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Friction at high normal pressures
Wear, 25 (1973) 225-244 IQ Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
FRICTION
225
AT HIGH NORMAL PRESSURES*
T. WANHEIM Department
of Mechanical
Technology, Technical University of Denmark, 2800 Lyngby (Denmark)
(Received March 26, 1973)
SUMMARY
Friction conditions between tool and workpiece in metal working are of the greatest importance to a number of factors such as force and mode of deformation, properties of the finished specimen and resulting surface roughness. It is shown, theoretically and experimentally, that the Amonton friction law expressed by r =w does not apply when normal pressure is higher than approximately the yield stress of the specimen; in this case it is necessary to consider the frictional stress as a function of normal pressure, surface topography, length of sliding, viscosity, and compressibility of the lubricant. The theoretical work was carried out by means of upper bound and slipline field analysis based on experiments with model surfaces in wax and metal. The theoretical model applied is one of multihole extrusion, the material beneath the valleys of the workpiece surface being extruded up towards the tool when the real area of contact exceeds a certain value. The effect of the trapped lubricant is to build up a back-pressure on the extrusion process. The experimental work was carried out with newly developed equipment enabling direct determination of the abovementioned function; construction and calibration of the equipment are described. The equipment allows determination of frictional stress on a surface with well-defined values of normal pressure, sliding length, and sliding velocity. The normal pressure can attain about 8 times the yield stress for commercially pure aluminium. The results obtained show reasonably good agreement between theory and experiment, and a dependence of the frictional stress on the sliding length, this dependence being a function of normal pressure.
1. INTRODUCTION
The two empirical laws most often applied to describe the tangential stresses between specimen and tool by metal working processes are the Amonton law
* Paper presented at the First World Conference on Industrial Tribology, New Delhi, December, 1972.
226
T. WANHEIhl
and the expression for stiction r=k
(2)
It is assumed that eqn. (1) is sufficient to describe the conditions until the product of friction coefficient p and normal stress q becomes higher than the yield stress in pure shear k of the material. The material will then stick to the tool and yielding take place in the interior of the material. If the transmitted friction stress z is plotted as a function of the normal pressure, a curve consisting of two straight lines results, Fig. I.
Fig. 1. Amonton friction and sticking friction US.normal pressure.
The well-known adhesion theory, Bowden and Tabor’ and others, based on the individual plastic deformation of the contact points, yields ,u=k/H
where H is the hardness of the workpiece. The ratio between real area of contact and apparent area is normally very small. If the normal pressure grows, the actual contact area will grow and approach the apparent one. This approach will probably be asymptotical, as very high pressures will be needed in the last phase. Since according to the adhesion theory, friction stress is proportional to the real area of contact, Fig. 2 can be drawn, where the ordinate is z and the asymptote has the equation z = k.
Fig. 2. Frictional stress US.normal pressure2. This curve resembles the broken curve (Fig. 1) but with no sharp line of division between the two regions’. With a lubricant, three models are used to describe its effect, viz. either hydrodynamic lubrication, boundary lubrication, or hydrostatic lubrication. In the latter, well-known in metal forming, the lubricant is trapped in numerous small
FRICTION
AT HIGH
NORMAL
227
PRESSURES
pockets formed in the surface of the specimen when tool and specimen meet. The pressure in these is sufficiently high to transmit the necessary normal stresses, whereas the transmissible tangential stress is modest3. 2. EXPERIMENTS
WITH
MODEL
SURFACE
IN MODEL
MATERIAL
To investigate the asymptotical approach of the real contact area towards the apparent one with increasing normal pressure, and to follow the deformation in detail, a plasticine model in plane strain with a painted grid was used. An idealized plane model of a surface was modelled using straight-sided asperities. The average slope on an average surface lies between 5” and 10”; however, 15” was chosen to obtain larger and more distinct deformations in the plasticine. Preventing surface expansion when pressing a tool down on it, the following two principal characteristics were noticed: ’ (a) Deformation of the grid penetrates far below the “surface”, presumably 0.3 of the distance between the tops in the last phases. (b) The slope of the asperities is apparently constant, or close to constant, although the bearing factor c1approaches 1, a being the ratio between real and apparent area of contact. Calculations with upper bounds clarify that the slipline solution of the problem might be analogous to the solution of inverted extrusion proposed by Hi114,where CIequals the extrusion reduction r. The development of the area of contact is shown in Fig. 3 (a-h) at its different stages. The starting point is Prandtl’s slipline solution of indenting a plane tool into a semi-infinite plate, modified to the present application.
b
Fig. 3. Proposed
development
of slipline field for asperity
deformation
at high contact
pressures.
T. WANHEIM
228
Figure 3(a) shows conditions for a==O.l. Figure 3(b) is the last state in this part of the solution where the isolated deformations touch. This value of a is very close to 0.33. It is probable that the slipline picture changes to the one shown in Fig. 3(c), which has been drawn on the basis of Hill’s extrusion solution. Figure 3(d) shows the field for cx=O.5, where it simply consists of triangles and circular arcs. This field leads the way to the conditions of c1>0.5 shown in Figs. 3(e-g). Figure 3(g) shows the last stage in this field; here the boundary slipline has reached horizontal. This occurs for a=O.88. Figure 3(h) has been constructed on the basis of a development of this field pointed out by Johnson5. This field can be applied for 0.88 < c1< 1. It must be emphasized that the slipline solution mentioned here only applies to a hard (smooth) tool and a soft (rough) specimen in static contact, and must be modified if the tool has a tangential movement relative to the specimen and r#O. For small z values, however, the changes will probably not be very drastic. An objection may be that the yield criterion is violated in the stiff region containing the valley, since the angle between the free surface and the bordering slipline is less than 45”. A slight degree of work-hardening could justify this error since the angle in most cases will exceed 40”. The slipline solution indicated here will be experimentally verified in the next section. 3. EXPERIMENTS
WITH MODEL
SURFACES IN METAL
A number of experiments were made with metallic model surfaces to investigate the variation of c( with the normal pressure both with and without a confined lubricant. 3.1. Equipment and experimental procedure
In a container consisting of two halves clamped together a specimen of commercially pure aluminium, 3@ x 40, was coined to get a surface consisting of concentric sloping ridges with 1.5 mm spacing. 7 coining tools were produced of hardened tool steel, 6 with slopes ~‘2.5”; 5.0”; 7.5”; 10.0”; 12.5”; and 15.0”, and one plane lapped tool. When the container is clamped together, the coining tools can move in it with a running fit. The experiments were carried out as follows: A specimen with plane end surfaces was placed in the container and coined at approx. 40 tons, using a thin layer of mineral oil as lubricant. After disassembling of the container the specimen was removed and annealed for 2 hours at 360°C. A diagram of the surface was obtained in a Perth-o-meter. The specimen was then placed in the container, flattened with the plane coining tool in a compression test machine to the force desired, removed, and another surface diagram obtained. Two diagrams of each specimen were drawn on the Perth-o-meter along two radii 180” apart, making measurement of the real area of contact at a given load possible. The specimen was replaced in the container, loaded to a higher force, removed, and diagrams obtained. There was no annealing between loadings. An example of the diagrams is shown in Fig. 4; (a) shows a diagram of the specimen after coining with a 7.5” tool, and (b) shows the specimen after loading to 10.0 tons. The aluminium used was from a single rod; the stress-strain curve of the
FRICTION
AT HIGH NORMAL
PRESSURES
229
230
“I’.WANHEIM
material was found to obey: CT= 15.98 f P-281 kp/mm* 3.2. Experiments 6 specimens were coined with the tools mentioned and loaded with the plane tool to 2.5; 5.0; 10.0; 15.0; 20.0; 25.0 tons, with intervenjng drawing of diagrams. The specimens to be loaded with trapped lubricant (castor oil and kerosene) were coined with the tool with profile angle 10.0”, and annealed. Then one specimen was loaded to 5 tons, No. 2 to 10 tons etc., until the 6th specimen which was loaded to 30 tons. Loading was made in the following way: After the container with the specimen had been placed in the testing machine, the lubricant was poured in to approx, 5 mm over the specimen. The plane tool was positioned and a load of 2.5 tons applied with a constant low speed. At 2.5 tons the load was held for 2 minutes, and then the required load was apphed at a constant low speed. This procedure prohibited variation in lubricant volume. This experimental technique can only be used for comparative measurement since the autermost ridge of the specimen, which contributes proportionally most to the area, does not have the same conditions as the others, having only lubricant under pressure at its inner side. consequently the real areas of contact measured for the fubricant experiments will be somewhat too large. The results from the diagram measurements from these experiments are shown in Fig. 5.
o-
o-D ----0 -of e-t0 - -
~~25' y" 5:0* y2 7-5' v=tE1D' v=lZ5" y=lS,O'
20
30
40
qkQ/ld
Fig. 5. Loading of model surfaces. Relatiw~ contact area as a function of normal ~E%suK.
The real area of contact without lubricant is seen to approach CC=1 asymptotically, with the smallest profile angles at the top. This effect is no doubt due to the strain-hardening, which will have most effect on the steepest ridges. Provided that the slipline solution stated apphes, the steepest ridges would be at the top for smal cr-values, where deformation on each top is considered separatefy, and when the extrusion mechanism has commenced, there would be no dependence on the profile angle. The strain hardening is apparently able to overshadow this effect.
FRICTION
AT HIGH NORMAL
PRESSURES
231
From the experiments with a confined lubricant, Fig. 5, this is of paramount importance to the bearing factor, probably mainly dependent on the topography of the surfaces and the viscosity and compressibility of the lubricant. At high pressure of approx. 4 x yield stress calculated for s=O.l the real area of contact is approx. 95% for dry deformation, approx. 55% with confined kerosene, and approx. 25% with confined castor oil. 3.3. Experimental verification of the slipline solution The assumed slipline solution has been verified in three ways by means of these experiments, namely by comparison of the calculated and measured forces, measuring the extension of the deformed zone by means of micro-hardness measurements, and measuring the profile angles for all deformation stages. 3.3.1. Comparison of calculated and measured forces The theoretical pressures for different values of a calculated for the slipline field were comparable with the measured values. By means of hardness measurements, it is possible to get an estimate of the yield stress and thereby the equivalent strain. Hardness measurements(Vickers 5 kp) were made on the undeformed material and on the flat tops of the deformed specimens after the last loading (25 tons). These last measurements were performed in the second ridge from the outside with 4 measurements on each of the six specimens deformed without lubricant. The mean of the four specimens with the steepest sides was calculated, while the two specimens with y = 2.5” and 5.0” were not included. Tabor’ has shown that the average strain during hardness indenting is very close to constant and equals 8%. Using this, the empirical stress-strain diagram of the material, and the proportionality between yield stress and hardness number, the equivalent strain after the last deformation can be calculated from the hardness measurements. It is also assumed that the equivalent strain under the asperities, as long as the deformation of these can be regarded individually, is the same as the equivalent strain during a hardness test, i.e. s=0.08, and that the equivalent strain Egrows linearly with ~1from the moment the extrusion mechanism becomes effective, i.e. a=O.338 for y= 11.25”. Thus one obtains: E = 0.08
for 0 < CI< 0.338
(4)
E = 0.073 a + 0.055 for 0.338 < u < 1 On this basis and using von Mises yield criterion, 2 k was calculated as a function of c1 for y= 11.25”, mean of the 4 slopes used, and in Fig. 6 q/2k is compared with the measured values. There is good agreement between calculated and measured values. 3.3.2. Measuring the size of deformation zone Figure 3 shows that even if the area of contact is changed, the deformed layer apparently keeps its thickness. Microhardness measurements must reveal this phenomenon on the aluminium specimens, as a rather well defined hardness plateau must be expected within the limitation of the sliplines if the solution holds.
232
T. WANHElM
Fig. 6. Theoretical and measured area of contact.
Fig. 7. Theoretical extension of slipline field.
From Fig. 7 it will be seen that the theoretical extension of the field relative to a fixed coordinate system can be divided into the extension of the field d measured from the surface to the deepest point and the movement of the too1 6. While d can be read from the slipline solutions for the individual cr-values, 6 is calculated by using volume constancy and the assumption that the profile angles keep constant during the whole process. It is obvious that this last assumption is closely connected with the slipline solution and is experimentally verified in section 3.3.3. If the distance between two asperities is called t, we get s/t = &x( 1 -&a) tg)J
(5)
For y = 15.0” and 0.365 < CI< 0.879 we have 0.368 < z/t < 0.410 where z = d + 6. Setting t= 1.5 mm, this defines a strain-hardened layer O-554.61 mm thick. To investigate this experimentally, a cut was made at right angles to the deformed surface after last loading in one of the specimens, Y”,,,,,= 15”. Under the 4th ridge from the outside a series of micro hardness measurements with 25 p load was made. Furthermore microhardness measurements were made in the undeformeb material far from the surface. In all approx. 80 microhardness measure’ments were made, 72 inside an area of 1.5 x 1.2 mm, placed in 12 rows and 6 columns. The 6 values in each row were reduced to three calculating averages
FRICTION AT HIGH NORMAL PRESSURES
233
Fig. 8. Micro hardness distribution under a ridge Battened to a=0305
of symmetrical points. The result is shown in Fig. 8 where the lines indicating points with the same hardness produce a well defined hardness plateau as predicted by slipline field theory. Theoretically determined points of the boundary correspond tolerably well to the boundaries of the plateau. Ex~riments show that the hardness plateau extends deeper into the material than predicted by theory. This can be explained by the strainhardening. To ensure that the hardness variation was not due to a small number of large grains in the material, the specimen was etched and photographed; only approx. 2-3 hardness measurements were observed in each grain. 3.3.3. The variation ofthe profile slope If the slipline field solution holds, the profile slope will be constant for all
181 16
02
U
0.6
w
x0
a
Fig. 9. Variation of profile slope with a.
234
T. WANHEIM
values of the factor of the area of contact CLThe diagrams shown in Fig. 4 and those analogous to the other profile slopes make a direct control of this possible. The profile slope was measured in 6 places on each diagram, mainly in the 3rd, 4th, and 5th valley from the outside. 12 angles were measured for each specimen after each load. Results are shown in Fig. 9, giving j! as a function of CC Although the average values of the angles measured in this way are too high compared to the nominal angles, the curves show that :’ is independent of M within the estimated measuring precision. These experiments are also in agreement with the “extrusion theory”. 3.4. Extended friction theory Using the “extrusion mechanism” described and the dependence of the area of contact on the normal pressure with and without confined lubricant, it is possible to establish a theoretical connection between friction conditions at low normal pressures and friction conditions at high normal pressures. Figure 10 shows a cut through a contact between tool and specimen. The dimension at right angles to the paper plane is large.
q
I
bbbbbbbbbbb
Fig. 10. Stresses in surface and lubricant
Vertical equilibrium gives:
If the pressure on the real area a A, which is necessary to bring the deformation to the value CI without confined liquid, is called pdr it is seen that Pd+Pf
(7)
=Pm
as the pressure in the liquid pf can be regarded as an extra hydrostatic pressure on the slipline field, or qd
=
a(Pm-Pf)
where qd is the pressure, averaged, on the entire area, necessary to bring the deformation to the value c1without confined liquid. From these equations we have Pf
=
(9)
q-qd
and pm=q+-
l-u a
qd
(10)
Figure 11 shows the determination of pf if a curve as Fig. 5 is established. Assuming that tangential sliding takes place following the Amonton law at the contact points and zero friction at the pockets, we have
235
FRICTION AT HIGH NORMAL PRESSURES
Fig. 11. Determination of hydrostatic pressure pr in trapped lubricant.
where as a condition contact:
we state the sticking criterion applied on the real area of (12)
as the yield stress of the material does not change by a su~rimposed pressure. One gets:
hydrostatic
(13)
r = /@q+(I-a)qJ expressing written
the connection
z= f(q) earlier mentioned.
The sticking criterion
is
or z = elk
(14)
It is now possible to calculate the expected z = f(q) for the three lubricant conditions: castor oil, kerosene, and dry, by applying the curves 5 or 6 and the expressions (13) and (14), which also demands knowledge of ,u and k. Theory will be compared with experimental measurements described in the next section. Calculations of k-values and friction coefficients p experimentally determined at low normal pressures (q <<2 k) with specially developed equipment show that o! k is aImost without exception smaller than ,a(aq+(l -a)%), which, if the same p is valid, means that for the three lubricating conditions examined there will be full stiction at the points of contact, and the friction is therefore determined by the area of contact CI,which is a function of the pressure, lubricant, and, as will later be shown, of the surface and sliding length. Attention is drawn to the similarity between eqn. (14) and the often used “proportional friction law” z = mk 0 < m < 1. In the next section the theory stated here will be verified experimentally. The author is aware of several of the great flaws in this “theory”. One of them, a slipline field solution allowing r#O in the surface, has been overcome. This solution will be published shortly.
236 4. HIGH
T. WANHEIM PRESSURE
FRICTION
EXPERIMENTS
Equipment was developed for measuring frictional stresses for normal pressures larger than the yield stress of the specimen when closely controlling the parameters normal pressure, sliding length, and surface roughness. 4.1. Equipment The principle of the experimental set-up is shown in Fig. 12. With the aid of tools 1 and 2, the specimen, 3, is pressed against a ring, 4, on which a ring, 5, is shrunk. With the aid of straingages, 6, mounted on the thick-walled ring, the surface pressure between specimen, 3, and inner ring, 4, may be measured. The pin, 7, centers tools, 1, and, 2, and specimen, 3. In the inner ring are eight pins, 8, which transfer the torque from the wire, 9, through the wire disk, 10, and disk, 11, and slip occurs between specimen, 3, and ring, 4.
L
’
+ Fig. 12. Principle
of measurement.
The experiments were performed on a dual-action hydraulic 320 ton deepdrawing press, where the main ram created the high normal pressure between specimen and ring. The wires, 9, run over two wheels and are attached to a frame, which is moved vertically by the auxiliary ram of the press. A potentiometer gives the position of the auxiliary ram and thereby the sliding length. Measurement of the frictional force is made possible by 4 straingage-equipped rings connecting four upward moving bars with the frame. Two cradles with ball joints are inserted to ensure zero sideforces on the specimen. The position of the wire-disk is controlled exactly by four ball bearings mounted independently on lightly springloaded supports. During the experiment the measurement signals were recorded on a u.v.recorder. The pressure ring is instrumented with 4 straingages active in the direction of the periphery. Between these, 4 dummygages are mounted on individual plates of material similar to the ring and ground to the same curvature as the ring. They are secured to the ring by rubber bands. The calibration of the pressure ring was performed with a rubber disk in the place of the specimen in the ring, the stress in the rubber disk in the chamber was assumed to be hydrostatic. The thickness of the rubber specimen under calibration was measured by means of a dial gage between the supports of the two tools. The elastic deformation of the tools was corrected for in a deadrun.
FRICTION
AT HIGH
Fig. 13. Calibration
NORMAL
of pressure
PRESSURES
237
ring with rubber.
Figure 13 shows the calibration curve obtained. As indicated there is good agreement between the points plotted with increasing and decreasing loading. However, the plot shows a discontinuity in slope at approx. 5,000 atm. The calibration was repeated a number of times with other rubber disks, with the same result. Bridgman* reports regarding compressibility measurements: “B. Synthetic and Natural Rubbers The measurement of these was undertaken at first for the entirely practical purpose of finding some way of predicting which might be expected to function best for high pressure packing on the piston of the apparatus for 30,000 kg/cm2. Ten different specimens of various synthetic rubbers were collected from dealers in the neighbourhood of Boston. Measurements disclosed an interesting feature, namely in some of them there was a rather sharp change in the direction of the curve of volume uersus pressure, or discontinuity of the second kind, at pressures roughly of the order of 5,000 kg/cm2”. It seems very likely that the discontinuity in compressibility discussed by Bridgman is caused by the same change in material properties shown in Fig. -13. To verify this point, additional calibrations were performed with soft PVC, which showed a change in slope at about 3,000 atm; the initial part of the curve had a slope similar to that of the calibration curve for rubber. The sensitivity of the transducer was then taken as the slope of the rubber curve for 0 < q < 5000 atm. 4.2. Materials and surfaces The specimen material contained Al 99.25x, Fe 0.55x, Cu 0.02x, Si 0.16%. The chamber in which the specimen is exposed to the high pressure is formed by two cylindrical tools and a ring. The plane surfaces of the tools are ground and then sandblasted to provide a better grip of the specimen. The ring against which the specimen slides, is manufactured from tool steel, heat-treated and annealed at 200°C. The composition of this material is: C 0.36x, Mn 0.70x, Si 0.30x, Cr 1.40x, Ni 1.40x, MO 0.20%. Two surface roughnesses were used in the ring. Surface R 1 is produced by grinding followed by honing and R 3 is obtained by sandblasting R 1. The R, and R,, values (pm) are obtained partly by direct measurements and partly
238
T. WANHEIM
TABLE I Surface
RI R3
Axial directiorl
Circumferential direction -~.__~..~_._~
R, R-x ____ ..~_._._
R*
R InBI
0.091 2.03
0.143 1.94
~ 13.0
~~ 14.2
TABLE I1 Surface
Manufacture
Feed
Nose
z-Direction
cp-Direction
(mmjrev)
R
R,
R,
R,.w
0.49
2.03
0.66
2.92
R,,,
(mm) -
EM 1 EM2
annealing + turning annealing + turning
0.065
1.3
1.08
0.3
0.5
6.0
._.
4.32 24.4
____..~
Turned surfaces: n= 1000 o/min. t=O.l mm Cutting fluid: alcohol
by measurements from a replica. The results are shown in Table I. EM 1 and EM 2 are machined with different feeds and tool radii. The surface roughnesses are listed in Table II. All surfaces have been documented in detailg. All specimens were degreased with triethylene chloride and then with acetone in a small vapor unit. The tools were carefully cleaned in benzol and acetone with clean paper towels at every change of lubricant. The lubricants were applied in excess by dipping the cleaned specimens in the lubricant immediately prior to performing the experiment. Similarly, the lubricant was applied in excess to the tool surfaces immediately after cleaning. The lubricants used were castor oil and kerosene. Some experiments used degreased specimens. 4.3. Experimental verification of the position of the zone of shear Each specimen had a slight scratch in the upper plane surface, and a mark painted on the inner cupshaped ring. This made it possible to control sliding that had taken place between specimen and ring and not between specimen and tool. However, this only refers to surface behaviour. Sub-surface metal may be unpredictably deformed by sticking friction. Before construction of the equipment this point had been examined partly by calculating the position of the zone of shear by an upper bound, and partly by carrying out plasticine experiments with simple equipment. All experiments confirmed calculations, that the most likely position of the zone of shear is between the cylindrical surface of the specimen and the ring, or just inside it in the case of stiction, the thickness of the zone getting larger for strain-hardening material and stiction. This behaviour was confirmed by a number of experiments with a split specimen, divided in a diameter plane. These experiments showed that if a lubricant is used, sliding takes place in
FRICTION
AT HIGH NORMAL
239
PRESSURES
the surface between specimen and ring in all cases. With degreased surfaces causing stiction, sliding takes place inside the material in a zone penetrating 2-3 mm radially into the material forming a fin or a “tail” on each half of the specimen. In this case the torque arm will therefore be misjudged by approx. 10%. The size of this sliding zone was independent of the press force in. all cases observed. Two split specimens with stiction have been used to calculate the value of the shear yield stress for the different sliding lengths. 10 mm from the end of the fin corresponding to 10 mm sliding length, the thickness was 2.17 mm, average of 4 tins. This permits the calculation of the shear strain rP after 10 mm sliding length, and by assuming yP linear with the sliding length, ypS,,,,,, and Thereby E,~ can be calculated, as “Yp Zmm can be calculated. &eq = l/X/%, and to this is added the strain originating from the slight forging of the specimen which is necessary to reach the ring. By means of the strain hardening curve of the material (3) o0 and thus k can be found. 4.4. Results Figures 14-20 show some of the results from the experiments with friction stress r as a function of the normal pressure CJ with the sliding length as parameter. For comparison, values calculated on the basis of the friction theory stated in section 3.4 are plotted. The sliding velocity for all experiments was calculated as I/= 3.2 mm/s.
Fig. 14. Dry friction, R l/EM 1 +EM
2.
Fig. 15. Kerosene, R 3/EM 1.
240
T. WANHEIM
Fig. 16. Kerosene, R l/EM 1.
Cm
Fig. 17. Kerosene, R l/EM 2.
t’
loo
m2
Fig. 18. Castor oil. R l/EM 1.
2000
LOU3
6000
Fig. 19. Castor oil, R l/EM 2.
Dry friction Only two series of experiments, R l/EM 1 and R l/EM 2 respectively were made. The results are shown in Fig. 14. It is seen that the surface roughness of the specimen has no influence on dry friction. The influence of the sliding length is considerable owing to strain hardening, and obviously the curves rise towards a horizontal asymptote for each sliding length. For comparison the theoretical
FRICTION
AT HIGH NORMAL
PRESSURES
241
800.
600
Fig. 20. Castor oil, R 3/EM
1
values have been drawn, the calculation of k appears from the previous section. Theory and experiments correspond very well in the shape of curves and in numerical values for pressures above 30004000 atm, whereas the theoretical values for small pressures are too low. Kerosene
R 3/EM 1 (Fig. 15): Stiction, compare with Fig. 14. Stiction results in a high sliding length effect owing to strain hardening. R l/EM 1: Corresponding high sliding length dependence in the pressure area 1000-2000 atm but much smaller sliding length dependence for lower and higher pressures. It will be noticed that in the area 1000-2000 atm, the results almost correspond to stiction, Fig. 16. R l/EM 2: Conditions here are of the same character, again highest sliding length dependence between normal pressure for 1000-2000 atm and small sliding length dependence for lower and higher pressures. It should be noticed that EM 1 here gives higher friction than EM 2, quite contrary to the results for castor oil, see later, Fig. 17. The theoretically calculated values are shown in Fig. 17 and must be compared to the I=0 curve. It appears that correlation is again good for higher pressures, from approx. 2000 atm and upwards, whereas for lower pressures there is an error up to a factor 2. Castor oil
R l/EM 1-R l/EM 2: The results have a common characteristic that the sliding length dependence is mainly seen in the first 2 mm. Increasing sliding length means decreasing friction contrary to the results from dry friction and kerosene. For a longer sliding path a certain further decrease of r occurs. The curves, however, differ for pressures under approx. 5000 atm, EM 1 gives a considerably lower friction than EM 2, contrary to the trend for kerosene, Figs. 18 and 19.
242
T. WANHEIM
Correlation with theory for EM 2 for pressures below approx. 4000 atm is not good, but the theoretical line has the right slope for pressures above this value. R 3/EM 1: Here conditions are quite different from sliding towards R 1. There is no decrease of r by initial sliding, but r increases steadily with sliding. Also in this case the sliding length effect depends on the pressure. It appears that high pressures reduce the sliding length effect as for the kerosene series, Fig. 20. 5. DISCUSSION
AND CONCLUSIONS
Friction coefficients found by low normal pressures with specially developed equipment’ are shown in the results from the experiments as a dotted, straight line through the O-point. The connections measured r = f(q) are seen to differ very considerably from this line by normal pressures over the yield stress; friction conditions at these high normal pressures are much too complicated to be described by the simple Amonton model. Any model of the type p=const. is too simple to describe conditions realistically even if “adjustment” of ,Dis applied. If the model is extended with the sticking friction so that two straight lines, p= const. and r = k, are used, conditions are somewhat improved. However, a complete description of the phenomenon cannot be obtained with this model. The theory stated in this paper describes conditions reasonably well for the cases investigated with only two important exceptions, the sliding length dependence measured, and the difference between theory and experiments by normal pressures under approx. 2000 kp/cm’. For the series made under stiction conditions this is fairly easy to explain. The sliding length dependence can directly be proved to be a consequence of strain-hardening of the tops and the material just under the surface. The difference between theory and experiment for normal pressures under 2000 kp/cm2 can be explained as follows: The theoretical curves shown as r =crk consistently lie below the experimental curves in this area. This is no doubt due to the fact that the influence of tangential stress on the area of contact has not been considered when determining the connection between the factor of area of contact tl and the normal pressure q in the theory stated. The proposed slipline solution is only valid exactly for z=O, for growing tangential stresses CIwill grow. The theoretical treatment of this, however, demands that the slipline solutions are extended to T#O in the surface. All experiments show that the lubricant by these high normal pressures to a great extent works by hydrostatic lubrication with trapped lubricant. Both the lubricant and the surface properties of specimen and tool are important to this mechanism. The lubricant properties which especially seem to influence the factor of the area of contact are viscosity and compressibility at different pressures. For a given surface geometry the viscosity of the lubricant is of importance to the quantity of lubricant which is trapped when the surfaces meet; higher contact velocity will probably trap more lubricant. When the pressure between the surfaces increases, the viscosity of the lubricant at the actual pressure will be a factor determining lubricant escaping and thereby the size of the area of contact. Compressibility of the lubricant will also influence the area of contact. When there is relative motion between the surfaces, the opening and closing
FRICTION
AT HIGH NORMAL
PRESSURES
243
of escape channels for the lubricant will be important, therefore, the sliding length is an important parameter comparable to the rheological properties of the lubricant, the surface geometry and the normal pressure. Sliding lengths applied in metal working may have very low values, e.g. slight reduction by cold rolling or forging, medium values as by wire drawing and higher values as between specimen and container by extrusion. Therefore the sliding length dependence will influence friction phenomena by these processes. These experiments have shown that z can increase as well as decrease with the sliding length. The first exists with dry friction, kerosene or a sandblasted tool. For dry friction the phenomenon can be explained by strainhardening. If a lubricant is used, the sliding length dependence is probably due either to strainhardening or to the fact that more lubricant escapes as sliding progresses causing the relative area of contact a and thereby r to grow. Figures 16, 17 and 20 show one common characteristic, the sliding length dependence is highest in the area 10004000 kp/cm2, whereas for low pressures (approx. 500 kp/cm2) there is apparently no sliding length dependence and for high pressures (>4-5000 kp/cm2) only a low dependence. A possible explanation is that at low pressures conditions are close to normal Amonton friction, the metallic contact consists of an open system of contact points, the lubricant can escape freely and does not carry any part of the load. Escape of lubricant therefore does not influence the real area of contact. With increased pressure larger quantities of lubricant are trapped in pockets, and the sliding length dependence can be explained by escape of the lubricant. The fact that the effect of this mechanism is reduced at high pressure values is probably caused by two phenomena: Firstly, the metal will seal the lubricant pockets better at increasing normal pressure, as seen by applying eqn. (7). Secondly, the viscosity of the lubricant will increase with pr, thus reducing the lubricant escape. Decreasing r by increasing sliding length is found in all series with castor oil and tool R 1. The explanation for this type of sliding length dependence may be chemical. The influence of the sliding length seems to manifest itself in the first two millimeters sliding length. However, a further decrease of r by sliding lengths exceeding this value may be traced. It can thus be established that friction conditions in the plastic area differ very considerably from friction conditions by normal pressures where the Amonton friction law can be applied. It will therefore hardly be rational to use the “friction coefficient”, which immediately leads to the idea of Amonton friction, but instead to use the more neutral “friction stress” and express this as a function of normal pressure, surface topography, sliding length, sliding velocity, striking velocity, compressibility, and viscosity. For each combination of metals and lubricant, surface-chemical phenomena will play a part, e.g. by adhesion between the metals and by formation of a metal soap with the lubricant. These chemical phenomena limit the extent to which the problem can be solved by applying mechanical model complexes such as plasticity theory and fluid mechanics on the friction and lubrication mechanisms, but a continued attack along these lines can widen the knowledge of these mechanisms considerably.
244
T. WANHEIM
REFERENCES 1 F. P. Bowden and D. Tabor, Bulletin 145. Comm. of Australia Council Sci, and Ind. Research, 1942. See also: Friction and Lubrication ofSolids, Clarendon Press, Oxford, 1950. 2 M. C. Shaw, A. Ber and P. A. Mamin, J. Basic Eng., 82 ( 1960) 342. 3 L. H. Butler, Metallurgia, April (1960) 167. 4 R. Hill. The Mathematical 7%eory of Plasticity, Clarendon Press, Oxford. 1950, p. 181. 5 W. Johnson, J. Mech. Phys. Solids, 4 (1956) 191. 6 E. H. Lee, J. Appl. Mech.. 19 (1952) 97. 7 D. Tabor. The Hardness of Metals, Oxford Univ. Press, London, 1951. 8 P. W. Bridgman, Collecred Experimental Papers, Vol. VIII, Harvard Univ. Press, Cambridge, Mass.. 1964. 9 T. Wanheim, Friktion ved hoje fladetryk, AMT-licentiatarbejde, Technical University of Denmark, 1969.
FRICTION
225
AT HIGH NORMAL PRESSURES*
T. WANHEIM Department
of Mechanical
Technology, Technical University of Denmark, 2800 Lyngby (Denmark)
(Received March 26, 1973)
SUMMARY
Friction conditions between tool and workpiece in metal working are of the greatest importance to a number of factors such as force and mode of deformation, properties of the finished specimen and resulting surface roughness. It is shown, theoretically and experimentally, that the Amonton friction law expressed by r =w does not apply when normal pressure is higher than approximately the yield stress of the specimen; in this case it is necessary to consider the frictional stress as a function of normal pressure, surface topography, length of sliding, viscosity, and compressibility of the lubricant. The theoretical work was carried out by means of upper bound and slipline field analysis based on experiments with model surfaces in wax and metal. The theoretical model applied is one of multihole extrusion, the material beneath the valleys of the workpiece surface being extruded up towards the tool when the real area of contact exceeds a certain value. The effect of the trapped lubricant is to build up a back-pressure on the extrusion process. The experimental work was carried out with newly developed equipment enabling direct determination of the abovementioned function; construction and calibration of the equipment are described. The equipment allows determination of frictional stress on a surface with well-defined values of normal pressure, sliding length, and sliding velocity. The normal pressure can attain about 8 times the yield stress for commercially pure aluminium. The results obtained show reasonably good agreement between theory and experiment, and a dependence of the frictional stress on the sliding length, this dependence being a function of normal pressure.
1. INTRODUCTION
The two empirical laws most often applied to describe the tangential stresses between specimen and tool by metal working processes are the Amonton law
* Paper presented at the First World Conference on Industrial Tribology, New Delhi, December, 1972.
226
T. WANHEIhl
and the expression for stiction r=k
(2)
It is assumed that eqn. (1) is sufficient to describe the conditions until the product of friction coefficient p and normal stress q becomes higher than the yield stress in pure shear k of the material. The material will then stick to the tool and yielding take place in the interior of the material. If the transmitted friction stress z is plotted as a function of the normal pressure, a curve consisting of two straight lines results, Fig. I.
Fig. 1. Amonton friction and sticking friction US.normal pressure.
The well-known adhesion theory, Bowden and Tabor’ and others, based on the individual plastic deformation of the contact points, yields ,u=k/H
where H is the hardness of the workpiece. The ratio between real area of contact and apparent area is normally very small. If the normal pressure grows, the actual contact area will grow and approach the apparent one. This approach will probably be asymptotical, as very high pressures will be needed in the last phase. Since according to the adhesion theory, friction stress is proportional to the real area of contact, Fig. 2 can be drawn, where the ordinate is z and the asymptote has the equation z = k.
Fig. 2. Frictional stress US.normal pressure2. This curve resembles the broken curve (Fig. 1) but with no sharp line of division between the two regions’. With a lubricant, three models are used to describe its effect, viz. either hydrodynamic lubrication, boundary lubrication, or hydrostatic lubrication. In the latter, well-known in metal forming, the lubricant is trapped in numerous small
FRICTION
AT HIGH
NORMAL
227
PRESSURES
pockets formed in the surface of the specimen when tool and specimen meet. The pressure in these is sufficiently high to transmit the necessary normal stresses, whereas the transmissible tangential stress is modest3. 2. EXPERIMENTS
WITH
MODEL
SURFACE
IN MODEL
MATERIAL
To investigate the asymptotical approach of the real contact area towards the apparent one with increasing normal pressure, and to follow the deformation in detail, a plasticine model in plane strain with a painted grid was used. An idealized plane model of a surface was modelled using straight-sided asperities. The average slope on an average surface lies between 5” and 10”; however, 15” was chosen to obtain larger and more distinct deformations in the plasticine. Preventing surface expansion when pressing a tool down on it, the following two principal characteristics were noticed: ’ (a) Deformation of the grid penetrates far below the “surface”, presumably 0.3 of the distance between the tops in the last phases. (b) The slope of the asperities is apparently constant, or close to constant, although the bearing factor c1approaches 1, a being the ratio between real and apparent area of contact. Calculations with upper bounds clarify that the slipline solution of the problem might be analogous to the solution of inverted extrusion proposed by Hi114,where CIequals the extrusion reduction r. The development of the area of contact is shown in Fig. 3 (a-h) at its different stages. The starting point is Prandtl’s slipline solution of indenting a plane tool into a semi-infinite plate, modified to the present application.
b
Fig. 3. Proposed
development
of slipline field for asperity
deformation
at high contact
pressures.
T. WANHEIM
228
Figure 3(a) shows conditions for a==O.l. Figure 3(b) is the last state in this part of the solution where the isolated deformations touch. This value of a is very close to 0.33. It is probable that the slipline picture changes to the one shown in Fig. 3(c), which has been drawn on the basis of Hill’s extrusion solution. Figure 3(d) shows the field for cx=O.5, where it simply consists of triangles and circular arcs. This field leads the way to the conditions of c1>0.5 shown in Figs. 3(e-g). Figure 3(g) shows the last stage in this field; here the boundary slipline has reached horizontal. This occurs for a=O.88. Figure 3(h) has been constructed on the basis of a development of this field pointed out by Johnson5. This field can be applied for 0.88 < c1< 1. It must be emphasized that the slipline solution mentioned here only applies to a hard (smooth) tool and a soft (rough) specimen in static contact, and must be modified if the tool has a tangential movement relative to the specimen and r#O. For small z values, however, the changes will probably not be very drastic. An objection may be that the yield criterion is violated in the stiff region containing the valley, since the angle between the free surface and the bordering slipline is less than 45”. A slight degree of work-hardening could justify this error since the angle in most cases will exceed 40”. The slipline solution indicated here will be experimentally verified in the next section. 3. EXPERIMENTS
WITH MODEL
SURFACES IN METAL
A number of experiments were made with metallic model surfaces to investigate the variation of c( with the normal pressure both with and without a confined lubricant. 3.1. Equipment and experimental procedure
In a container consisting of two halves clamped together a specimen of commercially pure aluminium, 3@ x 40, was coined to get a surface consisting of concentric sloping ridges with 1.5 mm spacing. 7 coining tools were produced of hardened tool steel, 6 with slopes ~‘2.5”; 5.0”; 7.5”; 10.0”; 12.5”; and 15.0”, and one plane lapped tool. When the container is clamped together, the coining tools can move in it with a running fit. The experiments were carried out as follows: A specimen with plane end surfaces was placed in the container and coined at approx. 40 tons, using a thin layer of mineral oil as lubricant. After disassembling of the container the specimen was removed and annealed for 2 hours at 360°C. A diagram of the surface was obtained in a Perth-o-meter. The specimen was then placed in the container, flattened with the plane coining tool in a compression test machine to the force desired, removed, and another surface diagram obtained. Two diagrams of each specimen were drawn on the Perth-o-meter along two radii 180” apart, making measurement of the real area of contact at a given load possible. The specimen was replaced in the container, loaded to a higher force, removed, and diagrams obtained. There was no annealing between loadings. An example of the diagrams is shown in Fig. 4; (a) shows a diagram of the specimen after coining with a 7.5” tool, and (b) shows the specimen after loading to 10.0 tons. The aluminium used was from a single rod; the stress-strain curve of the
FRICTION
AT HIGH NORMAL
PRESSURES
229
230
“I’.WANHEIM
material was found to obey: CT= 15.98 f P-281 kp/mm* 3.2. Experiments 6 specimens were coined with the tools mentioned and loaded with the plane tool to 2.5; 5.0; 10.0; 15.0; 20.0; 25.0 tons, with intervenjng drawing of diagrams. The specimens to be loaded with trapped lubricant (castor oil and kerosene) were coined with the tool with profile angle 10.0”, and annealed. Then one specimen was loaded to 5 tons, No. 2 to 10 tons etc., until the 6th specimen which was loaded to 30 tons. Loading was made in the following way: After the container with the specimen had been placed in the testing machine, the lubricant was poured in to approx, 5 mm over the specimen. The plane tool was positioned and a load of 2.5 tons applied with a constant low speed. At 2.5 tons the load was held for 2 minutes, and then the required load was apphed at a constant low speed. This procedure prohibited variation in lubricant volume. This experimental technique can only be used for comparative measurement since the autermost ridge of the specimen, which contributes proportionally most to the area, does not have the same conditions as the others, having only lubricant under pressure at its inner side. consequently the real areas of contact measured for the fubricant experiments will be somewhat too large. The results from the diagram measurements from these experiments are shown in Fig. 5.
o-
o-D ----0 -of e-t0 - -
~~25' y" 5:0* y2 7-5' v=tE1D' v=lZ5" y=lS,O'
20
30
40
qkQ/ld
Fig. 5. Loading of model surfaces. Relatiw~ contact area as a function of normal ~E%suK.
The real area of contact without lubricant is seen to approach CC=1 asymptotically, with the smallest profile angles at the top. This effect is no doubt due to the strain-hardening, which will have most effect on the steepest ridges. Provided that the slipline solution stated apphes, the steepest ridges would be at the top for smal cr-values, where deformation on each top is considered separatefy, and when the extrusion mechanism has commenced, there would be no dependence on the profile angle. The strain hardening is apparently able to overshadow this effect.
FRICTION
AT HIGH NORMAL
PRESSURES
231
From the experiments with a confined lubricant, Fig. 5, this is of paramount importance to the bearing factor, probably mainly dependent on the topography of the surfaces and the viscosity and compressibility of the lubricant. At high pressure of approx. 4 x yield stress calculated for s=O.l the real area of contact is approx. 95% for dry deformation, approx. 55% with confined kerosene, and approx. 25% with confined castor oil. 3.3. Experimental verification of the slipline solution The assumed slipline solution has been verified in three ways by means of these experiments, namely by comparison of the calculated and measured forces, measuring the extension of the deformed zone by means of micro-hardness measurements, and measuring the profile angles for all deformation stages. 3.3.1. Comparison of calculated and measured forces The theoretical pressures for different values of a calculated for the slipline field were comparable with the measured values. By means of hardness measurements, it is possible to get an estimate of the yield stress and thereby the equivalent strain. Hardness measurements(Vickers 5 kp) were made on the undeformed material and on the flat tops of the deformed specimens after the last loading (25 tons). These last measurements were performed in the second ridge from the outside with 4 measurements on each of the six specimens deformed without lubricant. The mean of the four specimens with the steepest sides was calculated, while the two specimens with y = 2.5” and 5.0” were not included. Tabor’ has shown that the average strain during hardness indenting is very close to constant and equals 8%. Using this, the empirical stress-strain diagram of the material, and the proportionality between yield stress and hardness number, the equivalent strain after the last deformation can be calculated from the hardness measurements. It is also assumed that the equivalent strain under the asperities, as long as the deformation of these can be regarded individually, is the same as the equivalent strain during a hardness test, i.e. s=0.08, and that the equivalent strain Egrows linearly with ~1from the moment the extrusion mechanism becomes effective, i.e. a=O.338 for y= 11.25”. Thus one obtains: E = 0.08
for 0 < CI< 0.338
(4)
E = 0.073 a + 0.055 for 0.338 < u < 1 On this basis and using von Mises yield criterion, 2 k was calculated as a function of c1 for y= 11.25”, mean of the 4 slopes used, and in Fig. 6 q/2k is compared with the measured values. There is good agreement between calculated and measured values. 3.3.2. Measuring the size of deformation zone Figure 3 shows that even if the area of contact is changed, the deformed layer apparently keeps its thickness. Microhardness measurements must reveal this phenomenon on the aluminium specimens, as a rather well defined hardness plateau must be expected within the limitation of the sliplines if the solution holds.
232
T. WANHElM
Fig. 6. Theoretical and measured area of contact.
Fig. 7. Theoretical extension of slipline field.
From Fig. 7 it will be seen that the theoretical extension of the field relative to a fixed coordinate system can be divided into the extension of the field d measured from the surface to the deepest point and the movement of the too1 6. While d can be read from the slipline solutions for the individual cr-values, 6 is calculated by using volume constancy and the assumption that the profile angles keep constant during the whole process. It is obvious that this last assumption is closely connected with the slipline solution and is experimentally verified in section 3.3.3. If the distance between two asperities is called t, we get s/t = &x( 1 -&a) tg)J
(5)
For y = 15.0” and 0.365 < CI< 0.879 we have 0.368 < z/t < 0.410 where z = d + 6. Setting t= 1.5 mm, this defines a strain-hardened layer O-554.61 mm thick. To investigate this experimentally, a cut was made at right angles to the deformed surface after last loading in one of the specimens, Y”,,,,,= 15”. Under the 4th ridge from the outside a series of micro hardness measurements with 25 p load was made. Furthermore microhardness measurements were made in the undeformeb material far from the surface. In all approx. 80 microhardness measure’ments were made, 72 inside an area of 1.5 x 1.2 mm, placed in 12 rows and 6 columns. The 6 values in each row were reduced to three calculating averages
FRICTION AT HIGH NORMAL PRESSURES
233
Fig. 8. Micro hardness distribution under a ridge Battened to a=0305
of symmetrical points. The result is shown in Fig. 8 where the lines indicating points with the same hardness produce a well defined hardness plateau as predicted by slipline field theory. Theoretically determined points of the boundary correspond tolerably well to the boundaries of the plateau. Ex~riments show that the hardness plateau extends deeper into the material than predicted by theory. This can be explained by the strainhardening. To ensure that the hardness variation was not due to a small number of large grains in the material, the specimen was etched and photographed; only approx. 2-3 hardness measurements were observed in each grain. 3.3.3. The variation ofthe profile slope If the slipline field solution holds, the profile slope will be constant for all
181 16
02
U
0.6
w
x0
a
Fig. 9. Variation of profile slope with a.
234
T. WANHEIM
values of the factor of the area of contact CLThe diagrams shown in Fig. 4 and those analogous to the other profile slopes make a direct control of this possible. The profile slope was measured in 6 places on each diagram, mainly in the 3rd, 4th, and 5th valley from the outside. 12 angles were measured for each specimen after each load. Results are shown in Fig. 9, giving j! as a function of CC Although the average values of the angles measured in this way are too high compared to the nominal angles, the curves show that :’ is independent of M within the estimated measuring precision. These experiments are also in agreement with the “extrusion theory”. 3.4. Extended friction theory Using the “extrusion mechanism” described and the dependence of the area of contact on the normal pressure with and without confined lubricant, it is possible to establish a theoretical connection between friction conditions at low normal pressures and friction conditions at high normal pressures. Figure 10 shows a cut through a contact between tool and specimen. The dimension at right angles to the paper plane is large.
q
I
bbbbbbbbbbb
Fig. 10. Stresses in surface and lubricant
Vertical equilibrium gives:
If the pressure on the real area a A, which is necessary to bring the deformation to the value CI without confined liquid, is called pdr it is seen that Pd+Pf
(7)
=Pm
as the pressure in the liquid pf can be regarded as an extra hydrostatic pressure on the slipline field, or qd
=
a(Pm-Pf)
where qd is the pressure, averaged, on the entire area, necessary to bring the deformation to the value c1without confined liquid. From these equations we have Pf
=
(9)
q-qd
and pm=q+-
l-u a
qd
(10)
Figure 11 shows the determination of pf if a curve as Fig. 5 is established. Assuming that tangential sliding takes place following the Amonton law at the contact points and zero friction at the pockets, we have
235
FRICTION AT HIGH NORMAL PRESSURES
Fig. 11. Determination of hydrostatic pressure pr in trapped lubricant.
where as a condition contact:
we state the sticking criterion applied on the real area of (12)
as the yield stress of the material does not change by a su~rimposed pressure. One gets:
hydrostatic
(13)
r = /@q+(I-a)qJ expressing written
the connection
z= f(q) earlier mentioned.
The sticking criterion
is
or z = elk
(14)
It is now possible to calculate the expected z = f(q) for the three lubricant conditions: castor oil, kerosene, and dry, by applying the curves 5 or 6 and the expressions (13) and (14), which also demands knowledge of ,u and k. Theory will be compared with experimental measurements described in the next section. Calculations of k-values and friction coefficients p experimentally determined at low normal pressures (q <<2 k) with specially developed equipment show that o! k is aImost without exception smaller than ,a(aq+(l -a)%), which, if the same p is valid, means that for the three lubricating conditions examined there will be full stiction at the points of contact, and the friction is therefore determined by the area of contact CI,which is a function of the pressure, lubricant, and, as will later be shown, of the surface and sliding length. Attention is drawn to the similarity between eqn. (14) and the often used “proportional friction law” z = mk 0 < m < 1. In the next section the theory stated here will be verified experimentally. The author is aware of several of the great flaws in this “theory”. One of them, a slipline field solution allowing r#O in the surface, has been overcome. This solution will be published shortly.
236 4. HIGH
T. WANHEIM PRESSURE
FRICTION
EXPERIMENTS
Equipment was developed for measuring frictional stresses for normal pressures larger than the yield stress of the specimen when closely controlling the parameters normal pressure, sliding length, and surface roughness. 4.1. Equipment The principle of the experimental set-up is shown in Fig. 12. With the aid of tools 1 and 2, the specimen, 3, is pressed against a ring, 4, on which a ring, 5, is shrunk. With the aid of straingages, 6, mounted on the thick-walled ring, the surface pressure between specimen, 3, and inner ring, 4, may be measured. The pin, 7, centers tools, 1, and, 2, and specimen, 3. In the inner ring are eight pins, 8, which transfer the torque from the wire, 9, through the wire disk, 10, and disk, 11, and slip occurs between specimen, 3, and ring, 4.
L
’
+ Fig. 12. Principle
of measurement.
The experiments were performed on a dual-action hydraulic 320 ton deepdrawing press, where the main ram created the high normal pressure between specimen and ring. The wires, 9, run over two wheels and are attached to a frame, which is moved vertically by the auxiliary ram of the press. A potentiometer gives the position of the auxiliary ram and thereby the sliding length. Measurement of the frictional force is made possible by 4 straingage-equipped rings connecting four upward moving bars with the frame. Two cradles with ball joints are inserted to ensure zero sideforces on the specimen. The position of the wire-disk is controlled exactly by four ball bearings mounted independently on lightly springloaded supports. During the experiment the measurement signals were recorded on a u.v.recorder. The pressure ring is instrumented with 4 straingages active in the direction of the periphery. Between these, 4 dummygages are mounted on individual plates of material similar to the ring and ground to the same curvature as the ring. They are secured to the ring by rubber bands. The calibration of the pressure ring was performed with a rubber disk in the place of the specimen in the ring, the stress in the rubber disk in the chamber was assumed to be hydrostatic. The thickness of the rubber specimen under calibration was measured by means of a dial gage between the supports of the two tools. The elastic deformation of the tools was corrected for in a deadrun.
FRICTION
AT HIGH
Fig. 13. Calibration
NORMAL
of pressure
PRESSURES
237
ring with rubber.
Figure 13 shows the calibration curve obtained. As indicated there is good agreement between the points plotted with increasing and decreasing loading. However, the plot shows a discontinuity in slope at approx. 5,000 atm. The calibration was repeated a number of times with other rubber disks, with the same result. Bridgman* reports regarding compressibility measurements: “B. Synthetic and Natural Rubbers The measurement of these was undertaken at first for the entirely practical purpose of finding some way of predicting which might be expected to function best for high pressure packing on the piston of the apparatus for 30,000 kg/cm2. Ten different specimens of various synthetic rubbers were collected from dealers in the neighbourhood of Boston. Measurements disclosed an interesting feature, namely in some of them there was a rather sharp change in the direction of the curve of volume uersus pressure, or discontinuity of the second kind, at pressures roughly of the order of 5,000 kg/cm2”. It seems very likely that the discontinuity in compressibility discussed by Bridgman is caused by the same change in material properties shown in Fig. -13. To verify this point, additional calibrations were performed with soft PVC, which showed a change in slope at about 3,000 atm; the initial part of the curve had a slope similar to that of the calibration curve for rubber. The sensitivity of the transducer was then taken as the slope of the rubber curve for 0 < q < 5000 atm. 4.2. Materials and surfaces The specimen material contained Al 99.25x, Fe 0.55x, Cu 0.02x, Si 0.16%. The chamber in which the specimen is exposed to the high pressure is formed by two cylindrical tools and a ring. The plane surfaces of the tools are ground and then sandblasted to provide a better grip of the specimen. The ring against which the specimen slides, is manufactured from tool steel, heat-treated and annealed at 200°C. The composition of this material is: C 0.36x, Mn 0.70x, Si 0.30x, Cr 1.40x, Ni 1.40x, MO 0.20%. Two surface roughnesses were used in the ring. Surface R 1 is produced by grinding followed by honing and R 3 is obtained by sandblasting R 1. The R, and R,, values (pm) are obtained partly by direct measurements and partly
238
T. WANHEIM
TABLE I Surface
RI R3
Axial directiorl
Circumferential direction -~.__~..~_._~
R, R-x ____ ..~_._._
R*
R InBI
0.091 2.03
0.143 1.94
~ 13.0
~~ 14.2
TABLE I1 Surface
Manufacture
Feed
Nose
z-Direction
cp-Direction
(mmjrev)
R
R,
R,
R,.w
0.49
2.03
0.66
2.92
R,,,
(mm) -
EM 1 EM2
annealing + turning annealing + turning
0.065
1.3
1.08
0.3
0.5
6.0
._.
4.32 24.4
____..~
Turned surfaces: n= 1000 o/min. t=O.l mm Cutting fluid: alcohol
by measurements from a replica. The results are shown in Table I. EM 1 and EM 2 are machined with different feeds and tool radii. The surface roughnesses are listed in Table II. All surfaces have been documented in detailg. All specimens were degreased with triethylene chloride and then with acetone in a small vapor unit. The tools were carefully cleaned in benzol and acetone with clean paper towels at every change of lubricant. The lubricants were applied in excess by dipping the cleaned specimens in the lubricant immediately prior to performing the experiment. Similarly, the lubricant was applied in excess to the tool surfaces immediately after cleaning. The lubricants used were castor oil and kerosene. Some experiments used degreased specimens. 4.3. Experimental verification of the position of the zone of shear Each specimen had a slight scratch in the upper plane surface, and a mark painted on the inner cupshaped ring. This made it possible to control sliding that had taken place between specimen and ring and not between specimen and tool. However, this only refers to surface behaviour. Sub-surface metal may be unpredictably deformed by sticking friction. Before construction of the equipment this point had been examined partly by calculating the position of the zone of shear by an upper bound, and partly by carrying out plasticine experiments with simple equipment. All experiments confirmed calculations, that the most likely position of the zone of shear is between the cylindrical surface of the specimen and the ring, or just inside it in the case of stiction, the thickness of the zone getting larger for strain-hardening material and stiction. This behaviour was confirmed by a number of experiments with a split specimen, divided in a diameter plane. These experiments showed that if a lubricant is used, sliding takes place in
FRICTION
AT HIGH NORMAL
239
PRESSURES
the surface between specimen and ring in all cases. With degreased surfaces causing stiction, sliding takes place inside the material in a zone penetrating 2-3 mm radially into the material forming a fin or a “tail” on each half of the specimen. In this case the torque arm will therefore be misjudged by approx. 10%. The size of this sliding zone was independent of the press force in. all cases observed. Two split specimens with stiction have been used to calculate the value of the shear yield stress for the different sliding lengths. 10 mm from the end of the fin corresponding to 10 mm sliding length, the thickness was 2.17 mm, average of 4 tins. This permits the calculation of the shear strain rP after 10 mm sliding length, and by assuming yP linear with the sliding length, ypS,,,,,, and Thereby E,~ can be calculated, as “Yp Zmm can be calculated. &eq = l/X/%, and to this is added the strain originating from the slight forging of the specimen which is necessary to reach the ring. By means of the strain hardening curve of the material (3) o0 and thus k can be found. 4.4. Results Figures 14-20 show some of the results from the experiments with friction stress r as a function of the normal pressure CJ with the sliding length as parameter. For comparison, values calculated on the basis of the friction theory stated in section 3.4 are plotted. The sliding velocity for all experiments was calculated as I/= 3.2 mm/s.
Fig. 14. Dry friction, R l/EM 1 +EM
2.
Fig. 15. Kerosene, R 3/EM 1.
240
T. WANHEIM
Fig. 16. Kerosene, R l/EM 1.
Cm
Fig. 17. Kerosene, R l/EM 2.
t’
loo
m2
Fig. 18. Castor oil. R l/EM 1.
2000
LOU3
6000
Fig. 19. Castor oil, R l/EM 2.
Dry friction Only two series of experiments, R l/EM 1 and R l/EM 2 respectively were made. The results are shown in Fig. 14. It is seen that the surface roughness of the specimen has no influence on dry friction. The influence of the sliding length is considerable owing to strain hardening, and obviously the curves rise towards a horizontal asymptote for each sliding length. For comparison the theoretical
FRICTION
AT HIGH NORMAL
PRESSURES
241
800.
600
Fig. 20. Castor oil, R 3/EM
1
values have been drawn, the calculation of k appears from the previous section. Theory and experiments correspond very well in the shape of curves and in numerical values for pressures above 30004000 atm, whereas the theoretical values for small pressures are too low. Kerosene
R 3/EM 1 (Fig. 15): Stiction, compare with Fig. 14. Stiction results in a high sliding length effect owing to strain hardening. R l/EM 1: Corresponding high sliding length dependence in the pressure area 1000-2000 atm but much smaller sliding length dependence for lower and higher pressures. It will be noticed that in the area 1000-2000 atm, the results almost correspond to stiction, Fig. 16. R l/EM 2: Conditions here are of the same character, again highest sliding length dependence between normal pressure for 1000-2000 atm and small sliding length dependence for lower and higher pressures. It should be noticed that EM 1 here gives higher friction than EM 2, quite contrary to the results for castor oil, see later, Fig. 17. The theoretically calculated values are shown in Fig. 17 and must be compared to the I=0 curve. It appears that correlation is again good for higher pressures, from approx. 2000 atm and upwards, whereas for lower pressures there is an error up to a factor 2. Castor oil
R l/EM 1-R l/EM 2: The results have a common characteristic that the sliding length dependence is mainly seen in the first 2 mm. Increasing sliding length means decreasing friction contrary to the results from dry friction and kerosene. For a longer sliding path a certain further decrease of r occurs. The curves, however, differ for pressures under approx. 5000 atm, EM 1 gives a considerably lower friction than EM 2, contrary to the trend for kerosene, Figs. 18 and 19.
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Correlation with theory for EM 2 for pressures below approx. 4000 atm is not good, but the theoretical line has the right slope for pressures above this value. R 3/EM 1: Here conditions are quite different from sliding towards R 1. There is no decrease of r by initial sliding, but r increases steadily with sliding. Also in this case the sliding length effect depends on the pressure. It appears that high pressures reduce the sliding length effect as for the kerosene series, Fig. 20. 5. DISCUSSION
AND CONCLUSIONS
Friction coefficients found by low normal pressures with specially developed equipment’ are shown in the results from the experiments as a dotted, straight line through the O-point. The connections measured r = f(q) are seen to differ very considerably from this line by normal pressures over the yield stress; friction conditions at these high normal pressures are much too complicated to be described by the simple Amonton model. Any model of the type p=const. is too simple to describe conditions realistically even if “adjustment” of ,Dis applied. If the model is extended with the sticking friction so that two straight lines, p= const. and r = k, are used, conditions are somewhat improved. However, a complete description of the phenomenon cannot be obtained with this model. The theory stated in this paper describes conditions reasonably well for the cases investigated with only two important exceptions, the sliding length dependence measured, and the difference between theory and experiments by normal pressures under approx. 2000 kp/cm’. For the series made under stiction conditions this is fairly easy to explain. The sliding length dependence can directly be proved to be a consequence of strain-hardening of the tops and the material just under the surface. The difference between theory and experiment for normal pressures under 2000 kp/cm2 can be explained as follows: The theoretical curves shown as r =crk consistently lie below the experimental curves in this area. This is no doubt due to the fact that the influence of tangential stress on the area of contact has not been considered when determining the connection between the factor of area of contact tl and the normal pressure q in the theory stated. The proposed slipline solution is only valid exactly for z=O, for growing tangential stresses CIwill grow. The theoretical treatment of this, however, demands that the slipline solutions are extended to T#O in the surface. All experiments show that the lubricant by these high normal pressures to a great extent works by hydrostatic lubrication with trapped lubricant. Both the lubricant and the surface properties of specimen and tool are important to this mechanism. The lubricant properties which especially seem to influence the factor of the area of contact are viscosity and compressibility at different pressures. For a given surface geometry the viscosity of the lubricant is of importance to the quantity of lubricant which is trapped when the surfaces meet; higher contact velocity will probably trap more lubricant. When the pressure between the surfaces increases, the viscosity of the lubricant at the actual pressure will be a factor determining lubricant escaping and thereby the size of the area of contact. Compressibility of the lubricant will also influence the area of contact. When there is relative motion between the surfaces, the opening and closing
FRICTION
AT HIGH NORMAL
PRESSURES
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of escape channels for the lubricant will be important, therefore, the sliding length is an important parameter comparable to the rheological properties of the lubricant, the surface geometry and the normal pressure. Sliding lengths applied in metal working may have very low values, e.g. slight reduction by cold rolling or forging, medium values as by wire drawing and higher values as between specimen and container by extrusion. Therefore the sliding length dependence will influence friction phenomena by these processes. These experiments have shown that z can increase as well as decrease with the sliding length. The first exists with dry friction, kerosene or a sandblasted tool. For dry friction the phenomenon can be explained by strainhardening. If a lubricant is used, the sliding length dependence is probably due either to strainhardening or to the fact that more lubricant escapes as sliding progresses causing the relative area of contact a and thereby r to grow. Figures 16, 17 and 20 show one common characteristic, the sliding length dependence is highest in the area 10004000 kp/cm2, whereas for low pressures (approx. 500 kp/cm2) there is apparently no sliding length dependence and for high pressures (>4-5000 kp/cm2) only a low dependence. A possible explanation is that at low pressures conditions are close to normal Amonton friction, the metallic contact consists of an open system of contact points, the lubricant can escape freely and does not carry any part of the load. Escape of lubricant therefore does not influence the real area of contact. With increased pressure larger quantities of lubricant are trapped in pockets, and the sliding length dependence can be explained by escape of the lubricant. The fact that the effect of this mechanism is reduced at high pressure values is probably caused by two phenomena: Firstly, the metal will seal the lubricant pockets better at increasing normal pressure, as seen by applying eqn. (7). Secondly, the viscosity of the lubricant will increase with pr, thus reducing the lubricant escape. Decreasing r by increasing sliding length is found in all series with castor oil and tool R 1. The explanation for this type of sliding length dependence may be chemical. The influence of the sliding length seems to manifest itself in the first two millimeters sliding length. However, a further decrease of r by sliding lengths exceeding this value may be traced. It can thus be established that friction conditions in the plastic area differ very considerably from friction conditions by normal pressures where the Amonton friction law can be applied. It will therefore hardly be rational to use the “friction coefficient”, which immediately leads to the idea of Amonton friction, but instead to use the more neutral “friction stress” and express this as a function of normal pressure, surface topography, sliding length, sliding velocity, striking velocity, compressibility, and viscosity. For each combination of metals and lubricant, surface-chemical phenomena will play a part, e.g. by adhesion between the metals and by formation of a metal soap with the lubricant. These chemical phenomena limit the extent to which the problem can be solved by applying mechanical model complexes such as plasticity theory and fluid mechanics on the friction and lubrication mechanisms, but a continued attack along these lines can widen the knowledge of these mechanisms considerably.
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REFERENCES 1 F. P. Bowden and D. Tabor, Bulletin 145. Comm. of Australia Council Sci, and Ind. Research, 1942. See also: Friction and Lubrication ofSolids, Clarendon Press, Oxford, 1950. 2 M. C. Shaw, A. Ber and P. A. Mamin, J. Basic Eng., 82 ( 1960) 342. 3 L. H. Butler, Metallurgia, April (1960) 167. 4 R. Hill. The Mathematical 7%eory of Plasticity, Clarendon Press, Oxford. 1950, p. 181. 5 W. Johnson, J. Mech. Phys. Solids, 4 (1956) 191. 6 E. H. Lee, J. Appl. Mech.. 19 (1952) 97. 7 D. Tabor. The Hardness of Metals, Oxford Univ. Press, London, 1951. 8 P. W. Bridgman, Collecred Experimental Papers, Vol. VIII, Harvard Univ. Press, Cambridge, Mass.. 1964. 9 T. Wanheim, Friktion ved hoje fladetryk, AMT-licentiatarbejde, Technical University of Denmark, 1969.