Skip to main content
SpringerLink
Log in

Impact of a Rigid Sphere with an Infinite Kirchhoff–Love Plate Taking into Account Bulk and Shear Relaxation

  • MECHANICS
  • Published:
Vestnik St. Petersburg University, Mathematics Aims and scope Submit manuscript

Abstract

The problem of the low-velocity normal impact of a rigid sphere with an infinite viscoelastic Kirchhoff–Love plate is considered. The dynamic behavior of a viscoelastic plate is described by a fractional-derivative standard linear solid model. The fractional parameter, which determines the order of the fractional derivative, takes into account the change in the viscosity of the plate material in the contact zone during impact. The plate buckling and the contact force are determined by the generalized Hertz theory. Using the algebra of Rabotnov operators and considering the effect of bulk and shear relaxation, we obtain an integral equation for the buckling of contacting bodies. An approximate solution of this equation makes it possible to find the time dependence not only for the contact buckling but also for the contact force.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. W. Goldsmith, Impact. The Theory and Physical Behaviour of Colliding Bodies (Arnold, London, 1960; Izd. Lit. po Stroit., Moscow, 1965).

  2. Yu. A. Rossikhin and M. V. Shitikova, “Transient response of thin bodies subjected to impact: Wave approach,” Shock Vib. Dig. 39, 273–309 (2007). https://doi.org/10.1177/0583102407080410

    Article  Google Scholar 

  3. C. Zener, “The intrinsic inelasticity of large plates,” Phys. Rev. 59, 669–673 (1941). https://doi.org/10.1103/PHYSREV.59.669

    Article  MATH  Google Scholar 

  4. J. W. Phillips and H. H. Calvit, “Impact of a rigid sphere on a viscoelastic plate,” J. Appl. Mech. 34, 873–878 (1967). https://doi.org/10.1115/1.3607850

    Article  Google Scholar 

  5. M. V. Shitikova, “Wave theory of impact and Professor Yury Rossikhin contribution in the field (a memorial survey),” J. Mater. Eng. Perform. 28, 3161–3173 (2019). https://doi.org/10.1007/s11665-018-3824-6

    Article  Google Scholar 

  6. M. V. Shitikova, “Fractional operator viscoelastic models in dynamic problems of mechanics of solids: A review,” Mech. Solids 57, 1–33 (2022). https://doi.org/10.3103/S0025654422010022

    Article  MATH  Google Scholar 

  7. M. V. Shitikova and A. I. Krusser, “Models of viscoelastic materials: A review on historical development and formulation,” Adv. Struct. Mater. 175, 285–326 (2022). https://doi.org/10.1007/978-3-031-04548-6_14

    Article  MathSciNet  MATH  Google Scholar 

  8. Yu. A. Rossikhin and M. V. Shitikova, “Fractional calculus in structural mechanics,” in Handbook of Fractional Calculus with Applications, Vol. 7: Applications in Engineering, Life and Social Sciences, Ed. by D. Baleanu and A. M. Lopes (De Gruyter, Berlin, 2019), Part A, pp. 159–192. https://doi.org/10.1515/9783110571905-009

  9. Yu. A. Rossikhin and M. V. Shitikova, “Classical beams and plates in a fractional derivative medium, impact response,” in Encyclopedia of Continuum Mechanics, Ed. by H. Altenbach and A. Öchsner (Springer-Verlag, Berlin, 2020), pp. 294–305. https://doi.org/10.1007/978-3-662-53605-6_86-1

    Book  Google Scholar 

  10. Yu. A. Rossikhin and M. V. Shitikova, “Fractional derivative Timoshenko beams and Uflyand–Mindlin plates, impact response,” in Encyclopedia of Continuum Mechanics, Ed. by H. Altenbach and A. Öchsner (Springer-Verlag, Berlin, 2020), pp. 962–971. https://doi.org/10.1007/978-3-662-53605-6_87-1

    Book  Google Scholar 

  11. Yu. A. Rossikhin and M. V. Shitikova, “Collision of two spherical shells. Fractional operator models,” in Encyclopedia of Continuum Mechanics, Ed. by H. Altenbach and A. Öchsner (Springer-Verlag, Berlin, 2020), pp. 324–332. https://doi.org/10.1007/978-3-662-53605-6_88-1

    Book  Google Scholar 

  12. Yu. A. Rossikhin and M. V. Shitikova, “Two approaches for studying the impact response of viscoelastic engineering systems: An overview,” Comput. Math. Appl. 66 (5), 755–773 (2013). https://doi.org/10.1016/j.camwa.2013.01.006

    Article  MathSciNet  MATH  Google Scholar 

  13. S. I. Meshkov, G. N. Pachevskaya, and V. S. Postnikov, “To the detection of volume relaxation during dynamic tests,” Uch. Zap. Erevan. Gos. Univ. 2, 10–14 (1968).

    Google Scholar 

  14. S. I. Meshkov and G. N. Pachevskaya, “Allowance for bulk relaxation by the method of internal friction,” J. Appl. Mech. Tech. Phys. 8, 47–48 (1967). https://doi.org/10.1007/BF00918032

    Article  Google Scholar 

  15. M. V. Shitikova, I. I. Popov, and Yu. A. Rossikhin, “Theoretical and experimental evidence of the bulk relaxation peak on the loss tangent versus frequency diagrams for concrete,” Mech. Adv. Mater. Struct. 30, 332–341 (2022). https://doi.org/10.1080/15376494.2021.2012857

    Article  Google Scholar 

  16. N. Makris, “Three-dimensional constitutive viscoelastic laws with fractional order time derivatives,” J. Rheol. 41, 1007–1020 (1997). https://doi.org/10.1122/1.550823

    Article  Google Scholar 

  17. Yu. A. Rossikhin and M. V. Shitikova, “Features of fractional operators involving fractional derivatives and their applications to the problems of mechanics of solids,” in Fractional Calculus: History, Theory and Applications, Ed. by R. Daou and M. Xavier (Nova Sci., New York, 2015), Ch. 8, pp. 165–226.

    MATH  Google Scholar 

  18. G. Alotta, O. Barrera, A. Cocks, and M. Di Paola, “On the behavior of a three-dimensional fractional viscoelastic constitutive model,” Meccanica 52, 2127–2142 (2017). https://doi.org/10.1007/s11012-016-0550-8

    Article  MathSciNet  MATH  Google Scholar 

  19. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives and Some of Their Applications (Nauka i Tekhnika, Minsk, 1987; Gordon & Breach, New York, 1993).

  20. Yu. A. Rossikhin and M. V. Shitikova, “Centennial jubilee of Academician Rabotnov and contemporary handling of his fractional operator,” Fract. Calculus Appl. Anal. 17, 674–683 (2014). https://doi.org/10.2478/s13540-014-0192-2

    Article  MathSciNet  MATH  Google Scholar 

  21. Yu. N. Rabotnov, “Equilibrium of an elastic medium with after-effect,” Fractional Calculus Appl. Anal. 17, 684–696 (1948). https://doi.org/10.2478/s13540-014-0193-1

    Article  MathSciNet  MATH  Google Scholar 

  22. K. L. Johnson, Contact Mechanics (Cambridge Univ. Press, Cambridge, 1985; Mir, Moscow, 1989).

  23. Yu. N. Rabotnov, Elements of Hereditary Solid Mechanics (Nauka, Moscow, 1977; Mir, Moscow, 1980).

Download references

Funding

This work was supported by the Ministry of Science and Higher Education of the Russian Federation, project no. FZGM-2020-0007.

Author information

Authors and Affiliations

  1. Moscow State University of Civil Engineering, 129337, Moscow, Russia

    M. V. Shitikova

  2. Voronezh State Technical University, 394006, Voronezh, Russia

    M. V. Shitikova

Authors
  1. M. V. Shitikova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. V. Shitikova.

Ethics declarations

I declare that I have no conflicts of interest.

Additional information

Translated by O. Pismenov

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shitikova, M.V. Impact of a Rigid Sphere with an Infinite Kirchhoff–Love Plate Taking into Account Bulk and Shear Relaxation. Vestnik St.Petersb. Univ.Math. 56, 107–118 (2023). https://doi.org/10.1134/S1063454123010119

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063454123010119

Keywords:

Search

Navigation