# Contact Mechanics and Friction: Is CAE Shedding Light on the Problem?

All movements on this planet involve contact mechanics and friction. Starting from simple walking, driving cars / bicycles to steam trains. Such physical problems involving contact mechanics are not only common in mechanical and civil engineering industries, as in Fig. 01, but also in environmental and medical applications like cardiovascular stents, hip replacements, pollution due to wear particles and more.

**Fig 01: Applications of contact. From left: car crash, sheet metal forming, eccentric loading on building foundation, roller bearing, impact on bridge columns (Source: Ref [2])**

## Studies of contact mechanics and friction through ages

The importance of friction and contact mechanics has been identified long ago. As shown in Fig. 02, the hieroglyphics show that Egyptians knew about lubrication and when they moved large stones, liquid used to be poured in front of the sledge. The ancient Indian civilization used hard stones like granite for constructions like Brihadeshwar temple. Granite is one of the hardest stones to cut and they knew that contact stresses could be exploited for cutting. Dry wooden wedges were inserted in adjacent holes, drilled from blunt chisels, and further watered. When the wedges expanded, the contact stresses cracked the granite.

**Fig 02: Knowledge of contact & friction in history: Egyptian hieroglyphics (left) (Source: Ref [2]); Cutting of granite in India (right) (Source: Quora)**

Contact mechanics and friction have been studied since the 15th century by great scientists starting from da Vinci, Coulomb, and Euler. Fig. 03 shows some of their devices or ideas regarding contact and friction.

**Fig 03: Historical devices of da Vinci’s experiments (left); Euler model for friction (middle) and Coulomb model of rough surfaces (right) (Source: Ref [2])**

Due to the nonlinear nature of contact mechanics, such problems are often approximated by special assumptions or boundary conditions. Contact mechanics and friction have been studied for several centuries and still remains one of the unsolved and toughest problems of solid mechanics. Due to the rapid growth of computational power, one can nowadays use the ideas of computational mechanics to numerically simulate applications of contact mechanisms for sufficient design accuracy. Yet, even to this day none of the standard finite element software is fully capable of solving contact problems, including friction, with robust algorithms. Hence, computational contact mechanics continues to remain a challenge for the **finite element** community.

## Contact Mechanics in SimScale

SimScale provides robust methods for enforcing contact constraints during FEM simulation. The numerics related to contact are brought through Code_Aster and CalculiX. Contact constraints are enforced through mathematical techniques called the Penalty method and Augmented Lagrangian method.

When a displacement or force is applied to a body during a finite element simulation, the body can be deformed into several possible configurations. Yet, it always results in the same unique configuration! The reason being that the deformation is governed by the principle of energy minimization. When force / displacement is applied, external work is done on the body. By conservation of energy, this energy is stored in the body as strain energy (stress x strain). Now, the strain (or deformation) is such that the total strain energy is as minimum as possible. For a simple linear elastic case, this can be written as:

Now, when two surfaces of the body (or multiple bodies) are in contact, then the surfaces are under compressive loading. Further on if the loading is increased, there can be an unphysical interpenetration and this should be avoided. Hence, when there is such an unphysical interpenetration, some extra energy is added. Since the goal is to look for a minimum energy, the system will try to avoid the addition of such energy contributions due to interpenetration. This way, a constraint is imposed on the system that the two surfaces can be in contact but cannot penetrate each other. With this additional energy contribution, now, owing to this contact (or actually when interpenetration happens), the total energy can be written as:

There are many approaches to writing this energy contribution from contact. In SimScale, Penalty and Augmented Lagrange approaches are available. Before discussing these approaches, it is imperative to understand standard terms like gap function and pressure used in this context.

### Kinematics of contact

Work or energy can be defined as force times displacement. During contact, the pressure (normal force) on the surface in contact is ** p** and tangential force, in the presence of friction, is

*The distance between the two surfaces along the normal direction is*

**tT**.**and along the tangential direction is**

*gN***as shown in Fig. 04.**

*gT***Fig 04: Definition of gap functions**

So for a simple normal contact, the work done can be written as** p x gN**. If there is no friction, there are no tangential stresses and hence no contribution from that. In the presence of friction, additional work contribution due to friction is added as

*tT*x

**gT**.The above is concisely written in terms of Signorini-Hertz-Moreau condition:

- When there is no contact, forces are zero but the gap is positive. Thus, no work is added.
- When there is contact, forces are non-zero but the gap is zero. Thus, no work is added.
- When there is interpenetration, the gap is negative and the pressure is non-zero and thus additional energy is added such that gap returns to zero in the subsequent iterations.

### Penalty method

When the penalty method is employed, as shown in Fig. 05, an ad-hoc penalty coefficient is assigned by the user.

**Fig 05: Input parameters for the penalty method**

The recommended value for the penalty coefficient is generally about 1 – 100 times the Young’s modulus of the material. Similarly, the friction penalty coefficient is assigned to the tangential direction. The penalty parameter is a scaling factor to calculate the normal pressure, only when there is contact or interpenetration, as

Normal pressure = Penalty coefficient x **gN**

Tangential force (or actually tangential traction) can similarly be calculated for sliding using the friction penalty coefficient. If there is an interpenetration, * gN* < 0 and this results in a compressive pressure. Now the change in energy contribution due to contact due to a small change in gap can be given as

and this is always greater than or equal to zero. Thus, the only possible minimum point is when the gap (* gN*) is zero. As the penalty parameter increases to infinity, the energy also increases, and the contact is imposed exactly. However, large penalty parameters lead to “ill-conditioning” and makes it harder to reach convergence. Thus, the recommended value for the penalty parameter is about 10 – 100 times the Young’s modulus of the material.

### Augmented Lagrangian Method

In order to circumvent the problem of ill-conditioning Lagrange multiplier method was explored. Here, the pressure (or the tractions) are considered as unknowns and iteratively solved as additional unknowns. Now the change in energy contribution due to contact due to a small change in gap can be given as

Here the pressure * p* is also considered unknown and solved for. So the total size of the problem increases and can add substantial costs depending on the number of elements that are in contact.

Contact is a nonlinear problem. This means that it is not known *a priori* if an interpenetration exists at a point. During the iteration, each point is checked for interpenetration and in such a case an extra constraint is added. This means that a point that was not undergoing interpenetration at an earlier iteration could be interpenetrating in this iteration. So each point on possible contact surface iterates between in-contact and no-contact. In this transition zone, Lagrange multiplier can demonstrate an oscillatory behavior.

Augmented Lagrange Method was formulated to bring the good of both penalty and Lagrange multiplier. It does not demonstrate an oscillatory behavior and has minimal interpenetration. The change in energy contribution for a small change in gap can be given as

where the variables are the same as described earlier.

**Fig 06: Inputs for Augmented Lagrange Multiplier option**

Here the earlier called penalty parameter is called “Augmentation coefficient”. The Lagrange multiplier “** p**” is an unknown and calculated during the solution process. The above have been explained for a normal contact. Similar arguments can be constructed for tangential contact involving friction. For a more detailed mathematical discussion on these topics the reader is referred to References [2] and [3].

## Numerics: Choosing the Best Option

Finally, getting to the hands-on part. SimScale offers exciting options to simulate contact in structural mechanics simulations as shown in Fig. 07.

**Fig 07: Global (left) and local (middle & right) options available in SimScale**

**Tip 01: Mesh refinement**

At present, SimScale uses a node-to-surface formulation as shown in Fig. 08. A perpendicular is dropped from each node on the slave surface to the master surface. This is used to distance between the slave node to the master surface. This formulation at presently is stable for small sliding only. Thus, the mesh at the contact surfaces needs to be well refined, as shown in the bolt connection example in Fig. 08.

**Fig 08: Slave node projected to master surface to determine gap function (left) (Source: http://www.code-aster.org/UPLOAD/DOC/Formations/13-contact-friction.pdf); Refinement of the mesh in the contact region (right)**

**Tip 02: Penalty or Augmented Lagrangian method?**

The global option is the selection of Penalty or augmented Lagrangian method as shown in Fig. 09.

**Fig 09: Global options that allow Penalty or Augmented Lagrange as method to impose contact constraints**

As indicated in the help tip, penalty method produces a small interpenetration while augmented Lagrange is almost exact. That apart, the Penalty method is generally sufficient for most problems of interest that involve small sliding. Augmented Lagrangian method could be used for improved accuracy for problems with less than 100,000 nodes. For larger problems, it would be necessary to note the number of elements / nodes in the contact region since this contributes to increase in the total equation size.

### Tip 03: **Friction**

Friction can be an important aspect in several physical applications and cannot be neglected. There are some applications where friction can be neglected and the obtained results are acceptable for preliminary design purposes.

SimScale presently offers consideration of friction through Coulomb friction law as shown in Fig. 10. Correspondingly, if friction is chosen, frictional penalty or frictional augmented method needs to be chosen in the local contact settings. If the global settings do not use friction, then frictionless options are the right ones to choose in the local settings.

**Fig 10: Global and local settings for imposition of friction between the contact surfaces**

### Tip 04: Contact smoothing

**Fig 11: Slave node being projected onto the master surface. Using first-order elements can result in surfaces where the normals are not smooth as shown**

As shown, the slave node when projected could be associated with the two different master surfaces. Such a situation could be common and lead to convergence issues when first-order elements are being used. Using the contact smoothing option allows for smoothing of the associated normals and prevent such ambiguities.

### Tip 05: Stop if interpenetration

SimScale offers one of the kind options that is generally not found in many commercial and academic codes. One of the very common reasons for the failure of contact simulations is the presence of interpenetration. When there is an interpenetration and it isn’t resolved after several iterations, the residual continues to remain beyond the threshold and the simulation crashes.

**Fig 12: Depiction of interpenetration at the contact surface**

In most occasions, the interpenetration is only at few nodes (as in Fig.12) and they would likely resolve in the next subsequent time step(s). In such a situation, it is likely that this option could be very helpful.

However, this option needs to be used with extreme caution. Like it is often said “With great power comes great responsibility!”. When this option is used, a critical assessment of the accuracy of the results needs to be considered.

### Tip 06: Verify norms

Just like the contact smoothing option, verifying the surface norms can be significantly helpful by ensuring that the normals are continuous. This is especially preferred when first-order meshes are used.

## Constraint between contact surfaces

In addition, to the above, there are situations when the contact surfaces do not undergo relative motion. For example, two surfaces could be glued or symmetry could exist or only sliding (but not normal) displacement could be allowed between the surfaces.

**Fig 13: Other contact options available in SimScale**

Cyclic symmetry exploits symmetry; bonded symmetry can be used when two surfaces, made of different solids in the assembly, are glued; sliding contact when there is no normal opening but sliding is allowed between the surfaces. These conditions can be used only when the sliding is small.

**Concluding remarks**

Computational contact mechanics remains one of the hardest problems of solid mechanics. The problem formulation is strongly nonlinear & beyond what is observed in most structural problems. SimScale offers several options that allow robust algorithms for structural simulations that involve contact. Long gone are the days when contact had to be approximated by special assumptions or boundary conditions.

#### References

[1] SimScale documentation: Physical contacts; Contact constraints

[2] Peter Wriggers, “Computational Contact Mechanics,” Springer-Verlag Berlin Heidelberg

[3] Alexander Konyokhov and Ridvan Izi, “Introduction to Computational Contact Mechanics: A Geometrical Approach,” John Wiley & Sons

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