Skip to main content
SpringerLink
Log in

Linear constrained Cosserat-shell models including terms up to \({O}(h^5)\): conditional and unconditional existence and uniqueness

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

In this paper, we linearise the recently introduced geometrically nonlinear constrained Cosserat-shell model. In the framework of the linear constrained Cosserat-shell model, we provide a comparison of our linear models with the classical linear Koiter shell model and the “best” first-order shell model. For all proposed linear models, we show existence and uniqueness based on a Korn’s inequality for surfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The terms of order \( O(x_3^3) \) are not relevant here, since we have taken a quadratic ansatz for the deformation.

  2. Observe that \(\mathrm{det}(\nabla y_0|n_0)=\sqrt{\det ([\nabla y_0]^T\nabla y_0)}\) is independent of \(n_0\).

  3. The definition of the admissible set \({\mathcal {A}}^{\mathrm{mod}}\) incorporates a weak reformulation of the imposed symmetry constraint \({\mathcal {E}}_{\infty } \in \mathrm{Sym}(3)\). We notice that the constraints \(U:= {Q}_{ \infty }^T (\nabla m|{Q}_{ \infty }Q_0.e_3)[\nabla \Theta ]^{-1} \in \mathrm{L^2}(\omega , \mathrm{Sym}^+(3))\) together with the compatibility conditions between \({Q}_{ \infty }\Big |_{\gamma _d}\) and the values of m on \({\gamma _d}\) will imply that \(Q_{\infty }\) and m are not independent variables and \( {Q}_{ \infty }=\mathrm{polar}\big [(\nabla m|n) [\nabla \Theta ]^{-1}\big ]\in \text {SO}(3), \) where \(n=\frac{\partial _{x_1} m\times \partial _{x_2} m}{\Vert \partial _{x_1} m\times \partial _{x_2} m\Vert }\) is the unit normal vector to the deformed midsurface. Assuming that the boundary data satisfy the conditions \({m}^*\in \mathrm{H}^1(\omega , {\mathbb {R}}^3)\) and \(\mathrm{polar}(\nabla {m}^*\,|\,n^*)\in \mathrm{H}^1(\omega , \mathrm{SO}(3))\), it follows that the admissible set is not empty.

References

  1. Adams, R.A.: Sobolev Spaces, Volume 65 of Pure and Applied Mathematics, 1st edn. Academic Press, London (1975)

    Google Scholar 

  2. Altenbach, H., Zhilin, P.A.: A general theory of elastic simple shells (in Russian). Uspekhi Mekhaniki 11, 107–148 (1988)

    MathSciNet  Google Scholar 

  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. Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch. Appl. Mech. 80, 73–92 (2010)

    Article  MATH  Google Scholar 

  5. Anicic, S.: A shell model allowing folds. In: Numerical Mathematics and Advanced Applications, pp. 317–326. Springer (2003)

  6. Anicic, S.: Du modèle de Kirchhoff-Love exact à un modèle de coque mince et á un modèle de coque pliée. PhD thesis, Université Joseph Fourier (Grenoble 1) (2001)

  7. Anicic, S., Léger, A.: Formulation bidimensionnelle exacte du modèle de coque 3D de Kirchhoff-Love. C. R. Acad. Sci. Paris Ser. Math. 329(8), 741–746 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bîrsan, M., Ghiba, I.D., Neff, P.: A linear Cosserat-shell model including terms up to \({O}(h^5)\). submitted, arXiv:2208.04574

  9. Bîrsan, M.: On Saint-Venant’s principle in the theory of Cosserat elastic shells. Int. J. Eng. Sci. 45, 187–198 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Bîrsan, M.: Thermal stresses in cylindrical Cosserat elastic shells. Eur. J. Mech. Solids A 28, 94–101 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Bîrsan, M., Altenbach, H.: On the dynamical theory of thermoelastic simple shells. Z. Angew. Math. Mech. 91, 443–457 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bîrsan, M., Neff, P.: Existence of minimizers in the geometrically non-linear 6-parameter resultant shell theory with drilling rotations. Math. Mech. Solids 19(4), 376–397 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bîrsan, M., Neff, P.: Shells without drilling rotations: A representation theorem in the framework of the geometrically nonlinear 6-parameter resultant shell theory. Int. J. Eng. Sci. 80, 32–42 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Blouza, A., Dret, H.L.: Sur le lemme du mouvement rigide. C. R. Acad. Sci. Paris Ser. Math. 319(9), 1015–1020 (1994)

    MathSciNet  MATH  Google Scholar 

  17. Blouza, A., Le Dret, H.: Existence and uniqueness for the linear Koiter model for shells with little regularity. Q. Appl. Math. 57(2), 317–337 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Budiansky, B., Sanders Jr., J.L.: On the “best” first-order linear shell theory. Progress in Applied Mechanics, The Prager Anniversary Volume (1963)

  19. Budiansky, B., Sanders Jr., J.L.: On the “best” first-order linear shell theory. Technical Report No. 14, Contract No. 1886(02). Division of Engineering and Applied Physics, Harvard University (1962)

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

    Google Scholar 

  21. Ciarlet, Ph.G.: Introduction to Linear Shell Theory. Series in Applied Mathematics, 1st edn. Gauthier-Villars, Paris (1998)

    Google Scholar 

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

    MATH  Google Scholar 

  23. Ciarlet, P.G.: An Introduction to Differential Geometry with Applications to Elasticity. Springer, Berlin (2005)

    MATH  Google Scholar 

  24. Ciarlet, Ph.G.: An introduction to differential geometry with applications to elasticity. J. Elast. 78(1–3), 1–215 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Cosserat, E., Cosserat, F.: Théorie des corps déformables. Hermann et Fils (reprint 2009), Paris (1909)

  26. Cosserat, E., Cosserat, F.: Sur la théorie des corps minces. C. R. Acad. Sci. 146, 169–172 (1908)

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  28. Do Carmo, M.G.: Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs (1976)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. Ghiba, I.D., Bîrsan, M., Lewintan, P., Neff, P.: The isotropic Cosserat shell model including terms up to \({O}(h^5)\). Part I: Derivation in matrix notation. J. Elast. 142, 201–262 (2020)

  31. Ghiba, I.D., Bîrsan, M., Lewintan, P., Neff, P.: The isotropic elastic Cosserat shell model including terms up to order \(O(h^5)\) in the shell thickness. Part II: Existence of minimizers. J. Elast. 142, 263–290 (2020)

  32. Ghiba, I.D., Lewintan, P., Neff, P.: On deformation measures in shell models (in preparation, 2022)

  33. Ghiba, I.D., Bîrsan, M., Lewintan, P., Neff, P.: A constrained Cosserat-shell model including terms up to \({O}(h^5)\). J. Elast. 146(1), 83–141 (2021)

    Article  MATH  Google Scholar 

  34. Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier-Stokes Equations, volume 749 of Lect. Notes Math. Springer, Heidelberg (1979)

  35. Koiter, W.T., Simmonds, J.G.: Foundations of shell theory. In: Theoretical and Applied Mechanics, pp. 150–176. Springer (1973)

  36. Leis, R.: Initial Boundary Value Problems in Mathematical Physics. Teubner, Stuttgart (1986)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  38. 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 

  39. 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)

    Article  MathSciNet  MATH  Google Scholar 

  40. Neff, P., Lankeit, J., Madeo, A.: On Grioli’s minimum property and its relation to Cauchy’s polar decomposition. Int. J. Eng. Sci. 80, 207–217 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  43. Toupin, R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 11, 385–413 (1962)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  45. Zhilin, P.A.: Applied Mechanics—Foundations of Shell Theory. State Polytechnical University Publisher, Sankt Petersburg (2006). ((in Russian))

    Google Scholar 

Download references

Acknowledgements

This research has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project No. 415894848: NE 902/8-1 (P. Neff).

Author information

Authors and Affiliations

  1. Department of Mathematics, Alexandru Ioan Cuza University of Iaşi, Blvd. Carol I, no. 11, 700506, Iasi, Romania

    Ionel-Dumitrel Ghiba

  2. Octav Mayer Institute of Mathematics of the Romanian Academy, Iaşi Branch, 700505, Iasi, Romania

    Ionel-Dumitrel Ghiba

  3. Head of Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127, Essen, Germany

    Patrizio Neff

Authors
  1. Ionel-Dumitrel Ghiba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patrizio Neff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ionel-Dumitrel Ghiba.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghiba, ID., Neff, P. Linear constrained Cosserat-shell models including terms up to \({O}(h^5)\): conditional and unconditional existence and uniqueness. Z. Angew. Math. Phys. 74, 47 (2023). https://doi.org/10.1007/s00033-023-01937-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-023-01937-7

Keywords

Mathematics Subject Classification

Search

Navigation