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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 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
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