On identification of characteristics of inhomogeneous viscoelastic bodies within the framework of a fractional order model
DOI:
https://doi.org/10.7242/1999-6691/2024.17.2.17Keywords:
viscoelasticity, fractional differential models, inhomogeneous materials, acoustic method, inverse problems, identification, regularizationAbstract
Nowadays, the development of models of viscoelastic materials with complex inhomogeneous structure is one of the hottest problems of continuum mechanics. Along with classical models, fractional-differential models of viscoelasticity have become increasingly popular. In this paper, we present a model for describing steady-state oscillations of inhomogeneous viscoelastic bodies in terms of fractional-order differential operators. Taking into account the fractional order of the model operators, the corresponding form of the complex module describing the material properties is constructed. The module includes four characteristics: instantaneous and long-term elastic moduli (in the case of material inhomogeneity, they are the functions of coordinates), relaxation time and fractionality parameter. The properties of the complex module are studied, and the ranges of the model parameters, at which the rheological properties are most pronounced, are identified. A general formulation of the inverse problem (IP) on the identification of function-parameters of the model based on the acoustic sounding data is proposed. In the framework of this formulation, the inverse problems for specific objects , namely, an inhomogeneous rod and a round plate, are considered. In both model problems, the influence of the fractionalization parameter on the amplitude-frequency characteristics is analyzed. It has been found that in the vicinity of viscoelastic resonances the fractionality parameter affects the parameters of the oscillatory process most significantly, which is typical for such problems. A solution to the nonlinear IPs under consideration is constructed based on the linearization method, which also serve as basis for the iterative processes supplemented with the elements of the projection approach. This allows one to determine corrections to the required functions in the specified classes of functions using regularization. For both IPs, a series of computational experiments were carried out. The analysis of the experimental results made it possible to formulate recommendations for the selection of optimal sensing modes.
Downloads
References
Shitikova M.V. Fractional Operator Viscoelastic Models in Dynamic Problems of Mechanics of Solids: A Review. Mechanics of Solids. 2021. Vol. 57, no. 1. P. 1–33. DOI: 10.31857/S0572329921060118.
Kieback B., Neubrand A., Riedel H. Processing techniques for functionally graded materials. Materials Science and Engineering: A. 2003. Vol. 362. P. 81–106. DOI: 10.1016/S0921-5093(03)00578-1.
Kerber M.L., Vinogradov V.M., Golovkin G.S. Polimernyye kompozitsionnyye materialy: struktura, svoystva, tekhnologiya / ed. by A. Berlin. St. Petersburg: Professiya, 2008. 560 p.
Vatulyan A.O. Koeffitsientnye obratnye zadachi mekhaniki. Moscow: Fizmatlit, 2019. 272 p.
Rossikhin Y.A. Reflections on Two Parallel Ways in the Progress of Fractional Calculus in Mechanics of Solids. Applied Mechanics Reviews. 2010. Vol. 63, no. 1. 010701. DOI: 10.1115/1.4000246.
Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity: AnI ntroduction to Mathematical Models. London: Imperial College Press, 2010. 368 p.
Bonfanti A., Kaplan J.L., Charras G., Kabla A. Fractional viscoelastic models for power-law materials. Soft Matter. 2020. Vol. 16, no. 26. P. 6002–6020. DOI: 10.1039/d0sm00354a.
Ogorodnikov E.N., Radchenko V.P., Ungarova L.G. Mathematical models of nonlinear viscoelasticity with operators of fractional integro-differentiation. PNRPU Mechanics Bulletin. 2018. No. 2. P. 147–161. DOI: 10.15593/perm.mech/2018.2.13.
Ungarova L.G. The use of linear fractional analogues rheological models in the problem of approximating the experimental data on the stretch polyvinylchloride elastron. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences. 2016. Vol. 20, no. 4. P. 691–706. DOI: 10.14498/vsgtu1523.
Nonnenmacher T.F., Glöckle W.G. A fractional model for mechanical stress relaxation. Philosophical Magazine Letters. 1991. Vol. 64, no. 2. P. 89–93. DOI: 10.1080/09500839108214672.
Pritz T. Analysis of Four-Parameter Fractional Derivative Model of Real Solid Materials. Journal of Sound and Vibration. 1996. Vol. 195. P. 103–115. DOI: 10.1006/jsvi.1996.0406.
Pritz T. Five-parameter fractional derivative model for polymeric damping materials. Journal of Sound and Vibration. 2003. Vol. 265, no. 5. P. 935–952. DOI: 10.1016/S0022-460X(02)01530-4.
Costa M.F.P., Ribeiro C.A modified fractional Zener model to describe the behaviour of a carbon fibre reinforced polymer. AIP Conference Proceedings. 2013. Vol. 1558. P. 606–609. DOI: 10.1063/1.4825564.
Wei L.F., Li W., Feng Z.Q., Liu J.T. Applying the fractional derivative Zener model to fitting the time-dependent material viscoelasticity tested by nanoindentation. Biosurface and Biotribology. 2018. Vol. 4. P. 58–67. DOI: 10.1049/bsbt.2018.0011.
Ciniello A.P.D., Bavastri C.A., Pereira J.T. Identifying Mechanical Properties of Viscoelastic Materials in Time Domain Using the Fractional Zener Model. Latin American Journal of Solids and Structures. 2017. Vol. 14. P. 131–152. DOI: 10.1590/1679-78252814.
Pawlak Z.M., Denisiewicz A. Identification of the Fractional Zener Model Parameters for a Viscoelastic Material over a Wide Range of Frequencies and Temperatures. Materials. 2021. Vol. 14, no. 22. 7024. DOI: 10.3390/ma14227024.
Carmichael B., Babahosseini H., Mahmoodi S.N., Agah M. The fractional viscoelastic response of human breast tissue cells. Physical Biology. 2015. Vol. 12, no. 4. 046001. DOI: 10.1088/1478-3975/12/4/046001.
Dai Z., Peng Y., Mansy H.A., Sandler R.H., Royston T.J. A model of lung parenchyma stress relaxation using fractional viscoelasticity. Medical Engineering & Physics. 2015. Vol. 37. P. 752–758. DOI: 10.1016/j.medengphy.2015.05.003.
Aleroev T.S., Erokhin S.V. Parametric Identification of the Fractional-Derivative Order in the Bagley–Torvik Model. Mathematical Models and Computer Simulations. 2019. Vol. 11. P. 219–225. DOI: 10.1134/S2070048219020030.
Vatulyan A.O., Yavruyan O.V., Bogachev I.V. Reconstruction of inhomogeneous properties of orthotropic viscoelastic layer. International Journal of Solids and Structures. 2014. Vol. 51, no. 11/12. P. 2238–2243. DOI: 10.1016/j.ijsolstr.2014.02.032.
Anikina T.A., Bogachev I.V., Vatulyan A.O., Dudarev V.V. Identification of properties of the inhomogeneous viscoelastic circular plate. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation. 2016. No. 2. P. 10–18.
Bogachev I.V., Vatulyan A.O., Dudarev V.V., Nedin R.D. The Investigation of the initial stress-strain state influence on mechanical properties of viscoelastic bodies. PNRPU Mechanics Bulletin. 2019. No. 2. P. 15–24. DOI: 10.15593/perm.mech/2019.2.02.
Vatulyan A.O., Varchenko A.A., Yurov V.O. Investigation of coefficient inverse problems taking into account rheology for functionally graded material. Bulletin of Higher Educational Institutions. North Caucasus Region. Natural Science. 2023. No. 3. P. 4–12. DOI: 10.18522/1026-2237-2023-3-4-12.
Bogachev I.V., Vatulyan A.O., Dudarev V.V. On the method of property identification of multilayer soft biological tissues. Russian Journal of Biomechanics. 2013. Vol. 13, no. 3. P. 37–48.
Bogachev I.V., Nedin R.D. On characteristic identification for prestressed human skin. Russian Journal of Biomechanics. 2021. Vol. 25, no. 3. P. 285–295. DOI: 10.15593/RJBiomech/2021.3.08.
Truesdell C.A. A first course in rational continuum mechanics. Baltimore, Maryland: The John Hopkins University, 1972. 417 p.
Christensen R.M. Theory of Viscoelasticity. An Introduction. New York, London: Academic Press, 1971. 267 p.
Abramovic M., Stigan I. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. U.S. Department of Commerce, National Bureau of Standards, 1964. 1046 p.
Fletcher C.A. Computational Galerkin methods. Springer-Verlag, 1984. 310 p.
Tikhonov A.N., Arsenin V.Y. Metody resheniya nekorrektnykh zadach. Moscow: Nauka, 1986. 288 p.
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Computational Continuum Mechanics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.