Wear, 95 (1984) 177 - 191 177

PROBAB~LIS~C MODEL FOR METAL-POLYMER

R. BASSANI, E. DI PASQUALE and C. VITAL1

FRI~ION

Dipartimento di Costruzioni Meccaniche e Nuclear& Universitd di Piss, 2-l-56100 Piss

@ta bf)

(Received August 4, 1983; accepted March 16,1984)

Summary

The contact and the friction in a polymer-mete pair are analysed. The metal surface is considered to be isotropic. The surface is described in terms of a random process model, assuming the distribution of the heights to be gaussian. The polymer is considered to be an elastic half-space. An expres- sion for the friction coefficient is obtained in terms of the r.m.s. deviation of the heights and of the correlation length of the profile. The results obtained form the basis for a subsequent experimental investigation of the influence of steel microtopography in the tribology of steel-polytetrafluoroethylene pairs.

1. Introduction

In recent years the influence of surface roughness in friction and wear phenomena has been investigated by several researchers (see in particular refs. 1 and 2).

In this connection it is necessary to characterize the surface mioro- topography by means of parameters which can be related to those pertinent to the tribological phenomenon (friction coefficient and wear factor).

Such an approach requires a method which provides as complete a description as possible of the microtopography of technical surfaces. This description must be related to the physics, the mechanics and the chemistry of the machining process and to the unctions behaviour of the surface.

The broad lines of research carried out are shown in the diagram of Fig. l(a), where the interactions between the various blocks may be seen. The model which will be used for the simulation is shown in Fig. l(b).

Starting from a mathematical model (threedimensional continuum) and from a tribological model, it was possible (as will be shown later) to obtain a number of similitude indices which characterize the behaviour of different friction pairs.

The tribological behaviour of metal-polymer pairs depends significantly on the radius of curvature of the peaks [ 1, 21; however, this parameter is

0 Elsevier ~quo~/~inted in The NetherI~ds

178

~

-siw&2tioo of 3 rrIWiqlcal phenomenon -

-expermenfsl analysis

(4

Fig. 1. (a) Block diagram of the research carried out;(b) block diagram for the simulation of a tribological system.

influenced to a great extent by the digitization process [3] and it is hard to obtain experimentally,

The geometrical-statistical analysis of isotropic gaussian surfaces used in this paper was introduced in refs. 4 and 5 and developed in ref. 6.

The contact and the friction between a metal and a thermoplastic polymer are analysed in the present paper with the aid of this model.

2. Simulation of the friction phenomenon

The interactions between a metallic surface and a thermoplastic polymer can be considered to be almost elastic. In addition, the deforma- tions are assumed to be concentrated in the polymer, whose elastic modulus is ten times or more less than that of the metal. It is also assumed that the tribological behaviour of a polymer-metal pair is influenced by the surface microtopography of the metal counterface alone.

Therefore a rigid rough surface indenting an elastic half-space is considered. The surfaces of various polymers, both thermoplastic and other- wise, can be idealized in this way because of the low rigidity and the well- known orientation phenomena of the surface structure in the direction of sliding [7].

179

In this paper we shall utilize the statistics of the peaks of an isotropic gaussian surface. The height distribution of a technical surface is often not gaussian [ 81, and the anisotropy of machined surfaces is more or less evident [ 91. For non-gaussian anisotropic surfaces, considerable analytical difficulties are encountered. It must be pointed out, however, that the assumptions of a gaussian surface and isotropy are verified to a good approximation [lo] for surfaces obtained by grinding, electrodischarge machining and honing, all of which are very interesting from a technical point of view.

On the basis of this assumption, the following quantities are obtained in ref. 6: the surface density Di of the peaks; the probability density pi(z) of the peak heights; the joint probability density pi@, k) of the heights and the curvatures of the peaks. These depend on the parameter CY, which gives the width of the power spectral density (PSD) of the profile and is defined by

mom4 (y=-

m22 mo, m2 and m4 being the non-zero moments of the PSD.

(1)

3. The mechanical model

The contact of each single asperity is assumed to be independent of the others. The contact between the rough surface and the elastic half-space is shown in Fig. 2. The contact between the tip of an asperity and the half- space is shown in Fig. 3(a). Because of the assumption of isotropy the tip is considered to be hemispherical.

The following dimensionless quantities are now defined: the dimension- less height is

i Z = 1/2

m0

the dimensionless compliance is

ii d=-

m0 l/2

Fig. 2. Contact between the rough surface (metal) and the elastic half-space (polymer).

180

I_ j \ ‘.., /

_:-’ ~_

.~~ lN~ ~, ,’ ‘,‘,/,,/,S ,I

<,,,,’ ,/ /,‘,,.,,/ ,, ;

,‘,/’ /,/ ,’ ,

, __,, L r ‘; 1 ,’ , I’ Lo 2z

li

(a) (b) Fig. 3. Force produced by the contact between an asperity and the elastic half-space.

the dimensionless curvature is

E k = _-

m4 l/2

The elementary (microscopic} normal load due to a single contact is 1111

4 mo314 (2 - cl}312 =- -

3 *rn41i4 k’/2

where

EP D=-_..--_

1 - VP2 The projection A, of the contact zone on the reference plane is

I/2 z-d =%G-

m4 k

(2)

(3)

in which a = (26F)1’2. The microscopic friction force is considered to result from the additive

contributions of the shearing (P,) and ploughing (P,) terms [ 121, i.e.

P, = P, + P, (4)

The shearing force is due to the adhesion processes between the polymer and the metal. If T is the interface strength, P, is given by

181

P, = TA,

*0 l/2 z-d

= 2nr- - *4

l/2 h

Although we consider here that the inter-facial strength is constant, it must be pointed out that this is only a rough approximation of the physical behaviour of a friction pair. In particular [ 131, r should depend on the con- tact pressure, thus becoming a function of r and z. The interfacial strength which appears in eqn. (2) is to be regarded as only an average in a proper range of values, depending on both the compliance and the surface micro- topography.

The ploughing force is due to the deformation processes produced in the polymer by the metal. For the sake of simplicity, the projection A, of the indented surface in the direction of sliding is considered to be roughly equal to the area of the inscribed isosceles triangle (Fig. 3(b)). This implies that, in dealing with ploughing phenomena, asperity tips are considered to be conical. We therefore obtain

= 2112

*4 l/4 kl/2

and

2 x 21’2 = 3x D*& - d)2

(6)

(7)

4. The model of surface microtopography

It is assumed that the autocorrelation function (ACF) (see Appendix A) is given by

R(x)= 02exp -i i 1 P

where u is the r.m.s. deviation and /.l* = 4.50 is the correlation length. If two samplings are /3* apart, they are statistically independent (R@*) = 0.01). Equation (8) is a good approximation of the ground surfaces commonly en- countered in actual friction pairs and of surfaces obtained by electrodis- charge machining [ 14 - 161.

The Fourier transform of the ACF, i.e. the PSD, is (see Appendix A)

S(w) = 02P a + (PN21 (9)

182

Such statistics imply a considerable fine structure, i.e. the contributions to the r.m.s. deviation from the high frequency components are not negligible. It is possible to apply the model if the components at a frequency higher than o, are cut off (Fig. 4).

The windowing of eqn. (9) causes eqn. (8) (Fig. 5) to beconvoluted with

W(x) = sin( 0,x)

XX (10)

Fig. 4. Wjndowing the PSD S(w) with a cut-off w, = ?r@.

Fig. 5. ACF R(x) and spatial window W(x).

Fig. 6. ACF resulting from the convolution of R(x) alid W(x) (obtained via a fast Fourier transform of the windowed PSD).

In order to maintain the shape of the ACF (Fig. 6), o, can be chosen such as to make the width of the principal peak of the spatial window (eqn. (lo)), i.e. the wavelength L = !&r/o, of the cut-off frequency, small com- pared with the correlation length.

183

A significant choice is suggested by the proposed mechanical model. The contacts between the asperities and the elastic half-space have been sup- posed to be independent of each other. In order to verify this assumption, it is convenient to indicate an estimate of the contact zone size that could be used as the cut-off wavelength.

If 2(x0, y,,) = z. > d, the expectation of the height at a point (xi, yi) a distance r from (x0, yo) is

JXZ(xi, ~i)lz(x~, yo) = 201 = 20 exp

Taking

(11)

r=Pln; 0

(12)

i.e. the distance at which the conditional expectation equals the compliance, we can compute the average in the range d < z < 00:

1 m F= -

@( 1 (27r)“2 d n : exp(--z2) dz

1 = a=o”Z,

@( 1 n i exp(--z2) dz

A plot of a uersus d is shown in Fig. 7. In particular, assuming d = 0.482, we obtain

1 a=-

71

71 w, = -

P

dmmsionless conplrnce d

Fig. 7. a vs. the dimensionless compliance d.

184

m 0 = 0.80~~

m2 = 1.20 5 P2

m4 = 5.39 5 P4

and

mom4 = 3.02 a=-

m22

(13)

(14)

(15)

(16)

The term (Y is now a constant. This means that the statistics of the peaks do not depend on the correlation length or on the r.m.s. deviation.

5. Statistical analysis

Friction forces can be expressed as sums of those generated in the single contacts.

The microscopic normal load depends on the two random variables z and k. In the contact conditions that were assumed above, the expected value is given [ 171 by integrating over the domain

d<.z<m

--oo<k<O

Thus we obtain

EtPn(z, k)) = j-h jkk kh(z, k) dh d --m

The density of contact spots is

n,=D, Pl@) dz

d

2 m4

= 67r X 31’2 G d s Plb) dz

(17)

from which the resultant (macroscopic) normal load F, is obtained by multiplying expressions (17) and (18) :

J’, = @(P,)

2 3/4 3/4 m

D m0 m4

= 9a x 31’2 m2

$ pi(z) da j%z j (’ ,,41”’ P&G k) dh d d -ce

(19)

185

For the macroscopic shearing force F, we similarly obtain

E{P,(z, It)) = jdz pp,k k)p,(z, k) dk (20)

and

d --m

1 l/2 m m4

l/2 -

=ii7TFx m2 d j-p&) d.z Idt j ~P,(G k) dk

d -0c (21)

and for the macroscopic ploughing force FP

Wpk k)) = jdz fpp(z, kh(z, k) dk d --m

(22)

and

F, = wW,)

p/2 mom4

D- f&J dz fk j+z -42~,k k) dk = 97rX31'2 m2 d

d -ca (23)

Combining eqns. (19), (21) and (23) with eqns. (13) - (15) we obtain

F,=B,a P

F, = B,

and

(25)

FP=BP f P2

(26)

where B,, B, and B, depend on the mechanical characteristics of the polymer, the contact conditions and the parameter (Y (constant).

Therefore the friction coefficients am as follows: the shearing friction coefficient is

4 P -- ” = B, o

the ploughing friction coefficient is

(27)

BP = pp=--

4, P (28)

the overall friction coefficient is

186

/J = l-b + Ilp (29)

In Fig. 8 the dependence of ps, pP and 1-1 on CJ, /_I and m2, m2 being the expected value of the squared slopes of the profile, is shown qualitatively. The values assumed for the various B in computing the curves in Fig. 8 are consistent with the experimental data.

Fig. 8. Relationship between the friction coefficients /A~, /_I~ and p and the parameters U, fl and m2.

6. Discussion

A mathematical model must be in agreement with previous experimen- tal studies. Various experiments [18] have already shown that the friction coefficient between steel and polytetrafluoroethylene behaves in a way similar to that obtained above (Fig. 9). Such behaviour can be explained on the basis of the assumption that adhesion controls the phenomenon at low r.m.s. deviations, although deformation processes prevail at high values of the r.m.s. deviation.

The model presented in this paper is able to simulate such behaviour, and it also shows the influence of the correlation length on the friction

\

: Ql 92 q5 1 I,5 4 10 17 50

Roughness - Width Cutoff (Rt )

Fig. 9. Friction coefficient us. roughness width cut-off (from ref. 17).

187

coefficient. It is obvious that highly correlated profiles will show higher adhesive effects.

Equations (2’7) and (28) for the friction coefficient were also obtained in terms of m2, so that the major role played by the asperity slopes in characterizing friction phenomena was pointed out. Such an effect has already been encountered [19, 201 in experimental conditions different from those assumed here.

The method adopted in selecting the cut-off ~velen~ is not only consistent with our assumptions about the physics of the problem but also in agreement with previous studies [21] which showed that some contact characteristics, such as the normal load, are not influenced by the fine struc- ture .

The effect of windowing the PSD is to neglect the influence of fine structure on friction. Justification of this neglect can be based on the assumption that, during the contact between an asperity and the elastic half- space, the fine structure does not affect the resultant of the generated normal and tangential loads.

The model proposed here can also be applied to simulate wear phenom- ena by assuming that the cont~butions of adhesive wear and abrasive wear can be added.

A diagram (Fig. 10) similar to that of Fig. 8 (showing wear rate uersus counterface r.m.s. roughness in a steel-polymer pair) was obtained in ref. 22.

,010 t

Fig. 10. Wear rate vs. r.m.s. roughness (from ref. 21).

In dealing with wear in polymer-metal pairs, several researchers [ 1,2] have suggested that fatigue mechanisms control the phenomenon. This implies that the microscopic stress state is hem much more important than in phenomena such as friction, where only the resultant of the contact forces between asperities can be assumed to be relevant.

Therefore the frequencies which affect a wear phenomenon are probably higher than those related to friction. The difference between the phenomena cannot be ~t~~h~ by the simplified theoretical analysis presented in this paper, where we have rest&ted ourselves to pointing out, qualitatively, the influence of certain parameters,

It might seem (Figs. 8 - 10) that a correlation existed between friction and wear, both depending on the same parameters. These parameters, how-

188

ever, must be measured over a different range of frequencies. An exper- imental analysis (Fig. 11) will indicate the frequencies related to any phenomenon, the parameters being obtained from filtered surfaces. Such an analysis is being carried out using a computer-~ded profilometer.

r----l Filter 3 r-l Filter b

Fig. 11. Block diagram for a subsequent experimental analysis.

7. Conclusions

Friction between a polymer (polytetrafluoroethylene) and a metal (steel) has been simulated, giving a ~~ationship between the friction coeffi- cient and two fundamental statistical parameters of the surface, the r,m.s. deviation and the correlation length. By taking the friction force as the sum of the shearing and ploughing components, the model shows the major role of adhesion at low r.m,s. deviations (high correlation) and of plough~g at high r.m.s. deviations (low correlation). The influence of surface slopes, which is already well known, is indicated by the mean of the similitude index o/p.

References

A. E. Holiander and J. K, Lancaster, An application of topographical analysis to the wear of polymers, Wear, 25 (1973) 155 - 170. S. R. Ainbinder, N. G. Andreeva and E. L. Tuinina, Counterbody roughness param- eters determining polyethylene wear resistance, Z’renie Iznoa Mash., 2 (I ) (1981) 12 . 21. D. J. Whitehouse and J. F. Archard, The properties of random surfaces of significance in their contact, Proc. R. Sot. London, 316 (1970) 97 - 121. M. S. Longuet-Higgins, The statistical analysis of a random moving surface, Philos. Trans. R. Sot. London, Ser. A, 249 (1957) 321 - 387. M, S. Longuet-Higgins, Statistical properties of an isotropic random surface, Philos. Trans. R. Sot. London, Ser. A, 250 (1957) 157 - 174. P. R. Nayak, Random process model of rough surfaces, J. Lubr. Technol., 93 (1971) 398 - 407. C. M. Pooley and D. Tabor, Friction and molecular structure: the behaviour of some thermoplastics, Proe. R. Sot. London, 329 (1972) 88 - 90.

189

8 D. J. Whitehouse, Beta functions for surface tipologie?, Ann. CZRP, 27 (1) (197%) 491- 49%.

9 J. Peters, P. Vanherck and M. Sastrodinoto, Assessment of surface tipologie analysis techniques, Ann. CZRP, 28 (2) (1979) 539 - 654.

10 T. R. Thomas and R. S. Sayles, Some problems in the tribology of rough surfaces, Z’ribol. Znt., 11 (3) (197%) 163 - 168.

11 S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1973.

12 H. Czichos, Tribology, Elsevier, Amsterdam, 1981. 13 B. J. Briscoe and D. Tabor, Shear properties of thin polymeric films, J. Adhes., 9

(197%) 145 - 155. 14 S. M. Pandit and M. C. Shaw, Characteristic shapes and wavelength decomposition of

surfaces in machining, Ann. CZRP, 30 (1) (1981) 487 - 492. 15 S. M. Pandit and K. P. Rajukar, Analysis of electrodischarge machining of cemented

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piece interaction in surface grinding, 9th North Am. Manufacturing Research Conf., 1981.

17 A. Papoulis, Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York, 1973.

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(1981) 235 - 246. 21 P. R. Nayak, Some aspects of surface roughness measurement, Wear, 26 (1973) 165 -

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Appendix A

Let us assume that the height of a surface over a reference (mean) plane is z(x, y), z being a random variable. Assuming that the function is sta- tionary and ergodic, the autocorrelation function (ACF) is defined as

1 L, L2 R(x,y) = lim -

ss k-*== 4.&J% -L _L WI, Y&(X + ~1, Y + ~1) &I dy,

L*-cW I 2

(AlI

The power spectral density (PSD) is

f%w, q) =~~W, Y) exp{--j&o, + YO,)} do dy -00

W)

190

When considering homogeneous isotropic surfaces the ACF depends on one variable only, and it can be computed along any profile:

The expression

S(o) = $ R(r) exp(-jwr) dr -co

(A4)

which is the Fourier transform of eqn. (A3), is the PSD of the profile. Its moment of order p will be

mp = _f

dS(w) do (As) --m

We thus obtain the following important relationships:

mo = E(2) (A6)

(As)

Appendix B : nomenclature

d

4

ED E

Fn FP FS k

m0,2,4

n,

PI@) P&, k)

dimensionless separation surface density of summits Young’s modulus of the polymer expected value resultant normal load per unit ama resultant ploughing force per unit area resultant shearing force per unit area dimensionless curvature moments of the power spectral density number of summit contacts per unit area probability density of the summit heights joint probability density of the heights and the curvatures of the peaks

Pf elementary friction force p* elementary normal load

PP elementary ploughing force

191

elementary shearing force distance between two points of the surface autocorrelation function power spectral density dimensionless height bandwidth parameter autocorrelation decay factor correlation length friction coefficient Poisson’s ratio of the polymer r.m.s. deviation of the heights wavenumber