International Journal of Modern Education and Computer Science (IJMECS)

ISSN: 2075-0161 (Print), ISSN: 2075-017X (Online)

Published By: MECS Press

IJMECS Vol.10, No.4, Apr. 2018

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

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Yaroslav Sokolovskyy, Maryana Levkovych

Index Terms

A mathematical model;derivatives of fractional order;rheological models; deformation processes


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 two-dimensional fractional-differential rheological models by Maxwell, Kelvin, and Voigt. Since the fractional-differential 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 fractional-differential 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.

Cite This Paper

Yaroslav Sokolovskyy, Maryana Levkovych, " Two-Dimensional Mathematical Models of Visco-Elastic Deformation Using a Fractional Differentiation Apparatus", International Journal of Modern Education and Computer Science(IJMECS), Vol.10, No.4, pp. 1-9, 2018.DOI: 10.5815/ijmecs.2018.04.01


[1]Machado, J.A.T., Galhano, A.M.S.F., “Fractional order inductive phenomena based on the skin effect” Non-linearly. Dyn. 68(1–2), pp. 107–115, 2012.

[2]Sheng, H., Chen, Y.Q., Qiu, T.S., “Fractional Processes and Fractional-Order Signal Processing”, Springer, New York, 2012.

[3]Mohamed A. E. Herzallah, Ahmed M. A. El-Sayed, Dumitru Baleanu, “On the fractional-order diffusion-wave process”, Rom. Journ. Phys., vol. 55, Nos. 3-4, pp. 274-284, 2010.

[4]F. A. Rihan, D. H. Abdel Rahman, S. Lakshmanan, and A. Alkhajeh, “A time delay model of tumour—immune system interactions: global dynamics, parameter estimation, sensitivity analysis,” Applied Mathematics and Computation, vol. 232, pp. 606–623, 2014.

[5]Y. Ferdri, “ Some applications of fractional order calculus to design digital filters for biomedical signal processing’, J. Mech. Med. Biol. 12(2), p. 13, 2012.

[6]Ammar Soukkou, Salah Leulmi,"Controlling and Synchronizing of Fractional-Order Chaotic Systems via Simple and Optimal Fractional-Order Feedback Controller", International Journal of Intelligent Systems and Applications(IJISA), Vol.8, No.6, pp.56-69, 2016. DOI: 10.5815/ijisa.2016.06.07

[7]Asim Kumar Das, Tapan Kumar Roy,"Fractional Order EOQ Model with Linear Trend of Time-Dependent Demand", IJISA, vol.7, no.3, pp.44-53, 2015. DOI: 10.5815/ijisa.2015.03.06

[8]Ammar Soukkou, M. C. Belhour, Salah Leulmi, "Review, Design, Optimization and Stability Analysis of Fractional-Order PID Controller", International Journal of Intelligent Systems and Applications(IJISA), Vol.8, No.7, pp.73-96, 2016. DOI: 10.5815/ijisa.2016.07.08

[9]O. Tolga Altinoz, A. Egemen Yilmaz, "Optimal PID Design for Control of Active Car Suspension System", International Journal of Information Technology and Computer Science(IJITCS), Vol.10, No.1, pp.16-23, 2018. DOI: 10.5815/ijitcs.2018.01.02

[10]V. Uchajkin, “Method of fractional derivatives”, Ulyanovsk: Publishing house «Artishok», p. 512, 2008.

[11]A. M. Nahushev, “Fractional calculus and its application”,  Moscow: Fizmatlit, p. 272, 2003.

[12]J. Tenreiro Machado, V. Kiryakova, F. Mainardi, “Recent history of fractional calculus”, Commun Nonlinear Science and  Numer Simulat., vol. 16.,  pp. 1140-1153, 2011.

[13]R. R. Nigmatullin, “Fractional integral and its physical interpretation”, Theoretical and Mathematical Physics, T. 90, No. 3., 1992, pp. 354-368.

[14]I. Podlubny, “ Fractional Differential Equations”, of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, vol. 198, p.340, 1999.

[15]S. V. Erokhin, T. S. Aleoev, L. Yu. Frishter, A. V. Kolesnichenko, “Parametric identification of the mathematical model of viscoelastic materials using fractional derivatives”, International Journal of Computational Civil and Structural Engineering, vol. 11, Issue 3,  pp. 82-85, 2015.

[16]S. V. Erokhin, “Models of creep and relaxation of materials using fractional derivatives”, Construction mechanics of engineering structures and structures, No. 6 Moscow, pp. 35-39, 2014.

[17]M. Javidi, B. Ahmad, “Advances in Difference Equations”, p. 375, 2013.

[18]N. W. Tschoegl, “The Phenomenological Theory of Linear Viscoelastic Behavior”, Berlin, Springer, 1989.

[19]E. N. Ogorodnikov, V. P. Radchenko, N. S. Yashagin, “Rheological models of a viscoelastic body with memory and differential equations of fractional oscillators”, Vestn. Himself. state tech un-that Sir Fiz. Mat. science, No. 1 (22), pp. 255-268, 2011.

[20]V. D. Beibalayev, “Mathematical model of heat transfer in mediums with fractal structure”, Mathematical modeling, vol. 21, No. 5, pp.55-62, 2009.

[21]V. D. Beybalaev, M. R. Shabanova, “Numerical method for solving the boundary value problem for a two-dimensional heat equation with derivatives of fractional order”, Vestn. Himself. state tech un-that Sir Fiz. Mat. Science, №5 (21), pp. 244-251, 2010.

[22]R. P. Meilanov, M. R. Shabanova, “The equation of thermal conductivity for media with fractal structure”, Modern science-intensive technologies, No. 8, pp. 84-85, 2007. 

[23]R. P.Meilanov, M. R. Shabanova, “Features the solution of the heat transfer transport equation in derivatives of fractional order”, Journal of Technical Physics, Vol. 8, Issue 7, pp. 1-6, 2011.

[24]A. K. Basayev, “Local-one-dimensional scheme for the equation of thermal conductivity with boundary conditions of the third kind”, Vladikavkaz Mathematical Journal, Vol. 13, Issue 1, pp. 3-12, 2011.

[25]Ya. Sokolowskyy and V. Shymanskyi, “Mathematical modeling of non-isothermal moisture transfer and rheological behavior in capillary-porous materials with fractal structure during drying”, Computer and Information Science, Canadian Center for Science and Education, Vol. 7, No. 4, pp. 111-122, 2014.

[26]Е. Ogorodnikov, V. Radchenko, L. Ugarova, “Mathematical modeling of hereditary deformational elastic body on the basis of structural models and of vehicle fractional integral-differentiation Riman-Liuvil”, Vest. Sam. Gos. Techn. Un-ty. Series. Phys.-math. sciences, tom 20, number 1, pp. 167-194, 2016.

[27]L. Kexue, P. Jigen, “Laplace transform and fractional differential equations”,  Appl. Math. Lett., 24, pp.  2019-2023, 2011. 

[28]N. J. Ford, A. C. Simpson, “The numerical solution of fractional differential equations: speed versus accuracy”, Numer. Algorithms, 26(4), pp. 333-346, 2001.

[29]K. Diethelm, “An algorithm for the numerical solution of differential equations of fractional order”, Electron. Trans. Numer. Anal. 5, pp. 1-6, 1997.

[30]G. M. Savin G., “Elements of the mechanics of hereditary environments”, Rheological bodies with the simplest law of linear deformation , No.  1, p.114, 1969. 

[31]G. M. Savin, “Elements of the mechanics of hereditary environments”, Rheological bodies with the simplest law of linear deformation, No.  2, p.137, 1970. 

[32]G. M. Savin, Ya. Ya. Ruschytsky, “Elements of the mechanics of hereditary environments”, The textbook for the mechanics and mathematics faculties of universities, K. -"High School", p. 251, 1976.

[33]Ya. I. Sokolovsky, M. V. Moskvitina, “Mathematical modeling of deformation-relaxation processes using derivatives of fractional order”, Bulletin of the National University "Lviv Polytechnic", Computer Science and Information Technologies. No. 826. - Lviv: NU "LP", pp. 175-184, 2015.

[34]V. V. Vasilyev, L. A. Simak, “Fractional calculus and approximation methods in the modeling of dynamic systems”, Scientific publication Kiev, National Academy of Sciences of Ukraine, p.256, 2008.

[35]S. G. Samko, A. A. Kilbas, O. I. Marichev, “Integrals and derivatives of fractional order and some of their’, Minsk: Science and Technology, p. 688, 1987. 

[36]Tong Liu, “Creep of wood under a large span of loads in constant and varying environments", Pt.1, Experimental observations and analysis, Holz als Roh- und Werkstoff 51,  pp. 400-405, 1993.