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On thin Timoshenko inclusions in elastic bodies with defects

  • Alexander KhludnevEmail author
Original
  • 19 Downloads

Abstract

The paper concerns an analysis of equilibrium problems for elastic bodies with elastic Timoshenko inclusion in the presence of defects. Defects are characterized by a positive damage parameter. This parameter is responsible for a connection between defect faces. Asymptotic properties of solutions are investigated with respect to the damage parameters as well as with respect to a rigidity parameter of the inclusions. Limit models are investigated; in particular, different equivalent problem formulations are proposed.

Keywords

Thin inclusion Timoshenko beam Defect Crack Delamination Non-penetration boundary condition Variational inequality 

Notes

Acknowledgements

This work was supported by RFBR (project 18-29-10007).

References

  1. 1.
    Almi, S.: Energy release rate and quasi-static evolution via vanishing viscosity in a fracture model depending on the crack opening. ESAIM: COCV 23(3), 791–826 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Dal Maso, G., Iurlano, F.: Fracture models as gamma-limits of damage models. Commun. Pure Appl. Anal. 12(4), 1657–1686 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Faella, L., Khludnev, A.M., Popova, T.S.: Junction problem for rigid and Timoshenko elastic inclusions in elastic bodies. Math. Mech. Solids 22(4), 737–750 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gao, Y., Ricoeur, A., Zhang, L.-L., Yang, L.-Z.: Crack solutions and weight functions for plane problems in three-dimensional quasicrystals. Arch. Appl. Mech. 84(8), 1103–1115 (2014)CrossRefGoogle Scholar
  5. 5.
    Gaudiello, A., Zappale, E.: Junction in a thin multidomain for a forth order problem. Math. Models Methods Appl. Sci. 16(12), 1887–1918 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gaudiello, A., Zappale, E.: A model of joined beams as limit of a 2D plate. J. Elast. 103(2), 205–233 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Itou, H., Khludnev, A.M.: On delaminated thin Timoshenko inclusions inside elastic bodies. Math. Methods Appl. Sci. 39(17), 4980–4993 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Khludnev, A.M., Kovtunenko, V.A.: Analysis of Cracks in Solids. WIT Press, Southampton-Boston (2000)Google Scholar
  9. 9.
    Khludnev, A.M.: Elasticity problems in non-smooth domains. Fizmatlit, Moscow (2010)Google Scholar
  10. 10.
    Khludnev, A.M., Leugering, G.R.: On Timoshenko thin elastic inclusions inside elastic bodies. Math. Mech. Solids 20(5), 495–511 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Khludnev, A.M.: On modeling elastic bodies with defects. Siberian Electronic Math. Reports. 15, 153–166 (2018)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Khludnev, A.M.: Shape control of thin rigid inclusions and cracks in elastic bodies. Arch. Appl. Mech. 83(10), 1493–1509 (2013)CrossRefzbMATHGoogle Scholar
  13. 13.
    Khludnev, A.M., Leugering, G.: Delaminated thin elastic inclusion inside elastic bodies. Math. Mech. Complex Syst. 2(1), 1–21 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Khludnev, A.M., Popova, T.S.: Junction problem for rigid and semi-rigid inclusions in elastic bodies. Arch. Appl. Mech. 86(9), 1565–1577 (2016)CrossRefGoogle Scholar
  15. 15.
    Faella, L., Khludnev, A.M., Popova, T.S.: Junction problem for rigid and Timoshenko elastic inclusions in elastic bodies. Math. Mech. Solids 22(4), 737–750 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kovtunenko, V.A., Leugering, G.: A shape-topological control problem for nonlinear crack—defect interaction: the anti-plane variational model. SIAM J. Control Optim. 54, 1329–1351 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kovtunenko, V.A.: Shape sensitivity of curvilinear cracks on interface to non-linear perturbations. Z. Angew. Math. Phys. 54(3), 410–423 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kozlov, V.A., Mazya, V.G., Movchan, A.B.: Asymptotic Analysis of Fields in a Multi-structure. Oxford University Press, New York (1999)Google Scholar
  19. 19.
    Lazarev, N.P., Rudoy, E.M.: Shape sensitivity analysis of Timoshenko plate with a crack under the nonpenetration condition. Z. Angew. Math. Mech. 94(9), 730–739 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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. 66(4), 2025–2040 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Le Dret, H.: Modeling of the junction between two rods. J. Math. Pures Appl. 68, 365–397 (1989)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Le Dret, H.: Modeling of a folded plate. Comput. Mech. 5, 401–416 (1990)CrossRefzbMATHGoogle Scholar
  23. 23.
    Panasenko, G.: Multi-scale Modelling for Structures and Composites. Springer, New York (2005)zbMATHGoogle Scholar
  24. 24.
    Perelmuter, M.: Nonlocal criterion of bridged cracks growth: weak interface. J. Eur. Ceram. Soc. 34(11), 2789–2798 (2014)CrossRefGoogle Scholar
  25. 25.
    Rudoy, E.M.: Asymptotic behavior of the energy functional for a three-dimensional body with a rigid inclusion and a crack. J. Appl. Mech. Tech. Phys. 52(2), 252–263 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rudoy, E.M.: Domain decomposition method for crack problems with nonpenetration condition. ESAIM: M2AN 50, 995–1009 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Saccomandi, G., Beatty, M.F.: Universal relations for fiber-reinforced elastic materials. Math. Mech. Solids 7(1), 99–110 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Shcherbakov, V.V.: Optimal control of rigidity parameter of thin inclusions in elastic bodies with curvilinear cracks. J. Math. Sci. 203(4), 591–604 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    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)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Titeux, I., Sanchez-Palencia, E.: Junction of thin plate. Europ. J. Mech. - A/Solids 19(3), 377–400 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Yao, J.: Instability of a composite reinforced with coated inclusions due to interface debonding. Arch. Appl. Mech. 85(4), 415–432 (2015)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Lavrentyev Institute of Hydrodynamics of RAS, and Novosibirsk State UniversityNovosibirskRussia

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