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The use of linear fractional analogues of rheological models in the problem of approximating the experimental data on the stretch polyvinylchloride elastron. (Russian. English summary) Zbl 1413.74033

Summary: We consider and analyze uniaxial phenomenological models of viscoelastic deformation based on fractional analogues of Scott Blair, Voigt, Maxwell, Kelvin and Zener rheological models. Analytical solutions of the corresponding differential equations are obtained with fractional Riemann-Liouville operators under constant stress with further unloading, that are described by a generalized (two-parameter) fractional exponential function and contains from two to four parameters depending on the type of model. A method for identifying the model parameters based on the background information for the experimental creep curves with constant stresses is developed. The nonlinear problem of parametric identification is solved by a two-step iterative method. The first stage uses the characteristic data points diagrams, and features in the behavior of the models under unrestricted growth of time and an initial approximation of the parameters are determined. At the second stage, the refinement of these parameters by coordinate descent (Hooke-Jeeves’s method) and minimizing the functional standard deviation for calculated and experimental values is made. The method of identification is realized for all the considered models on the basis of the known experimental data for uniaxial viscoelastic deformation of polyvinylchloride elastron at a temperature of 20\(^\circ\)C. The table-valued parameters for all models are given. The errors analysis of the constructed phenomenological models is made for the experimental data over the entire ensemble of curves viscoelastic deformation. It is found that the approximation errors for the Scott Blair fractional model is 14.17%, for the Voigt fractional model is 11.13%, for the Maxvell fractional model is 13.02%, for the Kelvin fractional model 10.56%, for the Zener fractional model is 11.06%. The graphs of the calculated and experimental dependences of viscoelastic deformation of polyvinylchloride elastron are compared.

MSC:

74D10 Nonlinear constitutive equations for materials with memory
26A33 Fractional derivatives and integrals
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