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Fictitious domain method for an equilibrium problem of the Timoshenko-type plate with a crack crossing the external boundary at zero angle

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Abstract

The equilibrium problems for homogeneous and inhomogeneous plates with a crack is considered. We impose the nonpenetration condition, which has the form of inequality (such as Signorini type condition), on the crack faces. In this paper, we deal with the cases that the crack intersects the external boundary at zero angle (on the mid-plane). Using the fictitious domain method, we establish the unique solvability of four equilibrium problems for different cases of non-Lipschitz domains. In these cases a precise connection between equilibrium problems for a plate contacting with a rigid obstacle and for a plate with a crack is identified.

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References

  1. Alekseev, G.V., Khludnev, A.M.: Crack in elastic body crossing the external boundary at zero angle. Vestnik Quart. J. Novosibirsk State Univ. Series: Math. Mech. Inform 9, 15–29 (2009). (in Russian)

    MATH  Google Scholar 

  2. Andersson, L.-E., Khludnev, A.M.: On crack crossing an external boundary. Fictitious domain method and invariant integrals. Sib. Zh. Ind. Mat. 11(3), 15–29 (2008). (in Russian)

    MathSciNet  MATH  Google Scholar 

  3. Baiocchi, C., Capello, A.: Variational and Quasivariational Inequalities: Application to Free Boundary Problems. Wiley, New York (1984)

    Google Scholar 

  4. Brusnikin, M.B.: Efficient algorithms for problems of the fictitious domain method in the multiply connected case. Dokl. Math. 66, 322–326 (2002)

    MATH  Google Scholar 

  5. Fichera, G.: Boundary value problems of elasticity with unilateral constraints. In: Flügge, S. (ed.) Handbuch der Physik, Band 6a/2. Springer-Verlag, Berlin (1972)

  6. Grisvard, P.: Singularities in Boundary Value Problems, Masson. Springer, Berlin (1992)

    Google Scholar 

  7. Itou, H., Kovtunenko, V.A., Tani, A.: The interface crack with Coulomb friction between two bonded dissimilar elastic media. Appl. Math. 56, 69–97 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Itou, H., Leugering, G., Khludnev, A.M.: Timoshenko thin inclusions in an elastic body with possible delamination. Dokl. Phys. 59(9), 401–404 (2014)

    Article  Google Scholar 

  9. Konovalov, A.N.: Fictitious domain method in twisting problems. Numer. Methods Solid Mech. 4, 109–115 (1973). (in Russian)

    Google Scholar 

  10. Kopchenov, V.D.: Approximate solutions of the Dirichlet problem by a fictitious domain method. Differ. Equ. 4, 151–164 (1968). (in Russian)

    MATH  Google Scholar 

  11. Khludnev, A.M.: Elasticity Problems in Nonsmooth Domains. Fizmatlit, Moscow (2010). (in Russian)

    Google Scholar 

  12. Knees, D., Schroder, A.: Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints. Math. Methods Appl. Sci. 35(15), 1859–1884 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lazarev, N.P.: Fictitious domain method in the equilibrium problem for a Timoshenko type plate contacting with a rigid obstacle. J. Math. Sci. 203, 527–539 (2014)

    Article  MathSciNet  Google Scholar 

  14. Lazarev, N.P.: An iterative penalty method for a monlinear problem of equilibrium of a Timoshenko-type plate with a crack. Numer. Anal. Appl. 4, 309–318 (2011)

    Article  Google Scholar 

  15. Lazarev, N.P.: An equilibrium problem for the Timoshenko-type plate containing a crack on the boundary of a rigid inclusion. J. Sib. Fed. Univ. Math. Phys. 6, 53–62 (2013)

    Google Scholar 

  16. Lazarev, N.P.: Shape sensitivity analysis of the energy integrals for the Timoshenko-type plate containing a crack on the boundary of a rigid inclusion. Z. Angew. Math. Phys. doi:10.1007/s00033-014-0488-4

  17. Lazarev, N.P., Rudoy, E.M.: Shape sensitivity analysis of Timoshenko’s plate with a crack under the nonpenetration condition. Z. Angew. Math. Mech. 94, 730–739 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nazarov, S.A.: Korn inequalities for elastic junctions of massive bodies, thin plates, and rods. Russ. Math. Surv. 63, 35–107 (2008)

    Article  MATH  Google Scholar 

  19. Rudoy, E.M.: Shape derivative of the energy functional in a problem for a thin rigid inclusion in an elastic body. Z. Angew. Math. Phys. doi:10.1007/s00033-014-0471-0

  20. Shcherbakov, V.V.: Existence of an optimal shape of the thin rigid inclusions in the Kirchhoff-Love plate. J. Appl. Ind. Math. 8(1), 97–105 (2014)

    Article  MathSciNet  Google Scholar 

  21. Stepanov, V.D., Khludnev, A.M.: The method of fictitious domains in the signorini problem. Sib. Math. J. 44, 1061–1074 (2003)

    Article  MathSciNet  Google Scholar 

  22. Vabishchevich, P.N.: The Method of Fictitious Domains in Problems of Mathematical Physics. Izdatel’stvo. Moskovskogo Univ, Moscow (1991). (in Russian)

    MATH  Google Scholar 

  23. Washizu, K.: Variational Methods in Elasticity and Plasticity, 3rd edn. Pergamon Press, New York (1982)

    MATH  Google Scholar 

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Acknowledgments

The work was partially supported by Russian Fund for Basic Research (Research Projects No. 13-01-00017).

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Authors and Affiliations

  1. North-Eastern Federal University, Belinsky, 58, Yakutsk, Russia, 677891

    N. P. Lazarev & N. V. Neustroeva

  2. Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan

    H. Itou

Authors
  1. N. P. Lazarev
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  2. H. Itou
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  3. N. V. Neustroeva
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Correspondence to N. P. Lazarev.

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Lazarev, N.P., Itou, H. & Neustroeva, N.V. Fictitious domain method for an equilibrium problem of the Timoshenko-type plate with a crack crossing the external boundary at zero angle. Japan J. Indust. Appl. Math. 33, 63–80 (2016). https://doi.org/10.1007/s13160-015-0200-x

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  • DOI: https://doi.org/10.1007/s13160-015-0200-x

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