- Email: [email protected]
Physical nature and properties of dynamic surface layers in friction
Tribology International 39 (2006) 426–430 www.elsevier.com/locate/triboint
Physical nature and properties of dynamic surface layers in friction V.L. Popova,*, S.G. Psakhieb a Technische Universita¨t Berlin, Institute of Mechanics, 10623 Berlin, Germany Institute of Strength Physics and Materials Science, Russian Academy of Sciences, Tomsk, Russia
b
Available online 8 June 2005
Abstract Simulations of the dynamic processes in micro contacts with the Method of Movable Cellular Automata (MCA) show that their common feature is formation of a boundary layer where intensive plastic deformation and mixing processes occur. The boundary layer is well localized and does not spread to deeper layers. We call this layer a ‘quasi-fluid layer’. The thickness of the boundary layer is roughly proportional to the viscosity of solid. This parameter thus should play an important role in determining the wear rate of materials in friction. To better understand the physical nature of the dynamic surface layers, we consider a simplified model of a solid consisting of many thin sheets, interacting with each other according to a ‘friction law’ of Coulomb type. A quasi-fluid layer is always developing if the ‘friction law’ does allow a bi-stability in some range of stresses with one static and one dynamic state at the same stress. The existence of the boundary layer motivates us to change the existing approach to calculating wear in frictional contacts. The wear should be understood not as ‘fracture’ but as ‘mass transport out of friction zone’. The process of stochastic transport of wear particles in the closed friction zone is at the same time the main mechanism of development of surface topography. A very important fact is that the conditions for appearance of a quasi-fluid layer depend on the minimal size of structural elements of the medium, which means that this effect cannot be principally described in the frame of a continuum model. q 2005 Elsevier Ltd. All rights reserved.
1. Introduction Most technical surfaces show roughness on different space scales. When pressed against each other, they initially come into contact only in a small number of micro contacts. The processes in micro contacts determine the friction forces between bodies and the wear rate. Their study is of great interest for understanding friction. In discussing the processes in micro contacts we encounter, however, the following problem. Fourier spectra of surface profiles of many technical surfaces are power functions in some interval of wave vectors [1]. In this interval the surfaces have the property of self similarity to scaling of wave vectors and amplitudes and can be referred to as fractal. This—at least partial—fractality leads to a difficulty, which was not known in contact mechanics several years ago: If we would measure the surface roughness and then solve the contact problem with the given roughness, than both the size of the contact region and pressure distribution in the contact * Corresponding author. Fax: C49 30 314 72575. E-mail address: [email protected] (V.L. Popov).
0301-679X/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2005.04.027
will depend on the accuracy to which the surface microrelief was determined. Thus, if we assume a model of absolutely smooth and elastic bodies (e.g. wheel and rail), we will have the solution first found by Hertz. In view of fractal structure of surfaces, however, the real contact occurs in regions far smaller than micrometer size. The transition to even lower (nanometer) scale level would cause further decrease in ‘real contact area’ and increase in ‘real pressure’ at nano-contacts. The exact mathematical result is that the real contact area of fractal surfaces in approximation of linear elasticity is exactly zero and the pressure in real contacts infinitely large [2]. This means physically, that if we have fractal surfaces, then there is always some space scale at which the solids should behave plastically. In our opinion, this is just this scale which is responsible for friction forces and wear in most tribological systems. Our present paper is devoted to the question what happens in the micro contacts on the scale where the pressures are already so high, that intensive processes of plastic deformation should occur. At this scale we await that processes of intensive plastic deformation, detaching of wear particles and their repeated welding into the surfaces due to the conditions of normal pressure combined with shear deformation occur.
V.L. Popov, S.G. Psakhie / Tribology International 39 (2006) 426–430
2. Numerical model
Table 1 Parameters of the modeled material
To model the processes in the surface layers, we use the method of Movable Cellular Automata (MCA) [4,5]. According to this method, a medium is represented as an ensemble of discrete elements—movable cellular automata characterized by continuous variables such as center of mass position, value of plastic deformation, rotation pseudovector as well as by the discrete variables characterizing connectivity of the neighboring automata. The principles of writing the equation of motion for a system of cellular automata and prescribing interactions between them are described in [4,5]. In the following simulations, a twodimensional version of MCA method has been used. In our case, a modeled object consisted of four parts (Fig. 1):
Young modulus Poisson ratio Density Elastic limit Yield stress Strain at the yield stress Ultimate strength Breaking deformation Viscosoity
– The upper layer of automata was an absolutely rigid, nondeformable body moving horizontally at velocities ranging between 1 and 10 m/s in different numerical experiments; – Two intermediate layers with initial roughness of nanometer range represented surface regions of the bodies in contact; – The lower layer was a fixed support. A constant normal force corresponding to the pressure range up to the 3/4 of the yield stress of the material acted upon all the elements of the upper layer. The diameter of the automata was from 2.5 to 10 nm in different numerical experiments. The elastic properties of the automata corresponded to the steel with Young’s modulus of 206 GPa and Poisson’s ratio of 0.3. The yield strength and ultimate strength with respect to tension were varied between 80 and 480 MPa and between 92 and 552 MPa, respectively. An important parameter determining stability of the plastic deformation processes and significantly affecting
427
EZ206 Gpa nZ0.3 rZ7800 kg/m3 sy1Z51–306 MPa sy2Z80–480 MPa 3y2Z0.015 s0Z92–552 MPa 3cZ0.04 hZ0.41 Pa s
the characteristic size of the surface region of severe plastic deformation occurs to be viscosity. Introduction of viscosity is necessary already from a formal consideration of providing stability to the calculation procedure. Phenomenological viscosity reflects dissipation processes under strain occurring due to electron and phonon excitation in a solid. It was assumed in the numerical model that viscous forces acting between unconnected but contacting automata are proportional to the relative velocity of motion. At the left and right fragment boundaries, periodic boundary conditions were used. The boundary conditions for the rigid upper automata layer were put as follows: constant velocity of the layer in the tangential direction and constant force in the normal direction. The initial roughness was specified explicitly. The material parameters used are listed in Table 1 (for determination of strength parameters see a graphic inset in Table 1).
3. Formation of a boundary ‘quasi-fluid’ layer
Fig. 1. Initial structure, dimension and loading conditions of the modeled fragment.
The numerical experiments show that within the first nanoseconds after onset of a relative tangential motion of bodies, the roughness of both surfaces is already severely deformed and fractured, and a dynamic equilibrium in the system is established at a temporal scale of about 100 ns. A pronounced boundary layer appears, where the processes of deformation, fracture, reconstruction of connectivity between elements, and intensive mixing take place. The motion in the layer resembles turbulent motion in fluids (Fig. 2). For this reason we refer to it as a quasi-fluid layer. Note, however, that this layer is not liquid in terms of thermodynamics. A quasi-fluid layer remains localized in
428
V.L. Popov, S.G. Psakhie / Tribology International 39 (2006) 426–430
Fig. 3. Multi-layer model for simulation of the quasi-fluid layer.
Fig. 2. ‘Instant picture’ of a quasi-fluid layer structure for a material of ultimate strength 552 MPa: pressure PZ382.5 MPa. The sliding velocity is 5 m/s.
the vicinity of the initial friction surface and does not propagate to deeper regions of the contacting bodies. A characteristic depth of the layer depends on the system parameters, first and foremost, on the effective viscosity of the system of automata. In our calculations, viscosity was used as a fitting parameter and was chosen so that the layer depth corresponded to the experimental values [5,6]. Initial roughness leaves the regularities of quasi-liquid layer formation unaffected. The dependence of the thickness of the quasi-fluid layer on the material parameters can be easily estimated with the following simple model. Let us assume, that the quasi-fluid layer has a thickness h and that it can be characterized by the dynamic viscosity h. The viscous stress in the layer is then sxhv/h, where v is the sliding velocity. In the stationary state it should be equal to the limiting shear stress s0, at which plastic yielding is setting on: s0xhv/h. Hence, h zh
v s0
(1)
4. A multi-layer model To better understand the physical nature of the dynamic surface layers, we consider a simplified model of a solid consisting of many thin sheets, modeling atomic layers of a crystalline solid. We assume, however, that the atomic layers do move relatively to each other as rigid sheets.
The only property of the atomic structure which we use in our model is the periodic interaction potential between sheets and a linear damping due to interactions with the phonon and electron subsystems of the solid. The equations of motion of all layers with exception of the first and the last layer are the following: m€x n C 2ax_n K ax_nC1 K ax_nK1 C f0 sin kðxn K xnK1 Þ K f0 sin kðxnC1 K xn Þ Z 0:
ð2Þ
The first is moved with a constant velocity V0 and for the last layer a different damping constant a1 is used to avoid reflection from the rigid non moving lower layer (Fig. 3). Numerical experiments with this model show, that in a under-damped systems a surface layer does appear, with a linear gradient of velocities (Fig. 4) and with dependence of the thickness on the velocity and material parameters as predicted by the above simple estimation (1). It is important to note, that the effective viscosity determining the thickness of the quasi-fluid layer is in this model exactly the same, as we would measure in the elastic state from the damping of elastic waves. A quasi-fluid layer is always developing if the ‘friction law’ between layers does allow a bi-stability in some range of stresses with one static and one dynamic state at the same stress. The physical nature of bi-stability is of no importance for the phenomenological picture of the process. For example, the above mentioned model is nothing but a modified Tomlinson model [7], which is just a rigid body moving in a periodic potential. In Tomlinson model, in the under damped case, the force necessary to start the motion is larger than the minimal force needed to maintain it. Thus, in a definite range of forces, the body can be either in the state of rest or in the state of motion. In the state of rest, only the conservative part of forces does play any role. In the state of motion, practically only the viscose part of force is important. In this sense we can refer to these states as to an ‘elastic’ and an ‘viscous’ one. If we ‘pile up’ the rigid
V.L. Popov, S.G. Psakhie / Tribology International 39 (2006) 426–430
Fig. 4. Schematic presentation of the simulation results with the multi-layer model. In the vicinity of the surface, there is a constant velocity gradient. The deeper layers are in the elastic state.
bodies with periodic interaction between them, the possibility of being either in an elastic or in viscous state is valid for each pair of them. This shows immediately the further possibility of one part of the system to be in a viscous state while the other in the elastic. Details of this behavior are presented in the paper by Geike & Popov in this issue. In the Tomlinson model, the bi-stability is due to the inertia effects. This is not the only possible mechanism of bistability. For example, if we consider an over damped many layer system of the above type, than it is not bi-stable. However, it becomes bi-stable if the deformation induced softening of material is taken into account. See for details the paper [8] and paper by Geike and Popov in this issue.
5. Wear The existence of the boundary layer motivates us to change the existing approach to calculating wear in frictional contacts. Wear is often understood as microfracture and detaching of surface particles. In the presence of a boundary layer, however, the wear is not directly connected with fracture and detaching of particles. Indeed, even if a particle with characteristic sizes of 10–100 nm has been broken from the surface, it remains for a long time (relative to the average time between two consequent impacts of the particle with surfaces) inside the friction zone. The separated particle suffers many ‘impacts’ from the surfaces and is reintegrated into the surfaces many times.
429
The wear should thus be understood not as ‘fracture’ but as ‘mass transport out of friction zone’. A real ‘act of wear’ occurs only when a wear particle comes close to the boundary of the friction zone where it is not in conditions of confined deformation anymore and, therefore, no reintegration of wear particles occurs. Let us estimate the wear rate due to stochastic transport of nano particles out of friction zone. There is only one characteristic length scale which characterizes the quasi-fluid layer—the thickness of this layer. Both detachment of particles in this layer and their reintegration occur at the distance Dxxh. The characteristic time of such elementary stochastic step has the order of magnitude Dtxh/v. The stochastic wandering of wear particles can be described as an effective diffusion with a diffusion coefficient D xDx2 =2t xhv=2. The average square of distance of a wear particle from its initial position is proportional to the time: L2Z2DtZhvt. Thus, the average time needed to reach the border of the friction zone is txL2/hv. In this time, most of the material of a quasi-fluid layer would be transported out of the friction zone. It thus would become thinner by h. The wear velocity can thus be estimated as vt xh=t xvðh=LÞ2 . This estimation is, however, not yet completely correct, as the above mentioned processes only occur in the regions of ‘real contact’. The portion of such contacts is, after Bowden and Tabor, approximately equal P/sy, where P is the apparent pressure and sy the yield stress. The final estimation of the wear velocity is thus vwear xv
P sy
2 h L
Experimental investigations of the surface layers of combustion engines carried out at the Institute of Applied Wear Research in Karlsruhe confirm the existence of dynamic nanolayers, in which intensive mass mixing effects do occur [9,6]. Experimental results show that a thin nanolayer (usually from 10 nm up to 70 nm) is formed on the surface of grey iron cylinder during the wear-in period (the first hours of work) in combustion engines. Layer thickness depends on conditions of preliminary running-in. Usually it decreases as running-in power increases. Once formed thin layers stay thin in future and demonstrate better wear resistance than thick layers. In our opinion, the mechanism of surface layer formation is the inhomogeneous deformation and mass mixing (mechanical alloying) in the quasi-fluid layer. Wear rate in good serial engines is in the range of 0.2 nm/h. Size of wear particles is distributed in the range from a few nanometers to micrometers. Experiments with nucleopore filters show that most wear particles have the size between 10 and 100 nm with nearly 50% having less than 40 nm in diameter [3,10]. All this is in a good qualitative agreement with the concept of quasi-fluid layer. The dependence of wear rate on the thickness of the layer with the chemically changed composition is also confirmed by experimental observations.
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V.L. Popov, S.G. Psakhie / Tribology International 39 (2006) 426–430
The process of stochastic transport of wear particles in a closed friction zone is in our opinion also the main mechanism of development of surface topography. The particles are detached and welded again into the surfaces in a random manner. This processes cause obviously not only the mass transport along the surfaces but also the material transfer from one contacting body to the other. This gives rise to a stochastic wandering of the position of the boundary between two bodies both in lateral and in normal directions. If we consider these processes of mass transport in a mathematical model, it occurs that they give rise to a stochastic surface with a region of fractal power spectrum and two cutting wave vectors—exactly as in the real technical surfaces. This mechanism will be discussed in detail in the paper by Schargott & Popov in this issue.
velocity v and density r can enter the relation for the friction coefficient only as rv2. Hence it follows that in the general case the friction coefficient may be thought of as a function of three independent combinations of the parameters (E, s0, r, v, P). Simulations of the friction coefficient with the method on Movable Cellular Automata show, however, that the friction coefficient does depend on the viscosity. This result is in contradiction with the assumption that the friction coefficient does depend only on the macroscopic parameters of the material. This means, that the microstructure of the material determines essentially the tribological properties of materials. This conclusion is confirmed by the consideration of many layer models, where the tribological properties also do depend on the discretization of the model and not only on the macroscopic material parameters (see paper by Geike & Popov for details).
7. Dependence of friction coefficient on loading and material parameters
8. Conclusion
6. Development of surface topography
Both wear and friction force are result of processes in micro contacts and specifically in the quasi-fluid layer. If we can model them, it should be possible to determine the dependence of the wear rate and the friction force on material constants and macroscopic loading conditions. This is indeed the main purpose of the friction physics: to be able to predict the tribological properties of materials and to design materials with desired properties. On what parameters do depend tribological properties of materials? We try to give the answer to this question at the example of the friction coefficient. Let us make a hypothesis, that the friction coefficient is a function only of macroscopic parameters. That means, that, for example the exact atomic lattice parameter should not have an effect on the tribological properties. Or, in other words, materials with different microscopic structure but the same macroscopic parameters have the same tribological properties. This assumption is made often implicitly without mentioning it. Under this assumption, certain conclusions on the friction coefficient as a function of material and loading parameters can be made from the analysis of dimensionality. Indeed, the friction coefficient is a dimensionless quantity and, hence, can depend only on dimensionless combinations of system parameters. It can be shown that no dimensionless combination involving viscosity can be made up from material and loading parameters (E, s0, r, v, P, h). This implies that the friction coefficient cannot depend on viscosity. The analysis of dimensionality shows that
Our numerical simulations show that dynamic processes of plastic deformation, micro fracture and micro welding at the nano level determine the friction force, wear and formation of surface topography in friction processes. Their further study is of utmost importance for tribology. The discrete structure of the matter at this scale essentially influences its tribological properties, which implies that the latter cannot in principle be determined in the frame of continuum mechanical models.
References [1] Persson BNJ, Bucher F, Chiaia B. Phys Rev 2002;B65:184106. [2] Persson BNJ. Elastoplastic contact between randomly rough surfaces. Phys Rev Lett 2001;87(11):116101. [3] Persson BNJ. Sliding friction. Physical principles and applications. 2nd ed. New York: Springer; 2000. [4] Popov VL, Psakhie SG. Physical mesomechanics 2001;4(1):15–25. [5] Popov VL, Psakhie SG, Gerve A, et al. Physical mesomechanics 2001;4(4):73–83. [6] Popov VL, Yu SI, Gerve´ A, Kehrwald B. Simulation of wear in combustion engines. Comput Mater Sci 2000;19(1–4):285–91. [7] Tomlinson GA. Philos Mag 1929;7. [8] Popov VL. A theory of the transition from static to kinetic friction in boundary lubrication layers. Solid State Commun 2000;115:369–73. [9] Scherge M, Shakhvorostov D, Pohlmann K. Fundamental wear mechanism of metals. Wear 2003;255:395–400. [10] B. Kehrwald, Untersuchung der Vorga¨nge in tribologischen Systemen wa¨hrend des Einlaufs, Dissertation, Universita¨t Karlsruhe, 1998.
Physical nature and properties of dynamic surface layers in friction V.L. Popova,*, S.G. Psakhieb a Technische Universita¨t Berlin, Institute of Mechanics, 10623 Berlin, Germany Institute of Strength Physics and Materials Science, Russian Academy of Sciences, Tomsk, Russia
b
Available online 8 June 2005
Abstract Simulations of the dynamic processes in micro contacts with the Method of Movable Cellular Automata (MCA) show that their common feature is formation of a boundary layer where intensive plastic deformation and mixing processes occur. The boundary layer is well localized and does not spread to deeper layers. We call this layer a ‘quasi-fluid layer’. The thickness of the boundary layer is roughly proportional to the viscosity of solid. This parameter thus should play an important role in determining the wear rate of materials in friction. To better understand the physical nature of the dynamic surface layers, we consider a simplified model of a solid consisting of many thin sheets, interacting with each other according to a ‘friction law’ of Coulomb type. A quasi-fluid layer is always developing if the ‘friction law’ does allow a bi-stability in some range of stresses with one static and one dynamic state at the same stress. The existence of the boundary layer motivates us to change the existing approach to calculating wear in frictional contacts. The wear should be understood not as ‘fracture’ but as ‘mass transport out of friction zone’. The process of stochastic transport of wear particles in the closed friction zone is at the same time the main mechanism of development of surface topography. A very important fact is that the conditions for appearance of a quasi-fluid layer depend on the minimal size of structural elements of the medium, which means that this effect cannot be principally described in the frame of a continuum model. q 2005 Elsevier Ltd. All rights reserved.
1. Introduction Most technical surfaces show roughness on different space scales. When pressed against each other, they initially come into contact only in a small number of micro contacts. The processes in micro contacts determine the friction forces between bodies and the wear rate. Their study is of great interest for understanding friction. In discussing the processes in micro contacts we encounter, however, the following problem. Fourier spectra of surface profiles of many technical surfaces are power functions in some interval of wave vectors [1]. In this interval the surfaces have the property of self similarity to scaling of wave vectors and amplitudes and can be referred to as fractal. This—at least partial—fractality leads to a difficulty, which was not known in contact mechanics several years ago: If we would measure the surface roughness and then solve the contact problem with the given roughness, than both the size of the contact region and pressure distribution in the contact * Corresponding author. Fax: C49 30 314 72575. E-mail address: [email protected] (V.L. Popov).
0301-679X/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2005.04.027
will depend on the accuracy to which the surface microrelief was determined. Thus, if we assume a model of absolutely smooth and elastic bodies (e.g. wheel and rail), we will have the solution first found by Hertz. In view of fractal structure of surfaces, however, the real contact occurs in regions far smaller than micrometer size. The transition to even lower (nanometer) scale level would cause further decrease in ‘real contact area’ and increase in ‘real pressure’ at nano-contacts. The exact mathematical result is that the real contact area of fractal surfaces in approximation of linear elasticity is exactly zero and the pressure in real contacts infinitely large [2]. This means physically, that if we have fractal surfaces, then there is always some space scale at which the solids should behave plastically. In our opinion, this is just this scale which is responsible for friction forces and wear in most tribological systems. Our present paper is devoted to the question what happens in the micro contacts on the scale where the pressures are already so high, that intensive processes of plastic deformation should occur. At this scale we await that processes of intensive plastic deformation, detaching of wear particles and their repeated welding into the surfaces due to the conditions of normal pressure combined with shear deformation occur.
V.L. Popov, S.G. Psakhie / Tribology International 39 (2006) 426–430
2. Numerical model
Table 1 Parameters of the modeled material
To model the processes in the surface layers, we use the method of Movable Cellular Automata (MCA) [4,5]. According to this method, a medium is represented as an ensemble of discrete elements—movable cellular automata characterized by continuous variables such as center of mass position, value of plastic deformation, rotation pseudovector as well as by the discrete variables characterizing connectivity of the neighboring automata. The principles of writing the equation of motion for a system of cellular automata and prescribing interactions between them are described in [4,5]. In the following simulations, a twodimensional version of MCA method has been used. In our case, a modeled object consisted of four parts (Fig. 1):
Young modulus Poisson ratio Density Elastic limit Yield stress Strain at the yield stress Ultimate strength Breaking deformation Viscosoity
– The upper layer of automata was an absolutely rigid, nondeformable body moving horizontally at velocities ranging between 1 and 10 m/s in different numerical experiments; – Two intermediate layers with initial roughness of nanometer range represented surface regions of the bodies in contact; – The lower layer was a fixed support. A constant normal force corresponding to the pressure range up to the 3/4 of the yield stress of the material acted upon all the elements of the upper layer. The diameter of the automata was from 2.5 to 10 nm in different numerical experiments. The elastic properties of the automata corresponded to the steel with Young’s modulus of 206 GPa and Poisson’s ratio of 0.3. The yield strength and ultimate strength with respect to tension were varied between 80 and 480 MPa and between 92 and 552 MPa, respectively. An important parameter determining stability of the plastic deformation processes and significantly affecting
427
EZ206 Gpa nZ0.3 rZ7800 kg/m3 sy1Z51–306 MPa sy2Z80–480 MPa 3y2Z0.015 s0Z92–552 MPa 3cZ0.04 hZ0.41 Pa s
the characteristic size of the surface region of severe plastic deformation occurs to be viscosity. Introduction of viscosity is necessary already from a formal consideration of providing stability to the calculation procedure. Phenomenological viscosity reflects dissipation processes under strain occurring due to electron and phonon excitation in a solid. It was assumed in the numerical model that viscous forces acting between unconnected but contacting automata are proportional to the relative velocity of motion. At the left and right fragment boundaries, periodic boundary conditions were used. The boundary conditions for the rigid upper automata layer were put as follows: constant velocity of the layer in the tangential direction and constant force in the normal direction. The initial roughness was specified explicitly. The material parameters used are listed in Table 1 (for determination of strength parameters see a graphic inset in Table 1).
3. Formation of a boundary ‘quasi-fluid’ layer
Fig. 1. Initial structure, dimension and loading conditions of the modeled fragment.
The numerical experiments show that within the first nanoseconds after onset of a relative tangential motion of bodies, the roughness of both surfaces is already severely deformed and fractured, and a dynamic equilibrium in the system is established at a temporal scale of about 100 ns. A pronounced boundary layer appears, where the processes of deformation, fracture, reconstruction of connectivity between elements, and intensive mixing take place. The motion in the layer resembles turbulent motion in fluids (Fig. 2). For this reason we refer to it as a quasi-fluid layer. Note, however, that this layer is not liquid in terms of thermodynamics. A quasi-fluid layer remains localized in
428
V.L. Popov, S.G. Psakhie / Tribology International 39 (2006) 426–430
Fig. 3. Multi-layer model for simulation of the quasi-fluid layer.
Fig. 2. ‘Instant picture’ of a quasi-fluid layer structure for a material of ultimate strength 552 MPa: pressure PZ382.5 MPa. The sliding velocity is 5 m/s.
the vicinity of the initial friction surface and does not propagate to deeper regions of the contacting bodies. A characteristic depth of the layer depends on the system parameters, first and foremost, on the effective viscosity of the system of automata. In our calculations, viscosity was used as a fitting parameter and was chosen so that the layer depth corresponded to the experimental values [5,6]. Initial roughness leaves the regularities of quasi-liquid layer formation unaffected. The dependence of the thickness of the quasi-fluid layer on the material parameters can be easily estimated with the following simple model. Let us assume, that the quasi-fluid layer has a thickness h and that it can be characterized by the dynamic viscosity h. The viscous stress in the layer is then sxhv/h, where v is the sliding velocity. In the stationary state it should be equal to the limiting shear stress s0, at which plastic yielding is setting on: s0xhv/h. Hence, h zh
v s0
(1)
4. A multi-layer model To better understand the physical nature of the dynamic surface layers, we consider a simplified model of a solid consisting of many thin sheets, modeling atomic layers of a crystalline solid. We assume, however, that the atomic layers do move relatively to each other as rigid sheets.
The only property of the atomic structure which we use in our model is the periodic interaction potential between sheets and a linear damping due to interactions with the phonon and electron subsystems of the solid. The equations of motion of all layers with exception of the first and the last layer are the following: m€x n C 2ax_n K ax_nC1 K ax_nK1 C f0 sin kðxn K xnK1 Þ K f0 sin kðxnC1 K xn Þ Z 0:
ð2Þ
The first is moved with a constant velocity V0 and for the last layer a different damping constant a1 is used to avoid reflection from the rigid non moving lower layer (Fig. 3). Numerical experiments with this model show, that in a under-damped systems a surface layer does appear, with a linear gradient of velocities (Fig. 4) and with dependence of the thickness on the velocity and material parameters as predicted by the above simple estimation (1). It is important to note, that the effective viscosity determining the thickness of the quasi-fluid layer is in this model exactly the same, as we would measure in the elastic state from the damping of elastic waves. A quasi-fluid layer is always developing if the ‘friction law’ between layers does allow a bi-stability in some range of stresses with one static and one dynamic state at the same stress. The physical nature of bi-stability is of no importance for the phenomenological picture of the process. For example, the above mentioned model is nothing but a modified Tomlinson model [7], which is just a rigid body moving in a periodic potential. In Tomlinson model, in the under damped case, the force necessary to start the motion is larger than the minimal force needed to maintain it. Thus, in a definite range of forces, the body can be either in the state of rest or in the state of motion. In the state of rest, only the conservative part of forces does play any role. In the state of motion, practically only the viscose part of force is important. In this sense we can refer to these states as to an ‘elastic’ and an ‘viscous’ one. If we ‘pile up’ the rigid
V.L. Popov, S.G. Psakhie / Tribology International 39 (2006) 426–430
Fig. 4. Schematic presentation of the simulation results with the multi-layer model. In the vicinity of the surface, there is a constant velocity gradient. The deeper layers are in the elastic state.
bodies with periodic interaction between them, the possibility of being either in an elastic or in viscous state is valid for each pair of them. This shows immediately the further possibility of one part of the system to be in a viscous state while the other in the elastic. Details of this behavior are presented in the paper by Geike & Popov in this issue. In the Tomlinson model, the bi-stability is due to the inertia effects. This is not the only possible mechanism of bistability. For example, if we consider an over damped many layer system of the above type, than it is not bi-stable. However, it becomes bi-stable if the deformation induced softening of material is taken into account. See for details the paper [8] and paper by Geike and Popov in this issue.
5. Wear The existence of the boundary layer motivates us to change the existing approach to calculating wear in frictional contacts. Wear is often understood as microfracture and detaching of surface particles. In the presence of a boundary layer, however, the wear is not directly connected with fracture and detaching of particles. Indeed, even if a particle with characteristic sizes of 10–100 nm has been broken from the surface, it remains for a long time (relative to the average time between two consequent impacts of the particle with surfaces) inside the friction zone. The separated particle suffers many ‘impacts’ from the surfaces and is reintegrated into the surfaces many times.
429
The wear should thus be understood not as ‘fracture’ but as ‘mass transport out of friction zone’. A real ‘act of wear’ occurs only when a wear particle comes close to the boundary of the friction zone where it is not in conditions of confined deformation anymore and, therefore, no reintegration of wear particles occurs. Let us estimate the wear rate due to stochastic transport of nano particles out of friction zone. There is only one characteristic length scale which characterizes the quasi-fluid layer—the thickness of this layer. Both detachment of particles in this layer and their reintegration occur at the distance Dxxh. The characteristic time of such elementary stochastic step has the order of magnitude Dtxh/v. The stochastic wandering of wear particles can be described as an effective diffusion with a diffusion coefficient D xDx2 =2t xhv=2. The average square of distance of a wear particle from its initial position is proportional to the time: L2Z2DtZhvt. Thus, the average time needed to reach the border of the friction zone is txL2/hv. In this time, most of the material of a quasi-fluid layer would be transported out of the friction zone. It thus would become thinner by h. The wear velocity can thus be estimated as vt xh=t xvðh=LÞ2 . This estimation is, however, not yet completely correct, as the above mentioned processes only occur in the regions of ‘real contact’. The portion of such contacts is, after Bowden and Tabor, approximately equal P/sy, where P is the apparent pressure and sy the yield stress. The final estimation of the wear velocity is thus vwear xv
P sy
2 h L
Experimental investigations of the surface layers of combustion engines carried out at the Institute of Applied Wear Research in Karlsruhe confirm the existence of dynamic nanolayers, in which intensive mass mixing effects do occur [9,6]. Experimental results show that a thin nanolayer (usually from 10 nm up to 70 nm) is formed on the surface of grey iron cylinder during the wear-in period (the first hours of work) in combustion engines. Layer thickness depends on conditions of preliminary running-in. Usually it decreases as running-in power increases. Once formed thin layers stay thin in future and demonstrate better wear resistance than thick layers. In our opinion, the mechanism of surface layer formation is the inhomogeneous deformation and mass mixing (mechanical alloying) in the quasi-fluid layer. Wear rate in good serial engines is in the range of 0.2 nm/h. Size of wear particles is distributed in the range from a few nanometers to micrometers. Experiments with nucleopore filters show that most wear particles have the size between 10 and 100 nm with nearly 50% having less than 40 nm in diameter [3,10]. All this is in a good qualitative agreement with the concept of quasi-fluid layer. The dependence of wear rate on the thickness of the layer with the chemically changed composition is also confirmed by experimental observations.
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The process of stochastic transport of wear particles in a closed friction zone is in our opinion also the main mechanism of development of surface topography. The particles are detached and welded again into the surfaces in a random manner. This processes cause obviously not only the mass transport along the surfaces but also the material transfer from one contacting body to the other. This gives rise to a stochastic wandering of the position of the boundary between two bodies both in lateral and in normal directions. If we consider these processes of mass transport in a mathematical model, it occurs that they give rise to a stochastic surface with a region of fractal power spectrum and two cutting wave vectors—exactly as in the real technical surfaces. This mechanism will be discussed in detail in the paper by Schargott & Popov in this issue.
velocity v and density r can enter the relation for the friction coefficient only as rv2. Hence it follows that in the general case the friction coefficient may be thought of as a function of three independent combinations of the parameters (E, s0, r, v, P). Simulations of the friction coefficient with the method on Movable Cellular Automata show, however, that the friction coefficient does depend on the viscosity. This result is in contradiction with the assumption that the friction coefficient does depend only on the macroscopic parameters of the material. This means, that the microstructure of the material determines essentially the tribological properties of materials. This conclusion is confirmed by the consideration of many layer models, where the tribological properties also do depend on the discretization of the model and not only on the macroscopic material parameters (see paper by Geike & Popov for details).
7. Dependence of friction coefficient on loading and material parameters
8. Conclusion
6. Development of surface topography
Both wear and friction force are result of processes in micro contacts and specifically in the quasi-fluid layer. If we can model them, it should be possible to determine the dependence of the wear rate and the friction force on material constants and macroscopic loading conditions. This is indeed the main purpose of the friction physics: to be able to predict the tribological properties of materials and to design materials with desired properties. On what parameters do depend tribological properties of materials? We try to give the answer to this question at the example of the friction coefficient. Let us make a hypothesis, that the friction coefficient is a function only of macroscopic parameters. That means, that, for example the exact atomic lattice parameter should not have an effect on the tribological properties. Or, in other words, materials with different microscopic structure but the same macroscopic parameters have the same tribological properties. This assumption is made often implicitly without mentioning it. Under this assumption, certain conclusions on the friction coefficient as a function of material and loading parameters can be made from the analysis of dimensionality. Indeed, the friction coefficient is a dimensionless quantity and, hence, can depend only on dimensionless combinations of system parameters. It can be shown that no dimensionless combination involving viscosity can be made up from material and loading parameters (E, s0, r, v, P, h). This implies that the friction coefficient cannot depend on viscosity. The analysis of dimensionality shows that
Our numerical simulations show that dynamic processes of plastic deformation, micro fracture and micro welding at the nano level determine the friction force, wear and formation of surface topography in friction processes. Their further study is of utmost importance for tribology. The discrete structure of the matter at this scale essentially influences its tribological properties, which implies that the latter cannot in principle be determined in the frame of continuum mechanical models.
References [1] Persson BNJ, Bucher F, Chiaia B. Phys Rev 2002;B65:184106. [2] Persson BNJ. Elastoplastic contact between randomly rough surfaces. Phys Rev Lett 2001;87(11):116101. [3] Persson BNJ. Sliding friction. Physical principles and applications. 2nd ed. New York: Springer; 2000. [4] Popov VL, Psakhie SG. Physical mesomechanics 2001;4(1):15–25. [5] Popov VL, Psakhie SG, Gerve A, et al. Physical mesomechanics 2001;4(4):73–83. [6] Popov VL, Yu SI, Gerve´ A, Kehrwald B. Simulation of wear in combustion engines. Comput Mater Sci 2000;19(1–4):285–91. [7] Tomlinson GA. Philos Mag 1929;7. [8] Popov VL. A theory of the transition from static to kinetic friction in boundary lubrication layers. Solid State Commun 2000;115:369–73. [9] Scherge M, Shakhvorostov D, Pohlmann K. Fundamental wear mechanism of metals. Wear 2003;255:395–400. [10] B. Kehrwald, Untersuchung der Vorga¨nge in tribologischen Systemen wa¨hrend des Einlaufs, Dissertation, Universita¨t Karlsruhe, 1998.