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An Optimal Size of a Rigid Thin Stiffener Reinforcing an Elastic Two-Dimensional Body on the Outer Edge

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Abstract

The equilibrium problem for a two-dimensional body with a crack is studied. We suppose that the body consists of two parts: an elastic part and a rigid thin stiffener on the outer edge of the body. Inequality-type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. For a family of variational problems, dependence of their solutions on the length of the thin rigid stiffener is investigated. It is shown that there exists a solution of an optimal control problem. For this problem, the cost functional is defined by a continuous functional on a solution space, while the length parameter serves as a control parameter.

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References

  1. Mura, T.: Micromechanics of Defects in Solids. Martinus Nijhoff, Dordrecht (1987)

    Book  MATH  Google Scholar 

  2. Dal Corso, F., Bigoni, D., Gei, M.: The stress concentration near a rigid line inclusion in a prestressed, elastic material. Part II. Implications on shear band nucleation, growth and energy release rate. J. Mech. Phys. Solids 56(3), 839–857 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Calvo-Jurado, C., Parnell, W.J.: The influence of two-point statistics on the Hashin–Shtrikman bounds for three phase composites. J. Comput. Appl. Math. 318, 354–365 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Annin, B.D., Kovtunenko, V.A., Sadovskii, V.M.: Variational and hemivariational inequalities in mechanics of elastoplastic, granular media, and quasibrittle cracks. In: Springer Proceedings in Mathematics and Statistics, vol. 121, pp. 49–56 (2015)

  5. Khludnev, A.M.: Optimal control of crack growth in elastic body with inclusions. Eur. J. Mech. A. Solids 29(3), 392–399 (2010)

    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 Nonsmooth Domains. Fizmatlit, Moscow (2010). (in Russian)

    Google Scholar 

  8. Kovtunenko, V.A., Leugering, G.: A shape-topological control problem for nonlinear crack-defect interaction: the antiplane variational model. SIAM J. Control Optim. 54(3), 1329–1351 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Khludnev, A.M.: Shape control of thin rigid inclusions and cracks in elastic bodies. Arch. Appl. Mech. 83(10), 1493–1509 (2013)

    Article  MATH  Google Scholar 

  10. Khludnev, A.M., Leugering, G.R.: On Timoshenko thin elastic inclusions inside elastic bodies. Math. Mech. Solids 20(5), 495–511 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kovtunenko, V.A.: Shape sensitivity of curvilinear cracks on interface to non-linear perturbations. Z. Angew. Math. Phys. 54(3), 410–423 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lazarev, N.P., Itou, H., Neustroeva, N.V.: Fictitious domain method for an equilibrium problem of the Timoshenko-type plate with a crack crossing the external boundary at zero angle. Jpn. J. Ind. Appl. Math. 33(1), 63–80 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lazarev, N.P.: Optimal control of the thickness of a rigid inclusion in equilibrium problems for inhomogeneous two-dimensional bodies with a crack. Z. Angew. Math. Mech. 96(4), 509–518 (2016)

    Article  MathSciNet  Google Scholar 

  14. Khludnev, A., Popova, T.: Junction problem for rigid and semirigid inclusions in elastic bodies. Arch. Appl. Mech. 86(9), 1565–1577 (2016)

    Article  Google Scholar 

  15. Khludnev, A.M., Popova, T.S.: Junction problem for Euler–Bernoulli and Timoshenko elastic inclusions in elastic bodies. Q. Appl. Math. 74(4), 705–718 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sherbakov, V.V.: The Griffith formula and J-integral for elastic bodies with Timoshenko inclusions. Z. Angew. Math. Mech. 96(11), 1306–1317 (2016)

    Article  MathSciNet  Google Scholar 

  17. Pyatkina, E.V.: Optimal control of the shape of a layer shape in the equilibrium problem of elastic bodies with overlapping domains. J. Appl. Ind. Math. 10(3), 435–443 (2016)

    Article  MathSciNet  Google Scholar 

  18. Leugering, G., Sokolowski, J., Zochowski, A.: Control of crack propagation by shapetopological optimization. Discrete Contin. Dyn. Syst. Ser. A 35(6), 2625–2657 (2015)

    Article  MATH  Google Scholar 

  19. Itou, H., Khludnev, A.M.: On delaminated thin Timoshenko inclusions inside elastic bodies. Math. Methods Appl. Sci. 39(17), 4980–4993 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Khludnev, A.M., Shcherbakov, V.V.: Singular invariant integrals for elastic bodies with thin elastic inclusions and cracks. Dokl. Phys. 61(12), 615–619 (2016)

    Article  Google Scholar 

  21. Bojczuk, D., Mróz, Z.: Topological sensitivity derivative and finite topology modifications: application to optimization of plates in bending. Struct. Multidiscip. Optim. 39(1), 1–15 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Savin, G.N., Fleishman, N.P.: Rib-reinforced plates and shells. Israel Program for Scientific Translation, Jerusalem (1967)

  23. Save, M., Prager, W.: Structural Optimization, vol. 1. Optimality Criteria Plenum Press, New York (1985)

    MATH  Google Scholar 

  24. Liu, Y., Shimoda, M.: Parameter-free optimum design method of stiffeners on thin-walled structures. Struct. Multidiscip. Optim. 49(1), 39–47 (2014)

    Article  MathSciNet  Google Scholar 

  25. Misseroni, D., Dal Corso, F., Shahzad, S., Bigoni, D.: Stress concentration near stiff inclusions: validation of rigid inclusion model and boundary layers by means of photoelasticity. Eng. Fract. Mech. 121–122, 87–97 (2014)

    Article  Google Scholar 

  26. Movchan, A.B., Nazarov, S.A.: Stress–strain state near the tip of a perfectly rigid three-dimensional spike introduced into an elastic body. Sov. Appl. Mech. 25(12), 1172–1180 (1989)

    Article  MATH  Google Scholar 

  27. Il’Ina, I.I., Silvestrov, V.: The problem of a thin interfacial inclusion detached from the medium along one side. Mech. Solids 40(3), 123–133 (2005)

    Google Scholar 

  28. Rudoy, E.M.: Numerical solution of an equilibrium problem for an elastic body with a thin delaminated rigid inclusion. J. Appl. Ind. Math. 10(2), 264–276 (2016)

    Article  MathSciNet  Google Scholar 

  29. Fedelinski, P.: Computer modelling and analysis of microstructures with fibres and cracks. J Achiev Mater Manuf Eng 54(2), 242–249 (2012)

    Google Scholar 

  30. Zohdi, T.I., Wriggers, P.: An Introduction to Computational Micromechanics, 2nd edn. Springer, Heidelberg (2008)

    MATH  Google Scholar 

  31. Fedeliński, P., Górski, R., Czyz, T., Dziatkiewicz, G., Ptaszny, J.: Analysis of effective properties of materials by using the boundary element method. Arch. Mech. 66(1), 19–35 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Salgado, N.K., Aliabadi, M.H.: The application of the dual boundary element method to the analysis of cracked stiffened panels. Eng. Fract. Mech. 54(1), 91–105 (1996)

    Article  Google Scholar 

  33. Hlavaček, I., Haslinger, J., Nečas, J., Lovišek, J.: Solution of Variational Inequalities in Mechanics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  34. Maz’ya, V.G.: Sobolev Spaces, Springer Series in Soviet Mathematics. Springer, Berlin (1985)

    Google Scholar 

  35. Mikhailov, V.P.: Partial Differential Equations. Mir, Moscow (1978)

    Google Scholar 

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Acknowledgements

This work has been supported by the Ministry of Education and Science of the Russian Federation within the framework of the base part of the state task (Project 1.7218.2017/6.7).

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Authors and Affiliations

  1. North-Eastern Federal University, Yakutsk, Russia

    Nyurgun Lazarev & Galina Semenova

Authors
  1. Nyurgun Lazarev
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  2. Galina Semenova
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Correspondence to Galina Semenova.

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Communicated by Jan Sokolowski.

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Lazarev, N., Semenova, G. An Optimal Size of a Rigid Thin Stiffener Reinforcing an Elastic Two-Dimensional Body on the Outer Edge. J Optim Theory Appl 178, 614–626 (2018). https://doi.org/10.1007/s10957-018-1291-8

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  • DOI: https://doi.org/10.1007/s10957-018-1291-8

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