LICENTIATE T H E S I S

Luleå University of TechnologyDepartment of Applied Physics and Mechanical Engineering

Division of Machine Elements

2008:16|: 102-1757|: -c -- 08 ⁄16 --

2008:16

Rough surfaces in contact - artificial intelligence and

boundary lubrication

Universitetstryckeriet, Luleå

Marco Pietro Rapetto

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Preface

This licentiate thesis is the result of the work conducted during the last two years at the Division of Machine Elements at Luleå University of Technology. The project has been carried out in close cooperation with SKF Engineering and Research Centre at Nieuwegein (The Netherlands), which I visited in January and February 2008, and is a part of a larger SKF project. The financial support for this work has been provided by ‘the Marie Curie Early Stage’ research program through the ‘AI4IA’ consortium and by the Swedish Science Council (VR). Their support is gratefully acknowledged.

I would like to thank Prof E. Ioannides, Technical Director Product R&D from SKF for his kind permission to publish this paper. I thank my supervisors prof. Roland Larsson and prof. Piet Lugt for their guidance and encouragement in my work. I thank Dr. Andreas Almqvist and his family for his excellent help through all the discussions we have had and for their warm welcome, especially during my first months in Luleå. I also send my gratitude to all my colleagues at the Division of Machine Elements for providing a pleasant and enjoyable environment. The free time would not have been so nice without my friends who deserve to be thanked. Finally I thank my family who always supported and encouraged me.

Marco Rapetto Luleå, May 2008

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Abstract

Interacting surfaces are found in mechanical systems and components. Since engineered surfaces are not perfectly smooth, only a fraction of the nominal surface area is actually in contact. This fraction is denoted as the real area of contact, Ar, and is formed by the sum of the contact spots between the two touching surfaces. If these contacting surfaces are sliding, then friction and wear occur in these actual contacts. Friction and wear may be controlled by lubrication: depending on the operating conditions different types of lubrication regime exist. When the surfaces are completely separated by the fluid film and load is carried by hydrodynamic action, contacts operate in hydrodynamic regime. When the load is carried by the lubricating fluid and asperity contact, the regime becomes mixed lubrication. In boundary lubrication, surfaces are in contact and the load is carried by surface asperities. In many cases this is the critical lubrication regime that governs the life of the components. Due to the complexity of thin film boundary lubrication, design of lubricated interfaces is still a trial-and-error process. The mechanism of formation and rupture of oxide layers and boundary layers is not completely known and a reliable model for rough surfaces in boundary lubrication is currently lacking. This study focuses on boundary lubrication regime: the effect of surface roughness on the real area of contact is investigated and a numerical model for the sliding interaction between two asperities in sliding contact is developed.

Numerical simulations of normal, dry, friction free, linear elastic contact of rough surfaces are performed. A variational approach is followed and the FFT-technique is used to speed up the numerical solution process. Five different steel surfaces are measured using a Wyko optical profilometer and several 2-D profiles are taken. The real area of contact and the pressure distribution over the contact length are calculated for all the 2-D profiles. A new slope parameter is defined. An artificial neural network is applied to determine the relationship between the roughness parameters and the real area of contact.

Boundary lubrication mechanism is usually controlled by the additives present in the oil that form low friction, protective layers on the wearing surfaces. Chemical reactions between the lubricant molecules and the asperity surface may take place. These reactions are activated by certain values of pressure and temperature. Fundamental research on the influence of surface roughness on contact conditions is hence required and is a key

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factor in understanding the wear mechanism in boundary lubrication condition since pressure distribution, shear stresses, frictional heating, mechanical wear highly depends on surface topography. Modelling boundary lubrication requires knowledge in many fields: contact mechanics, thermodynamics, surface chemistry etc, thus different sub-models interacting each other must be created. It is complicated and may be not feasible within a foreseeable time period to take into account all the different parameters and evaluate them. Artificial intelligence is a way to overcome the problem and determine the relationship between input parameters and desired outputs.

An elasto-plastic analytical model is used to determine the variation of pressure distribution and shear stress during the collision process of two asperities in sliding contact. The outputs of the elasto-plastic model are inputs of the thermal model that calculates the temperature rise during the collision process. The desorption of the adsorbed layer is determined by using existing adsorption theories and finally the probability of wear is computed at each time step of the collision process. Different results obtained using different adsorption theories and different input parameters are compared.

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Appended papers

Paper A: M. P. Rapetto, A. Almqvist, R. Larsson, P. M. Lugt, ‘On the influence of surface roughness on real area of contact in normal, dry, friction free, rough contact by using a neural network’, presented at the 11th

International Conference on Metrology & Properties of Engineering Surfaces, In Proceedings of the 11th International Conference on Metrology & Properties of Engineering Surfaces, Huddersfield, U.K., July 2007, accepted for publication on a special edition of Wear.

Paper B: M. P. Rapetto, R. Larsson, P. M. Lugt, ’A numerical model for the interaction between two asperities in sliding contact’, to be submitted for publication.

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Contents

Preface ……………………………………………………………... i

Abstract ............................................................................................. iii

Appended papers ………………………………………………….. v

Nomenclature ……………………………………………………... ix

Chapter 1 …………………………………………………………... 1 Introduction ………………………………………………………… 1 1.1 Boundary Lubrication ………………………………………... 2 1.2 Neural networks ……………………………………………… 4 1.3 Research objectives …………………………………………... 6

Chapter 2 …………………………………………………………. 7 The research project ………………………………………………... 7 2.1 Summary of appended papers ………………………………... 7 2.2 Conclusions ………………………………………………….. 8 2.3 Future works ………………………………………………… 9

A Paper A ………………………………………………………. 13

B Paper B ………………………………………………………. 27

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ix

Nomenclature

a Semi-width of contact [m] Ar Real area of contact d Interference [m] h Planck constant [J·s] dH Heat of adsorption [kJ·mol-1] E Young modulus [GPa] E* Combined Young modulus [GPa] he Oxide layer thickness [m] hc Oxide layer critical thickness [m] h0 Initial thickness of the oxide layer [m] H Hardness [GPa] L Load [N] Κ Thermal conductivity [W·m-1

·K-1] Kb Boltzman constant [J·K-1] m Lubricant / additive molecular weight [amu] NA Avogadro constant [mol-1] p Pressure [Pa] q Heat flux [N·m-1

·s-1] r Radius of the asperity [m] R Effective radius [m] Sa Fractional film defect So Probability of contact protected by oxide Sn Probability of metal contact t Time [s] T Temperature [K] v Sliding velocity [m·s-1] w Wear rate [m·s-1] W Load per unit length [N·m-1] Y Yield stress [Pa] ξ wear constant ν Poisson ratio κ Thermal diffusivity [m2

·s-1] τ Shear stress [Pa] τ0 Shear strength [Pa] η Fluid viscosity [Pa·s] μ Friction coefficient

x

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Chapter 1

Introduction

Tribology is the science of friction, wear and lubrication. The scientific study of tribology has a long history and some basic laws of friction are though to have been developed by Leonardo da Vinci in the 15th century. However the understanding of friction and wear languished for several centuries and it can be said that tribology is a very new field of science and most of the knowledge were gained after the Second World War. The word tribology was defined in 1967 by a committee of the Organization for Economic Cooperation and Development and derives from the Greek word ‘tribos’ meaning rubbing or sliding [1].

In machines used in everyday life, many components parts are in relative motion with respect to each other: these parts can be in contact, rolling, sliding or rubbing over one another. During their interactions forces are transmitted, mechanical energy converted, physical and chemical natures of the interacting materials are altered. The nature and consequence of the interactions that take place at the interface determine friction and wear. Friction and wear are not material properties; they are system properties and depend on the materials used and the operational conditions. Friction and wear may be controlled by lubrication: depending on the operating conditions different types of lubrication regime exist (fig. 1.1). In boundary lubrication (BL) condition surfaces are in contact and the load is carried by the surface asperities; in mixed lubrication (ML) the load is carried by both the lubricant film and the asperities in contact; in the hydrodynamic lubrication (HL) a full film of lubricant carries the load and the surfaces are not in contact. Elastohydrodynamic lubrication (EHL) is the type of hydrodynamic lubrication in which the elastic deformations of the contacting surfaces cannot be neglected. This is often the case of non-conformal contacts, for example the contact between the roller and the raceway in a typical roller bearing.

Chapter 1 Introduction

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The fast improvements of computers allow for complex tribological problems to be solved numerically. Numerical simulations may reduce expensive tribological testing and allow more detailed investigations of the influence of specific parameters in order to increase the understanding of the physical problem considered [2].

In this thesis fundamental research is carried out to study the interaction between asperities in contact in boundary lubrication condition.

Figure 1.1: Stribeck curve representing the different lubrication regime.

1.1 Boundary lubrication

Lubrication can be defined as any means capable of controlling friction and wear of interacting surfaces in relative motion under load. Boundary lubrication often occurs under high load and low speed conditions in bearings, gears, piston rings, transmissions etc. In many cases it is the critical lubrication mechanism that determines the life of the component subjected to wear.

When the load is high or the velocity is low the hydrodynamic pressure

1.1 Boundary lubrication

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Figure 1.2: A typical rolling bearing.

may not be sufficient to fully support the load, and the surfaces come into contact. The contact occurs at the peaks of the surfaces, called asperities. The amount and the extent of the asperity contact depend on many factors: surface roughness, fluid film pressure, normal load, hardness, elasticity of the asperities, etc. When the average fluid film thickness falls below the average relative surface roughness, surface contact becomes a major part of the load supporting system. Under boundary lubrication conditions, interactions between the two surfaces take place in the form of asperities colliding with each other. These collisions produce a wide range of consequences at the asperity level, from elastic deformation to plastic deformation, up to fracture. These collisions produce friction, heat, and sometimes wear. Chemical reactions between the lubricant molecules and the asperity surface, due to frictional heating may take place [3]. Boundary lubrication mechanism is usually controlled by the additives present in the oil. It involves the formation of low friction, protective layers on the wearing surfaces. If a low shear-strength layer can be formed on a hard substrate, then low coefficient of friction may be achieved.

In order to model boundary lubrication different models are needed: a contact mechanic model have to be used to calculate the pressure distribution and shear stress between the asperities and the transition between elastic, elasto-plastic and plastic contact conditions; a thermal model to determine the flash temperature caused by frictional heating; a chemical model to describe the chemical reactions that may take place between the lubricant and the surface, and the consequent change of friction coefficient, a wear model to calculate the wear of the colliding asperities. Many contact models exist. The contact between two rough surfaces may be studied by taking the measured profiles and computing the real pressure distribution, shear stresses, area of contact at each contact

Chapter 1 Introduction

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spot [4]. Other researchers prefer analytical models which consider each asperity as a sphere or ellipsoid: these models allow to model the transition between elastic, elasto-plastic and plastic deformation [5]. Many thermal models were developed in literature: the flash temperature distribution is calculated by solving the heat conduction equation, the Fourier law or the equations proposed by Jaeger [6]. Researchers derived also analytical solutions of interface flash temperature for general sliding contact case [7]. Some attempts were made to model boundary lubrication but at the moment a reliable model for rough surfaces in contact in boundary lubrication condition is lacking [8].

1.2 Neural networks

Modelling asperities collision in boundary lubrication regime is a complex problem since it requires knowledge in contact mechanics, thermodynamics, surface chemistry, etc. The number of parameter that must be taken into account is huge: material properties, lubricant properties, friction coefficient, sliding velocity, initial temperature at which the component is working, etc. It is complicated and may be not feasible within a foreseeable time period to take into account all the different parameters and evaluate them. Artificial intelligence is a way to overcome the problem and determine the relationship between input parameters and desired outputs.

Figure 1.3: Example of application of neural networks.

An artificial neural network (ANN) is a non-linear method of regression that has the advantage that the form of the regression equation has not to be

1.2 Neural networks

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specified before the analysis. It has the ability of learning relationships through repeated presentation of data and is capable of generalizing to new, previously unseen data. The central idea of the neural network method is this: given examples of a relationship between the inputs xi and a target t, the neural network is expected to learn a model of the relationship between xi and t. A well trained network will give, for any given xi, an output y that is close to the target value t. 'Training' the network means to optimize the weights wi that produce a function that fits the provided training data well. An artificial neural network is composed by several processing elements called neurons arranged in layers. A neuron simply multiplies each input xi by a random weight wi and sums the products together with a constant b to produce the scalar value ( )∑ +⋅= bxwn ii . This value n is the argument of a non linear function, usually a hyperbolic tangent function or a logistic function to produce the neuron output. This output can be the input to other neurons in the following layer. A layer that produces the network output is called an output layer; all the other layers are called hidden layers. A problem connected to neural networks is overfitting which leads to poor generalization. This problem is overcome by dividing the dataset in three groups: a training data set used to build the model and optimize the weights, a validation dataset used to monitor the error during the training and a testing dataset used to check the generalization ability of the neural network. The training is stopped when the validation error increases for a specified number of iterations. The number of hidden layers, the number of neurons in each hidden layer and the training algorithm, used by the ANN to find the best set of weights, have to be optimised in order to reach the best performance. The error can be reduced by using the average of predictions from a number of different ANN models, that is, a committee of models [9], [10], [11].

Figure 1.4: Representation of a single neuron.

xn

x1

x2

w1

w2

wn

tanh y = tanh(ΣΣΣΣwixi + b))))

Chapter 1 Introduction

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1.3 Research objective

The objectives of the research work are the following:

� To investigate the influence of roughness parameters on the real area of contact in rough contacts.

� To evaluate the use of neural networks in solving tribological problems.

� To develop a model describing the interaction process between two sliding asperities in boundary lubrication condition.

The study of the influence of surface roughness on contact conditions must be the first step in developing a model describing the behaviour of rough surfaces in contact in boundary lubrication condition. Friction and wear occur in the actual contact; in case of a small real area of contact the load is distributed over a relatively small number of asperities which will result in high pressure peaks. The real area of contact influences the heat flow per unit area and flash temperatures between the interacting asperities. Depending on the distance between contact spots, each hot spot may thermally influence the neighbouring asperities [12]. The chemical reactions between the lubricant and the surfaces require a certain value of temperature and pressure that depend on the contact condition.

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Chapter 2

The research project

2.1 Summary of appended papers

Paper A

In the engineering practice, the specification of the expected performance with respect to friction, wear and life of machine element surfaces is done through (standardized) surface roughness parameters. Paper A aims to investigate the influence of standard roughness parameters on the real area of contact in 2D normal, dry, friction free, elastic rough contacts using a neural network. A new slope parameter is also introduced and its effect on the real area of contact is studied. The model developed in [1] is used in order to perform numerical simulations of normal, dry, friction free, linear elastic contact of rough surfaces. A variational approach is followed and the FFT-technique is used to speed up the numerical solution process. Five different steel surfaces are measured using a Wyko optical profilometer and several 2-D profiles are taken. The real area of contact and the pressure distribution over the contact length are calculated for all the 2-D profiles. A new slope parameter is defined. An artificial neural network is applied to determine the relationship between the roughness parameters and the real area of contact. The trained model is able to capture the dependence of the real area of contact on the roughness parameters. The ability of the neural network to generalize on unseen data is tested. The neural network is able to prove the correlation between the roughness parameters and the real area of contact.

Chapter 2 The research project

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Paper B

In this paper a numerical model for the interaction between two sliding asperities in boundary lubrication condition is developed. It is assumed that the asperities are covered by an oxide layer over which lies an adsorbed film. The main idea is to investigate if a single collision is enough to produce a temperature high enough to generate the desorption of adsorbed layers and start the wear process. An analytical elasto-plastic contact model is developed. The flash temperature is calculated by solving the heat conduction equation coupled with the Fourier law. Three different existing theories are used to calculate the fractional film defect and results are compared. Different initial temperatures are also considered. The probabilities of protection given by the adsorbed layer, oxide layer and the probability of metal contact are calculated at each time step of the collision process. A wear model similar to that one used in [5] is introduced to calculate the wear of oxide layers.

2.2 Conclusions

The overall conclusions from this study are the following:

� It is possible to predict the real area of contact by knowing the mean slope, kurtosis and skewness of a profile for a given load.

� The most important parameter is the mean slope and a smaller effect is given by kurtosis and skewness.

� The linear correlation between the new slope parameter and the real area of contact is higher than the standard one but when the new slope parameter is combined with kurtosis and skewness does not bring any improvement in the analysis.

� Neural network models are powerful tools to identify trends in datasets and for identification of a minimal set of input parameters analysing the contributions of the input parameters.

� Existing adsorption theories have identified the lubricant and tribological parameters involved in the adsorption process but they behave in different ways. The theory proposed by Stolarski showed the importance of the sliding velocity. It is concluded that adsorption theory must be studied further and that adsorption models must be validated experimentally.

2.3 Future works

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� Flash temperatures in asperity collisions are low at low velocities (0.1 m/s) and will hardly cause desorption of additives. At higher velocities, the flash temperature will be high enough to change the coverage of adsorbed molecules.

� The initial temperature is an important parameter controlling the onset of desorption. Even moderate increase of the initial temperature may radically deteriorate the protection of the surface.

2.3 Future works

Future works will be towards the development of a multi-asperity sliding contact model in boundary lubrication regime. A reliable adsorption model is needed and must be developed. The formation and rupture of oxide layer and boundary layer must be better understood and modelled. Exploratory experimental work may have to be performed. Further contact mechanic simulations must be performed in order to determine if the highest peeks are in contact for a long time or always in contact and to better understand if and when there are thermal interactions between different hot spots. A 3D model can also be developed. In the field of artificial intelligence it may be of interest to evaluate the efficiency of genetic programming in solving complex tribological problems.

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Bibliography

[1] G. W. Stachowiak. Engineering tribology, Butterworth – Heinemann, 2000. [2] F. Sahlin. Hydrodynamic lubrication of rough surfaces, Licentiate thesis, Luleå University of Technology, online, 2005. [3] B. Bhushan. Modern tribology handbook, Boca Raton, Fla: CRC Press, 2001. [4] A. Almqvist, F. Sahlin, R. Larsson, S. Glavatskih. On the dry elasto-plastic contact of nominally flat surfaces, Tribology International,40(4):574-579, 2007. [5] J. Jamari. Running in of rolling contact, Ph.D. thesis, University of Twente (The Netherlands), online, 2006. [6] Carlslaw, Jaeger. Conduction of heat in solids, Oxford Press, 1959. [7] X. Tian, F. Kennedy. Maximum and average flash temperatures in sliding contacts, Journal of Tribology, 116:167-174,1994. [8] H. Zhang, L. Chang, M. N. Webster, A. Jackson. A micro-contact model for boundary lubrication with lubricant surface physiochemistry, Transaction of the ASME,125:8-15,2003. [9] H. Demuth and M. Beale. Neural network toolbox for use with matlab user’s guide version 3.0, The Math Works Inc. [10] H. K. D. H. Bhadeshia. Neural networks in material science, ISIJ International 39:966-979, 1999. [11] R. Kemp, G. A. Cottrell, H. K. D. H. Bhadeshia, G. R. Odette, T. Yamamoto, H. Kishimoto. Neural network analysis of irradiation hardening in low activation steels, Journal of nuclear materials, 348:311-328, 2006. [12] Gecim, Winer. Transient temperatures in the vicinity of an asperity contact, Journal of Tribology, 107:333-342, 1985. [13] S. Cupillard. Lubrication of conformal contacts with surface texturing,Licentiate thesis, Luleå University of Technology, online, 2007. [14] A. Almqvist. Rough surface elastohydrodynamic lubrication and contact mechanics, Licentiate thesis, Luleå University of Technology, online, 2004.

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Paper A

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On the influence of surface roughness on real area of contact in normal, dry, friction

free, rough contact by using a neural network

M. P. Rapettoa, A. Almqvista, R. Larssona P. M. Lugta,b

aDivision of Machine Elements, Luleå University of Technology, SE-97187 Sweden bSKF Engineering and Research Centre, P.O. 2350, 3430 DT Nieuwegein, The Netherlands

Abstract

A model previously developed at LTU was used in order to perform numerical simulations of normal, dry, friction free, linear elastic contact of rough surfaces. A variational approach was followed and the FFT-technique was used to speed up the numerical solution process. Five different steel surfaces were measured using a Wyko optical profilometer and several 2-D profiles were taken. The real area of contact and the pressure distribution over the contact length were calculated for all the 2-D profiles. A new slope parameter was defined. An artificial neural network was applied to determine the relationship between the roughness parameters and the real area of contact. The trained model was able to capture the dependence of the real area of contact on the roughness parameters. The ability of the neural network to generalize on unseen data was tested. The neural network was able to prove the correlation between the roughness parameters and the real area of contact.

Keywords: Dry contact, surface roughness, real area of contact, neural networks.

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1. Introduction

Engineered bearing surfaces are not perfectly smooth. Therefore only a fraction of the nominal surface area (An) is actually in contact. This fraction is denoted as the real area of contact, Ar, and is formed by the sum of the contact spots between the two touching surfaces. If these contacting surfaces are sliding then friction and wear occur in these actual contacts. In case of a small real area of contact, the load is distributed over a relatively small number of asperities which will result in high pressure peaks and thus an increased risk of failure for the component. In addition, frictional heating between the interacting highly loaded asperities can cause flash temperatures of several hundreds of degrees Celsius. A decrease of the real area of contact results in an increase of the heat flow per unit area with again results in very high temperature gradients in the roughness summits [1]. To prevent this severe contact, the surfaces of machine elements are usually, at least partically, separated by a lubricant film generated by hydrodynamic action. The effectiveness of this separation is mainly determined by the lubricant viscosity.

Many tribological problems are influenced by the real area of contact, thus it is very important to study how the surface topography influences the contact conditions and the real area of contact in order to produce surface topographies that have a positive influence on contact conditions. There are two approaches to model the contact between surfaces: statistical models and deterministic models. The first statistical model is due to Greenwood and Williamson [2]. This model was then improved by many researchers; see for example [3] and [4]. These models made it possible to calculate the real contact pressure and the real area of contact. Tian and Bhushan [5] based their theoretical model on a variational principle, see Kalker [6]. Recently Persson [7] introduced a new theory on contact mechanics for randomly rough surfaces based on fractals. Jamari and Schipper [8] described asperities as well-defined features but developed a deterministic model and implemented a theoretical model for the elastic-plastic contact of ellipsoid bodies.

There are many deterministic numerical techniques that may be applied to solve the mechanisms of dry: multilevel techniques [9], FFT [10], [11], moving grid method [12]. Almqvist et al [13] developed a deterministic model to be used for numerical simulation of the contact of linear elastic perfectly plastic rough surfaces. They based their model on variational principles and they used FFT-technique to facilitate the numerical solution process. In the engineering practise, the specification of the expected

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performance with respect to friction, wear and life of machine element surfaces is done through (standardised) surface roughness parameters.

The present paper aims to investigate the influence of standard roughness parameters on the real area of contact in 2D normal, dry, friction free, elastic rough contacts using a neural network. It means that by knowing the value of these parameters it is possible to predict the value of Ar for a given load. A new slope parameter is also introduced and its effect on the real area of contact is studied.

An artificial neural network is a non-linear method of regression that has the advantage that the form of the regression equation has not to be specified before the analysis. It has the ability of learning relationships through repeated presentation of data and is capable of generalizing to new, previously unseen data. A description of the method is given in [14].

2. Methodology

Five different bearing steel surfaces which have been run in on a ball-on-disk machine were measured using a Wyko optical profilometer. From each 3D surface several 2D profiles were taken. 1046 profiles were analyzed. The following standard roughness parameters were calculated for each profile: centre line average Ra, root mean square Rq, kurtosis Rku, skewness Rsk, mean slope Rdq, core roughness depth Rk, reduced peak height Rpk. The contact between each profile and a smooth flat plane was considered and the real area of contact Ar was calculated for each profile using the model described in [13] while applying a load W=104 N/m. The contact mechanics problem was solved by applying the minimum potential energy theory with an equation of force balance assuming that the contact is frictionless, isothermal and linear elastic. The pressure distribution over the contact length was calculated and the real area of contact was computed as a percentage of the nominal contact area. A linear correlation analysis (Neurosolution 5.0) was performed between the inputs and between inputs and output by calculating the sample correlation coefficient (Table 2). The mean slope Rdq is the highest correlated input parameter toward the output Ar. A high linear correlation exists also between Rdq and Ra , Rdq and Rq, Rdq and Rk . Ra, Rq and Rk are also correlated. There is also a moderate correlation between Rk and Rpk. Andberg et al [15] showed that within the Rk family, the only genuinely independent pair of parameters are Rpk and Rvk and outlying peaks cause a large increase in Rpk. Due to these

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cross-correlation, the number of inputs parameters for the ANN can be reduced to Rdq, Rku, Rsk. The output was the Ar.

Table 1: Input parameters.

Table 2: Linear correlation analysis.

3. Neural network model

A multilayer perceptron neural network was built. The number of hidden layers, the number of neurons in each hidden layer and the training algorithm were optimised in order to reach the best performance. The performance criterion was the mean squared error (MSE) between the ANN output y and the target t: MSE = [Σ(t-y)2]/N. The best ANN was found to be made up of 2 hidden layers: the first hidden layer consisted of 12 hyperbolic tangent functions, the second hidden layer had 4 hyperbolic tangent functions (Fig.1). The output layer was a linear layer. The best training algorithm was shown to be the resilient backpropagation

Input minimum maximumRa 0.02 [μm] 0.47 [μm]Rq 0.03 [μm] 0.64 [μm]Rku 1.8 10.3 Rsk -2.5 2.0 Rdq 0.002 [rad] 0.11 [rad]Rk 0.05 [μm] 1.07 [μm]Rpk 0.004 [μm] 1.55 [μm]

Rdq Rdq Ra Rq Rku Rsk Rk Rpk Ar

Rdq 1 Ra 0.83 1 Rq 0.80 0.99 1 Rku 0.01 0.32 0.38 1 Rsk -0.49 -0.58 -0.60 -0.33 1 Rk 0.93 0.90 0.87 0.08 -0.50 1 Rpk 0.47 0.63 0.64 0.25 0.01 0.50 1 Ar -0.77 -0.48 -0.45 0.16 0.04 -0.65 -0.34 1

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algorithm. Both the inputs and output were first normalized in order to have zero mean and unity standard deviation. The data order was then randomized and the data were divided in training dataset and testing dataset. A total of 100 networks were trained. The models were ranked using the mean squared error value in the testing dataset. It is possible that a committee of models can make a more reliable prediction than an individual model [16], [17] and [18].

6 8 10 12 14 16 18 200.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

N. neurons first hidden layer

MS

E

2 neur. 2nd h.l.

4 neur. 2nd h. l.

6 neur. 2nd h. l.8 neur. 2nd h. l.

10 neur. 2nd. h. l.

1 hidden layer

Figure 1: Optimum number of layers and neurons.

4. Results

The agreement between desired values of Ar and ANN output was higher by using a committee of 2 models and calculating the Ar as the average of the prediction of the two ANN models. Figure 2 shows the result of the training and Fig. 3 represents the result of the testing. For small values of Ar the predictions are better, while for high values of Ar the error is higher. The correlation coefficient 'r' is around 0.98 meaning that there is a high linear correlation between the desired Ar and the predicted values. A sensitivity analysis was conducted to provide a measure of the relative

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importance among the inputs of the neural network model and to illustrate how the model output varied in response to variation of an input. The first input was varied between its mean +/- its standard deviations while all other inputs were fixed at their respective means. The network output was computed for 50 steps above and below the mean. This process was repeated for each input. The sensitivity of each input was the standard deviation of the output divided by the standard deviation of the input which was varied to create the output (Table 3). A plot was created for each input showing the network output over the range of the varied input (Fig. 4, 5, and 6). The most important parameter is the mean slope Rdq since its sensitivity value is the highest. This parameter can explain a large amount of the variation in Ar. A small contribution is given by the skewness Rsk and the kurtosis Rku. The kurtosis Rku is the least important parameter. Figures 4, 5 and 6 show that the real area of contact decreases when the values of the roughness parameters increase. The significance of each of the input variables was also investigated by comparing the ANN models with one of the input parameters removed in each case (Table 4). This analysis confirmed the result of the sensitivity analysis and showed that all the three parameters are needed to make a good prediction.

0 5 10 15 20 25 300

5

10

15

20

25

30

Desired values %

AN

N v

alu

es %

MSE = 0.65

r = 0.99

Figure 2: Training result: desired Ar versus ANN results.

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0 5 10 15 20 250

5

10

15

20

25

Desired values %

AN

N v

alue

s %

MSE = 0.92

r = 0.98

Figure 3: Testing result: desired Ar versus ANN results.

2 2.5 3 3.5 4 4.5 5 5.51.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

Varied input kurtosis

AN

N o

utp

ut %

Figure 4: Sensitivity plot: varied input Rku.

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Table 3: Sensitivity analysis.

Table 4: Significance of the input parameters.

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.42

2.5

3

3.5

4

4.5

5

5.5

6

Varied input skewness

AN

N o

utp

ut %

Figure 5: Sensitivity plot: varied input Rsk.

Parameter SensitivityRku comm. 0.32 Rsk comm. 1.26 Rdq comm. 45.85

Method MSE R ANN only Rdq 5.86 0.89ANN Rdq and Rku 6.73 0.87ANN Rdq and Rsk 2.27 0.96

23

0.02 0.03 0.04 0.05 0.06 0.071

2

3

4

5

6

7

Varied input mean slope

AN

N o

utp

ut %

Figure 6: Sensitivity plot: varied input Rdq.

5. A new slope parameter

For two rough surfaces in contact, contact initially occurs at a small number of asperities (the highest), the definition of mean slope may, therefore be restricted by considering only the surface above a certain height., say ⋅n zmax, where zmax is the highest asperity on the surface. As a result of an optimization process where the correlation coefficient between the new slope parameter and the Ar has been maximized, n was found to be 0.4 with a correlation coefficient of -0.83. The neural network model was used to investigate the significance of Rdq04. The new parameter Rdq04 did not bring any improvement in the analysis (MSE = 1.11; r = 0.98). A reason may be the high correlation between Rdq04 and Rdq (r = 0.81).

24

6. Conclusions

A neural network model was used to predict the real area of contact between rough 2D surfaces and a rigid plane for a given load and to investigate the influence of standardised roughness parameters on the real area of contact. The analysis demonstrates the possibility to predict Ar by knowing the mean slope Rdq, kurtosis Rku and skewness Rsk of a 2D profile. The most important parameter was shown to be the mean slope and a smaller effect (but still important) was given by Rku and Rsk. A new definition of slope was also introduced. Even if the linear correlation coefficient between Rdq04 and Ar was higher, this parameter did not bring any improvement in the analysis. The present study showed that ANN models are a powerful tool to identify trends in datasets and for identification of a minimal set of input parameters analyzing the contributions of the input parameters. This paper should be viewed as a first step in a long-term effort in modelling of tribological phenomena using neural networks and artificial intelligence techniques as simulation tools. The next step in this study is to investigate if these conclusions apply to 3D surfaces as well.

Acknowledgments

The authors would like to thank Prof E. Ioannides, Technical Director Product R&D from SKF for his kind permission to publish this paper, Dr S. Velickov from SKF for his help in the ANN modelling and the AI4IA consortium for financing this project.

References

[1] A. van Beek. Advanced engineering design, TU Deft, 2006. [2] J. A. Greenwood and J. B. P. Williamson. Contact of nominally flat

surfaces, Proc. Roy. Soc (London), Series A. 295:300-319, 1966. [3] J. A. Greenwood and J. H. Tripp. The contact of two nominally rough

flat surfaces, Proc. Ist. Mech. Eng., 185:625-633, 1971.

25

[4] A. W. Bush and R. D. Gipson, and T. R. Thomas. The elastic contact of rough surfaces, Wear 35:87-111, 1975.

[5] X. Tian and B. Bhushan, A numerical three dimensional model for the contact of rough surfaces by variational principle. ASME Journal of Tribology, 118:33-42, 1996.

[6] J. J. Kalker. Variational principles in contact mechanics, J. Ist. Math. Appl., 20:199-219, 1977.

[7] B. N. J. Persson. Contact mechanics for randomly rough surfaces, Surface Science Report, 61:201-227, 2006.

[8] J. Jamari and D. J. Schipper. An elastic-plastic contact model of ellipsoid bodies, Tribology Letters, 21:262-271, 2006.

[9] A. A. Lubrecht and E. Ioannides. A fast solution of the dry contact problem and the associated sub-surface stress field, ASME Journal of Tribology, 113:128-133, 1989.

[10] Y. Ju and T. N. Farris. Spectral analysis of two-dimensional contact problems, ASME Journal of Tribology, 118:320-328, 1996.

[11] H. M. Stanley and T. Kato. An FFT-based method for rough surface contact, ASME Journal of Tribology 119:481-485, 1997.

[12] N. Ren and S. Lee. Contact simulation of three dimensional rough surfaces using moving greed method, ASME Journal of Tribology,115: 597-601, 1993.

[13] A. Almqvist, F. Sahlin, R. Larsson, S. Glavatskih. On the dry elasto-plastic contact of nominally flat surfaces, ASME Tribology International 40:574-579, 2007.

[14] H. K. D. H. Bhadeshia. Neural networks in material science, ISIJ International, 39:966-979, 1999.

[15] C. Andberg P. Pawlus, B. G. Rosen and T. R. Thomas. Comparative discrimination of honed surface structures, NT2006-XX-YY.

[16] D. J. C. MacKay. Bayesian non-linear modeling with neural networks, online Available: www.inference.phy.cam.ac.uk/mackay, 1995.

[17] S. M. Bateni, D. S. Jeng, B. W. Melville. Bayesian neural network for prediction of equilibrium and time-dependent scour depth around bridge piers, Advances in Engineering Software, 38:102-111, 2007.

[18] R. Kemp, G. A. Cottrell, H. K. D. H. Bhadeshia, G. R. Odette, T. Yamamoto, H. Kishimoto. Neural network analysis of irradiation hardening in low activation steels, Journal of nuclear materials348:311-328, 2006.

[19] B. Bhushan. Introduction to tribology, Wiley, New York 2002

26

[20] H. Demuth and M. Beale. Neural network toolbox for use with matlab, user’s guide version 3.0, The Math Works Inc.

27

Paper B

28

29

A numerical model for the interaction between two asperities in sliding contact

M. P. Rapettoa, R. Larssona P. M. Lugta,b

aDivision of Machine Elements, Luleå University of Technology, SE-97187 Sweden bSKF Engineering and Research Centre, P.O. 2350, 3430 DT Nieuwegein, The Netherlands

Abstract

As a step towards modelling the tribological behaviour of rough surfaces in sliding contact in the boundary lubrication regime, a numerical model for the interaction between two cylindrical asperities in sliding contact has been analyzed. For a given initial interference the variation of contact pressure, shear stress and flash temperature has been calculated during the contact time. The desorption of an adsorbed layer has been calculated using existing adsorption theories. Different results are compared. This work is intended as a first step in developing a model describing the formation and rupture of oxide layers, boundary layers and micro-wear of asperities in contact and defining a regime map for asperities interactions.

Keywords: asperity contact, boundary lubrication, adsorption

30

1. Introduction

It is well known that the contact between surfaces of machine components is made by asperity interaction. These asperities contacts play a significant role in the tribological performance of mechanical systems under boundary lubricated conditions. In order to study the behaviour of two asperities in sliding contact under boundary lubrication condition different models are needed: a contact mechanic model has to be used to calculate the pressure distribution between the asperities and the transition between elastic, elasto-plastic and plastic contact conditions; a thermal model to determine the flash temperature caused by frictional heating; a wear model to predict the desorption of adsorbed layer and consequent change of friction coefficient and wear of oxide layers. Kingsbury [1] proposed an expression for the fractional film defect in a micro-contact. He related the friction coefficient to the temperature, sliding velocity and the adsorption energy of a base lubricant. Sullivan [2] developed a theoretical model of oxidational wear under boundary lubrication condition based on Kinsbury’s adsorption model. A system for wear prediction in lubricated sliding contact was developed by Stolarski [3]. The rate of wear was predicted in probabilistic terms and the adsorption theory was used to represent the influence of a lubricant on the wear process. Lee and Cheng [4] related the scuffing to the breakdown of the adsorbed film by using the Langmuir adsorption theory. They defined the asperity flash temperature as a criterion to determine the scuffing. Shen et al. [5] modelled the competition between the oxide formation and its mechanical wear. Zhang et al. [6] developed a numerical model to study the behaviour of sliding micro-contacts. With their model they calculated the friction force, load carrying capacity and flash temperature of the micro-contact by relating them to the desorption of the adsorbed film, removal of oxide layer and evolution of the deformation state during the collision.

The aim of this study is to develop a numerical model for the interaction between two asperities in sliding contact. The pressure and flash temperature are calculated at each time step of the interaction process. The fractional film defect approach proposed by Stolarski [3] is followed to calculate the desorption of the adsorbed layers. The Langmuir adsorption isotherm and the Volmer adsorption isotherm [6] are also used and results are compared. The main idea is to investigate if a single interaction process between two asperities is enough to produce a temperature high enough to generate the desorption of adsorbed layers and start the wear process.

31

2. Mechanical model

A line contact is considered. The two asperities are cylindrical having radii r1 and r2 and maximum interference d0. The bottom asperity is fixed while the upper asperity is translated laterally. Their bases are held at a fixed distance h.

The relationship between h and d0 is:

021 drrh −+= (1) The initial value of the tangential distance between the two asperities at the point of initial contact is:

2221 )( hrrLi −+= (2)

At every point of the interaction process the relationship between the local interference d and the maximum interference d0 is:

2021

221 )( drrLrrd −++−+= (3)

The model developed by Johnson and Shercliff [7] is used to study the variation of elastic deformation, pressure distribution and contact length during the interaction process. The contact between the asperities is assumed to be a Hertzian strip of semi-width a. An expression given by Johnson [8] for the compression d of a cylinder of radius r relative to its

x h

y

d0

r1

r2

Li

v

Fig. 1: Sliding asperity contact.

32

centre due to a Hertzian load W per unit length is used to define the elastic compression of the two contacting asperities. The total compression of the two contacting asperities is given by:

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

−−

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

−−

= 14

log2)1(

14

log2)1( 2

2

21

1

2

a

r

E

W

a

r

E

Wd

π

ν

π

ν (4)

where E is the Young modulus, ν the Poisson ratio. For an Hertzian line contact of maximum pressure pmax :

2

max2

2

4⎟⎠⎞

⎜⎝⎛

==∗∗ E

p

R

a

RE

W

π (5)

where E* is the combined elastic modulus and R is the effective radius. Hence:

⎥⎦

⎤⎢⎣

⎡−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=⎥

⎦

⎤⎢⎣

⎡−⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

=∗

∗1

4log21

8log2

2 max

2

max

2

p

E

E

p

a

R

R

a

R

d (6)

The variation of a and pmax is given by equations (3), (5), (6). The Von Mises criterion is used to determine the limit of elastic deformation:

222 26 Yp =+ τ (7)

where Y is the yield stress. The elastic limit is reached when:

22

_ 32

τ+==p

Yp ecr (8)

The Von Mises equation states that any stress combination can be applied to an element of material and the material will remain elastic until the proper summation of all stresses equals 2Y2. By combining equation (8) and equation (6), a critical interference for elastic deformation is found:

Rp

E

E

pd

cr

crcr ⎥

⎦

⎤⎢⎣

⎡−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛= 1

4log2

*2

* (9)

If the load of the contacting asperities is increased significantly, such that

33

the deformations become irreversibly, the contact operates in the fully plastic contact region. For a frictional contact the transition point from elasto-plastic to plastic contact may not be easily defined. A critical pressure for plastic deformation is derived based on Tabor’s yielding equation for asperity junction:

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

20

2

2_

1

1

H

p pcr

τ

μ (10)

where τ = μp is the shear stress, μ is the friction coefficient, τ0 is the substrate shear strength, p is the pressure and H is the hardness. For frictionless contact in the fully plastic contact regime, the mean contact pressure equals the value of the hardness. A critical interference for plastic deformation may also be determined.

Zhang et al. [9] performed a FEM analysis to study the effects of friction on the contact and deformation behaviour in sliding asperity contact. They found a curve describing the variation of critical interference for plastic deformation as a function of the friction coefficient. For a frictionless contact the interference causing the onset of fully plastic deformation of the contact is about 40 times of dcr_e, which is in the same order of magnitude of the value obtained using other models (see for example [10]). That graph (reproduced in fig. 2) is used in the present model to find a value for the critical interference for plastic deformation dcr_p.

When the contact is fully plastic the mean contact pressure is kept constant to pcr_p and the semi-contact width is equal to:

dRa 2= (11)

An approach similar to that one used by Jamari [10] and Zhao [11] is used to model the elasto-plastic regime. Thus:

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−−−=

)log()log(

)log()log()(

__

____

ecrpcr

pcrecrpcrpcr dd

ddpppp (12)

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−+⎟

⎟⎠

⎞⎜⎜⎝

⎛

−

−−−+=

2

__

_

3

__

_ 32)7.02(7.0ecrpcr

ecr

ecrpcr

ecr

dd

dd

dd

ddRdRdRda (13)

34

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

5

10

15

20

25

30

35

40

Friction coefficient

dcr p /

dcr

e

Figure 2: dcr_p / dcr_e as a function of coefficient of friction from [9].

-4 -2 0 2 4 6 8 10 12 14

x 10-5

0

1

2

3

4

5

x 109

Position [m]

Con

tact

pre

ssur

e [P

a]

2.29e-5 s9.14e-5 s2.29e-4 s4.57e-4 s6.86e-4 s9.14e-4 s0.0011 s

Figure 3: Variation of pressure distribution with time. The pressure decreases when the friction coefficient starts to increase because of the desorption process.

35

v

3. Thermal model

The temperature rise due to frictional heating is modelled as follows. The asperities are considered flat. The contact length varies in size with times from zero to a maximum to zero again and the size is given by the mechanical model previously described. The time of contact is given by 4amax / v.

Figure 4: Shape of the asperities for thermal analysis.

It is assumed that all the heat generated by friction is conducted inside the bodies. Outside the contact region the surfaces are assumed to be insulated. In roller bearings the sliding velocity is low, stationary conditions may be assumed since the dimensionless parameter (Peclet

number) κ2

va < 0.1 for all the sliding velocities considered [12]. The

temperature distribution is calculated using the heat conduction equation combined with the Fourier law. The frictional heat flux is equal to:

q = μpv (14)

where v is the sliding velocity, μ the friction coefficient and p the pressure in the contact. The friction coefficient is considered constant but its value changes depending on the desorption of the adsorbed layer (see [6]):

μ = Saμa + S0μ0 + Snμm

Sa + S0 + Sn = 1

where Sa is the probability of protection given by an adsorbed layer, So is the probability of protection given by the oxide layer, Sn is the probability of metal contact. A heat partition is needed since part of the heat will go to one surface and part to the second surface. Assuming that the two

36

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-3

10

15

20

25

30

35

40

Time of contact [s]

Tem

pera

ture

ris

e [°

C]

jaeger equation

heat conduction eq.

Figure 5: Comparison with Jaeger solution. Stationary constant heat supply on infinite strip.

1.5 2 2.5 3160

180

200

220

240

260

280

Velocity v [m/s]

Tem

pera

ture

ris

e [°

C]

present modelQiu

Figure 6: Comparison with Qiu model. The data used are the same used in [15].

asperities have the same bulk temperature, are made of the same material and neglecting the effect of velocity (stationary condition) the heat partition can be set to 0.5 and only the temperature on one surface need to be studied. Thus the equations to be solved are:

37

t

T

y

T

x

T

∂

∂=

∂

∂+

∂

∂

κ

12

2

2

2

(15)

y

TKq

∂

∂−= on the surface (16)

On the surface of the body the Fourier law is used. The heat conduction equation is solved by means of the finite differences method with Dirichlet boundary conditions. Fig. 5 shows a comparison with an analytical equation due to Jaeger [13] for a stationary heat supply on infinite strip, when the model is set to the same condition. The two solutions diverge when the time increases but the reason is that the formula given by Jaeger is only valid for transient cases. The temperature continues to increase and never reaches a steady state. This behaviour is given by an exponential integral. Fig. 6 is a comparison with Qiu and Cheng result [14] (and also Gao and Lee [15] which performed the same simulation using FFT technique and obtained the same result) when the same conditions are used. They considered the Hertzian contact between two spheres with smooth surfaces. One sphere was moving while the other one was stationary. The maximum contact pressure was 1.45 GPa and the nominal Hertzian contact width was 2.26 10-4 m. The temperature obtained with the present model is in the same order of magnitude. Fig. 7 shows the variation of surface temperature distribution with time.

0 10 20

x 10-5

10

20

30

40

50

60

70

80

90

Position [m]

Tem

pera

ture

ris

e [°

C]

1.14e-4 s1.60e-4 s2.28e-4 s4.57e-4 s6.86e-4 s9.14e-4 s0.001 s

Figure 7: Variation of surface temperature distribution with time.

38

4. Existing adsorption theories

A formula proposed by Stolarski [3] is used to determine the fractional film defect. The fractional film defect is defined as the ratio of the number of sites on the contact area not occupied by lubricant molecules to the total number of sites available on the contact area. Two opposing contact asperities covered by adsorbed lubricant molecules coming into contact are considered. According to Stolarski the sliding velocity have to be taken into account since at slow rates of approach, the adsorbed molecules will have more time to desorb. The expression of fractional film defect proposed by Stolarski has been slightly modified to be able to compare with other models. It must be mentioned that formula (6) in [3] contains some printing errors. The fractional film defect is given by:

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛ −

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ⋅−=

RT

dH

vm

TS m

a exp1095.2

exp5.0

5.04

(17)

where Tm is the melting temperature of the lubricant, v is the sliding velocity; m is the molecular weight, dH the heat of adsorption, R the gas constant, T the temperature. The Langmuir adsorption isotherm and the Volmer adsorption isotherm may also be used to determine the desorption of the adsorbed layer. In the Langmuir adsorption theory the concentration of the adsorbed lubricant molecules is related to the surface temperature and the surrounding lubricant pressure by the following formula [4]:

⎭⎬⎫

⎩⎨⎧−

+

=

RT

dH

h

TmK

h

TKF

FS

bba

exp2

2

π (18)

TmK

pF

bπ2=

Where Kb is the Boltzmann constant, h Plank’s constant. The Volmer adsorption isotherm is given by:

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−

=⎟⎠⎞

⎜⎝⎛

a

a

a

a

S

S

S

S

RT

dHpb

1exp

1exp0 (19)

39

12/3

20 1000

2−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛=

hN

TmKTKb

A

bb

π

with NA Avogadro’s constant, Sa fractional film defect.

5. Wear model

A wear model is defined to calculate the wear of oxide layers. The wear rate is given in the following form according to the wear theory proposed by Archard:

vH

ptw ξ=)( (20)

During a small time step dt the oxide layer thickness he decreases to he-w(t)dt. During the same time step the desorption of the adsorbed layer brings out new oxide exposure with a probability Sa –Sa’ where Sa’ is the probability of adsorbate coverage at t+dt. The new average thickness is given by:

a

aaaee S

SShSwdthh

'1

)'()1)((' 0

−

−+−−= (21)

Following an approach proposed by Zhang [6], the probability of oxide layer protection is written as:

)1(00 aSSS −= (22)

where 0S is given by:

⎪⎪⎩

⎪⎪⎨

⎧

=

<<

≥

=

00

0

1

0

e

cec

e

ce

h

hhh

h

hh

S

When the oxide layer is between 0 and hc 0S is approximated by a linear function. This linear function is followed based on the argument of the

40

growth of oxide on asperities being continuously interrupted by removal [16].

6. Results

The model developed is used to simulate a sample problem of asperity collision. It has been shown, by considering statistical distributions of heights of surface asperities, that the average size of a micro contact is almost constant and independent of load. Surfaces in contact will always have some asperities elastically deformed and some plastically deformed. The interaction between 2 asperities is a transient problem, thus the length of contact is an important parameter. Five bearing surfaces were measured using a Wyko optical profilometer. The maximum measured radius of curvature was around 100μm. The sample problem considers two equal asperities of 100μm radius. The upper asperity is sliding with a velocity of 0.1 m/s. The bottom asperity is fixed. The material properties and data used are shown in Table 1. The maximum interference was set to the critical interference for plastic deformation, thus the two asperities deform elastically at the beginning of the interaction process, then elasto-plastically and they reach plastic deformation at the position of maximum interference. Two initial contact temperatures are considered: 20 oC and 80 oC. The fractional film defect is calculated using the three formula (17), (18), (19) at each time step and the friction coefficient is updated for the next time step. Results are compared. The temperature that may cause the desorption of the adsorbed layer is given by the sum of the bulk temperature Tsurr and the flash temperature dT produced during the contact. Fig. 7 and Fig. 8 show the temperature rise dT over the initial temperature in the contact area. The low temperature rise is due to the low velocity and the short contact time. For an initial temperature of 20 0C, if the Volmer adsorption isotherm is applied, a minimum fractional film defect of 0.89 is reached, as a consequence the friction coefficient increases and the temperature is higher than that one obtained using the formula (17). When formula (17) is used to determine desorption of the adsorbed layer, the additive does not desorb (Sa min= 0.99). Near the end of the contact the temperature decreases since the contact length is reduced. Fig. 8 and Fig. 9 show the temperature rise and the variation of fractional film defect as a function of time when the initial temperature is set to 80 °C. In this case higher temperatures are reached if formula (17) is used. The explanation is

41

found in Fig. 10 which explains the behaviour of the three different adsorption theories. The curve proposed by Stolarski is steeper and a slightly increase in temperature is significant. The influence of initial temperature is investigated when formula (17 ) is used (Figs. 11 to 14). The oxide layer does not give any protection due to the high contact pressure. Fig. 15 shows the temperature rise, the increase of coefficient of friction due to desorption, the variation of contact pressure and shear stress during the interaction process when the initial temperature is set to 80 oC. Due to increase of friction the asperity looses its load carrying capacity.

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

0

5

10

15

20

25

30

35

40

Time [s]

Tem

pera

ture

ris

e [°

C]

Volmer

Stolarski

Figure 7: Temperature rise when the initial temperature is set to 20 °C.

42

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

0

10

20

30

40

50

60

70

Time [s]

Tem

pera

ture

ris

e [°

C]

Stolarski

Volmer

Figure 8: Temperature rise when the initial temperature is set to 80 °C.

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

Fra

ctio

nal f

ilm d

efec

t

Langmuir

StolarskiVolmer

Figure 9: Fractional film defect as a function of time of contact.

43

200 300 400 500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Temperature [K]

Fra

ctio

nal f

ilm d

efec

t

Langmuir

VolmerStolarski

Figure 10: Fractional film defect as a function of temperature. dH = 40 kJ/mol, m = 800 amu, Tm = 290 K, p = 6.64 GPa.

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

Fra

ctio

nal f

ilm d

efec

t

To = 20 °C

To = 40 °C

To = 60 °C

To = 80 °C

Figure 11: Variation of fractional film defect during the time of contact for different initial temperature To.

44

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time [s]

Pro

babi

lity

of o

xide

laye

r pr

otec

tion

To = 20 °C

To = 40 °CTo = 60 °C

To = 80 °C

Figure 12: Probability of oxide layer protection for different initial temperature T0.

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

Pro

babi

lity

of m

etal

con

tact

To = 20 °C

To = 40 °CTo = 60 °C

To = 80 °C

Figure 13: Probability of metal contact.

45

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

0

10

20

30

40

50

60

70

80

90

100

Time [s]

Tem

pera

ture

ris

e [°

C]

To = 20 °C

To = 40 °C

To = 60 °C

To = 80 °C

Figure 14: Temperature rise for different initial temperatures.

0 0.5 1 1.5

x 10-3

0

50

100

Tem

pera

ture

ris

e [°

C]

Time [s]0 0.5 1 1.5

x 10-3

0

0.2

0.4

0.6

0.8

Fric

tion

coef

ficie

nt

Time [s]

0 0.5 1 1.5

x 10-3

2

3

4

5

6x 10

9

Time [s]

Ave

rage

pre

ssur

e [P

a]

0 0.5 1 1.5

x 10-3

0

0.5

1

1.5

2x 10

9

Time [s]

She

ar s

tres

s [P

a]

Figure 15: Temperature rise, friction coefficient, mean contact pressure, shear stress versus for T0 = 80 °C.

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7. Comparison between different adsorption models

The following graphs compare the different adsorption models. When the velocity is increased (fig. 16), the curve describing the adsorption theory of Stolarski moves to the right: higher temperatures are needed to cause desorption. An explanation could be that at low sliding velocities, adsorbed molecules have ample time to desorb [17]. This is in agreement with the experiments performed by Hirst and Stafford [18]. For further explanation see [3]. The Volmer and Langmuir isotherm are not influenced by velocity. Fig. 19 shows the influence of the contact pressure. In Fig. 17 the product pressure times velocity pv has been kept constant in order to obtain the same heat flow q. Thus if the velocity increase, the pressure must decrease to keep the same heat flow q = μpv (assuming the same friction coefficient). It is interesting to see that the Stolarski adsorption isotherm behaves in a different way than the Volmer and Langmuir adsorption theories. Increasing the velocity, the Stolarski curve moves to the right, wiz higher temperatures are needed for desorption, while the other curves move to the left. Since q is constant, the temperature rise produced by frictional heating is the same. The different theories predict different probability of fractional film defect, which means different coefficient of friction and different temperature rise in the next time steps. Figures 18, 20 and 21 show how the heat of adsorption, molecular weight and melting point of a lubricant influence the fractional film defect according to the three different theories.

8. Discussion

The existing adsorption models behave in different ways. Stolarski [3] claims that his model agrees well with some experimental results obtained by Frewing [19], and Matveevsky [20]. In all models adsorption is limited to a monolayer. A reliable adsorption model is needed since the desorption process is a key factor in the evolution of the asperity collision process. It has been shown that due to a low velocity, a low initial friction coefficient and a short time of contact the temperature rise is low. This is confirmed also in [21] and can be easily verified by applying other thermal models using the appropriate data for an asperity collision. In this situation the

47

200 300 400 500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Temperature [K]

Fra

ctio

nal f

ilm d

efec

t

LangmuirVolmer

Stolarski v = 0.1 m/s

Stolarski v = 0.2 m/s

Stolarski v = 0.3 m/s

Stolarski v = 0.4 m/sStolarski v = 1 m/s

Figure 16: Influence of velocity. dH = 40 kJ/mol, m = 800 amu, p = 6.64 GPa.

200 300 400 500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Temperature [K]

Fra

ctio

nal f

ilm d

efec

t

v =0.1 m/s

v = 0.2 m/s

v = 0.3 m/s

Figure 17: Fractional film defect versus temperature when the product pv = constant. dH = 40 kJ/mol, m = 800 amu, pv = 6.64 108 N/(m·s). dashdot line = Langmuir, dashed line = Volmer, solid line = Stolarski adsorption theory.

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200 300 400 500 600 700 800 900 1000 11000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Temperature [K]

Fra

ctio

nal f

ilm d

efec

tm = 800 amu

m = 600 amu

m = 400 amu

Figure 18: Influence of molecular weight. dH = 40 kJ/mol, p = 6.64GPa, v = 0.1 m/s.

200 300 400 500 600 700 800 900 1000 11000

0.1

0.2

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0.6

0.7

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0.9

1

Temperature [K]

Fra

ctio

nal f

ilm d

efec

t

Langmuir p = 6.64 GPaVolmer p = 6.64 GPaStolarskiLangmuir p = 4 GPaVolmer p = 4 GPa

Figure 19: Influence of pressure.

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200 300 400 500 600 700 800 900 1000 11000

0.1

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0.7

0.8

0.9

1

Temperature [K]

Fra

ctio

nal f

ilm d

efec

t

290 K270 K250 K320 K390 K

Figure 20: Influence of melting point.

200 300 400 500 600 700 800 900 1000 11000

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0.7

0.8

0.9

1

Temperature [K]

Fra

ctio

nal f

ilm d

efec

t

dH = 40 kJ/mol

dH = 30 kJ/mol

Figure 21: Influence of heat of adsorption.

initial temperature is an important parameter to start the desorption process. When it starts, the change of friction coefficient causes higher temperatures. Higher temperatures can be reached for longer time of contact. Simulations of rough surfaces in contact are needed to determine if the highest peeks are in contact for longer time or always in contact. It has

50

been shown [22] and it is easy to show with the present model [23] that if the distance between two asperities is larger than 10 times the contact size, there will be no thermal interaction between the asperities. Gao [15] developed a transient flash temperature model for 3D rough surfaces in contact: his temperature profile appeared surprisingly gradual although the pressure profile had extremely sharp peeks. This is an indication of significant cross-heating activity between the asperities. Lee and Cheng [4] claim that the assumption that asperities are far apart from each other is unreasonable since the asperities are closely packed together and the temperature rise on one asperity cause the surrounding asperities to heat up. It must also be mentioned that the cooling down process is very rapid [24] and takes a time interval which is about equal to the time of asperity contact [23]. The conclusion is that surface roughness is an important parameter and its influence on frictional heating should be investigated. Wear of oxide layers is determined by using Archard’s wear theory. A constant wear coefficient was used. Quinn developed a theory for oxidational wear. His theory may be used to find the right value of the wear coefficient. The type of oxide on the surface must be known. The tribological values of the Arrhenius constant and activation energy for parabolic oxidation have to be known. The original oxidational wear theory is based on an interpretation of the K-factor in the Archard wear law and assumed that a total number of 1/K passes are needed for the whole of the dominant oxidized plateau to build up to a critical thickness at which it becomes unstable and breaks away from its substrate. Experiments are used to find the tribological values of the Arrhenius constant and activation energy [25].

9. Conclusions

A model describing the interaction process between two sliding asperities in boundary lubrication was developed. Different adsorption theories were compared. It was shown that:

� Existing adsorption theories have identified the lubricant and tribological parameters involved in the adsorption process but they behave in different ways. The theory proposed by Stolarski showed the importance of the sliding velocity. It is concluded that

51

adsorption theory must be studied further and that adsorption models must be validated experimentally.

� Flash temperatures in asperity collisions are low at low velocities (0.1 m/s) and will hardly cause desorption of additives. At higher velocities, the flash temperature will be high enough to change the coverage of adsorbed molecules.

� The initial temperature is an important parameter controlling the onset of desorption. Even moderate increase of the initial temperature may radically deteriorate the protection of the surface.

Table 1: Material properties and input data used in the calculations.

Symbol description Value Unit c Specific heat 500 JKg-1K-1

dH Heat of adsorption 40 kJ/mol d0 Max interference = dcr_p m E Young modulus 210 GPa H hardness 7.4 GPa h Planck constant 6.62608 ·

10-34 Js

hc Critical thickness oxide layer 0.25·10-9 mhe Initial thickness oxide layer 1·10-9 mK Thermal conductivity 47 Ns-1K-1

Kb Boltzman constant 1.38066 · 10-23

JK-1

m Lubricant / additive molecular weight 800 amu NA Avogadro constant 6.0221367·

1023 mol-1

r Asperity radius 100·10-6 mR Gas constant 8.31451 v Sliding velocity 0.1 ms-1

T0 Initial temperature 293 - 353 K ν Poisson ratio 0.3 μa Friction coeff. of contact covered by adsorbed

film 0.1

μ0 Friction coeff. of contact covered by oxide layer 0.3 μn Friction coeff. with no boundary protection 0.8 ξ Wear constant 2·10-4

τ0 Shear strength 0.28H GPa ρ Mass density 7875 Kg m-3

κ Thermal diffusivity K/(cρ) m2s-1

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Acknowledgments

The authors would like to thank Prof E. Ioannides, Technical Director Product R&D from SKF for his kind permission to publish this paper. We also want to thank the Swedish Science Council (VR) and the Marie Curie AI4IA consortium for financing this project.

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