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Zhuravkov M.A., Doctor of Physical and Mathematical Sciences, Professor, Vice-Rector, Head of the Department "Theoretical and applied mechanics", Belarusian State University, Minsk, Republic of Belarus

Pleskachevsky Yu.M., Corresponding Member of the NAS of Belarus, Doctor of Technical Sciences, Professor, Chairman of the Presidium of the Gomel branch of the NAS of Belarus, Gomel, Republic of Belarus, This email address is being protected from spambots. You need JavaScript enabled to view it.

Romanova N.S., Head of the Department of Youth Programs and Projects, Competitor of the Department of Theoretical and Applied Mechanics, Mechanical and Mathematical Faculty of the Belarusian State University, Minsk, Republic of Belarus

Year 2012 Issue 3-4 Pages 158-165
Type of article RAR Index UDK 539.2/.6+612.76+519.68:[5/6+3] Index BBK  

Classical development and the formation of some new modern branches in mechanics require continuous improvement and modification of mathematical models and methods to solve the model problems. In this article we consider some developments in mechanics and mathematical models for the description of the mechanical state and different material behaviour and perspectives of fractional calculus for the use in mechanics, and describe the conjugate biophysical and biomechanical problems, the mathematical models of material deformation taking into account their complex heterogeneous structure.


mechanical and mathematical models of fractional order, generalized model of Maxwell, Kelvin-Voigt, Zener, models of viscoelasticity, heterogeneous environments

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  • Samko S.G. Kilbas A.A., Marichev O.I. Integraly i proizvodnye drobnogo porjadka i nekotorye prilozhenija [Integrals and derivatives of fractional order and some applications]. Minsk, Nauka i tehnika, 1987. 687 p.
  • Podlubny I. Fractional differential equations. Mathematics in Sciences and Engineering. San Diego, 1999. 198 p.
  • Bagley R.L., Torvik P.J. On the fractional calculus model of viscoelastic behavior. Journal of Rheology, 1986, vol. 30, pp. 133-155.
  • Caputo M. Vibrations of an infinite plate with a frequency independent Q. Journal of the Acoustical Society of America, 1976, vol. 60(3), pp. 634-639.
  • Caputo M., Mainardi F. A new dissipation model based on memory mechanism. Pure and Applied Geophysics, 1971, vol. 91(1), pp. 134-147.
  • Rouse P.E. The theory of the linear viscoelastic properties of dilute solutions of coiling polymers. Journal of Chemical Physics, 1953, vol. 21, pp. 1272-1280.
  • Compte A. Stochastic foundations of fractional dynamics. Phys. Rev.E., 1996, vol. 53(4), pp. 4191-4193.
  • Zhuravkov M.A., Starovojtov Je.I. Mehanika sploshnyh sred. Teorija uprugosti i plastichnosti [Continuum Mechanics. The theory of elasticity and plasticity]. Minsk, BGU, 2011. 543 p.
  • Koeller R.C. Applications of fractional calculus to the theory of viscoelasticity. Journal of Applied Mechanics, 1984, vol. 51(2), pp. 299-307.
  • Friedrich C. Relaxation and retardation functions of the Maxwell model with fractional derivatives. Rheologica Acta, 1991, vol. 30, pp. 151-158.
  • Nonnenmacher T.F., Glockle W.G. A fractional model for mechanical stress relaxation. Phil. Mag. Lett., 1991, vol. 64(2), pp. 89-93.
  • Heymans N., Bauwens J.-C. Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheologica Acta., 1994, vol. 33(3), pp. 210-219.
  • Padovan J., Sawicki J.T. Diophantinized fractional representations for nonlinear elastomeric media. Computers and Structures, 1998, vol. 66, pp. 613-626.
  • Schiessel H. [et al.]. Generalized viscoelastic models: Their fractional equations with solutions. Journal of Physics A: Mathematical and General, 1995, vol. 28, pp. 6567-6584.
  • Hwang J.S., Wang J.C. Seismic response prediction of high damping rubber bearings fractional derivatives Maxwell model. Engineering Structures, 1998, vol. 20(9), pp. 849-856.
  • Park S.W. Analytical modeling of viscoelastic dampers for structural and vibration control. Int.J. Solids Struct., 2001, vol. 38(44-45), pp. 8065-8092.
  • Makris N., Constantinou M.C. Fractional derivative Maxwell model for viscous dampers. Journal of Structural Engineering ASCE, 1991, vol. 117(9), pp.2708-2724.
  • Rabotnov Ju.N. Polzuchest' jelementov konstrukcij [Creep of structural elements]. Moscow, Nauka, 1966. 753 p.
  • Meshkov S.I., Pachevskaja G.N., Shermergor T.D. K opisaniju vnutrennego trenija pri pomoshhi drobno-jeksponencial'nyh jader [The description of internal friction using fractional exponential kernels]. PMTF, 1966, no. 3.
  • Zelenev V.M., Meshkov S.I., Rossihin Ju.A. Zatuhajushhie kolebanija uprugo nasledstvennyh sistem so slabosinguljarnymi jadrami [Damped oscillations of elastic hereditary systems with weakly singular kernels]. PMTF, 1970, no. 2, pp. 104-108.
  • Sasso M., Palmieri G., Amodio D. Application of fractional derivative models in linear viscoelastic problems. Mech. Time-Depend Mater, 2011, vol. 15, pp. 367-387.
  • Can Meral F., Thomas J. Royston Surface response of a fractional order viscoelastic halfspace to surface and subsurface sources. J. Acoust. Soc. Am., vol. 126(6), 2009, pp. 3278-3285.
  • Warlus S., Ponton A., Leslous A. Dynamic viscoelastic properties of silica alkoxide during the sol-gel transition. Eur. Phys. Journal E., 2003, vol. 12, pp. 275-282.
  • Dietrich L., Turski K. Analysis of identification methods for the viscoelastic properties of materials. Eng. Trans., 1992, vol. 40, pp. 501-523.
  • West B.J., Deering W. Fractal physiology for physicists: Levy statistics. Phys. Rep., 1994, vol. 246, pp. 1-100.
  • West B.J. Fractal probability density and EEF/ERP time series (Chapter 10). In Fractal Geometry in Biological Systems. Boca Raton, CRC, 1995, pp. 267-316.
  • West B.J., Bologna M., Grigolini P. Physics of Fractal Operators. New York, Springer, 2003.
  • Fung Y.C. Biomechanics: Mechanical properties of living tissues. Springer-Verlag, New York, 1981.
  • Iomin A. Fractional transport of cancer cells due to self-entrapment by fission. Mathemat. Modeling of biological systems. Modeling and simulation in science, engineering and technology, 2007, part IV, pp. 193-203.
  • Palocaren A., Drapaca C. Biomechanical modeling of tumor growth: its relevance to glioma research. Intern. Journ. Of Num. Analysis and Modeling, Series B, 2012, vol. 3(1). - pp. 94-108.
  • Metzler R., Klafter J. The Random walker's guide to anomalous diffusion: a fractional dynamics approach. Phys. Reports, 2000, vol. 339, pp. 1-77.
  • Weiss M., Hashimoto H., Nilsson T. Anomalous protein diffusion in living cells as seen by fluorescence correlation spectroscopy. Biophys. Journ., 2002, vol. 84, pp. 4043-4052.
  • Craiem D.O. [et al.]. Fractional calculus applied to model arterial viscoelasticity. Latin American Applied Research, 2008, vol. 38, pp. 141-145.
  • Djordjevic V.D. [et al.]. Fractional derivatives embody essential features of cell rheological behavior. Ann. Biomed. Eng., 2003, vol. 31, pp. 692-699.
  • Kiss M.Z., Varghese T., Hall T.J. Viscoelastic characterization of in vitro canine tissue. Phys. Med. Biol., 2004, vol. 49, pp. 4207-4218.
  • Hardung, V. Method for measurement of dynamic elasticity and viscosity of caoutchouc-like bodies, especially of blood vessels and other elastic tissues. Helv. Physiol. Pharmacol. Acta., 1952, vol. 10, pp. 482-498.
  • Pleskachevskij Ju.M., Shil'ko S.V. Auksetiki: modeli i prilozhenija [Auxetics: models and applications]. Vesti NAN Belarusi [News of the NAS of Belarus], 2003, no. 4, pp. 58-68.
  • Zhuravkov M.A., Prohorov P.A. Deformirovanie blochno-sloistyh massivov gornyh porod v okrestnosti podzemnyh sooruzhenij [Deformation of block-layered rock mass in the vicinity of underground structures]. Gornaja mehanika [Rock mechanics], 2008, no. 2, pp. 3-13.
  • Zhuravkov M.A., Makaeva T.A. Mehaniko-matematicheskie modeli povedenija deformiruemyh tverdyh uprugih sred s uchetom ih vnutrennej struktury [Mechanics and mathematical models for behavior of the deformable solid and elastic mediums with regard to their internal structure]. Mehanika mashin, mehanizmov i materialov [Mechanics of machines, mechanisms and materials], 2012, no. 1, pp. 29-38.