The role of insulating surface films in frictionally excited thermoelastic instabilities

Wear, 24 (1973) 189-198 0 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

189

THE ROLE OF INSULATING SURFACE FILMS IN FRICTIONALLY EXCITED THERMOELASTIC INSTABILITIES

R. A. BURTON ~or~~wes~e~ Uniwrsity, Evanston, III. 60201 (U.S.A.) (Received December 21, 1972)

SUMMARY

The growth of pressure disturbances in a zone of sliding contact is investigated for bodies which carry theimally insulating surface films. It is shown that in some circumstances thin films totally alter the stability behavior, giving rise to an instability which would otherwise be absent. Solutions are offered for the cases where (a) the thermal boundary layer is thinner than the surface film and (b) where the film is thin relative to thermal boundary layer. A suggested interpolation is proposed between these two bracketing cases. Films of interest might include natural oxidation products, dry lubricants, hard coatings, decorative coatings.

INTRODUCTION

In ref. 1 a seal-like configuration has been investigated where sliding occurs parallel to the line of contact between the edges of two plates or thin tubes. It has been shown that, even for nominally uniform contact, pressure ~sturban~s will sometimes grow and lead to locally increased heating and stress. It has also been shown that two limiting cases for such contacts lend themselves to simple analysis leading to direct criteria for instability. These two cases are (1) a given material sliding on its own kind, and (2) a good thermal conductor sliding on a good thermal insulator. In the first case it is shown that, for a range of selected n terials, stability is assured for all reasonable values of the friction coefficient. In the second case it is shown that there will always be an instability if speed is raised sufficiently high. In addition, computer solutions are carried out for the more complex case where conductivities differ but are not widely separated, and it is shown that the limiting solution for insulator on conductor is not an unrealistic first approximation even for the less extreme differences in properties. The limiting solution becomes doubly interesting because of a singular feature of this kind of contact. When a thermal insulator slides on a conductor under conditions of instability, the disturbance in that contact zone tends to remain

190

R. A. BURTON

stationary in theconductor and, therefore, to be moving at the sliding velocity relative to the insulator. This leads to deep thermal penetration into the conductor and corresponding thermal expansion; it leads to shallow penetration into the insulator which plays the role only of “frictional heater” of the conductor. Because of this shallow penetration the insulating body does not have to be made wholly of insulating material in order to play its role. The body may even be made of a highly conducting substrate with a thin insulating film and its effect will be that of an insulator so long as the film thickness i is near the effective thickness 6 of the thermal boundary layer. For high sliding speeds and highly insulating film material this thickness /z need only be 0.001 in. or less. On the other hand if the disturbance wave moves slowly relative to a surface covered by thin film, the role of the film becomes relatively unimportant. One may ask, then, how it comes about that a disturbance will arise at an appropriate speed relative to a ~lrn-covered surface so as to cause it to act as a solid insulator. In answer, stability analysis of such systems assumes that random disturbances are continually supplied varying in velocity and spatial distribution. Eventually a component of this general disturbance will appear having the characteristics of a growing disturbance (if, indeed, a growing type of disturbance can exist), and this small component will then grow. Actually the analysis discussed here is concerned only with a zero growth rate dist~ban~e, hence one which neither grows or decays. Within the frame of reference discussed such a disturbance lies on the boundary between a decaying and a growing disturbance and hence may be said to lie on the “threshold of instability”. In a closely related system the full dynamics of such disturbances have been explored in considerable detai12, hence one may have confidence in this interpretation and need not complicate the present analysis by enlarging it beyond seeking the disturbance threshold. In terms of the above discussion a second question may be asked as to how two bodies, each with insulating films, behave. In answer we note that if the disturbance moves slowly relative to one body and rapidly relative to the other, the net effect is that the film will have little effect on the body where the wave is moving slowly and the role of the body as a good conductor is unaffected. If the film is thick enough on the other body the fast moving thermal wave will not penetrate it, and it will act as an insulator. Therefore all of the circumstances are present for one body to act as insulator and the other to act as conductor. It is arbitrary which body performs as which, and, indeed, if no major nonlinearities enter the interactions, each body can simultaneously play each role relative to separate disturbances. That is, each can have a relatively stationary growing disturbance on it while acting as insulator relative to the disturbance on the other body. In the paragraphs which follow these concepts will be dealt with quantitatively. TEMPERATURE

DISTURBANCES

It has been shown’ that a solution to Fourier’s heat flow equation can be written as T = lToJe-by sin(wx-wct+ay) where

(1)

EFFECT OF SURFACE FILMS IN THERMOELASTIC

INSTABILITY

191

This corresponds to a temperature wave moving in the direction x along the boundary y=O of a semi-infinite body, when the temperature deep in the material is zero (i.e. ambient). This solution may be thought of as a sinusoidal disturbance wave or as one harmonic component of a more complex wave. The disturbance decays exponentially with depth below the surface and the thickness of this thermal boundary layer is effectively S = l/b. For cases where c/k is quite large relative to w 2k+ 6 1 = bA ( oc )

(2)

Note that w is a measure of spatial periodicity of the disturbance and is typically of the order of unity. For example, for a 1 in. diameter seal, and a disturbance completing a single cycle on its perimeter, U)X=R and x = rcD, hence o = 1. If the seal is to be supported on 3 lobes of the disturbance w = 3 for the 1 in. seal. In any event w is of the order of unity. The quantity K is usually of the order of IO-‘, and c is usually not of interest unless it is, say, 10 in& or greater. Consequently eqn. (2) provides a realistic estimate of 6. Figure 1 illustrates typical values of 6 versus c for Sic and glass two representative insulating materials taken from Table I of ref. 1. For film thickness of 0.01 in. the entire thermal layer may be thought of as lying in the film for quite modest disturbance velocities. If the film is 0.001 in. quite high disturbance velocities would be called for to reduce S to this value; although 1000 in/s is not outside the range sliding speeds encountered in machinery. Holding in mind that a disturbance may contain higher periodicity components, it is possible for such a component to be associated with a thermal layer thinner than that 0.001 in. film even at low velocities. Consideration of thermal layer thickness alone provides useful insight, but further discussion requires an estimation of the influence of the surface film on heat transfer when the film is thin relative to 6 as calculated in eqn. (2), but may still have a partially insulating effect. THIN FILM SOLUTION

When the film is thin relative to 6 its energy storage is negligible and it becomes simply a thermal resistance. For a given heat flow per unit of surface area q, the temperature drop across the film would be

where K, is the conductivity of the film. If a sinuoidal temperature distribution travels on the surface of the film, a corresponding but weakened distribution will travel along the film-substrate interface. The temperature solution for the sub-

192

R. A. BURTON

strate may still be described by eqn. (1) where IT’] is replaced by IZJ the modulus ofthe interface temperature wave. For this distribution the heat flux into the substrate is = -KIT,1 [q COS(OX--~X~)-~~ sin(ox-act)]

(4)

Restricting consideration to high sliding speed a,+bS+1/6,, where S, is calculated according to eqn. (2) using substrate properties. Using eqns. (3) and (4) one may write IA&l _=-J2

ITI

W

(5)

W,

but since

letting (C,,/C,, pr)* = cp, where rp is typically near unity, or

(6) The modulus of the entire temperature drop 1AT’,,,] is, taking account of the 7r/4phase angle between ATr and T,,

IGetI lOI -=lTplJT ITI

1 + 1= 1+ 5 0K,

* J

sq f

(7)

A useful chart in Fig. 4 of ref. 1 shows the effect that the hypothetically reduced conductivity of one sliding body has on the sliding speed at the threshold of instability. This chart may be used to obtain insight into the present problem

Thermal

Boundary

Layer

Thickness,

8 (inches)

Fig. 1. The effect of the velocity of a disturbance insulating materials.

on thermal boundary layer penetration

in two

EFFECT OF SURFACE

FILMS IN THERMOELASTIC

193

INSTABILITY

by finding an equivalent K,/K that would produce the same diminution of heat flow as is caused by the film under consideration. To draw an analogy to reduced conductivity we again refer to eqn. (8), of ref. 1 and retain the assumption of high sliding speed, in which case

1 (y)”1

(8)

141 = KT

Since K also appears in the diffusivity term k=K/c,p, we may say that, for a hypothetical substance where K varies but specific heat and density remain fixed,

(9)

141CCV’IK+ Hence for a hypothetical

reduced K which will be called Kh, and for a fixed q 00)

An analogy can now be drawn with the film by letting

IG,,,III T,I = I U/l T,I or, (11)

Table I shows typical effects of thin films on the apparent conductivity of an aluminum substrate (see ref. 1 Table I for properties). Note that for the sliding speeds chosen thermal penetration 6, extends from 55 microinches to 0.1 in., and the effect of the 0.001 in. film on the aluminum varies from moderate to an almost complete blockage of heat flow. THICK FILM BEHAVIOR

Since the assumption

of zero energy storage breaks down unless A/&< 1,

TABLE I EFFECTOF THIN INSULATING FILMS ON APPARENT CONDUCTIVITY SUBSTRATE WHERE o = 1,1 =O.OOl in., cp= 1 Disturbance velocity, q (in./s)

w,

Sic

Glass

70,000 20,000 7,000 2,000 700 200 70 20 7 2

3500 1000 350 100 35 10 3.5 1 0.35 0.1

KhIK

Sic 0.000055 0.0003 1 0.00055 0.0031 0.0055 0.01 0.017 0.03 0.055 0.01

1.82 1.0 0.588 0.323 0.182 0.10 0.0588 0.0323 0.0182 0.01

0.018 0.049 0.106 0.218 0.372 0.55 0.68 0.805 0.88 0.915

Glass 0.0011 0.0035 0.0091 0.026 tV0.0655 0.148 0.266 0.435 0.60 0.74

OF AN ALUMINUM

R. A. BURTON

194

the thin film equation, eqn. (ll), should not be expected to be valid beyond, say, &G,=O.l. To provide a solution valid at the other extreme end of the performance range where d/&r> 1 it is possible to solve the heat flow equation for a film with a sinusoidal temperature wave on one surface and a constant temperature on the other surface. This would correspond to a film in contact with a perfect conductor and should not be far from a film on a real, “good” conductor for n/s,= 1. Under the assumptions previously applied (rapidly moving wave, etc.} the temperature equation becomes T = AemaYsin(wx-- ct + ay) - %eQYsin (ox - wet - ay + Y) where Y is a phase shift to be determined by boundary conditions. Taking T = 0 at ay=&‘lS, =n, and letting 6=(0xwet) O=~e-nsin(~+~)-%e~sin(~-~+~)

(12)

If !P=22n, then A eWzn= % The temperature

(13)

gradient, which determines heat flow, becomes

dT = aAe~"Y~-sin(8+uy)+c0s(8~uy)f aY

-u%e”Y[sin(8-ay+cp)-cos(8-ay+cp)]

(14

At y=O dT &)=

(15)

or substituting for b aT -= -Au,/2[sin/B-~)-e~2”sin(l)-~+ ay If(ee*“sin Y)‘-=$1

IdyI 3T

=

IhJ21(1

+e-‘” cos cp)

+]

(16)

(17)

y=o

Also for e- 2n small ITI = ~A~(l-e-2”~~s2n)

(18)

or

ITI

IAl= (1-ee2”

cos 2n)

Returning to eqn. (17) and remembering a = l/&

Since ~-CD for a slab it follows that

(19)

EFFECT OF SURFACE

FILMS IN THERMOELASTIC

195

INSTABILITY

[l +eT2”‘* cos A./i&] [l -e-2z’+ cos &4,]

Id lqsinhl

(21)

As before, since an equivalent heat reduction from lowering conductivity

-=I41

-Kh *

(>

is

(22)

hslahl Kf

To convert this into terms of the apparent conductivi~ KS we write

of the conductive substrate

(23) It follows, then, that KS K, = K, Kh

Table it is expected On the other where 1/S, is

1

1+ ezA’+cos (&/SC) - e- 21/drcos (&G,)

II shows typical results for the thick film calculations. For large J./a, to be high since it is based upon zero resistance of the substrate. hand the purpose of the calculation is to establish an “anchor” valid large or at least near unity.

TABLE II THICK FILM CALCULATION FOR APPARENT SUBSTRATE, WHERE w= 1,1=0.001, AND p= 1

CONDUCTIVITY

IXsrurbance velocity,

Sic

Glass

Glass

&m

2oo:ooo 10;OOO 7,000 2,000

ALUMINIUM

Kh/K,

Tcmoo 70,000 20,000

AN

c

(h/s) SIC

OF

3,500 1,000 350 100

EFFECT OF INSULATING

00.00017

;8

1

0.08 1

0.0039

0.0003 1 0.00055 0.001 0.0017 0.003 1

3.23 1.82 1.00 0.588 0.323

1.05 0.9 0.8 1.3 3.6

0.09 0.066 0.052 0.135 1.05

0.0043 0.0032 0.0025 0.0065 0.05 1

FILMS ON THE THRESHOLD

OF INSTABILITY

It is possible to take the hypothetical conductivity and use this in conjunction with the stability plot of ref. 1 (Figs. 2 and 3). Typical results are shown in Fig. 2, for the 0.001 in. fdm thickness of SC on alu~num. For plotting, the thick film solution is shown as a solid line up to h/6 =0.588, and the thin film solution is shown as a solid line to ~,G=O.l. An extension of the thin film solution is offered as a bridge between solutions, and must not be far from the true solution.

196

R. A. BURTON

0

I

2

3

4

5

6

7

B

9

IO

K / K for Alummum

Fig. 2. Effect of 0.001 in. thick film of SE on Al. Intercepts with “film solution” friction coefficient give critical disturbance speed in the lesser conductive body. Fig. 3. Effect of a 0.001 in. glass film on the critical

disturbance

velocity

and

curves

for

in aluminum.

In Fig. 3 a similar plot is shown for an 0.001 in. film of glass on aluminum. Here the thick film solution is restricted to a narrow zone at the left and the practicality of using the extended thin film solution is again apparent. Of special interest are the intercepts of the equivalent conductivity curves for the films and the stability threshold curves for the various friction coefficients and conductivity ratios. The intersection of a pair of curves would correspond to the threshold of instability for the given film and the given friction coefficient, and it would be associated with a characteristic velocity cl, the rate of movement of the disturbance on the insulating body. Figures 4 and 5 show the results of plotting the disturbance velocity c1 against friction coefficient corresponding to the intercepts. Additional curves are shown for different film thickness and were calculated in the same manner. In each case it is seen that increased thickness or lowered conductivity leads to lowered stability threshold speeds. Even for p = 0.2 or less, as with boundary lubrication or solid film lubrication, a 0.003 in. film of the SIC shows a threshold speed of 200 in+. When one considers that typical solid lubricant films have thermal properties similar to Sic it is apparent that the lubricant film itself can lower the threshold of instability even while reducing friction. In the case of the glass film the most interesting observation is that very, very thin films can bring the stability threshold into the ordinary operating range. Since glass-like films form naturally on some substances, these observations indicate that

EFFECT

OF SURFACE

FILMS

IN THERMOELASTIC

197

INSTABILITY

t

\

\

\ ; 1.0

20

20

Fig. 4. Effect of thin glass films on Al as to critical Fig. 5. Effect of thin Sic films on Al as to critical

K,/K

for

Cost

disturbance

Coefflaent,

Friction

Coefficient,

p

velocity. velocity.

Iron

Fig. 6. Effect of 0.001 in. insulating Fig. 7. Comparison

disturbance

Fmtmn

films on cast iron.

of the effect of 0.001 in. films on iron and aluminum.

p

198

.R. A. BURTON

their effects should not be ignored even when the films are very thin. Additional comparisons involving iron substrates are shown in Figs. 6 and 7. Turning now to the question of films lying on both contacting bodies we note in ref. 1 that for ~=0.2, the ratio of cz/cl ~0.05, where this is the wave velocity in the conductive body in ratio to wave velocity in the insulating body. Let us now refer back to Fig. 2 and assume that disturbance speed is approximately 1000 in./s on one body, then for a 0.001 in. SIC film the hypothetical conductivity would be about 0.35 K,. If the disturbance is moving at an estimated speed of 50 in./s in the second body the apparent conductivity of this body is 0.73 K,. Hence the body playing the role of insulator has less than half the apparent conductivity of the body playing the role of “conductor”. Because of this, even though both bodies are physically identical, they may behave as insulator or conductor at appropriate disturbance needs. It is obviously not reliable to extend this analysis further for quantitative estimates when cI/cl is larger; instead a complete analysis for film covered bodies is being developed. The results above, however show that such an analysis should be of considerable value in making quantitative these newly discovered stability phenomena. CONCLUSION

The above derivations and plots show how an insulating film can drastically alter the stability threshold for frictionally excited thermoelastic instabilities. The derivations have drawn upon the effect of the film in producing an equivalent reduction in conductivity of the sliding material and serve as an instructive, approximately quantitative treatment of the problem. Phase shifts produced- by the film were not completely taken into account and the derivations were not expected to be highly accurate over part of the range covered. Nevertheless we may say that the basic physics of the interactions have been made apparent by these derivations.

REFERENCES 1 R. A. Burton, V. Nerlikar and S. R. Kilaparti, Thermoelastic instability in a seal-like configuration, Wear, 24 (1973) 177-188. 2 T. A. Dow and R. A. Burton, The role of wear in the initiation of thermoelastic instabilities in sliding contact, J. Lubric. Technol., 95 (1973) 71-75; ASME Paper 72-Lub-45.