Skip to main content
SpringerLink
Log in

A simple triangular finite element for nonlinear thin shells: statics, dynamics and anisotropy

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

This work presents a simple finite element implementation of a geometrically exact and fully nonlinear Kirchhoff–Love shell model. Thus, the kinematics are based on a deformation gradient written in terms of the first- and second-order derivatives of the displacements. The resulting finite element formulation provides \(C^1\)-continuity using a penalty approach, which penalizes the kinking at the edges of neighboring elements. This approach enables the application of well-known \(C^0\)-continuous interpolations for the displacements, which leads to a simple finite element formulation, where the only unknowns are the nodal displacements. On the basis of polyconvex strain energy functions, the numerical framework for the simulation of isotropic and anisotropic thin shells is presented. A consistent plane stress condition is incorporated at the constitutive level of the model. A triangular finite element, with a quadratic interpolation for the displacements and a one-point integration for the enforcement of the \(C^1\)-continuity at the element interfaces leads to a robust shell element. Due to the simple nature of the element, even complex geometries can be meshed easily, which include folded and branched shells. The reliability and flexibility of the element formulation is shown in a couple of numerical examples, including also time dependent boundary value problems. A plane reference configuration is assumed for the shell mid-surface, but initially curved shells can be accomplished if one regards the initial configuration as a stress-free deformed state from the plane position, as done in previous works.

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

Notes

  1. Therein the author introduced the double-cross product: \(.\)

References

  1. Antman S (1995) Nonlinear problems of elasticity. Springer, New York

    Book  MATH  Google Scholar 

  2. Ball J (1977a) Convexity conditions and existence theorems in non-linear elasticity. Arch Ration Mech Anal 63:337–403

    Article  MATH  Google Scholar 

  3. Ball J (1977b) Constitutive inequalities and existence theorems in nonlinear elastostatics. In: Knops RJ (ed) Symposium on non-well posed problems and logarithmic convexity. Springer-lecture notes in mathematics, vol 316

  4. Balzani D, Neff P, Schröder J, Holzapfel G (2006) A polyconvex framework for soft biological tissues. Adjustment to experimental data. Int J Solids Struct 43(20):6052–6070

    Article  MathSciNet  MATH  Google Scholar 

  5. Balzani D, Gruttmann F, Schröder J (2008) Analysis of thin shells using anisotropic polyconvex energy densities. Comput Methods Appl Mech Eng 197:1015–1032

    Article  MathSciNet  MATH  Google Scholar 

  6. Balzani D, Schröder J, Neff P (2010) Applications of anisotropic polyconvex energies: thin shells and biomechanics of arterial walls. In: Poly-, quasi- and rank-one convexity in applied mechanics. CISM course and lectures, vol 516. Springer, New York

  7. Başar Y, Diing Y (1997) Shear deformation models for large-strain shell analysis. Int J Solids Struct 34:1687–1708

    Article  MATH  Google Scholar 

  8. Başar Y, Grytz Y (2004) Incompressibility at large strains and finite-element implementation. Acta Mech 168:75–101

    Article  MATH  Google Scholar 

  9. 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 130:57–79

    Article  MathSciNet  MATH  Google Scholar 

  10. Bischoff M, Ramm E (1997) Shear deformable shell elements for large strains and rotations. Int J Numer Methods Eng 40(23):4427–4449

    Article  MATH  Google Scholar 

  11. Boehler JP (1978) Lois de comportement anisotrope des milieux continus. J Méc 17(2):153–190

    MathSciNet  MATH  Google Scholar 

  12. Boehler J (1979) A simple derivation of representations for non-polynomial constitutive equations in some cases of anisotropy. Z angew Math Mech 59:157–167

    Article  MathSciNet  MATH  Google Scholar 

  13. Boehler JP (1987) Introduction to the invariant formulation of anisotropic constitutive equations. In: Boehler JP (ed) Applications of tensor functions in solid mechanics, vol 292. International Centre for Mechanical Sciences, Springer, Vienna, p 13–30. ISBN 978-3-211-81975-3

  14. Bonet J, Gil AJ, Ortigosa R (2016) On a tensor cross product based formulation of large strain solid mechanics. Int J Solids Struct. doi:10.1016/j.ijsolstr.2015.12.030. ISSN 0020-7683

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

    Article  MATH  Google Scholar 

  16. Campello EMB, Pimenta PM, Wriggers P (2011) An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Comput Mech 48:195–211

    Article  MathSciNet  MATH  Google Scholar 

  17. Ciarlet PG (1988) Mathematical elasticity: three-dimensional elasticity, vol I. Elsevier, Amsterdam

    MATH  Google Scholar 

  18. Ciarlet PG (1993) Mathematical elasticity: three-dimensional elasticity, vol 1. Elsevier, Amsterdam

    MATH  Google Scholar 

  19. Cirak F, Long Q (2011) Subdivision shells with exact boundary control and non-manifold geometry. Int J Numer Methods Eng 88(9):897–923

    Article  MathSciNet  MATH  Google Scholar 

  20. Cirak F, Ortiz M (2001) Fully C1-conforming subdivision elements for finite deformation thin-shell analysis. Int J Numer Methods Eng 51(7):813–833

    Article  MATH  Google Scholar 

  21. Dacorogna B (1989) Direct methods in the calculus of variations. Applied mathematical sciences, vol 78, 1st edn. Springer, Berlin,

  22. de Boer R (1982) Vektor-und Tensorrechnung für Ingenieure. Springer, Berlin

    Book  MATH  Google Scholar 

  23. Ebbing V, Balzani D, Schröder J, Neff P, Gruttmann F (2009) Construction of anisotropic polyconvex energies and applications to thin shells. Comput Mater Sci 46(3):639–641

    Article  Google Scholar 

  24. Hughes TJR, Tezduyar TE (1981) Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element. J Appl Mech 48(3):587–596

  25. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195

    Article  MathSciNet  MATH  Google Scholar 

  26. Ivannikov V, Tiago C, Pimenta PM (2014) Meshless implementation of the geometrically exact Kirchhoff–Love shell theory. Int J Numer Methods Eng 100:1–39

    Article  MathSciNet  MATH  Google Scholar 

  27. Ivannikov V, Tiago C, Pimenta PM (2015) Generalization of the C1 TUBA plate finite elements to the geometrically exact Kirchhoff–Love shell model. Comput Methods Appl Mech Eng

  28. Kiendl J, Bletzinger K-U, Linhard J, Wüchner R (2009) Isogeometric shell analysis with Kirchhoff–Love elements. Comput Methods Appl Mech Eng 198(49–52):3902–3914

    Article  MathSciNet  MATH  Google Scholar 

  29. Kiendl J, Bazilevs Y, Hsu M-C, Wüchner R, Bletzinger K-U (2010) The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches. Comput Methods Appl Mech Eng 199(37–40):2403–2416

    Article  MathSciNet  MATH  Google Scholar 

  30. Korelc J (1997) Automatic generation of finite-element code by simultaneous optimization of expressions. Theor Comput Sci 187(1):231–248

    Article  MATH  Google Scholar 

  31. Korelc J (2002) Multi-language and multi-environment generation of nonlinear finite element codes. Eng Comput 18:312–327

    Article  Google Scholar 

  32. Korelc J, Wriggers P (2016) Automation of finite element methods, 1st edn. Springer. doi:10.1007/978-3-319-39005-5

  33. Malkus D, Hughes TJR (1978) Mixed finite element methods—reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng 15(1):63–81

    Article  MATH  Google Scholar 

  34. Marsden JE, Hughes TJR (1983) Mathematical foundations of elasticity. Dover Publications, Inc., New York

    MATH  Google Scholar 

  35. Millan D, Rosolen A, Arroyo M (2010a) Nonlinear manifold learning for meshfree finite deformation thin-shell analysis. Int J Numer Methods Eng 93(7):685–713

  36. Millan D, Rosolen A, Arroyo M (2010b) Thin shell analysis from scattered points with maximum-entropy approximants. Int J Numer Methods Eng 85(6):723–751

  37. Ojeda R, Prusty BG, Lawrence N, Thomas G (2007) A new approach for the large deflection finite element analysis of isotropic and composite plates with arbitrary orientated stiffeners. Finite Elem Anal Des 43:989–1002

    Article  Google Scholar 

  38. Oliveira IP, Campello EMB, Pimenta PM (2006) Finite element analysis of the wrinkling of orthotropic membranes. In: III European conference on computational mechanics: solids, structures and coupled problems in engineering: book of abstracts. Springer, Dordrecht, p 661–673

  39. Ota N, Wilson L, Gay A, Neto S, Pellegrino S, Pimenta PM (2016) Nonlinear dynamic analysis of creased shells. Finite Elem Anal Des 121:64–74

    Article  MathSciNet  Google Scholar 

  40. Pimenta PM (1993) On the geometrically-exact finite-strain shell model. In: Proceeding of the third Pan-American congress on applied mechanics, PACAM III, January 1993. Escola Politecnica da Universidade de Sao Paulo, Sao Paulo, p 616–619

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Pimenta PM, Almeida Neto ES, Campello EMB (2010) A fully nonlinear thin shell model of Kirchhoff–Love type. In: New trends in thin structures: formulation, optimization and coupled problems. Springer Vienna, Vienna, p 29–58

  43. Rabczuk T, Areias PMA, Belytschko T (2007) A meshfree thin shell method for non-linear dynamic fracture. Int J Numer Methods Eng 72:524–548

    Article  MathSciNet  MATH  Google Scholar 

  44. Samanta A, Mukhopadhyay M (1999) Finite element large deflection static analysis of shallow and deep stiffened shells. Finite Elem Anal Des

  45. Sansour C, Kollmann F (2000) Families of 4-node and 9-node finite elements for a finite deformation shell theory. An assessment of hybrid stress, hybrid strain and enhanced strain elements. Comput Mech 24:435–447

    Article  MATH  Google Scholar 

  46. Schröder J (2010) Anisotropic polyconvex energies. In: Poly-, quasi- and rank-one convexity in applied mechanics. Springer, Vienna, p 53–105

  47. Schröder J, Neff P (2001) On the construction of polyconvex anisotropic free energy functions. In: Miehe C (ed) Proceedings of the IUTAM Symposium on computational mechanics of solid materials at large strains. Kluwer Academic Publishers, Dordrecht, p 171–180

  48. Schröder J, Neff P (2003) Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. Int J Solids Struct 40:401–445

    Article  MathSciNet  MATH  Google Scholar 

  49. Schröder J, Neff P (eds, 2010) Poly-, quasi- and rank-one convexity in applied mechanics, CISM book, vol 516. Springer, Wien

  50. Schröder J, Neff P, Ebbing V (2008) Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors. J Mech Phys Solids 56(12):3486–3506

    Article  MathSciNet  MATH  Google Scholar 

  51. Schröder J, Balzani D, Stranghöner N, Uhlemann J, Gruttmann F, Saxe K (2011) Membranstrukturen mit nicht-linearem anisotropem materialverhalten - aspekte der materialprüfung und der numerischen simulation 86:381–389

  52. Silhavy M (1997) The mechanics and thermodynamics of continuous media. Springer, Berlin

    Book  MATH  Google Scholar 

  53. Simo JC, Fox DD (1989) On a stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization. Comput Methods Appl Mech Eng 72:267–304

  54. Simo JC, Rifai MS (1990) A class of assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29:1595–1638

    Article  MathSciNet  MATH  Google Scholar 

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

  56. Simo JC, Fox DD, Rifai MS (1990) On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory. Comput Methods Appl Mech Eng 79:21–70

  57. Truesdell C, Noll W (2004) The non-linear field theories of mechanics, 3rd edn. Springer, Berlin

    Book  MATH  Google Scholar 

Download references

Acknowledgments

The first and third authors gratefully acknowledge support by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 “Novel finite elements for anisotropic media at finite strain” under the Project “Reliable Simulation Techniques in Solid Mechanics, Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis” (SCHR 570/23-1). In addition to that the author P. M. Pimenta expresses his acknowledgement to the Alexander von Humboldt Foundation for the Georg Forster Award that made possible his stays at the Universities of Duisburg-Essen and Hannover in Germany in the triennium 2015–2017 as well as to the French and Brazilian Governments for the Chair CAPES-Sorbonne that made possible his stay at Sorbonne Universités during the year of 2016 on a leave from the University of São Paulo.

Author information

Authors and Affiliations

  1. Institut für Mechanik, Fachbereich für Ingenieurwissenschaften / Abtl. Bauwissenschaften, Universität Duisburg-Essen, Universitätsstr. 15, 45117, Essen, Germany

    Nils Viebahn & Jörg Schröder

  2. Department of Structural and Geotechnical Engineering, Polytechnic School at University of Sao Paulo, São Paulo, Brazil

    Paulo M. Pimenta

Authors
  1. Nils Viebahn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paulo M. Pimenta
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jörg Schröder
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nils Viebahn.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Viebahn, N., Pimenta, P.M. & Schröder, J. A simple triangular finite element for nonlinear thin shells: statics, dynamics and anisotropy. Comput Mech 59, 281–297 (2017). https://doi.org/10.1007/s00466-016-1343-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-016-1343-6

Keywords

Search

Navigation