Parametric homogeneity and non-classical self-similarity. I: Mathematical background. II: Some applications
The concept of parametric homogeneity, which includes parametric homogeneous (PH) and parametric-quasi-homogeneous (PQH) functions, PH-sets and corresponding transformations, is a kind of non-classical self-similarity. Introduction and study of the concept is important for understanding scaling properties of natural phenomena. PH-functions and PQH-functions are natural generalizations of concepts of homogeneous and quasi-homogeneous functions when the discrete group of coordinate dilations is considered instead of the continuous group. Here, we study some properties of PH- and PQH-functions and some of their applications. Several ways of constructing PH-function of arbitrary degree are described. It is shown that PH-functions often have fractal graphs and can be nowhere differentiable. Such common for fractal geometry objects as the von Koch curve, the Weierstraß-Mandelbrot and Takagi-Hopsen functions, the Cantor staircase and set possess PH-properties. However, the fractal properties of an object and the property of parametric-homogeneity are independent of each other, and the considered scaling law (PH-law) is not a kind of fractal scaling. A nowhere differentiable self-similar function and some associated PH-functions are introduced as examples of non-fractal scaling. In part II, we present some applications of PH-functions to nonlinear problems of solid mechanics. First, 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. It is shown that log-periodic functions can be useful in the description of the experimental data concerning seismic activation and can be used in earthquake predictions. 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.
contact between punch and deformable half-space
scientific article; zbMATH DE number 1270090
scientific article