Advertisement

On an optimal control problem for the shape of thin inclusions in elastic bodies

  • V. V. Shcherbakov
Article

Abstract

An optimal control problem is considered for a two-dimensional elastic body with a straight thin rigid inclusion and a crack adjacent to it. It is assumed that the thin rigid inclusion delaminates and has a kink. On the crack faces the boundary conditions are specified in the form of equalities and inequalities which describe the mutual nonpenetration of the crack faces. The derivative of the energy functional along the crack length is used as the objective functional, and the position of the kink point, as the control function. The existence is proved of the solution to the optimal control problem.

Keywords

crack thin rigid inclusion nonlinear boundary conditions optimal control derivative of energy functional 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. F. Morozov,Mathematical Questions of Crack Theory (Nauka, Moscow, 1984) [in Russian].Google Scholar
  2. 2.
    A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids (WIT Press, Southampton, Boston, 2000).Google Scholar
  3. 3.
    A.M. Khludnev, Elasticity Theory Problems in Nonsmooth Domains (Fizmatlit, Moscow, 2010) [in Russian].Google Scholar
  4. 4.
    G. P. Cherepanov, Mechanics of Brittle Fracture (Nauka, Moscow, 1974; McGraw-Hill, New York, 1979).Google Scholar
  5. 5.
    E. M. Rudoi, “Differentiation of Energy Functionals in Two-Dimensional Elasticity Theory for Solids With Curvilinear Cracks,” Zh. Priklk. Mekh. Tekhn. Fiz. 45(6), 83–94 (2004) [J. Appl.Mech. Tech. Phys. 45 (6), 843–852 (2004)].MathSciNetMATHGoogle Scholar
  6. 6.
    E. M. Rudoi, “Differentiation of Energy Functionals in a Problem on Curvilinear Crack With Possible Contact of the Faces,” Mekh. Tverd. Tela No. 6, 113–127 (2007).Google Scholar
  7. 7.
    N. V. Banichuk, Optimization of Elastic Body Forms (Nauka, Moscow, 1980) [in Russian].Google Scholar
  8. 8.
    N. V. Banichuk, Optimization of Structure Components of Composites (Mashinostroenie, Moscow, 1988) [in Russian].Google Scholar
  9. 9.
    V. G. Litvinov, Optimization in Elliptic Boundary Value Problems With Application to Mechanics (Nauka, Moscow, 1987) [in Russian].Google Scholar
  10. 10.
    J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape Design: Theory and Applications (Wiley, New York, 1988; Mir, Moscow, 1992).Google Scholar
  11. 11.
    K. A. Lur’e, Optimal Control in Problems of Mathematical Physics (Nauka, Moscow, 1975) [in Russian].Google Scholar
  12. 12.
    A. M. Khludnev and J. Sokolowski, Modelling and Control in SolidMechanics (Birkhauser, Basel, 1997).Google Scholar
  13. 13.
    A. M. Khludnev and G. Leugering, “Optimal Control of Cracks in Elastic Bodies With Thin Rigid Inclusions,” Z. Angew.Math.Mech. 91(2), 125–137 (2011).MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    A. M. Khludnev, “Optimal Control of Crack Growth in Elastic Body With Inclusions,” Europ. J. Mech. A/Solids 29(3), 392–399 (2010).MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. Khludnev, G. Leugering, and M. Specovius-Neugebauer, “Optimal Control of Inclusion and Crack Shapes in Elastic Bodies,” J. Optim. Theory Appl. 155(1), 54–78 (2012).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    A. M. Khludnev and M. Negri, “Optimal Rigid Inclusion Shapes in Elastic Bodies With Cracks,” Z. Angew. Math. Phys. 64(1), 179–191 (2012).MathSciNetCrossRefGoogle Scholar
  17. 17.
    G. Leugering and A. M. Khludnev, “On the Equilibriumof Elastic Bodies Containing Thin Rigid Inclusions,” Dokl. Akad. Nauk 430(1), 47–50 (2010) [Dokl. Phys. 55 (1), 18–22 (2010)].MathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Lavrent’ev Institute of HydrodynamicsNovosibirskRussia

Personalised recommendations