Kinetics of particle detachment: contribution of a granular model

'lransient Processes in Tribology G. Dalmaz et al. (Editors) 9 2004 Elsevier B.V. All fights reserved

63

K i n e t i c s of particle d e t a c h m e n t : c o n t r i b u t i o n of a g r a n u l a r m o d e l N. Fillot, I. Iordanoff and Y. Berthier Laboratoire de M6canique des Contacts et des Solides, INSA de Lyon, Bat Jean d'Alembert, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France This work is based on the third body concept, from which wear can be defined as the complex competition between third body flows inside a contact. Experimental observations have shown that particle detachments are linked to their definitive ejection from the contact. This paper presents a (DEM) numerical model and proposes a study of the life of a contact, by treating the creation of particles and their ejection separately in order to identify their influence on each other. This work focuses in particular on modelling particle detachments, when the ejection of the particle is imposed. Important information is obtained by analysing the stresses in the skin of a degradable granular material. 1.

INTRODUCTION

After attempts to explain friction and wear in terms of volume and surfaces, Godet proposed the concept of third body in the 1970's [1,2]. In brief, the third body can be defined as the interfacial media between two solids in contact. It turned out to be a fundamental tool for solid lubrication that does not have any general equations, such as the Reynolds equation for fluid lubrication. Continuing along these lines, Berthier proposed the "Tribological Circuit" [3] to explain wear as a complex competition between flows of solid particles inside a contact. Figure 1 shows a source flow of third body (Qs), since particles can detach from one of the solids in contact. Qi represents the quantity of third body that circulates in the contact. The ejection flow Qe splits in two ways: either the third body particles enter the contact again (Qr) or they are definitively lost, thereby constituting the wear flow (Qw).

First body 1 Qs .....

Qi I First b ~

!

Qr Figure 1. The Tribological Circuit

Experimental observations [4] have shown that the activation of the source flow corresponds to a wear flow of the third body. In this case, production and ejection of third body particles are linked, but some questions remain: What controls what? Does the source flow produce "too many" particles for the contact, resulting in the ejection of the particles? Or does the internal flow of the third body decrease due to the wear flow, thereby activating the source flow? In order to complete the experimental approach and understand what happens inside a contact, a numerical model has been built to simulate the third body flows [5,6]. Although these numerical simulations cannot give quantitative information, they have the great advantage of providing a dynamic view of the third body, with the possibility of measuring flow, speed and stress. With a numerical model, third body flows can be dissociated, as can the mechanical and physicochemical actions. In this paper, particular attention is given to modelling the source flow of the third body, both without a wear flow and with an imposed wear flow, with the removal of particles at the interface. This approach, completed by the analysis of the stresses near the surface of the material, provides qualitative descriptions and explanations of the contact behaviour.

64 2.

P R E S E N T A T I O N OF THE M O D E L

2.1. The choices Two models must be chosen: one for the first bodies, which are constituted by the materials in contact, and the other for the third body particles at the interface. Three kinds of models for solid third bodies have been proposed in the past: the quasi-hydrodynamic approach [7-9], kinetic models [10-12] and the Discrete Element Method (DEM) [13-15], based on Molecular Dynamics. The first two models have proven to be very efficient in the specific case of solid lubrication, where the solid third body is sufficiently abundant to be considered as a continuous medium. However in the general case, the third body is discontinuous and DEM must be used. DEM, developed by Cundall and Strack for geotechnical applications [16,17], has been widely used to understand the behaviour of third body particles in a dry contact. The particles are governed by dynamic equations. Inter-particle forces are set as force-displacement laws and trajectories are calculated by using Newton's second law. The first bodies rubbing together are often considered as continuous materials and the Finite Element Method is therefore used accurately [18]. However, in order to model wear mechanisms, many authors [19-21] simply remove parts of the surfaces in contact and do not take into account the effects of the detached particles on further degradations. The aim here is therefore to build (numerically) a degradable material capable of releasing particles. DEM, which has already been used to model third body particles, is extended to the materials of first bodies. A simplified degradable material is then defined as an assembly of particles stuck together. This approach allows the same model to take into account both the first and third bodies. The simulation time required to observe the phenomena can be very long. When compared with a granular study of flow under stabilized conditions, the simulation times are 10 to 100 times longer. A 2D model was preferred to a 3D model in order to carry out a parametric study on the transient phenomenon of degradation. 2.2. The model for the third body To model the internal flow of the third body, Sbve and al. [22,23] proposed a semi-2D model: the particles constituting the third body are elastic spheres and the first bodies are two rigid assemblies

of spheres located along a defined profile. As a consequence, contact between the third body particles and contact between the third body particles and the surfaces can be computed in the same way (see figure 2). In this paper, the radii of these spheres are chosen randomly in a range of 50% around a mean value R. Each plate has a single degree of freedom that allows translation. The upper plate is subjected to a constant normal pressure PN and is free to move vertically while the lower one moves horizontally with a constant velocity V o. Boundary conditions are periodic in order to imitate an infinite flow field.

Figure 2. Granular solid third body sheared between two first bodies. The progression of the sample requires cyclic calculations. At a given time, the position of each particle is known. A search for particles in contact is carried out. If contact exists, the overlapping of the particles is determined and the resulting force is calculated by using interaction laws. Three types of normal interactions are represented during the contact between two spheres. Repulsion is represented by a linear spring, whose stiffness is defined by K while the force F~ corresponding to an overlap 6is: Fr = K * 8 9

Adhesion is a constant 7q8 :

F,,=~

65 The dissipation of energy in the contact is due to viscous damping and the damping force takes the classical form:

where c~rB is the damping coefficient (<1), 8 the impact speed and M o the equivalent mass of the contact (when two grains of mass mi and mj collide): 1

1

1

Mo

mi

mj

The sum of the normal interaction forces is: F,, = fir + fi, + fid

F,, = - K.5 + ~'m - 2~

4 K.M o "~"

where F and g are algebraic values. This type of law is commonly used in DEM simulations. One of the main advantages is that it provides a direct link between the viscous damping coefficient and the coefficient of restitution used in kinetics models [24-

26]. A tangential interaction is also often used in the literature [26,27], for example, in the form of a Coulomb friction force. The tangential force Ft is then directly linked to the normal force Fn. For numerical needs, the law is regularised to make the initial step softer (see figure 3). In this paper, v (i.e. the regularisation parameter) is fixed at 100 (dimensionless value, see chapter 3).

velocities and positions. See [5] for more details of this model. A 3D expression of this model has been performed by Saulot et al. [29], with the possibility of providing a wear flow of the third body, since the particles can exit from the contact from the "sides". Their approach is complementary to this study. This work based on the source flow of the third body is carried out in 2D in order to control the wear flow and reduce calculation time. Particles can not exit the contact and this study focuses on the process of particle detachment. 2.3. The model of the first body As explained in section 2.1, the material of the first body is modelled with DEM, as is the third body. Composed of an assembly of grains, this material can release particles. The explicit algorithm that manages third body movements is applied in the same way to the first body grains. Only the interaction forces are different. Delenne and al. [30] proposed a DEM model for the mechanical behaviour of a cohesive granular media. The main idea is to define hooked points where the granules constituting the material are linked. The contact between two grains is managed with the distance between the hooked points of each grain. This distance can be understood as the length of an elastic joint of adhesive. The normal contribution Fn of the adhesive is proportional to the projection of this distance along the axis that links the centres of the spheres (dn). The projection dt (figure 4) along a tangential axis gives the tangential force contribution Ft. F,, = K,,.d,,

jk, Ft

F, = K,.dt

PTBFN _a3Ut~ -

Ut

-~ITBFN

Kn and Kt are the normal and tangential stiffnesses. The distances are algebraic values and the forces can act in both directions.

Figure 3. Friction force between grains. Ut is the tangential sliding speed between two grains.

dt

Adding all the contact forces acting on a particle, its new acceleration is obtained by using Newton's second law. Finally, by using a velocity Verlet algorithm [28], the integration of the acceleration is performed over a small time step to give new

Figure 4. Bonded spheres with hooked points. Representation of dn and dt.

66

The granular material must release grains to constitute the third body. The bonds must be capable of breaking. With Fn* and Ft* two critical values for the normal and tangential contributions, a rupture criterion, adapted from Delenne et al., is defined as:

When ~ < 0, cohesion takes place. When ~ = 0, the bond breaks. With this form for the rupture criterion, the bond cannot break with compression only. The third body particles produced by the breaking of a first body bond (which corresponds to a crack in our model), can be constituted by a single sphere, or a group of spheres that are still stuck together. For the sake of convenience, the ratio: "total number of third body spheres produced over the initial number of spheres in the first body" will be used to represent the state of degradation. In the following simulations, for the sake of simplicity, only one of the first bodies is constructed with bonded spheres. The other is still a rigid plate as described in section 2.2. A diagram of the system is presented in figure 5.

Two mechanical phases coexist. The third body is a dynamic phase where the characteristic parameters are the velocities, flows, etc. With the granular material, which represents the degradable first body, a quasi-static phase is obtained. The interesting data are the forces, stresses, etc. In this study, particular care is taken to measure and analyse the stresses in the "skin" of the material, i.e. on an approximate depth of 1 or 2 grains below the rubbing surface (defined by a rectangle in figure 5). By summing all the contacts ~, in this volume V, with l being the distance between the centres of two grains and f their interaction force, the average stress tensor is defined [31-33] as:

~ij =v~a liaf/ " The behaviour of the maximum shear stress in this region was interesting. With (Yl and ~2 being the Eigen Values of 5 , it is defined as: 1

max The ratio "Cmax/ PN will be used in the following charts. e

Figure 5. A granular material rubs along a rigid plate. V is the domain where the stresses are calculated.

EVOLUTION OF THE CONTACT, ACCORDING TO THE MODEL

Descartes carried out experiments [4,34,35], measuring the flows of the third body under a wide range of mechanical and physicochemical conditions. The use of this model constitutes a continuation of this qualitative experimental work, providing help for the comprehension of physical phenomena that are not easily measurable in experiments. Numerical simulations were carried out with different operating conditions, the granular material and the third body particles (see table 1). This section explains the phenomena that occur in each step of the contact history in all these simulations. The values presented in Table 1 are dimensionless. The reference distance is R (the mean radius of the spheres). The reference mass is the mean mass M of the spheres and the reference stiffness is K = Kn, the normal stiffness in a contact between two (material or third body) grains. The dimensionless tangential stiffness is Kt = 0.3. The time step is defined as 0.1

67 Table 1 The numerical simulations for the different mechanisms, material and third body parameters. Simulation #7 is a reference simulation and the changes of parameters in the other simulations are framed. Simulation # Mechanism 1st bod~, 3 ra body

16 2

10 .4

~10"---'~~

10 -4

10 .3

10 -3

10 -3 10 -3

5.10 -4 10-5

10 -3

10 .4

7

10 .3

10 -4

8 9 10

10-3 10 -3

10-4 10-4

10 .3

10 -4

11

10 -3

10 -4

12

10 -3

10 -4

13

10 -3

10 -4

14

10 -3

10 -4

0.01 - 0.004 0.01 -0.004 0.01 0.004 0.01 0.004 0.01 0.004 0.02 0.008 0.01 0.004 0.01 - 0.004 0.01 0.004 0.01 0.004 0.01 0.004 0.01 0.004 0.01 0.004 0.01 - 0.004 i

of the mean contact oscillation period. In the granular domain, that has a length of 20 grains, the granular material contains 520 detachable spheres. There were no third body particles at the beginning of the simulations. Reference papers [35-37] defined three stages in the life of a tribological contact. At first, thin oxide films ("screens") protect the surfaces temporarily. Initiation of cracks, formation of TTS (Tribologically Transformed Structure, see [38]) and detachments of particles are observed in the second phase: the "birth" of the contact. In the third phase, i.e. the "proper life" of the contact, the generation of particles is equal to their ejection from the contact, thus reaching a stationary state. The mechanical model does not enable the observation of "screens" and the life of the contact begins here during the phase in which particles are generated. 3.1. "Birth" of the Contact

At the beginning of the simulations, the granular material slides directly along a rigid plate. However a certain time is necessary before the first detachment of particles can be observed. Figure 6, shows the evolution of relevant parameters for simulation #7, and is a simplified example that shows a very general trend.

10 -3

0 0 0 0 0 0 0 0.001 0.01 0 0 0 0 0

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.005 0.3

10 .3 10 .3 10 -3 10 .3 10 .3 10 .3 10 .3 10 -3 ......

3 . 1 0 .3 10 -3 10 -3 10 -3

The maximum shear stress calculated inside the skin of the material increases until the first "cracks" appear in the granular material (rupture of mechanical bounds, as explained in section 2). The increase in the number of cracks is thus directly linked to the decrease of the maximum shear stress. When the first detachment of particles occurs, the stresses have already decreased, but cracks are present in the material. 100

.........

9~ .

' ....

~I

90 80

-

,~.......

=

"

:

Jr

........

: ............

~

............

i

Detached grains / Nb grains initial material (%)

-

Number of cracks in the material Min thickness of third body (dimensionless)

.

7O 6O 50 40

,

30 ~ 20

r

0

;

/ i

![ ~

~:-:~

.:" :

?

~:,,

..............

10 2o 30 40 Sliding distance (dimensionless)

Figure 6. Progression of different parameters for simulation #7 during the birth of the contact.

5O

68 The detachments (initially marginal, as in figure 6) multiply very quickly. A steep degradation "slope" occurs, signifying that the source flow of the third body is very big. Then the degradations stop or slow down considerably. This great amount of detached particles during the birth of the contact has been shown by Descartes [35]. The simulations also show (see figure 6) that this phase ends as soon as a layer of third body is formed, separating the surfaces completely. Among the three categories of parameters (mechanism, material, third body) that can, a priori, control this first degradation "slope", figure 7 shows that the third body parameters have no influence on the time when this occurs. Note that the "third body parameters" also include the properties for the interactions between the two surfaces. The birth of the contact is thus greatly influenced by mechanical parameters such as normal pressure, sliding speed and the resistance of the granular material.

Stable Contact

Figure 8 shows the progression of simulation #7, as an example for the situation in which, after the first phase of severe degradation (what we call "birth" of the contact), no more degradation takes place (see the first part of the chart). This situation is usually encountered with small I.tTB, 7TB or C~xB. Descartes showed [34] an experimental example of this kind of behaviour with materials having rough surfaces. The maximum shear stress calculated in the skin of the granular material (not drawn) stays relatively low (= 0.13 * PN). Enough third body is formed in the contact, providing a speed accommodation with a low stress level. [6] shows that the level of the third body layer (the quantity of third body required) depends on the operating conditions, the granular material and the third body.

8 0

" ............. : " 1

.2 L_.

70

:::: : , " - - . ~ : : : ~ : : : : - -

, ;::::::. . . . . . . . -.::::~- ~ :~

: : : : : : - - ,~ --

-r

i

,

i 9149 9 ,.; - pr~uced"3id~b~E>dy(Qs)- I - - - . Active 3rd body (Qi) f ........ Eliminated 3rd body_~.(Q.u)._}

60 E

60,

: .........

.......

,,-.,

F '~'~" ! / /

i ,1

.E .E 40 .0

:7

,/

...

30

j / /

-'

,,,, / f #,,,#

:,"

!

:

(/)

.__q C~ _Q

z

.~ 20

0

..1...

0

~ "to -5 z

Sliding distance (dimensionless)

Figure 7. Progression of the source flow of the third body for several simulations, involving different parameters. 3.2. "Proper life" of the contact Without the ejection of particles, once a layer of third body is formed, preventing direct interactions between the first bodies, two scenarios can occur: either the degradations stop (stable contact), or certain degradations still occur (contact in which degradations continue). In this case the properties of the interface (laxa, ~'TB, ~TB), predominate for discrimination between these two cases.

1O0

J ......

i .........

,

:

.

200 300 400 500 600 700 Sliding distance (dimensionless)

800

Figure 8. Internal, source and wear flow along the sliding distance in simulation #7. A wear flow is imposed from 500 DSD (dimensionless sliding distance). To be closer to the experimental cases where competition between the source flow and wear flow take place, a wear flow has to be added. A natural wear flow (for example, at the sides of the contact) cannot be produced in 2D. To do this, see the work of Saulot et al. [29]. However an artificial wear flow can be computed, making it possible to remove grains of the third body. In this case the wear flow is imposed. The second part of the chart in figure 8 shows the progression of the internal flow and the source flow, by imposing a wear flow (from a dimensionless

69 sliding distance - DSD - of 500), removing 1 grain (0.2% of the initial number of grains in the material) every DSD. It is shown that the introduction of this artificial wear flow reactivates the source flow. Moreover, the source flow produces as many third body particles as the wear flow removes. The source flow is thus only a consequence of the wear flow. By focusing on the behaviour of the internal third body quantity Qi, it can be seen that it decreases as soon as the wear flow removes grains. The source flow, however, is only activated when the quantity of particles at the interface reaches a critical value. When Qi is "big enough", the third body production stops, etc. Qi is therefore stationary in the sense that it oscillates between two critical values: one representing the level where degradation must occur, while the other is the level where degradations are no longer necessary. The upper level for the internal third body quantity corresponds to the stable level observed with no wear flow.

it

10 5 0

650

/ 700 750 800 Sliding distance (dimensionless)

Remark: In a fluid lubricated contact, the shear stress in the Newtonian third body is defined [39] as: ~U 'l~xy fluid -- 17

Oy

where r/is the fluid viscosity and i)u/i)y = j, is the shearing rate (u is the speed of the flow and y is the normal direction). Considering that the fluid adheres to the surfaces, the shear stress in the fluid equals the shear stress in the skin of the material. With a linear velocity gradient, it becomes:

VP 'xy material = 77

H

with Vp being the sliding speed and H the height of third body. Applying this expression to the case of our solid third body, it is possible to define the height of the third body required, for a given sliding speed and third body properties (characterised by r/here), to reach a shear stress in the material that prevents degradations (i.e. providing "small enough" forces compared to Fn* and Ft*). In a general way, it is also necessary to take into account the normal pressure imposed when considering a maximum shear stress. Of course, solid lubrication is not so simple but this simple model and the numerical simulations appear to share similar trends.

. . . . Maximum shear . ......... stress /. Pn (%) ..... . 30 ! _ ~ Nb 3rd body grains / Nb grains initial material (~

35Ii

occurs. A critical value for Qi corresponds to a critical stress for the material.

850

Figure 9. Progression of the stresses in the material and the internal source flow for simulation #7, when a wear flow is imposed. The progression of Qi can be compared with the progression of the maximum shear stress near the rubbing surface. Figure 9 shows that the maximum shear stress increases as soon as the level of third body decreases. When the level of third body is low, the stresses in the material are high, thereby explaining the degradations. When degradations occur, the quantity of third body increases and the stresses decrease. With low stresses, no degradation

Contact with remaining degradations With another set of simulations (corresponding to high ~tTB,YTBand O~TB),degradations still occur when the surfaces are fully separated, even without wear flow. Experimentally, Descartes has shown [4,35] that a more cohesive and ductile third body provides more degradations. The "slope" of degradation is far shallower but can subsist. Simulation #9 is an example of these situations. In figure 10, the progression of the maximum shear stress near the surface is compared to the cumulated source flow of the third body (reflecting the number of third body particles in the contact).

70 After the first step where the degradation level is high, the stresses in the material remain high (compared to the previous case). Degradations occur, each time modifying the level of stresses. When the stresses reach a "sufficiently low" value, the degradations stop. More third body means lower stresses and less source flow. This explains the general shape for the chart of the third body source flow (without wear flow).

With a "small" imposed wear flow of the same order of magnitude as the "natural" degradation in the "proper life" (figure 11) of the contact, the resulting source flow is relatively unchanged: it produces a third body, but no more than without a wear flow. In this case, the source flow is independent. The third body quantity is therefore the consequence of both the source flow and the wear flow, oscillating around a given value. 100 ............... ,........................

,

......

,

"'

,

..-.. 0 . 8

--

~ ..............

i. . . . . . .

~

..............

~ ,.._...

;hear stress / Pn y grains / Nb grains initial material .................................................

0.7

90

' ---~.~,-~ .... .... .....

.,z,_80 E 7O

Qs (Qw = O) Qs (Qw imposed) Qi (Qw imposed) Qw imposed

06 ~r " 60 05

~ 5o

04

!

('~

40 c~

z -- 3C r"

~ 2C ~1C z 0 0

0

.................

500

i

looo

0

2ooo

3o0o

4000

~c;oo

6000

SlJding dJstance ( d i m e n s i o n l e s s ) 1000 1500 2000 2500 3000 Sliding d i s t a n c e ( d i m e n s i o n l e s s )

3500

4000

Figure 11. Third body flows for simulation #14, with and without an imposed wear flow of 1% for 520 DSD (= 1 grain/100 DSD).

Figure 10. Progression of the source flow and the maximum shear stress in the skin of the material for simulation #9.

I O0 ...-..

One of the general trends (for a weakly cohesive 3 ~a body), expressed by Saulot et al. [29] with a 3D model that provides a natural wear flow, is that the wear flow is greater when there are more particles at the interface. This remark can be applied to the case where the source flow is still active in the "proper life" of the third body and producing increasing quantities of it. One might expect that in a real contact, increasing the thickness of the third body would generate a wear flow. In this case, the wear flow is the consequence of the source flow.

90 _e (],) .4,

8O

E 7O :-,-2 60 c-

5o 40 _13

z --- 30 (/) r-

~

20

U~

-Q 10 z ~

It is also interesting to study the effect of a wear flow on the contact equilibrium. An artificial wear flow is then added to our model. In simulation #14, the second phase of degradation lasts a particularly long time. Figures 11 and 12 show two different behaviours according to two different imposed wear flows, each time compared to the situation without wear flow.

t

- - - - Qs (Qw = O) i-Qs (Qw imposed) . . . . Qi (Qw imposed) ..-._-~- Q w imPose - d

..........

t

1000

t

.......................

!

i

..........

!

2000 3000 4000 5000 Sliding d i s t a n c e ( d i m e n s i o n l e s s )

,,

,

6000

Figure 12" Third body flows for simulation # 14, with and without an imposed wear flow of 1% for 104 DSD (= 1 grain/20 DSD).

71 With a higher wear flow (Fig 12), particles are removed faster than they are produced. Qi reaches a critical value and the source flow, small at first, increases to exactly counterbalance the imposed wear flow. Then, the same phenomena occurs, as described above. Note that the birth of the contact is not affected by the presence of the imposed wear flow. There is initially no third body (or very little) so it is impossible to remove it. Then the speed of production of the particles is much higher than their exit from the contact. 4.

ASSESSMENT AND PROSPECTS

This numerical study reproduces experimental observations where the source flow of the third body corresponds to the wear flow of the third body, i.e. the production and ejection of particles are counterbalanced. It provides the possibility of discriminating the cases where the wear flows control the source flow and where the source flow controls the wear flow, by dissociating the flows and analysing the relationship between the stresses near the rubbing surfaces and the quantity of third body in the contact. Depending on the mechanism (imposed pressure and sliding speed), particles are produced. The degradations slow down when a layer of third body is formed and efficiently separates the surfaces. Then slight degradations can take place until the stresses in the material are low enough. An imposed wear flow can contribute to the reactivation of the source flow. Controlling wear (to extend the life of a contact, avoiding interminable degradations) entails that the quantity of detached particles should be high enough so that the stresses in the skin of the material can progress to an acceptable point while being low enough to prevent the third body being ejected from the contact. Limiting the ejection of particles, the model allowed us to study the mechanisms of particle detachments. The next step is to build a 3D model that includes a degradable material and the possibility for the particles to be ejected from the sides of the contact. This complete model is necessary in order to get closer to the reality of the experiments.

REFERENCES

[ 1] Godet, M., 1984, "The Third Body Approach: a Mechanical View of Wear", Wear, 100, pp. 437-452. [2] Godet, M., 1990, "Third-Bodies in Tribology", Wear, 136, pp. 29-45. [3] Berthier, Y., 1995, "Maurice Godet's Third Body", 22 nd Leeds-Lyon Symposium on Tribology, Tribology series, 31, pp. 21-30. [4] Descartes, S., Berthier, Y., 2001, "Competition Between Mechanical and Physico-chemical Actions: Consequence on Friction. The Case of a Contact Lubricated by a Third Body Stemming from a Coating of MOS1.6", Proceedings of the Second World Tribology Congress, Scientific achievements, industrial applications, future challenges, F. Franek, W.J. Bartz, A. Pauschitz, eds., Vienna, pp. 67-71. [5] Iordanoff, I., S~ve B., Berthier, Y., 2002, "Solid Third Body Analysis Using a Discrete Approach: Influence of Adhesion and Particle Size on the Macroscopic Behavior of the Contact", ASME J. Tribol., 124, pp. 530-538. [6] Fillot, N., Iordanoff, I., Berthier, Y., 2002, "Stable Contact and Degradation of Materials: A Model for Third Body Source Flow", Proc. 6th Austrib International Tribology Conference, G.W. Stachowiak, eds., Perth, Vol. 2, pp. 709716. [7] Heshmat, H., 1992, "The Quasi-hydrodynamic Mechanism of Powder Lubrication, Part i: Lubricant Flow Visualisation", Lubrication Engineering, 48 (2), pp. 96-104. [8] Heshmat, H., 1992, "The Quasi-hydrodynamic Mechanism of Powder Lubrication, Part ii: Lubricant Film Profile", Lubrication Engineering, 48 (5), pp. 373-383. [9] Heshmat, H., 1995, "The Quasi-hydrodynamic Mechanism of Powder Lubrication, Part iii: On Theory and Rheology of Triboparticles", Tribol. Trans., 38 (2), pp. 269-276. [10]Savage, S.B., 1987, "A Simple Theory for Granular Flow of Rough, Inelastic, Spherical Particles", J. Appl. Mech., 54, pp. 47-53. [11]McKeague K.T., Khonsari, M.M., 1996, "Generalized Boundary Interactions for Powder Lubricated Couette Flows", ASME J. Tribol., 118,580-588. [12]Yu, C.M., Craig, K., Tichy, J., 1996, "Granular Collision Lubrication", J. Rheol., 38 (4), pp. 921-936.

72 [13]Dubujet, P., Ghaoudi, A., Chaze, M. and Sidoroff, F., 1995, "Particulate and Granular Simulation of Third Body Behaviour", 22na Leeds-Lyon Symposium on Tribology, Elsevier Tribology series, 31, pp. 355-365. [ 14] Elrod H.G., 1995, "Numerical Experiments with Flows of Elongated Granules-Part II", 22nd Leeds-Lyon Symposium on Tribology, Elsevier Tribology series, 31, pp. 347-354. [15]Lubrecht, A.A., Chan Tien, C.E., Berthier, Y., 1995, "A Simple Model for Granular Lubrication, Influence of Boundaries", 22nd Leeds-Lyon Symposium on Tribology, Elsevier Tribology series, 31, pp. 377-385. [16] Cundall, P.A., Strack, O.D.L., 1979, "A Discrete Numerical Model for Granular Assemblies", Geotechnique, 29 (1), pp. 47-65. [ 17] Cundall, P.A., 1987, "Distinct Element Modes of Rock and Soil Structure", Analytical and computational methods in engineering rock mechanics, E.T. Brown, eds. , Imperial College of Science and Technology, Londres, pp. 129163. [18]Linck, V., Baillet, L., Berthier, Y., 2003, "Modeling the Consequences of Local Kinematics of the First Body on Friction and on Third Body Sources in Wear", Wear, 255, pp. 299-308. [19]Ele6d, A., A., Devecz, J., Balogh, T., 2000, "Numerical Modeling of the Mechanical Process of Particle Detachment by Finite Element Method", Periodica polytechnica Ser Transp. Eng., 28, pp. 77-90. [20]0qvist, M., 2001,"Numerical simulations of mild wear using updated geometry with different step size approaches", Wear, 249, pp. 6-11. [21]PSdra, P., Andersson, S., 1997, "Wear simulation with the Winkler surface model", Wear, 207, pp. 79-85. [22]S~ve, B., Iordanoff, I., Berthier, Y., 2000, "Using Discrete Models to Simulate Solid Third Bodies: Influence of the Inter-granule Forces on the Macroscopical Behaviour", 27 na Leeds-Lyon Symposium on Tribology, Elsevier Tribology series, 39, pp. 361-368. [23] S~ve, B., Iordanoff, I., Berthier, Y., Jacquemard, P., 2000, "The Granular Approach: A Tool for Understanding Solid Third Body Behavior in Contact", Proc. of the 5 th International Tribology Conference, Nagasaki, Vol. 2, pp. 1103-1108.

[24]Iordanoff, I., Khonsari, M.M., 2003, "Granular Lubrication: Towards an Understanding of the Transition between Kinetic and Fluid Regime", submitted to ASME J. Tribol. [25]Mishra, B.K., Murty, C.V.R., 2001, "On the Determination of Contact Parameters for Realistic DEM Simulations of Ball Mills", Powder Technology, 115, pp. 290-297. [26] Cleary, P.W., 1998, "Predicting Charge Motion, Power Draw, Segregation and Wear in Ball Mills using Discrete Element Methods", Minerals Engineering, 11 (11), pp. 1061-1080. [27] Elperin, T., Golshtein, E., 1997, "Comparison of Different Models for Tangential Forces Using the Particle Dynamics Method", Physica A, 242, pp. 332-340. [28]Verlet, L., 1967, "Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules", Phys. Rev., 159, pp. 98-103. [29] Saulot, A., Iordanoff, I., Safon, C., Berthier, Y., 2003, "Numerical Study of the Wear Flows in a Plane Contact. Effect of Third Body Rheology", 30th Leeds-Lyon Symposium on Tribology. [30] Delenne, J.-Y., Moulay, S. E. Y., B6net, J.-C., 2002, "Comportement M6canique et Rupture de Milieux Granulaires Coh6sifs", C.R. Mecanique, 330, pp. 475-482. [31]Love, A. E. H., 1927, "A Treatise of Mathematical Theory of Elasticity", Cambridge University Press, Cambridge. [32]Christoffersen, J., Mehrabadi, M. M., NematNasser, S., 1981,"A Micromechanical Description of Granular Material Behavior", ASME J. Appl. Mech., 48, pp. 339-344. [33] Kruyt, N. P., Rothenburg, L., 1996, "Micromechanical Definition of Strain Tensor for Granular Materials", ASME J. Appl. Mech., 118, pp. 706-711. [34] Descartes, S., Berthier, Y., 2001, "Frottement et Usure Etudi6s ~ Partir de la Rh6ologie et des D6bits de 3e Corps Solide : Cas d'un 3e Corps Issu d'un Rev~tement de MoSx", Mat6riaux & Techniques n ~1-2, pp. 3-14. [35]Descartes, S., 1997, "Lubrification Solide ?~ Partir d'un Rev&ement de MoSx: Cons6quences de la R6ologie et des D6bits de Troisi~me Corps sur le Frottement ", Ph.D. Thesis, INSA Lyon, Lyon, France. [36] Hurriks, P.L., 1970, "the Mechanism of Fretting - A Review", Wear, 15, pp. 389-409.

73 [37]No11, N., 1997, "Conception et Naissance d'un Contact Tribologique", Ph. D. Thesis, INSA Lyon, Lyon, France. [38]Sauger, E., Fouvry, S., Ponsonnet, L., Kapsa, Ph., Martin, J.M., Vincent, L., 2000,

"Tribologically Transformed Structure in Fretting", Wear, 245, pp. 39-52. [39]Frene, J., Nicolas, D., Degueurce, B., Berthe, D., Godet, M., 1 9 9 0 , "Hydrodynamic Lubrication- Bearings and Thrust Bearings", In Elsevier, Tribology Series 33, Dowson D., eds.