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A finite element method for contact using a third medium

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Abstract

The numerical simulation of contact problems is nowadays a standard procedure in many engineering applications. The contact constraints are usually formulated using either the Lagrange multiplier, the penalty approach or variants of both methodologies. The aim of this paper is to introduce a new scheme that is based on a space filling mesh in which the contacting bodies can move and interact. To be able to account for the contact constraints, the property of the medium, that imbeds the bodies coming into contact, has to change with respect to the movements of the bodies. Within this approach the medium will be formulated as an isotropic/anisotropic material with changing characteristics and directions. In this paper we will derive a new finite element formulation that is based on the above mentioned ideas. The formulation is presented for large deformation analysis and frictionless contact.

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Notes

  1. More complex material models (e.g. finite plasticity) will not change the approach followed within this paper. They will only yield more complicated equations to describe the material behaviour of the contacting bodies and thus are omitted for clarity of the new ideas.

  2. Another possibility could be to interpolate between both strain energy functions \(W^M=(1-\gamma ) W^M_\mathrm{{iso}}+ \gamma \,W^M_\mathrm{{aniso}}\), where \(\gamma \) is connected to the approach (gap), thus \(\gamma =\gamma (g_n)\), with the idea to omit the isotropic part once the gap is small.

  3. In case of friction two additional different structural tensors have to be defined that are associated with the tangents at the deformed surface of bodies \(\mathcal B ^\alpha \), e.g, \(\mathbf{M}_1^\alpha = \varvec{\varphi }^\alpha _{,1}\otimes \varvec{\varphi }^\alpha _{,1}\) and \(\mathbf{M}_2^\alpha = \varvec{\varphi }^\alpha _{,2}\otimes \varvec{\varphi }^\alpha _{,2}\) are defined.

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Authors and Affiliations

  1. Institute for Continuum Mechanics, Leibniz Universität Hannover, Appelstraße 11, 30167 , Hannover, Germany

    P. Wriggers

  2. Institute for Mechanics, University of Duisburg-Essen, Universitätsstraße 15, 45117 , Essen, Germany

    J. Schröder & A. Schwarz

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  1. P. Wriggers
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  2. J. Schröder
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Correspondence to P. Wriggers.

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Wriggers, P., Schröder, J. & Schwarz, A. A finite element method for contact using a third medium. Comput Mech 52, 837–847 (2013). https://doi.org/10.1007/s00466-013-0848-5

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