Modeling tribological contacts at multiple scales

By Jane Marie Andrew, Contributing Editor | TLT Cover Story December 2023

Capturing the full complexity of tribological contact requires careful integration of disparate phenomena.

Tribological contact is inherently one of the most complex engineering scenarios that modelers face.
Models must correctly represent the interactions among phenomena operating at different scales and according to very different physics. 
Getting accurate and physically valid results depends on first clearly identifying what you want to predict, then thinking about the physics involved, and only then choosing a modeling framework appropriate to the scale of the problem.

For most of history, humans who wanted to understand the physical world were limited to what they could explore with their own senses. Although our senses are vastly extended by powerful scientific instruments, the idea of experimentation is essentially the same as it has been for millennia. Today, however, researchers also have a completely different mode of exploration: computer modeling. By representing a physical system mathematically through a computer program, researchers can simulate phenomena that would be impossible to observe even with the most sensitive instruments.

Modeling is important in tribology because everything interesting is happening on the tiny, tiny patches where surfaces are in contact. (Even apparently flat surfaces are in contact only on a small area.) These contact patches are measured in micrometers or even nanometers. What happens in these tiny patches dominates the behavior of the entire system, whether that be blood cells sliding along blood vessels or bearings turning in the gearbox of a huge, 10-megawatt wind turbine. The problem is that, in most cases, we can’t observe the contact patches directly. 

Understanding how contacts work is important because tribological interfaces are central to the design of virtually every product and manufacturing technology. They are hard to observe, however, so engineers have traditionally had to spend large amounts of time and money on physical prototyping to achieve correct performance. Today, by using models of contact phenomena, engineers can replace physical prototyping with simulated performance testing. 

There’s a catch, though. Although tribological behavior begins within a small contact domain, what happens in that domain is influenced by what’s happening in its vicinity, for example, how well the lubricant flows or whether the interface is externally heated (see Why Modeling Even Simple Contact Is Complex). “The broad scientific challenge is both conceptual and empirical. A holistic understanding of interfaces requires knowledge of their shape, structure and chemistry from the atomic to the macroscale, as well as their dynamic evolution under a range of often extreme operational conditions—temperature, pressure, shear, chemical reactivity, strain and so on,” says STLE member professor Daniele Dini, head of the tribology group in the Department of Mechanical Engineering, Imperial College London (see Phenomena at a Tribological Contact). This constellation of influences means that contact modeling is considered a “multiphysics” problem; that is, significantly different governing equations must be solved. This problem is compounded by the fact that the field of tribology spans the most disparate of applications, from biology to aerospace. Thus, what is recognized as an appropriate modeling solution in one area cannot be easily translated to another area. Furthermore, very different approaches may be needed when treating, for example, frictional interactions in soft tissues at a cellular level (which requires biological as well and engineering knowledge) versus in bearings used in a wind turbine in the presence of harsh environmental conditions. 

Why modeling even simple contact is complex 
Michael Watson, postdoctoral researcher at the University of Sheffield, gives some examples of the factors that make contact modeling so difficult: 

For the simplest example of frictionless, dry, elastic contact, while the contact patch might spread over millimeters, individual asperities [rough areas] might be contacting on a micrometer length scale. Because of that, we need to discretize the relatively large contact into very small points. This puts the solution out of reach of normal finite element solvers and means we have to use a trick, namely, pretending that the body is a flat half space [extending infinitely from a flat boundary],1 to solve the stress equations. Some¬times this works well, but sometimes it doesn’t. In particular, any phenomenon that depends on the true 3D shape of the surface, such as a spiky asperity getting flattened, needs to be modeled by a fully 3D solution to the stress equations, but such a solution is out of reach for current technology. 

Another problem we often face is pushing materials beyond their linear region. For example, most dry contacts involve some sort of plastic deformation, but even for the simplified flat case, deformation is extremely costly to calculate. Also, the flat-case simplification often won’t capture the sort of behavior most people think about when talking about plastic deformation in a contact (e.g., ploughing or flattening of asperities). 

Phenomena at a tribological contact
“Tribological contact is inherently one of the most complex engineering scenarios that modelers face,” says Daniele Dini, head of the tribology group, Department of Mechanical Engineering, Imperial College London. Here are just a few of the phenomena that must be considered:
Origin of friction
Surface roughness 
Evolution of tribofilms (solid surface films)
Evolution of material microstructure 
Evolution of damage near surfaces 
For lubricated contacts: Fluid rheology 
Changes in fluid film behavior.

If such broader influences aren’t considered, simulation results may be flawed—which could prove costly. Thus, a key challenge in modeling a tribological contact is the need to account for phenomena that occur at vastly different orders of magnitude in both size and time and that involve very different physics, all within the same model—an approach known as multiscale modeling. 

The multiscale problem 
“It’s best to see ‘modeling in tribology’ as a continuum from trying to predict the friction coefficient of some simple system (think diamonds in a vacuum) from first principles to using a statistical model to describe the results of a big, dirty experiment. In the middle are mixed models that use first principles to describe stresses, or empirical models based on experimental data,” says Michael Watson, postdoctoral researcher at the University of Sheffield. One approach to solving the multiscale problem is to connect models designed for different scales. 

In a comprehensive technical review of contact modeling, Vakis and colleagues describe multiscale modeling as “a technique in which two (or more) different models related to different [scales] (or different matter descriptions) interact, i.e., exchange data, in a way that enhances the information that can be obtained about the model phenomenon.”2 Figure 1 illustrates some modeling techniques that are used to study contacts. Various contact modeling solutions promising to overcome the scale issue are already available, although they tend to be optimized for solving specific problems. They span a vast range in complexity, cost and functionality and have been used in many practical applications to solve real world problems, for example to study friction in train wheels3 (see also One Perspective: Wheel-Rail Contact).

Figure 1. Tribological systems are simulated with different techniques at different scales. An area of current interest is linking molecular dynamics (MD) simulations to smaller scales through quantum mechanical descriptions (ab initio or first principles MD models and density functional theory [DFT] models) and to larger scales through continuum models (including computational fluid dynamics [CFD] models). Figure courtesy of Daniele Dini, from Ref. 4.

One perspective: Wheel-rail contact
The contact between the wheel of a train and a rail is extremely important for train safety. This contact scenario illustrates the complexity and importance of contact modeling. The friction, wear and damage behavior must be predicted, but conditions at the contact are influenced strongly by multiple (and sometimes unpredictable) factors. 

The following is an edited interview with Klaus Six, who has worked extensively on the problem of wheel-rail contact, initially with Siemens and currently as key researcher in the Department of Rail Systems, Virtual Vehicle Research Center, Graz, Austria.

What are you working on?
My research involves everything about wheel-rail tribology. One aspect of our work is to build up deep physical understanding, that is, what physical mechanisms and phenomena are responsible for certain problems. In other projects, we are trying to bring the developed models to application. We really try to have a wide range of projects from fundamental to applied research.

What don’t people understand about your work?
When I say I work on railways and wheel-to-rail contact, people think, “Well, railways have worked for more than 150 years; what is still unknown?” Just think though: you have a contact patch the size of a fingernail, but for a locomotive, say, you have loads in the range of 10 tons and more on one wheel. That scale is interesting.

What else is special about the wheel-rail interface?
You have stresses that are far above the yield stress of typical steels, caused by a huge normal loading in combination with a huge tangential loading because of the motion of the wheel relative to the rail. You always have plastic deformation in the near-surface layer. You get continual change in the wheel-rail surface roughness. There’s heat, because of the huge relative motions and the huge loads. You have huge frictional power going into the interface, which can cause material phase change. You have cracks starting and growing. And finally, it’s a very uncontrolled environment, meaning you have everything you can imagine on the wheel-rail surface: iron oxides, wear particles, leaves, water. You also have to consider operational and maintenance needs. For example, in certain circumstances you want a high friction value to transmit traction and braking forces. Under different scenarios, for example, in a tight curve, you want low friction to reduce the wear effects, so you apply lubricants and friction modifiers. An oil layer on the rail surface can bring in fluid dynamics effects. Everything that a tribologist wants to investigate!

I would summarize it this way: you have huge loads, uncontrolled conditions, and everything is changing all the time. That’s what makes it really difficult. I don’t know any other example from mechanical engineering that has that combination of complexity. And all that complexity means that you cannot use standard tools to predict all phenomena occurring at the wheel-rail interface When is it okay to rely solely on models? 

At my previous work, we did a lot of vehicle dynamics simulations. The goal was to design a vehicle to be comfortable and safe, that is, to ensure derailment doesn’t happen. We used multibody dynamics simulation software packages that included some models of wheel-rail contact at a macroscopic level. You needed to calculate the contact patch size, assume smooth surfaces and prescribe a certain coefficient of friction, and then the model was ready to use. This kind of model is perfectly okay for this kind of investigation because from experience everybody knows that under dry conditions you will get a coefficient of friction in the range of 0.3 to 0.4 and 0.2 under wet conditions. Those values are well validated for this kind of investigation. 

But when it comes to the best type of sand to use for the sanding system [for improving friction] or what to do when you have a leaf layer on the rail surface? No models exist for that. So you start to develop a model. You need to choose a modeling environment. You have to define the scale at which you want to model it. For example, we do particle-based modeling, where we consider each sand grain individually. There is no experiment available for that. If you build up such new models, where you’re trying to describe a very detailed physical phenomenon, then of course you need new experiments. That’s the thing I personally like the most. You start to think how to model a certain phenomenon, and in parallel you need to think about which kind of experiments you need. At the end, hopefully, the model is well calibrated and well validated, and when you apply it to other conditions, it should work. 

What’s the most promising approach to modeling wheel-rail interfaces at this point? 
I really like first of all to separate the physical phenomena occurring in the wheel-rail interface as clearly as possible. For example, how does roughness in the wheel-rail interface evolve during braking or traction? What is the influence of sand? When you put one sand grain into the wheel-rail interface, how does it interact with the wheel-rail surfaces? How does it improve friction? 

I think the best way to proceed is to separate the phenomena and then to inves¬tigate them on the right scale with the right modeling approach. I have the feeling that in the past, many people started somewhere in the macroscopic scale because of calculation time, and put a lot of effort there, but they did not consider the local effects in the wheel-rail interface, and so the basis of the models is very weak. 

When you really separate the effects, define clear research questions, have the right models, do the right experiments—small-scale, full-scale—then at the end you have one phenomenon well understood and the right model to simulate it with. Then you do this step by step for the different phenomena. Just putting models together does not make sense. Very different questions—for example, vehicle dynamics studies versus crack initiation/propagation studies for maintenance planning—will need very different models, in my opinion. 

What are some examples of modeling issues for particular phenomena? 
These are all examples where first of all you need to understand which phenomena are responsible for a certain damage or change. Based on this knowledge, you decide which model to use. 

White etching layers. One poorly understood effect is the development of so-called white etching layers in the wheel-rail interface. This is a martensite-like structure that appears as a white layer when you do metallographic etching of the surface. It’s undesired because it represents an initial damage that can lead to certain damage patterns (e.g., squats, a type of defect that can lead to a catastrophic rail break) but can’t be prevented. We know it is related to sliding effects, from braking, for example, but nobody knows how it forms We can speculate that traction and braking and the resulting high slip rates cause high temperatures in the wheel-rail interface. Then because of the heating and fast cooling, you get a phase change in the material microstructure, leading to a martensite-like structure hundreds of microns thick, in the near-surface layer. This sounds good, but nobody has really done the fundamental work to investigate how this white etching comes about. To model this process, you must go very deeply into materials science. You will need a temperature model; you will need to know something about the microstructure. 

Sanding systems. Many rail vehicles have sanding systems. They blow sand into the wheel-rail interface under low-adhesion conditions—for example, when there are leaves on the rail surface. The sand improves the friction. You can just use a macroscopic model and say the coefficient of friction has increased from 0.05 to 0.15, for example. The model would nicely reproduce, for example, braking distances. But with this approach you have no idea what is going on in the wheel-rail interface. We decided to use a discrete element method, which is particle-based modeling where we really follow one sand grain entering the contact patch, getting crushed, indenting into the wheel-rail surfaces, getting sheared, breaking again and so on. That was why we chose this particle-based approach: it was the right scale because we want to follow the sand grain and we want to describe how it affects wheel-rail surfaces and how it produces the increased friction. 

Crack initiation and growth. We know from a lot of experiments and from our modeling work that huge plastic deformations occur in the near-surface layer of wheels and rails, and they really change the microstructure and then define the path of surface-initiated cracks. From our physical understanding, we know that a crack initiation model needs to consider severe plastic deformation. With this knowledge, we know we need a very detailed material model, and we know that a standard finite element model will not be able to describe this huge plastic shear deformation. 

Everybody who starts to work on crack modeling in wheel-rail interfaces starts by using a finite element model. I’m convinced that this is not very sound. They will inevitably get very frustrated. In my opinion, it’s just not the right tool, for two reasons. First, it’s a matter of calculation time. You need to use the elastoplastic materials law, which means that calculating even one over-rolling will take a long time, even with good computer resources, but then you have to account for millions of over-rollings. Second, finite element meshes aren’t able to describe the kind of physics happening in such a huge shear deformation on the surface. You may add in some kind of crack initiation model, but it will likely be derived from microscopic experiments that don’t include plastic deformation. True, there are meshless finite element approaches, but the key is that this huge plastic deformation changes the microstructure. You need to know the effects on the microstructure, and that goes beyond just having a nice finite element model. 

In principle, feeding results from one scale to another should yield more accurate results: for example, if information about molecular interactions that determine the formation of protective tribofilms (calculated from quantum mechanics and/or molecular dynamics simulations) is passed to models that predict friction under lubricated conditions at the macroscale. In fact, when the processes at different scales are essentially separate, this type of “hierarchical” model can be relatively accurate and problem-free. That isn’t automatically the case, however, for a couple of reasons. First, the required computation time may be impractical (or outright impossible). According to Mahdi Mohammadpour, associate professor/senior lecturer, Loughborough University, UK, and founder of TriboDENS, “The multi-scale problem has been tackled by incorporating different simulation assumptions or computational accelerators (e.g., multigrid modeling). These solutions have resulted in computational enablers in tribology, but a reasonably realistic simulation still requires high-performance computing facilities. If multiphysics complexity is added (e.g., tribo-dynamics in rotating machinery), the solution may be completely unfeasible.” Second, although the compromises or assumptions mentioned by Mohammadpour make computation practical, they can introduce new problems. When models are later connected, assumptions made at one scale may, when used beyond their original scope, distort the calculations at another scale.

“Use with caution”
Although multiscale models are being used successfully in engineering applications, good physical judgment also is required. A simulation may spit out a result that looks useful, but such results cannot be used and trusted blindly; they need to be evaluated and interpreted by experts. As Vakis and coauthors point out, “An important question in multiscale modeling is the following: how to identify which spatial and temporal scales and mechanisms are relevant for understanding the phenomena to be modeled.” They note that a “simple recipe” is to introduce additional scales, not only in terms of dimensions but also in the type of computational approach. They go on to pinpoint a key weakness, however: “[T]his simple recipe can often be ineffective as it depends on the ability of the ‘user’ to add the right details at the right scale, and may lead to the neglect of important information flow across the scales.”2 Any engineer or researcher using such models should, therefore, think carefully about the cumulative impact of assumptions made at various scales.

Integration into design
Such models have given us a better understanding of the interactions that govern the response of engineering systems, but Dini identifies some pitfalls for designers: “These models are often problem-specific and fail to enable full integration in design strategies. Furthermore, they often consider only snapshots. We need a simulation strategy and a platform that allows engineers a convenient way to put tribology at the center of the design strategy. There is a need for flexible, predictive models of interfaces that describe their operando evolution.”

Watson raises another concern: “Each point along the scale continuum has its place and its own specific challenges, but one challenge that hits every point but that I rarely see discussed is validation.” Because “first principles” models (e.g., molecular dynamics simulations or Hertzian contact mechanics) aren’t expected to get the right answer the first time, they can’t be fully depended on in any situation. “That’s why people commonly add empirical models that they think capture the physics at play, but they typically stop tweaking the model when it fits the data on hand,” Watson says. As a result, the correlation between models and the real world appears better than is warranted by the underlying representation. “In essence a model is a theory of how something works, and, as a theory, its test should be its predictive power, not how well it fits to the data available when it was made,” Watson cautions.

A further difficulty, Watson says, is that often these models are not fully released to the research community, or they turn out to have issues when tested outside the original parameters, so comparing reported results remains difficult. Still, there are notable exceptions. Watson cites open source, cutting-edge codes such as Tamaas5 and says that some lubrication models are approaching good estimates of friction from first principles, even for elastohydrodynamic lubrication.

Asking the right question
The issue is sometimes more fundamental, as noted by Klaus Six, key researcher, Virtual Vehicle Research Center, Graz, Austria. “In doing a lot of reviewing work for journals and for doctoral theses, one of the main problems I see is that people start to work without defining a clear research question. First of all, you need to make clear what phenomenon you want to understand or predict, and then start thinking about modeling and testing. I very often have the feeling that people work the other way around,” Six says. Just using a complex model that takes a lot of calculation time does not guarantee more accurate results. “If the model is not able to describe relevant physical phenomena, it’s useless. First describe clearly what you want to predict, then think carefully about what the main physical effects are, and only then start thinking about what modeling approach is right,” Six advises.

Future models
Progress on this complex problem will require developments along several avenues, including changing how models are created and shared, exploring how modeling interacts with experimentation and integrating new computational tools.

Democratization and open source
Mohammadpour points to a challenge that is both technical and social: “In my view, the greatest challenge of computational tribology is lack of democratization in this field. Computational methods often require highly skilled tribologists to be successfully deployed. It is widely accepted that a tribology simulation will never be as accessible as a structural finite element analysis, but the current level of accessiblity is not good enough. There are examples of attempts to resolve this problem to allow engineers even at the graduate level to perform tribological simulations.”

One such solution is a free Python package for tribology called Slippy,6 developed at the University of Sheffield and the University of Leeds. The code provides a framework for a common approach to equations across multiple scales, according to STLE member Tom Slatter, professor, Department of Mechanical Engineering, the University of Sheffield. Slippy makes training in contact modeling more accessible, even to undergraduates, because the user doesn’t need to start with actual surface data. A student can generate a random surface and then explore it with other parts of the Slippy package. “It’s an application that’s meant to help people just getting into tribology,” Slatter says.

While it is important to improve access to training in tribological modeling, it also is important to improve the accessibility of both the models themselves and the data that are generated from the models or used to validate their implementation and verify their predictive capabilities. As Watson notes, “I believe a large part of future modeling must be open source. I have implemented many models from papers that claim good results, only to be disappointed either by the performance of the model itself or by the paper missing out hidden factors or nontrivial solution details. Normal peer review is not enough to catch small mathematical typos, but these can render a paper almost unusable. If the modeling code for a paper exists and runs, we as a community should require its release as the true object of peer review.”

A replacement for experiments?
For some purposes, models and simulations are good enough to replace physical testing. This success has led some to speculate that experiments will soon become largely unnecessary. Of course, models are only as accurate as their inputs, so data will still be required for material, lubricant, surface and so on. Therefore, experiment in this sense never goes away. However, “if by experiment we refer to a physical test for validation of the design or simulation,” Mohammadpour says, “this need will ultimately be eliminated in many subjects, especially those which are well established. This is already happening in big marine systems, for example, as the unique engines and transmissions of these systems do not give opportunity for prototyping. Therefore, the developers rely on simulation and calculations, which work surprisingly well. The problems may happen when new physics are introduced. An example is the electric field present in modern electrified powertrains, which leads to different behavior of the tribofilm and the lubricant and to novel failure modes.”

Watson likewise thinks this speculation is overstated: “I don’t believe that simulation will fully replace experiments in my lifetime, but I do think we will see an increase in models driving design and informing experiments.”

Dini agrees, saying, “Given the complexity of the mechanisms involved in tribological interfaces and the variety of applications, it will be probably impossible to ever replace experiemental testing completely. What I believe will happen is that computational models will be used for investigations that are extremely difficult to perform, especially at the smallest of scales, but there will always be a need for targeted experiments to complement and validate models, especially when new solutions and products are developed.” Data also will continue to be needed at the beginning of the modeling process as well, Dini says, as the foundation for developing “digital twins”—paired real and virtual processes or components that are each used iteratively to improve the performance of the other.

Advances in experimental and characterization techniques can change the equation too, Dini notes. An example is the use of multiple techniques to probe a single sample, whether at in-house labs or at synchrotron X-ray facilities; such techniques can provide data on previously inaccessible aspects of interfacial behavior.

Artificial intelligence and other computational strategies
New tools are emerging on the computational side, too. “The next step will be significant enhancement of simulation speed using artificial intelligence. This will break barriers and allow coupling of multiphysics and multiscale simulations that is currently impossible,” says Mohammadpour.7 Dini agrees: “Machine learning can speed up the integration between models and experimental data as well as contribute to the development of new disruptive data-driven modeling tools.”  

Watson agrees; his view is that speed can be achieved by treating models more statistically in general, both with machine learning and with other statistical approaches. “In my opinion, the future of tribology modeling is embracing statistical solutions to questions we would traditionally answer numerically. Great progress has been made in graphics simulation problems by implementing machine learning models for a special formulation of the Navier-Stokes equations. These models allow the equations to be approximately solved extremely quickly. Such models would unlock solutions to a host of truly 3D behaviors that remain out of reach of traditional solvers,” Watson says. 

A roadmap for progress 
According to Dini, taking tribological modeling and simulations to the next level will require progress in three areas. Researchers and engineers need to: 
1. Embed various machine learning and data-handling tools into all stages of the process of designing new tribological solutions. Tasks for such tools could range from the analysis of experimental data to autonomous design of molecules and materials. 
2. Integrate the traditionally distinct stages of modeling and experiment. With the two more tightly coupled, synergies will be found between information available across scales. 
3. Develop accurate virtual replicas of experiments, components and engineering systems. Such replicas will support end-to-end optimization of tribological interfaces with respect to key performance targets (including time-evolution, efficiency and long-term reliability).  

Inevitably, understanding a complex problem like tribological contact will require multifaceted solutions. As shiny new modeling strategies arrive and computing facilities become ever more powerful, the challenge for modelers and engineers is to remember to return, over and over, to the fundamental processes occurring at interfaces.

1. “An elastic half space is an elastic material that extends infinitely in all directions including depth with the surface at the top considered as the boundary,” in S.-V. Kontomaris and A. Malamou (2020), “Small oscillations of a rigid sphere on an elastic half space: A theoretical analysis,” European Journal of Physics, 41, 055004,
2. Vakis, A.I., Yastrebov, V.A., Scheibert, J., et al. (2018), “Modeling and simulation in tribology across scales: An overview,” Tribolology International, 125, pp. 169-199,
3. Fukagai, S., Watson, M., Brunskill, H.P., Hunter, A.K., Marshall, M.B. and Lewis, R. (2021), “In situ evaluation of contact stiffness in a slip interface with different roughness conditions using ultrasound reflectometry,” Proceedings of the Royal Society A: Mathematical, Physical, and Engineering Sciences, 477, 20210442,
4. Ewen, J.P., Heyes, D.M. and Dini, D. (2018), Advances in nonequilibrium molecular dynamics simulations of lubricants and additives, Friction, 6, pp. 349-386,
5. Tamaas — A High-performance Library for Periodic Rough Surface Contact, hosted by École Polytechnique Fédérale de Lausanne, Laboratoire de Simulation en Mécanique des Solides,
6. Slippy: A Python Package for Tribologists, developed by the University of Sheffield and the University of Leeds under the “Friction: The Tribology Enigma” Programme Grant, funded by the UK Engineering and Physical Sciences Research Council (EP/R001766/1),
7. Walker, J., Questa, H., Raman, A., Ahmed, M., Mohammadpour, M., Bewsher. S.R. and Offner, G. (2023), “Application of tribological artificial neural networks in machine elements,” Tribology Letters, 71, 3,

Jane Marie Andrew is a science writer based in the Chicago area. You can reach her at