Third medium contact method
The third medium contact (TMC) is an implicit formulation for contact mechanics. Contacting bodies are embedded in a highly compliant medium (the third medium), which becomes increasingly stiff under compression. The stiffening of the third medium allows tractions to be transferred between the contacting bodies when the third medium between the bodies is compressed. In itself, the method is inexact; however, in contrast to most other contact methods, the third medium approach is continuous and differentiable, which makes it applicable to applications such as topology optimization.<ref name=":0" /><ref>Frederiksen, Andreas Henrik; Sigmund, Ole; Poulios, Konstantinos (2023-10-07). "Topology optimization of self-contacting structures". Computational Mechanics. arXiv:2305.06750. doi:10.1007/s00466-023-02396-7. ISSN 1432-0924.</ref>
The method was first proposed by Peter Wriggers et al. where a St. Venant-Kirchhoff material was used to model the third medium.<ref>Wriggers, P.; Schröder, J.; Schwarz, A. (2013-03-30). "A finite element method for contact using a third medium". Computational Mechanics. 52 (4): 837–847. Bibcode:2013CompM..52..837W. doi:10.1007/s00466-013-0848-5. ISSN 0178-7675. S2CID 254032357.</ref> This approach requires explicit treatment of surface normals. A simplification to the method was offered by Bog et al. by applying a Hencky material with the inherent property of becoming rigid under ultimate compression.<ref>Bog, Tino; Zander, Nils; Kollmannsberger, Stefan; Rank, Ernst (October 2015). "Normal contact with high order finite elements and a fictitious contact material". Computers & Mathematics with Applications. 70 (7): 1370–1390. doi:10.1016/j.camwa.2015.04.020. ISSN 0898-1221.</ref> This property has made the explicit treatment of surface normals redundant, thereby transforming the third medium contact method into a fully implicit method. The addition of a new void regularization by Bluhm et al. further extended the method to applications involving moderate sliding, rendering it practically applicable.<ref name=":0">Bluhm, Gore Lukas; Sigmund, Ole; Poulios, Konstantinos (2021-03-04). "Internal contact modeling for finite strain topology optimization". Computational Mechanics. 67 (4): 1099–1114. arXiv:2010.14277. Bibcode:2021CompM..67.1099B. doi:10.1007/s00466-021-01974-x. ISSN 0178-7675. S2CID 225076340.</ref>
Methodology
A material with the property that it becomes increasingly stiff under compression is augmented by a regularization term. In terms of strain energy density, this may be expressed as
<math>\Psi(u) = W(u) + \mathbb{H}(u)\vdots\mathbb{H}(u)</math>,
where <math display="inline"> \Psi(u)</math> represents the augmented strain energy density in the third medium, <math display="inline">\mathbb{H}(u)\vdots\mathbb{H}(u)</math> is the regularization term representing the inner product of the spatial Hessian by itself, and <math display="inline">W(u)</math> is the underlying strain energy density of the third medium, e.g. a Neo-Hookean solid or another hyperelastic material. The term<math display="inline">\mathbb{H}(u)\vdots\mathbb{H}(u)</math> is sometimes referred to as HuHu-regularization or the Hessian-Hessian (HH) regularization.
