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Low friction coefficient estimation for model material experiments
303
,1-Elsevier Sequoia S.A., Lausanne
Printed in The Netherlands
LOW FRICTION COEFFICIENT MATERIAL EXPERIMENTS
ESTIMATION
FOR MODEL
D. C. FRICKER and T. WANHEIM Division
of Mechanical
(Received October 5,
Processing
of Materials,
A.M.T.,
Technical
Uniuersity
of Denmark
(Denmark)
1973)
SUMMARY
A previously suggested method of estimating low friction coefficients from the bulk deformation of long rectangular laminae during plastic compression, was used for wax specimens with various lubricants to enable the choice of the best lubricant for experiments with model materials. This method was also used for aluminium specimens with liquid and solid lubricants. Existing theory was extended to cover both the Amonton friction model and that of constant friction stress. Friction coefficients for certain waxes Iubricated with Vaseline were found to be as low as 0.01. With molybdenum disulphide grease on aluminium, the friction coefficient was found to decrease as compression proceeded. Friction coefficients with lead foil were found to agree with their independently known values. Lubrication breakdown at the periphery of specimens was observed in some tests, causing overestimation of the friction coefficient.
1. INTRODUCTION
Model materials, chosen to satisfy known simulation criteria, can be economically very useful in the design of forming processes and tooling. When conducting plane strain model material tests between two rigid parallel boundaries, it is convenient to be able to see the deformation pattern of the surface of a specimen so that the mechanics of a process can be analysed. It is important to minimise interfacial friction between the specimen and the transparent boundaries in order to achieve plane strain conditions i.e. no variations of stress and strain through the thickness of the specimen. Such tests have indicated that the friction present can be very low and so an accurate method of estimating this friction is needed to enable the choice of the best lubricant. This paper uses the method described by Hill’ to obtain values of the friction coefficient, from the comparison of theoretical predictions and experimental results of the increase in length of a long rectangular lamina, during compression between overlapping parallel platens. This method is sensitive for low values of friction coefficient, whereas the method of Male and Cockcroft employing the deformation of a ring of material is less sensitive in this range. These two methods are therefore complementary.
304
D. C. FRICKER,
T. WANHEIM
Since the specimen geometry is chosen such that the ratio of length:width: thickness is initially 2bo: 24,: h, = 20: 2: 1, variations in stress through the thickness and across the width may be neglected in comparison with variations along the length. This makes the deformation amenable to analysis. The theoretical situation is thus one of plane stress since there is no external pressure on the edges of the lamina. For liquid lubricants employed in the experiments described later, it seems reasonable to express the friction stress relationship in terms of the Amonton law of sliding friction r = c1P
(1)
where h is the usual coefficient of friction and p is the normal pressure between the sliding surfaces. However, there may well be a dependence of friction stress z on sliding length, surface topography and pressure-variant lubricant properties, as described by Wanheim3. A sufficiently thin foil of some softer solid acts as an efficient lubricating agent, the friction stress being equal to the yield stress in shear of the foil. The friction stress can be expressed as r = .$_i, = constant
(2)
Theory has been pursued along both these lines of friction stress relationship and the simplified plasticity condition used by Hill has been replaced by the more accurate von Mises criterion. 2. THEORY
(i) The lamina is homogeneous and isotropic, (ii) The platens are rigid and parallel, lamtna(iii) Th e frtc ti on coefficient is constant at all points on the surfaces of the (iv) Th e variations of stress and strain through the thickness are negligible compared with variations along the length i.e. the interfacial friction stress is transmitted equally through the thickness (v) The variations of stress across the width are negligible compared with variations along the length, (vi) The uniaxial yield stress of the lamina Y is constant along the length i.e. no work-hardening. Assumptions (iv) and (v) give a plane stress situation during compression. Because of symmetry, only half the length of the lamina need be considered analytically. Figure 1 shows coordinates, velocities and incremental displacements of any point within the lamina. For an infinitesimally thin section in the left-hand half of the lamina, the stresses are as shown in Fig. 2. Assuming the deformation is slow such that inertia forces may be neglected, the equilibrium equation for this element reduces to
MEASUREMENT
305
FRICTION
OF LOW
Fbsitim of my pint
Velocity
dqlx q+;ij;
”
dh da
t/
Diip4acement
db
Fig. I. Location
of reference
Fig. 2. Stresses
axes within
on an elemental
the lamina.
section
Since the compression is plastic, the von Mises yield criterion must be satisfied, and in this present case 5 & p, so that pz+q2-pq
Differentiation
dq
= Y2
(4)
with respect to x gives
q-3
2q-p TX=----
dp
(5)
dx
The relevant Levy-Mises plastic stress/strain relationships for increments in strain in the x- and z-directions can be combined to give
dE,---
2q-P
d&
2P-q
Now ds,=dh/h
(6)
and ds,=(du/dx)dt
where db=udt.
So at any point
(7) It can be seen from eqn. (6) that the longitudinal strain increment vanishes when becomes equal to p. If a section at distance x’ exists where this plane strain condition holds, a central region of the lamina of length 2 (b-x’) will undergo no extension and, since there will be no relative motion of the surfaces, p and q will have their maximum values of 2Y/J3 and Y/,/3 respectively.
2q
2.2. Friction equations 2.2.1. For a liquid lubricant, where eqn. (1) applies, eqn. (3) becomes hg
= 2pp==
[q+2
(Y2 -
y)
lfromeqn.(4)
Integration by substitution, from one end to a distance x, gives
(8)
306
D. C. FRICKER, T. WANHEIM
px - = i [J3 h
arcsin $$
since q=O at x=0. Plane strain occurs in a central region of the lamina the compression if, substituting the limiting value q= Y/J3,
pbo3
_ 4
from the beginning
of
0.263
From eqns. (7) and (8) d2b 1 _--_-------_ dx dh Integrating
dp
--
dx 2pp
over the current
db,=-gln$
:,
$(ln
half-length
P)
b
since p=Yatx=O.
When pbo/ho 20.263, equal to 2/J3, so
1
the ratio pJY
remains
constant
throughout
the compression,
d&,+&)0.0719 P
Integration
over the compression
gives (9)
In comparison,
Hill predicts
” = (;j bo
* (1 - f)/(l
- 2)
when
,ub/h<0.288
and db,=
(-dh)
y
when pb/h > 0.288
In the present experiments, the nominal specimen dimensions were be/ho = 10. When p~O.0263, pb increases as the compression proceeds and it becomes difficult to determine the theoretical relationship between the fractional increase in length (b/b, - 1) and the fractional decrease in thickness (1 - h/h,). It has been shown that the approximation used by Hill, in place of the von Mises yield criterion, produces little error when p>O.O263. For convenience, his theoretical predictions when p=O.Ol and p-0.02 have been used in this present paper. The relationship when p=O is the well-known parabolic curve. Figures 5-8 show the predictions for various values of p. 2.2.2. When eqn. (2) applies,
z can be expressed
in terms of Y as
s=jY
where i can be considered as the friction of foil and specimen material.
coefficient
for a specific
combination
MEASUREMENT
OF LOW FRICTION
307
Equation (3) becomes h$
=2[Y
so 2iY q=hX From eqn. (7) d2b 1 ---_--dx dh
dp 1 dx 2{Y
Integrating over the current half-Iength b
If there exists a central region of plane strain from the beginning of the compression, then from eqn. (10) 5b,> 1 , i.e. for b~/~~= 10, [ >Oo.029 &I 2J3 In this case, &, will be constant throughout the compression Integration of eqn. (11) gives
and equal to 2Y/,/3.
This theoretical relationship is shown in Figs. 9 and 10. Since the experiments performed were for [z=-0.029, no relationship was derived for { <0.029. Assumption (vi) was made for the sake of theoretical simplicity. When the material is work-hardening, there arises a further term in eqn. (5) involving dY/dx. This is the only departure from ideal plastic theory, which work-hardening causes and the Iamina test materials were such that the variation of Y along the length at any time produced only a negligible error. Assumption (iii) may not hold true in experiments. If this is the case, experimental results will indicate an equivalent friction coefficient which can be estimated from the gradient of the curve of fractionaf increase in length against fractional reduction in thickness, at any time during the compression. 3. EXPERIMENTS
Both wax and metal laminae were used for experiments; five varieties of wax were used and the metal employed was aluminium of commercial purity. Hill shows that the theory becomes less accurate at large reductions in thickness and so all tests were terminated when a reduction of approxjmately 50% had been reached. Measurements of the length and thickness of a lamina after various reductions form the basis of experimental results. Theoretical predictions are based on an initial specimen geometry of b,/h,= IO. In general, the actual geometry departed slightly from this value and so a correction was always applied to
D. C. FRICKER, T. WANHEIM
308
experimental results. By taking measurements when the specimen was unloaded, any elastic deformation which occurred during the compression, was not included in the measurements. Before each test the platens and, in the case of Al, the specimen were thoroughly degreased. 3.1. Wax laminae The varieties of waxes used in tests are listed below, together with their compositions (where known). Name of wax
FiEia
The exact composition is not known (Filia % Indramic (micro- is a registered trade crystalline wax) mark), but it is likely % Harpiks (natural that it has 5&-60% resin) kaolin added to a paraffin wax. “/, Kaolin
A
B
c
I)
20
20
60
40
55
40
0
60
25
40
40
0
~5~~~ositi5n
These waxes were chosen because of their known characteristics4.
and different
stress/strain
Two halves of a specimen were cut from a homogeneous block of cast wax, of nominal thickness 19.8 mm. The two halves were butted end to end and seam-welded to produce a lamina whose nominal initial length and width were 400 mm and 40 mm respectively and whose faces were parallel to within 0.1 mm. Figure 3 shows the press used for testing wax specimens. A hand-operated press was modified to allow the specimen 1 to be compressed between two 5 mm thick acrylic plastic platens 2 and 3, which were kept parallel during compression by two rectangular-hollow-section steel beams 4 and 5, of section 76 x 51 x 5 mm and of length 600 mm. The press and beams were sufficiently rigid to keep the faces of all Iamina parallel to within 0.2 mm. The increase in length of lamina was measured using dial gauges 6 and 7, whose anvils 8 and 9 were in contact with the ends of the lamina. The internal springs of the gauges had been removed so that errors resulting from imposed end-forces and from the anvils being pressed into the ends of the wax, were avoided. The decrease in thickness was measured using a dial gauge 10.
Fig. 3. Apparatus for testing wax Iaminae
MEASUREMENT
OF LOW FRICTION
309
After lubricant had been applied to all surfaces, the specimen was positioned centrally between the platens, the platens and specimen were centred in relation to the press and the test was carried out immediately. This urgency was required because some of the waxes exhibited lubricant-absorbing characteristics. All waxes tested showed notable creep and so after each step in the compression and the removal of the applied load (including the weight of the upper beam), the gauges were read, as nearly as possible, immediately. It was impossible to renew the lubricant between steps because any disturbance of the wax would have resulted in unaccountable deformations. Three lubricants were employed in tests with Filia e;iz. vaseline, Molykote DX paste and a silicone spray. The effects of surface roughness were studied by performing tests where the surfaces of the lamina had been knurled and also where lateral slits had been cut in the surfaces. Only one lubricant was used in tests with waxes A, B, C and D. Results with Filia showed that Vaseline was the best lubricant of those tested and so this was the choice for further tests. Some of the tests indicated that the built-up edges formed in cutting a specimen, might influence the results considerably. In such cases, additional tests were performed after careful removal of these edges. 3.2. Aluminium laminae Specimens were machined from rolled strips of thickness 5 mm, 4 mm, and 3 mm, such that their dimensions were in the nominal ratio of 1ength:width: thickness= 20:2: 1 and their anisotropic behaviour was minimised. Laminae faces were not machined and were initially parallel to within 0.02 mm. Experiments were performed in a 60 tonne universal testing machine fitted with an automatic speed regulator which allowed deformation to be carried out at constant speed. Rectangular hardened-steel platens were used and these had been lapped to produce a surface finish which was measured to be R, =0.094 pm and R,,, = 1.32 pm. Specimens were positioned centrally on the platens to ensure uniform thickness reduction. Length measurements were taken along the centre-line of the specimen and thickness measurements were taken at different sections. Usual speeds of reduction were chosen, such that the strain rate would be the same in all tests, as follows: Initial nominal thickness mm Speed mm/s
5 0.025
4 0.02
3 0.015
Laminae were tested both with their original rolled surface finish and also after their surfaces had been sandblasted. This enabled a conclusion to be drawn about the influence of surface finish on the deformation of a lamina. Two lubricants were employed in tests with Al specimens viz. (1) a liquid lubricant, Molypan N (2) a solid lubricant, lead foil. 3.2.1. Molypan N This lubricant was a molybdenum disulphide high-pressure grease and was applied in excess to all surfaces, after the specimen had been removed from the platens to enable its dimensions to be measured. When the laminae were machined
310
D. C. FRICKER,
‘I’. WANHEIM
from the strip, built-up edges were formed, of greater thickness than the bulk of the specimen. To investigate this factor, a 4 mm specimen having a rolled surface finish, was tested after these edges had been removed by hand with a flat grindstone. An additional test at a speed of 0.18 mm/s was performed with a 4 mm specimen, to investigate the influence of speed on the deformation of a lamina lubricated with a liquid. 3.2.2. Lead foil Three sheets of lead foil were roiled to thicknesses of 0.18, 0.1 and 0.05 mm. Tests were carried out using specimens of 4 mm and 3 mm thickness and after measurements of the dimensions of a specimen had been taken, a new piece of overlapping foil was used. It was found that the foil adhered strongly to a sandblasted surface, so that the foi1 could not be replaced during a test. Furthermore, measurement of the thickness’ of the specimen was overestimated because of the presence of the foil and a correction based on volume constancy of both the specimen and the foil was applied. One test at a speed of ten times the usual speed was carried out on a specimen of thickness 4 mm. Two continuous tests were performed on 3 mm specimens at speeds of 0.15 mm/s and 0.015 mm/s. These tests investigated the influence of stepwise deformation.
6-
VUXA
Filia
a1
CL2
oI3
Fig. 4. Compressive Fig. 5. Comparison friction is assumed.
o,fl
a5
stress-strain of calculated
0.6
0.7 E
curves of three varieties and observed
values
of waxes. of the extension
of Filia
laminae;
Amonton
MEASUREMENT
As cast
OF LOW
311
FRICTION
and cut
6
‘-t;;
1-e
Fig. 6. Comparison of calculated and observed values of the extension lubricated with Vaseline; Amonton friction is assumed.
of laminae
of waxes
Fig. 7. Comparison of calculated and observed values of the extension lubricated with Vaseline; Amonton friction is assumed.
of laminae
of waxes C and
4. RESULTS
A and B
D
AND DISCUSSION
In all tests, the curvature of the edges of laminae was so small that no significant error was introduced into the results by this effect. This affords evidence of the validity of the theoretical assumption (iv). 4.1. Wax ~amina~ The uniaxial compression curves of three varieties of waxes used in experiments, are given in Fig. 4. The results of tests with wax under various conditions are shown in Figs. 5, 6 and 7. It can be seen that all the results show that the friction coefficient is constant until a certain point in the compression has been reached, whereupon the coefficient steadily increases as the compression is continued. Since it was impossible to relubricate the surfaces during a test, lubricant escaped gradually from the interface, both at the periphery and also into any cracks which were formed. Furthermore, the surface area increased as compression proceeded, so that breakdown in lubrication occurred at some time. Further compression increased the number and area of these regions of breakdown, causing an increase in friction coefficient. From Fig. 5, it can be seen that Vaseline possesses a much lower friction
312
D. C. FRICKER,
T. WANHEIM
coefficient than the other lubricants and breakdown occurs at a later point in the reduction. This is possibly explained by the semi-solid nature of Vaseline, tending to separate the specimen and platens better than did the other lubricants. Longitudinal cracks in the surfaces of Filia laminae after compression were evident, the size of cracks after a given compression increasing as the original quantity of the lubricant increased. Such cracks are formed by the pressurised lubricant forcing its way into existing depressions; since the internal compressive stress is less in the width than in the length direction, it is easier for specimen material to flow in the width direction, forming a longitudinal crack. Therefore results shown are for a minimally thick application of lubricant. Figure 5 also shows that the friction coefficient is initially unaffected by the surface topography but that the break-point occurs earlier in the compression when escape paths exist in the original surface. The knurled surface indentations were sufftciently large to be able to receive lubricant squeezed from the interface and so no additional cracks were apparent. The pitch of the lateral slits was not small enough to accommodate all escaping lubricant and cracks were formed between the slits. Lubrication breakdown at the periphery of specimens, resulting in some direct contact between the lamina and platens, was observed. It is worth considering the effects of this peripheral contact zone on the deformation of a lamina lubricated with a liquid. When this zone is formed, it effectively seals the major escape path of the lubricant still present on the surface. Apart from escaping into any cracks which may be formed in the surface, the lubricant must remain separating the surface of the lamina from the platen. Because of the relatively high interfacial friction in this zone, additional compressive stresses will be exerted on the bulk of the lamina, which are not considered in the analysis. Since the friction stress is assumed to be dependent on the normal pressure over the greater part of the surface, the presence of this zone would produce higher friction stresses than the lubrication conditions suggest. A slight decrease in friction coefficient can be seen when the built-up edges were removed, and this agrees with the observation that the width of the peripheral contact zone was less in this case. This effect can be seen strikingly in Fig. 6. Wax A showed no cracking whatsoever and no signs of wax/platen contact at the periphery of the laminae. Wax B showed no surface cracks but wax/platen contact was apparent. The friction coefficients are lower and the break-points occur later in the compression with these waxes, as compared to Filia. Figure 7 again shows this built-up edge effect. Wax C exhibited many small surface cracks and this accounts for its break-point occurring sooner than with waxes A and B. Wax/platen contact was not observed, however. Wax D showed no surface cracks and the wax/platen contact zone was much smaller in the test where the built-up edges had been removed, than when they had not. The coefficient of friction is seen to be less in the former case and its break-point to occur later in the reduction. These results lend weight to the argument that the existence of peripheral contact between the lamina and the platens affects the bulk deformation of a lamina, and hence the magnitude of the apparent friction coefficient. Further evidence of this effect can be seen from results with Al laminae.
MEASUREMENT
OF LOW FRICTION
313
The yield stress of typical Al specimens was inferred from hardness tests on the bulk metal; these values are shown below: Initial thickness of specimen (rolled surface) mm percentage reduction Y kp/mm2
3 zero 15.6
3 250 >21
4 zero 12
4 “50 20.5
In all tests with Al laminae it was found impossible to prevent lubrication breakdown at the periphery of the specimens. This phenomenon has also been observed by van Rooyen and Backofen”.
Figure 8 shows that the results for 4 mm specimens suggest a slightly higher value of friction coefficient than those for 3 mm and 5 mm laminae. Although no definite explanation is apparent, this discrepancy is possibly due to the combined effects of different size and of different work-hardening characteristics of the specimens. It can be seen that increasing the speed of the compression slightly increases the friction coefficient. This is possibly due to the breakdown of peripheral lubrication being retarded by the higher speed of the escaping lubricant, so that less lubricant is trapped when the contact zone is finally formed. Figure 8 also shows that sandblasting the surface of a specimen increases the 04
swfoG? sp?dmm& r5mm Rolled 0,025 ohm -II0.02 035 r3mm -**0.015 w3mm Sandblasted aa o4mm -I0.02 n4mm Rolled 0.18 03 i x4mm Rolled i built0.02 %rn% 0.25 ~
0.2 -
Fig. 8. Comparison of calculated and observed values of the Molypan N: Amonton friction is assumed.
extension
of Al
lubricated with
314
D. C. FRICKER, T. WA~HEI~
friction coefficient. Microscopic examination of rolled and sandblasted surfaces after compression, revealed that both surfaces have cracks in the longitudinal direction. The sandblasted surface appeared pitted and possessed deep cracks, whereas the rolled surface had less deep cracks and appeared smoother. Since these cracks form an escape path for the liquid lubricant, it follows that a sandblasted surface would produce a worse lubrication situation than a rolled surface. The only apparent explanation for the significantly lower friction coefficient indicated in one test with a 3 mm specimen is the effect of the peripheral contact zone: the width of this zone was much less in this test than in all other Al tests. It can be seen that the results of Al tests with and without the built-up edges having been removed, are identical. On the lamina with these edges removed, lubrication breakdown still occurred and a zone was formed. The trend of tests with Molypan indicates that the apparent friction coefficient decreases as the compression progresses. Van Rooyen and Backofen have also observed decreasing friction coefficient, although their experimental conditions were sufficiently different from those here to limit the applicability of their explanation of this effect to the present results. It is likely that the liquid lubricant possesses pressure-dependent characteristics but this cannot fully explain these results. One inaccuracy of the theory, described fully by Hill, caused by the gradual inapplicability ofassumption (v), does not explain this trend. Even after 50% reduction, the length/width ratio was still about 6. Although the results with a 5 mm specimen agree with other results, the specimen showed pronounced anisotropic deformation. It is thought that the large surface of this specimen caused non-uniform elastic deformation in the platens. With 4 mm and 3 mm specimens, the elastic deformation was small and the faces of laminae were parallel to within 0.02 mm. The results to demonstrate, however, that this experimental method is not particularly sensitive to such anisotropic behaviour. 4.2.2. Lead foil A direct check on the theory is possible when the frictional properties of a lubricant are known independently. So long as extrusion in the width dire&on does not occur, the deformation of a lamina should indicate a friction coefficient [ = s/Y 2: 0.8 kp/mm2/14 kp/mm2 = 0.06 at the start of compression 2: 0.8/20 = 0.04 at the end of compression where s is the limiting yield stress in shear of the foil. The trend of results with lead foil, showjn in Figs. 9 and 10 indicates that the friction coefficient decreases as the compression progresses. This agrees excellently with theory. Results from Hill indicate the same effect. A comparison of Figs. 9 and 10 shows that the inferred friction Goefficient for 3 mm specimens with rolled surfaces is slightly lower than for 4 mm specimens and this is in agreement with their different yield stresses. Surface finish and the speed of compression are seen to have negligible influence on the results. Also continuous compression and compression interrupted to take measurements and replace the foil, give negligibly different results. A peripheral contact zone was also observed with lead foil as lubricant. However, its effect on the deformation of the laminae would be expected to be less than with a liquid lubricant. Microscopic examination of laminae with rolled surfaces revealed surface
MEASUREMENT
OF LOW
315
FRICTION
v
Fig. 9. Comparison of calculated lubricated with lead foil; constant
O~lmm
Rolled
and observed values of the extension friction stress is assumed.
of 4 mm thick
Al laminae
Fig. 10. Comparison of calculated and observed values of the extension lubricated with lead foil; constant friction stress is assumed.
of 3 mm thick
Al laminae
cracks after compression similar to those on laminae lubricated with Molypan. Examination of sandblasted surfaces was not immediately possible because of the adhesion of the lead foil to the surfaces of the specimens. A further check on the theory is provided by a comparison of the observed and calculated length of that part of the specimen which is deformed in plane strain throughout the compression. Since only tests with lead foil indicate that plane strain conditions apply throughout, the profile of one specimen after compression was measured. The observed length was 30 mm, whilst the calculated length was 34 mm. Agreement is excellent. Also the final maximum width was 16.0 mm, whilst the width calculated on the assumption of plane strain was 16.3 mm. Hill found similar agreement when this check was applied to his results with a liquid lubricant. Table I summarises all results, giving typical numerical estimations of friction coefficient under various conditions. 5. CONCLUSION
Friction coefficients less than 0.05 can be estimated from the bulk deformation of a long rectangular lamina compressed plastically between platens. Because the method relies on bulk deformation, the results only indicate an equivalent friction coefficient in the case where the frictional conditions change between one point and another on the surface. However, this is usually sufficient for practical purposes. Variations in this equivalent friction coefficient during the compression can be estimated. The method is simple in preparation, instrumentation and procedure and the results show remarkably good repeatability. Errors arising from curvature of the edges of laminae, work-hardening and anisotropy do not invalidate the results. The Amonton friction model and that of constant friction stress are adequate descriptions of the frictional behaviour with liquid and solid lubricants respectively, under test conditions viz. low friction coefficient and normal pressure of the order of the yield stress of the lamina material. In some cases, it was found impossible to avoid peripheral lamina/platen contact and the results of such tests indicated a higher friction coefficient than
I
Molypan
p = 0.03
zero
Surface cracks lamina/platen contact zone unavoidable
Lubricant
Typical friction coefficient
Percentage reduction
Comments
N
50
p:
steel
0.02
and
hardened
Lapped
Platen material
strip
RESULTS
Rolled aluminium
OF NUMERICAL
Lamina material
SUMMARY
TABLE
Al work hardening causes friction coefficient to change
<25
p z 0.02
Molykote
No cracks but edge effect
<45
<40
< 10
No cracks or edge effect
/,l
Wax B 20-40-40
/I-o.01
Vaseline
Wax A 20-55-25
/I = 0.024
Silicone
Surface cracks unavoidable; built-up edge effect apparent
<35
50
zero
p=O.O16
5 = 0.04
i = 0.06
plastic
Vaseline
Acrylic
Filia wax
COEFFICIENTS
Lead foil
OF FRICTION
<30
/I
Wax D 40-604
Many small No cracks, no cracks edge effect but edge effect
._~~..__
$30
,” = 0.01
Wax C 60-o-40
D
z
P 0
MEASUREMENT
317
OF LOW FRICTION
was actually present over the greater part of the surface. However, this contact zone will also manifest itself in practice and so test results are applicable to practical situations. The disposition of specimen material to form surface cracks during compression, is an important parameter in the frictional behaviour because the formation of cracks implies a worsening of the lubrication conditions. With certain waxes, the coeMicient of friction can be as low as 0.01. The increase in the coefficient of friction observed at large compression is unlikely to occur in plane strain model material tests because the expansion of the surfaces adjacent to the transparent boundaries is close to zero in such tests, and so any increase in friction caused by the lubricant having to cover a growing surface area, will not occur. REFERENCES 1 2 3 4
R. A. T. H. be 5 G.
Hill, Phil. Msg., 7 (41) (1950) 733. T. Male and M. G. Cockcroft, J. Inst. Met., 93 (1964) 38. Wanheim, Wear, 25 (1973) 225. B. Pedersen and T. Wanheim, Development of model materials for metal published. T. van Rooyen and W. A. Backofen, Inr. J. Me&. Sci., 1 (1960) 1.
forming
analysis,
to
,1-Elsevier Sequoia S.A., Lausanne
Printed in The Netherlands
LOW FRICTION COEFFICIENT MATERIAL EXPERIMENTS
ESTIMATION
FOR MODEL
D. C. FRICKER and T. WANHEIM Division
of Mechanical
(Received October 5,
Processing
of Materials,
A.M.T.,
Technical
Uniuersity
of Denmark
(Denmark)
1973)
SUMMARY
A previously suggested method of estimating low friction coefficients from the bulk deformation of long rectangular laminae during plastic compression, was used for wax specimens with various lubricants to enable the choice of the best lubricant for experiments with model materials. This method was also used for aluminium specimens with liquid and solid lubricants. Existing theory was extended to cover both the Amonton friction model and that of constant friction stress. Friction coefficients for certain waxes Iubricated with Vaseline were found to be as low as 0.01. With molybdenum disulphide grease on aluminium, the friction coefficient was found to decrease as compression proceeded. Friction coefficients with lead foil were found to agree with their independently known values. Lubrication breakdown at the periphery of specimens was observed in some tests, causing overestimation of the friction coefficient.
1. INTRODUCTION
Model materials, chosen to satisfy known simulation criteria, can be economically very useful in the design of forming processes and tooling. When conducting plane strain model material tests between two rigid parallel boundaries, it is convenient to be able to see the deformation pattern of the surface of a specimen so that the mechanics of a process can be analysed. It is important to minimise interfacial friction between the specimen and the transparent boundaries in order to achieve plane strain conditions i.e. no variations of stress and strain through the thickness of the specimen. Such tests have indicated that the friction present can be very low and so an accurate method of estimating this friction is needed to enable the choice of the best lubricant. This paper uses the method described by Hill’ to obtain values of the friction coefficient, from the comparison of theoretical predictions and experimental results of the increase in length of a long rectangular lamina, during compression between overlapping parallel platens. This method is sensitive for low values of friction coefficient, whereas the method of Male and Cockcroft employing the deformation of a ring of material is less sensitive in this range. These two methods are therefore complementary.
304
D. C. FRICKER,
T. WANHEIM
Since the specimen geometry is chosen such that the ratio of length:width: thickness is initially 2bo: 24,: h, = 20: 2: 1, variations in stress through the thickness and across the width may be neglected in comparison with variations along the length. This makes the deformation amenable to analysis. The theoretical situation is thus one of plane stress since there is no external pressure on the edges of the lamina. For liquid lubricants employed in the experiments described later, it seems reasonable to express the friction stress relationship in terms of the Amonton law of sliding friction r = c1P
(1)
where h is the usual coefficient of friction and p is the normal pressure between the sliding surfaces. However, there may well be a dependence of friction stress z on sliding length, surface topography and pressure-variant lubricant properties, as described by Wanheim3. A sufficiently thin foil of some softer solid acts as an efficient lubricating agent, the friction stress being equal to the yield stress in shear of the foil. The friction stress can be expressed as r = .$_i, = constant
(2)
Theory has been pursued along both these lines of friction stress relationship and the simplified plasticity condition used by Hill has been replaced by the more accurate von Mises criterion. 2. THEORY
(i) The lamina is homogeneous and isotropic, (ii) The platens are rigid and parallel, lamtna(iii) Th e frtc ti on coefficient is constant at all points on the surfaces of the (iv) Th e variations of stress and strain through the thickness are negligible compared with variations along the length i.e. the interfacial friction stress is transmitted equally through the thickness (v) The variations of stress across the width are negligible compared with variations along the length, (vi) The uniaxial yield stress of the lamina Y is constant along the length i.e. no work-hardening. Assumptions (iv) and (v) give a plane stress situation during compression. Because of symmetry, only half the length of the lamina need be considered analytically. Figure 1 shows coordinates, velocities and incremental displacements of any point within the lamina. For an infinitesimally thin section in the left-hand half of the lamina, the stresses are as shown in Fig. 2. Assuming the deformation is slow such that inertia forces may be neglected, the equilibrium equation for this element reduces to
MEASUREMENT
305
FRICTION
OF LOW
Fbsitim of my pint
Velocity
dqlx q+;ij;
”
dh da
t/
Diip4acement
db
Fig. I. Location
of reference
Fig. 2. Stresses
axes within
on an elemental
the lamina.
section
Since the compression is plastic, the von Mises yield criterion must be satisfied, and in this present case 5 & p, so that pz+q2-pq
Differentiation
dq
= Y2
(4)
with respect to x gives
q-3
2q-p TX=----
dp
(5)
dx
The relevant Levy-Mises plastic stress/strain relationships for increments in strain in the x- and z-directions can be combined to give
dE,---
2q-P
d&
2P-q
Now ds,=dh/h
(6)
and ds,=(du/dx)dt
where db=udt.
So at any point
(7) It can be seen from eqn. (6) that the longitudinal strain increment vanishes when becomes equal to p. If a section at distance x’ exists where this plane strain condition holds, a central region of the lamina of length 2 (b-x’) will undergo no extension and, since there will be no relative motion of the surfaces, p and q will have their maximum values of 2Y/J3 and Y/,/3 respectively.
2q
2.2. Friction equations 2.2.1. For a liquid lubricant, where eqn. (1) applies, eqn. (3) becomes hg
= 2pp==
[q+2
(Y2 -
y)
lfromeqn.(4)
Integration by substitution, from one end to a distance x, gives
(8)
306
D. C. FRICKER, T. WANHEIM
px - = i [J3 h
arcsin $$
since q=O at x=0. Plane strain occurs in a central region of the lamina the compression if, substituting the limiting value q= Y/J3,
pbo3
_ 4
from the beginning
of
0.263
From eqns. (7) and (8) d2b 1 _--_-------_ dx dh Integrating
dp
--
dx 2pp
over the current
db,=-gln$
:,
$(ln
half-length
P)
b
since p=Yatx=O.
When pbo/ho 20.263, equal to 2/J3, so
1
the ratio pJY
remains
constant
throughout
the compression,
d&,+&)0.0719 P
Integration
over the compression
gives (9)
In comparison,
Hill predicts
” = (;j bo
* (1 - f)/(l
- 2)
when
,ub/h<0.288
and db,=
(-dh)
y
when pb/h > 0.288
In the present experiments, the nominal specimen dimensions were be/ho = 10. When p~O.0263, pb increases as the compression proceeds and it becomes difficult to determine the theoretical relationship between the fractional increase in length (b/b, - 1) and the fractional decrease in thickness (1 - h/h,). It has been shown that the approximation used by Hill, in place of the von Mises yield criterion, produces little error when p>O.O263. For convenience, his theoretical predictions when p=O.Ol and p-0.02 have been used in this present paper. The relationship when p=O is the well-known parabolic curve. Figures 5-8 show the predictions for various values of p. 2.2.2. When eqn. (2) applies,
z can be expressed
in terms of Y as
s=jY
where i can be considered as the friction of foil and specimen material.
coefficient
for a specific
combination
MEASUREMENT
OF LOW FRICTION
307
Equation (3) becomes h$
=2[Y
so 2iY q=hX From eqn. (7) d2b 1 ---_--dx dh
dp 1 dx 2{Y
Integrating over the current half-Iength b
If there exists a central region of plane strain from the beginning of the compression, then from eqn. (10) 5b,> 1 , i.e. for b~/~~= 10, [ >Oo.029 &I 2J3 In this case, &, will be constant throughout the compression Integration of eqn. (11) gives
and equal to 2Y/,/3.
This theoretical relationship is shown in Figs. 9 and 10. Since the experiments performed were for [z=-0.029, no relationship was derived for { <0.029. Assumption (vi) was made for the sake of theoretical simplicity. When the material is work-hardening, there arises a further term in eqn. (5) involving dY/dx. This is the only departure from ideal plastic theory, which work-hardening causes and the Iamina test materials were such that the variation of Y along the length at any time produced only a negligible error. Assumption (iii) may not hold true in experiments. If this is the case, experimental results will indicate an equivalent friction coefficient which can be estimated from the gradient of the curve of fractionaf increase in length against fractional reduction in thickness, at any time during the compression. 3. EXPERIMENTS
Both wax and metal laminae were used for experiments; five varieties of wax were used and the metal employed was aluminium of commercial purity. Hill shows that the theory becomes less accurate at large reductions in thickness and so all tests were terminated when a reduction of approxjmately 50% had been reached. Measurements of the length and thickness of a lamina after various reductions form the basis of experimental results. Theoretical predictions are based on an initial specimen geometry of b,/h,= IO. In general, the actual geometry departed slightly from this value and so a correction was always applied to
D. C. FRICKER, T. WANHEIM
308
experimental results. By taking measurements when the specimen was unloaded, any elastic deformation which occurred during the compression, was not included in the measurements. Before each test the platens and, in the case of Al, the specimen were thoroughly degreased. 3.1. Wax laminae The varieties of waxes used in tests are listed below, together with their compositions (where known). Name of wax
FiEia
The exact composition is not known (Filia % Indramic (micro- is a registered trade crystalline wax) mark), but it is likely % Harpiks (natural that it has 5&-60% resin) kaolin added to a paraffin wax. “/, Kaolin
A
B
c
I)
20
20
60
40
55
40
0
60
25
40
40
0
~5~~~ositi5n
These waxes were chosen because of their known characteristics4.
and different
stress/strain
Two halves of a specimen were cut from a homogeneous block of cast wax, of nominal thickness 19.8 mm. The two halves were butted end to end and seam-welded to produce a lamina whose nominal initial length and width were 400 mm and 40 mm respectively and whose faces were parallel to within 0.1 mm. Figure 3 shows the press used for testing wax specimens. A hand-operated press was modified to allow the specimen 1 to be compressed between two 5 mm thick acrylic plastic platens 2 and 3, which were kept parallel during compression by two rectangular-hollow-section steel beams 4 and 5, of section 76 x 51 x 5 mm and of length 600 mm. The press and beams were sufficiently rigid to keep the faces of all Iamina parallel to within 0.2 mm. The increase in length of lamina was measured using dial gauges 6 and 7, whose anvils 8 and 9 were in contact with the ends of the lamina. The internal springs of the gauges had been removed so that errors resulting from imposed end-forces and from the anvils being pressed into the ends of the wax, were avoided. The decrease in thickness was measured using a dial gauge 10.
Fig. 3. Apparatus for testing wax Iaminae
MEASUREMENT
OF LOW FRICTION
309
After lubricant had been applied to all surfaces, the specimen was positioned centrally between the platens, the platens and specimen were centred in relation to the press and the test was carried out immediately. This urgency was required because some of the waxes exhibited lubricant-absorbing characteristics. All waxes tested showed notable creep and so after each step in the compression and the removal of the applied load (including the weight of the upper beam), the gauges were read, as nearly as possible, immediately. It was impossible to renew the lubricant between steps because any disturbance of the wax would have resulted in unaccountable deformations. Three lubricants were employed in tests with Filia e;iz. vaseline, Molykote DX paste and a silicone spray. The effects of surface roughness were studied by performing tests where the surfaces of the lamina had been knurled and also where lateral slits had been cut in the surfaces. Only one lubricant was used in tests with waxes A, B, C and D. Results with Filia showed that Vaseline was the best lubricant of those tested and so this was the choice for further tests. Some of the tests indicated that the built-up edges formed in cutting a specimen, might influence the results considerably. In such cases, additional tests were performed after careful removal of these edges. 3.2. Aluminium laminae Specimens were machined from rolled strips of thickness 5 mm, 4 mm, and 3 mm, such that their dimensions were in the nominal ratio of 1ength:width: thickness= 20:2: 1 and their anisotropic behaviour was minimised. Laminae faces were not machined and were initially parallel to within 0.02 mm. Experiments were performed in a 60 tonne universal testing machine fitted with an automatic speed regulator which allowed deformation to be carried out at constant speed. Rectangular hardened-steel platens were used and these had been lapped to produce a surface finish which was measured to be R, =0.094 pm and R,,, = 1.32 pm. Specimens were positioned centrally on the platens to ensure uniform thickness reduction. Length measurements were taken along the centre-line of the specimen and thickness measurements were taken at different sections. Usual speeds of reduction were chosen, such that the strain rate would be the same in all tests, as follows: Initial nominal thickness mm Speed mm/s
5 0.025
4 0.02
3 0.015
Laminae were tested both with their original rolled surface finish and also after their surfaces had been sandblasted. This enabled a conclusion to be drawn about the influence of surface finish on the deformation of a lamina. Two lubricants were employed in tests with Al specimens viz. (1) a liquid lubricant, Molypan N (2) a solid lubricant, lead foil. 3.2.1. Molypan N This lubricant was a molybdenum disulphide high-pressure grease and was applied in excess to all surfaces, after the specimen had been removed from the platens to enable its dimensions to be measured. When the laminae were machined
310
D. C. FRICKER,
‘I’. WANHEIM
from the strip, built-up edges were formed, of greater thickness than the bulk of the specimen. To investigate this factor, a 4 mm specimen having a rolled surface finish, was tested after these edges had been removed by hand with a flat grindstone. An additional test at a speed of 0.18 mm/s was performed with a 4 mm specimen, to investigate the influence of speed on the deformation of a lamina lubricated with a liquid. 3.2.2. Lead foil Three sheets of lead foil were roiled to thicknesses of 0.18, 0.1 and 0.05 mm. Tests were carried out using specimens of 4 mm and 3 mm thickness and after measurements of the dimensions of a specimen had been taken, a new piece of overlapping foil was used. It was found that the foil adhered strongly to a sandblasted surface, so that the foi1 could not be replaced during a test. Furthermore, measurement of the thickness’ of the specimen was overestimated because of the presence of the foil and a correction based on volume constancy of both the specimen and the foil was applied. One test at a speed of ten times the usual speed was carried out on a specimen of thickness 4 mm. Two continuous tests were performed on 3 mm specimens at speeds of 0.15 mm/s and 0.015 mm/s. These tests investigated the influence of stepwise deformation.
6-
VUXA
Filia
a1
CL2
oI3
Fig. 4. Compressive Fig. 5. Comparison friction is assumed.
o,fl
a5
stress-strain of calculated
0.6
0.7 E
curves of three varieties and observed
values
of waxes. of the extension
of Filia
laminae;
Amonton
MEASUREMENT
As cast
OF LOW
311
FRICTION
and cut
6
‘-t;;
1-e
Fig. 6. Comparison of calculated and observed values of the extension lubricated with Vaseline; Amonton friction is assumed.
of laminae
of waxes
Fig. 7. Comparison of calculated and observed values of the extension lubricated with Vaseline; Amonton friction is assumed.
of laminae
of waxes C and
4. RESULTS
A and B
D
AND DISCUSSION
In all tests, the curvature of the edges of laminae was so small that no significant error was introduced into the results by this effect. This affords evidence of the validity of the theoretical assumption (iv). 4.1. Wax ~amina~ The uniaxial compression curves of three varieties of waxes used in experiments, are given in Fig. 4. The results of tests with wax under various conditions are shown in Figs. 5, 6 and 7. It can be seen that all the results show that the friction coefficient is constant until a certain point in the compression has been reached, whereupon the coefficient steadily increases as the compression is continued. Since it was impossible to relubricate the surfaces during a test, lubricant escaped gradually from the interface, both at the periphery and also into any cracks which were formed. Furthermore, the surface area increased as compression proceeded, so that breakdown in lubrication occurred at some time. Further compression increased the number and area of these regions of breakdown, causing an increase in friction coefficient. From Fig. 5, it can be seen that Vaseline possesses a much lower friction
312
D. C. FRICKER,
T. WANHEIM
coefficient than the other lubricants and breakdown occurs at a later point in the reduction. This is possibly explained by the semi-solid nature of Vaseline, tending to separate the specimen and platens better than did the other lubricants. Longitudinal cracks in the surfaces of Filia laminae after compression were evident, the size of cracks after a given compression increasing as the original quantity of the lubricant increased. Such cracks are formed by the pressurised lubricant forcing its way into existing depressions; since the internal compressive stress is less in the width than in the length direction, it is easier for specimen material to flow in the width direction, forming a longitudinal crack. Therefore results shown are for a minimally thick application of lubricant. Figure 5 also shows that the friction coefficient is initially unaffected by the surface topography but that the break-point occurs earlier in the compression when escape paths exist in the original surface. The knurled surface indentations were sufftciently large to be able to receive lubricant squeezed from the interface and so no additional cracks were apparent. The pitch of the lateral slits was not small enough to accommodate all escaping lubricant and cracks were formed between the slits. Lubrication breakdown at the periphery of specimens, resulting in some direct contact between the lamina and platens, was observed. It is worth considering the effects of this peripheral contact zone on the deformation of a lamina lubricated with a liquid. When this zone is formed, it effectively seals the major escape path of the lubricant still present on the surface. Apart from escaping into any cracks which may be formed in the surface, the lubricant must remain separating the surface of the lamina from the platen. Because of the relatively high interfacial friction in this zone, additional compressive stresses will be exerted on the bulk of the lamina, which are not considered in the analysis. Since the friction stress is assumed to be dependent on the normal pressure over the greater part of the surface, the presence of this zone would produce higher friction stresses than the lubrication conditions suggest. A slight decrease in friction coefficient can be seen when the built-up edges were removed, and this agrees with the observation that the width of the peripheral contact zone was less in this case. This effect can be seen strikingly in Fig. 6. Wax A showed no cracking whatsoever and no signs of wax/platen contact at the periphery of the laminae. Wax B showed no surface cracks but wax/platen contact was apparent. The friction coefficients are lower and the break-points occur later in the compression with these waxes, as compared to Filia. Figure 7 again shows this built-up edge effect. Wax C exhibited many small surface cracks and this accounts for its break-point occurring sooner than with waxes A and B. Wax/platen contact was not observed, however. Wax D showed no surface cracks and the wax/platen contact zone was much smaller in the test where the built-up edges had been removed, than when they had not. The coefficient of friction is seen to be less in the former case and its break-point to occur later in the reduction. These results lend weight to the argument that the existence of peripheral contact between the lamina and the platens affects the bulk deformation of a lamina, and hence the magnitude of the apparent friction coefficient. Further evidence of this effect can be seen from results with Al laminae.
MEASUREMENT
OF LOW FRICTION
313
The yield stress of typical Al specimens was inferred from hardness tests on the bulk metal; these values are shown below: Initial thickness of specimen (rolled surface) mm percentage reduction Y kp/mm2
3 zero 15.6
3 250 >21
4 zero 12
4 “50 20.5
In all tests with Al laminae it was found impossible to prevent lubrication breakdown at the periphery of the specimens. This phenomenon has also been observed by van Rooyen and Backofen”.
Figure 8 shows that the results for 4 mm specimens suggest a slightly higher value of friction coefficient than those for 3 mm and 5 mm laminae. Although no definite explanation is apparent, this discrepancy is possibly due to the combined effects of different size and of different work-hardening characteristics of the specimens. It can be seen that increasing the speed of the compression slightly increases the friction coefficient. This is possibly due to the breakdown of peripheral lubrication being retarded by the higher speed of the escaping lubricant, so that less lubricant is trapped when the contact zone is finally formed. Figure 8 also shows that sandblasting the surface of a specimen increases the 04
swfoG? sp?dmm& r5mm Rolled 0,025 ohm -II0.02 035 r3mm -**0.015 w3mm Sandblasted aa o4mm -I0.02 n4mm Rolled 0.18 03 i x4mm Rolled i built0.02 %rn% 0.25 ~
0.2 -
Fig. 8. Comparison of calculated and observed values of the Molypan N: Amonton friction is assumed.
extension
of Al
lubricated with
314
D. C. FRICKER, T. WA~HEI~
friction coefficient. Microscopic examination of rolled and sandblasted surfaces after compression, revealed that both surfaces have cracks in the longitudinal direction. The sandblasted surface appeared pitted and possessed deep cracks, whereas the rolled surface had less deep cracks and appeared smoother. Since these cracks form an escape path for the liquid lubricant, it follows that a sandblasted surface would produce a worse lubrication situation than a rolled surface. The only apparent explanation for the significantly lower friction coefficient indicated in one test with a 3 mm specimen is the effect of the peripheral contact zone: the width of this zone was much less in this test than in all other Al tests. It can be seen that the results of Al tests with and without the built-up edges having been removed, are identical. On the lamina with these edges removed, lubrication breakdown still occurred and a zone was formed. The trend of tests with Molypan indicates that the apparent friction coefficient decreases as the compression progresses. Van Rooyen and Backofen have also observed decreasing friction coefficient, although their experimental conditions were sufficiently different from those here to limit the applicability of their explanation of this effect to the present results. It is likely that the liquid lubricant possesses pressure-dependent characteristics but this cannot fully explain these results. One inaccuracy of the theory, described fully by Hill, caused by the gradual inapplicability ofassumption (v), does not explain this trend. Even after 50% reduction, the length/width ratio was still about 6. Although the results with a 5 mm specimen agree with other results, the specimen showed pronounced anisotropic deformation. It is thought that the large surface of this specimen caused non-uniform elastic deformation in the platens. With 4 mm and 3 mm specimens, the elastic deformation was small and the faces of laminae were parallel to within 0.02 mm. The results to demonstrate, however, that this experimental method is not particularly sensitive to such anisotropic behaviour. 4.2.2. Lead foil A direct check on the theory is possible when the frictional properties of a lubricant are known independently. So long as extrusion in the width dire&on does not occur, the deformation of a lamina should indicate a friction coefficient [ = s/Y 2: 0.8 kp/mm2/14 kp/mm2 = 0.06 at the start of compression 2: 0.8/20 = 0.04 at the end of compression where s is the limiting yield stress in shear of the foil. The trend of results with lead foil, showjn in Figs. 9 and 10 indicates that the friction coefficient decreases as the compression progresses. This agrees excellently with theory. Results from Hill indicate the same effect. A comparison of Figs. 9 and 10 shows that the inferred friction Goefficient for 3 mm specimens with rolled surfaces is slightly lower than for 4 mm specimens and this is in agreement with their different yield stresses. Surface finish and the speed of compression are seen to have negligible influence on the results. Also continuous compression and compression interrupted to take measurements and replace the foil, give negligibly different results. A peripheral contact zone was also observed with lead foil as lubricant. However, its effect on the deformation of the laminae would be expected to be less than with a liquid lubricant. Microscopic examination of laminae with rolled surfaces revealed surface
MEASUREMENT
OF LOW
315
FRICTION
v
Fig. 9. Comparison of calculated lubricated with lead foil; constant
O~lmm
Rolled
and observed values of the extension friction stress is assumed.
of 4 mm thick
Al laminae
Fig. 10. Comparison of calculated and observed values of the extension lubricated with lead foil; constant friction stress is assumed.
of 3 mm thick
Al laminae
cracks after compression similar to those on laminae lubricated with Molypan. Examination of sandblasted surfaces was not immediately possible because of the adhesion of the lead foil to the surfaces of the specimens. A further check on the theory is provided by a comparison of the observed and calculated length of that part of the specimen which is deformed in plane strain throughout the compression. Since only tests with lead foil indicate that plane strain conditions apply throughout, the profile of one specimen after compression was measured. The observed length was 30 mm, whilst the calculated length was 34 mm. Agreement is excellent. Also the final maximum width was 16.0 mm, whilst the width calculated on the assumption of plane strain was 16.3 mm. Hill found similar agreement when this check was applied to his results with a liquid lubricant. Table I summarises all results, giving typical numerical estimations of friction coefficient under various conditions. 5. CONCLUSION
Friction coefficients less than 0.05 can be estimated from the bulk deformation of a long rectangular lamina compressed plastically between platens. Because the method relies on bulk deformation, the results only indicate an equivalent friction coefficient in the case where the frictional conditions change between one point and another on the surface. However, this is usually sufficient for practical purposes. Variations in this equivalent friction coefficient during the compression can be estimated. The method is simple in preparation, instrumentation and procedure and the results show remarkably good repeatability. Errors arising from curvature of the edges of laminae, work-hardening and anisotropy do not invalidate the results. The Amonton friction model and that of constant friction stress are adequate descriptions of the frictional behaviour with liquid and solid lubricants respectively, under test conditions viz. low friction coefficient and normal pressure of the order of the yield stress of the lamina material. In some cases, it was found impossible to avoid peripheral lamina/platen contact and the results of such tests indicated a higher friction coefficient than
I
Molypan
p = 0.03
zero
Surface cracks lamina/platen contact zone unavoidable
Lubricant
Typical friction coefficient
Percentage reduction
Comments
N
50
p:
steel
0.02
and
hardened
Lapped
Platen material
strip
RESULTS
Rolled aluminium
OF NUMERICAL
Lamina material
SUMMARY
TABLE
Al work hardening causes friction coefficient to change
<25
p z 0.02
Molykote
No cracks but edge effect
<45
<40
< 10
No cracks or edge effect
/,l
Wax B 20-40-40
/I-o.01
Vaseline
Wax A 20-55-25
/I = 0.024
Silicone
Surface cracks unavoidable; built-up edge effect apparent
<35
50
zero
p=O.O16
5 = 0.04
i = 0.06
plastic
Vaseline
Acrylic
Filia wax
COEFFICIENTS
Lead foil
OF FRICTION
<30
/I
Wax D 40-604
Many small No cracks, no cracks edge effect but edge effect
._~~..__
$30
,” = 0.01
Wax C 60-o-40
D
z
P 0
MEASUREMENT
317
OF LOW FRICTION
was actually present over the greater part of the surface. However, this contact zone will also manifest itself in practice and so test results are applicable to practical situations. The disposition of specimen material to form surface cracks during compression, is an important parameter in the frictional behaviour because the formation of cracks implies a worsening of the lubrication conditions. With certain waxes, the coeMicient of friction can be as low as 0.01. The increase in the coefficient of friction observed at large compression is unlikely to occur in plane strain model material tests because the expansion of the surfaces adjacent to the transparent boundaries is close to zero in such tests, and so any increase in friction caused by the lubricant having to cover a growing surface area, will not occur. REFERENCES 1 2 3 4
R. A. T. H. be 5 G.
Hill, Phil. Msg., 7 (41) (1950) 733. T. Male and M. G. Cockcroft, J. Inst. Met., 93 (1964) 38. Wanheim, Wear, 25 (1973) 225. B. Pedersen and T. Wanheim, Development of model materials for metal published. T. van Rooyen and W. A. Backofen, Inr. J. Me&. Sci., 1 (1960) 1.
forming
analysis,
to