Computer modeling of local tribological contacts by the exam

doi:10.1016/j.physme.2008.03.008

Computer modeling of local tribological contacts by the example

of the automotive brake friction pair

A.I. Dmitriev, A.Y u. Smolin, S.G. Psakhie, W. ?sterle 1, H. Kloss 1, and V .L. Popov 2

Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634021, Russia 1

Federal Institute for Materials Research and Testing (BAM), Berlin, 12203, Germany 2

Institute of Mechanics, Berlin Technical University, Berlin, 10623, Germany

In the paper the method of discrete modeling (movable cellular automata method) and combined discrete-continuous description of the simulated medium are used to analyze processes occurring in the local contact of the automotive brake system. The characteristic size of the considered region is 1.5 P m. The following contact situation is simulated: steel fiber coated by an iron oxide film as the brake pad and pearlitic steel also coated by an iron oxide layer as the disc. On the assumption of oxide layer wearing we simulate the iron oxide iron oxide, iron oxide metal and metal metal contacts.

The calculation results for the friction coefficient for various contact situations give quite adequate values. For example, for the oxide oxide system the calculated coefficient is approximately equal to 0.4, while for the metal metal contact the obtained value varies from 0.7 to 0.9. Analysis of a set of the obtained results allows concluding that oxide is formed more rapidly than the sliding layer, which in turn makes the friction coefficient value stabilized.

Copyright ? 2008 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV . All rights reserved.1. Introduction

The disc pad friction pair in the modern automotive brake is a complex multicomponent system. It is obvious that the quality of its performance can be governed by vary-ing the composition of the pad composite. Over 90 % of modern automotive brake pads are made of polymer ma-trix composites with a great variety of components bonded by phenolic resin [1]. Along with the requirement of good braking performance, however, account should be taken of the environmental aspects that become more and more im-portant. The manufacturers of brake systems have to re-place environmentally hazardous constituents, such as as-bestos, heavy metals and inorganic fibers. Besides, the di-versity of the matrix and filler materials used for pads and discs is due to the difficulty of simultaneously meeting the requirements imposed on friction pair materials [2]:

1) the average friction coefficient μ should be ~

0.45for brake systems of passenger cars, higher than 0.5 for sports cars and 0.35 for railway transport;

2) the friction coefficient variation 'μ during the bra-king cycle should be minimum and tend to zero in the limi-ting case;

3) guaranteed and safe braking in dry and wet condi-tions;

4) low rate of the disc and pad wear as well as mini-mum number of wear particles;

5) the raw materials used for pad production should be cheap, but environmentally safe;

6) brake squeal should be avoided, etc.

Modern technology allows producing pads that meet high performance requirements, but nevertheless there is still lack of knowledge about the influence of certain para-meters, e.g., chemical composition and microstructure of the system, on its properties. The micromechanisms of fric-tion and wear that in many respects define the macroscopic behavior of the friction pair are little understood as well.Tribological systems are studied with both experimen-tal and theoretical methods. The experimental investiga-tion is often time and money consuming. To study the in-fluence of an elementary friction or wear mechanism on strength and wear resistance of the entire material is very difficult because of small spatial and temporal parameters as well as complex formulation of used composites. The actual contact area is still hard to reach in experiments,especially immediately during the tests. In this connection,computer simulation as a kind of theoretical investigation can be effectively applied to study various aspects of fric-tion and wear [3 7]. The simulation results can help in pre-dicting material behavior at contact interaction and make for further improvement of tribotechnical properties of the studied materials.

When choosing computer simulation methods, one should take into account that friction and wear occur toge-ther with intense damage formation and accumulation in the contact region, material mixing, spalling and other pro-cesses related to continuity violation [8, 9]. In papers [10 14] the behavior features of the surface layer in tribologi-cal contact are studied with the movable cellular automata

(ì?à) method. It is proved to be good for the solution of such problems (simulation of material behavior features in the contact area between the railway wheel and rail surface [11], interaction of the surface layers of the cylinder walls and piston in combustion engines [12], surface profile modification induced by friction and wear [13]). Owing to the introduced notion of the state of a pair of automata and the specified criteria of linked-to-unlinked state switching and vice versa, the MCA-method allows one to directly simulate processes related to continuity violation in the ma-terial: from microdamage formation to macrocrack genera-tion and mixing of material fragments. Through varying the state switching criterion of a pair of automata one can also simulate both the failure of the existing and formation of a new surface profile in the course of different processes occurring at the microlevel, which accompany friction and wear.

Note that the power of modern computing systems still hinders a detailed description of extended objects due to a limited number of elements of the medium at its discrete representation. In the present paper we hence apply, along with the MCA-method, a so-called combined approach that uses both discrete and continuous representation of the simulated medium [15, 16]. The continuous representation is used to describe the material areas that undergo mainly elastic deformation and the probability of damage forma-tion in them is very low. The MCA-method describes the behavior of most strongly affected regions with the pro-bability of failure and material mixing in them. The ap-plication of such a combined approach is most justified for the solution of various tribological problems when the be-havior of extended objects should be described, with the contact area where damage is formed and evolves being just a narrow layer close to the surface of the interacting bodies.

The aim of the present paper is to simulate some typi-cal situations occurring at local contacts in the disc pad friction pair of the automotive brake, using the MCA-me-thod and combined discrete-continuous approach.

2. Experimental findings put at the basis of simulation

In papers [17, 18] the focused-ion-beam (FIB) tech-nique is applied to analyze the surface layers of automo-tive brake pads and discs, which have been formed at bra-king. The cross-sections of different surface regions and at different composition of the pad structure are analyzed. It is found that even at the micrometer scales one can often see the formation of only small areas of the friction layer with steel fibers as a base, which comprise one of the pad components. Figure 1(a) gives an electron-microscopic ima-ge of a pad surface region with a steel fiber clearly ob-served on its surface. The energy dispersive X-ray (EDX) analysis (Fig. 1(b)) and subsequent study with transmis-sion electron microscopy (TEM) (Fig. 1(c)) show that the

friction layer mainly contains nanocrystalline iron oxide

c

Deformed layer

,

)(3)()(2)()(2)(2)()(int ij j i y j i x j i y j i x j i W V V V V V c c c c (1)

where V and W are respectively the normal and tangential stress calculated in a local coordinate system (Fig. 3) with regard to the influence of particles surrounding the consi-dered pair.

If the distance between the unlinked automata becomes less than the sum of their radii, such automata are assumed to be in contact.

The notion of the relative overlap ij H (deformation in the normal direction) and normal stress ij V (specific force) are introduced in the MCA-method for each automa-ton as follows:

°°ˉ°°?- V

H .,22

ij

n ij i i ij ij S F d d q (2)Here, ij q is the distance between the center of mass of automaton i and point of its contact with automaton j , i d is the size (diameter) of automaton i , ij n F is the normal force,and ij S is the contact area (Fig. 3(a )).

[18]. Based on the TEM results, we may distinguish a thin layer at the material surface whose structure differs from that of the bulk. As evident from Fig. 1(c ), the surface fric-tion layer is characterized by more smeared and darker struc-tural elements, whereas the metal surface by coarser and lighter inclusions.

Since ordinary brake discs are made of cast iron, their internal microstructure presents carbon flakes in the pear-litic matrix. According to the Raman spectroscopy results (Fig. 2), the surface of both the disc and pad contains car-bon. The worn-out disc surfaces were investigated in order to identify transparent phases of the friction layer. A typi-cal spectrum of the disc surface which is observed in grey light in an optical microscope is illustrated in Fig. 2(b ).The smeared maximum with wave number 670 cm 1 can be attributed to the mostly pronounced magnetite phase band [19]. The high peaks at 1 340 and 1 598 cm 1 correspond to carbon. The intensity ratio and peak broadening point to a high misorientation degree as compared to pyrolytic gra-phite. It is found from the ratio of the two peaks that the grain size of nanocrystalline carbon is about 5 nm.3. Brief description of the used simulation methods 3.1. Method of movable cellular automata

As has been said in the introduction, the behavior fea-tures of materials in the contact area of the brake pad and disc are studied with the MCA-method. Its main aspects are described elsewhere [10, 20]. In the method the simula-ted system is represented as an ensemble of interacting par-ticles (automata) of finite size.

One of the key differences of the MCA-method from the classical cellular automata is the introduction of an addi-tional type of state, namely, the state of a pair of automata.In the simplest case, these are two states linked and un-linked; the transition from the first one to second allows simulating material failure. The failure criterion, i.e. the rule of linked-to-unlinked state switching for the pair of automata i and j , is determined in the given paper on the basis of stress intensity calculation in the pair by the expression

a

b

b

a

°°ˉ

°°?- W ' J W .,ij ij ij ij S ij S F r t V (3)In expression (3) we use the relative velocity of tangen-tial displacement determined as

,

eff ji

j ij i ij ij ij s q q r V Z Z Z where ij ij x ij r V Z eff is the effective rate of the pair rota-tion around the center of mass of automaton i , and ij F W is the tangential force.

3.2. Combination of the MCA-method and continuous

description of the simulated medium

The technique of combining the discrete and continu-ous methods as well as each of them separately are de-scribed in detail elsewhere [15, 16]. The combination im-plies that we define a conjugate boundary between the dis-crete and continuous regions and assume it to belong to both the regions. In this case, to every calculation grid node lying on the boundary corresponds an automaton (MCA-method element). The motion of the boundary nodes to-gether with conjugate automata located in them is carried out in the grid method. In a general case, several cellular automata can be located between two neighboring grid nodes lying on the boundary. The coordinates of the auto-mata located between the two neighboring boundary nodes are calculated through the interpolation of displacements of the corresponding grid nodes.

The motion continuity at the interface is provided at the stage of calculating the velocity of the boundary nodes in the continuous region. Independently of the interpreta-tion and used discretization approach finite-difference or finite-element when constructing a difference analog of equations their shape and meaning of terms remain the same. The left side contains the product of mass by the component of grid node acceleration, and the right side contains the component of the resultant force applied to the node:

,,1| N

i i x F x m

,,1| N

i i y F y m (4),i i xy i i xx i x x y F 'V 'V .

i i xy i i yy i y y x F 'V 'V For all internal nodes of the continuous region the ex-pression for the total force acting on the center of the scheme illustrated in Fig. 4 will be written as:

),

()(1342312413423124IV

IV III III II II I I IV IV III III II II I I IV

I,x x x x y y y y F xy xy xy xy xx xx xx xx i i x V V V V

V V V V | ).

()(312413423124IV

13IV III 42III II II I I IV IV III III II II I I IV

I,y y y y x x x x F xy xy xy xy yy yy yy yy i i y V V V V

V V V V | For all boundary nodes the terms for the missing cells

(dashed line in Fig. 4) are replaced by forces acting on corre-sponding automata from the side of the MCA-region. The motion of such nodes is calculated on the basis of equation (4) that, depending on the interface geometry, would in-clude forces found in the MCA-region. After the displace-ment of the automaton inseparably linked with a grid node the corresponding force is transferred from the grid region to the internal MCA-region.

4. Results of friction simulation with the movable cellular automata method 4.1. Numerical model of friction

To construct a numerical model of friction for local con-tacts in the brake disc pad system, we should take into account the characteristic scale at which processes respon-sible for the nature of friction in the surface layers of the contacting bodies occur.

Based on the analysis of microscopic and microanalyti-cal observation results, we consider the following contact situations typical for the local contacts of the disc pad system: steel fiber coated by a nanocrystalline friction layer as the pad, and pearlitic steel, which is the grey cast iron matrix, also coated by the friction layer as the disc. The modeled local contact on the brake disc surface is schema-tically illustrated in Fig. 5(a ), and the region simulated with the movable cellular automata method is represented in Fig. 5(b ).

Different layers of the simulated region initially consis-ted of linked automata. The layers directly involved in fric-tion interaction consisted mainly of automata with para-meters corresponding to the physico-mechanical proper-ties of iron oxide .4O Fe 3 To simulate the experimentally observed mixing of iron oxide with solid lubricant compo-nents of the pad, a small amount of inclusions were added to these friction layers, among which of particular interest are nanocrystalline graphite inclusions being the major solid lubricant component in the disc pad system. The experi-

mentally observed size of the inclusions is 10 nm [17, 18].The characteristics size of the fragmented grain structure in steel beneath the surface oxide layer is of the same or-der. Hence all automata of the simulated region have the same size 10 nm.

Owing to the lack of accurate experimental data on the mechanical properties of the sliding layer materials, we used various combinations of property ratios for the automata of the matrix and inclusions. The concentration of inclu-sions varied, and damage in the iron oxide matrix was given explicitly by breaking links in individual pairs of automata and removing some automata. The mechanical properties used to set the interaction parameters for the model materi-als (Fig. 6) are listed in Table 1.

According to the failure criterion used to switch the state of a pair of automata, two initially linked automata became unlinked in case if stress intensity (1) in the pair attained the strength of a softer material. For a reverse transition from the unlinked to linked state the normal stress value in the contacting pair of automata was compared to the stress varying in the interval 1y V s V (Fig. 6) for different com-binations of materials in the pair of automata. The properties of adhesion between the graphite inclusion automata and iron oxide matrix automata as well as between iron oxide automata and automata of the steel layers were imitated using an additional parameter that allows increasing or de-creasing the link strength.

The loading scheme used in the model (Fig. 5(b )) imi-tated real conditions on the brake disc pad contact. As schematically shown, initially the automata corresponding to the pad and disc were separate. Then, constant velocity V was applied in the horizontal direction to all automata of the lower layer of the block imitating the disc. The pulsed action was smoothed using the procedure of linear velocity growth from zero to the maximum value 10 m/s. After the lower block automata acquired the maximum velocity, the automata in the upper layer of the upper block imitating the pad were subjected to a constant force acting vertically down. In this case, we also used the procedure of linear force growth from zero to maximum. The maximum force value varied for different problems in a range correspon-ding to pressures P from 20 to 200 MPa. The vertical ex-tension of the simulated blocks was imitated with a special procedure of energy dissipation by the viscoelastic law in the upper block for the upper layer automata and in the lower block for the lower layer automata. In the horizontal direction periodic boundary conditions were used, which allowed repetition of the simulated region along the

??

y V MPa

45052029015,2y V MPa

50080030535,s V MPa 550920340452y H 0.040.040.0080.05s

H 0.106

0.106

0.009

0.15

Ferritic steel Perlitic steel Iron oxide

Fig. 5. Schematic image of the brake disc local contact (a ); loading scheme and initial structure of the simulated region (b ). The composition of the surface layer of the disc and pad: Fe 3O 4 iron oxide nanoparticles mixed with carbon particles acting as a solid lubricant

D i s c

P a d

F r i c t i o n l a y e r

axis, namely: the automaton leaving the simulated region through the right boundary was instantly transported to the left for the length of the region and entered the system through the left boundary. Thus, the total number of auto-mata was unchanged in the calculations. We should point out a correlation between the applied periodic boundary conditions and real situation on the brake disc surface as the latter is also periodically involved in friction due to its rotation, though with a different period and linear size of the region. The initial roughness of the simulated surfaces was specified in an explicit form. The calculation results have shown, however, that its peculiarities exerted no sig-nificant effect on the final simulation findings.

The automaton size and elastic properties of the model materials determine the integration step of the numerical scheme, which is 2.5

10 13 s. The value is by two orders of magnitude higher than the time step usually used in the molecular dynamics method.

4.2. Simulation results

Analysis of the surface structure of interacting materi-als shows that the deformed layer formed at the relative motion of the bodies has the structure and properties strong-ly different from those in the material bulk. This layer un-dergoes severe deformation and failure, roughness smoo-thing, rupture of the existing and generation of new inter-automata links as well as mixing of materials of the contac-ting bodies. The formation of a boundary sliding layer can be considered as the formation of a third body caused by friction processes. A very important fact is that the sliding layer remains spatially localized close to the surfaces of the interacting bodies and does not propagate deeper into the material.

The friction coefficient value was estimated during simu-lation through calculating components of the normal and tangential force acting on automata of the disc boundary layer. The instantaneous friction coefficient was found as a ratio of the resultant forces acting on all boundary layer automata at the current time step. Despite the fact that the friction coefficient was calculated already at the stage of steady-state behavior, its instantaneous values correspond-ing to different moments of time were widely scattered due to the generation of various instantaneous configurations on the surfaces of the contacting bodies. The calculated friction coefficients were compared to macroscopic data through averaging over two thousand adjacent instantaneous values. Despite such a wide scatter of the instantaneous values, the found average friction coefficients were quite adequate to the simulated contact situations.

Let us consider in more detail two most probable situa-tions on a local contact, which play a crucial role at brak-ing.

4.2.1. Simulation of the local iron oxide iron oxide contact

The most probable situation is the contact of steel sub-strates with nanocrystalline oxide films and carbon inclu-sions, on the part of both the disc and pad. Figure 7 depicts structural evolution at relative motion of two such surfaces.According to the model parameters (Table 1), iron oxide particles are harder than carbon particles pointed by ar-rows in Fig. 7(a ).

In the figures one can clearly see the formed layer in which the materials of the both surfaces mix. Regarding the system behavior, the so-called stages of run-in and steady-state friction can be distinguished. The comparison

Fig. 7. Simulated structure for the iron oxide iron oxide contact for different moments of time: 0.0125 (a ); 0.15 (b ); 0.25 (c ); 0.375 P s (d )

Graphite inclusions

d

Graphite

of Fig. 7(a , b ) and Fig. 7(c , d ) shows that after 0.2 P s from the beginning of relative motion the sliding layer structure mostly turns to homogeneous.

The damage pattern in the sliding layer was analyzed through constructing configurations of interautomata links at different time intervals. Figure 8(a ) illustrates a fragment of the structure of interautomata bond distribution in the sliding layer at steady-state motion. The dots denote the centers-of-mass of automata, and the lines between the cen-ters-of-mass indicate the state type of a pair. The presence of the line means the linked state of a pair, and the absence of the line unlinked state. The friction of two brittle materials is clearly seen to result in the formation of a third body consisting mainly of free nanosized particles, which pertain to the materials of the both surfaces, and a small number of more complex conglomerates of different shape and size. At later simulation stages the oxide layer under-goes fracture manifested as a network of cracks generated at the interface between the oxide matrix and graphite in-clusions. The surface examination with a scanning elec-tron microscope confirms that the weak interfacial region

between graphite particles and oxide is often the place of multiple damage formation.

Based on the experimental data and behavior features of layers in the performed calculations, the criterion for the transition of a pair from the unlinked to linked state was chosen so that the generation of new linked pairs for oxide particles would occur rarely. This explains the mainly gra-nulated structure in the friction layer (Fig. 8(a )). Figure 8(b )shows the time dependences of the instantaneous friction coefficient and its average value. The average friction co-efficient value for the oxide oxide contact interaction re-mains almost constant with time and varies from 0.3 to 0.4,i.e. agrees quantitatively well with the experimental data for the automotive brake system [4, 21].

We have also analyzed the influence of graphite inclu-sions, namely, their content and characteristic size, on the oxide layer behavior during friction. Two systems similar to the shown in Fig. 5(b ), but with a different content and configuration of automata of graphite-like inclusions, were additionally generated. Figure 9 gives simulation results for the case of high content of graphite inclusions (volume frac-

a

b

a

tion ~17 %) consisting mainly of individual 10 nm particles and a small number of carbon conglomerates with a cha-racteristic size of no more than 30 nm. Figure 10 corres-ponds to the case of a relatively low content of graphite inclusions in the following configuration: a predominant number of graphite conglomerates ~ 50 nm in size and a small number of inclusions in the form of individual auto-mata with graphite properties. In the both cases, the loa-ding conditions were the same and corresponded to the problem depicted in Fig. 5(b).

The comparison of the simulation results shows that the character of damage formation in the oxide layer for the considered cases differs noticeably. For a high concen-tration of graphite inclusions (Fig. 9), multiple damages occur at the contact boundaries of the oxide matrix and graphite inclusions. Some of them coalesce and form a kind of a crack that propagates through the oxide layer to the contact interface with the metal substrate. In the case of low content of graphite inclusions (Fig. 10), one can see strong chipping of the oxide layer near the graphite inclu-sions, which are close to the free surfaces, and almost no damage in internal oxide layers.

Despite the difference of damage evolution in the ox-ide layer, the calculated average friction coefficient value remains in the range 0.3 0.4 for the both cases, which agrees well with experiment (Figs. 9(b) and 10(b)).

Since, as noted above, there are no accurate experimen-tal data on the mechanical properties of iron oxide used in the model, the influence of strength characteristics on the failure of oxide layers was evaluated through analogous calculations on the assumption of a lower-strength oxide. The investigation has revealed that strength variation exerts almost no effect on the friction coefficient and failu-re peculiarities of the oxide layer, with the difference that for the lower-strength material the oxide layer failure is more intense and the formation of the friction layer con-sisting of wear particles is more rapid.4.2.2. Simulation of the metal metal contact

One more typical local contact situation is the interac-tion of the disc metal base with the pad metal fiber. This contact can particularly form as a result of oxide layer fai-lure on the indispensable condition that wear particles are removed from the interaction region of the contacting sur-faces. Such contact situation is quite often, which is substan-tiated by numerous results of focused-ion-beam scanning the disc and pad surfaces after braking. The investigation findings point to the presence of individual regions subject to more intense wear. Figure 11(a) demonstrates a typical structure of interautomata links in the central region of the simulated system at steady-state motion for the case of the metal metal contact.

When simulating the interaction of two metal bodies, we assume the possibility of both active rupture of existing bonds (linked states) and the formation of new links be-tween individual automata belonging to both the same and different bodies. This transition can be interpreted as nano-welding in contact patches. In the given calculations the criterion of switching a pair of unlinked automata to the linked state corresponds to the point 1y V on the response function of the model material (Fig. 6(b)). In the metal metal contact simulation a damaged region is formed near the free surfaces, little mutual transfer of automata belong-ing to the opposite bodies occurs, and there are almost no wear particles. As shown in Fig. 11(a), new surface pro-files are generated during friction, which consist of par-ticles belonging to the both interacting bodies. One can clearly see in the figure that the thickness of the friction layer formed in such contact is much larger than the cha-racteristic size of the initial surface roughness, but much smaller than the friction pair size.

It is significant that the friction coefficient value calcu-lated for the metal metal contact falls in the range be-tween 0.8 and 0.9, which also agrees well with the experi-

mental values [21]. The time dependences of the instanta-

neous and average friction coefficient are given in Fig. 11(b ). The average friction coefficient value slightly fluctuates, which is evidently due to the absence of wear particles in contrast to the oxide oxide contact.

Similar friction coefficient values were found not only for the contact of steels with different mechanical para-meters (ferritic steel with pearlitic), but also for the contact of steels with the same properties. The damaged regions near the free surfaces, which are observed in Fig. 11(a ),can be described as plastically deformed. The obtained data agree with the results of paper [6] which testifies that local pressures in contact patches can greatly exceed nominal ones and are sufficient for the plastic deformation of the surface layers.

5. Simulation of friction with a combined discrete and continuous description of the material

5.1. Description of the simulated system

Before we proceed to the description of the numerical model, notice that the combination of the discrete and conti-nuous methods for the description of a friction pair solves,along with purely technical problems arising at computing time optimization, another important problem. It is obvi-ous that the friction force and wear depend on processes occurring at different scales from atomic to macroscopic.As noted in [13], for an accurate simulation of contact inter-action the surface should be discretized with a step consi-derably smaller than the wavelength of the applied force.The representation of the whole body as a set of so small structural elements makes it nearly impossible to solve nu-merically dynamic friction problems and requires the in-troduction of specific boundary conditions, e.g., viscoelas-tic, like in the previous calculations for the upper and lower boundary layer. In the combined approach rather large cells can be used for the numerical method of continuum mecha-nics. This allows explicitly simulating the whole volume of

the contacting bodies and taking into account the depth of force action from the surface with a wavelength from the size of one automaton to the width of the simulated contact region.

In the discrete-continuous description of the simulated medium processes in the surface layers of the contacting bodies were studied with the use of the system depicted in Fig. 12. The metal substrate layers (upper for the pad and lower for the disc) subjected mainly to elastic strains were simulated with the grid method of continuum mechanics.

The contact region subject to the greatest structural changes

was simulated using the movable cellular automata method (MCA-method). For the motion continuity condition to be strictly fulfilled at the contact boundaries, the upper part of the upper block and lower part of the lower MCA block were given by automata with parameters of corresponding steel: ferritic steel for the disc and pearlitic steel for the pad.

The loading conditions were the same as for the system shown in Fig. 5(b ), i.e. for the lower nodes of the grid simu-lating the disc we set constant displacements with velocity 10 m/s, and for the upper nodes of the grid simulating the pad we specified stresses that imitated the action of con-stant external pressure, whose value varied in a certain range. To minimize the wave effects related to the dynamic problem statement, the displacement and stress values were gradually increased to maximum. Like in the case of simu-lating with solely the movable cellular automata method,the external pressure was given only after the lower block achieved the maximum constant tangential velocity. In the ?? axis direction on the boundaries of the both methods periodic boundary conditions were used. The size of a mo-vable cellular automaton was 10 nm and the grid step in the both directions was 50 nm, i.e. the contact boundary between two grid nodes contained 5 automata.

Like in the previous calculations with the aid of the movable cellular automata method, in the near-surface re-gion of the contacting bodies we initially generated a sur-face layer consisting mainly of automata with iron oxide properties and individual automata-inclusions with the me-chanical properties of graphite. The properties of the model materials are given in Table 1.

To reveal the evolution features of processes in the con-tact region of the friction pair within the combined approach,we also considered separately the contact of a steel region of the disc surface and a steel region of the pad as well as similar situation with regard to the presence of a graphite lubricant layer.

5.2. Results of simulation within the combined approach The simulation results for the local contact interaction of two oxide layers with graphite inclusions show that, like in analogous situations represented in Figs. 8 10, a fric-tion layer is formed in the contact region, in which par-ticles belonging to the both interacting surfaces mix. This is well seen in Fig. 13(a ) depicting the configuration of automata of the central fragment at the stage of steady-state motion. Figure 13(b ) gives the time dependence of the instantaneous and average friction coefficient. Despite the instantaneous friction coefficient values strongly fluc-tuate, the average value remains almost unchanged and falls in the range from 0.3 to 0.4, which also agrees well with the experimental data and previous calculations.

The simulation of contact between two metal surfaces within the combined approach has revealed some diffe-rences of the material behavior in the friction contact re-gion. As before, a damaged region is formed near the free surfaces, which can be considered as plastically deformed,and an account for the possibility of link recovery between unlinked pairs of automata gives a negligible number of wear particles. At the same time, within the combined ap-proach a more intense mixing of materials of the opposite surfaces is observed, which is more pronounced at a higher external pressure. Figure 14(a , c ) illustrates the structure of interautomata links of the central fragment at local pres-sures 115 and 230 MPa, respectively. The region involved in material mixing is larger for the case of high pressure.The average friction coefficient value therewith remains almost the same.

As for the average friction coefficient value, it reduces down to 0.6 in contrast to the case depicted in Fig. 11. By comparing the configurations of interautomata links we may suggest that a decrease in the force of tangential motion resistance and consequently in the friction coefficient is due to the formation of a layer in which mass mixing takes

place. A distinctive feature of evolution of the average fric-

tion coefficient for the case of two interacting metal sur-faces is still the fluctuating behavior during the calculation time intervals.

To simulate the peculiarities of contact between two metal surfaces with regard to the presence of a solid lubri-cant, a layer consisting of automata with graphite proper-ties was explicitly given on the surface of the lower block (disc). The layer thickness comprised 150 nm. The calcula-tion results have shown that during interaction of the two surfaces the graphite layers contacting with the metal body fail almost completely (Fig. 15(a )). The rupture of links between graphite particles and their neighbors causes the formation of a layer (solid lubricant) in which solely auto-mata with graphite properties move. This significantly re-duces the average friction coefficient down to 0.25 and fluc-tuation amplitude of the instantaneous friction coefficient

a

b

d

a

b

(Fig. 15(b)). Note that a two-fold increase of normal pres-sure had a negligible effect on the average friction coeffi-cient.

6. Conclusions

Based on comparing the results of simulation within two approaches with experimental data, the following sug-gestion can be made concerning the dynamics of some pro-cesses occurring on the surface of the contacting bodies. Since nanocrystalline iron oxide is the key component of the surface layers formed during friction, the tribooxidation processes play a dominant role in stabilizing the operation of the friction system on the whole. The fine microstruc-ture and homogeneous chemical composition of friction layers on the disc and pad indicate that iron oxide contains solid lubricant inclusions only as individual nanoparticles. Surface examination using transmission optical and scan-ning electron microscopy shows that friction layers on pads are not discontinuous, but appear as islands separated by carbon or more worn-out regions. The disc surface often has dark strips covered by the friction layer. The focused-ion-beam scanning of surface sections reveals that even at the micron and lower scales only part of the surface of pri-mary metal contacts is covered by an oxide layer, while other regions make up the surface of deformed metal. Con-sequently, the iron oxide earlier formed on the surface un-dergoes rapid failure during friction.

This hypothesis is justified by simulation results for short times of braking. If we assume that part of nanosized wear particles is removed from the friction contact region, the friction layer will disappear for a short period of time, leav-ing the two metal surfaces in contact in the considered re-gion. As the simulation results demonstrate, the local con-tact of friction interaction between two oxide layers is cha-racterized by the friction coefficient close to the average friction coefficient of real brake systems. This suggests that oxidation processes rapidly restore the worn-out friction layer and thus prevent the formation of local contacts of metal surfaces with the friction coefficient close to 1 for pure metals.

The formation of a sliding layer stabilizes the friction coefficient and decreases the resistance to the relative mo-tion of the interacting bodies in the friction pair.

The results obtained indicate that it is possible and use-ful to apply both the movable cellular automata method and combined continuous and discrete description for the solution of various tribological problems. At the same time, the simulation data are rather qualitative. The construction of quantitative dependences requires additional investiga-tion with systematic approach to the choice of the simu-lated structure and substantiation by experimental data.

The work has been carried out at the financial support of INTAS (Grants YS No. 04-83-3544, OS 77/9-1), Ger-man Research Foundation (DFG), Fundamental Research Program of the RAS Department of Energy, Engineering, Mechanics and Control Processes No. 4.12 (Project No. 1), and RFBR (Grant No. 05-08-33530-a).

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