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Tangential differential calculus and the finite element modeling of a large deformation elastic membrane problem

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Abstract

We develop a finite element method for a large deformation membrane elasticity problem on meshed curved surfaces using a tangential differential calculus approach that avoids the use of classical differential geometric methods. The method is also applied to form finding problems.

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Notes

  1. Henceforth, we use the notation \(\{\bullet \}\) to indicate an implicit function.

  2. Note that the first Piola Kirchhoff stress tensor is a two-point tensor with “legs” in both current and reference configurations.

  3. For illustration, we here give the explicit component forms in the orhogonal \(XYZ\) and \(\xi \eta \zeta \) systems, respectively.

  4. Note that \(\varvec{L}_{NN}\) is of second order.

References

  1. Ahmad S, Irons BM, Zienkiewicz OC (1970) Analysis of thick and thin shell structures by curved finite elements. Intern J Numer Methods Eng 2(3):419–451

    Article  Google Scholar 

  2. Betsch P, Gruttmann F, Stein E (1996) A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains. Comput Methods Appl Mech Eng 130(1–2):57–79

    Article  MATH  MathSciNet  Google Scholar 

  3. Bletzinger K-U, Wüchner R, Daoud F, Camprubí N (2005) Computational methods for form finding and optimization of shells and membranes. Comput Methods Appl Mech Eng 194(30–33):3438–3452

    Article  MATH  Google Scholar 

  4. Bonet J, Wood RD (2008) Nonlinear continuum mechanics for finite element analysis, 2nd edn. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  5. Campello EMB, Pimenta PM, Wriggers P (2003) A triangular finite shell element based on a fully nonlinear shell formulation. Comput Mech 31(6):505–518

  6. Chapelle D, Bathe KJ (2003) The finite element analysis of shells-fundamentals. Computational fluid and solid mechanics. Springer, Berlin

    Book  Google Scholar 

  7. Chien D (1995) Numerical evaluation of surface integrals in three dimensions. Math Comput 64(210):727–743

    Article  MATH  MathSciNet  Google Scholar 

  8. Ciarlet PG (2000) Mathematical elasticity. Vol. III, volume 29 of studies in mathematics and its applications. North-Holland Publishing Co., Amsterdam

    Google Scholar 

  9. Delfour MC, Zolésio J-P (1995) A boundary differential equation for thin shells. J Differ Equ 119(2):426–449

    Article  MATH  Google Scholar 

  10. Delfour MC, Zolésio J-P (1997) Differential equations for linear shells: comparison between intrinsic and classical models. In: Advances in mathematical sciences: CRM’s 25 years (Montreal, PQ, 1994), volume 11 of CRM Proceedings, Lecture Notes, American Mathematical Society, Providence, pp 41–124

  11. Dziuk G (1991) An algorithm for evolutionary surfaces. Numer Math 58(6):603–611

    MATH  MathSciNet  Google Scholar 

  12. Monteiro EE, He Q-C, Yvonnet J (2011) Hyperelastic large deformations of two-phase composites with membrane-type interface. Int J Eng Sci 49(9):985–1000

    Article  MATH  Google Scholar 

  13. Hansbo P, Larson MG (2014) Finite element modeling of a linear membrane shell problem using tangential differential calculus. Comput Methods Appl Mech Eng 270:1–14

    Article  MATH  MathSciNet  Google Scholar 

  14. Hansbo P, Larson MG, Larsson K (2014) Variational formulation of curved beams in global coordinates. Comput Mech 53(4):611–623

    Article  MATH  MathSciNet  Google Scholar 

  15. Hughes TJR, Liu WK (1981) Nonlinear finite element analysis of shells: part I. Three dimensional shells. Comput Methods Appl Mech Eng 26(3):331–362

    Article  MATH  MathSciNet  Google Scholar 

  16. Ibrahimbegović A, Gruttmann F (1993) A consistent finite element formulation of nonlinear membrane shell theory with particular reference to elastic rubberlike material. Finite Elem Anal Des 13(1):75–86

    Article  MATH  Google Scholar 

  17. Klinkel S, Govindjee S (2002) Using finite strain 3D-material models in beam and shell elements. Eng Comput 19(8):902–921

    Article  MATH  Google Scholar 

  18. Larsson F, Runesson K (2004) Modeling and discretization errors in hyperelasto-(visco-)plasticity with a view to hierarchical modeling. Comput Methods Appl Mech Eng 193(48–51):5283–5300

    Article  MATH  MathSciNet  Google Scholar 

  19. Pimenta PM, Campello EMB (2009) Shell curvature as an initial deformation: a geometrically exact finite element approach. Intern J Numer Methods Eng 78(9):1094–1112

    Article  MATH  MathSciNet  Google Scholar 

  20. Simo JC, Fox DD, Rifai MS (1989) On a stress resultant geometrically exact shell model. II. The linear theory; computational aspects. Comput Methods Appl Mech Eng 73(1):53–92

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This research was supported in part by the Swedish Foundation for Strategic Research Grant No. AM13-0029 and the Swedish Research Council Grants No. 2011-4992 and No. 2013-4708.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Jönköping University, 551 11, Jönköping, Sweden

    Peter Hansbo

  2. Department of Mathematics and Mathematical Statistics, Umeå University, 901 87, Umeå, Sweden

    Mats G. Larson

  3. Department of Applied Mechanics, Chalmers University of Technology, 412 96, Göteborg, Sweden

    Fredrik Larsson

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  1. Peter Hansbo
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  2. Mats G. Larson
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  3. Fredrik Larsson
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Correspondence to Peter Hansbo.

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Hansbo, P., Larson, M.G. & Larsson, F. Tangential differential calculus and the finite element modeling of a large deformation elastic membrane problem. Comput Mech 56, 87–95 (2015). https://doi.org/10.1007/s00466-015-1158-x

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