On Solvability of Boundary Value Problems for Elastic Micropolar Shells with Rigid Inclusions

Abstract—

In the framework of the linear theory of micropolar shells, existence and uniqueness theorems for weak solutions of boundary value problems describing small deformations of elastic micropolar shells connected to a system of absolutely rigid bodies are proved. The definition of a weak solution is based on the principle of virial movements. A feature of this problem is non-standard boundary conditions at the interface between the shell and solids.

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REFERENCES

  1. 1

    I. I. Vorovich, “On the existence of solutions in the nonlinear theory of shells,” Izv. Akad.Nauk SSSR Ser. Mat. 19 (4), 173–186 (1955).

    MathSciNet  MATH  Google Scholar 

  2. 2

    I. I. Vorovich, “On the existence of solutions in the nonlinear theory of shells,” Dokl. Akad. Nauk SSSR (N.S.) 117 (2), 203–206 (1957).

  3. 3

    I. I. Vorovich, Mathematical Problems of the Nonlinear Theory of Shallow Shells (Nauka, Moscow, 1989) [in Russian].

    Google Scholar 

  4. 4

    L. P. Lebedev and V. A. Eremeyev, “Academician Iosif I. Vorovich,” ZAMM 91 (6), 429–432 (2011).

    ADS  MathSciNet  Article  Google Scholar 

  5. 5

    P. G. Ciarlet, Mathematical Elasticity, Vol. III: Theory of Shells (Elsevier, Amsterdam, 2000).

  6. 6

    L. P. Lebedev, “On the solution of a dynamic problem for viscoelastic shells,” Dokl. AN SSSR, 267 (1), 62–64 (1982).

    MathSciNet  Google Scholar 

  7. 7

    V. A. Eremeyev and L. M. Zubov, Mechanics of Elastic Shells (Nauka, Moscow, 2008) [in Russian].

    Google Scholar 

  8. 8

    V. A. Eremeyev, L. P. Lebedev, and H. Altenbach, Foundations of Micropolar Mechanics (Springer, Heidelberg, 2013).

    Google Scholar 

  9. 9

    V. A. Eremeyev, M. J. Cloud, and L. P. Lebedev, Applications of Tensor Analysis in Continuum Mechanics (World Scientific, New Jersey, 2018).

    Google Scholar 

  10. 10

    A. Libai and J. G. Simmonds, The Nonlinear Theory of Elastic Shells (Cambridge Univ. Press, Cambridge, 1998).

    Google Scholar 

  11. 11

    V. Konopińska and W. Pietraszkiewicz, “Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells,” Int. J. Solids Struct. 44 (1), 352–369 (2007).

    Article  Google Scholar 

  12. 12

    J. Chróścielewski, A. Sabik, B. Sobczyk, and W. Witkowski, “2-D constitutive equations for orthotropic Cosserat type laminated shells in finite element analysis,” Compos. Part B: Eng. 165, 335–353 (2019).

    Article  Google Scholar 

  13. 13

    V. A. Eremeyev and L. P. Lebedev, “Existence theorems in the linear theory of micropolar shells,” ZAMM 91 (6), 468–476 (2011).

    ADS  MathSciNet  Article  Google Scholar 

  14. 14

    V. A. Eremeyev and L. P. Lebedev, “Existence of weak solutions in elasticity,” Math. Mech. Solids 18 (2), 204–217 (2013).

    MathSciNet  Article  Google Scholar 

  15. 15

    V. A. Eremeyev, F. dell’ Isola, C. Boutin, and D. Steigmann, “Linear pantographic sheets: existence and uniqueness of weak solutions,” J. Elast. 132 (2), 175–196 (2018).

    Article  Google Scholar 

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Funding

This work was supported by the Russian Foundation for Basic Research (grant no. 20-08-00450А).

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Correspondence to V. A. Eremeev or L. P. Lebedev.

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Translated by I. K. Katuev

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Eremeev, V.A., Lebedev, L.P. On Solvability of Boundary Value Problems for Elastic Micropolar Shells with Rigid Inclusions. Mech. Solids 55, 852–856 (2020). https://doi.org/10.3103/S0025654420050052

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Keywords:

  • micropolar shells
  • weak solutions
  • existence and uniqueness
  • rigid inclusions