Definitions
Damped vibrations of such elastic thin bodies as plates and circular cylindrical shells embedded into a viscoelastic medium, the rheological features of which are described by fractional derivatives, are considered in this entry.
Backgrounds
Interest in fractional calculus has quickened profoundly in the past few decades, resulting in a large body of articles devoted to this challenge, which is clearly emphasized in a set of review papers published in the field (Rossikhin and Shitikova, 1997a, 2010; Gaul, 1999; Shimizu and Zhang, 1999). The most recent state-of-the-art article Rossikhin and Shitikova (2010) is devoted to the analysis of new trends and recent results carried out during the last decade in the field of fractional calculus application to mechanics of materials and dynamic problems of structural mechanics, while the historical survey about two parallel ways in the...
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References
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Rossikhin YA (2010) Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids. Appl Mech Rev 63(1), Article ID 010701. https://doi.org/1.1115/1.4000246
Rossikhin YA, Shitikova MV (1997a) Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems. Acta Mech 120(1–4):109–125. https://doi.org/10.1007/BF0114319
Rossikhin YA, Shitikova MV (1997b) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev 50(1):15–67. https://doi.org/10.1115/1.3101682
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Rossikhin YA, Shitikova MV (2006) Analysis of damped vibrations of linear viscoelastic plates with damping modeled with fractional derivatives. Signal Proc 86:2703–2711. https://doi.org/10.1016/j.sigpro.2006.02016
Rossikhin YA, Shitikova MV (2010) Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl Mech Rev 63(1), Article ID 010801. https://doi.org/1.1115/1.4000563
Rossikhin YA, Shitikova MV (2012) Analysis of damped vibrations of thin bodies embedded into a fractional derivative viscoelastic medium. J Mech Behav Mater 21(5–6):155–159. https://doi.org/10.1515/jmbm-2013-0002
Shimizu N, Zhang W (1999) Fractional calculus approach to dynamic problems of viscoelastic materials. JSME Int J Ser C 42(4):825–837. https://doi.org/10.1299/jsmec.42.82
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Rossikhin, Y.A., Shitikova, M.V. (2018). Thin Bodies Embedded in Fractional Derivative Viscoelastic Medium, Dynamic Response. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_90-1
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