Skip to main content
SpringerLink
Log in

Shape sensitivity analysis of the energy integrals for the Timoshenko-type plate containing a crack on the boundary of a rigid inclusion

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

An equilibrium problem for an elastic Timoshenko-type plate containing a rigid inclusion is considered. On the interface between the elastic plate and the rigid inclusion, there is a vertical crack. Inequality-type boundary conditions are imposed at the crack faces to guarantee mutual nonpenetration. By using a sufficiently smooth perturbation determined in the middle plate plane, the variation of plate geometry is specified. The formula of the derivative of the plate energy functional with respect to the perturbation parameter is deduced.

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

Similar content being viewed by others

References

  1. Cherepanov G.P.: Mechanics of Brittle Fracture. McGraw-Hill, New York (1979)

    MATH  Google Scholar 

  2. Parton V.Z., Morozov E.M.: Mechanics of Elastic-Plastic Fracture. Hemisphere Publishing Corp., Washington (1989)

    MATH  Google Scholar 

  3. Khludnev A.M.: Elasticity Problems in Non-smooth Domains. Fizmatlit, Moscow (2010)

    Google Scholar 

  4. 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 

  5. Khludnev A.M., Kovtunenko V.A.: Analysis of Cracks in Solids. WIT Press, Boston (2000)

    Google Scholar 

  6. Khludnev A.M., Negri M.: Optimal rigid inclusion shapes in elastic bodies with cracks. ZAMP 64(1), 179–191 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Kovtunenko V.A.: Primal-dual methods of shape sensitivity analysis for curvilinear cracks with nonpenetration. IMA J. Appl. Math. 71, 635–657 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hintermueller M., Kovtunenko V.A.: From shape variation to topology changes in constrained minimization: a velocity method based concept. Optim. Methods Softw. 26, 513–532 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Shcherbakov V.V.: Optimal control of a rigidity parameter of thin inclusions in elastic bodies with curvilinear cracks. Vestn. Novosib. State Univ. (Math. Mech. Inf.) 13(1), 135–149 (2013)

    MATH  Google Scholar 

  10. 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(1), 53–62 (2013)

    Google Scholar 

  11. 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(9), 730–739 (2014)

    Article  MathSciNet  Google Scholar 

  12. Rudoy E.M.: The Griffith formula and Cherepanov–Rice integral for a plate with a rigid inclusion and a crack. J. Math. Sci. 186(3), 511–529 (2012)

    Article  MathSciNet  Google Scholar 

  13. Rudoy E.M.: The shape derivative of the energy integral in elasticity theory for bodies with rigid inclusions and cracks. J. Math. Sci. 198(5), 608–620 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rudoy E.M.: Asymptotics of the energy functional for a fourth-order mixed boundary value problem in a domain with a cut. Sib. Math. J. 50(2), 341–354 (2009)

    Article  MathSciNet  Google Scholar 

  15. Pelekh B.L.: Theory of Shells with Finite Shear Rigidity. Naukova Dumka, Kiev (1973)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Delfour M.C., Zolesio J.-P.: Shapes and Geometries. Metrics, Analysis, Differential Calculus, and Optimization. SIAM, Philadelphia (2011)

    MATH  Google Scholar 

  18. Sokolowski J., Zolesio J.-P.: Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  19. Kovtunenko V.A.: Invariant energy integrals for the non-linear crack problem with possible contact of the crack surfaces. J. Appl. Math. Mech. 67(1), 99–110 (2003)

    Article  MathSciNet  Google Scholar 

  20. 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 (2014)

Download references

Author information

Authors and Affiliations

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

    N. Lazarev

Authors
  1. N. Lazarev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Lazarev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lazarev, N. 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. 66, 2025–2040 (2015). https://doi.org/10.1007/s00033-014-0488-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00033-014-0488-4

Mathematics Subject Classification

Keywords

Search

Navigation