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.[1]
The method was first proposed by, Jörg Schröder, and Alexander Schwarz where a St. Venant-Kirchhoff material was used to model the third medium.[2] 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.[3] This property has made the explicit treatment of surface normals redundant, thereby transforming the third medium contact method into a fully implicit method, which is a contrast to the more widely used Mortar methods or Penalty methods. The addition of a new regularization by Bluhm et al. to stabilize the third medium further extended the method to applications involving moderate sliding, rendering it practically applicable.[4]
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
\Psi(u)=W(u)+H(u)\boldsymbol{\scriptstyle{\vdots}}H(u)
where represents the augmented strain energy density in the third medium, is the regularization term representing the inner product of the spatial Hessian by itself, and is the underlying strain energy density of the third medium, e.g. a Neo-Hookean solid or another hyperelastic material. The term is commonly referred to as HuHu-regularization.