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On thin inclusions in elastic bodies with defects

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Abstract

An equilibrium problem for a 2D elastic body with thin inclusions and defects is analyzed. The presence of defects means that the problem is formulated in a non-smooth domain. The defects are characterized by a positive damage parameter. Nonlinear boundary conditions at the defect faces are imposed to prevent a mutual penetration between the faces. An existence of solutions is proved, and different formulations of the problem are proposed. We study an asymptotics of solutions with respect to the damage parameter and analyze the limit models. Moreover, we study the dependence of the solution on the rigidity parameter of the inclusions. In particular, passages to infinity and to zero of the rigidity parameter are investigated.

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References

  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, 791–826 (2017)

    MathSciNet  MATH  Google Scholar 

  2. Gaudiello, A., Zappale, E.: Junction in a thin multidomain for a forth order problem. Math. Models Methods Appl. Sci. 16, 1887–1918 (2006)

    Article  MathSciNet  Google Scholar 

  3. Gaudiello, A., Monneau, R., Mossino, J., Murat, F., Sili, A.: Junction of elastic plates and beams. ESAIM: Control Optim. Calc. Var. 13, 419–457 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Hilchenbach, C.F., Ramm, E.: Optimization of multiphase structures considering damage. Struct. Multidiscip. Optim. 51, 1083–1096 (2015)

    Article  MathSciNet  Google Scholar 

  5. Itou, H., Khludnev, A.M.: On delaminated thin Timoshenko inclusions inside elastic bodies. Math. Methods Appl. Sci. 39, 4980–4993 (2016)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  8. Khludnev, A.M., Faella, L., Perugia, C.: Optimal control of rigidity parameters of thin inclusions in composite materials. Z. Angew. Math. Phys. 68, 47 (2017)

    Article  MathSciNet  Google Scholar 

  9. Khludnev, A.M., Leugering, G.: Delaminated thin elastic inclusion inside elastic bodies. Math. Mech. Complex Syst. 2, 1–21 (2014)

    Article  MathSciNet  Google Scholar 

  10. Khludnev, A.M.: Rigidity parameter identification for thin inclusions located inside elastic bodies. J. Optim. Theory Appl. 172, 281–297 (2017)

    Article  MathSciNet  Google Scholar 

  11. Khludnev, A.M.: On modeling elastic bodies with defects. Sib. Electron. Math. Rep. 15, 153–166 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Khludnev, A.M., Shcherbakov, V.V.: Singular path-independent energy integrals for elastic bodies with Euler–Bernoulli inclusions. Math. Mech. Solids 22, 2180–2195 (2017)

    Article  MathSciNet  Google Scholar 

  13. 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)

    Article  MathSciNet  Google Scholar 

  14. Kovtunenko, V.A.: Shape sensitivity of curvilinear cracks on interface to non-linear perturbations. Z. Angew. Math. Phys. 54, 410–423 (2003)

    Article  MathSciNet  Google Scholar 

  15. Kozlov, V.A., Mazya, V.G., Movchan, A.B.: Asymptotic Analysis of Fields in a Multi-structure. Oxford Mathematical Monographs. Oxford University Press, New York (1999)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. 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, 2025–2040 (2015)

    Article  MathSciNet  Google Scholar 

  18. Panasenko, G.: Multi-scale Modelling for Structures and Composites. Springer, New York (2005)

    MATH  Google Scholar 

  19. Perelmuter, M.: Nonlocal criterion of bridged cracks growth: weak interface. J. Eur. Ceram. Soc. 34, 2789–2798 (2014)

    Article  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. 66, 1923–1937 (2015)

    Article  MathSciNet  Google Scholar 

  21. Rudoy, E.M.: Domain decomposition method for crack problems with nonpenetration condition. ESAIM: M2AN 50, 995–1009 (2016)

    Article  MathSciNet  Google Scholar 

  22. Saccomandi, G., Beatty, M.F.: Universal relations for fiber-reinforced elastic materials. Math. Mech. Solids 7, 99–110 (2002)

    Article  MathSciNet  Google Scholar 

  23. Shcherbakov, V.V.: Shape optimization of rigid inclusions in elastic plates with cracks. Z. Angew. Math. Phys. 67, 71 (2016)

    Article  Google Scholar 

  24. Shcherbakov, V.V.: Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions. Z. Angew. Math. Phys. 68, 26 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Lavrentyev Institute of Hydrodynamics of SB RAS, Novosibirsk, Russia, 630090

    A. M. Khludnev

  2. Novosibirsk State University, Novosibirsk, Russia, 630090

    A. M. Khludnev

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  1. A. M. Khludnev
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Correspondence to A. M. Khludnev.

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Khludnev, A.M. On thin inclusions in elastic bodies with defects. Z. Angew. Math. Phys. 70, 45 (2019). https://doi.org/10.1007/s00033-019-1091-5

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  • DOI: https://doi.org/10.1007/s00033-019-1091-5

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