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A study of transitional friction behavior at scaled micro contacts
99
Wear, 172 (1994) 99-105
A study of transitional
friction behavior at scaled micro contacts
K.S. Lee Depament
Y.-M. Mechanical
(Received
of Mechanical
Chen
Engineering,
University
of Usan,
L&an, 680-749
(Korea)
and J.W. Dally
Engineering
Department,
August 6, 1993; accepted
University
October
of Maryland,
College
Park, MD 20742
(USA)
21, 1993)
Abstract Two-dimensional photoelastic experiments were conducted to study the transitional behavior of friction at multipoint contacts. The study was performed using a photoelastic plate to sense both normal and frictional forces at each contact. The fringe patterns, which varied continuously under the contact points of the scaled-up asperities and the photoelastic plate, were recorded by a camcorder. The spatial data from the fringe patterns was sufficient to calculate the normal and frictional forces at each contact point during the transition between static and kinetic friction. The generally accepted fact is that the difference between the coefficients of static and kinetic friction is small and that the coefficient of friction at each micro contact is the same. However, the photoelastic results show that the difference between the coefficients of static and kinetic friction is quite large and that the coefficient of friction at each contact point varies from one point to another and changes during sliding. A local coefficient
of friction also is different significantly from the global coefficient of friction.
1. Introduction It is common practice in determining the frictional force Q from the normal force I’, to employ Amonton’s law of friction given by
Q=cLp
(1)
where g is the coefficient of friction. This simple form of the friction law is based on the global force measurements of P and Q, which represent the average contribution of many local friction forces. These forces occur at each micro contact made when many asperities interact with a counter surface. Two assumptions are necessary to invoke Amonton’s law [l]. (a) During sliding, the resistive force per unit area of contact is constant. (b) The real area of contact is proportional to the normal load. These two assumptions imply the generally accepted belief that the local coefficient of friction associated with each micro contact is the same as the global coefficient of friction. In the study of friction, many models describing friction mechanisms have been proposed using this assumption. Although this assumption has not been challenged, because of the difficulty in measuring the local frictional force at each micro contact, the fundamental question regarding the magnitude of the coefficient of friction at each micro contact still remains.
0043-1648/94/$07.00 0 1994 Elsevier SSDI 0043-1648(93)06370-J
Sequoia. All rights reserved
With regard to this question, Dally and Chen [2] investigated the magnitude of the coefficient of static friction at each micro contact using photoelastic measurements of the local forces. The authors simulated micro contacts by using scaled-up asperities made from 12.7 mm radius steel rounds. A photoelastic plate with a flat surface as a counter surface was used to sense the local forces. They determined the contact forces (normal and frictional) and the associated coefficient of static friction at each contact from the isochromatic fringe patterns. While they showed that the coefficient of static friction at each micro contact is different from the global coefficient of friction, the study was limited to friction measurements prior to slippage. In this study, the coefficients of kinetic friction as well as static friction at each contact are investigated. A simplified two-dimensional model, which is composed of the artificially scaled up asperities and a photoelastic plate with a flat surface, is employed to simulate the typical contacting surfaces. Photographs of the fringe patterns, that are changing continuously under the contact points during the transition period from static to kinetic friction, are recorded with the camcorder. These dynamic fringe patterns are analysed using a similar method to Dally and Chen’s [2], to determine the normal and frictional forces and the associated coefficient of friction at each contact point. Finally, the
validity of the two assumptions inherent in Amonton’s law of friction is examined.
2. Experimental
method
To simulate the typical contacting surfaces shown in Fig. l(a), a simplified two-dimensional model was constructed as shown in Fig. l(b). The arbitrarily shaped asperities of the upper body are modeled by a series of cylindrically shaped asperities, and the lower body is modeled by a half plane with a flat surface. A schematic diagram of the experimental apparatus is presented in Fig. 2, and a photograph of the actual equipment is given in Fig. 3. The scaled-up asperities are fabricated from 12.7 mm radius rounds for the fivecontact-point experiment and 6.35 mm radius rounds
(4
\
LAAAAJ
\
(b) Fig. 1. (a) Typical dimensional model
Fig.
2. Schematic
contacting surfaces and of contacting surfaces.
diagram
of the experimental
(b)
simplified
apparatus.
two-
Fig. 3. Photograph
of the
TABLE carbonate
1. Elastic
constants
Modulus Poisson’s Material
of elasticity ratio (v) fringe value
(E) cf,)
actual and
experimental material
set-up.
fringe
value
of poly-
2480 MPa 0.38 7.0 kN/(m-fringe)
for the seven-contact-point experiment. The aluminum rounds are attached to a straight steel bar with screws. The surfaces of the asperities were roughened using number 400 grit abrasive to increase the friction. The photoelastic plate was fabricated from a polycarbonate sheet, 6.3 mm thick, cut into a 250 X 165 mm rectangle. The model was supported on three sides by an aluminum frame. The elastic constants and the material fringe value for the polycarbonate model material are presented in Table 1. To prevent parasitic loading of the model due to the irregularities on the surface of the model frame, cardboard shims were installed at both the sides and bottom of the polycarbonate plate [3]. The model frame is attached to a steel plate and can move freely in the horizontal direction over rollers which support the steel plate. To measure the global frictional force, a loadcell is attached to the steel bar that supports the asperities. The loadcell is aligned with the contact surface to avoid the couple moment caused by any difference in elevation between the contact surface and the loadcell. The normal load was applied by placing the weights on the hanger attached to a loading lever arm. The applied normal load was kept constant at 942 N for both the five and seven-contact point experiments. After applying the normal load, the tangential load was applied by placing the weights on the second hanger until motion occurred. The sliding distance of the model relative to the asperities was approximately 40 mm before a stop terminated motion. The relative velocity increased with time and position as the friction force
101
K.S. tee et al. / Transitional friction behavior at scaled micro contacts
decreased. A maximum velocity of 0.43 m s-’ was observed just before the photoelastic model engaged the stop. As tangential load was increased, the isochromatic fringe patterns were recorded with a camcorder operated at 30 frames s-‘. The video images were transferred to a VCR which was connected to the camcorder. To avoid blurring of the fringe pattern images during sliding, a fast shutter speed (l/1000 s) was employed. The loadcell output during the transition period (from static to kinetic) was recorded on a digital storage oscilloscope at a sampling rate of 1 ms point-‘.
3. Analysis of the isochromatic fringe patterns A time-sequence of isochromatic fringe patterns for the seven-contact-point experiment is illustrated in Figs. 4(a)-4(e). The isochromatic fringe pattern corresponding to static friction (just before the model begins to
move) is shown in Fig. 4(a). The other fringe patterns, presented in Figs. 4(b)-4(e), represent the state of kinetic friction. The time interval between each fringe pattern is appro~mate~y 33.3 ms. The fringe patterns were transferred from the tape of the VCR to RAM storage in a personal computer by using a frame grabber system. These stored images of the fringe patterns were processed to extract the centerline of each fringe using a thinning algorithm 141. After the thinning process, the centerline of each fringe was digitized to give spatial data for a numerical analysis to determine the normal and frictional force at each contact point. The number of data points varies from 1300 to 1700 for each fringe pattern. A typical centerline display obtained from the static iso~hromatic fringe pattern is shown in Fig. 5. From the stress optic law of photoelasticity [S], the fringe order N is given in terms of the stresses as
(2) where h is the thickness of the model and f0 is the material fringe value. The Cartesian stress components at an arbitrary point A&, z), illustrated in Fig. 6, are given by the Flamant solution [6] as u = xx
2 _ 21~-+J’~~i~ + Qi(x -xoJ’l p{(x -xoij2 -t-2-y
j-1
u?g-P-NX-x,l)” +-e 2z2{Piz + Qi(x -x,J”}
(3)
2(x -x&(p,z + Qi(x -xOt>‘} p((x x$ f 22)2
-
where x,‘s define the position of each contact point and n is the number of contact points. Let us combine eqns. (2) and (3) and define a function G, as
Fig. 5. Extracted in Fig. 4(a).
of the js~hromatic
fringes shown
A(w)
(e)
Fig. 4. A series of isochromatic fringe patterns for the sevencontact-point experiment. (a) static,(b) At = 33 ms, velocity= 0.024 m s-‘, (c) A~=67 ms, velocity=0.048 m s-‘, (d) bt=lOO ms, velocity=O.167 m SC’, (e) k-133, velocity=O.286 m s-l.
centerlines
Fig. 6. Coordinate
system for the two-dimensional
model.
102
KS.
Lee et al. / Transitional
friction
behavior
at scoled
micro
contacts
two statistical measures of error were defined
where
The normal and frictional forces and the associated coefficients of friction that were determined are summarized in Table 2 and Table 3 for the five-contactpoint experiment and the seven-contact-point experiment, respectively. To verify the accuracy of the values,
where Ni is the input fringe order for a given data point whose position is (xm, z,,J. NC is the calculated fringe order using the values in Tables 2 and 3 for the unknown quantities. The quantity m is the average of the fringe order deviation (AN), and (SD), is the standard deviation of the fringe order deviation. Results for m and (SD),, summarized in Table 4, indicate that the average deviation and the standard deviation are quite small. Reconstructed fringe patterns generated by using the solution for the normal and frictional forces for the seven-contact-point experiment are presented in Fig. 7. The actual fringe pattern is shown in Fig. 4(a). Comparing the re~nst~cted fringe pattern with the ~~es~nding actuai fringe pattern reveals that the two patterns are in good agreement. Since the magnitudes of the average and the standard deviation of the fringe order deviation are quite small and because the reconstructed fringe patterns agree well with the
TABLE
coefficients
Bij =xX;
-&Y;
+G)
fZ4 + 4xixjz2
Ci~=P~Pj~-tPiQ~~j+PjQi~i+Q,QjX~Xj In eqn. (4), the subscript m identifies the point at position (x,, z,,,) with a fringe order N,. The unknowns Pi, Qi and X, were solved by empioying an overdeterministic method [7,8], where the number of data points greatly exceed the number of unknowns. 4. Results and discussion
2. Calculated Number contact point
fa)
of
normal
and frictional
forces and associated
Normal force (N)
Frictional force (N)
Coefficient of friction
tpi)
cQi>
(/.d
of friction for five-contact-point
experiment
Frictional force (N)
Coefficient of friction
P=LP;
Q=xQi
P&= Q/f’
941.9
489.1
0.519
Normal force (N)
1
277.6
121.1
0.436
2 3 4 5
168.4 165.7 122.8 207.4
81.1 75.3 63.8 147.8
0.481 0.454 0.519 0.713
(bb)
1 2 3 4 5
267.1 161.9 159.7 122.2 217.4
103.7 60.8 44.6 39.9 65.3
0.388 0.376 0.279 0.326 0.301
928.3
314.4
0.339
(c)
1 2 3 4 5
277.1 170.5 164.6 134.7 230.3
90.4 49.2 32.9 34.2 47.8
0.326 0.289 0.200 0.254 0.207
977.2
254.5
0.260
(d)
1 2 3 4 5
268.9 154.6 150.7 131.8 221.5
77.8 39.3 27.1 26.4 26.3
0.289 0.254 0.180 0.200 0.119
927.4
197.0
0.212
(ef
1 2 3 4 5
263.0 156.1 151.9 143.3 228.5
74.9 32.1 23.1 24.3 26.3
0.285 0.205 0.152 0.169 0.115
942.8
180.6
0.192
103
KS. Lee et al. / Transitional friction behavior at scaled micro contacts TABLE
3. Calculated
and frictional
farces and associated
coefficients
of friction
for seven-intact-lint
experiment
Normal force (N) P=cPi
Frictional force (N)
Coefficient of friction
Q=EQi
~8
0.580 0.396 0.620 0.577 0.559 0.446 0.612
916.7
503.5
0.549
109.1 32.2 24.2 32.9 63.9 15.9 48.5
0.529 0.228 0.381 0.495 0.512 0.240 0.236
874.0
326.7
0.374
197.9 142.2 58.8 74.2 144.6 100.7 243.9
99.6 25.9 32.0 30.6 55.8 21.0 21.0
0.503 0.182 0.543 0.413 0.386 0.208 0.086
962.4
285.8
0.297
1 2 3 4 5 6 7
192.4 139.4 44.4 65.4 131.6 105.3 228.2
84.6 39.0 18.2 31.4 50.6 10.1 21.4
0.440 0,280 0.410 0.480 0.384 0.100 0.094
901.8
255.2
0.283
1 2 3 4 5 6 7
209.6 151.7 46.7 49.3 116.8 84.0 217.3
65.7 29.2 27.1 31.9 37.8 6.7 3.5
0.313 0.192 0.579 0.649 0.324 0.079 0.016
875.3
201.8
0.23’1
Normal force (N)
Frictional force (N)
Coefkient of friction
tpi)
(Qi)
(Eli)
1 2 3 4 5 6 7
199.0 134.7 62.9 71.2 133.4 79.3 236.1
115.5 53.4 39.0 41.1 74.6 35.4 144.5
1 2 3 4 5 6 7
206.3 140.9 63.6 66.5 124.8 66.1 205.7
1 2 3 4 5 6 7
Number contact point
TABLE viation
normal
of
4. Average and standard
deviation
= QP
of fringe order de-
Five-contact-point
Seven-contact-point
Average Xi?
Standard deviation (SD),
Average N
Standard deviation (SD),
0.017 0.004 - 0.005 0.002
0.132 0.146 0.163 0.174
0.019 0.010 - 0.011 0.012
0.094 0.145 0.180 0.191
-0.004
0.173
-0.009
0.189
actual fringe patterns, we have confidence in the accuracy of the determination of the normal and frictional forces.
Fig. 7. Reconstructed fringe pattern associated with the calculated normal and frictional forces for Fig. 4(a).
The variation of the global frictional force with time is shown in Fig. 8. In this figure, the solid line represents the measured values from the loadcell and the symbols represent the sarne of the local frictional forces determined from the photoelastic analysis. The bracketed letter below each symbol indicates the corresponding fringe pattern in Fig. 4. Since the time for each fringe pattern relative to the oscilloscope record was not
104
KS’. Lee et ul
i Transitional
friction
behavior
at scaled
micro
confects
0.2 -
o, 0
60
150
100
200
250
o.o! 0
100
60
Time (ms)
t
1
150
200
Time (ms)
(a) 600
l.ol
-
Loadcell A
Calculation 0.6
c
100
os t
0
50
“.I 150
100
200
50
Time (ms) (bl
0
50
100
150
Time (ma) (W
Fig. 8. Variation of the global frictiona force with time during the transition period for (a) Eve-contact-lint experiment and (b) seven-contact-point experiment.
Fig. 9. Variations of the coefficient of friction at each contact point with time for (a} five-contact-point experiment and (b) seven-contact-point experiment.
known, the symbols in Fig. 8 were moved as a group along the time axis to match with the loadcell output values. It is evident that the sum of the local frictional forces agree well with the measured values from the loadcell. Contrary to the belief that the coefficient of kinetic friction is less than that of static friction by a small amount, it is clear from Fig. 8 that the difference is quite large. The kinetic coefficient of friction at each contact point is shown as a function of time in Fig. 9. In this figure, the solid line represents the global coefficient of friction pB and the symbols represent the coefficient of friction at each contact point. The differences between the coefficient of friction at each contact point and the global coefficient of friction are significant. In this work, the materials of the contacting bodies are dissimilar, i.e. the asperities are metal (aluminum) and the counter
surface is plastic (polycarbonate). The frictional force in this model is mainly due to the deformation. We expect that the magnitude of the frictional force will depend on the shape of the asperity (e.g. angle of attack) and the depth of penetration, which is proportional to the normal load. In these experiments, the shape of all the asperities is the same, but the angle of attack varied with the depth of penetration which was p~~rtional to the normal load. The ratio of the frictional force to the normal force, i.e. the coefficient of friction, will have a different value at each contact point, depending on the magnitude of the normal force. The coefficient of friction is proportional to the ratio of the depth of penetration (d) to the radius of asperity (r) [9]. The magnitude of the ratio of the penetration depth to the radius of asperity (d/r) in the present work varies from 0.028 to 0.042 for the five-contact-
KS. Lee et al. i Tra~itio~l
friction behavior at sculed micro contacts
point experiment and from 0.01 to 0.056 for the sevenintact-lint experiment. Since the d/r ratio of the asperities was always less than 0.1, the findings of Suh [9] indicate that the variation of the coefficient of friction at each contact point is expected to be very small. However, it is clear from Fig. 9, that the coefficient of friction at each contact point varies significantly. Furthermore, it can be observed in Tables 2 and 3 that in many instances there is no relation between Pi and fii. The variations in the local coefficient of friction appear to be independent of the local normal force. The reason for this behavior is not clear and more study is required before the phenomena can be understood.
References R.D. Ameli, P.B. Davies, J. Halling and T.L. Whomes, Tribology: principles and Design Applications, Springer, New York, 1991, p. 28. J.W. Dally and Y.M. Chen, A photoelastic study of friction at multipoint contacts, Exp. Me&., 31 (2) (1991) 144-149. R.B. Heywood, Designing by Ph~~i~t~~~, Chapman and Hail, London, 1952, pp. 96-98. T.Y. Zhang and C.Y. Suen, A fast parallel algorithm for thinning digital patterns, Commun. ACM, 27 (1984) 239-239. J.W. Dally and W.F. Riley, Ekperimentnl Stress Analysk, McGraw-Hill, New York, 2nd edn., 1978, pp. 406-482. L.E. Malvem, Introduction to the Mechanics of Continuous Medic, Prentice-Hall, Englewood CJifTs, NJ, 1969, pp. 539-546. R.J. Sanford and J.W. Dally, A general method for determining mixed-mode stress intensity factors from isochromatic fringe patterns, Eng. &act. Mech., II (1979) 621-633. R.J. Sanford, Application of the least-squares method to photoelastic analysis, Exp. Me&., 20 (1980) 192-197. N.P. Suh, T~~p~ysics, Prentice-Hap, Eagfewood Cliffs, NJ, 1986, pp. 295-304.
5. Summary and conclusions Two-dimensional photoelastic experiments were conducted to obtain dynamic &chromatic fringe patterns under the contact points between scaled-up asperities and a photoelastic half plane. The experiments were conducted during the transition period from static to kinetic friction. The normal and frictional forces and the associated coefficient of friction at each contact point were determined with an overdeterministic numerical method by using the digitized input data from the extracted centerlines of the obtained fringe patterns. By comparing the reconstructed fringe patterns with the actual fringe patterns and also by determining some statistical quantities, the accuracy of the calculated normal and frictional forces at each contact point was verified. Contrary to the generally accepted fact that the difference between the coefficients of static and kinetic friction is small and the coefficient of friction at each micro contact asperity is the same, these results show that the difference between the coefficients of static and kinetic friction is quite large and the coefficient of friction varies greatly between contact points and also is significantly different from the global coefficient of friction
The authors wish to thank the National Science Foundation and the University of Ulsan for supporting this research.
105
Appendix A: Nomencla~~
; M : Ni N, LW AIV P
Q
r (SD), -G 2
depth-of -penetration material fringe value (KN m-l) thickness of the photoelastic plate (mm) number of input data points number of contact points fringe order input fringe order calculated fringe order fringe order deviation average of fringe order deviation normal force (N) frictional force (N) radius of curvature of asperity standard deviation of m Cartesian coordinates
Greek symbols coefficient of friction CL coefficient of friction at each contact point cti global coefficient of friction 4X u.uf @YzzY Cartesian stress components (MPa) rxL
Wear, 172 (1994) 99-105
A study of transitional
friction behavior at scaled micro contacts
K.S. Lee Depament
Y.-M. Mechanical
(Received
of Mechanical
Chen
Engineering,
University
of Usan,
L&an, 680-749
(Korea)
and J.W. Dally
Engineering
Department,
August 6, 1993; accepted
University
October
of Maryland,
College
Park, MD 20742
(USA)
21, 1993)
Abstract Two-dimensional photoelastic experiments were conducted to study the transitional behavior of friction at multipoint contacts. The study was performed using a photoelastic plate to sense both normal and frictional forces at each contact. The fringe patterns, which varied continuously under the contact points of the scaled-up asperities and the photoelastic plate, were recorded by a camcorder. The spatial data from the fringe patterns was sufficient to calculate the normal and frictional forces at each contact point during the transition between static and kinetic friction. The generally accepted fact is that the difference between the coefficients of static and kinetic friction is small and that the coefficient of friction at each micro contact is the same. However, the photoelastic results show that the difference between the coefficients of static and kinetic friction is quite large and that the coefficient of friction at each contact point varies from one point to another and changes during sliding. A local coefficient
of friction also is different significantly from the global coefficient of friction.
1. Introduction It is common practice in determining the frictional force Q from the normal force I’, to employ Amonton’s law of friction given by
Q=cLp
(1)
where g is the coefficient of friction. This simple form of the friction law is based on the global force measurements of P and Q, which represent the average contribution of many local friction forces. These forces occur at each micro contact made when many asperities interact with a counter surface. Two assumptions are necessary to invoke Amonton’s law [l]. (a) During sliding, the resistive force per unit area of contact is constant. (b) The real area of contact is proportional to the normal load. These two assumptions imply the generally accepted belief that the local coefficient of friction associated with each micro contact is the same as the global coefficient of friction. In the study of friction, many models describing friction mechanisms have been proposed using this assumption. Although this assumption has not been challenged, because of the difficulty in measuring the local frictional force at each micro contact, the fundamental question regarding the magnitude of the coefficient of friction at each micro contact still remains.
0043-1648/94/$07.00 0 1994 Elsevier SSDI 0043-1648(93)06370-J
Sequoia. All rights reserved
With regard to this question, Dally and Chen [2] investigated the magnitude of the coefficient of static friction at each micro contact using photoelastic measurements of the local forces. The authors simulated micro contacts by using scaled-up asperities made from 12.7 mm radius steel rounds. A photoelastic plate with a flat surface as a counter surface was used to sense the local forces. They determined the contact forces (normal and frictional) and the associated coefficient of static friction at each contact from the isochromatic fringe patterns. While they showed that the coefficient of static friction at each micro contact is different from the global coefficient of friction, the study was limited to friction measurements prior to slippage. In this study, the coefficients of kinetic friction as well as static friction at each contact are investigated. A simplified two-dimensional model, which is composed of the artificially scaled up asperities and a photoelastic plate with a flat surface, is employed to simulate the typical contacting surfaces. Photographs of the fringe patterns, that are changing continuously under the contact points during the transition period from static to kinetic friction, are recorded with the camcorder. These dynamic fringe patterns are analysed using a similar method to Dally and Chen’s [2], to determine the normal and frictional forces and the associated coefficient of friction at each contact point. Finally, the
validity of the two assumptions inherent in Amonton’s law of friction is examined.
2. Experimental
method
To simulate the typical contacting surfaces shown in Fig. l(a), a simplified two-dimensional model was constructed as shown in Fig. l(b). The arbitrarily shaped asperities of the upper body are modeled by a series of cylindrically shaped asperities, and the lower body is modeled by a half plane with a flat surface. A schematic diagram of the experimental apparatus is presented in Fig. 2, and a photograph of the actual equipment is given in Fig. 3. The scaled-up asperities are fabricated from 12.7 mm radius rounds for the fivecontact-point experiment and 6.35 mm radius rounds
(4
\
LAAAAJ
\
(b) Fig. 1. (a) Typical dimensional model
Fig.
2. Schematic
contacting surfaces and of contacting surfaces.
diagram
of the experimental
(b)
simplified
apparatus.
two-
Fig. 3. Photograph
of the
TABLE carbonate
1. Elastic
constants
Modulus Poisson’s Material
of elasticity ratio (v) fringe value
(E) cf,)
actual and
experimental material
set-up.
fringe
value
of poly-
2480 MPa 0.38 7.0 kN/(m-fringe)
for the seven-contact-point experiment. The aluminum rounds are attached to a straight steel bar with screws. The surfaces of the asperities were roughened using number 400 grit abrasive to increase the friction. The photoelastic plate was fabricated from a polycarbonate sheet, 6.3 mm thick, cut into a 250 X 165 mm rectangle. The model was supported on three sides by an aluminum frame. The elastic constants and the material fringe value for the polycarbonate model material are presented in Table 1. To prevent parasitic loading of the model due to the irregularities on the surface of the model frame, cardboard shims were installed at both the sides and bottom of the polycarbonate plate [3]. The model frame is attached to a steel plate and can move freely in the horizontal direction over rollers which support the steel plate. To measure the global frictional force, a loadcell is attached to the steel bar that supports the asperities. The loadcell is aligned with the contact surface to avoid the couple moment caused by any difference in elevation between the contact surface and the loadcell. The normal load was applied by placing the weights on the hanger attached to a loading lever arm. The applied normal load was kept constant at 942 N for both the five and seven-contact point experiments. After applying the normal load, the tangential load was applied by placing the weights on the second hanger until motion occurred. The sliding distance of the model relative to the asperities was approximately 40 mm before a stop terminated motion. The relative velocity increased with time and position as the friction force
101
K.S. tee et al. / Transitional friction behavior at scaled micro contacts
decreased. A maximum velocity of 0.43 m s-’ was observed just before the photoelastic model engaged the stop. As tangential load was increased, the isochromatic fringe patterns were recorded with a camcorder operated at 30 frames s-‘. The video images were transferred to a VCR which was connected to the camcorder. To avoid blurring of the fringe pattern images during sliding, a fast shutter speed (l/1000 s) was employed. The loadcell output during the transition period (from static to kinetic) was recorded on a digital storage oscilloscope at a sampling rate of 1 ms point-‘.
3. Analysis of the isochromatic fringe patterns A time-sequence of isochromatic fringe patterns for the seven-contact-point experiment is illustrated in Figs. 4(a)-4(e). The isochromatic fringe pattern corresponding to static friction (just before the model begins to
move) is shown in Fig. 4(a). The other fringe patterns, presented in Figs. 4(b)-4(e), represent the state of kinetic friction. The time interval between each fringe pattern is appro~mate~y 33.3 ms. The fringe patterns were transferred from the tape of the VCR to RAM storage in a personal computer by using a frame grabber system. These stored images of the fringe patterns were processed to extract the centerline of each fringe using a thinning algorithm 141. After the thinning process, the centerline of each fringe was digitized to give spatial data for a numerical analysis to determine the normal and frictional force at each contact point. The number of data points varies from 1300 to 1700 for each fringe pattern. A typical centerline display obtained from the static iso~hromatic fringe pattern is shown in Fig. 5. From the stress optic law of photoelasticity [S], the fringe order N is given in terms of the stresses as
(2) where h is the thickness of the model and f0 is the material fringe value. The Cartesian stress components at an arbitrary point A&, z), illustrated in Fig. 6, are given by the Flamant solution [6] as u = xx
2 _ 21~-+J’~~i~ + Qi(x -xoJ’l p{(x -xoij2 -t-2-y
j-1
u?g-P-NX-x,l)” +-e 2z2{Piz + Qi(x -x,J”}
(3)
2(x -x&(p,z + Qi(x -xOt>‘} p((x x$ f 22)2
-
where x,‘s define the position of each contact point and n is the number of contact points. Let us combine eqns. (2) and (3) and define a function G, as
Fig. 5. Extracted in Fig. 4(a).
of the js~hromatic
fringes shown
A(w)
(e)
Fig. 4. A series of isochromatic fringe patterns for the sevencontact-point experiment. (a) static,(b) At = 33 ms, velocity= 0.024 m s-‘, (c) A~=67 ms, velocity=0.048 m s-‘, (d) bt=lOO ms, velocity=O.167 m SC’, (e) k-133, velocity=O.286 m s-l.
centerlines
Fig. 6. Coordinate
system for the two-dimensional
model.
102
KS.
Lee et al. / Transitional
friction
behavior
at scoled
micro
contacts
two statistical measures of error were defined
where
The normal and frictional forces and the associated coefficients of friction that were determined are summarized in Table 2 and Table 3 for the five-contactpoint experiment and the seven-contact-point experiment, respectively. To verify the accuracy of the values,
where Ni is the input fringe order for a given data point whose position is (xm, z,,J. NC is the calculated fringe order using the values in Tables 2 and 3 for the unknown quantities. The quantity m is the average of the fringe order deviation (AN), and (SD), is the standard deviation of the fringe order deviation. Results for m and (SD),, summarized in Table 4, indicate that the average deviation and the standard deviation are quite small. Reconstructed fringe patterns generated by using the solution for the normal and frictional forces for the seven-contact-point experiment are presented in Fig. 7. The actual fringe pattern is shown in Fig. 4(a). Comparing the re~nst~cted fringe pattern with the ~~es~nding actuai fringe pattern reveals that the two patterns are in good agreement. Since the magnitudes of the average and the standard deviation of the fringe order deviation are quite small and because the reconstructed fringe patterns agree well with the
TABLE
coefficients
Bij =xX;
-&Y;
+G)
fZ4 + 4xixjz2
Ci~=P~Pj~-tPiQ~~j+PjQi~i+Q,QjX~Xj In eqn. (4), the subscript m identifies the point at position (x,, z,,,) with a fringe order N,. The unknowns Pi, Qi and X, were solved by empioying an overdeterministic method [7,8], where the number of data points greatly exceed the number of unknowns. 4. Results and discussion
2. Calculated Number contact point
fa)
of
normal
and frictional
forces and associated
Normal force (N)
Frictional force (N)
Coefficient of friction
tpi)
cQi>
(/.d
of friction for five-contact-point
experiment
Frictional force (N)
Coefficient of friction
P=LP;
Q=xQi
P&= Q/f’
941.9
489.1
0.519
Normal force (N)
1
277.6
121.1
0.436
2 3 4 5
168.4 165.7 122.8 207.4
81.1 75.3 63.8 147.8
0.481 0.454 0.519 0.713
(bb)
1 2 3 4 5
267.1 161.9 159.7 122.2 217.4
103.7 60.8 44.6 39.9 65.3
0.388 0.376 0.279 0.326 0.301
928.3
314.4
0.339
(c)
1 2 3 4 5
277.1 170.5 164.6 134.7 230.3
90.4 49.2 32.9 34.2 47.8
0.326 0.289 0.200 0.254 0.207
977.2
254.5
0.260
(d)
1 2 3 4 5
268.9 154.6 150.7 131.8 221.5
77.8 39.3 27.1 26.4 26.3
0.289 0.254 0.180 0.200 0.119
927.4
197.0
0.212
(ef
1 2 3 4 5
263.0 156.1 151.9 143.3 228.5
74.9 32.1 23.1 24.3 26.3
0.285 0.205 0.152 0.169 0.115
942.8
180.6
0.192
103
KS. Lee et al. / Transitional friction behavior at scaled micro contacts TABLE
3. Calculated
and frictional
farces and associated
coefficients
of friction
for seven-intact-lint
experiment
Normal force (N) P=cPi
Frictional force (N)
Coefficient of friction
Q=EQi
~8
0.580 0.396 0.620 0.577 0.559 0.446 0.612
916.7
503.5
0.549
109.1 32.2 24.2 32.9 63.9 15.9 48.5
0.529 0.228 0.381 0.495 0.512 0.240 0.236
874.0
326.7
0.374
197.9 142.2 58.8 74.2 144.6 100.7 243.9
99.6 25.9 32.0 30.6 55.8 21.0 21.0
0.503 0.182 0.543 0.413 0.386 0.208 0.086
962.4
285.8
0.297
1 2 3 4 5 6 7
192.4 139.4 44.4 65.4 131.6 105.3 228.2
84.6 39.0 18.2 31.4 50.6 10.1 21.4
0.440 0,280 0.410 0.480 0.384 0.100 0.094
901.8
255.2
0.283
1 2 3 4 5 6 7
209.6 151.7 46.7 49.3 116.8 84.0 217.3
65.7 29.2 27.1 31.9 37.8 6.7 3.5
0.313 0.192 0.579 0.649 0.324 0.079 0.016
875.3
201.8
0.23’1
Normal force (N)
Frictional force (N)
Coefkient of friction
tpi)
(Qi)
(Eli)
1 2 3 4 5 6 7
199.0 134.7 62.9 71.2 133.4 79.3 236.1
115.5 53.4 39.0 41.1 74.6 35.4 144.5
1 2 3 4 5 6 7
206.3 140.9 63.6 66.5 124.8 66.1 205.7
1 2 3 4 5 6 7
Number contact point
TABLE viation
normal
of
4. Average and standard
deviation
= QP
of fringe order de-
Five-contact-point
Seven-contact-point
Average Xi?
Standard deviation (SD),
Average N
Standard deviation (SD),
0.017 0.004 - 0.005 0.002
0.132 0.146 0.163 0.174
0.019 0.010 - 0.011 0.012
0.094 0.145 0.180 0.191
-0.004
0.173
-0.009
0.189
actual fringe patterns, we have confidence in the accuracy of the determination of the normal and frictional forces.
Fig. 7. Reconstructed fringe pattern associated with the calculated normal and frictional forces for Fig. 4(a).
The variation of the global frictional force with time is shown in Fig. 8. In this figure, the solid line represents the measured values from the loadcell and the symbols represent the sarne of the local frictional forces determined from the photoelastic analysis. The bracketed letter below each symbol indicates the corresponding fringe pattern in Fig. 4. Since the time for each fringe pattern relative to the oscilloscope record was not
104
KS’. Lee et ul
i Transitional
friction
behavior
at scaled
micro
confects
0.2 -
o, 0
60
150
100
200
250
o.o! 0
100
60
Time (ms)
t
1
150
200
Time (ms)
(a) 600
l.ol
-
Loadcell A
Calculation 0.6
c
100
os t
0
50
“.I 150
100
200
50
Time (ms) (bl
0
50
100
150
Time (ma) (W
Fig. 8. Variation of the global frictiona force with time during the transition period for (a) Eve-contact-lint experiment and (b) seven-contact-point experiment.
Fig. 9. Variations of the coefficient of friction at each contact point with time for (a} five-contact-point experiment and (b) seven-contact-point experiment.
known, the symbols in Fig. 8 were moved as a group along the time axis to match with the loadcell output values. It is evident that the sum of the local frictional forces agree well with the measured values from the loadcell. Contrary to the belief that the coefficient of kinetic friction is less than that of static friction by a small amount, it is clear from Fig. 8 that the difference is quite large. The kinetic coefficient of friction at each contact point is shown as a function of time in Fig. 9. In this figure, the solid line represents the global coefficient of friction pB and the symbols represent the coefficient of friction at each contact point. The differences between the coefficient of friction at each contact point and the global coefficient of friction are significant. In this work, the materials of the contacting bodies are dissimilar, i.e. the asperities are metal (aluminum) and the counter
surface is plastic (polycarbonate). The frictional force in this model is mainly due to the deformation. We expect that the magnitude of the frictional force will depend on the shape of the asperity (e.g. angle of attack) and the depth of penetration, which is proportional to the normal load. In these experiments, the shape of all the asperities is the same, but the angle of attack varied with the depth of penetration which was p~~rtional to the normal load. The ratio of the frictional force to the normal force, i.e. the coefficient of friction, will have a different value at each contact point, depending on the magnitude of the normal force. The coefficient of friction is proportional to the ratio of the depth of penetration (d) to the radius of asperity (r) [9]. The magnitude of the ratio of the penetration depth to the radius of asperity (d/r) in the present work varies from 0.028 to 0.042 for the five-contact-
KS. Lee et al. i Tra~itio~l
friction behavior at sculed micro contacts
point experiment and from 0.01 to 0.056 for the sevenintact-lint experiment. Since the d/r ratio of the asperities was always less than 0.1, the findings of Suh [9] indicate that the variation of the coefficient of friction at each contact point is expected to be very small. However, it is clear from Fig. 9, that the coefficient of friction at each contact point varies significantly. Furthermore, it can be observed in Tables 2 and 3 that in many instances there is no relation between Pi and fii. The variations in the local coefficient of friction appear to be independent of the local normal force. The reason for this behavior is not clear and more study is required before the phenomena can be understood.
References R.D. Ameli, P.B. Davies, J. Halling and T.L. Whomes, Tribology: principles and Design Applications, Springer, New York, 1991, p. 28. J.W. Dally and Y.M. Chen, A photoelastic study of friction at multipoint contacts, Exp. Me&., 31 (2) (1991) 144-149. R.B. Heywood, Designing by Ph~~i~t~~~, Chapman and Hail, London, 1952, pp. 96-98. T.Y. Zhang and C.Y. Suen, A fast parallel algorithm for thinning digital patterns, Commun. ACM, 27 (1984) 239-239. J.W. Dally and W.F. Riley, Ekperimentnl Stress Analysk, McGraw-Hill, New York, 2nd edn., 1978, pp. 406-482. L.E. Malvem, Introduction to the Mechanics of Continuous Medic, Prentice-Hall, Englewood CJifTs, NJ, 1969, pp. 539-546. R.J. Sanford and J.W. Dally, A general method for determining mixed-mode stress intensity factors from isochromatic fringe patterns, Eng. &act. Mech., II (1979) 621-633. R.J. Sanford, Application of the least-squares method to photoelastic analysis, Exp. Me&., 20 (1980) 192-197. N.P. Suh, T~~p~ysics, Prentice-Hap, Eagfewood Cliffs, NJ, 1986, pp. 295-304.
5. Summary and conclusions Two-dimensional photoelastic experiments were conducted to obtain dynamic &chromatic fringe patterns under the contact points between scaled-up asperities and a photoelastic half plane. The experiments were conducted during the transition period from static to kinetic friction. The normal and frictional forces and the associated coefficient of friction at each contact point were determined with an overdeterministic numerical method by using the digitized input data from the extracted centerlines of the obtained fringe patterns. By comparing the reconstructed fringe patterns with the actual fringe patterns and also by determining some statistical quantities, the accuracy of the calculated normal and frictional forces at each contact point was verified. Contrary to the generally accepted fact that the difference between the coefficients of static and kinetic friction is small and the coefficient of friction at each micro contact asperity is the same, these results show that the difference between the coefficients of static and kinetic friction is quite large and the coefficient of friction varies greatly between contact points and also is significantly different from the global coefficient of friction
The authors wish to thank the National Science Foundation and the University of Ulsan for supporting this research.
105
Appendix A: Nomencla~~
; M : Ni N, LW AIV P
Q
r (SD), -G 2
depth-of -penetration material fringe value (KN m-l) thickness of the photoelastic plate (mm) number of input data points number of contact points fringe order input fringe order calculated fringe order fringe order deviation average of fringe order deviation normal force (N) frictional force (N) radius of curvature of asperity standard deviation of m Cartesian coordinates
Greek symbols coefficient of friction CL coefficient of friction at each contact point cti global coefficient of friction 4X u.uf @YzzY Cartesian stress components (MPa) rxL