different fractal dimensions have been constructed with successive random addition algorithm, the applicability of various dimension determination methods at nanometer scale has been studied. As to the metallic fractures, owing to the limited number of slit islands in a slit plane or limited datum number at nanometer scale, it is difficult to use the area-perimeter method or power spectrum method to determine the fractal dimension. Simulation indicates that box-counting method can be used to determine the fractal dimension at nanometer scale. The dimensions of fractures of valve steel 5Cr21Mn9Ni4N have been determined with STM. Results confirmed that fractal dimension varies with direction at nanometer scale. Our study revealed that, as to theoretical profiles, the dependence of frsctal dimension with direction is simply owing to the limited data set number, i.e. the effect of boundaries. However, the dependence of fractal dimension with direction at nanometer scale in real metallic fractures is correlated to the intrinsic characteristics of the materials in addition to the effect of boundaries. The relationship of fractal dimensions with the mechanical properties of materials at macrometer scale also exists at nanometer scale. Project supported by the National Natural Science Foundation of China (Grant Nos. 59771050 and 59872004) and the Foundation Fund of Ministry of Metallurgical Industry.