A physically stabilized and locking-free formulation of the (SHB8PS) solid-shell element
DOI:
https://doi.org/10.13052/REMN.16.1037-1072Keywords:
SHB8PS solid-shell, hourglass, shear and membrane locking, assumed strain method, orthogonal projectionAbstract
In this work, the formulation of the SHB8PS finite element is reviewed in order to eliminate some persistent membrane and shear locking phenomena. This is a solid-shell element based on a purely three-dimensional formulation. In fact, the element has eight nodes as well as five integration points, all distributed along the “thickness” direction. Consequently, it can be used for the modeling of thin structures, while providing an accurate description of the various through-thickness phenomena. The reduced integration has been used in order to prevent some locking phenomena and to increase computational efficiency. The spurious zero-energy modes due to the reduced integration are efficiently stabilized, whereas the strain components corresponding to locking modes are eliminated with a projection technique following the Enhanced Assumed Strain (EAS) method.
Downloads
References
Abed-Meraim F., Combescure A., Stabilité des éléments finis sous-intégrés, Rapport interne
n° 247, LMT de Cachan, janvier 2001.
Abed-Meraim F., Combescure A., “SHB8PS a new intelligent assumed strain continuum
mechanics shell element for impact analysis on a rotating body”, First M.I.T. Conference
on Comput. Fluid and Solid Mechanics, 12-15 June 2001, U.S.A.
Abed-Meraim F., Combescure A., “SHB8PS – a new adaptative, assumed-strain continuum
mechanics shell element for impact analysis”, Computers & Structures, vol. 80, 2002,
p. 791-803.
Ahmad S., “Analysis of thick and thin shell structures by curved finite elements”,
International Journal for Numerical Methods and Engineering, vol. 2, 1970, p. 419-451.
Alves de Sousa R.J., Cardoso R.P.R., Fontes Valente R.A., Yoon J.W., Gracio J.J.,
Natal Jorge R.M., “A new one-point quadrature enhanced assumed strain (EAS) solidshell
element with multiple integration points along thickness: Part I – Geometrically
linear applications”, International Journal for Numerical Methods and Engineering,
vol. 62, 2005, p. 952-977.
Alves de Sousa R.J., Cardoso R.P.R., Fontes Valente R.A., Yoon J.W., Gracio J.J.,
Natal Jorge R.M., “A new one-point quadrature enhanced assumed strain (EAS) solidshell
element with multiple integration points along thickness: Part II – Nonlinear
applications”, International Journal for Numerical Methods and Engineering, vol. 67,
, p. 160-188.
Alves de Sousa R.J., Yoon J.W., Cardoso R.P.R., Fontes Valente R.A., Gracio J.J., “On the
use of a reduced enhanced solid-shell (RESS) element for sheet forming simulations”,
International Journal of Plasticity, vol. 23, 2007, p. 490-515.
Ayad R., Batoz J.L., Dhatt G., « Un élément quadrangulaire de plaque basé sur une
formulation mixte-hybride avec projection en cisaillement », Revue européenne des
éléments finis, vol. 4, 1995, p. 415-440.
Ayad R., Dhatt G., Batoz J.L., “A new hybrid-mixed variational approach for Reissner-
Mindlin plates: The MiSP model”, International Journal for Numerical Methods and
Engineering, vol. 42, 1998, p. 1149-1179.
Bathe K.-J., Dvorkin E.N., “Four-node plate bending element based on Mindlin/Reissner
plate theory and a mixed interpolation”, International Journal for Numerical Methods
and Engineering, vol. 21, 1985, p. 367-383.
Bathe K.-J., Dvorkin E.N., “Formulation of general shell elements – The use of mixed
interpolation of tensorial components”, International Journal for Numerical Methods and
Engineering, vol. 22, 1986, p. 697-722.
Batoz J.L., Dhatt G., Modelling of structures by finite elements, Hermès, Paris, 1992.
Belytschko T., Ong J.S.-J., Liu W.K., Kennedy J.M., “Hourglass control in linear and
nonlinear problems”, Computer Methods in Applied Mechanics and Engineering, vol. 43,
, p. 251-276.
Belytschko T., Wong B.L., Stolarski H., “Assumed strain stabilization procedure for the 9-
node Lagrange shell element”, International Journal for Numerical Methods and
Engineering, vol. 28, 1989, p. 385-414.
Belytschko T., Bindeman L.P., “Assumed strain stabilization of the eight node hexahedral
element”, Computer Methods in Applied Mechanics and Engineering, vol. 105, 1993,
p. 225-260.
Buechter N., Ramm E., Roehl D., “Three-dimensional extension of non-linear shell
formulation based on the enhanced assumed strain concept”, International Journal for
Numerical Methods and Engineering, vol. 37, 1994, p. 2551-2568.
Buragohain D.N., Ravichandran P.K., “Modified three-dimensional finite element for general
and composite shells”, Computers & Structures, vol. 51, 1994, p. 289-298.
Chapelle D., Bathe K.-J., “Fundamental considerations for the finite element analysis of shell
structures”, Computers & Structures, vol. 66, 1998, p. 19-36.
Chapelle D., Bathe K.-J., The finite element analysis of shells – Fundamentals, Springer,
Berlin, 2003.
Chen Y.-I., Wu G.-Y., “A mixed 8-node hexahedral element based on the Hu-Washizu
principle and the field extrapolation technique”, Structural Engineering and Mechanics,
vol. 17, n° 1, 2004, p. 113-140.
Cheung Y., Chen W.-J., “Refined hybrid method for plane isoparametric element using an
orthogonal approach”, Computers & Structures, vol. 42, 1992, p. 683-694.
Cho C., Park H.C., Lee S.W., “Stability analysis using a geometrically nonlinear assumed
strain solid shell element model”, Finite Elements in Analysis and Design, vol. 29, 1998,
p. 121-135.
Domissy E., Formulation et évaluation d’éléments finis volumiques modifiés pour l’analyse
linéaire et non linéaire des coques, Thèse de doctorat, UT Compiègne, 1997.
Dvorkin E.N., Bathe K.-J., “Continuum mechanics based four-node shell element for general
non-linear analysis”, Engineering Computations, vol. 1, 1984, p. 77-88.
Flanagan D.P., Belytschko T., “A uniform strain hexahedron and quadrilateral with
orthogonal hourglass control”, International Journal for Numerical Methods and
Engineering, vol. 17, 1981, p. 679-706.
Graf W., Chang T.Y., Saleeb A.F., “On the numerical performance of three-dimensional thick
shell elements using a hybrid/mixed formulation”, Finite Elements in Analysis and
Design, vol. 2, 1986, p. 357-375.
Hallquist J.O., Theoretical manual for DYNA3D, UC1D-19401 Lawrence Livemore National
Lab., University of California, 1983.
Hauptmann R., Schweizerhof K., “A systematic development of solid-shell element
formulations for linear and non-linear analyses employing only displacement degrees of
freedom”, International Journal for Numerical Methods and Engineering, vol. 42, 1998,
p. 49-69.
Hauptmann R., Doll S., Harnau M., Schweizerhof K., “Solid-shell elements with linear and
quadratic shape functions at large deformations with nearly incompressible materials”,
Computers & Structures, vol. 79, 2001, p. 1671-1685.
Jirousek J., Wroblewski A., Qin Q.H., He X.Q., “A family of quadrilateral hybrid-Trefftz pelements
for thick plate analysis”, Computer Methods in Applied Mechanics and
Engineering, vol. 127, 1995, p. 315-344.
Kim Y.H., Lee S.W., “A solid element formulation for large deflection analysis of composite
shell structures”, Computers & Structures, vol. 30, 1993, p. 269-274.
Kim K.D., Liu G.Z., Han S.C., “A resultant 8-node solid-shell element for geometrically
nonlinear analysis”, Computational Mechanics, vol. 35, 2005, p. 315-331.
Klinkel S., Gruttmann F., Wagner W., “A continuum based three-dimensional shell element
for laminated structures”, Computers & Structures, vol. 71, 1999, p. 43-62.
Klinkel S., Wagner W., “A geometrical non-linear brick element based on the EAS-method”,
International Journal for Numerical Methods and Engineering, vol. 40, 1997, p. 4529-4545.
Korelec J., Wriggers P., “Consistent gradient formulation for a stable enhanced strain method
for large deformations”, Engineering Computations, vol. 13, 1996, p. 103-123.
Legay A., Combescure A., “Elastoplastic stability analysis of shells using the physically
stabilized finite element SHB8PS”, International Journal for Numerical Methods and
Engineering, vol. 57, 2003, p. 1299-1322.
Lemosse D., Eléments finis isoparamétriques tridimensionnels pour l’étude des structures
minces, Thèse de doctorat, Ecole Doctorale SPMI/INSA-Rouen, 2000.
Liu W.K., Guo Y., Tang S., Belytschko T., “A multiple-quadrature eight-node hexahedral
finite element for large deformation elastoplastic analysis”, Computer Methods in Applied
Mechanics and Engineering, vol. 154, 1998, p. 69-132.
MacNeal R.H., Harder R.L., “A proposed standard set of problems to test finite element
accuracy”, Finite Elements in Analysis and Design, vol. 1, 1985, p. 3-20.
Onate E., Castro J., Derivation of plate based on assumed shear strain fields, New Advances
in Computational Structural Mechanics, Ladevèze and Zienkiewicz (eds.), Elsevier,
Amsterdam, 1992, p. 237-288.
Pian T.H.H., Tong P., “Relations between incompatible model and hybrid stress model”,
International Journal for Numerical Methods and Engineering, vol. 22, 1986, p. 173-181.
Puso M.A., “A highly efficient enhanced assumed strain physically stabilized hexahedral
element”, International Journal for Numerical Methods and Engineering, vol. 49, 2000,
p. 1029-1064.
Reese S., Wriggers P., Reddy B.D., “A new locking-free brick element technique for large
deformation problems in elasticity”, Computers & Structures, vol. 75, 2000, p. 291-304.
Reese S., “A large deformation solid-shell concept based on reduced integration with
hourglass stabilization”, International Journal for Numerical Methods and Engineering,
vol. 69, 2007, p. 1671-1716.
Simo J.C., Hughes T.J.R., “On the variational foundations of assumed strain methods”,
Journal of Applied Mechanics, ASME, vol. 53, 1986, p. 51-54.
Simo J.C., Fox D.D., Rifai S., “On a stress resultant geometrically exact shell model. Part II:
The linear theory; Computational Aspects”, Computer Methods in Applied Mechanics
and Engineering, vol. 73, 1989, p. 53-92.
Sze K.Y., Ghali A., “Hybrid hexahedral element for solids, plates, shells and beams by
selective scaling”, International Journal for Numerical Methods and Engineering,
vol. 36, 1993, p. 1519-1540.
Sze K.Y., Yao L.Q., “A hybrid stress ANS solid-shell element and its generalization for smart
structure modelling. Part I-solid-shell element formulation”, International Journal for
Numerical Methods and Engineering, vol. 48, 2000, p. 545-564.
Timoshenko S., Woinowsky-Krieger S., Theory of plates and shells, McGraw-Hill, London,
Vu-Quoc L., Tan X.G., “Optimal solid shells for non-linear analyses of multilayer
composites. I. Statics”, Computer Methods in Applied Mechanics and Engineering,
vol. 192, 2003, p. 975-1016.
Wall W.A., Bischoff M., Ramm E., “A deformation dependent stabilization technique,
exemplified by EAS elements at large strains”, Computer Methods in Applied Mechanics
and Engineering, vol. 188, 2000, p. 859-871.
Wriggers P., Reese S., “A note on enhanced strain methods for large deformations”,
Computer Methods in Applied Mechanics and Engineering, vol. 135, 1996, p. 201-209.
Xu X., Cai R., “A new plate shell element of 16 nodes and 40 degrees of freedom by relative
displacement method”, Communications in Numerical Methods in Engineering, vol. 9,
, p. 15-20.
Zhu Y.Y., Cescotto S., “Unified and mixed formulation of the 8-node hexahedral elements by
assumed strain method”, Computer Methods in Applied Mechanics and Engineering,
vol. 129, 1996, p. 177-209.
