Computational & Technology Resources
an online resource for computational,
engineering & technology publications
Civil-Comp Proceedings
ISSN 1759-3433
CCP: 75
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping and Z. Bittnar
Paper 74

Contact Formulation for Solid-Shell Elements undergoing Large Deformations

M. Harnau and K. Schweizerhof

Institute for Mechanics, University of Karlsruhe, Germany

Full Bibliographic Reference for this paper
M. Harnau, K. Schweizerhof, "Contact Formulation for Solid-Shell Elements undergoing Large Deformations", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Sixth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 74, 2002. doi:10.4203/ccp.75.74
Keywords: solid-shell elements, large deformations, mixed interpolations, linear and quadratic finite elements, penalty method, contact problems.

Summary
Efficient computation for example in sheet metal forming is obtained usually by using finite elements based on standard shell theory assumptions [5] [9] instead of fully three-dimensional continuum based elements. However, many requirements in such investigations, as strains and stresses in thickness direction, in particular, when looking at edges and special situations like large stretching and bending with small radii, cannot be provided by 'classical' shell formulations. Therefore in [2] a so-called 'Solid-Shell' formulation, following similar developments in [4] [6] [7], was proposed. The 'Solid-Shell' formulation is based solely on displacement degrees of freedom belonging to the upper and lower shell surfaces and thus the use of rotational degrees of freedom can be avoided. As no kinematical assumption is applied beyond standard 3D continuum theory also general three-dimensional material laws can be provided and a combination with standard three-dimensional solid elements is easily possible.

In particular, to achieve a better geometric approximation beyond 'Solid-Shell' elements with bilinear in-plane shape functions also biquadratic in-plane shape functions are considered. To overcome the locking problems, which appear for both types of elements, different schemes are used (see also [1]) and almost locking free element formulations can finally be presented.

A special application for the 'Solid-Shell' elements are sheet metal forming problems. To describe such kind of problems as free bending or deep drawing, contact formulations are necessary to introduce the contact condition of the metal sheets against the rigid tools. To describe the rigid contact surfaces analytical functions are used, thus simple surface geometries can easily be described. In this paper the penalty method will be used without taking friction into account as a first step in our developments.

With usual nodal contact formulations as in [8] the problem of weighting the single nodal contribution appears. The weight depends on the geometry as well as on the kind of node - edge, corner or center - for the biquadratic elements. To overcome this problem contact interface elements are developed with four resp. nine nodes similar to the 'Solid-Shell' element types. Therefore instead of evaluating the contact conditions at the nodal points, the contact forces are integrated over the element area [3] and the penetration function is evaluated at the Gauss points of the contact segment, as it is also apparently done in [10]. Alternative strategies as e.g. described in [8] are based on sampling points with area weighting. To describe contact against plane surfaces the usual number of Gauss points is used in the contact interface elements. For non plane and partial overlapping contact geometries increasing the number of Gauss points can be necessary to fulfill the contact condition.

On some numerical examples the performance of the developed algorithms is demonstrated and some recommendations are made concerning the discretization.

References
1
R. Hauptmann, S. Doll, M. Harnau, and K. Schweizerhof. "'Solid-Shell' elements with linear and quadratic shape functions at large deformations with nearly incompressible materials". Comp. Struct., 79(18):1671-1685, 2001. doi:10.1016/S0045-7949(01)00103-1
2
R. Hauptmann and K. Schweizerhof. "A systematic development of solid-shell element formulations for linear and nonlinear analyses employing only displacement degrees of freedom". Int. J. Numer. Methods Engng., 42:49-70, 1998. doi:10.1002/(SICI)1097-0207(19980515)42:1<49::AID-NME349>3.3.CO;2-U
3
T.A. Laursen. "Computational contact and impact mechanics", Springer, 2002.
4
C. Miehe. "A theoretical and computational model for isotropic elastoplastic stress analysis in shells at large strains". Comput. Methods Appl. Mech. Engrg., 155:193-234, 1998. doi:10.1016/S0045-7825(97)00149-7
5
P.M. Naghdi. "Foundations of Elastic Shell Theory", volume 4 of Progress in Solid Mechanics. North-Holland Publ. Comp., 1963.
6
H. Parisch. "A continuum-based shell theory for non-linear applications". Int. J. Numer. Methods Engng., 38:1855-1883, 1995. doi:10.1002/nme.1620381105
7
H. Schoop. "Oberflächenorientierte Schalentheorien endlicher Verschiebungen", Ingenieur-Archiv, 56:427-437, 1986. doi:10.1007/BF00533829
8
P. Wriggers, C. S. Han, and A. Rieger. "H-adaptive finite element methods for contact problems", In J. Whiteman, editor, Mathematics of Finite Elements and Applications, pages 117-142. Proceedings of MAFLAP 99, Elsevier, Amsterdam, 2000. doi:10.1016/B978-008043568-8/50007-9
9
W. Zerna. "Mathematisch strenge Theorie elastischer Schalen", ZAMM, 42:333-341, 1964. doi:10.1002/zamm.19620420707
10
Y. Zhu. "ANSYS nonlinear contact analysis technology", In Course Notes Contact. ANSYS, Inc. Corporated, 2001.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £125 +P&P)