RESISTANCE OF THE SURFACE MATERIAL TO MICRO-SHEAR, REMAINING STATIONARY WHEN THE NORMAL PRESSURE CHANGES

I. R. Konyakhin

(Presented by the Department of Strength of Materials)

The article shows that, in the Amontons–Coulomb law of friction,

$P = A + \psi_s N$,

the constant quantity $A$ may vary for various reasons. The value of $A$ is influenced by the substance located between the contacting surfaces, as well as by its microfriction.

The many experiments we carried out with static friction always confirmed a direct proportionality between the force of static friction and the normal load.

As an example, one may cite one of such experiments, carried out on specimens: steel on steel. The tested specimens, placed one upon the other, were loaded from above by a normal pressure of a definite magnitude. After this, the upper body was displaced over the lower one by a horizontal force, until separation of the surfaces occurred, that is, until the transition to their ordinary relative sliding. Then the same procedure was repeated, but at some other value of the normal load, and so on several times. From the obtained values of the separating forces and the normal loads acting in this case, the graph shown in Fig. 1 was plotted.

On the graph one sees a straight line passing through the origin of coordinates.

Thus, the dependence obtained is

$$ P = \psi_s N, $$

where $\psi_s$ is the coefficient of static friction.

The obtained dependence expresses the Amontons–Coulomb law. The coefficient of static friction is determined from the slope of the straight line of the graph,

$$ \psi_s = \frac{P}{N} = 0.62. $$

As experiments have shown, for hard materials, for example steel, copper, the straight line of the graph passes through the origin of coordinates; for soft materials, however, for example zinc, the straight line of the graph passes above the indicated point.

The passage of the straight line on the graph above the origin is observed, moreover, also in the case where the contacting surfaces of refractory metals are covered with some foreign substance.

Fig. 1. Dependence between the static friction force and the normal pressure.

For example, graphite introduced between hardened steel surfaces causes the straight line of the graph to pass above the origin, whereas in the absence of graphite no such phenomenon is observed with these surfaces.

In the presence of graphite, in this case the equation of the straight line of the graph is obtained in the form

\[ P = A + \varphi N, \]

where the constant term of the equation \(A\) does not depend on the magnitude of the normal pressure.

Some liquid lubricants introduced between steel surfaces cause the appearance of the quantity \(A\), while some other lubricants do not possess this property.

For similar tests, crocus was used in the form of a fine powder. At first the coefficient of friction was found to be \(\varphi_s = 0.1\); here the crocus acted as a lubricant. After the specimens had been run in under pressure, the static coefficient of friction increased to \(\varphi_s = 0.48\). This circumstance indicates that the crocus film remaining on the surfaces in the form of a reddish deposit was greatly reduced in thickness and became a quasi-solid body. In addition, it contributed to the appearance of the force \(A\), which was absent with clean surfaces.

Subsequent experiments with crocus were carried out in the same manner as in the preceding case, i.e., the static friction force was determined by displacing the upper specimen from its position under the action of normal loads of different magnitudes.

In all, three experiments were carried out. The first of them was immediately after running-in of the specimens, the second after two days of preliminary contact, and the third immediately after the second. In the last case, preliminary separation of the surfaces was first performed under a pressure of \(45\) kg, and then the friction force of the specimens loaded by normal forces \(N = 5, 15, 25, 35\) kg was determined.

The graph shown in Fig. 2 presents the results of all three experiments in the form of inclined straight lines. Straight line 1 represents the relationship between the friction force and the normal pressure after the specimens had been run in against one another in the presence of a crocus film on their surfaces.

Fig. 2. Change in the magnitude of \(A\) for various causes: 1 — after running-in, 2 — after two days of contact, 3 — after tearing away under a pressure of 45 kg.

In this case the value \(A_1 = 3\) kg (straight line 1). Line 2 is the friction graph for the same specimens after 2 days of stationary contact; the value \(A\) increased to \(A_2 = 5\) kg.

Straight line 3 is the friction graph after preliminary tearing away of the surfaces at a pressure of 45 kg. In the last experiment the value of the constant force \(A\) increased to \(A_3 = 10\) kg.

Obviously, in both of the latter cases, that is, during stationary contact and in the process of preliminary tearing away of the surfaces, there occurred a decrease in the thickness of the crocus film, as well as its strengthening as a quasi-solid body, which also contributed to an increase in the constant force \(A\).

As for the coefficient of static friction, determined by the slope of the straight line of the graph, it remained almost unchanged in all 3 experiments.

In the experiments described, the force of static friction was determined by tearing away the surfaces; consequently, the greatest displacement force was found. However, before starting from rest the body first acquires a micromotion which, accelerating, leads to tearing away. If, however, a pulling force of such magnitude is applied to the specimen that it is incapable of producing micromotion, but under its action true rest is possible, such a force will be smaller in magnitude than the tearing-away force. There is a difference in magnitude between the latter and the pulling force at which micromotion is only just beginning to appear but has not yet occurred.

Let the force under true rest be denoted by \(P_s\), in distinction from the tearing-away force \(P\); then \(P_s < P\).

If an experiment is carried out to determine the relationship between the force of true rest \(P_s\) and the normal load, then one also obtains

linear dependence. \(P_s=\mu N\), or else \(P_s=A+\mu N\), depending on the experimental conditions, as in the preceding cases.

Here the quantity \(\mu\), which is the coefficient of displacement, is smaller than the value of the coefficient of friction \(\mu_s\) determined by the tearing-away of surfaces,

\[ \mu<\mu_s . \]

If, under a tractive force \(P_s\) that is not yet capable of producing micromotion, the normal load is reduced, then the specimen begins to move forward with a microscopically small velocity, which can be observed on our instrument, giving an apparent magnification of the displacement of 250,000 times; that is, it is possible to notice a displacement of the order of 0.005 micron.

However, after reducing the normal force, one may also correspondingly reduce the magnitude of the tractive force to such an extent that the specimen remains in place. If the tractive force is reduced by too large an amount, not corresponding to the normal load, the specimen will shift backward, i.e., in the direction opposite to the direction of shear under a tractive force greater than \(P_s\).

Consequently, it is possible simultaneously to reduce both the normal pressure and the tractive force in such a way that the specimen remains in place, shifting neither forward nor backward. In this case the ratio of the tractive force to the normal load remains unchanged.

Let us show one such experiment, carried out on the materials: hardened steel on hardened steel. Under the action of a normal load \(N_c=45\) kg, a displacement was produced by a force \(P_s=25.5\) kg. Then both forces, the normal as well as the tangential, were reduced simultaneously in such a way that the specimen shifted neither forward nor backward. The graph of this experiment is shown in Fig. 3, where a straight line is seen with a slope to the abscissa axis equal to the coefficient of displacement \(\mu=0.60\).

Visible graph labels: vertical axis: \(P_s\), “Displacement force”; horizontal axis: \(N\), “Normal pressure.”

Fig. 3. Simultaneous reduction of the tractive force and the normal pressure with the specimen immobile.

The ability of the specimen to undergo displacement as a result of a reduction of the normal load is manifested not only at a tractive force equal to \(P_s\), that is, close to the tearing-away force, but also at any smaller value of it. In this case—

case, the specimen, of course, does not tend to tear away, but moves forward by a certain amount proportional to the decrease in pressure. A decrease in the pulling force at constant pressure always leads to elastic return, that is, to motion of the upper specimen along the lower one in the direction opposite to the direction of the displacements produced by the pulling force.

In view of all that has been said, a simultaneous decrease of the pulling force and the normal load may be carried out after displacement by a pulling force of any magnitude.

A similar experiment was carried out on specimens: hardened steel on hardened steel, with displacement forces of 23 kg, 17 kg, 10 kg, and 7 kg. The force \(P_c = 23\) kg was close to the force of tearing away, while the remaining values were considerably smaller than it.

The graphs of the experiment are given in Fig. 4, where all four straight lines are shown. The first of them corresponds to the displacement force \(P_s = 23\) kg, the second to the force \(P_c = 17\) kg, the third to the pulling force at displacement \(P_c = 10\) kg, and the fourth was obtained with the pulling force \(P_c = 7\) kg. The forces 17, 10, and 7 kg are considerably smaller than the maximum force of true rest \(P_s = 23\) kg.

In the graph of Fig. 4 the straight lines do not pass through the origin of coordinates. Evidently, on the steel surfaces of the specimens there was a film of foreign substance because of insufficient cleaning of the surfaces before the experiment.

Fig. 4. Simultaneous decrease of the normal load and the pulling force with the specimen stationary.

Such an adsorbed film is always present on the working surfaces of specimens. In the experiment of Fig. 3 the specimens were rubbed against each other, and therefore the indicated film was destroyed by running-in.

In the present experiment (Fig. 4), however, the specimens were not run in; consequently, the adsorbed film remained intact.

In the last experiment (Fig. 4), straight line 4, the magnitude of the force \(A\) decreased somewhat in comparison with straight lines 1, 2, and 3. Similar decreases were also observed in other experiments.

In addition, a phenomenon was observed in which the force \(A\) is at first absent, and then appears, despite the fact that no foreign substances in the form of films were introduced between the working surfaces of the specimens.

The experiment was carried out on the following materials: hardened steel on zinc. The friction force was determined by tearing the surfaces apart at small normal loads within the range from 1 to 6 kg. In all, three analogous experiments were performed; their graphs, in the form of straight lines 1, 2, and 3, are shown in Fig. 5. They all have different slopes to the abscissa axis; straight line 3 cuts off on the ordinate axis the value \(A = 1\) kg.

Fig. 5. Change in friction force as a function of preliminary loading and preliminary displacement.

In the first experiment the normal pressure was applied successively from 1 to 6 kg, and in each case the specimen was sheared from its position by a horizontal force of a certain magnitude, equal to the friction force. In the graph of Fig. 5 this is represented by straight line 1. From its slope to the abscissa axis, the coefficient of static friction is determined: \(\psi_s = 0.54\).

In the second experiment the specimens were preliminarily loaded with a normal pressure \(N = 45\) kg, released from it, and then the forces of static friction were determined at \(N = 1, 2, 3, 4, 5\), and 6 kg, i.e., in the same way as in the preceding case. In the graph of Fig. 5, straight line 2 was obtained, situated with a greater slope to the abscissa axis than curve 1. The coefficient of static friction is equal to \(\psi_s = 0.78\). In this experiment the increase in the coefficient of friction was caused by the preliminary pressing of the specimens.

In the third experiment, a preliminary displacement was made by the force \(P_c = 20\) kg at the normal load \(N = 45\) kg. The specimens were then freed from these forces, and the experiment was subsequently carried out analogously to the preceding one.

As a result, straight-line graph 3 in Fig. 5 was obtained. The coefficient of static friction increased to \(\psi_s = 0.94\).

In addition, a constant force \(A = 1\) kg appeared, which is evident from the fact that straight line 3 does not pass through the origin.

In the present case it cannot be assumed that the force \(A\) is due to adhesion, since its appearance was preceded only by a preliminary displacement, which cannot be the cause of the appearance of sticking of the surfaces. The appearance of the quantity \(A\) is due to some other cause, which we shall clarify below.

First it is necessary to set forth certain propositions following from work [1].

After a preliminary shear of a specimen under pressure, an internal force appears in the layers of the contact material, directed oppositely to the preliminary displacement. When the displacing force is removed, it remains within the material as if in a braked state. If the specimen is now displaced in the reverse direction, this occurs with a reduced tensile force, since the internal force now assists the pulling force. The magnitude of the internal force can be determined by the selection method [1]. If, after the preliminary displacement, the pulling force is removed and then the normal pressure is gradually reduced, the internal force is released and the specimen tends to move back. This is a phenomenon of return of the second kind.

If, however, an opposing force is applied to the specimen and increased so that the specimen does not move, then the internal force will be taken up by the dynamometer of the instrument and can be determined from its magnitude.

The force holding the specimen in place and equal to the part of the internal force released when a portion of the normal pressure is removed is called the supporting force [1].

We shall next show an experiment illustrating the significance of the internal force of microdisplacement. The experiment was carried out on the materials: hardened steel on copper.

Fig. 6. Axis of immobility — 1 and axis of support — 2. Difference of their ordinates lies on straight line 3.

A preliminary displacement was made by a force \(P_c = 14\ \mathrm{kg}\) at a normal load \(N_c = 45\ \mathrm{kg}\). After this, both forces \(P_c\) and \(N_c\) were simultaneously and smoothly decreased while their ratio remained unchanged. The specimen remained immobile. As a result, the curve in Fig. 6 was obtained. Straight line 1, obtained with the immobile

the specimen by removing both loads, we shall call the line of immobility.

Next, the same displacement was made as in the preceding case, i.e., \(P_c = 14\ \text{kg}\) and \(N_c = 45\ \text{kg}\). After removal of the tensile force, the normal pressure was gradually decreased. In order that the specimen should not undergo a return of the second kind, a tensile force directed opposite to its motion was gradually increased from zero, that is, a bracing operation was carried out.

In the graph of Fig. 6, line 2 was obtained; it is the bracing line. Both lines are directed toward one point on the ordinate axis and intercept on it the value

\[ A = 3.5\ \text{kg}. \]

In the graph of Fig. 6, line 2 was obtained; it is the bracing line. Both lines are directed toward one point on the ordinate axis and intercept on it the value \(A = 3.5\ \text{kg}\).

If from the ordinates of the immobility graph—1 one subtracts the corresponding ordinates of the bracing line—2, then a straight line—3 is obtained, directed toward the origin of coordinates,

\[ P_s = \psi_s N, \]

but since

\[ P_s = P_c - P_t, \]

i.e., precisely the indicated difference of ordinates, the equation of line 3 of the graph is written as

\[ P_c - P_t = \psi_s N, \tag{1} \]

where \(\psi_s\) is the static coefficient of friction, and \(P_t\) is the internal force. For line 2 in its straight segment

\[ A - P_t = \psi_t N, \tag{2} \]

here \(\psi_t\) is the coefficient of internal resistance, which is the proportionality coefficient of line 2. From the simultaneous solution of the obtained expressions 1 and 2 we find

\[ \frac{P_c - P_t}{A - P_t} = \frac{\psi_s}{\psi_t} = a, \]

i.e., some constant quantity.

After some transformations we obtain

\[ A = \frac{P_c}{a} - \left(\frac{1}{a} + 1\right)P_t. \tag{3} \]

The last equation gives the dependence of the force \(A\) on the magnitude of the tensile force and the internal resistance \(P_t\).

For

\[ P_t = 0 \]

\[ A = \frac{P_c}{a} = P_c \frac{\psi_t}{\psi_s}, \]

but since in this case \(\psi_t\), which is the angular coefficient of line 2, is also equal to zero, \(A = 0\).

Thus, the experiment shows that in the present case the magnitude of the force \(A\) is caused by internal braking of the contact elements in its stressed state, and not by adhesion, as might have been supposed.

But adhesion also manifests itself in other cases, and therefore the law of displacement must be expressed by the equation

\[ P_c = A_a + A_c + \psi_s N, \]

where \(P_c\) is the force of static displacement; \(A_a\) is a constant due to adhesion; \(A_c\) is a constant due to the internal braking of the material caused by the normal load; \(\psi_s\) is the coefficient of static friction, and \(N\) is the magnitude of the normal pressure in kilograms.

REFERENCES

1. I. R. Konchakhin. Constrained microdeformation of shear of the surface material. Izvestiya vuzov, MVSSO, Mechanics and Machine Building, No. 7, 1961.