Journal of Nonlinear Science

, Volume 6, Issue 2, pp 169–199 | Cite as

A nonlinear extensible 4-node shell element based on continuum theory and assumed strain interpolations

  • P. Betsch
  • E. Stein


A quadrilateral continuum-basedC 0 shell element is presented, which relies on extensible director kinematics and incorporates unmodified three-dimensional constitutive models. The shell element is developed from the nonlinear enhanced assumed strain (EAS) method advocated by Sino & Armero [1] and formulated in curvilinear coordinates. Here, the EAS-expansion of the material displacement gradient leads to the local interpretation of enhanced covariant base vectors that are superposed on the compatible covariant base vectors. Two expansions of the enhanced covariant base vectors are given: first an extension of the underlying single extensible shell kinematic and second an improvement of the membrane part of the bilinear element. Furthermore, two assumed strain modifications of the compatible covariant strains are introduced such that the element performs well even in the case of very thin shells.


Thickness Strain Transverse Shear Strain Assumed Strain Enhanced Assumed Strain Tangent Matrix 
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  1. [1]
    J. C. Simo and F. Armero, Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes,Int. J. Num. Meth. Eng. 33 (1992) 1413–1449.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    E. Ramm, Geometrisch nichtlineare Elastostatik und finite Elemente, Habilitation, Bericht Nr. 76-2, Institut für Baustatik, Universität Stuttgart, (1976).Google Scholar
  3. [3]
    T. J. R. Hughes and W. K. Liu, Nonlinear finite element analysis of shells: Part I. Threedimensional shells.Comp. Meth. Appl. Mech. Eng. 26 (1981) 331–362.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    J. C. Simo, D. D. Fox and M. S. Rifar, On a stress resultant geometrically exact shell model. Part III: Computational aspects of the nonlinear theory.Comp. Meth. Appl. Mech. Eng. 79 (1990) 21–70.MATHCrossRefGoogle Scholar
  5. [5]
    F. Gruttmann, W. Wagner, P. Wriggers, A nonlinear quadrilateral shell element with drilling degrees of freedom,Archive Appl. Mech. 62 (1992) 474–486.MATHCrossRefGoogle Scholar
  6. [6]
    Y. Basar, Y. Ding and W. B. Krätzig, Finite-rotatation shell elements via mixed formulation,Comp. Mech. 10, (1992) 289–306.MATHCrossRefGoogle Scholar
  7. [7]
    J. C. Simo, M. S. Rifai and D. D. Fox, On a stress resultant geometrically exact shell model. Part IV: Variable thickness shells with through-the-thickness stretching.Comp. Meth. Appl. Mech. Eng. 81, (1990) 91–126.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    N. Büchter and E. Ramm, 3d-extension of nonlinear shell equations based on the enhanced assumed strain concept,Comp. Meth. Appl. Sci. (1992) 55–62.Google Scholar
  9. [9]
    N. Büchter, E. Ramm and D. Roehl, Three-dimensional extension of nonlinear shell formulations based on the enhanced assumed strain concept.Int. J. Num. Meth. Eng. 37 (1994) 2551–2568.MATHCrossRefGoogle Scholar
  10. [10]
    P. Steinmann, P. Betsch and E. Stein, FE plane stress analysis incorporating arbitrary 3d large strain constitutive models, submitted toEng. Comp. (1994).Google Scholar
  11. [11]
    P. Betsch, F. Gruttmann and E. Stein, A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains, to appear inComp. Meth. Appl. Mech. Eng. (1994).Google Scholar
  12. [12]
    C. Sansour, A theory and finite element formulation of shells at finite deformations including thickness change: Circumventing the use of a rotation tensor.Arch. Appl. Mech. 65 (1995) 194–216.MATHCrossRefGoogle Scholar
  13. [13]
    A. Kühhorn and H. Schoop, A nonlinear theory for sandwich shells including the wrinkling phenomenon.Arch. Appl. Mech. 62, (1992) 413–427.MATHCrossRefGoogle Scholar
  14. [14]
    H. Parisch, A continuum-based shell theory for non-linear applications,Int. J. Num. Meth. Eng.,38 (1995) 1855–1883.MATHCrossRefGoogle Scholar
  15. [15]
    H. Schoop, A simple nonlinear flat element for large displacement structures,Computers & Structures 32 (1989) 379–385.MATHCrossRefGoogle Scholar
  16. [16]
    J. C. Simo and M. S. Rifai, A class of mixed assumed strain methods and the method of incompatible modes,Int. J. Num. Meth. Eng. 29 (1990) 1595–1638.MATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    E. N. Dvorkin and K.-J. Bathe, A continuum mechanics based four-node shell element for general nonlinear analysis,Eng. Comput. 1 (1984) 77–88.Google Scholar
  18. [18]
    K. J. Bathe and E. N. Dvorkin, A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation,Int. J. Num. Meth. Eng. 21 (1985) 367–383.MATHCrossRefGoogle Scholar
  19. [19]
    N. Stander, A. Matzenmiller and E. Ramm, An assessment of assumed strain methods in finite rotation shell analysis,Eng. Comput. 6 (1989) 58–66.Google Scholar
  20. [20]
    H. Parisch, An investigation of a finite rotation four-node assumed strain shell element,Int. J. Num. Meth. Eng. 31, (1991) 127–150.MATHCrossRefGoogle Scholar
  21. [21]
    P. Betsch and E. Stein, An assumed strain approach avoiding artificial thickness straining for a nonlinear 4-node shell element.Comm. Num. Meth. Eng. 11 (1995) 899–909.MATHCrossRefGoogle Scholar
  22. [22]
    J. C. Simo and T. J. R. Hughes, On the variational foundations of assumed strain methods,J. Appl. Mech. 53 (1986) 51–54.MATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    B. D. Reddy and J. C. Simo, Stability and convergence of a class of enhanced strain methods, preprint (1994).Google Scholar
  24. [24]
    P. M. Naghdi, The theory of plates and shells, inHandbuch der Physik, Band VIa/2 (Springer, Berlin, 1972) 425–640.Google Scholar
  25. [25]
    E. L. Wilson, R. L. Taylor, W. P. Doherty and J. Ghaboussi, Incompatible Displacement Models, Numerical and Computer Models in Structural Mechanics, eds. S. J. Fenves, N. Perrone, A. R. Robinson and W. C. Schnobrich, New York: Academic Press (1973) 43–57.Google Scholar
  26. [26]
    J. C. Simo, F. Armero and R. L. Taylor, Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems,Comp. Meth. Appl. Mech. Eng. 110 (1993) 359–386.MATHMathSciNetCrossRefGoogle Scholar
  27. [27]
    A. E. Green and W. Zerna,Theoretical Elasticity (Oxford University Press, London, 1954).MATHGoogle Scholar
  28. [28]
    T. J. R. Hughes,The Finite Element Method (Prentice Hall, Englewood Cliffs, NJ, 1987).MATHGoogle Scholar
  29. [29]
    O. C. Zienkiewicz and R. L. Taylor,The Finite Element Method, Volume 1,Basic Formulation and Linear Problems (McGraw-Hill, London, 1989).Google Scholar
  30. [30]
    C. Miehe, Aspects of the formulation and finite element implementation of large strain isotropic elasticity,Int. J. Num. Meth. Eng. 37 (1994) 1981–2004.MATHMathSciNetCrossRefGoogle Scholar
  31. [31]
    J. C. Simo, R. L. Taylor and K. S. Pister, Variational and projection methods for the volume constraint in finite deformation elasto-plasticity,Comp. Meth. Appl. Mech. Eng. 51 (1985) 177–208.MATHMathSciNetCrossRefGoogle Scholar
  32. [32]
    R. H. MacNeal and R. L. Harder, A proposed standard set of problems to test finite element accuracy,Finite Elements in Analysis and Design 1 (1985) 3–20.CrossRefGoogle Scholar
  33. [33]
    J. C. Simo, D. D. Fox and M. S. Rifai, On a stress resultant geometrically exact shell model. Part II: The linear theory; Computational aspects,Comp. Meth. Appl. Mech. Eng. 73 (1989) 53–92MATHMathSciNetCrossRefGoogle Scholar
  34. [34]
    T. J. R. Hughes and E. Carnoy, Nonlinear finite element shell formulation accounting for large membrane strains,Comp. Meth. Appl. Mech. Eng. 39 (1983) 69–82.MATHCrossRefGoogle Scholar
  35. [35]
    H. Parisch, Efficient non-linear finite element shell formulation involving large strains,Eng. Comput. 3, (1986) 121–128.Google Scholar
  36. [36]
    K. Schweizerhof and E. Ramm, Displacement dependent pressure loads in nonlinear finite element analyses,Comput. Struct. 18 (1984) 1099–1114.MATHCrossRefGoogle Scholar
  37. [37]
    F. Gruttmann and R. L. Taylor, Theory and finite element formulation of rubberlike membrane shells using principal stretches,Int. J. Num. Meth. Eng. 35 (1992) 1111–1126.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • P. Betsch
    • 1
  • E. Stein
    • 1
  1. 1.Institut für Baumechanik und Numerische MechanikUniversität HannoverGermany

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