Skip to main content
SpringerLink
Log in

Wrinkling of Orthotropic Membranes: An Analysis by the Polar Method

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

In the present paper, a simple membrane model based on the wrinkle strain approach is revisited with the aim of examining how the material elastic constants affect the static response of anisotropic membranes when wrinkling is taken into account.

Employing the polar method, we analyze the role played by the polar moduli, which enable expressing the elasticity matrix components of an anisotropic material in terms of its invariant quantities. With reference to orthotropic materials, we first address the issue of membrane susceptibility to wrinkling by investigating the influence of the three polar parameters characterizing the anisotropic part of the constitutive law. The stress and strain states at any given point in a wrinkled membrane are analyzed by searching for explicit expressions for the principal values of stress and wrinkle strain.

Finally, a comparison between our results and those obtained by a numerical solution available in the literature is made in the basic case of a membrane subjected to a pure shear strain state.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Wong, Y.W., Pellegrino, S.: Wrinkled membranes—Part III: numerical simulations. J. Mech. Mater. Struct. 1, 61–93 (2006)

    Google Scholar 

  2. Reissner, E.: On tension field theory. In: Proc. 5th Int. Cong. Appl. Mech, pp. 88–92 (1938)

    Google Scholar 

  3. Wu, C.H.: Plane linear wrinkle elasticity without body force. Report Dept. Materials Engineering, University of Illinois, Chicago (1974)

  4. Pipkin, A.C.: The relaxed energy density for isotropic elastic membranes. IMA J. Appl. Math. 50, 225–237 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Pipkin, A.C.: Relaxed energy densities for large deformations of membranes. IMA J. Appl. Math. 52, 297–308 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Epstein, M.: On the wrinkling of anisotropic elastic membranes. J. Elast. 55, 99–109 (1999)

    Article  MATH  Google Scholar 

  7. Barsotti, R., Ligarò, S.S.: Stress distributions in partly wrinkled anisotropic membranes. In: Proc. WCCM V, Fifth World Congress on Computational Mechanics, Vienna, Austria, July 7–12, 2002

    Google Scholar 

  8. Verchery, G.: Les invariants des tenseurs d’ordre 4 du type de l’élasticité. In: Proc. of Colloque Euromech 115, Villard-de-Lans, pp. 93–104. Editions du CNRS, Paris (1979)

    Google Scholar 

  9. Roddeman, D.G., Drucker, J., Oomens, C.W.C.: The wrinkling of thin membranes: part I theory. J. Appl. Mech. 54, 884–887 (1987)

    Article  ADS  MATH  Google Scholar 

  10. Vannucci, P.: Plane anisotropy by the polar method. Meccanica 40, 437–454 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Vannucci, P.: The polar analysis of a third order piezoelectricity-like plane tensor. Int. J. Solids Struct. 44, 7803–7815 (2007)

    Article  MATH  Google Scholar 

  12. Vannucci, P., Verchery, G.: Anisotropy of plane complex elastic bodies. Int. J. Solids Struct. 47, 1154–1166 (2010)

    Article  MATH  Google Scholar 

  13. Klein, F.: The Mathematical Theory of the Top. Princeton, USA (1896)

  14. Michell, J.H.: The inversion of plane stress. Proc. Lond. Math. Soc. 34, 134–142 (1902)

    MathSciNet  MATH  Google Scholar 

  15. Kolosov, G.V.: On an Application of Complex Function Theory to a Plane Problem of the Mathematical Theory of Elasticity. Yuriev, Russia (1909)

  16. Muskhelishvili, N.J.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Gröningen (1953). English translation by J.R.M. Radok

    MATH  Google Scholar 

  17. Green, A., Zerna, W.: Theoretical Elasticity. Clarendon, Oxford (1954)

    MATH  Google Scholar 

  18. Milne-Thomson, L.M.: Plane Elastic Systems. Springer, Berlin (1960)

    Book  MATH  Google Scholar 

  19. Vannucci, P.: A special planar orthotropic material. J. Elast. 67, 81–96 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Vannucci, P.: On special anisotropy of paper. J. Elast. 99, 75–83 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Vannucci, P.: Influence of invariant material parameters on the flexural optimal design of thin anisotropic laminates. Int. J. Mech. Sci. 51, 192–203 (2009)

    Article  MATH  Google Scholar 

  22. Pedersen, P.: On optimal orientation of orthotropic materials. Struct. Optim. 1, 101–106 (1989)

    Article  Google Scholar 

  23. Vincenti, A., Desmorat, B.: Optimal orthotropy for minimum elastic energy by the polar method. J. Elast. 102, 55–78 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Catapano, A., Desmorat, B., Vannucci, P.: Invariant formulation of phenomenological failure criteria for orthotropic sheets and optimization of their strength. Math. Methods Appl. Sci. (2012). doi:10.1002/mma.2530

    MathSciNet  MATH  Google Scholar 

  25. Hornig, J., Schoop, H.: Closed form analysis of wrinkled membranes with linear stress–strain relation. Comput. Mech. 30, 259–264 (2003)

    Article  MATH  Google Scholar 

  26. Woo, K., Igawa, H., Jenkins, C.H.: Analysis of wrinkling behavior of anisotropic membrane. Comput. Model. Eng. Sci. 6, 397–408 (2004)

    MATH  Google Scholar 

  27. Jarasjarungkiat, A., Wuchner, R., Bletzinger, K.-U.: A wrinkling model based on material modification for isotropic and orthotropic membranes. Comput. Methods Appl. Mech. Eng. 197, 773–788 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Civil Engineering, University of Pisa, Largo L. Lazzarino 1, 56126, Pisa, Italy

    Riccardo Barsotti

  2. UFR des Sciences, Université de Versailles Saint-Quentin-en-Yvelines, 45 avenue des Etats-Unis, 78035, Versailles, France

    Paolo Vannucci

Authors
  1. Riccardo Barsotti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paolo Vannucci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Riccardo Barsotti.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barsotti, R., Vannucci, P. Wrinkling of Orthotropic Membranes: An Analysis by the Polar Method. J Elast 113, 5–26 (2013). https://doi.org/10.1007/s10659-012-9408-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-012-9408-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation