Formation Of Self-similar Structures Research Articles

Features of Spatial-Temporal Hierarchical Structures Formation

The degree of ordering of the structure of technologically important materials formed as a result of the evolution of complex physicochemical systems determines their physical properties, in particular optical. In this regard, the primary task for the theoretical study of methods for obtaining materials with predetermined physical properties is to develop approaches to describe the evolution of fractal (scale-invariant) objects in the formation of self-similar structures in systems exhibiting chaotic behavior. The paper forms an idea of the processes of evolution in materials formed as a result of stochastic processes. It is established that the conduct of ultrametrics in time space allows to characterize the time of the evolutionary process of fractal dimension, which is calculated either theoretically or model. The description of evolutionary processes in a condensed medium, accompanied by topological transformations, is significantly supplemented by the method of describing the stages of evolution of structures, which makes it possible to analyze a wide range of materials and can control their properties, primarily optical. It is shown that the most large-scale invariant structures, due to the investigated properties, can be used as information carriers. It is demonstrated that the presence in physical systems of fractal temporal dimension and generates a self-similar (consisting of parts in a sense similar to the whole object) evolutionary tree, which, in turn, generates spatial objects of non-integer dimension, observed in real situations. On the other hand, temporal fractality provides analysis of systems with dynamic chaos, leading to universal relaxation functions. In particular, in systems with a large-scale invariant distribution of relaxation characteristics, an algebraic law of relaxation is manifested, which leads to rheological models and equations of states, which are characterized by fractional derivatives. It is argued that the fractal dimension of time hierarchies stores information that determines the process of self-organization. Developed in the paper ideas about the processes of building the structure of materials, which lead to the fractal geometry of objects, can be used to predict their properties, in particular, optical.

Grinfeld instability as a mechanism of the formation of self-similar structures on aluminum single-crystal foils under cyclic tension

Based on the analysis of our results and data available in the literature, it has been shown that the formation of self-similar tweed structures on (100)[001] aluminum single-crystal foils during constrained cyclic tension occurs under conditions of the Grinfeld instability. This is confirmed by the good agreement between the theoretical estimates obtained for the period of tweed structures in terms of the Grinfeld instability model and the experimentally measured values. It has been demonstrated that the Grinfeld instability manifests itself in different boundary conditions associated with the specific features of the elasto-plastic deformation of a two-layer aluminum foil-specimen system, which is responsible for the self-similarity of tweed structures. It has been assumed that the material redistribution on the surface of foils is due to the migration of point defects that are formed during cyclic tension and exhibit a sufficient mobility at room temperature.

Formation of self-similar structures on{100}〈00l〉 aluminum single-crystal foils under cyclic tension

We study the relief formed on {100}〈001〉 aluminum single-crystal foils attached to thick aluminum alloy specimens under low-cycle fatigue deformation. The formation of specific surface structures on the aluminum foils is shown to relate to the “foil - specimen” and “aluminum foil - its surface layer” interfaces. Particularly, the formation of longitudinal macroscopic bands on the aluminum foil is explained by periodic distribution of compressive stresses arising at the “foil - specimen” interface. At the interface between the macroscopic bands and tweed structure with spacing ∼2.8 μm a fine tweed substructure of the submicron range is found. The periodic 2D lattices of various scale formed on the foils are self-similar in the linear size range from fractions to hundreds of microns.

Discussion

In the section of the paper titled “The Formation of Self-Similar Structures,” the authors describe the essential aspects of Constructal Theory. However, Professor Adrian Bejan from Duke University has been working for a long time in putting together the ideas and the formalism of this theory—he actually gave it the appropriate name of “Constructal” 1, and has analyzed many examples in nature and engineering within the framework of this theory, extending it to a variety of situations, with his original contributions, most of them nicely described in his book 2. After having reviewed the book written by Professor Bejan 3 and having closely followed his publications during the last decade, I found that the core analysis and synthesis methods of Constructal Theory are totally based on the author’s own research experience during many years, previously published in specialized journals and books, and supported by many observed facts and results, fully documented in the open literature. The Constructal Theory, established by Bejan, explains how certain basic elements, individually and collectively optimized to form an arrangement, are employed to construct more complex natural systems, within the specific constrains imposed by the physics in every case.The paper by Chai and Shoji is full of ideas and expressions of Constructal Theory that were first contributed and published by Professor Bejan, especially the discussion about academic and industrial implications, in the “Concluding Remarks.”

Self-oscillatory modes of fatigue fracture and the formation of self-similar structures at the fracture surface

The jump-like extension of the fatigue crack as described by P. J. E. Forsyth is considered. It is shown that such a peculiar extension regime of the fatigue crack is connected with two features: the presence of fatigue kinetic diagram discontinuity and the appearance of the self-similar intermediate asymptotic crack propagation regime with scaling properties. A model of self-similar jump-like extension of fatigue crack and the critical parameters which control such crack extension are presented.