Definitions
The problem on impact of a viscoelastic spherical impactor upon an elastic Bernoulli-Euler beam and Kirchhoff-Love plate in a viscoelastic medium have been formulated for the case, when the viscoelastic features of a viscoelastic impactor are described by the fractional derivative standard linear solid model, while the damping features of the surrounding medium are modelled by the fractional derivative Kelvin-Voigt model.
Backgrounds of Fractional Calculus in Mechanics and Tautochrone Problem
History of Fractional Calculus is presented in detail in many monographs and surveys, among them (Samko et al., 1993; Debnath, 2013; Valerio et al., 2014), while its earliest applications in viscoelasticity and mechanics of solids are described in Rossikhin (2010), and Mainardi (2012). This history begins from the work by N.H. Abel.
Consider the problem on finding in the vertical plane (Ï„, s) such an absolutely...
This is a preview of subscription content, log in via an institution to check access.
References
Debnath L (2013) A brief historical introduction to fractional calculus. Int J Math Educ Science Tech 35(4):487–501. https://doi.org/10.1080/00207390410001686571
Mainardi F (2012) An historical perspective on fractional calculus in linear viscoelasticity. Frac Calc Appl Anal 15(4):712–717. https://doi.org/10.2478/s13540-012-0048-6
Rabotnov YN (1977) Elements of Hereditary Solid Mechanics. Nauka, Moscow (Engl. transl. by Mir Publishers, Moscow, 1980)
Rabotnov YN (2014) Equilibrium of an elastic medium with after-effect. Frac Calc Appl Anal 17:684–696. https://doi.org/10.2478/s13540-014-0193-1 (Eng. transl. from PMM, 1948)
Rossikhin YA (2010) Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids. Appl Mech Rev 63, paper ID 010701. https://doi.org/10.1115/1.4000246
Rossikhin YA, Shitikova MV (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev 50:15–67. https://doi.org/10.1115/1.3101682
Rossikhin YA, Shitikova MV (2010) Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl Mech Rev 63, paper ID 010801. https://doi.org/10.1115/1.4000563
Rossikhin YA, Shitikova MV (2013) Two approaches for studying the impact response of viscoelastic engineering systems: an overview. Comput Math Appl 66:755–773. https://doi.org/10.1016/j.camwa.2013.01.006
Rossikhin YA, Shitikova MV (2014) Centennial jubilee of Academician Rabotnov and contemporary handling of his fractional operator. Frac Calc Appl Anal 17(3):674–683
Rossikhin YA, Shitikova MV (2015) Features of fractional operators involving fractional derivatives and their applications to the problems of mechanics of solids. In: Zeid Daou RA, Moreau X (eds) Fractional calculus: applications. NOVA Publishers, New York, pp 165–226
Rossikhin YA, Shitikova MV, Estrada Meza MG (2016a) Modeling of the impact response of a beam in a viscoelastic medium. Appl Math Sci 10(49):2471–2481. https://doi.org/10.12988/ams.2016.66186
Rossikhin YA, Shitikova MV, Trung PT (2016b) Application of the fractional derivative Kelvin-Voigt model for the analysis of impact response of a Kirchhoff-Love plate. WSEAS Trans Math 15:498–501
Rossikhin YA, Shitikova MV, Trung PT (2017) Impact of a viscoelastic sphere against an elastic Kirchhoff-Love plate embedded into a fractional derivative Kelvin-Voigt medium. Int J Mech 11:58–63
Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives (theory and applications). Gordon and Breach, Switzerland
Valerio D, Machado JT, Kiryakova V (2014) Some pioneers of the applications of fractional calculus. Frac Calc Appl Anal 17(2):552–578. https://doi.org/10.2478/s13540-014-0185-1
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2018 Springer-Verlag GmbH Germany, part of Springer Nature
About this entry
Cite this entry
Rossikhin, Y.A., Shitikova, M.V. (2018). Classical Beams and Plates in a Fractional Derivative Medium, Impact Response. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_86-1
Download citation
DOI: https://doi.org/10.1007/978-3-662-53605-6_86-1
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-53605-6
Online ISBN: 978-3-662-53605-6
eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering