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On the elementary theory of friction
135
Wear,45 (1977) 135 - 138 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
Short Communication
On the elementary
theory
of friction
S. N. POSTNIKOV* Physical-Technical
Department,
Gorky Polytechnic Institute, Gorky, 603024
(U.S.S.R.)
Z. ELIEZER Department of Mechanical Engineering and Materials Science Laboratories, of Texas at Austin, Austin, Texas 78712 (U.S.A.)
The University
(Received March 21, 1977)
An attempt is made in this paper to prove the validity of the elementary theory of friction. The relationships between the friction force, the absorption of energy and the sliding speed are discussed in terms of available experimental data. The elementary theory of friction was developed by analogy with the classical theory of light dispersion [l] **. The model of the resonanceselective mechanism of energy dissipation, which is different from the model proposed in ref. 2, forms the basis of this theory. On considering the relative sliding of the frictional couple components as the principal factor which stipulates the dynamic resonance of equivalent oscillators (atoms or groups of atoms), it was shown that the friction force can be given by the following equation : Wf
whereQi, mi, Woi, Pi, Qi(k), ci and ni are the charge, mass, natural frequency, damping coefficient, quality factor, period of identity of position and surface concentration of the oscillators, respectively; Z. is the initial distance between the rubbing surfaces, N the normal load, b the coefficient of closest approach of the surfaces, S the real area of contact and ok the excitation frequency; the symbol i(k) indicates the functional interrelation between Qi and wk (wk = 2nvk = 27ru/ai, u being the speed of sliding). If the rubbing surfaces are considered as two sliding atomic planes, eqn. (1) reduces to *Currently visiting scientist with the University of Texas at Austin. **The English version of ref. 1 is in preparation and will be published by Van Nostrand Reinhold Company in 1978.
136 F=
nq*s 8n&naQ(Z,
- bN)*((w;
- a’)2
+ 4132w25”2
(2)
Equation (Z), as well as eqn. (l), is the expanded form of the equation 2lAW J'=lJT
(3)
where Z A W is the total fraction of energy losses taking place over a period T = a/u in the two symmetrical (with respect to the in~rface) systems of oscillators. In this sense, the elementary theory of friction is the logical continuation of earlier work [3]. The resemblance between the models of external friction and of frequency-dependent internal friction of the relaxation-resonance type made it possible to introduce the concept of a frictional absorption spectrum Q&?t(~)w A W/W, similar to the mechanical spectrum of a solid Q&:((Y)[l] . Obviously, F-
(4)
Since the mechanism by which oscillation energy is dissipated comprises phenomena identical in their physical essence for the external and internal friction, the qualitative dependence of F on Q-r must be the same regardless of what type of friction is under consideration. In the absence of direct experimental data connecting F and Q-l the functional dependence expressed by eqn. (4) can be verified only if a third parameter, on which both F and Q-l depend simultaneously, could be found. A convenient way to vary Q-l at constant frequency is, for instance, by nuclear irradiation of the material. Experimental results [4] show that with increasing time of irradiation the absorption of energy decreases (Fig. 1). At the same time, the friction force is known to decrease drastically when a surface is irradiated by accelerated particles [ 51. Since both the absorption of energy and the friction force depend on the time of ~diation in the same manner, a tentative conclusion can be reached that the functional dependence F(Q-’ ) as described by eqn. (4) is valid at least in this case. The disturbance of the periodicity of the lattice by irradiation is, in the authors’ opinion, the physical basis for the similar variation exhibited by F and Q-l. This lattice disturbance impedes free movement of dislocations, resulting in hardening of the material with the corresponding decrease in F [S] . Furthermore, for the same reason, Q-l should also decrease as discussed in [4] . The physical nature of the dependence of friction force on sliding speed can also be explained in terms of the F( Q-l) relationship. As follows from eqns. (1) and (2), the frictional absorption spectrum, and consequently the function F(v), can have a number of alternating maxima and minima similar to the curve for the scattering of light in the presence of several absorption bands. Accordingly, as was emphasized in ref. 7, any formal models made on the assumption that the friction force was independent of
137
I
HIGH
TIME
PURITY
OF IRRADIATION,
COPPER
HOURS
Fig. 1. The variation of Q-l (at constant frequency) with the time of irradiation of a copper specimen with cobalt gamma. The data were taken from Fig. 3 of ref. 4.
sliding speed are acceptable only for steady state regimes. They do not reflect the periodicity of the pair interactions of particles forming condensed phases. The positions of the extrema on the F(u) curve depend upon the initial physico-chemical state of the surface layers and upon its changes as the frictional interaction develops. Thermal transition from the mainly elastic deformations to plastic displacements, formation of the screening oxide films which have lubricating functions, different secondary effects determined by the presence of polar lubricants etc. - all these factors can significantly influence the dependence of the friction force on speed. This is why such a dependence is one of the most complicated problems in tribology [ 81. A clear example of the influence of thermal effects on the F(u) curve is furnished by experiments in which the sliding speed was up to 1000 m s-l (w = 1013 s-l) [9] . An intense energy absorption due to the dynamic resonance of the atomic oscillators results in melting of the surface layers with a considerable reduction .in friction force. In the range of sliding speeds most often encountered in practice, however, the characteristic frequency wc of the atomic oscillators is well above their excitation frequency w determined by the speed u of sliding through the near-surface force fields carried by the opposed surfaces. Therefore, the expression for the friction force (eqn. (2)) becomes F=
nq4S 8n&$maQ(Z,,
-
21iV)~
(5)
Considering that S and b do not depend upon the speed of sliding, eqn. (5) can be written F = kQ-l
where k is constant
(6)
for a given material.
138
Since Q-l is known to decrease with frequency [4], F is expected to decrease with sliding speed. Such a tendency was in general found to be true for real materials [lo] , at least in a certain range of velocities. Conclusions Experimental results qualitatively support the assumption made in the elementary theory of friction that the friction force varies proportionally to 9-l which characterizes the absorption of energy in sliding friction. The relationship between F and 9-l makes it possible to predict the variation of F with sliding speed.
References 1 S. N. Postnikov, Electrical Phenomena in Friction and Cutting, Volgo - Vyatsk. Knizhn. Izd., Gorky, 1975, p. 26 (in Russian). 2 V. A. Bufeev, Proc. Conf. Electrochemical Processes in Friction and their Use in Wear Prevention, Odessa, 1973, p. 7 (in Russian). 3 E. Adirovich and D. Blokhintsev, J. Phys. USSR, 7 (1) (1943) 29. 4 A. V. Granato, Internal friction damping and cyclic plasticity, Am. Sot. Test. Mater. Spec. Tech. Publ., STP, 378 (1965) 93. 5 E. A. Dukhovskoy, V. S. Onischenko, A. N. Ponomarev, A. A. Silin and V. L. Talroze, Dokl. Akad. Nauk SSSR, 189 (6) (1969) 1211 (in Russian). 6 F. P. Bowden and D. Tabor, Friction and Lubrication of Solids, Vol. 1, Oxford Univ. Press: Clarendon Press, London, 1964, p. 98. 7 S. N. Postnikov and Ya. K. Navrotskaya, On the dependence of friction on sliding speed, Proc. Conf. Theory of Friction and Wear, Tashkent, 1975, p. 22 (in Russian). 8 A. S. Akhmatov, Molecular physics of boundary lubrication, Israel Programme for Scientific Translations, Jerusalem, 1966, p. 337. 9 F. P. Bowden and E. H. Freitag, The friction of solids at very high speeds, Proc. R. Sot. London, Ser. A, 248 (1958) 350. 10 E. Rabinowicz, Friction and Wear of Materials, Wiley-Interscience, New York, 1965, p. 60.
Wear,45 (1977) 135 - 138 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
Short Communication
On the elementary
theory
of friction
S. N. POSTNIKOV* Physical-Technical
Department,
Gorky Polytechnic Institute, Gorky, 603024
(U.S.S.R.)
Z. ELIEZER Department of Mechanical Engineering and Materials Science Laboratories, of Texas at Austin, Austin, Texas 78712 (U.S.A.)
The University
(Received March 21, 1977)
An attempt is made in this paper to prove the validity of the elementary theory of friction. The relationships between the friction force, the absorption of energy and the sliding speed are discussed in terms of available experimental data. The elementary theory of friction was developed by analogy with the classical theory of light dispersion [l] **. The model of the resonanceselective mechanism of energy dissipation, which is different from the model proposed in ref. 2, forms the basis of this theory. On considering the relative sliding of the frictional couple components as the principal factor which stipulates the dynamic resonance of equivalent oscillators (atoms or groups of atoms), it was shown that the friction force can be given by the following equation : Wf
whereQi, mi, Woi, Pi, Qi(k), ci and ni are the charge, mass, natural frequency, damping coefficient, quality factor, period of identity of position and surface concentration of the oscillators, respectively; Z. is the initial distance between the rubbing surfaces, N the normal load, b the coefficient of closest approach of the surfaces, S the real area of contact and ok the excitation frequency; the symbol i(k) indicates the functional interrelation between Qi and wk (wk = 2nvk = 27ru/ai, u being the speed of sliding). If the rubbing surfaces are considered as two sliding atomic planes, eqn. (1) reduces to *Currently visiting scientist with the University of Texas at Austin. **The English version of ref. 1 is in preparation and will be published by Van Nostrand Reinhold Company in 1978.
136 F=
nq*s 8n&naQ(Z,
- bN)*((w;
- a’)2
+ 4132w25”2
(2)
Equation (Z), as well as eqn. (l), is the expanded form of the equation 2lAW J'=lJT
(3)
where Z A W is the total fraction of energy losses taking place over a period T = a/u in the two symmetrical (with respect to the in~rface) systems of oscillators. In this sense, the elementary theory of friction is the logical continuation of earlier work [3]. The resemblance between the models of external friction and of frequency-dependent internal friction of the relaxation-resonance type made it possible to introduce the concept of a frictional absorption spectrum Q&?t(~)w A W/W, similar to the mechanical spectrum of a solid Q&:((Y)[l] . Obviously, F-
(4)
Since the mechanism by which oscillation energy is dissipated comprises phenomena identical in their physical essence for the external and internal friction, the qualitative dependence of F on Q-r must be the same regardless of what type of friction is under consideration. In the absence of direct experimental data connecting F and Q-l the functional dependence expressed by eqn. (4) can be verified only if a third parameter, on which both F and Q-l depend simultaneously, could be found. A convenient way to vary Q-l at constant frequency is, for instance, by nuclear irradiation of the material. Experimental results [4] show that with increasing time of irradiation the absorption of energy decreases (Fig. 1). At the same time, the friction force is known to decrease drastically when a surface is irradiated by accelerated particles [ 51. Since both the absorption of energy and the friction force depend on the time of ~diation in the same manner, a tentative conclusion can be reached that the functional dependence F(Q-’ ) as described by eqn. (4) is valid at least in this case. The disturbance of the periodicity of the lattice by irradiation is, in the authors’ opinion, the physical basis for the similar variation exhibited by F and Q-l. This lattice disturbance impedes free movement of dislocations, resulting in hardening of the material with the corresponding decrease in F [S] . Furthermore, for the same reason, Q-l should also decrease as discussed in [4] . The physical nature of the dependence of friction force on sliding speed can also be explained in terms of the F( Q-l) relationship. As follows from eqns. (1) and (2), the frictional absorption spectrum, and consequently the function F(v), can have a number of alternating maxima and minima similar to the curve for the scattering of light in the presence of several absorption bands. Accordingly, as was emphasized in ref. 7, any formal models made on the assumption that the friction force was independent of
137
I
HIGH
TIME
PURITY
OF IRRADIATION,
COPPER
HOURS
Fig. 1. The variation of Q-l (at constant frequency) with the time of irradiation of a copper specimen with cobalt gamma. The data were taken from Fig. 3 of ref. 4.
sliding speed are acceptable only for steady state regimes. They do not reflect the periodicity of the pair interactions of particles forming condensed phases. The positions of the extrema on the F(u) curve depend upon the initial physico-chemical state of the surface layers and upon its changes as the frictional interaction develops. Thermal transition from the mainly elastic deformations to plastic displacements, formation of the screening oxide films which have lubricating functions, different secondary effects determined by the presence of polar lubricants etc. - all these factors can significantly influence the dependence of the friction force on speed. This is why such a dependence is one of the most complicated problems in tribology [ 81. A clear example of the influence of thermal effects on the F(u) curve is furnished by experiments in which the sliding speed was up to 1000 m s-l (w = 1013 s-l) [9] . An intense energy absorption due to the dynamic resonance of the atomic oscillators results in melting of the surface layers with a considerable reduction .in friction force. In the range of sliding speeds most often encountered in practice, however, the characteristic frequency wc of the atomic oscillators is well above their excitation frequency w determined by the speed u of sliding through the near-surface force fields carried by the opposed surfaces. Therefore, the expression for the friction force (eqn. (2)) becomes F=
nq4S 8n&$maQ(Z,,
-
21iV)~
(5)
Considering that S and b do not depend upon the speed of sliding, eqn. (5) can be written F = kQ-l
where k is constant
(6)
for a given material.
138
Since Q-l is known to decrease with frequency [4], F is expected to decrease with sliding speed. Such a tendency was in general found to be true for real materials [lo] , at least in a certain range of velocities. Conclusions Experimental results qualitatively support the assumption made in the elementary theory of friction that the friction force varies proportionally to 9-l which characterizes the absorption of energy in sliding friction. The relationship between F and 9-l makes it possible to predict the variation of F with sliding speed.
References 1 S. N. Postnikov, Electrical Phenomena in Friction and Cutting, Volgo - Vyatsk. Knizhn. Izd., Gorky, 1975, p. 26 (in Russian). 2 V. A. Bufeev, Proc. Conf. Electrochemical Processes in Friction and their Use in Wear Prevention, Odessa, 1973, p. 7 (in Russian). 3 E. Adirovich and D. Blokhintsev, J. Phys. USSR, 7 (1) (1943) 29. 4 A. V. Granato, Internal friction damping and cyclic plasticity, Am. Sot. Test. Mater. Spec. Tech. Publ., STP, 378 (1965) 93. 5 E. A. Dukhovskoy, V. S. Onischenko, A. N. Ponomarev, A. A. Silin and V. L. Talroze, Dokl. Akad. Nauk SSSR, 189 (6) (1969) 1211 (in Russian). 6 F. P. Bowden and D. Tabor, Friction and Lubrication of Solids, Vol. 1, Oxford Univ. Press: Clarendon Press, London, 1964, p. 98. 7 S. N. Postnikov and Ya. K. Navrotskaya, On the dependence of friction on sliding speed, Proc. Conf. Theory of Friction and Wear, Tashkent, 1975, p. 22 (in Russian). 8 A. S. Akhmatov, Molecular physics of boundary lubrication, Israel Programme for Scientific Translations, Jerusalem, 1966, p. 337. 9 F. P. Bowden and E. H. Freitag, The friction of solids at very high speeds, Proc. R. Sot. London, Ser. A, 248 (1958) 350. 10 E. Rabinowicz, Friction and Wear of Materials, Wiley-Interscience, New York, 1965, p. 60.