Third medium contact method

[1][2][3][4][5][6] The method was first proposed in 2013 by Peter Wriggers [de], Jörg Schröder, and Alexander Schwarz, where a St. Venant-Kirchhoff material was used to model the third medium.

[7] This approach required explicit treatment of surface normals and continued to be used[8][9][10] until 2017, when Bog et al. simplified the method by applying a Hencky material with the inherent property of becoming rigid under ultimate compression.

However, at this stage, the third medium contact method could only handle very small degrees of sliding, and a friction model for TMC had yet to be developed.

[9][10] Consequently, TMC was abandoned at an early stage and remained largely unknown in contact mechanics.

In 2021, the method was revived when Gore Lukas Bluhm, Ole Sigmund, and Konstantinos Poulios worked on nonlinear buckling problems and realized that a highly compliant void material could transfer forces in a topology optimization setting.

Bluhm et al. added a new regularization to stabilize the third medium, enabling the method to contact problems involving moderate sliding and thus making it practically applicable.

[12] The use of TMC in topology optimization was refined in subsequent work and applied to more complex problems.

[13][6][4] In 2024, Frederiksen et al.[3] proposed a crystal plasticity-inspired scheme to include friction.

This involved adding a term to the material model to contribute to high shear stresses in the contact interface, along with a plastic slip scheme to release shear stresses and accommodate sliding.

During the same period, new regularization methods were proposed,[4][14][15] and the method was extended to thermal contact by Dalklint et al.[5] and utilized for pneumatic actuation by Faltus et al.[14] TMC relies on a material model for the third medium, which stiffens under compression.

The most commonly applied material models are of a neo-Hookean type, characterized by a strain energy density function:

approaches zero, this material model exhibits the characteristic of becoming infinitely stiff.

To address this, regularization techniques are applied to the strain energy density function.

is the underlying strain energy density of the third medium, e.g. a neo-Hookean solid or another hyperelastic material.

[4] The LuLu term is designed to mitigate the penalization of bending and quadratic compression deformations while maintaining the penalization of excessive skew deformations, thus preserving the stabilizing properties of the HuHu regularization.

This reduced penalization on bending deformations enhances the accuracy of modeling curved contacts, particularly beneficial when using coarse finite element meshes.

A later improvement by Wriggers et al.[15] directly utilizes the rotation tensor

[14] The integration of friction into the TMC method represents a significant advancement in simulating realistic contact conditions, addressing the previous limitations in replicating real-world scenarios.

When a neo-Hookean material model is used to represent the third medium, it exhibits much greater stiffness in compression compared to shear during contact.

To address this and provide shear resistance, an anisotropic term is incorporated into the neo-Hookean material model.

This modification rapidly builds up shear stress in compressed regions of the third medium, which is crucial for accurately modeling frictional contact.

In this formulation, the extended strain energy density expression with the added shear term is:

To release the shear stresses at the onset of sliding, a framework inspired by crystal plasticity is employed.

This includes a yield criterion specifically designed to replicate the effects of Coulomb friction.

This framework allows the model to simulate the onset of sliding when the shear stress, provided by the added anisotropic term, exceeds a certain threshold, effectively mimicking real-world frictional behavior.

The yield criterion, based on the Coulomb friction model, determines when sliding occurs, initiating once the shear stress surpasses a critical value.

One of the key advantages of TMC is that it eliminates the need to explicitly define surfaces and contact pairs, thereby simplifying the modeling process.

In topology optimization, TMC ensures that sensitivities are properly handled, enabling gradient-based optimization approaches to converge effectively and produce designs with internal contact.

[13][14] TMC has also been extended to applications involving frictional contact and thermo-mechanical coupling.

[3][5] These advancements enhance the method’s utility in modeling real-world mechanical interfaces.