Skip to main content
SpringerLink
Log in

Equilibrium of a Three-Layer Plate with a Crack

  • Published:
Journal of Applied and Industrial Mathematics Aims and scope Submit manuscript

Abstract

We study the problem of equilibrium for a three-layer plate clamped at the outer edge and containing a through vertical crack. The three-layer plate consists of two structural layers considered as anisotropic Kirchhoff–Love plates and a soft layer between them. The nonpenetration condition is imposed at the crack edges in the structural layers. The passage to the limit as the width of the soft layer tends to zero and its reduced stiffness tends to infinity is considered. The unique solvability is shown and variational and differential statements are presented for both problems.

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

Fig. 1.

Similar content being viewed by others

REFERENCES

  1. V. V. Bolotin and Yu. N. Novichkov, Mechanics of Multilayered Structures (Mashinostroenie, Moscow, 1980) [in Russian].

    Google Scholar 

  2. E. Reissner, “Contributions to the problem of structural analysis of sandwich-type plates and shells,” in Theory and Practice of Sandwich Construction in Aircraft. A Symposium, Preprint No. 165 (Inst. Aeron. Sci., 1948), pp. 21–48.

  3. E. I. Grigolyuk, “Equation of three-layer sandwich shells with a light packing,” Izv. Akad. Nauk SSSR. Otd. Tekh. Nauk (1), 77–84 (1957).

  4. E. I. Grigolyuk and P. P. Chulkov, “On the general theory of three-layer shells with a big deflection,” Dokl. Akad. Nauk SSSR 150 (5), 1012–1014 (1963).

    Google Scholar 

  5. E. I. Grigolyuk and G. M. Kulikov, “Generalized model of mechanics of thin-walled structures made of composite materials,” Mekh. Kompoz. Mater. (4), 698–704 (1988).

  6. V. I. Korolev, Elastoplastic Deformation of Shells (Mashinostroenie, Moscow, 1971) [in Russian].

    Google Scholar 

  7. Yu. I. Dimitrienko, “Asymptotic theory of multilayer thin plates,” Vestn. MGTU im. N. E. Baumana. Ser. Estesv. Nauki (3), 86–99 (2012).

  8. Yu. I. Dimitrienko, E. A. Gubareva, and Yu. V. Yurin, “Variational equations of asymptotic theory of multilayer thin plates,” Vestn. MGTU im. N. E. Baumana. Ser. Estesv. Nauki (4), 67–87 (2015).

  9. E. I. Grigolyuk and G. M. Kulikov, “Development of the theory of elastic multilayered plates and shells,” Vestn. TSTU 11, 439–448 (2005).

    Google Scholar 

  10. A. R. Rzanitsyn, Built-Up Bars and Plates (Stroiizdat, Moscow, 1986) [in Russian].

    Google Scholar 

  11. A. M. Khludnev, “On the contact of two plates one of which contains a crack,” J. Appl. Math. Mech. 61 (5), 851–862 (1997).

    Article  MathSciNet  Google Scholar 

  12. A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids (WIT Press, Southampton–Boston, 2000).

  13. E. M. Rudoy, “Differentiation of energy functionals in two-dimensional elasticity theory for solids with curvilinear cracks,” J. Appl. Mech. Tech. Phys. 45 (6), 843–852 (2004).

    Article  MathSciNet  Google Scholar 

  14. A. M. Khludnev, Elasticity Theory Problems in Nonsmooth Domains (Fizmatlit, Moscow, 2010) [in Russian].

    Google Scholar 

  15. N. P. Lazarev, “The problem of equilibrium of a Timoshenko-type plate containing a through-thickness crack,” Sib. Zh. Ind. Mat. 14 (4), 32–43 (2011).

    MathSciNet  MATH  Google Scholar 

  16. V. V. Shcherbakov, “On an optimal control problem of thin inclusions shapes in elastic bodies,” Sib. Zh. Ind. Mat. 16 (1), 138–147 (2013) [J. Appl. Ind. Math. 7 (3), 435–443 (2013)].

    Article  MathSciNet  Google Scholar 

  17. E. M. Rudoy, N. A. Kazarinov, and V. Yu. Slesarenko, “Numerical simulation of the equilibrium of an elastic two-layer structure with a crack,” Sib. Zh. Ind. Mat. 20 (1), 77–90 (2017) [Numer. Anal. Appl. 10 (1), 63–73 (2017)].

    Article  MathSciNet  Google Scholar 

  18. Y. Beneveniste and T. Miloh, “Imperfect soft and stiff interfaces in two-dimensional elasticity,” Mech. Mater. 33, 309–323 (2001).

    Article  Google Scholar 

  19. A. M. Khludnev, “On modelling elastic bodies with defects,” Sib. Electron. Math. Rep. 15, 153–166 (2018).

    MATH  Google Scholar 

  20. I. V. Fankina, “On the equilibrium of a two-layer elastic structure with a crack,” Sib. Zh. Ind. Mat. 22 (4), 107–120 (2019) [J. Appl. Ind. Math. 13 (4), 629–641 (2019)].

    Article  MathSciNet  Google Scholar 

  21. I. V. Fankina, “On the equilibrium problem for a two-layer structure with the upper layer covering a defect tip,” Sib. Electron. Math. Rep. 17, 141–160 (2020).

    MathSciNet  MATH  Google Scholar 

  22. E. Rudoy, “Asymptotic modelling of bonded plates by a soft thin adhesive layer,” Sib. Electron. Math. Rep. 17, 615–625 (2020).

    MathSciNet  MATH  Google Scholar 

  23. A. Furtsev and E. Rudoy, “Variational approach to modelling soft and stiff interfaces in the Kirchhoff–Love theory of plates,” Int. J. Solid Struct. 202, 562–574 (2020).

    Article  Google Scholar 

Download references

Funding

This work was supported by the Russian Foundation for Basic Research, project no. 18-29-10007.

Author information

Authors and Affiliations

  1. Lavrent’ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    E. V. Pyatkina

Authors
  1. E. V. Pyatkina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. V. Pyatkina.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pyatkina, E.V. Equilibrium of a Three-Layer Plate with a Crack. J. Appl. Ind. Math. 16, 122–135 (2022). https://doi.org/10.1134/S1990478922010124

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478922010124

Keywords

Search

Navigation