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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2016, Volume 20, Number 4, Pages 691–706 (Mi vsgtu1523)

Mechanics of Solids

The use of linear fractional analogues of rheological models in the problem of approximating the experimental data on the stretch polyvinylchloride elastron

L. G. Ungarova

Samara State Technical University, Samara, 443100, Russian Federation

Abstract: We considere and analyze the 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 written by the 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 was developed. Nonlinear problem of parametric identification is solved by 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 the initial approximation of parameters are determined. At the second stage, the refinement of these parameters by coordinate descent (the Hooke–Jeeves's method) and minimizing the functional standard deviation for calculated and experimental values is made. Method of identification is realized for all the considered models on the basis of the known experimental data uniaxial viscoelastic deformation of Polyvinylchloride Elastron at a temperature of 20$^\circ$C and five the tensile stress levels. Table-valued parameters for all models are given. The errors analysis of constructed phenomenological models is made to experimental data over the entire ensemble of curves viscoelastic deformation. It was 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 submitted.

Keywords: fractional rheological models, parameter identification, Riemann–Liouville operator, experimental data, polyvinylchloride elastron, approximation error.

DOI: https://doi.org/10.14498/vsgtu1523

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Bibliographic databases:

UDC: 539.313:517.968.72
MSC: 74D10, 26A33
Original article submitted 18/XI/2016
revision submitted – 04/XII/2016

Citation: L. G. Ungarova, “The use of linear fractional analogues of rheological models in the problem of approximating the experimental data on the stretch polyvinylchloride elastron”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:4 (2016), 691–706

Citation in format AMSBIB
\Bibitem{Ung16} \by L.~G.~Ungarova \paper The use of linear fractional analogues of rheological models in the problem of~approximating the experimental data on~the stretch polyvinylchloride elastron \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2016 \vol 20 \issue 4 \pages 691--706 \mathnet{http://mi.mathnet.ru/vsgtu1523} \crossref{https://doi.org/10.14498/vsgtu1523} \zmath{https://zbmath.org/?q=an:06964664} \elib{https://elibrary.ru/item.asp?id=28862963} 

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This publication is cited in the following articles:
1. E. N. Ogorodnikov, V. P. Radchenko, L. G. Ungarova, “Mathematical models of nonlinear viscoelasticity with operators of fractional integro-differentiation”, PNRPU Mechanics Bulletin, 2018, no. 2, 147–161 (In Russian)
2. D. V. Ivanov, “Identification of linear dynamic systems of fractional order with errors in variables based on an augmented system of equations”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 25:3 (2021), 508–518
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