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On the Coercivity of Strain Energy Functions in Generalized Models of 6-Parameter Shells

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Sixty Shades of Generalized Continua

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 170))

Abstract

In this paper we consider geometrically nonlinear 6-parameter shell models. We establish some existence proofs by the direct methods of the calculus of variations. In contrast to more classical approaches, we also investigate models up to order h5 in the shell thickness, where the form of the equations is determined by a dimensional descent from a three-dimensional Cosserat model.

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References

  1. Love A (1944) A Treatise on the Mathematical Theory of Elasticity, Dover, New York.

    Google Scholar 

  2. Novozhilov V (1959) The Theory of Thin Shells, Noordhoff, Groningen.

    Google Scholar 

  3. Koiter W (1966) On the nonlinear theory of thin elastic shells, Proc K Ned Akad Wet B 69:1-54.

    Google Scholar 

  4. Ciarlet P (2000) Mathematical Elasticity, Vol. III: Theory of Shells, North-Holland, Amsterdam.

    Google Scholar 

  5. Steigmann D (2013) Koiter’s shell theory from the perspective of three-dimensional nonlinear elasticity, J Elasticity 111:91-107.

    Google Scholar 

  6. Steigmann D, Bîrsan M, Shirani M (2023) Lecture Notes on the Theory of Plates and Shells, Series Solid Mechanics and Its Applications, Springer, to appear.

    Google Scholar 

  7. Reissner E (1974) Linear and nonlinear theory of shells, In: YC Fung, EE Sechler (Eds) Thin Shell Structures, pp 29-44, Prentice-Hall, Englewood Cliffs, New Jersey.

    Google Scholar 

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

    Google Scholar 

  9. Altenbach H, Zhilin P (2004) The theory of simple elastic shells, In: R Kienzler, H Altenbach, I Ott (Eds) Theories of Plates and Shells. Critical Review and New Applications, Euromech Colloquium 444, Lecture Notes in Applied and Computational Mechanics, vol. 16, pp 1-12, Springer, Heidelberg

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  12. Eremeyev V, Zubov L (2008) Mechanics of Elastic Shells (in Russ), Nauka, Moscow.

    Google Scholar 

  13. Ciarlet P (1998) Introduction to Linear Shell Theory, Gauthier-Villars, Paris.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  16. Ghiba I, Bîrsan M, Neff P (2022) A linear isotropic Cosserat shell model including terms up to O(h5). Existence and uniqueness, (submitted), arXiv:2208.04574v1.

  17. Neff P (2004) 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, Cont Mech Thermodynamics 16:577-628.

    Google Scholar 

  18. Neff P (2007) A geometrically exact planar Cosserat shell-model with microstructure: Existence of minimizers for zero Cosserat couple modulus, Math Mod Meth Appl Sci 17:363-392.

    Google Scholar 

  19. Neff P, Bîrsan M, Osterbrink F (2015) Existence theorem for geometrically nonlinear Cosserat micropolar model under uniform convexity requirements, J Elasticity 121:119-141.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  22. Bîrsan M (2020) Derivation of a refined six-parameter shell model: descent from the three-dimensional Cosserat elasticity using a method of classical shell theory, Math Mech Solids 25:1318-1339.

    Google Scholar 

  23. Eremeyev V, Pietraszkiewicz W (2004) The nonlinear theory of elastic shells with phase transitions, J Elasticity 74:67-86.

    Google Scholar 

  24. Pietraszkiewicz W (2011) Refined resultant thermomechanics of shells, Int J Engng Sci 49:1112-1124.

    Google Scholar 

  25. Bîrsan M, Neff P (2017) Analysis of the deformation of Cosserat elastic shells using the dislocation density tensor, In: F dell’Isola, M Sofonea, D Steigmann (Eds) Mathematical Modelling in Solid Mechanics, Advanced Structured Materials, vol. 69, pp 13-30, Springer Nature, Singapore.

    Google Scholar 

  26. Altenbach H, Eremeyev V (2009) On the linear theory of micropolar plates, Z Angew Math Mech 89:242-256.

    Google Scholar 

  27. Altenbach H, Eremeyev V (2013) Cosserat-type shells, In: H Altenbach, V Eremeyev (Eds) Generalized Continua - from the Theory to Engineering Applications, CISM Courses and Lectures, vol. 541, pp 131-178, Springer, Wien.

    Google Scholar 

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

    Google Scholar 

  29. Bîrsan M, Ghiba I, Martin R, Neff P (2019) Refined dimensional reduction for isotropic elastic Cosserat shells with initial curvature, Math Mech Solids 24:4000-4019.

    Google Scholar 

  30. Ghiba I, Bîrsan M, Lewintan P, Neff P (2020) The isotropic Cosserat shell model including terms up to O(h5). Part I: Derivation in matrix notation, J Elasticity 142:201-262.

    Google Scholar 

  31. Ghiba I, Bîrsan M, Lewintan P, Neff P (2020) The isotropic Cosserat shell model including terms up to O(h5). Part II: Existence of minimizers, J Elasticity 142:263-290.

    Google Scholar 

  32. Sander O, Neff P, Bîrsan M (2016) Numerical treatment of a geometrically nonlinear planar Cosserat shell model, Comput Mech 57:817-841.

    Google Scholar 

  33. Bîrsan M (2011) Alternative derivation of the higher-order constitutive model for six-parameter elastic shells, Z Angew Math Phys 72:50.

    Google Scholar 

  34. Ghiba I, Bîrsan M, Lewintan P, Neff P (2021) A constrained Cosserat-shell model including terms up to O(h5), J Elasticity 146:83-141.

    Google Scholar 

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Acknowledgements

This research has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project no. 415894848: BI 1965/2-1 (M. Bîrsan) and NE 902/8-1 (P. Neff).

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University Duisburg-Essen, Thea-Leymann Str. 9, 45127, Essen, Germany

    Mircea Bîrsan

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

    Mircea Bîrsan

  3. Lehrstuhl für Nichtlineare Analysis und Modellierung, Faculty of Mathematics, University Duisburg-Essen, Thea-Leymann Str. 9, 45127, Essen, Germany

    Patrizio Neff

Authors
  1. Mircea Bîrsan
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  2. Patrizio Neff
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Correspondence to Mircea Bîrsan .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Fakultät für Maschinenbau, Otto-von-Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

    Holm Altenbach

  2. Tallinn University of Technology, Tallinn, Estonia

    Arkadi Berezovski

  3. Università degli Studi dell’Aquila, L'Aquila, Italy

    Francesco dell'Isola

  4. Institute of Problems of Mechanical Engineering, Saint Petersburg, Russia

    Alexey Porubov

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Bîrsan, M., Neff, P. (2023). On the Coercivity of Strain Energy Functions in Generalized Models of 6-Parameter Shells. In: Altenbach, H., Berezovski, A., dell'Isola, F., Porubov, A. (eds) Sixty Shades of Generalized Continua. Advanced Structured Materials, vol 170. Springer, Cham. https://doi.org/10.1007/978-3-031-26186-2_6

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