The role of surface asperities in transmitting tangential forces between metals

THE ROLE

OF SURFACE

TANGENTIAL

ASPERITIES

FORCES

J. J. O’<:C)NNOK

IN TRANSMITTING

BETWEEN

AND

IC. I.,.

METALS

JOHNSON

Zteparimcnf oj Aewraautlcs and Engineer& Mechanics, U&versity of Minnesota, Minn. f W.S.A .) aad ~e~a~~~~~~ of EngineeGzg, Cambridge University (Great Britaira) (Received

:\ugust 20, rghr; accepted November

SUM-M.4

LO, 1962)

RY

When metallic bodies are placed in contact under normal and tangential forces, the material in the region of the contact surface is deformed. The object of this investigation was to compare the relative magnitudes of the deformation of the surface asperities with the deformation of the bulk of the bodies when tangential forces less than limiting friction are applied. The tangential compliance of bodies of different hardness and surface roughness was measured. Of the specimens examined, there was evidence of full plastic deformation of the asperities only on a soft, artificially rough, surface during the first application of the tangential force. Such deformation made only a small contribution to the total tangential compliance of the bodies. During subsequent applications of the tangential force, the asperities behaxred entirely elastically and their deformations were then negligible. It is concluded that an elastic theory, ignoring surface roughness, can be used to predict accurately the behavior of typical engineering surfaces in contact under normal and tangential forces. INTRODUCTION

The situation is considered where two metallic bodies are pressed together by a force normal to their surfaces of contact and acted upon subsequently by a tangential force tending to cause one to slide upon the other. In the theory of such contact problems, it is usual to assume the bodies to be perfectly etastic and to ignore the effect of surface roughness, i.e. to consider the contact to be smooth and continuous. However, the work of BOWDEN and TABOR in developing the adhesion theory of friction has shown that the behavior of individual junctions at the true points of contact, and therefore the detailed mechanism of surface interaction, is predominantly a plastic phenomenon. It is the purpose of this paper to expain why a purely elastic model, which ignores the plastic behavior of individual surface asperities, can be used to predict accurately the behavior of metal surfaces in contact under normal and tangential forces. SUlllMARY OF EXISTING EVIDENCE

The adhesion

theory of friction

When two metallic bodies are placed in contact, true contact occurs only at the crests of the surface asperities. TABOR’ has produced considerable evidence to show that such local regions of contact, being few in number, suffer plastic deformation under the normal load and form cold-welded junctions between the metal surfaces. wear, 6 (‘963)

I I&--13’)

TANGENTIALFORCES BETWEEN METALS

119

NOMENCLATURE N T A Ao AfZ Y E G V

“d P 6 I

h ;

PO a a’ b Y

i

Normal force Tangential force Real area of contact Real area of contact under normal force Apparent area of contact Yield strength in tension Young’s Modulus (for steel taken = 30. 106 lb./sq. in.) Modulus of rigidity (for steel = 11.5. 106 lb./sq. in.) Poisson’s ratio (for steel = 0.3) Constant Tangential force coefficient (= T/N) Coefficient of limiting friction Tangential displacement Wavelength of surface asperities Peak to peak amplitude of surface asperities Shear traction Normal pressure Yield pressure Radius of contact circle Inside radius of slip annulus Halfwidth of contact strip Radius of point within contact circle Amplitude of slip Radius of curvature

The strength in shear of these junctions permits the transmission of tangential force between the surfaces; the limiting friction force is that required to rupture the junctions. MCFARLANE AND TABOR~, COURTNEY-PRATT AND EISNER~, and TABOR~ have examined the behavior of what amounted to a single asperity under the action of normal and tangential forces up to the point of sliding. In the experiments detailed in ref. 3, a spherical slider was pressed against a flat specimen with a normal force N sufficiently large to cause plastic flow of the material in the region of contact; the real and apparent contact areas were then equal and given by (1) where PO is the yield pressure of the material ($0 * 2.8Y where Y is the yield strength in tension). A subsequently applied tangential force T resulted in a considerable increase in the real contact area (deduced from electrical conductivity measurements) and large plastic displacements in the tangential direction before sliding took place. Figures Ia and b, taken from ref. 4, are typical. Figure Ia, for platinum specimens, shows the tangential displacement plotted against the tangential force coefficient $($ = (T/N)) during the first application of tangential force. The growth in junction size, expressed as the ratio of the real area at any load A to the initial area under normal load alone Ao, is plotted against 4 in Fig. Ib. Wear, 6 (1963)

118-139

From a simple criterion of plastic flow it is shown in rei. L that the shear stre+ transmitted by an individual junction must result in an increase in the real contact area according to a relationship of the form

where ocis a constant in the range 3 to 30 depending on the geometry of the junction and the elastic-plastic properties of the materials. The junction-growth measurements are shown in Fig. rb to correlate closely with eqn. (2) if cx is chosen to havt the value 12. 0.6

0.6

0.5

2

0.5

0.4

.v E 8

0.4

z 2

0.3

e 5

0.3

z E

0.2

s -$ 2 $j ”

F F

1 2

c” 0.1 4

0.1

s:

0

0 0

5

6: Tangential

10

o

15

dlsplacemWt (a)

Fig.

0.2

F

20 (Cm

25 x 10m4)

1.0

12 &

14

Growth

1.6 of

real

lB

21)

contact

22

2.4

area

&f

Results of Courtney-Pratt and Eisner for single plastic junction (platinum). 0 Lubricated. Unlubricated. (a) Force/displacement results. - - ..- Initial elastic slope. (b) Junction growth results. ----Eqn. (I), (x = II. I.

It was found that the first application of load to the junction was essentially irreversible ; reducing the tangential force reduced the displacement by a small amount and made virtually no difference to the real contact area. The system, once established, behaved elastically when the tangential force was oscillated. The plastic deformation process continued from where it had left off when the tangential force was subsequently increased, until the point of sliding was reached. The value of # at which sliding occurred was determined by the cleanliness of the surfaces and the presence or otherwise of oxide films. It should also be mentioned here that ARCHARIP has demonstrated that the regions of real contact in a rnult~-j~~~t~o~ system can remain predominantly elastic under a normal load. This conclusion is supported by the work reported here. The elastic theory of a sphere and a plane in contact tinder normal and tatigential forces

The Hertz theory of contact under normal force (ref. 6, p. 375) is well known. and MINDLIN et at.8 examined the problem where a tangential force less than limiting friction is subsequently applied. The following assumptions were made in this work: (I) the materials are ideally elastic ; (2) the surfaces are smooth, giving continuous contact (no surface asperities) ; (3) the frictional properties of the surfaces are such that there is no relative movement (slip) between the surfaces at a point where the tangential traction s is less than the MINDLIN’

Wcur,G (1963) 118-139

I21

TANGENTIAL FORCESBETWEEN METALS product of the normal pressure p and a constant coefficient

of friction p, while slip

does occur wherever s 2 ,I.+. IFrom the Hertz theory it is possible to predict the extent of the contact

area

between the bodies, the distribution of stresses within the bodies, and the normal compliance of the contact. The MINDLIN theory showed that the addition of a tangential force to the system results in slip over an annulus at the edge of the contact circle, while the surfaces remain locked together over a circle at the center of the contact. It enabled the surface distribution of shear traction and the tangential compliance of the bodies to be calculated, and predicted the extent of the slip region and, when the tangential force is oscillated at a given amplitude, the magnitude of the loss of frictional energy per cycle*. The question arises : How closely does such an idealized theory predict the behavior of real engineering surfaces? Experimental evidence The most recent confirmation of the Hertz theory of contact was given by BOWIE (see ref. 10). JOHNSON’SQexperimental evidence largely confirmed the accuracy of MINDLIN’S theory when applied to hard steel surfaces. The more recent work of GOODMAN AND RRoWN~Q has shown that an apparent size effect in JOHNSON’s energy loss results was due to variations in the coefficient of friction from experiment to experiment. The two sets of experiments show, however, the damping capacity of the contact to be less than the predictions of the theory which may be said to provide an upper limit to the experimental results. JOHNSON11 has recently shown that this small discrepancy between theory and experiment is consistent with a variation of the coefficient of friction across the contact circle, an effect which could be produced by repeated slip at the periphery of the contact circle. Experiments support the accuracy of the theory.

by KLINT~~ also

There is therefore good agreement between the body of experimental evidence available and an idealized elastic theory of the behavior of metallic contacts. In all the experiments mentioned in this section, measurements were made only after several cycles of tangential force had been applied to the contact so that the effects reported in refs. 3 and 4 may have gone unnoticed. OBJECTSOF THIS INVESTIGATION An apparent discrepancy exists between the results of the two approaches to the study of the interaction of metal surfaces. The behavior of a single metallic junction was found to be predominantly plastic during the first application of forces. Multijunction contacts, on the other hand, were found to behave closely in accordance with an elastic theory. In order to reconcile these two approaches, the behavior of surfaces with different degrees of hardness and roughness was examined. The contribution of asperity distortion to the overall tangential compliance of the contact was established during the first application of loads, when plastic effects could be present, and during subsequent loadings when the asperities would be expected to remain predominantly elastic. * A summary of the theoretical results is given in Appendix I. Wear, 6 (1963)

118-139

J. J. O’CONXOH,

I22

THE

K. L. JOHSSON

APPARATUS

Following the previous work, the contact of a spherical surface and a plane surfacca was examined. Figure 2 shows a diagram of the essential components of the apparatus. The flat ends of a cylindrical block A were used as the plane surfaces. The surfaces of the two blocks B in contact with A had a radius of curvature of about 30 in. ; the material being steel throughout, contact circles approximately 0.5 in. in diameter were obtained when the normal load N was 13,440lb.

A

B’

6,

t 27’

Fig. 2. Diagram of experimental

system and applied loads.

The spherical blocks were hard and lapped while the hardness and surface texture of the flat-ended block was varied. The relative tangential displacement of the system under a tangential force T per contact was measured by means of a pneumatic displacement head which was attached to blocks B and rested against block A at points in the apparatus reasonably remote from the contact regions. The normal load was applied through a small loading frame; the apparatus was placed in a hydraulic testing machine for the application of steady and oscillating tangential forces. The details of the loading devices are described in ref. 13. THE

BEHAVIOR

OF ROUGH

SURFACES

UNDER

NORMAL

FORCES

The choice of the experimental surfaces was guided by a study of a model of a rough surface under normal forces. Alz idealized rough surface under normal forces Consider a surface with a sinusoidal profile given by h y

=

-cos a.?

where h is the peak-to-peak amplitude of the asperities and J the wavelength. This surface is placed in contact with a flat surface of the same material. Real contact is established at the points x = o, + I, -&z;l, etc. If the average pressure between the surfaces is p, the force per unit length on one asperity is pa. The radius of curvature Iz of the sinusoidal profile at the points of real contact is 12/23&. The width 2b of the real contact strips for purely elastic deformation is found from the Hertz theory (ref. 6, p. 382):

wear,

0 (‘903)

I r8-139

TANGENTIAL Hence the ratio of real to apparent A*

zb

contact

123

METALS

areas is

4

-=-ZT-

A,

FORCES BETWEEN

(41

n

1

It is assumed

that the elastic limit is reached when the maximum

the principal

stresses in the bodies

asperity)

becomes

Hence, for elastic*

TABOR’

equal to the yield strength

(p. 50) suggests

figure would

difference

Hertz

between

pressure

on an

Y.

behavior

that 3-dimensional

some IOO times that required Therefore,

(0.6 times the maximum

asperities

to cause initial

yield.

become

fully plastic

at a load

In this case the corresponding

be about 75.

for fully plastic

behavior A

EP

(6)

I, < o.or7 yz

The experimental swfaces The experiments (b)

hard

representative surfaces

were performed

and ground, values

scratched.

of the roughness

and are compared

behavior

with three types of surface:

(c) soft and in Table

From

the limits

given by eqns. (5) and (6), for a normal

appropriate

maximum

Hertzian

pressure

ground Cc) soft scratched

surface

CONTACT

w

0.258

96.544

-

7oo

356,160

0.278

83.328

30

145

73.698

0.348

53,088

5

would

undergo

fully

elastic.

plastic

I, eqn. 2).

p,?.,

35% 169

predominantly

and fully

AREAS

700

remain

3)

I

APPARENT

(P.S.i.)

the soft surface would

AND

{Appendix

a (in.)

hardness

(a) hard polished Cb) hard

for elastic

Y (Psi.)

Vickers

Swface

Whereas

ROUGHNESS

(Fig.

for the two rough

load of 13,440 lb. from which the

was calculated

TABLE SURFACE

(a) hard and polished, measurements

(I/h) were estimated

factor I with

profile

plastic

8653641 etasiic

-

deformation

No plastic

plastic

deformation

-

>25

co.33

>374

(5

the ground

hard

of the polished

surface would be expected. 1t is worth

mentioning

at this point that it was necessary

* This elastic limit should not be confused

with the “limit

to go to the extreme

of elastic behavior”

mentioned

of by

KLINT~~.

Wear, 6 (1963)

x18-r3g

manufacturing full plastic

an artificially-

rough soft surface

in order to realize the condition

01

flow [(A/h) < 5

(a)

Polished

hard

surface

(b)

Ground

hard

surface;

Scratched

soft

&5

0.0; in.

*

(cl

average

3

surface;

average

$-=?I

Fig. 3. Profile records of the specimen surfaces.

Application

of normal

loads

A normal load of 13,440 lb. was applied to each of the specimens and removed again. The surfaces were then examined visually. (a) Apparent contact areas. The radius of the apparent contact area produced in each case is given in Table I, compared with the value 0.266 calculated from Hertz. interference with the tips of the larger surface irregularities accounts for the increased apparent contact areas observed with the rougher surfaces. (b) Real contact areas. For perfectly elastic conditions, the ratio of real to apparent contact areas for the case of a smooth sphere in contact with a rough plane is easily calculated from eqn. (4) by substituting the Hertzian pressure for @. (Appendix I, eqn. (2)) Similarly for fully plastic conditions, from eqn. (I) the value of the ratio at a radius r across a diameter of the contact circle perpendicular to the orientation of the asperities is (7) Wrw

0 (ry63) 11%13’)

TANGENTIAL

,

FORCES

0.8

BETWEEN

0.6

METALS

125

a2

0.4

0

f Fig. 4. Measurement of real contact area. Soft scratched surface. + After normal force only. o After tangential force to point of sliding. A After tangential force up to # = 0.167. x After 4 cycles, amplitude (b = & 0.167. 0 After zo,ooo cycles, amplitude (p = -& 0.~5.

0.6

a5

M

0.3

a2

0.1

0

0

1

2

3

4

6, Taogentki displacement km x IO-‘) Fig. 5. Results of Courtney-Pratt and Eisner for single plastic junction (steel). Predicted relationship between 6 and the overall 4 when the junction is in the centre of a multi-junction contact. - - - Initial elastic slope. If‘-&%‘,6 (1963)

118-139

120

J.

J.

O’tWiNOR,

K.

L. JOHNSON

Examination of the three specimen surfaces under a microscop? after loading showed that the asperities on the soft surface had been permanently flattened by thc~ action of interfacial pressure (Fig. 8); on the other two surfaces, which by eqn. (3) should be elastic, no signs of permanent deformation could be observed either directI>, or from profile records. It was possible to measure the widths of the flattened asperity tips on the soft surface. The surfaces of the contact strips could simultaneously be brought sharply into focus, using a microscope with a depth of focus of 3. IO--~ inches. Measurements of ratio ho/An were made by traversing along a diameter at right-angles to the scratches with a micrometer-microscope. Figure 4 shows the distribution of Aoj-ll. across the radius of the contact circle. (The further results in this figure, lying above the normal load results, will be discussed later.) The values plotted are averaged from both ends of the diameter and both ends of the specimen. The observations follow eqn. (7) if $0 = 186,370p.s.i. The value of It’ deduced from hardness measurements was 73,700 p.s.i., from which $0 = 2.53V. The value of the coefficient 2.53 is in reasonable agreement with theory and experiment’. Since the distribution of ,40/A, follows the theory quite well, it appears that the distribution of elastic stresses in the bulk of the solids is not greatly affected by the presence of surface asperities.

THE BEHAVIOR OF ROUGH SURFACES UNDER NORMAL AND TANGENTIAL FORCES

When a tangential force less than limiting friction is applied to a sphere in contact with a plane, the resulting tangential displacement may be considered to consist of two parts. The first of these is the elastic distortion of the bulk of the bodies due to the distributed shear traction on the interface; the second portion is the displacement of the asperities themselves near the center of the contact circle. Where contact is maintained through a large number of smallregular asperities the evidence of the previous section suggests that the distribution of elastic stresses in the bulk of the solids is not very different from that which would exist if the contact were smooth and continuous. The bulk displacement would then be given by the elastic theory. (Appendix I, eqn. 5). The magnitude of the asperity displacements on the other hand would depend on whether the initial normal load produced either elastic, elastic-plastic or fully plastic deformation in the regions of real contact; the displacements would be largest when fully plastic deformation occurs. The tangential displacement of an asperity which is at the center of a multijunction contact between a steel sphere and a steel plane can be calculated, using the data of COURTNEY-PRATT AND EISNER for a steel junction (Fig. 5) in conjunction with the HERTZ-MINDLIN theory (Appendix I). It is the displacement of the central junctions which determines the magnitude of the second portion of the overall displacement mentioned above. It is emphasized that the initial tangential displacements of even a fully plastic junction are elastic in magnitude: compare the experimental results for small values of 4 (Figs. ra and 5) with the initial elastic slopes (dashed lines) calculated from the elastic theory (Appendix I, eqn. 65).

TANGENTIAL FORCES BETWEEN METALS The initial tangential force produces growth in junction size.

127

plastic flow only in the normal direction

with a resulting

In a multi-junction contact, the ratio of tangential to normal forces applied to any junction is controlled by the distribution of elastic stresses in the bulk of the material.

0

2

0

4

6

8

12

10

6, Tangentialdisplacement

(in. x 10-5

compliance measurements. First loading. + Polished hard surface. l Ground hard surface. -__ Elastic theory. - - - Elastic theory + asperity displacement.

Fig. 6. Tangential

The contribution which asperity deformation makes to the overall tangential compliance of the system is therefore determined by the ratio of elastic tangential traction to normal pressure at the center of the contact circle. From Appendix I, eqns. (2) and (3) (8)

Substituting

for alla from Appendix I, eqn. (4) at

r=

a

0,

$j =

T/A’

” = P

I +

(I -

T//d+

+

(I -

(9)

T/pN)t

Equation (9) gives the relation between ti applied to a central junction and TIN applied to the whole contact. The experimental points in Fig. 5 give the relation between $ applied to any junction and the resulting tangential displacement of the junction. Combining the two sets of results, the relation between T/N applied to the whole contact and the resulting portion of the overall displacement due to deformation of the asperities is obtained and is shown as a solid line in Fig. 5. Wear, 6 (1963)

118-139

It will he seen that the displacement of even a fully plastic junction grows vcr\’ slowly with ?‘/i\runtil the point of sliding is approached. The rate of growth of s]i, at Y/U= o with ?‘/N is only 4_for small values of 7’/K (eqn. (9)) so that the junction displacements remain elastic virtually up to the point of sliding. It is concluded from this discussion that the tangential compliance of a multijunction contact is primarily controlled by the elastic deformation of the bulk of the material near the contact surfaces. The distortion of the surface asperities would remain small until the point of sliding is approached.

When a normal load of 13,4.~+0lb. had been applied, the specimen surfaces were each subjected to the foll~)wing types of tangential loadings: (i) Tangential force up the point of sliding. (ii) T an gen ti‘a 1 force up to a maximum value less than limiting friction. (iii) Tangential force up to the point of sliding four times, the normal load being removed and re-applied between each loading, the specimens remaining untouched in the apparatus. After each experiment the system was re-aligned to within 0.002 in. (iv) Four complete static cycles of oscillating tangential force at an amplitude less than limiting friction. (v) Same as (iv), followed by ZO,OOO cycles at the same amplitude applied at 5oo cycles/min, followed by four static cycles at the same amplitude. Experiments (iv) and (v) were performed on the soft surface only. Before each type of experiment, the specimen surfaces were freshly prepared; the results should therefore include effects due to plastic flow of the asperities. The contact surface was cleaned by washing with benzene and ether. Tangential force-tangential displacement curves were obtained from each experiment. The surfaces were subsequently examined visually and, where possible, measurements made of the real contact areas. Results (u) Comy%ance measurements. Figure 6 shows the load deflection curves obtained from series (i), compared with the theoretical curve calculated from Appendix I, eqn. (5), based on a contact radius a = 0.26 in. and the appropriate value of p. The dashed line in this figure includes the addition due to asperity displacements as previously calculated. Until the point of sliding is approached, the three surfaces behaved closely alike and in accordance with the elastic theory. In series (iii), it was found in each case that the coefficient of friction rose progressively from test to test up to a value of 0.5-0.6. The measured tangential displacements during the subsequent loadings followed the elastic theory up to the previous maximum load. The results of the fourth loading in each case are plotted in Fig. 7 and compared with the elastic theory. The three surfaces behaved almost identically virtually up to the point of sliding and in close accordance with the elastic theory. There is a slight discrepancy in both figures between the measured results and the theoretical curves near the point of sliding. The location of the theoretical curve near the point of sliding depends on the value used for UN, the limitina friction force. The detection of the exact point of sliding is a matter of some ex&zrimental difficulty. These experiments were conducted on an hydraulic testing machine so that the tangential force could be increased very slowly and continuously. When the contact could sustain no further increase in the tangential force, the rupture of the remaining metallic junctions near the center of Wear, b (1963) 11%13g

TANGENTIAL

FORCES BETWEEN

METALS

129

the contact was accompanied by a sharp metallic noise; simultaneously, the level of the displacement gage jumped off the scale, confirming that sliding had taken place. The value of the tangential force recorded at the instant the noise was heard was taken to be the limiting friction force FN. It is probable that the limiting friction force was slightly overestimated in this way, which would account for the slight discrepancy. An elegant method of detecting the point of sliding in a contact subjected to oscillating tangential forces is described in ref. IO.

-.-

0.5

z

$

0.4

5 8 ”

8 L. 0.3 e $7 s P t_m0.2 a‘

0.1

0

0

2

4

6

6

IO

8, Tangential displacement

12

14

16

18

(in.xlO-‘)

Fig. 7. Tangential compliance measurements. Fourth loading. + Polished hard surface. *Ground hard surface. 0 Soft scratched surface. Elastic theory.

(b) Area of contact measurements. Examination of the hard ground surface after these tests revealed that some permanent deformation had taken place. For instance, after series (i), measurements of the ratio A/A., though less precise than on the soft surface, showed that the real contact area in the center of the circle had a value of about 0.~23of the apparent. The theoretical value under normal load is 0.2 assuming perfectly elastic compression and 0.02 assuming purely plastic compression. A growth in the real contact area from 0.02 to 0.28 under the action of the tangential force is quite untenable since it would demand a value of a in eqn. (2) of about 1750. More likely, this surface had a real contact area of about 0.2 under normal force, corresponding to elastic behavior, and the restricted plastic flow produced by the tangential force gave rise to a relatively small amount of junction growth. Figure 8 shows the appearance of the soft surface after different experiments. In Fig. 8a, the edge of the contact circle is shown, to demonstrate the appearance of the undefo~ed surface. The flattened asperities at the center of the contact circle after normal loading are shown in Fig. 8b. The further increase in the real area of contact after the addition of a tangential force to the point of sliding will be seen by comparing Fig. 8b with Fig. SC. fi’&ZV, 6

(1963)

118-139

The complete set of averaged values of the ratio A/Au. on the soft surface after each of the experiments is shown in Fig. 4, compared with the values obtained for the normal load alone. After the first test, the real area A had increased to about double AO and followed the same distribution. In this case, the point of sliding had been reached so that A/A,) Il~?nJ~.0 (rqfI3) 1 rs- rjq

TANGENTIAL

should be constant

FOXES

across the contact;

in Fig. 4 was obtained by two. Substituting

BETWEEN

the theoretical

simply by multiplying the observed

METALS

131

curve marked

the ordinates

$ = ,u = 0.33

of the normal force curve

results # = ,LJ = 0.33 and A/A0

= 2 into eqn. (2)

yields a value of 27 for a. While this is larger than TABOR’S value based on platinum, it is of the correct order of magnitude. The value of A/A0 obtained in this experiment indicates that the degree of plastic deformation which took place was of the same order as in the experiment of COURTNEYPRATT AND EISNER on the platinum specimens. The results of the second test will be seen to lie between the normal load results and those obtained

when sliding took place. The value of the tangential

force coeffi-

cient varies across the contact (eqn. (8)). An estimate of the resulting variation in junction growth was made, using eqn. (2) in conjunction with eqn. (8). The result of this calculation is shown in Fig. 4 for T/pN = 0.75, p = 0.33 and (x = 27. The observed values do not follow this curve very closely, probably because of the lower coefficient of limiting friction (approx 0.25) in this particular experiment. The surfaces were extensively damaged after the third series of tests; the asperities on the soft surface had been smeared into each other so that contact must have been almost complete and continuous. Using cy = 27, 4 = p = 0.31, and (from the normal load test) Ao/Aa = 0.28, it is found that A/A, in the center of the soft surface would be expected to be about 0.8 after the fourth tangential loading. After series (iv), the real contact area was found to have grown in the slip region at the outer edge of the contact circle, relative to the center. This effect is thought to be due to an increase in the coefficient of friction in the slip region as a result of the four cycles of alternating slip. The increase in the real contact area in the slip region after 20,000

cycles of load

is even more marked. The distribution in the center follows the theoretical curve for T/pN = 0.75 quite well, but considerably exceeds the theory (based on a uniform value of p = 0.33) in the slip region compared with the center. For example, the observed value of A/Aa = 0.53 at Y/a = 0.75, when substituted into eqn. (2) gives ,u = 0.5 compared with the overall value of 0.33. (c) Energy loss measurements. In series (v), the load deflection curves had the form of the familiar hysteresis loops which arise from the dissipation of energy in the contacts~g,iO. The measured areas of the loops were found to decrease with time; the typical

variation

of the area of the loops is shown in Table TABLE

II.

II

VALUES OF ENERGY LOSS PER CYCLE (in. lb.) T/N

= 0.15

The decrease of loop area with time is shown to be consistent with a growth in ,u with time.

1.gr

zo,oooth loop

2nd loop

rst loop Measured

Measured

Calculated, p = 0.33

Measured

Calculated, p = 0.5

1.16

1.16

0.57

0.65

The second loop was considerably thinner than the first but the third and fourth loops coincided sensibly with the second, the area of these loops agreeing closely with MINDLIN’S theory (Appendix I, eqn. (7)) based on a value of ,u = 0.33. The loops Wear, 6 (1963) 1r8-139

1. _I. 0 C’ONNOII, Ii. I.. JOHSSOS

19 observed

after

siderably

thinner

series

zo,ooo

agrees

reasonably zo,ooo

several accounts

recent

loops. value

with

of load;

value,

were

the growth

between

presented

each

other

but were

measurements

had reached

area

so that

based

con

for the

a value

of 0.5

on this value

the reduction

in loop

of ,U area

in ,U in the slip region. made

after

the

contact

in ,U in the slip region

his results

and the elastic

had

sustained

almost

certainly

theory.

(JOHNSON’s

this conclusion.)

area of the first loop supports

growth

with

of contact

of the loop

a growth

measurements

work11 supports

area

that p in the slip region

is consistent cycles

identical The

the observed

for the discrepancy

The larger junction

with

damping

thousand

virtually

the theoretical well

cycles

were

the earlier suggested

cycles;

JOHNSON’S”

more

cycles

than

(v) experiments

after after

zo,ooo

by

the picture

COCRTNEY-PRATT

of irreversible

plastic

flow and

AND EISNEK.

ELASTIC COMPLIA1U’CE OF THE ASPERITIES Appreciable between ity would thereby

elastic

two contacting serve modify

to the

that

was

the

(see Appendix

II).

in the slip annulus, (Appendix

dated

distortion

by elastic

of the slip annulus is likely

between

of shear made

surface

at the periphery

traction

In the sphere traction

to estimate tractions This

. 10-6 in. for the situation

displacements

edge

attributed

bodies.

the distinction

analysis

fact z

is often

the distribution

areas of contact the order

metal

to blur

A theoretical due

flexibility

to the irregularities the regions between

are applied

considered

and that

the surfaces.

in this paper.

rise from zero at Y = in only

the modification

elastic

displacement

discontinuously

displacement

I, eqn. (8)). It is therefore of the asperities

such flexibil-

of slip and no slip and

the additional

additional

however,

at the interface

and plane problem,

at the

turned

real

out to be of

The relative a’ to 130

surface

. 10-6 in.

clear

that

slip can be accomo-

a very

thin

region

to the theoretical

at the inside distribution

of

to be slight.

(a) Section of contact

circle.

Fig. Q. Fretting on a hard ground steel surface Wear, 6 (1963) I IS- 139

TANGENTIAL

FORCES BETWEEN

METALS

I33

After zo,ooo cycles, each of the three specimen surfaces showed extensive fretting damage on an annulus at the periphery of the contact circle. The dimensions of the fretted annulus and of the theoretical annulus of slip (Appendix I, eqn. (4)) agreed quite closely, as would be expected from JOHNSON’S workQJr. No fretting was visible on the inner circle of radius a’, the no-slip region. On each surface there was a thin region separating the undamaged from the fully damaged regions (Figs. 9, IO and II). It is thought that the relative surface displacements in this thin transition region are sufficiently small to be accomodated by elastic distortion of the asperities so that full fretting damage is not developed on the asperities in this region. On the soft regular surface (Fig. IO), the inside edge of the slip region was sharpIy defined by the boundary between undamaged and damaged surface and the transition region was indeed very thin (less than 0.003 in.). On the ground hard surface, the less regular height of the asperities contributed to rather more blurring of the inside edge of the slip annulus (Fig. 9). However, individual asperities clearly showed a transition from slin to no slin over a distance of about 0.005 in. It will be seen that the development of fretting on an individual asperity follows 1

I

(b) Fretted annulus.

the same pattern as on the contact surlace as a whole. Slip and fretting initiate at the edges and gradually spread across the whole surface of the asperity. It is estimated that the amplitude of slip on the surface of an asperity required to produce the first visible trace of fretting must be of the order IO--~ in.

TArjGENTIAL

FORCES

BETWEEN

135

METALS

(b) Fretted

annulus

CONCLUSIONS %%en turo m&&c bodies are pressed together and subjected to ~~gen~~~ forces, it is concluded that, in spite of possible plastic deformation of the surface irregularities, the overall compliance of the system is closely given by the bulk elastic deformation of the bodies, neglecting the existence of surface roughness. However the inv~stigati~n has clarified the role of surface asperities in transmitting normal and tangential farces.

UP?,?‘,6

(1963)

I X8-

139

Normal

force

It has been shown that the surface irregularities of typical engineering surface> are deformed elastically or, at least, do not reach the condition of full plasticity. Under these conditions, the area of real contact is determined by the surface topography (ratio of I/h) as well as the hardness of the material, and is appreciabl!, greater than a purely plastic analysis would suggest.

Provided the surface is sufficiently soft and rough to permit full plastic deformation of the asperities under a normal load, the application of a subsequent tangential force causes a growth in the real area of contact in the manner described by TABOH et al. The contribution of the displacements of the individual asperities to the overall compliance of a multi-junction contact is small, even when plastic flow of the junctions takes place in the first loading. In subsequent loadings, the asperity displacements are elastic and negligible. A purely elastic theory which assumes continuous contact is entirely adequate to predict the tangential compliance of multi-junction contacts.

Under the action of oscillating tangential forces, repeated interfacial slip at the edge of the apparent contact area causes, through some cleaning mechanism, the coefficient of friction and the true area of contact to rise locally in this region. The assumption of a constant coefficient of limiting friction in MINDLIN’Selastic analysis of the sphere in contact with a plane is then not justified and accounts for the overestimatian of the energy dissipated by slip in cyclic loading. The theory may then be said to provide an upper limit to the damping capacity of the contact. ACKNOWLEDGEMENT

The authors wish to acknowledge, with thanks, BRAND with the experiments.

the help given by Mr. I&HARD

TANGENTIAL

FORCES BETWEEN

APPENDIX

137

METALS

I

Sztmmary of the results of the ~er~z-~~~dL~n tlteory

of contact

An elastic sphere of radius R is pressed against a plane of the same material with a normal force N. A tangential force T (less than ,uN) is subsequently applied. The radius of the contact circle isa (4 and the distribution of pressure across the contact circle is6

The distribution of tangential traction across the contact circle is7 (3)

where a’, the radius of the “locked” circle at the center of the contact circle is7 a’ = a

(I - 5))”

(4)

Slip takes place over the annulus a’ < Y < a. The relative tangential displacement of the two bodies is7

The initial elastic slope of the force deflection curve is therefore da aT=

2-v OS=

(6)

-&a

When the tangential force is oscillated at an amplitude T*(< of frictional energy due to slip is 8 ilE =

,uN), the resulting

4 JLb~zNa[I-~I-~)5~~~(I+~)~] 9(2 -

loss

(7)

The amplitude of relative movement of points on both surfaces in the slip annulus is9 3ildN e=16Ga.(z-v)

Y 2 *-~,i*-l(~)][1-*(~)‘]+~.~ . n: 0a f[ APPENDIX

[I--(s)‘]‘)

18)

II

Calculatiom of the elastic flexibility of surface asperities In calculating the elastic compliance of a rough surface, the shape of the surface is used to determine the location and extent of the real contact area; the surface is then considered to be flat, the loads being applied through the real contact regions, since the height of asperities on most typical surfaces is small compared with the wavelength. It is not likely that bending of the asperities, for instance, would be appreciable. WWY,

6 (1963)

I X8-139

138

.I_ J. O’CONNOR, Ii. 1.. JOHNSON

An assessment of the tangential compliance of asperities has therefore been made by calculating the difference between the surface tangential displacements produced by (a) a continuous distribution of traction s per unit length and (b) a discontinuous distribution of traction S per unit length applied over contact strips of width ah at x = o, i_ A, 3 22, etc., (2 is the wavelength of the asperities). It will be seen that the difference between the surface displacements due to the two distributions of traction may be obtained by considering an elastic half-space subjected to the following form of traction, periodic with 1: (1)

with the condition zSb = sil The net force applied to the surface is

The rectangular wave-form of the surface loading can be expressed as a Fourier series

where

0’ 0

a2

Ratio

Fig,rz.

0.4

cs

&(

contact v

G

Plot of -f-l-A s

a8

1.0

areas

I-V

against AlAo. 3c= Weav, b (‘963)

I IX- 13’)

TANGENTIAL

Then for rzy of the form

FORCES BETWEEN

METALS

I39

(I)

an = EK*

sin (3cnK)

(3)

Now the displacement in the x direction of a point in the elastic half-space subjected to a concentrated surface tangential force P per unit length applied at the origin is

viw)=-2

(I-~)10g(X"+y~)+[

where B is a constant From (4), a distribution of surface traction cements at the origin given by z, -

(r-&V)A;

a(l_

R-1 %

y2 1+B

x* -t y*

of the form

(r-&j.

(2)

produces

surface displa-

m sin(nnK) SL Ix 2.8% n-1

(5)

Equation (5) gives an expression for the elastic tangential displacement of in a contact between two rough surfaces transmitting an average shear per unit length. The normal displacement of an asperity due to an average per unit length has the same form. 1% is shown plotted The quantity (v/A) . (G/s) . {+/(I Y)], t o within

an asperity traction s pressure s in Fig. IZ.

Wear, 6 (x963)

I IS- 139