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Surface and Coatings Technology, 68/69 (1994) 582—590
Comparative tribological study of chromium coatings with different specific hardness A. DarbeIdaa, J. von Stebuta, M. Bartholeb, P. Belliardc, L. Le1ait~’,G. Zacharied aLaboratoire de Science et Genie des Surfaces, (URA CNRS 1402), École des Mines, Parc de Saurupt, F-54042 Nancy, France bPSA Peugeot—Citroën, F-781 40 Vélizy Villacoublay, France cRenault CIB Sce.937, 8—10 Av. E. Zola, F-92109 Boulogne-Billancourt, France dElectricité de France, Centre de Recherches des Renardières, BP 1, F-77252 Moret-sur-Loing, France
Abstract The wear resistance in dry friction of two electrolytic and two PVD hard chromium coatings deposited on construction steel substrates is studied by means of standard pin-on-disc and unidirectional multipass scratch testing. For both of these friction modes, low cycle—high load operation with cemented carbide pins leads to essentially coating-hardness-controlled, abrasive wear. For the well adhering commercial coatings, critical loads (both for through-thickness cracking and for spalling failure) assessed by standard scratch testing are inadequate for quality ranking with respect to wear resistance. Steady state friction corresponds to a stabilized third body, essentially composed of chromium oxide.
1. Introduction Hard chromium plating is a widespread technique to enhance the wear and corrosion resistance of technical parts. To overcome problems of intrinsic cracks always resulting from the standard, electrolytic d.c. method, a pulsed d.c. method has been proposed [1,2]. Physical vapor deposition (PVD) is another “environmentally friendly” technique to produce crack-free, high hardness chromium coatings [3,4]. In this case, the hardness can be tailored by doping with small concentrations (about 1 wt.%) of carbon. This is achieved either by sputtering already doped, sintered targets or else, in reactive mode, with undoped targets. The present work is a comparative study on the mechanical strength (adhesion, cohesion and wear resistance) of such perfectly adhering coatings of identical thickness (15 lIm), deposited under identical conditions on identical, medium hard (HY = 3.5 GPa) construction steel substrates. In an earlier paper [5], a detailed comparison of high load indentation, scratch testing and WC pin-on-disc wear testing showed the superior wear resistance of the carbon-doped PVD coating. It also showed that the wear resistance quality ranking is different from such a ranking in indentation and scratch testing. In a second paper [6], we focused on multipass scratch testing as a way of producing wear with practically no recycling of loose debris. For the specific case of standard, electrolytic chromium plate, debris generated during the wear process was seen to have a pronounced abrasive action.
0257—8972/94/57.00 SSDI 0257-8972(94)08032-T
In the present paper, the triboscopic approach of Ref. [6] for on-line tracking of friction and wear of the two dry sliding partners is extended to three hard chromium coatings prepared by more recent deposition techniques [1—4]. Special attention is paid to the wear mechanisms in the run-in regime.
2. Experimental details 2.1. Surface mechanical testing As in Ref. [7], high load Vickers indentation and scratch testing were carried out, respectively, in the static and the dynamic modes of a CSEM Revetest automatic scratch tester. Pin-on-disc wear testing was realized in dry sliding on a rig also manufactured by CSEM. The multipass scanning friction rig is derived from a standard three-dimensional scanning stylus profilometer [6]. A relative horizontal scanning motion of the specimen flat is obtained by means of stepping motorpowered translation tables. As for most of the standard pin-on-disc rigs, the normal contact load on the stationary pin is applied by a dead weight on a balanced lever. A piezoelectric transducer ensures high stiffness in the friction force monitoring. The profilometer stylus is in a “horseback” position on top of the sliding pin. This configuration allows for continuous monitoring during the wear process of the relative vertical displacement stored continuously, together with the corresponding friction force signal (for details, see Ref. [6]). The friction rig is entirely automatic. A standard PC system handles
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Elsevier Science S.A. All rights reserved
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all the mechanical steering functions (specimen translation, pin “lift off” and “landing”), as well as data acquisition and logging (the pin’s scanning coordinates and vertical position, together with the associated friction force). The ongoing friction signal is monitored on-line by the software. At the end of a complete friction experiment, this software gives access to practically any data reduction, as well as three-dimensional topographic or triboscopic presentation [6], as familiar from atomic force microscopy. The contact load was 10 N and the dry sliding speed was 1 mm s’ for both friction rigs, while the sliding distance per cycle was 4 mm in the multipass and 30 mm in the pin-on-disc operation mode. All the friction experiments were run in a standard laboratory environment (21°Cand (50 ±5)% relative humidity). 2.4. Wear quant~ficationand wear mechanism analysis Wear quantification was achieved by means of threedimensional stylus profilometry [8], by interrupting the friction experiment. With the multipass rig, it could also be tracked on-line during the entire experiment, both for the specimen and for the pin. For metallography, reflected light and scanning electron microscopies (RLM and SEM) were used. For surface chemical analysis within the wear track and on the pin contact area, we adopted electron probe microanalysis (EPMA). 2.3. Pin material and geometry In the present study, standard sintered WC Brinell balls (0 1.58 mm, Ra ~i0.1 j.tm) were used. Because of the high hardness (HV ~ 15 GPa), pin wear becomes very low, thus preserving a practically constant pin geometry during the whole friction experiment. Nevertheless, for these balls, friction levels similar to those with steel pins are obtained, 2.4. Specimens The coated specimens for surface mechanical testing were square flats, 60 mm wide and 8 mm thick. The substrates were made of AISI 4135 construction steel, heat treated to HV = 3.5 GPa. Standard grinding would produce a surface finish prior to coating of Ra ~ 0.3 p.m. The coatings 15 p.m thick produced by industrial suppliers, only slightly degraded the overall surface roughness. The detailed specimen characteristics of contact mechanical interest are compiled in Table 1.
3. Results and discussion 3.1. Coating morphology Fig. 1 gives an insight into the surface morphology of the coatings. In all cases, the effect of substrate surface grinding marks on the growth mechanisms is clearly
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perceptible. The coatings all have a non-uniform surface morphology with distinct growth defects, most of which are aligned along the grinding direction. This is evidence that they must have been nucleated at grinding-induced surface flaws. The cracks in the standard hard chromium plate cover the whole surface in an irregular network. With the dominant crack direction being perpendicular to the grinding marks, this indicates directional anisotropy in the internal stress. 3.2. Intrinsic coating mechanical properties For the two PVD coatings, the coating-specific Young’s modulus could be assessed by free flexural vibrations (see Ref. [9] for information on the measurement technique). We measured 293 GPa for the doped (1% C) and 250 GPa for the undoped PVD coating, with an experimental error of ±25 GPa. The coating hardness (Table 1) is clearly dependent on the deposition technique. The doped PVD coating is by far the hardest and can be expected to have the best wear resistance when wear is entirely abrasive. This high hardness is undoubtedly related to the doping level rather than the deposition technique because the undoped PVD coating is considerably softer, ranking last on the hardness scale. For the electrolytically deposited coatings, the pulsed d.c. technique lowers the hardness level. The average internal coating stress has been assessed from the bending strain resulting when the substrate material is thinned away underneath the coating [10]. These data confirm general knowledge on the compressive nature of internal stresses in the PVD coatings, and the tensile nature in electrolytic hard chromium plates. When comparing with the absolute values of hardness, it is striking that, for similar numerical values, the sign of the corresponding internal stress state can change. In particular, the comparison of internal stresses for the two electrolytically deposited coatings explains why the pulsed d.c. technique yields crack-free coatings. The strain energy release rate (G) has been computed in the well-established way [14], from the length of radial cracks in high load Vickers indentation. The undoped PVD chromium is seen to be perfectly plastic. Strikingly enough, the doped PVD behaves best, combining high hardness and high toughness. Another astonishing result is the lower brittleness of the standard electrolytic hard chromium plate as compared with the coating deposited by the pulsed current technique. Close inspection shows that, in the pulsed technique, radial cracks propagate very far from the indent corners, whereas the pre-existing cracks act as propagation barners in the standard procedure. Detailed information on scratch testing results can be found in Ref. [5]. The major results were critical loads for the first through-thickness cracking of L~1~ 10 N for
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Table 1 Friction specimen features Specimen code
Deposition technique
Coating nature’
Coating roughness R~(j.tm)
ECC ECP PVD.D
Standard electrolytic d.c. method Electrolytic pulsed d.c. method Non-reactive magnetron sputtering
PVD.N
Intrinsic coating 1’ hardness H 0025 (GPa)
Average coating internal stress’
Strain energy release rate
(GPa)
G” (J m2)
Composite macrohardness Hv 1’ (GPa)
Cr with cracks
0.42 ±0.03
9 ±0.3
+0.607 ±0.007
13
3.8 ±0.3
Cr without cracks
0.25 ±0.02
8 ±0.3
+0.198 ±0.007
8
3.8 ±0.4
1 wt.% C-doped Cr, columnar Undoped Cr, columnar
0.53 ±0.03
19 ±0.3
1.007 ±0.007
75
7.4 ±0.4
0.42 ±0.03
6 ±0.5
No radial cracks
3.8 ±0.2
—
—0.136 ±0.007
‘General coating thickness, 15 ±0.5 im. b General substrate hardness, HV~025 = 3.5 GPa. ‘Assessed from specimen bending strain (thinning substrate from underneath). d Computed from the length of radial cracks in high load (FN = 10 N) Vickers indentation. ‘Under maximum load, FN =10 N.
PVD.D.
PVD.N.
,~
I___
~~~—JIIIIII~
I
~~__j-
~
ECC.
.
ECP.
____
____
____
wIy~
~~dH
—11
‘~
U”
_
~
Fig. 1. SEM images of chromium coatings obtained by two different techniques: electrolytic, standard d.c. method (11CC); pulsed d.c. method (ECP) and magnetron sputtering method; undoped (PvD.N); 1 wt.% C-doped chromium (PVD.D).
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all the coatings, with ±5N experimental scatter in the
stabilization; the synchronism between friction and
numerical values. For the carbon-doped PVD coating, the critical load for gross spalling was L~2= 40 N, while
height oscillations is obvious even in this phase of
underline the analogy between the three-dimensional “friction surface” (Fig. 2(a)) and its corollary of the three-dimensional “wear surface” (Fig. 2(b)), as filtered by the pin, both plotted over sliding distance and pass number. One clearly distinguishes three different friction
statistically fluctuating high amplitude variations in the run-in transition region (Figs. 2(c) and 2(d); pass number 70); (3) stage III (beyond 200 passes)—a stabilized final regime after “run-in” (stage I + II), where friction is practically constant both vs. the sliding distance and vs. the pass number. The analogous “ripples” in the “friction surface” and the “wear surface” indicate that the corresponding fniction and height oscillations are closely related, and correspond for phase Ito a temporary and for phase III to a stabilized local morphology of the specimen itself. In contrast, when zooming over the transition region in stage II, the synchronized high amplitude friction and height oscillations are uncorrelated in time (no groovelike features in this part of the friction surface and the wear surface). This strongly suggests stick—slip phen-
and wear regimes:
omena related to repeated build-up and shearing of
we found L~2>100 N for the three remaining specimens. Thus, quality ranking of the coatings based on scratch testing alone would clearly disqualify the carbon-doped PYD coating. 3.3. Friction and wear behaviour 3.3.1. Multipass, triboscopic approach Fig. 2 shows a typical triboscopic analysis of the “friction and wear life” for the specimen prepared by the pulsed current technique, which is similar to its d.c.
counterpart discussed in detail in Ref. [6]. Let us
(1) stage I (Figs. 2(a) and 2(b); 0—10 passes)—with
transfer debris on the slider, as conjectured in Ref. [6].
low initial friction (J1~0.1)and rapid, wear-induced variation of “height” (owing to changes in contact geom-
etry, i.e. ploughing into the coating and wear of the ball) as monitored by the stylus pick-up; (2) stage 11(10—200 passes)—a transition region with very important friction and height oscillations before
3.3.2. Reduced standard friction and wear approach vs. time
When taking the friction and wear data as a function of the pass number after 2 mm of sliding, the plots correspond to a traditional, tribological presentation. MULTI-PASS - TOPOGRAPHY
MULTI-PASS - FRICTION
4
1.00
+4.00
(~)
L(mm)
7fP~ass
7cP~Pass
[jim]
I.’
~ 0
___
2
4mm
0
2
4mm
Fig. 2. Triboscopic projection of (a) friction and (b) topography (vertical displacement of the pin) during multipass operation with a cemented carbide ball 1.58 mm in diameter under a normal load of 10 N for the specimen prepared by the pulsed current technique (ECP), plotted vs. sliding distance (L = 4mm) and pass number (N ~ 500 passes). (c), (d) Individual scans of friction and vertical displacement during the 70th pass (Stage II).
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This can be seen in Figs. 3(a) and 3(b) for all four coatings studied in this work. The difference in friction of the four coatings is clear: the initial stage I regime is quite short and depends considerably on the specimen; the intermediate stage II or run-in regime is practically identical for the two electrolytic coatings, it is considerably reduced, both in amplitude and in duration, for the doped PVD coating, and it continues up to the end for the soft, undoped PVD coating; the final, stabilized stage III regime corresponds to practically the same friction level for all specimens, except the lowest hardness undoped PVD coating, where the persistence of stage II oscillations suggests severe adhesive wear. For all the specimens, the “height” vs. pass number plots essentially confirm the corresponding general friction tendencies. They underline the considerable difference in associated coating wear. The stability of friction in stage III is accompanied by practically zero wear (constant height parameter) for all but the undoped PVD coatings. In this latter case, wear becomes increasingly important beyond 250 passes, which is in agreement with the severe friction level observed in this region. 3.3.3. Comparison of conventional pin-on-disc and multipass friction In Fig. 3(c), we present the average friction, as monitored in equivalent pin-on-disc operation, vs. the number of cycles under an identical normal contact load (FN= 10 N). Again, three different friction stages are observed, with a low initial level lasting over a short period only, an intermediate, high friction stage II, and a final, stabilized, lower friction regime, whose level is distinctly higher than in corresponding multipass operation. For the four hard chromium coatings, there is a reasonable resemblance in the apparent overall fiction histories in Fig. 3(a) and 3(c), even though stage II extends over a somewhat longer period in Fig. 3(c) for conventional d.c. or pulsed d.c. coatings. The comparison of the friction data for the two operation modes suggests a distinct difference, which must be related to the way in which wear debris and transfer layer formation occur for the two rigs (debris recycling for pin-on-disc and debris accumulation at the track end for multipass operation). 3.3.4. Comparison of pin-on-disc and multipass wear after 1000 cycles Fig. 4 shows typical selected portions of the wear tracks of all the specimens after 1000 consecutive cycles, The difference in wear behaviour is especially obvious for the PVD coatings: practically no wear for the doped and severe wear with coating perforation for the undoped
Tribology of Cr coatings
specimen. In this case, the difference in overall wear for the two operation modes, only slightly perceptible for the other coatings, becomes very pronounced both for the wear volume itself and for the shape of the wear track. As a by-product, this three-dimensional comparative presentation also reveals the size of the growth defects of the standard electrolytic hard chromium plate (ECC, first line). These profilometric representations can be conveniently exploited by computing in a standard routine the corresponding integrated specific wear volumes per unit sliding distance. Such numerical wear data are plotted in Fig. 5 as a function of the coating hardness. For all the specimens, the lower wear rate for multipass operation is clear. In addition, the coating hardness has a clear effect on wear, as expected in the case of abrasive wear. It should be underlined that these results are in complete contradiction to the above ranking based on scratch testing. In that case, the doped PVD coating had been placed at the end of the quality scale, while it clearly ranks at the top in the present dynamic tribological situation (pin-on-disc as well as multipass). The most important wear volume variations are measured for coating hardnesses below 9 GPa (hardness of the standard hard chromium plate). When assuming abrasive wear to be the dominant mechanism, these data can be reasonably well explained. They are in agreement with the steep decrease in wear rate when the average hardness ratio of the abraded body (the coated specimen) and the abrasive particles (wear debris) becomes bigger than unity [11]. Most of the debris is likely to be highly work hardened and oxidized chromium from the coating, and thus sufficiently hard to become an abrasive third body. Three-dimensional wear volume assessment of the pin tips after identical sliding lengths reveals that there is practically no shape modification of the pins in multipass operation, while all the tips are truncated with approximately the same tip wear volume for all coatings for the pin-on-disc geometry. Therefore, the third body in the case of pin-on-disc operation should comprise WC partides from the cemented carbide pin, in addition to the work-hardened chromium debris. This should partly account for the difference in wear rates in Fig. 5 for the two operation modes. 3.4. Metallographic and tribochemical analysis ofthe friction partners for standard hard chromium plate A full understanding of the difference in the friction and wear mechanisms in the present study can only come from analysis of the corresponding “third bodies” (adhering transfer layers and mobile wear debris). This has been done in the present study by means of scanning electron microscopy (SEM) and EPMA for the conven-
A. Darbeida et a!.
1.00
::
MULTI-PASS FRICTION -
,
I,
II
iii
ECC.
~=—~-——
ECP.
/
587
Tribology of Cr coatings
MULTI-PASS
+6.00
n
::
-
TOPOGRAPHY III
® ECC.
_____________________________
+6.00
ECP.
~
~ PVD.D
PVD.D.
[pm] 0.00
0.50
0.00 ____________________________________________
PVD.N
0.0
250
500 Passes
-6.00 ___________________________________________ +6.00 PVD.N.
0.0
250
500 Passes
PIN ON DISC FRICTION -
1.00
ECC.
ECP.
PVD.D :50/\._
0.00
_________________________________________
PVD.N 0.50
0.00 _________________________________________
0.0
250
Soocycles
-
Fig. 3. Plot of (a) local friction and (b) corresponding topography (height Z) extracted from Figs. 2(a) and 2(b) vs. pass number for the four different chromium coatings during multipass operation at sliding distance of 2 mm from the beginning of the wear track (L = 2 mm). (c) Average friction as monitored in pin-on-disc operation vs. the number of cycles under identical conditions (FN = 10 N, v = 1 mm s 1, cemented carbide ball 1.58 mm in diameter).
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Tribology of Cr coatings
SEM images of the wear track corresponding to stages I,
PIN ON DISC
ECC.
ECC.
-
1
~
‘~
/
“
ECP.
ECP. _______
~
______________________
/
t
1 —— -
-~i~vi~i.
PVD.N.
~ __________
_______
/
2.05 _____________________
_________________________ ____________________
Fig. 4. Scanning stylus, three-dimensional profilometric representations of wear tracks after 1000 passes in multipass and pin-ondisc operation.
_____________________________________________ Multi pass operation
~4OOO
2~
PVD.N. (6GPa)
ECP.
ECC.
(5GPa)(9GPa)
Coating hardness
~. PVD.D.
(19GPa)
Fig. 5. Variation of wear volume (calculated from the threedimensional profilometric data of Fig. 4) per unit sliding distance as a function of coating hardness in multipass and pin-on-disc operation after 1000 passes.
tional hard chromium plate only (forreasons of economy in experimental effort). Fig. 6 shows such an analysis at the different steps of wear life in multipass operation. The first line shows
II and III. Loose debris is practically absent in this
multipass mode. One observes the progressive build-up of an adhering transfer layer in the friction track. EPMA shows this layer to be essentially made up of oxide (most likely Cr2O3), and rare traces of cobalt and tungsten from the pin, localized essentially along the track borders. The second and the third rows of Fig. 6 show the corresponding energy-dispersive X-ray images of the pin over stages I—Ill. The progressive build-up of a more or less adhering, clearly brittle transfer layer is evident. From this inspection, we infer that the stable, almost continuous transfer layer in the track at the end of stage III most likely corresponds to a redeposition of primary transfer present in dynamic equilibrium on the pin. increase Within in friction framework between stages interpretation, I and theFig. steep 3) should the correspond to of thethis build-up of II a (see chromium transfer layer on the pin after disruption of the initial oxide and/or contaminant layer on the coating. This situation is synonymous with severe, adhesive friction, fluctuations in the real contact area, and important oscillations in the friction force, as observed experimentally. In stage III, the stabilized, low friction regime can consolidated chromium oxide layer in the friction track. then be understood by the simultaneous presence of a For pin-on-disc operation in stages I and II, the qualitative general mechanisms of initial oxide and/or contaminant film disruption on the coating, and chromium transfer layer build-up are similar to the above situation for multipass operation. However, in stage III, Fig. 7 shows a distinct difference—only isolated adhesive transfer layer redeposition events are observed and there is more pronounced evidence of the abrasive action of wear particles. (it is noteworthy that multipass operation initially introduced by Bull and Rickerby [12] as a “model abrasive wear mode” turns out to be less abrasive than the well-known pin-on-disc mode). The stabilized, apparently similar stage III friction regimes in pin-on-disc and multipass operation correspond to the two two tribologically situations. the first case, first bodiesdifferent are separated by aInlayer of mobile wear particles localizing the wear activity; in the second case, low friction is the result of a practically stable oxide transfer film redeposited on the coating. This is not just an academic difference. When looking at the wear volumes of the undoped PVD coatings (Fig. 4), the considerably more severe wear in pin-ondisc operation is obvious. The practical tribological implication of this result should be underlined. Dry pin-on-disc friction with automatic recycling of wear debris is often inconsiderately applied to simulate general wear behaviour, even when wear particles are evacuated owing to environmental conditions (lubrication, gravitational force
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Tribology of Cr coatings
STAGE II
589
STAGE III
I
Fig. 6. SEM and energy-dispersive X-ray images of wear tracks and film transfer on the WC—Co balls during the three different stages offriction in multipass operation for standard d.c. electrolytic chromium coating (ECC).
etc.). This may lead to considerable difference in the basic wear mechanism to be simulated.
4. Conclusion
We have shown that multipass friction is a powerful tool to study third-body generation and related wear mechanisms. This method, coupled with on-line topographical tracking, allows for convenient simultaneous study of ongoing wear. Triboscopic representation of friction is another useful tool to obtain a valid diagnosis of the prevailing wear modes. Among the commercial hard chromium coating studied in this work, the carbon-doped PVD coating,
with an excellent compromise in high hardness and fracture toughness, is seen to have by far the best wear resistance in dry sliding under a normal load of 10 N on a cemented carbide ball 1.58 mm in diameter. Apparently, the presence of pre-existing cracks and/or friction fatigue-induced cracks does not have any detrimental effect on the wear life in dry friction. This might well be different in the case of lubricated operation and a corrosive environment.
Acknowledgements This work was accomplished as part of a joint contract with Electricité de France, PSA Peugeot—Citroen, and
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Tribology of Cr coatings
Renault. It is part of the dissertation [13] of the principal author. The assistance of J.P. Haeussler in the EPMA, and J. Poirson in the SEM and energy-dispersize X-ray work are gratefully appreciated.
—
I—
References 1
~ -- -
~j
-
I
‘~
—
-
*
.
—
Fig. 7. SEM image of the wear tracks in multipass and pin-on-disc operation in the stage III stabilized friction regime. Homogeneous film transfer in multipass over the entire contact area; local spalling of the transfer layer at the end of the track.
[1] J. Reby, G. Vasseur and B. Sutter, Proc. 12th Congres Mondial des Traitements de Surface, Vol. II, AITE, Paris, 1988, pp. 369—379. [2] T. Pearson and J.K. Dennis, Proc. 12th Congrés Mondial des Traitements de Surface, Vol. II, AITE, Paris, 1988, pp. 407—4 16. [3] J.L. Hyung. J. APP!. Phys., 57 (1) (1985) 4037—4039. [4] J.W. Patten, Thin Solid Films, 63 (1979) 121—129. [5] A. Darbeida, J. von Stebut, R. Rezakhanlou, M. Barthole, P. Belliard and L. Lelait, Proc. 20th Leeds—Lyon Symp. on Tribology, September 1993, in press. [6] A. Darbeida, J. von Stebut, M. Assoul, mt. J. Mach. Tool. Manuf., in press. [7] A. Darbeida, A. Saker, A. Billard and J. von Stebut, Surf. Coat. Technol., 60 (1993) 434—440. [8] J. von Stebut, A. Darbeida, A. Saker, A. Billard and R. Rezakhanlou, Surf. Coar. Technol., 60 (1993) 434—440. [9] F. Moussu and M. Nivoit, J. Sound Vibr., 165(1) (1993)149—163. [10] P.M. Ramsey, H.W. Chandler and T.F. Page, Surf. Coat. Technol., 43—44 (1990) 223. [11] H. Czichos, Tribology. A systems approach to the science and technology of friction, lubrication and wear, Tribol. Ser., 1, Elsevier, Amsterdam, 1978. [12] S.J. Bull and D.S. Rickerby, Thin Solid Films, 181 (1989) 545—553. [13]A. Darbeida, Misc au point d’une démarche expérimentale de caractérisation de la tenue mécanique des dépôts durs protecteurs. Mécanismes d’endommagement sous sollicitations avec contact en relation avec les caractéristiques microstructurales, Ph.D. Thesis, Institut National Polytechnique de Lorraine, Nancy, 1994. [14] AG. Evans and T.R. Wilshaw, Quasi-static solid particle damage in brittle solids—I: Observations, analysis and implication, Acta Metal!., 24 (1976) 939.