МАТЕМАТИЧЕСКИЕ МОДЕЛИ НЕЛИНЕЙНОЙ ВЯЗКОУПРУГОСТИ С ОПЕРАТОРАМИ ДРОБНОГО ИНТЕГРО-ДИФФЕРЕНЦИРОВАНИЯ

Abstract

Based on the method of structural modelling and the Boltzmann-Volterra hypothesis of the hereditary elasticity deformable solid, the article considers linear and nonlinear fractional analogues of classical rheological models, such as Newton (the so-called model of Scott Blair), Voigt, Maxwell, Kelvin and Zener using the tool of fractional integro-differentiation of Riemann Liouville. The classes of nonlinear mathematical models are distinguished, for which the solution of the creep problem can be obtained explicitly in terms of known special functions. A technique for identifying the parameters of the proposed mathematical models is developed on the basis of known experimental data on uniaxial stretching of samples at different and constant load levels. In the presence of explicit solutions to the creep problem, the parameters of mathematical models are determined from the solution of the problem of approximating the experimental values of deformation using the method of least squares with subsequent refinement by the coordinate-wise descent method at all time points for all stress values in a series of experiments. For nonlinear mathematical models of viscoelastic deformation, which do not allow to find the solution of the creep problem in an explicit form, a method for determining the model parameters based on the coordinate-wise descent method with inversion at each step to the numerical solution of the defining integral equation has been developed. The method of identifying model parameters with operators of fractional integro-differentiation is realized on the example of creep of polyvinylchloride plastic compound. The values of the parameters for all the models studied are given, their adequacy to the experimental data is checked, and errors in the deviation of the calculated data from the experimental values are analyzed. As an example, a comparative analysis of the relative error in approximating the experimental creep curves by theoretical deformation values is made in the framework of a linear, nonlinear integrable and nonlinear nonintegrable fractional analog of the Kelvin model. The article oulines the appropriateness of using the viscoelastic deformation models with the operators of fractional integro-differentiation, based on the comparison of calculations of the considered models with the calculations of the viscoelasticity models having integral operators of integro-differentiation.