Mathematical Modeling of Nonequilibrium Physical Processes, Taking into Account the Memory Effects and Spatial Correlation

Abstract

Modern heterogeneous composite materials are characterized by fractal structure. The fractality of medium typically reflects self-similarity of the heterogeneous structure and reorganization of the structural elements of the material. The use of a fractional integro-differential apparatus makes it possible to take into account these properties of material as well as the effects of memory, the deterministic chaos, the complex nature of spatial correlations, the variability of rheological properties and self-organization. This paper investigates and develops mathematical models of visco-elastic deformation processes in media with fractal structure. The rheological models have been obtained in an integral form, and identified were the creep and relaxation kernels and thermodynamic characteristics, which enables to determine the elastic and residual stresses of the material. Adapted was the method of disintegration of two-dimensional fractional-differential creep kernels for rheological models, as well as an algorithm for identifying fractal parameters of creep functions was developed. The dynamics of changes in temperature, moisture content and components of the stress-strain state of capillary-porous materials with consideration for the fractal structure is analyzed, which allows for taking into account the effects of memory, self-organization, residual deformation depending on the type of material and the direction of anisotropy.