Skip to main content

Dynamic Sliding Contact for a Thin Elastic Layer

  • 452 Accesses

Part of the Advanced Structured Materials book series (STRUCTMAT,volume 151)

Abstract

The contribution is concerned with dynamics of a thin elastic layer, subject to sliding contact. Both one- and two-sided sliding contact are studied, revealing the presence of the fundamental vibration modes. First, mixed boundary conditions modelling two-sided sliding are addressed, allowing a factorisation of the dispersion relation. Then, the asymmetric problem of one-sided sliding contact is tackled, with mixed conditions along the contact surface and prescribed normal stress on the opposite face. Using symmetry, this problem is found to be related to that for a layer of a double thickness, with classical boundary conditions in terms of stresses. In this case, the fundamental mode of interest coincides with the zero-order Rayleigh-Lamb symmetric wave. A long-wave low-frequency perturbation scheme is implemented for the forced problem.

Keywords

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Achenbach J (1973) Wave Propagation in Elastic Solids. North Holland, Amsterdam

    Google Scholar 

  • Aghalovyan LA (2015) Asymptotic Theory of Anisotropic Plates and Shells. World Scientifict, Singapore

    Google Scholar 

  • Argatov I, Mishuris G (2011) Frictionless elliptical contact of thin viscoelastic layers bonded to rigid substrates. Applied Mathematical Modelling 35(7):3201–3212

    Google Scholar 

  • Argatov I, Mishuris G (2016) Contact Mechanics of Articular Cartilage Layers. Springer, Cham

    Google Scholar 

  • Argatov I, Mishuris G (2018) Cylindrical lateral depth-sensing indentation of anisotropic elastic tissues: Effects of adhesion and incompressibility. The Journal of Adhesion 94(8):583–596

    Google Scholar 

  • Barnett DM, Gavazza SD, Lothe J, Chadwick P (1988) Slip waves along the interface between two anisotropic elastic half-spaces in sliding contact. Proceedings of the Royal Society of London A Mathematical and Physical Sciences 415(1849):389–419

    Google Scholar 

  • Belyaev AK, Morozov NF, Tovstik PE, Tovstik TP (2019) Two-dimensional linear models of multilayered anisotropic plates. Acta Mechanica 230(8):2891–2904

    Google Scholar 

  • Belyaev AK, Morozov NF, Tovstik PE, Tovstik TP (2021) Applicability ranges for four approaches to determination of bending stiffness of multilayer plates. Continuum Mechanics and Thermodynamics 33(4):1659–1673

    Google Scholar 

  • Borodich FM (2014) The Hertz-Type and Adhesive Contact Problems for Depth-Sensing Indentation. In: Bordas SPA (ed) Advances in Applied Mechanics, Elsevier, vol 47, pp 225–366

    Google Scholar 

  • Borodich FM, Galanov BA, Perepelkin NV, Prikazchikov DA (2019) Adhesive contact problems for a thin elastic layer: Asymptotic analysis and the JKR theory. Mathematics and Mechanics of Solids 24(5):1405–1424

    Google Scholar 

  • Darinskii AN,WeihnachtM(2005) Interface waves on the sliding contact between identical piezoelectric crystals of general anisotropy. Wave Motion 43(1):67–77

    Google Scholar 

  • Erbaş B, Yusufoğlu E, Kaplunov J (2011) A plane contact problem for an elastic orthotropic strip. Journal of Engineering Mathematics 70(4):399–409

    Google Scholar 

  • Erbaş B, Kaplunov J, Nobili A, Kılıç G (2018) Dispersion of elastic waves in a layer interacting with a winkler foundation. The Journal of the Acoustical Society of America 144(5):2918–2925

    Google Scholar 

  • Erbaş B, Kaplunov J, Elishako_ I (2021) Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation. Mathematics and Mechanics of Solids DOI 10.1177/10812865211023885

    Google Scholar 

  • Goldenveizer AL, Lidsky VB, Tovstik PE (1979) Free Vibrations of Thin Elastic Shells (in Russ.). Nauka, Moscow

    Google Scholar 

  • Kaplunov J, Prikazchikov DA, Rogerson GA (2005) On three-dimensional edge waves in semiinfinite isotropic plates subject to mixed face boundary conditions. Acoustical Society of America Journal 118(5):2975–2983

    Google Scholar 

  • Kaplunov J, Prikazchikov D, Sultanova L (2018) Justification and refinement of Winkler-Fuss hypothesis. Zeitschrift für Angewandte Mathematik und Physik 69(3):80

    Google Scholar 

  • Kaplunov J, Prikazchikov D, Sultanova L (2019) Elastic contact of a stiff thin layer and a halfspace. Zeitschrift für Angewandte Mathematik und Physik 70(1):22

    Google Scholar 

  • Kaplunov JD (1995) Long-wave vibrations of a thinwalled body with fixed faces. The Quarterly Journal of Mechanics and Applied Mathematics 48(3):311–327

    Google Scholar 

  • Kaplunov JD, Nolde EV (2002) Long-wave vibrations of a nearly incompressible isotropic plate with fixed faces. The Quarterly Journal of Mechanics and Applied Mathematics 55(3):345–356

    Google Scholar 

  • Kaplunov JD, Kossovich LY, Nolde EV (1998) Dynamics of ThinWalled Elastic Bodies. Academic Press, San Diego

    Google Scholar 

  • Kaplunov JD, Rogerson GA, Tovstik PE (2005) Localized vibration in elastic structures with slowly varying thickness. The Quarterly Journal of Mechanics and Applied Mathematics 58(4):645–664

    Google Scholar 

  • Kudish II, Pashkovski E, Volkov SS, Vasiliev AS, Aizikovich SM (2020) Heavily loaded line EHL contacts with thin adsorbed soft layers. Mathematics and Mechanics of Solids 25(4):1011–1037

    Google Scholar 

  • Kudish II, Volkov SS, Vasiliev AS, Aizikovich SM (2021) Characterization of the behavior of different contacts with double coatings. Mathematics and Mechanics of Complex Systems 9(2):179–202

    Google Scholar 

  • Lashhab MI, Rogerson GA, Prikazchikova LA (2015) Small amplitude waves in a pre-stressed compressible elastic layer with one fixed and one free face. Zeitschrift für Angewandte Mathematik und Physik 66(5):2741–2757

    Google Scholar 

  • Le KC (1999) Vibrations of Shells and Rods. Springer, Berlin

    Google Scholar 

  • Mikhasev GI, Tovstik PE (2020) Localized Dynamics of Thin-Walled Shells. Chapman & Hall/CRC Monographs and Research Notes in Mathematics, CRC Press. Taylor & Francis

    Google Scholar 

  • Moukhomodiarov RR, Pichugin AV, Rogerson GA (2010) The transition between neumann and dirichlet boundary conditions in isotropic elastic plates. Mathematics and Mechanics of Solids 15(4):462–490

    Google Scholar 

  • Nolde EV, Rogerson GA (2002) Long wave asymptotic integration of the governing equations for a pre-stressed incompressible elastic layer with fixed faces. Wave Motion 36(3):287–304

    Google Scholar 

  • Nolde EV, Prikazchikova LA, Rogerson GA (2004) Dispersion of small amplitude waves in a prestressed, compressible elastic plate. Journal of Elasticity 75(1):1–29

    Google Scholar 

  • Prikazchikova L, Aydın YE, Erba¸s B, Kaplunov J (2020) Asymptotic analysis of an anti-plane dynamic problem for a three-layered strongly inhomogeneous laminate. Mathematics and Mechanics of Solids 25(1):3–16

    Google Scholar 

  • Rayleigh L (1888) On the free vibrations of an infinite plate of homogeneous isotropic elastic matter. Proceedings of the London Mathematical Society s1-20(1):225–237

    Google Scholar 

  • Rogerson GA, Sandiford KJ, Prikazchikova LA (2007) Abnormal long wave dispersion phenomena in a slightly compressible elastic plate with non-classical boundary conditions. International Journal of Non-Linear Mechanics 42(2):298–309, special Issue in Honour of Dr. Ronald

    Google Scholar 

  • Vinh PC, Ngoc Anh VT (2014) Rayleigh waves in an orthotropic half-space coated by a thin orthotropic layer with sliding contact. International Journal of Engineering Science 75:154–164

    Google Scholar 

Download references

TS acknowledges support by the European Union, European Regional Development Fund, within the scope of the framework of the programme for investments in growth and jobs 2014-2020, contract No. C3330-18-952007 (EAGLE), in Sect. 9.2. JK and DAP acknowledge support from the Russian Science Federation, Grant No. 20-11-20133, in Sects. 9.1, 9.3 and 9.4.

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kaplunov, J., Prikazchikov, D.A., Savšek, T. (2022). Dynamic Sliding Contact for a Thin Elastic Layer. In: Altenbach, H., Bauer, S., Eremeyev, V.A., Mikhasev, G.I., Morozov, N.F. (eds) Recent Approaches in the Theory of Plates and Plate-Like Structures. Advanced Structured Materials, vol 151. Springer, Cham. https://doi.org/10.1007/978-3-030-87185-7_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-87185-7_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-87184-0

  • Online ISBN: 978-3-030-87185-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics