Parametric homogeneity and non-classical self-similarity. II. Some applications

Abstract

Some problems of mechanics are considered from the standpoint of the parametric homogeneity concept. The mathematical background of the concept was studied in the first part of the paper. First, some applications of PH-functions to nonlinear problems of solid mechanics are considered, namely the contact between a punch, whose shape is described by a positive PH-function, and a deformable half-space is considered using a similarity approach. Then, the popular concept of log-periodicity (complex exponent) is considered as a particular case of parametric homogeneity. The cases when the concept is useful in describing non-smooth self-similar phenomena are described. It is shown that PH-functions and in particular log-periodic functions can be useful in the description of the experimental data concerning seismic activation and can be used in earthquake predictions. Some natural phenomena and their models which have PH-features are also considered, and some examples of the appearance of PH-functions in solutions of some differential equations are given. The fractal and parametric homogeneous descriptions of phenomena are also discussed. Finally, a self-similar problem of multiple fracture is studied, namely a discrete self-similar problem of stick-slip crack propagation when a main crack is surrounded by defects and its extension is discontinuous consisting of a sequence of finite growth steps.