Mathematical modeling of physical processes in composition media

Abstract

A continuous medium, which is quite common in the industrial sphere, consisting of a set of layers (phases), i.e. a layered unidirectional composite medium (composites), and physical processes in the layers, i.e. transfer processes, wave processes, and changes in the stress-strain state of this this medium, is considered. A mathematical description of the structure of the compositional medium in terms of the layered domain is realized, a Sobolev space of functions with a carrier in the layered domain (together with auxiliary spaces) is constructed to describe the quantitative characteristics of the layers, and the weak solvability of the corresponding boundary value problems is established. At the same time, in the places of mutual adjacency of the layers, the conditions describing the regularities of the transfer process and the wave process, as well as changes in the stress-strain state and the displacement of layer points were determined. The work consists of three parts. The first part contains a description of the structure of the compositional medium, the basic concepts, and a description of the classical spaces of functions with a carrier in the layered domain. The second part is devoted to the construction of auxiliary spaces for the mathematical description of boundary problems of the processes of transfer and wave process, and to obtaining sufficient conditions for their solvability. The third part contains a description of the elastic properties of the composite medium, the problem of stress-strain is formulated for which the space of admissible solutions satisfying the relations describing the laws of displacement of points at the junctions of layers is constructed, the conditions of weak solvability of the specified problem are established. The results of the work are used in the analysis of problems of optimization of physical processes and phenomena in composite materials.