Two-Dimensional Mathematical Models of Visco-Elastic Deformation Using a Fractional Differentiation Apparatus

Abstract

In this paper, using fractional differential and integral operators, constructed are two-dimensional mathematical models of viscoelastic deformation, which are characterized by memory effects, spatial non-locality, and self-organization. The fractal rheological models by Maxwell, Kelvin and Voigt, their structural properties and the influence of the fractional integro-differential operator on the process of viscoelasticity are investigated. Using the Laplace transform method and taking into account the properties of the fractional differential apparatus, analytical relations are obtained in the integral form for describing the stresses of generalized twodimensional fractional-differential rheological models by Maxwell, Kelvin, and Voigt. Since the fractionaldifferential parameters of fractal models allow describing deformation-relaxation processes more perfectly than traditional methods, algorithmic aspects of identification of structural and fractal parameters of models are presented in the work. Explicit expressions have been obtained to describe the deformation process for one-dimensional fractionaldifferential models by Voigt, Kelvin, and Maxwell. The results of identification of structural and fractal parameters of the Maxwell and Voigt models are presented. The estimates of the accuracy of the obtained identification results were found using the statistical criterion based on the correlation coefficient. The influence of fractional-differential parameters on deformation-relaxation processes is investigated.