Existence Theorems in the Geometrically Non-linear 6-Parameter Theory of Elastic Plates

Abstract

In this paper we show the existence of global minimizers for the geometrically non-linear equations of elastic plates, in the framework of the general 6-parameter shell theory. A characteristic feature of this model for shells is the appearance of two independent kinematic fields: the translation vector field and the rotation tensor field (representing in total 6 independent scalar kinematic variables). For isotropic plates, we prove the existence theorem by applying the direct methods of the calculus of variations. Then, we generalize our existence result to the case of anisotropic plates.

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References

  1. 1.

    Aganović, I., Tambača, J., Tutek, Z.: Derivation and justification of the models of rods and plates from linearized three-dimensional micropolar elasticity. J. Elast. 84, 131–152 (2006)

    MATH  Article  Google Scholar 

  2. 2.

    Aganović, I., Tambača, J., Tutek, Z.: Derivation and justification of the model of micropolar elastic shells from three-dimensional linearized micropolar elasticity. Asymptot. Anal. 51, 335–361 (2007)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Altenbach, H., Zhilin, P.A.: The theory of simple elastic shells. In: Kienzler, R., Altenbach, H., Ott, I. (eds.) Theories of Plates and Shells. Critical Review and New Applications, Euromech Colloquium, vol. 444, pp. 1–12. Springer, Heidelberg (2004)

    Google Scholar 

  4. 4.

    Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (1995)

    Google Scholar 

  5. 5.

    Bîrsan, M.: Inequalities of Korn’s type and existence results in the theory of Cosserat elastic shells. J. Elast. 90, 227–239 (2008)

    MATH  Article  Google Scholar 

  6. 6.

    Bîrsan, M.: On the dynamic deformation of porous Cosserat linear-thermoelastic shells. Z. Angew. Math. Mech. 88, 74–78 (2008)

    MATH  Article  Google Scholar 

  7. 7.

    Bîrsan, M., Altenbach, H.: A mathematical study of the linear theory for orthotropic elastic simple shells. Math. Methods Appl. Sci. 33, 1399–1413 (2010)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Bîrsan, M., Neff, P.: On the equations of geometrically nonlinear elastic plates with rotational degrees of freedom. Ann. Acad. Rom. Sci. Ser. Math. Appl. 4, 97–103 (2012)

    MathSciNet  Google Scholar 

  9. 9.

    Chróścielewski, J., Kreja, I., Sabik, A., Witkowski, W.: Modeling of composite shells in 6-parameter nonlinear theory with drilling degree of freedom. Mech. Adv. Mater. Struct. 18, 403–419 (2011)

    Article  Google Scholar 

  10. 10.

    Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statics and Dynamics of Multifold Shells: Nonlinear Theory and Finite Element Method. Wydawnictwo IPPT PAN, Warsaw (2004) (in Polish)

    Google Scholar 

  11. 11.

    Chróścielewski, J., Pietraszkiewicz, W., Witkowski, W.: On shear correction factors in the non-linear theory of elastic shells. Int. J. Solids Struct. 47, 3537–3545 (2010)

    Article  Google Scholar 

  12. 12.

    Ciarlet, P.G.: Mathematical Elasticity, Vol. II: Theory of Plates, 1st edn. North-Holland, Amsterdam (1997)

    Google Scholar 

  13. 13.

    Ciarlet, P.G.: Mathematical Elasticity, Vol. III: Theory of Shells, 1st edn. North-Holland, Amsterdam (2000)

    Google Scholar 

  14. 14.

    Cosserat, E., Cosserat, F.: Théorie des corps déformables. Librairie Scientifique A. Hermann et Fils (engl. translation by D. Delphenich 2007), reprint 2009 by Hermann Librairie Scientifique, ISBN 978 27056 6920 1, Paris (1909)

  15. 15.

    Davini, C.: Existence of weak solutions in linear elastostatics of Cosserat surfaces. Meccanica 10, 225–231 (1975)

    MATH  Article  Google Scholar 

  16. 16.

    Eremeyev, V.A., Lebedev, L.P.: Existence theorems in the linear theory of micropolar shells. Z. Angew. Math. Mech. 91, 468–476 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Eremeyev, V.A., Pietraszkiewicz, W.: The nonlinear theory of elastic shells with phase transitions. J. Elast. 74, 67–86 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Eremeyev, V.A., Pietraszkiewicz, W.: Local symmetry group in the general theory of elastic shells. J. Elast. 85, 125–152 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Eremeyev, V.A., Pietraszkiewicz, W.: Thermomechanics of shells undergoing phase transition. J. Mech. Phys. Solids 59, 1395–1412 (2011)

    MathSciNet  ADS  Article  Google Scholar 

  20. 20.

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

    Google Scholar 

  21. 21.

    Fox, D.D., Simo, J.C.: A drill rotation formulation for geometrically exact shells. Comput. Methods Appl. Mech. Eng. 98, 329–343 (1992)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  22. 22.

    Green, A.E., Naghdi, P.M., Wainwright, W.L.: A general theory of a Cosserat surface. Arch. Ration. Mech. Anal. 20, 287–308 (1965)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Kreja, I.: Geometrically Non-linear Analysis of Layered Composite Plates and Shells. Monographs of Gdansk University of Technology, Gdańsk (2007)

  24. 24.

    Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  25. 25.

    Makowski, J., Pietraszkiewicz, W.: Thermomechanics of shells with singular curves. Zeszyty Naukowe IMP PAN Nr 528(1487), Gdańsk (2002)

  26. 26.

    Naghdi, P.M.: The theory of shells and plates. In: Flügge, S. (ed.) Handbuch der Physik. Mechanics of Solids, vol. VI a/2, pp. 425–640. Springer, Berlin (1972)

    Google Scholar 

  27. 27.

    Neff, P.: A geometrically exact Cosserat-shell model including size effects, avoiding degeneracy in the thin shell limit. Part I: Formal dimensional reduction for elastic plates and existence of minimizers for positive Cosserat couple modulus. Contin. Mech. Thermodyn. 16, 577–628 (2004)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  28. 28.

    Neff, P.: The Γ-limit of a finite strain Cosserat model for asymptotically thin domains versus a formal dimensional reduction. In: Pietraszkiewiecz, W., Szymczak, C. (eds.) Shell-Structures: Theory and Applications, pp. 149–152. Taylor and Francis Group, London (2006)

    Google Scholar 

  29. 29.

    Neff, P.: A geometrically exact planar Cosserat shell-model with microstructure: existence of minimizers for zero Cosserat couple modulus. Math. Models Methods Appl. Sci. 17, 363–392 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Neff, P., Chełmiński, K.: A geometrically exact Cosserat shell-model for defective elastic crystals. Justification via Γ-convergence. Interfaces Free Bound. 9, 455–492 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Neff, P., Forest, S.: A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elast. 87, 239–276 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Neff, P., Hong, K.-I., Jeong, J.: The Reissner-Mindlin plate is the Γ-limit of Cosserat elasticity. Math. Models Methods Appl. Sci. 20, 1553–1590 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Neff, P., Jeong, J., Münch, I., Ramezani, H.: Linear Cosserat elasticity, conformal curvature and bounded stiffness. In: Maugin, G.A., Metrikine, V.A. (eds.) Mechanics of Generalized Continua. One Hundred Years After the Cosserats. Advances in Mechanics and Mathematics, vol. 21, pp. 55–63. Springer, Berlin (2010)

    Google Scholar 

  34. 34.

    Paroni, R.: Theory of linearly elastic residually stressed plates. Math. Mech. Solids 11, 137–159 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Paroni, R., Podio-Guidugli, P., Tomassetti, G.: The Reissner-Mindlin plate theory via Γ-convergence. C. R. Acad. Sci. Paris, Ser. I 343, 437–440 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Pietraszkiewicz, W.: Refined resultant thermomechanics of shells. Int. J. Eng. Sci. 49, 1112–1124 (2011)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Reissner, E.: Linear and nonlinear theory of shells. In: Fung, Y.C., Sechler, E.E. (eds.) Thin Shell Structures, pp. 29–44. Prentice-Hall, Englewood Cliffs (1974)

    Google Scholar 

  38. 38.

    Rubin, M.B.: Cosserat Theories: Shells, Rods and Points. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  39. 39.

    Sansour, C., Bufler, H.: An exact finite rotation shell theory, its mixed variational formulation and its finite element implementation. Int. J. Numer. Methods Eng. 34, 73–115 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Simo, J.C., Fox, D.D.: On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization. Comput. Methods Appl. Mech. Eng. 72, 267–304 (1989)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  41. 41.

    Sprekels, J., Tiba, D.: An analytic approach to a generalized Naghdi shell model. Adv. Math. Sci. Appl. 12, 175–190 (2002)

    MathSciNet  Google Scholar 

  42. 42.

    Zhilin, P.A.: Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12, 635–648 (1976)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Zubov, L.M.: Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin (1997)

    Google Scholar 

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Acknowledgements

The first author (M.B.) is supported by the German state grant: “Programm des Bundes und der Länder für bessere Studienbedingungen und mehr Qualität in der Lehre”. Useful discussions with Professor V.A. Eremeyev are gratefully acknowledged.

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Correspondence to Patrizio Neff.

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Bîrsan, M., Neff, P. Existence Theorems in the Geometrically Non-linear 6-Parameter Theory of Elastic Plates. J Elast 112, 185–198 (2013). https://doi.org/10.1007/s10659-012-9405-2

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Keywords

  • Elastic plates
  • Geometrically non-linear plates
  • Shells
  • Existence of minimizers
  • 6-parameter shell theory
  • Cosserat plate

Mathematics Subject Classification (2010)

  • 74K20
  • 74K25
  • 74G65
  • 74G25