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Continuum Mechanics and Thermodynamics

, Volume 31, Issue 1, pp 189–207 | Cite as

An asymptotic membrane model for wrinkling of very thin films

  • Antonio BattistaEmail author
  • Aziz Hamdouni
  • Olivier Millet
Original Article
  • 39 Downloads

Abstract

In this work, a formal deduction of a two-dimensional membrane theory, similar to Landau–Lifshitz model, is performed via an asymptotic development of the weak formulation of the three-dimensional equations of elasticity. Some interesting aspects of the deduced model are investigated, in particular the property of obtaining a hyperbolic equation for the out-of-plane displacement under a certain class of boundary conditions and loads. Some simple cases are analyzed to show the relevant aspects of the model and the phenomenology that can be addressed. In particular, it is shown how this mathematical formulation is capable to describe instabilities well known as wrinkling, often observed for the buckling of very thin membranes.

Keywords

Asymptotic methods Dimensional analysis Plate theory Membrane theory Hyperbolic problem 

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References

  1. 1.
    Friedrichs, K.O., Dressler, R.F.: A boundary-layer theory for elastic plates. Commun. Pure Appl. Math. 14(1), 1–33 (1961)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ciarlet, P.G., Destuynder, P.: Une justification du modèle bi-harmonique en théorie des plaques. C. R. Acad. Sci. Paris 285, 851–854 (1977)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, 3rd edn. Pergamon Press, Oxford, UK (1986)zbMATHGoogle Scholar
  4. 4.
    Vandeparre, H., Piñeirua, M., Brau, F., Roman, B., Bico, J., Gay, C., Bao, W., Lau, C.N., Reis, P.M., Damman, P.: Wrinkling hierarchy in constrained thin sheets from suspended graphene to curtains. Phys. Rev. Lett. 106(22), 224301 (2011)ADSGoogle Scholar
  5. 5.
    Hure, J., Roman, B., Bico, J.: Stamping and wrinkling of elastic plates. Phys. Rev. Lett. 109(5), 054302 (2012)ADSGoogle Scholar
  6. 6.
    Takei, A., Brau, F., Roman, B., Bico, J.: Stretch-induced wrinkles in reinforced membranes: From out-of-plane to in-plane structures. EPL (Europhys. Lett.) 96(6), 64001 (2011)ADSGoogle Scholar
  7. 7.
    Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49(12), 1294–1301 (2011)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Altenbach, H., Eremeyev, V.A., Morozov, N.F.: Linear theory of shells taking into account surface stresses. In Doklady Physics 54(12), 531 (2009). (SP MAIK Nauka/Interperiodica)ADSGoogle Scholar
  9. 9.
    Autieri, C.: Antiferromagnetic and xy ferro-orbital order in insulating SrRuO3 thin films with SrO termination. J. Phys. Condens. Matter 28(42), 426004 (2016)ADSGoogle Scholar
  10. 10.
    Giorgio, I., Corte, A.Della, dell’Isola, F., Steigmann, D.J., Steigmann, D.J.: Buckling modes in pantographic lattices. C. R. Mecanique 344(7), 487–501 (2016)ADSGoogle Scholar
  11. 11.
    Giorgio, I., Grygoruk, R., dell’Isola, F., Steigmann, D.J.: Pattern formation in the three-dimensional deformations of fibered sheets. Mech. Res. Commun. 69, 164–171 (2015)Google Scholar
  12. 12.
    Alibert, J.J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Seppecher, P., Alibert, J.J., dell’Isola, F.: Linear elastic trusses leading to continua with exotic mechanical interactions. J. Phys. Conf. Ser. 319, 012018 (2011)Google Scholar
  14. 14.
    Madeo, A., Ferretti, M., dell’Isola, F., Boisse, P.: Thick fibrous composite reinforcements behave as special second-gradient materials: three-point bending of 3D interlocks. Zeitschrift für angewandte Mathematik und Physik 66(4), 2041–2060 (2015)ADSMathSciNetzbMATHGoogle Scholar
  15. 15.
    Millet, O., Hamdouni, A., Cimetière, A., Elamri, K.: Analyse dimensionnelle de l’équation de navier et application à la théorie des plaques minces. Journal de Physique III 7(10), 1909–1925 (1997)ADSGoogle Scholar
  16. 16.
    Millet, O., Hamdouni, A., Cimetière, A.: Dimensional analysis and asymptotic expansions of equilibrium equations in nonlinear elasticity. Part I: the membrane model. Arch. Mech. 50(6), 953–973 (1998)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Millet, O., Hamdouni, A., Cimetière, A.: Dimensional analysis and asymptotic expansions of equilibrium equations in nonlinear elasticity. Part II: the two-dimensional von karman model. Arch. Mech. 50(6), 975–1001 (1998)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Millet, O., Hamdouni, A., Cimetière, A.: Construction d’un modèle eulérien de plaques en grands déplacements par méthode asymptotique. Comptes Rendus de l’Académie des Sciences-Series IIB-Mechanics-Physics-Chemistry-Astronomy 325(5), 257–261 (1997)zbMATHGoogle Scholar
  19. 19.
    Millet, O., Hamdouni, A., Cimetière, A.: A classification of thin plate models by asymptotic expansion of non-linear three-dimensional equilibrium equations. Int. J. Non-Linear Mech. 36(1), 165–186 (2001)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hamdouni, A., Millet, O.: Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Part I: the shallow shells. Arch. Mech. 55(2), 135–176 (2003)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hamdouni, A., Millet, O.: Classification of thin shell models deduced from the nonlinear three-dimensional elasticity. Part II: the strongly curved shells. Arch. Mech. 55(2), 177–220 (2003)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Audoly, B., Roman, B., Pocheau, A.: Secondary buckling patterns of a thin plate under in-plane compression. Eur. Phys. J. B Condens. Matter Complex Syst. 27(1), 7–10 (2002)Google Scholar
  23. 23.
    Audoly, B.: Stability of straight delamination blisters. Phys. Rev. Lett. 83(20), 4124 (1999)ADSGoogle Scholar
  24. 24.
    Hutchinson, J.W., He, M.Y., Evans, A.G.: The influence of imperfections on the nucleation and propagation of buckling driven delaminations. J. Mech. Phys. Solids 48(4), 709–734 (2000)ADSzbMATHGoogle Scholar
  25. 25.
    Audoly, B., Boudaoud, A.: Buckling of a stiff film bound to a compliant substrate-Part I: formulation, linear stability of cylindrical patterns, secondary bifurcations. J. Mech. Phys. Solids 56(7), 2401–2421 (2008)ADSMathSciNetzbMATHGoogle Scholar
  26. 26.
    Audoly, B., Boudaoud, A.: Buckling of a stiff film bound to a compliant substrate-Part II: a global scenario for the formation of herringbone pattern. J. Mech. Phys. Solids 56(7), 2422–2443 (2008)ADSMathSciNetzbMATHGoogle Scholar
  27. 27.
    Audoly, B., Boudaoud, A.: Buckling of a stiff film bound to a compliant substrate-Part III: herringbone solutions at large buckling parameter. J. Mech. Phys. Solids 56(7), 2444–2458 (2008)ADSMathSciNetzbMATHGoogle Scholar
  28. 28.
    Rivlin, R.S.: Plane strain of a net formed by inextensible cords. In: Collected Papers of RS Rivlin, pp. 511–534. Springer, New York (1997)Google Scholar
  29. 29.
    dell’Isola, F., Della Corte, A., Greco, L., Luongo, A.: Plane bias extension test for a continuum with two inextensible families of fibers: a variational treatment with lagrange multipliers and a perturbation solution. Int. J. Solids Struct. 81, 1–12 (2016)Google Scholar
  30. 30.
    Placidi, L., Greco, L., Bucci, S., Turco, E., Rizzi, N.L.: A second gradient formulation for a 2D fabric sheet with inextensible fibres. Zeitschrift für angewandte Mathematik und Physik 67(5), 114 (2016)ADSMathSciNetzbMATHGoogle Scholar
  31. 31.
    Placidi, L., Andreaus, U., Della Corte, A., Lekszycki, T.: Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Zeitschrift für angewandte Mathematik und Physik 66(6), 3699–3725 (2015)ADSMathSciNetzbMATHGoogle Scholar
  32. 32.
    Placidi, L., Andreaus, U., Giorgio, I.: Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J. Eng. Math. 103, 1–21 (2016)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Alibert, J.J., Della Corte, A.: Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Zeitschrift für angewandte Mathematik und Physik 66(5), 2855–2870 (2015)ADSMathSciNetzbMATHGoogle Scholar
  34. 34.
    Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für Angewandte Mathematik und Physik 67(4), 1–28 (2016)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Turco, E.: Discrete is it enough? The revival of Piola–Hencky keynotes to analyze three-dimensional Elastica. Contin. Mech. Thermodyn. pp. 1–19 (2018).  https://doi.org/10.1007/s00161-018-0656-4

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire des Sciences de l’Ingénieur pour l’Environnement UMR - 7356 CNRSUniversité de La RochelleLa RochelleFrance
  2. 2.M&MoCSUniversità degli Studi dell’AquilaL’AquilaItaly

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