PARAMETRIC HOMOGENEITY AND NONCLASSICAL SELF-SIMILARITY - II - SOME APPLICATIONS

Some problems of mechanics are considered from the standpoint of the p arametric homogeneity concept. The mathematical background of the conc ept was studied in the first part of the paper. First, some applicatio ns of PH-functions to nonlinear problems of solid mechanics are consid ered, namely the contact between a punch, whose shape is described by a positive PH-function, and a deformable half-space is considered usin g a similarity approach. Then, the popular concept of log-periodicity (complex exponent) is considered as a particular case of parametric ho mogeneity. The cases when the concept is useful in describing non-smoo th self-similar phenomena are described. It is shown that PH-functions and in particular log-periodic functions can be useful in the descrip tion of the experimental data concerning seismic activation and can be used in earthquake predictions. Some natural phenomena and their mode ls which have PH-features are also considered, and some examples of th e appearance of PH-functions in solutions of some differential equatio ns are given. The fractal and parametric homogeneous descriptions of p henomena are also discussed. Finally, a self-similar problem of multip le fracture is studied, namely a discrete self-similar problem of stic k-slip crack propagation when a main crack is surrounded by defects an d its extension is discontinuous consisting of a sequence of finite gr owth steps.