Third medium contact method

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

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.

History

The method was first proposed in 2013 by Peter Wriggers, Jörg Schröder, and Alexander Schwarz where a St. Venant-Kirchhoff material was used to model the third medium . This approach, which requires explicit treatment of surface normals, continued to be used until a simplification to the method was offered in 2017 by Bog et al. by applying a Hencky material with the inherent property of becoming rigid under ultimate compression. This property 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. At this stage, however, the third medium contact method was only able to handle very small degrees of sliding. Additionally, a friction model for TMC had yet to be developed. The rising popularity of Mortar methods, which emerged in the same period, having a rigorous mathematical foundation and a rapid development and adoption, overshadowed the TMC method. Thus, TMC was abandoned at an early stage and remained a largely unknown method for contact mechanics.

In 2021 the method was revived when Lukas Bluhm, Ole Sigmund, and Konstantinos Poulios rediscovered the method as they realised that a highly compliant void material could transfer forces between in a topology optimization setting. 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. The use of TMC in topology optimization was refined in later work.

In 2024, Frederiksen et al. proposed a crystal plasticity inspireed scheme to include friction. This involved adding a term to the material model, contributing to high shear stresses in the contact interface, in addition to plastic slip scheme to release shear stresses and accomodate sliding. In the same period, new regularization methods were proposed and the method was extended to thermal contact by Dalklint et al., and utilized for pneumatic actuation by Faltus et al..

Principles

Notation conventions

${\mathbf {a}}\cdot {\mathbf {b}}=a_{i}b_{i}$	Inner product
${\mathbf {a}}\otimes {\mathbf {b}}=a_{i}b_{j}$	Outer product
${\mathbf {A}}:{\mathbf {B}}=A_{ij}B_{ij}$	Double contraction
${\boldsymbol {\mathcal {A}}}\,{\boldsymbol {\scriptstyle {\vdots }}}\,{\boldsymbol {\mathcal {B}}}={\mathcal {A}}_{ijk}{\mathcal {B}}_{ijk}$	Triple contraction
$||{\mathbf {A}}||={\sqrt {{\mathbf {A}}:{\mathbf {A}}}}$	Frobenius norm
$\mathbb {H} {\mathbf {a}}={\dfrac {\partial ^{2}a_{i}}{\partial X_{j}\partial X_{k}}}$	Hessian of a vector field ${\mathbf {a}}$
$\mathbb {L} {\mathbf {a}}={\dfrac {\partial ^{2}a_{i}}{\partial X_{j}\partial X_{j}}}$	Laplacian of a vector field ${\mathbf {a}}$

Material model

Increasing strain energy density of neo-Hookean solid under uniaxial compression.

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 having a strain energy density function:

$W({\mathbf {u}})={\frac {K}{2}}({\text{ln}}|{\mathbf {F}}|)^{2}+{\frac {G}{2}}\left(|{\mathbf {F}}|^{-2/3}||{\mathbf {F}}||^{2}-3\right)$,

where $K$ is the bulk modulus, $G$ is the shear modulus, and ${\mathbf {F}}=\nabla {\mathbf {u}}+{\mathbf {I}}$ is the deformation gradient tensor of the displacement field ${\mathbf {u}}$. This material model exhibits the characteristic of becoming infinitely stiff as the current material volume $|{\mathbf {F}}|$ approaches zero. Consequently, when the third medium is compressed, its volume remains positive and finite. Under sufficient compression, the third medium attains a stiffness comparable to the solids between which it is compressed.

Regularization

For contact without sliding the neo-Hookean material model is often sufficiently stable to serve as a third medium by itself. However, when sliding is involved the stability deteriorates. This has motivated the application of different regularization techniques to stabilise the third medium.

Regularizations are commonly applied by adding a regularization term directly to the strain energy density function of the material model. A common approach is the so-called HuHu regularization, expressed as:

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

where ${\textstyle \Psi ({\mathbf {u}})}$ represents the augmented strain energy density of the third medium, ${\textstyle \mathbb {H} {\mathbf {u}}\,{\boldsymbol {\scriptstyle {\vdots }}}\,\mathbb {H} {\mathbf {u}}}$ is the regularization term representing the inner product of the spatial Hessian by itself, and ${\textstyle W({\mathbf {u}})}$ is the underlying strain energy density of the third medium, e.g. a Neo-Hookean solid or another hyperelastic material.

The HuHu regularization was the first regularization developed specifically for TMC. A later improvement of this regularization is known as the HuHu-LuLu regularization and is expressed as

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

where $\mathbb {L} {\mathbf {u}}$ is the Laplacian of the displacement field ${\mathbf {u}}$ and ${\text{Tr}}({\mathbf {I}})$ is the trace of the identity matrix corresponding to the dimension of the problem, i.e. 2D or 3D . The LuLu term is designed to reduce penalization on bending and quadratic compression deformations while maintaining penalization on excessive skew deformations, thus retaining the stabilizing properties of HuHu. The reduced penalization of bending deformations allows for more accurate modeling of curved contacts, which is particularly beneficial when the domain discretization used in finite element analysis is coarse. Similarly, the reduced penalization on quadratic compression is especially advantageous in topology optimization applications, where finite elements with varying material densities experience non-uniform deformation under compression.

An alternative and more complex regularization approach is the penalization of volume change and rotations proposed by Faltus et al., which remains to be extended to applications in 3D. A later improvement of this approach is offered by Wriggers et al. , who utilized the rotation tensor ${\mathbf {R}}$ directly, rather the approximation thereof used in .

Friction

A gap between a bulge and a solid bed is filled with a third medium ($\Omega _{v}$). The direction normal ${\mathbf {n}}_{0}$ and parallel ${\mathbf {s}}_{0}$ to the contact interface are defined based on the surface of the solid bed. Left and right boundaries are periodic and a displacement is enforced on $\Gamma _{D}$.

Sliding bulge problem solved with TMC friction (Blue) and a Lagrange multiplier approach (Black) for a friction coefficient of 0.3.

The integration of friction into TMC marks a significant advancement in simulating realistic contact conditions, addressing the previous limitation of the TMC method in replicating real-world scenarios. Currently, only one approach for adding friction is available . This approach adds shear stress to the contact and releases it through plastic slip if the contact is sliding.

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 if compressed regions of the third medium are subjected to, this is crucial for accurately modeling frictional contact.

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

$W_{ext}({\mathbf {u}})=W({\mathbf {u}})+{\dfrac {\beta }{2}}\left({\mathbf {C}}_{e}:({\mathbf {s}}_{0}\otimes {\mathbf {n}}_{0})\right)^{2}$,

where $\beta$ is a scaling parameter, ${\mathbf {s}}_{0}$is a unit vector parallel to the direction of sliding, ${\mathbf {n}}_{0}$ is a unit vector perpendicular to the contact interface, and ${\mathbf {C}}_{e}={\mathbf {F}}_{e}^{T}\cdot {\mathbf {F}}_{e}$ is the right Cauchy-Green tensor of the elastic deformation. The shear extension works by penalising the contribution in ${\mathbf {C}}_{e}$ associated with shear in the slip direction ${\mathbf {s}}_{0}$.

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

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