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Optimal Control of Parameters for Elastic Body with Thin Inclusions

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Abstract

In this paper, an equilibrium problem for 2D non-homogeneous anisotropic elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. A connection between the inclusions at a given point is characterized by a junction stiffness parameter. The elastic inclusion is delaminated, thus forming an interfacial crack with the matrix. Inequality-type boundary conditions are imposed at the crack faces to prevent interpenetration. Existence of solutions is proved; different equivalent formulations of the problem are discussed; junction conditions at the connection point are found. A convergence of solutions as the junction stiffness parameter tends to zero and to infinity as well as the rigidity parameter of the elastic inclusion tends to infinity is investigated. An analysis of limit models is provided. An optimal control problem is analyzed with the cost functional equal to the derivative of the energy functional with respect to the crack length. A solution existence of an inverse problem for finding the junction stiffness and rigidity parameters is proved.

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References

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Khludnev, A.M., Faella, L., Perugia, C.: Optimal control of rigidity parameters of thin inclusions in composite materials. ZAMP 68, 47 (2017)

    MathSciNet  MATH  Google Scholar 

  4. Hintermueller, M., Kovtunenko, V.A., Kunisch, K.: Obstacle problems with cohesion: a hemi-variational inequality approach and its efficient numerical solution. SIAM J. Optim. 21, 491–516 (2011)

    Article  MathSciNet  Google Scholar 

  5. 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 

  6. 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 

  7. Lazarev, N.P., Semenova, G.: An optimal size of a rigid thin stiffener reinforcing an elastic two-dimensional body on the outer edge. J. Opt. Theory Appl. 178, 614–626 (2018)

    Article  MathSciNet  Google Scholar 

  8. Rudoy, E.M.: Asymptotics of energy functional for elastic body with a crack and rigid inclusion: 2D case. Appl. Math. Mech. 75, 719–729 (2011)

    Google Scholar 

  9. Shcherbakov, V.V.: Optimal control of rigidity parameter of thin inclusions in elastic bodies with curvilinear cracks. J. Math. Sci. 203, 591–604 (2014)

    Article  MathSciNet  Google Scholar 

  10. Faella, L., Khludnev, A.M.: Junction problem for elastic and rigid inclusions in elastic bodies. Math. Methods Appl. Sci. 39, 3381–3390 (2016)

    Article  MathSciNet  Google Scholar 

  11. Khludnev, A.M., Faella, L., Popova, T.S.: Junction problem for rigid and Timoshenko elastic inclusions in elastic bodies. Math. Mech. Solids 22, 737–750 (2017)

    Article  MathSciNet  Google Scholar 

  12. Khludnev, A.M., Popova, T.S.: Junction problem for Euler–Bernoulli and Timoshenko elastic inclusions in elastic bodies. Quart. Appl. Math. 74, 705–718 (2016)

    Article  MathSciNet  Google Scholar 

  13. Khludnev, A.M.: On modeling thin inclusions in elastic bodies with a damage parameter. Math. Mech. Solids. 24, 2742–2753 (2019)

    Article  MathSciNet  Google Scholar 

  14. Ciarlet, P.G., Le Dret, H., Nzengwa, R.: Junctions between three dimensional and two dimensional linearly elastic structures. J. Math. Pures Appl. 68, 261–295 (1989)

    MathSciNet  MATH  Google Scholar 

  15. 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 

  16. Gaudiello, A., Zappale, E.: A model of joined beams as limit of a 2D plate. J. Elast. 103, 205–233 (2011)

    Article  MathSciNet  Google Scholar 

  17. Le Dret, H.: Modeling of the junction between two rods. J. Math. Pures Appl. 68, 365–397 (1989)

    MathSciNet  MATH  Google Scholar 

  18. Le Dret, H.: Modeling of a folded plate. Comput. Mech. 5, 401–416 (1990)

    Article  Google Scholar 

  19. Titeux, I., Sanchez-Palencia, E.: Junction of thin plates. Eur. J. Mech. A Solids 19, 377–400 (2000)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Pasternak, I.M.: Plane problem of elasticity theory for anisotropic bodies with thin elastic inclusions. J. Math. Sci. 186, 31–47 (2012)

    Article  MathSciNet  Google Scholar 

  22. Vynnytska, L., Savula, Y.: Mathematical modeling and numerical analysis of elastic body with thin inclusion. Comput. Mech. 50, 533–542 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Mallick, P.K.: Fiber-Reinforced Composites Materials, Manufacturing, and Design. Marcel Dekker Inc, New York (1993)

    Google Scholar 

  25. 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 

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Acknowledgements

The authors have been supported by the gruppo nazionale per l’analisi matematica, la probabilità e le loro applicazioni (GNAMPA) of the Istituto nazionale di alta matematica (INdAM). The first author was also supported by RFBR (19-51-50004).

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Authors and Affiliations

  1. Lavrentyev Institute of Hydrodynamics of the RAS, Novosibirsk State University, Novosibirsk, Russia, 630090

    Alexander Khludnev

  2. DIEI Dipartimento di Ingegneria Elettrica e dell’Informazione “M. Scarano”, Universita di Cassino e del Lazio Meridionale, 03043, Cassino, Italy

    Antonio Corbo Esposito & Luisa Faella

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  1. Alexander Khludnev
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  2. Antonio Corbo Esposito
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  3. Luisa Faella
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Correspondence to Alexander Khludnev.

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Khludnev, A., Esposito, A.C. & Faella, L. Optimal Control of Parameters for Elastic Body with Thin Inclusions. J Optim Theory Appl 184, 293–314 (2020). https://doi.org/10.1007/s10957-019-01620-w

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  • DOI: https://doi.org/10.1007/s10957-019-01620-w

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