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.

Sliding contact of solids (black) through a third medium (white) using the third medium contact method with HuHu-regularization.

The method was first proposed by Peter Wriggers, Jörg Schröder, and Alexander Schwarz where a St. Venant-Kirchhoff material was used to model the third medium. 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. 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.

Principles

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

$\Psi (u)=W(u)+\mathbb {H} (u)\,{\boldsymbol {\scriptstyle {\vdots }}}\,\mathbb {H} (u)$,

where ${\textstyle \Psi (u)}$ represents the augmented strain energy density in the third medium, ${\textstyle \mathbb {H} (u)\,{\boldsymbol {\scriptstyle {\vdots }}}\,\mathbb {H} (u)}$ is the regularization term representing the inner product of the spatial Hessian by itself, and ${\textstyle W(u)}$ is the underlying strain energy density of the third medium, e.g. a Neo-Hookean solid or another hyperelastic material. The term ${\textstyle \mathbb {H} (u)\,{\boldsymbol {\scriptstyle {\vdots }}}\,\mathbb {H} (u)}$ is commonly referred to as HuHu-regularization. The HuHu-regularization is the first regularization developed specifically for TMC. A later improvement of this regularization is known as the HuHu-LuLu-regularization and is expressed as

$\Psi (u)=W(u)+\mathbb {H} (u)\,{\boldsymbol {\scriptstyle {\vdots }}}\,\mathbb {H} (u)-{\dfrac {1}{\mathrm {Tr} (I)}}\mathbb {L} (u)\cdot \mathbb {L} (u)$,

where $\mathbb {L} (u)$ is the Laplacian of the displacement field $u$ .

Applications

TMC is commonly used in computational mechanics and topology optimization, primarily for its ability to model contact mechanics in a differentiable and fully implicit manner. A key advantage of TMC is that it eliminates the need to explicitly define surfaces and contact pairs, simplifying the modeling process. The following are notable applications of the TMC method:

1. Topology Optimization

In topology optimization, TMC ensures that sensitivities are properly handled, enabling gradient-based optimization approaches to converge effectively and yield design with internal contact. Notable designs obtained by this approach are compliant mechanisms such as hooks, bending mechanisms, and self-contacting springs .

The design of metamaterials is a common application for topology optimization, where TMC has extended the domain for possible designs .

2. Integration with Advanced Material Models

TMC has been extended to applications involving frictional contact and thermo-mechanical coupling . These approaches expand the method’s utility in modelling real-world mechanical interfaces.

3. Soft Robotics

Soft springs and pneumatically activated systems, which can be used in the design of soft robots, have been modelled using TMC .

See also

• Contact mechanics
• Hyperelastic materials
• Topology optimization