Research and Mathematical Modeling of Fractional-Differential Rheological Models

Abstract

Deformation processes in media with fractal structure have been studied. At present, research on the construction of mathematical methods and models of interconnected deformation-relaxation and heat-mass transfer processes in environments with a fractal structure is at an early stage. There are a number of unsolved problems, in particular, the problem of correct and physically meaningful setting of initial and boundary conditions for nonlocal mathematical models of nonequilibrium processes in environments with fractal structure remains unsolved. To develop adequate mathematical models of heat and mass transfer and viscoelastic deformation in environments with fractal structure, which are characterized by the effects of memory, self-organization and spatial nonlocality, deterministic chaos and variability of rheological properties of the material, it is necessary to use non-traditional approaches. -differential operators. The presence of a fractional derivative in differential equations over time characterizes the effects of memory (eridity) or non-marking of modeling processes. The implementation of mathematical models can be carried out by both analytical and numerical methods. In particular, in this paper the integral form of fractional-differential rheological models is obtained on the basis of using the properties of the non-integer integral-differentiation operator and the Laplace transform method. The obtained analytical solutions of mathematical models of deformation in viscoelastic fractal media made it possible to obtain thermodynamic functions, creep nuclei and fractal-type relaxation. Developed software to study the effect of fractional differentiation parameters on the rheological properties of viscoelastic media.