Paper VI (ii) Elastic behaviour of coated rough surfaces

I57

Paper VI (ii)

Elastic behaviour of coated rough surfaces J.I. McCool

Numerical solutions developed by Chen and Engel for the elastic deformation of layered elastic half spaces are recast in the form of correction factors. These correction factors apply to the load and area at a circular microcontact computed using the Hertz equations when both contacting bodies are composed entirely of the coating material. The values thus corrected are the load and area applicable to the composite material consisting of the coating and the substrate. The correction factors depend on the ratio of the coating thickness to the uncorrected microcontact radius. The amount of the correction thus varies with the height of individual asperities. Using approximating functions for the correction factors and the Greenwood-Williamson microcontact model, a simulation scheme is outlined for determining the total microcontact load and area for coated surfaces. Results are obtained using representative surface characteristics and four values of the ratio of the elastic modulus of the coating to the elastic modulus of the substrate. The values of these ratios are E1/E2 = 1/10, 1/3, 3 and 10, and span the range from very soft to very hard coatings.

1

INTRODUCTION AND SUMMARY

Microcontact models relate surface finish to the stresses and deformation at the asperities which contact when rough surfaces mesh. They are thus useful for interpreting surface finish characteristics in terms which have engineering significance in a given application. Extant microcontact models differ in the assumptions they invoke about the form and stochastic characteristics of the asperity geometry. It was shown in [l], that the Greenwood-Williamson model [Z], one of the oldest and simplest microcontact models is nevertheless quite accurate when compared to models which relax many of its assumptions. The Greenwood-Williamson model postulates spherically capped asperities having a non-random sphere radius and Gaussian distributed height. The Hertz equations governing elastic contact of spheres and half spaces are used to compute the load, stress, and contact area on a deformed Greenwood-Williamson asperity. In this paper, results from the literature on the axisymmetric contact of layered bodies are used to develop a Greenwood-Williamson type microcontact model for coated surfaces. Contact problems for layered media have been considered by many authors using differing assumptions, boundary conditions, and methods of solution. Burmister [3], considered the stresses and deformations in layered media due to a prescribed surface stress distribution. Tu and Gazis [4], considered the problem of a

plate pressed between spheres of dissimilar elastic properties. Finite difference solutions have been developed by Kennedy and Ling [ 5 ] , and Chiu and Hartnett [6], for the elastic indentation of layered media. Gupta and Walowit [7], consider the contact of an elastic cylinder and a layered elastic solid. El-Sherbiney and Halling [8],consider the contact of an elastic sphere and a thin elastic layer attached to a rigid substrate. They employ two solutions from the Russian literature; one, based on the method of Tu and Gazis, is valid when the coating layer thickness "tIq is less than half the contact radius rrall. A second solution is valid for t/a > 1.7. Interpolation is used for the intermediate range (0.5 < t/a < 1.7) and approximate corrections are presented which allow for an elastic substrate. Halling in [9], suggests that the results in [8] can be used to compute an "effective" elastic modulus for application of the Greenwood-Williamson model to the contact of coated rough surfaces but does not elaborate on how this computation is carried out. Chen and Engel [lo], have developed a solution method for the axisymmetric contact of a rigid indenter and elastic multi-layered media. In their methodology, the interfacial pressure distribution is taken as a linear combination of a set of "base" functions with the coefficients determined so as to best match the surface displacement. A key advantage of their approach is the ease with which it generalizes to account for an elastic indenter. Section 2 of this paper is a synopsis of Chen and Engel's methodology.

158

In Section 3 i t is shown that Chen and Engel's results can be recast to express the load, contact radius, and area of a coated asperity contacting a coated elastic half plane with a prescribed interference, in terms of correction factors which apply to the ordinary Hertzian values of these quantities, computed as if both bodies were made entirely of the coating material. The correction factors are functions of the ratio of the coating thickness and the contact radius "a". The magnitude of the correction thus varies from asperity to asperity with the random asperity height. Numerical values of the correction factors were computed using results given in [lo], for a Poisson's ratio of 1/3 and four values of the ratio E /E of the elastic modulus of the coatinglad substrate. Approximat ion formulas were developed to express the corrections as functions of coating thickness and contact radius. Section 4 describes the logical basis for constructing a simulation model to compute the distribution of asperity load and area for coated Greenwood-Williamson surfaces. Section 5 describes the results of 160 runs made with the simulation model to explore the effect of coating thickness, coating modulus, separation and finish on the mean real pressure and mean area of microcontacts. It is found that as the coating thickness increases, the mean real asperity pressure approaches its asymptotic value more slowly than does mean asperity area. In addition it is found that soft coatings and smooth surfaces result in faster asymptotic convergence with coating thickness than hard coatings and rough surfaces. Section 6 explains how the simulation results may be extended to compute the contact density and joint stiffness for coated Greenwood-Williamson surfaces. 2

TEE CONTACT OF COATED AXISMMETRIC BODIES

The methodology developed by Chen and Engel [lo] for determining the load and deflection of contacting axisymmetric coated bodies, is adopted. They consider an elastic half space with Young's Modulus and Poisson's ratio of E2 and v2, coated with a layer of thickness t of a material having corresponding properties El and vl* An axially symmetrical rigid indenter whose profile shape is S(r) is pressed by an amount A into the coated half space causing contact to take place over a circular region of radius a. (cf. Figure 1) The displacement in the z direction is a function of the distance r from the contact center and the depth z, and is denoted uz(r,z). The surface displacement at a general position r within the contact, uz(r,O), satisfies the relation: uz(r,O)

=

A - S(r)

; 0 5 r 5 a

(1)

At the interface of the coating and substrate the stresses and displacements are constrained to be equal, giving the conditions:

(3)

(1) urz(r,t) (1)

=

(2) urz (r,t) (2)

uz z (r,t) = uzz(r,t)

(4

(5

where ur(r,z) is the radial displacement at coordinate position r,z and the superscripts refer to the coating and the substrate. uzz denotes the normal stress in the z direction and urz denotes the shear stress. For a specified shape S(r) and contact radius a, the objective is to find the pressure distribution q(r) and the penetration A. The resultant load is then: P

a 2 n J q(r)rdr

(6) 0 For a spherical punch of radius R, with R >> a, the surface displacement is well approximated by, 2 uz(r,O) = A - r /2R (7) =

Chen and Engel approximate the pressure distribution as a linear combination of the functions q.(r) as follows:

Where the pi are n unknown coefficients and the functions qi(r) are defined as: 2 i-1/2 q.(r) = (1 - (r/a) ] (9) With this representation the total load P given by Eq (6) becomes:

P

nE,

=

2 l+Vl

n

c

pi/( 1 + 2i) i=l

In approximating q(r) by Eq (8) the choice n = 5 was made in the numerical results given in [lo]. The surface displacement caused by the application of the pressure distribution [E1/2(1 + v)]qi(r) to the coated body is denoted Xi('). From Eq(7) and the principle of superposition, the displacement due to q(r) is thus , u (r,O)

.n

=

o - C piXi(r) a i=l

(11)

From Eqs (7) and (11) the error in the surface displacement due to the approximation is

159

&(r)

=

u (r,O)

uz (r,O)

-

=

2

A - r /2R

-

p*

=

2 3n(l - vl) n 3(1 - vl )PR ._.. - --8 b --E pi/(l+2i) 4Ela3 i=l (20)

and, The relative error, i.e., the error as a fraction of the maximum surface displacement A is: c(r)

a

d(r)/A

=

2 1" 1 - r /2RA - - C piXi(r) (13) a i=l

Introducing the parameter

3

CORRECTION FACTORS FOR AREA AND LOAD

When a rigid spherical indenter of radius R contacts hombgeneous elastic half plane, with a maximum penetration A, the load P and contact radius a are given by the ?allowing expressions due Hertz: (cf. Johnson [12])

a

2

@ = a /2RA

80

PH = 74j E,R1/2 A3/2

The mean square relative error over the contact area is expressible as: I

l aJ re2(r)dr 'a o

= -

where,

(16)

E'

=

E/(1 - vL)

aH

=

[RA]1 / 2

(23)

and,

The mean square error is thus a function of D (through the parameter @) and the n coefficients pi (i = 1, ...n). To find the values giving the smallest mean square relative error, one sets

is:

The area AH of the circular contact region

A H - n aH 2 = m n

(25)

This leads to the set of n + 1 linear equations 12 a 3 - 2 8 - - J r3 piwi(r)dr = 0 (17) a4 o i=l

The same expressions apply if both bodies are elastic if A is taken as the total approach of points on the two bodies which are remote from the region of contact and E' is taken to be Eft,given by

7O rw.(r) J

(1 - v 2 )/E + (1 - vI 2)/EI (26) P P wherein the subscripts p and I refer to the plane and the indenter respectively.

[l - @,'/a2

-

i=l

p.w.(r)]dr 1 1

= 0;

i=l,..n where,

1/E"

Using E q ( 2 0 ) to express P in terms of P* gives,

P In [ l o ] Chen and Engel solve for the values of w.(r) corresponding to the functions q.(r) for i'= 1, 2...5 and satisfying the bAundary conditions (Eqs. (2)-(5)), using a numerical procedure developed by Chen [ll]. They performed the computations for 13 choices of t/a and four values o f the ratio E /E2 with the assumption that v1 = v2 = 113. They then determined the values of pi and @ by numerically integrating the functions of r in Eqs. (17) and (la), and solving the resultant set of simultaneous linear equations for p and i B. They do not list the individual values of p. and 6. Instead they give for each value of tya and E /E the values of the quantities: 1

=

2

4/3 E'

=

From E q ( 2 1 ) 'a

a3 P*/R

(27)

*

RA/S

=

and thus using (27) in (26), p

=

4/3 El R1/2A3/2 (P* 1 6*3/2) 1

(29)

Comparing (22) and (29) leads to P = CP

H'

(30)

P is the load corresponding to deformation A i? the half space is made entirely of the coating material. Cp defined as,

cp =

P*/6

*3/2

(31)

160

may be regarded as a correction factor that accounts for the fact that the surface coating has finite thickness t. By analogy, if the indenter is also a coated, elastic body made of the same material and having the same coating thickness as the half plane, one simply uses

so that

- E t 1 aH3

PH

When t -+ 0 , i.e. the substrate material governs the coated contact behavior, the contact radius a should be the value given by

- (P/Er2)1/3 S

as From Eq(27) and (23): a

*

(RA/6 )

=

1/2 =

(33)

aH cR

But as we have seen, when t P

where CR

(1/6* ) 1/2

I

(34)

Like Cp, C may be viewed as a correction factor whereby ?he Hertz contact radius aH is adjusted to reflect the effect of the finite coating thickness and the substrate material. n

The area A is computed as naL so that from Eq(28) and (25): (35) where

cA

-=

=

cR

2

.

(t/a) (a/aH)

=

(t/a) CR

(37)

Using the tabled values in [lo] the factors CA, C and C were computed as functions of !/aH an5 are listed in Table 1. Figure 2 is a plot of C against t/aH for all 4 It is seen from values of the ratio E Fig. 2 that for smal11t,2C converges to E2/E1 signifying that the materig1 acts like the substrate when the coating thickness is small. Correspondingly, for large t, CP approaches unity indicating that the materlal behaves like the coating when the coating thickness is sufficiently large.

/E .

Figure 3 shows CR plotted against t/aH for E1/E2 = 1/3 and 3. CR is unity for small and large t values. For soft coatings CR exceeds unity at intermediate coating thicknesses. For hard coatings CR is less than unity indicating a smaller contact area than when the coating is thick. The behavior of CR at small t is explainable as follows: In general, from Eqs(22) and (24) aH and

PH

A1/2

- E'l

A3/2

EZ'/E1'PH

Thus CR

=

1 as t

-+

'P /E ' E ']'I3 H 1 2

=

aH

(42)

0.

A series of nonlinear curve fits were made for CA and Cp as functions of t/aH. These expressions are listed in Table 2. The maximum error is less than 7% for any of the fits. Where possible, the expressions were devised to give the required asymptotic behavior with small and large values of t discussed above. l4ICROCONTACT SIMULATION MODEL

A simulation model comparable to that described

notftd, Cheg and Engel give tabled values of P and 6 as a function of t/a. This is less useful than to have the correction factors expressed in terms of t/a To transform one uses Eq(33) and wriaes =

=

aS =[E2

4

* 1/6

As

t/aH

C P P H and therefore =

= 0,

in [13], was developed to analyze the microcontact behavior of contacting coated rough surfaces using the Greenwood-Williamson Model adapted to account for the presence of a coating on both surfaces. Under the Greenwood-Williamson Model the contact of two isotropic rough surfaces is treated as the contact of a single equivalent rough surface and a smooth plane. The asperities are regarded as spheres of radius R whose height x above the mean plane is a Gaussian distributed random variable with standard deviation as.

Figure 4 shows a number of such summits schematically for the case where the coated smooth plane is held at the distance d parallel to the summit area plane. The summits for which x > d become microcontacts and deform by the amount A = x - d

(43)

At a prescribed value of the separation d the objective is to determine the distribution of the asperity load, and the microcontact area and radius. The input to the model is the coating thickness t, the elastic modulus of the coating, El, the separation d, the appropriate ratio of coating to substrate modulus (E /E = 1/10, 1/3, 3, lo), the summit radius R a!id $he standard deviation oS of the summit height distribution. The simulation encompasses the following steps:

(39)

1. An asperity height value x is drawn from the Gaussian distribution of asperity heights

161

If A = x - d is negative, step 1 is repeated. If A is positive, P and aH are computed using Eqs(22) and (247.

Figure 5 is a plot of mean real pressure RP against coating thickness for each value E /E of the modulus ratio of coating to shbszrate using the smooth specimen (301) results. The plot is drawn for a dimensionless separation d/us = 1.0.

2.

3. t/a is computed and C and CP are evaluate! as a function of +/a using the fitted equations from Table 2 !or the appropriate ratio E1/EZ. 4. The asperity area and load are computed using A = CA

AH

(44)

and P = CP H'

(45)

The contact radius a is computed as a

=

(A/n) 1/2

(46)

5. Steps 1-4 are repeated as many times as desired and then, 6. The means and standard deviations of A, P and a are computed and histograms are compiled.

PARAMETRIC STUDIES

5

Roughness measurements were made on flat M50 steel specimens that were manufactured for use as test specimens in pin-on-disk tests of coated surfaces. The measurements comprised five repeated traces in three directions spaced at 4 5 O apart. The equivalent values of the RMS profile height Rq and slope Aq were determined as proposed by Sayles and Thomas [14]. Using these values the Greenwood-Williamson model parameters R and uS were computed by means of the spectral estimation approach proposed in [15] as implemented using the program RUFFIAN, described in [16]. In this determination the assumption was made that the mating surface is smooth. The results are listed in Table 3 for two specimens, Nos. 101 and 301 which span the roughness range to be explored in the planned pin-on-disk tests. Also shown, for reference, are the values of the estimated spectral exponent k (cf. (151) and the summit density DSUM.

A total of 4 x 4 x 5 x 2 = 160 simulation runs were made corresponding to all combinations of: 4 values of the modulus ratio, E1/E2 = 1/10) 1/3, 3 and 10 4 values of the coating thickness CT = 0, 0.5, 1.0 and 1.5 (w) 5 values of the dimensionless scaled separation d/os = 1, 1.5, 2, 2.5 and 3.0 2 roughness levels corresponding to specimens no. 101 and 301 For each run, 1000 microcontacts were simulated. The average value of the microcontact area and the microcontact load P as determined by the simulation were diviced to approximate the real contact pressure RP =P/A. RP is an important indicator of the severity of the surface loading.

All the curves on Fig. 8 have the same point of origin since at CT = 0, the material responds as the substrate steel. When the coating is sufficiently thick the substrate will cease to matter and the material will respond as if it were made entirely of the coating material. For the soft coatings a thickness of 1.5 pm appears to be as a large as necessary since the plots have leveled off. For the hard coatings, the asymptote appears to be beyond 1.5 pm. Figure 6 shows mean real pressure RP plotted for each coating thickness against dimensionless separation d/us for the 301 specimen with the hardest coating. Separation is seen to have a comparatively small effect on RP with the thinnest coatings but a larger effect with the thicker coatings. Figure 7 is comparable to Figure 6 but is drawn for the softest coating. The curves for coating thickness of 1 and 1.5 pin are quite close, confirming the observation that for soft coatings the behavior becomes asymptotic for thicknesses of 1-1.5 pm. Figure 8 shows average microcontact area A plotted for each coating modulus as a function of coating thickness. The plot is drawn for specimen 301 at a dimensionless separation of d/o = 3. With respect to contact area, unlike S RP, the asymptotic behavior with coating thickness sets in earlier. For the soft coatings there is negligible change in area beyond a coating thickness of 0.5 pm. The hard coatings have not quite leveled off at a thickness of 1.5 pm but the area is not changing greatly. Figure 9 shows RP plotted against the elastic modulus of the coating for the four coating thickness values. The plot is drawn for specimen No. 301 at a standardized separation of d/us = 1.0. When there is no coating, (CT = 0 ) ) the substrate determines the response so there is no variation in the plot with coating modulus. The plot suggests that linear interpolation could reasonably be used to compute the real pressure at values of the elastic modulus intermediate to the four values used in this study. Figure 10 is comparable to Figure 9 but shows the variation of microcontact area with modulus. As noted previously the differences in area due to coating thickness are small compared to the real pressure differences. Figure 11 compares the effect of the thickness of the hardest coating (E1/E2 = 10) on RP at d/us = 1.0, for the two specimens. RP is seen to increase at a faster rate with coating thickness for the rougher specimen (No. 101). For the soft coating, (not shown) RP also decreases more rapidly with coating thickness for the rougher than for the smoother

162

surface. At a fixed d/os, microcontact area is larger for the rougher surface but varies with coating thickness in approximately the same way. 6

DETERMINING TEE SEPARATION D/us

In elastohydrodynamic lubricated contact the separation may reasonably be talcen to be determined by the film thickness h, which is interpreted as the distance between the roughness mean planes of the contacting bodies. The relation between d/us and h involves the roughness spectrum and is given in [15]. For dry contact d/u may be determined as the separation at which '?he mean load Q over the nominal area of contact .A is equal to the load externally applied to the contact. To use the simulation results to make this determination one multiplies the mean asperity load P at the separation d/us by the expected number of microcontacts. The expected number of microcontacts is the expected number of summits per unit area DSUM, multiplied by the nominal area .A and by the probability that a summit is a microcontact. This probability is:

P[summit=microcontact] P[summit height x > d]

= =

l-+(d/os)

=

P

DSUM

*

[l-+(d/os)]

*

.A

=

DSUM[l-+(d/us)]

P

1. 2.

3.

4.

5.

7.

(48)

The load per unit nominal area, i.e. the nominal pressure, is thus: Q /Ao

References

6.

The mean load supported over the nominal contact area is

8.

(49) 9.

Figure 12 is a plot of the logarithm (base 10) of Q/AO against the dimensionless separation d/oS for the hard coating applied at a thickness of 1.5 pm. For a given applied load per unit nominal area, the dimensionless separation is seen to be smaller for the smoother surface (Spec. NO. 301) than for the rougher surface (Spec. No. 101). As an example, a load of Q2= 10 N applied over an area of .A = 1 mm giving log 10(Q/Ao) = 1.0, results in d/u cz 2.2 for specimen No. 301 and d/os = 2.7 for specimen no. 101. The absolute mean separation is computed from the individual us values in Table 2:

d301

=

2.2 x 0.052

=

0.114 pm

dlOl

=

2.7 x 0.254

=

0.686 pm

10.

11. 12. 13.

14. 15.

Repeating this calculation for different Q values will yield the Q vs. d relationship characterizing the stiffness of coated joints. Figure 13 shows the effect of coating type on the load/ separation relationship for specimen No. 301 with a 1.5 pm coating thickness.

ACKNOWLEDGMENT

This investigation was supported by DOE-ECUT under DOE Contract No.DE-AC02-87CE90001.AOOO. It was performed within the Tribology Program managed by Mr. David Mello. Dr. Fred Nichols, manager of the Tribology Project, was the technical monitor. This support is gratefully acknowledged.

(47)

where + is the standard normal cumulative distribution function.

Q

7

16.

McCool, J. I., "Comparison of Models for the Contact of Rough Surfaces", Wear, 107, pp. 37-60, (1986). Greenwood, J., and Williamson, J., "Contact of Nominally Flat Surfaces", Proc. R. Soc. London, Series A, 295, pp. 300-319, (1966). Burmister, D. M., "The General Theory of Stresses and Displacements in Layered Systems", Jnl. Appl. Physics, Vol. 16, pp. 89-94, Feb. (1945). Tu, Y., and Gazis, D., "The Contact Problem of a Plate Pressed Between Two Spheres", ASME Trans., Jnl. Appl. Mech., pp. 659-666, Dec. (1964). Kennedy, F., and Ling, F., "Elasto-Plastic Indentation of a Layered Medium", ASME Trans., Jnl. Engr. Matls. and Tech., pp. 97-103, April (1974). Chiu, Y., and Hartnett, M., "A Numerical Solution for Layered Solid Contact Problems with Application to Bearings", ASME Trans., Jnl. Lub. Tech., Vol. 105, pp. 585-590, Oct. (1983). Gupta, P., and Walowit, J., "Contact Stress Between an Elastic Cylinder and a Layered Elastic Solid", ASME Trans., Jnl. of Lub. Tech., pp. 250-257, April (1974). El-Sherbiney, M., and Halling, J., "The Hertzian Contact of Surfaces Covered with Metallic Films", Wear, Vol. 40, pp. 325-337, (1976). Halling, J., "The Tribology of Surface Coatings, Particularly Ceramics", Proc. Inst. Mech. Engrs., Vol. 200, No. C1, pp. 31-40, (1986). Chen, W., and Engel, P., "Impact and Contact Stress Analysis in Multilayer Media", Int. J. Solids Structures, Vol. 8, pp. 1257-1281, (1972). W. T. Chen, "Computation of Stresses and Displacements in Layered Media", Int. J. Engng Sci., 9, pp. 775-800 (1971). Johnson, K. L., Contact Mechanics, Cambridge University Press, 1985. McCool, J. I., and John J., "Flash Temperature on the Asperity Scale and Scuffing", ASME Transactions, Jnl. of Tribology, Vol. 110, No. 4, pp. 659-663, October (1988). Sayles, R. S. and Thomas T. R., "Thermal Conductance of a Rough Elastic Contact", Appl. Energy, 1, pp. 249-267 (1976). McCool, J. I., "Relating Profile Instrument Measurements to the Functional Performance of Rough Surfaces", ASME Transactions, Journal of Tribology, Vol. 9, No. 2, pp. 264-270, (April 1987). McCool, J. I., "Predicting the Upper Percentiles of Flash Temperature at Microcontacts", Surface Topography, Vol. 1, No. 3, pp. 343-355, (September, 1988).

163

El/E2=1/3 t/aH CA CR CP -----------_--___-____ --------

0 0.117476 0.180841 0.244851 0.373081 0.499766 0.745414 1.206660 1.745012 2.262380 4.277158 8.275616 16.27156

1 1.380072 1.453488 1.498801 1.546551 1.561037 1.543448 1.456028 1.353363 1.279591 1.143380 1.070091 1.034233

1.17, 765 1.20! 507 1.224255 1.243604 1.249414 1.242356 1.206660 1.163341 1.131190 1.06928" 1.034452 1.016973

-t/aH -------

CA

CR

0 0.095695 0.14 1127 0.185615 0.271707 0.354663 0.515102 0.835017 1.259260 1.709714 3.625295 7.582342 15.60697

1 0.915751 0.885191 0.861326 0.820277 0.786164 0.737028 0.697253 0.704771 0.730780 0.821423 0.898:11 0.951475

1 0.956949 0.949846 0.923077 0.903692 0.886659 0.858504 0.835017 0.839507 0.854857 0.906324 0.947793 0.975436

I

0 0.106600 0.162593 0.219476 0.335075 0.451610 0.711218 1.129817 1.661648 2.177131 4.192909 8.194468 16.19304

10 7.278482 6.368164 5.652272 4.609759 3.895887 3.005700 2.162781 1.724623 1.515346 1.2 31162 1.107954 1.052838

0.3333 0.345010 0.350038 0.355083 0.365739 0.377457 0.403752 0.462048 0.532376 0.592040 0.738!70 0.850476 0.928010

TABLE 1

-

0.091762 0.132614 0.171499 0.243524 0.309261 0.428878 0.656193 0.970594 1.329146 3.012376 6.768974 14.60593

1 1.136364 1.174950 1.204239 1.247505 1.274697 1.405086 1.276487 1.227144 1.184975 1.098780 1.049208 1.024275

1 1.066004 1.083951 1.097378 1.116918 1.129025 1.185363 1.129817 1.107766 1.088565 1.048227 1.024308 1.012065

CA

CR

1 0.842034 0.781616 0.735294 0.658935 0.597764 0.510934 0.430589 0.418690 0.441657 0.567151 0.715922 0.833333

1 0.917624 0.884091 0.857493 0.811748 0.773152 0.714796 0.656193 0.647063 0.664573 0.75?0L;4 0.846122 0.912871

Correctign Factors CA, CR and C

3 2.773669 2.670842 2.573500 2.395744 2.238909 2.241148 1.659825 1.446389 1.330942 1 7573 1.075357 1.037670

___---------_-_ CP

vs.

0.1 0.106088 0.108283 0.110591 0.115750 0.121595 0.134947 0.166450 0.211642 0.258615 0.418917 0.604243 0.760498

t/aH

Modulus Ratio E1/E2 0.10

IF T<.7454 TEEN CP=1 + 9 *EXP(-T/.4274)-.769) ELSE CP=1.1226+4.264*EXP(-T/.8944))

IF T<.49997 THEN CA=LEXP(-T/.7518)'.369) 0.33

IF T<.7112 THEN CA=Z-EXP(-(T/2.095)'.66)

ELSE CA=lr.561037*EXP(-(T-.49997)/2.8817) ELSE CA=1+.405*EXP(-(T-.7112)/2.161)~.6021)

IF T<.7112 THEN CP=ltZ*EXP(-(T/1.72)'.6855) 3.0

IF T<.835 THEN CAE.6641+.3359*EXP( -(T/. 3475)'. 9817) ELSE CA-1-. 411*EXP(-(T/5)'. 7296) IF T<.835 TEEN CP-1-.667*EXP(-(T/2.818)'1.274)

10.0

ELSE CP=1.095+2.737*EXP(-(T/.7881))

ELSE CP=1-1.179*EXP(-(T/l.472)~.4412)

IF T<.9706 TEEN CA=.4002+.5998*EXP(-(T/.2751)'1.121)

ELSE CA=1-.8236*EXP(-(T/6,341)*.5805)

IF T<.9706 THEN CP=l-.9*EXP(-(T/3.962)*1.436) ELSE CP=1-1.172*EXP(-(T/5.906)'.5179)

TABLE 2.

BnS height R

Fitted Equation: for Correction Factors CA and C (Tmt/a8)

BnS slope

Spectral exponent

Asperity Radius R

h S i ty

BnS height

(non-dim)

0

Q!!.F2

0

k

Summit

DSW

U

Specimfm No.

Q m J -

Aq (radius)

101

0.256

0.048

1.97

9.42

6.6184

0.254

301

0.0~3

0.0122

1.85

35.8

7.10E4

0.052

Table 3

-

Measured and Computed Roughness Characteristics far Tvo Test Specimens

164

10000 BOO0 “n

2 6000

:

-

4000

d

Flgure 1.0 Rlgld Indenter and Layered Substrate

3

2000 0 0 I

10.000

B

A LOAO FACTOR

CP VS.

0.2

0.4

NORMALIZED COATING THICKNESS

0.6

1.2

1.0

0.6

1.6

1.4

Coating Thickness (u)

t,/E2

- 0.1

Figure 6.

1.000

U e m R e d Prellsurs

Vm.

Dimenmionlase Separation Spaeiman No. 301. EllEz

-

10.

10000

o’i$o.0!30

3.433

i 3 . k

iO.bO0

6 $57

16.657

20

0

T/AH

A.

3 Figure 2.

Load Correction Factor

YE.

P

Dimensionless Coating Thickness.

2000 0 0.0

I - = - f - I

0.5

1.0

1.5

I

I

2.5

2.0

7

: 5

3.0

Dimensionleas Separation. dla 8.000

RADIUS FACTOR CR VS. NORMALIZE0 COATING THICKNESS

-8-

4 c 7 - 0

x

CT-0.5

CT.

+

1.0

CT.

l.5

1.700

i .40D U EIIEZ

1.100

-

113

0.5004

Figure 3.

~ e a lpressure vm. D h o s i o n l o s s Separation Specimen

Figure 7.

0 . BOO

3.333

6.667

Radius Correction Factor

i0.06U T/AH YE.

i3.333

16.667

20

301.

NO.

El/e,

I0

Dimensionless Coating Thickness.

0.0

I

0.5

I

1.0

I

1.5

Dimmsiooleas Separation.

Flgufe4 - Coated Greenwood-WllllamsonAsperltler Contacting a Smooth Plane

I

I

2.5

2.0

I

3.0

3.5

dies

+

C T - I.5

-

1/10

165

Figure 8.

Hean Contact Art* VB. Costina Thickness Specimen No. 301. dlna

-

Uam b.1

11.

?lgur.

3.

Premmura 0.. C a t i n 8 ?hickner. E,/E,

-

10, d h l

400001

-

1.0.

I

30000 20000 10000

0.51

0.oi 0.0

4

I

0.2

I

,

0.4

I

I

, I (

0.6

0.8

coating Thickness

I

I

1.0

I

I

1.2

I

I

1.4

N !

1. 6

0

0.4

0.2

0.0

0.6

1.0

0.6

1.2

1.4

b . )

ioooo 1 fi

, I

I

8000

I

v

I6000 t

2

4000

1 2000

0.0

I

I

1.0

0.5

I

1.5

S m d A t d i l l d S.p~.~ie~

0

0

500000

iOOO000

1KIOOOO

2000000

2500000

Coating Uodulua (Nlmm2)

+el.

-4-rr.0

4

n

-

o

,.I

pI,.*_

"0.

-

to,

4 ss.z_

L..

-

11

I

2.0

I

2.5 ,dk,

I

3.0

I

3.5

1.6