STATIC MICRODISPLACEMENT OF ONE BODY RELATIVE TO ANOTHER

I. R. Konyakhin

(Presented by the scientific seminar of the Department of Strength of Materials)

This article describes various kinds of preliminary displacements, differing in their character, that occur before the sliding of one body relative to another. The character of these displacements may vary depending on the conditions in which the microprotrusions of rough surfaces located at the points of contact of two bodies find themselves. The article is written on the basis of experiments carried out by microdeforming a rough area by means of another body having a comparatively smooth working surface.

Introduction

In the literature devoted to the question of preliminary displacement, there is only some very scant information on the character of deformation of the contact material.

A detailed study of preliminary displacement reveals that there is a certain variety in the forms of microshear.

Preliminary displacements may be divided into two types: static, occurring independently of time, and kinetic, connected with time by a definite dependence.

In the present work only static displacements are considered, studied by means of experiments carried out on an apparatus that is a certain improvement of the apparatus of A. V. Verkhovsky [1].

A characteristic feature of the microdeformation of shear of material located at the points of contact of two bodies is that microshear cannot be produced without the pressure of one body on the other, since the tensile force during displacement of a body is transmitted by it to the other body through static friction.

The action of the normal load during shear produces in the contact a special state, the essence of which is as follows.

The microprotrusions of a rough surface, having different heights, do not all come into contact at once, but gradually, one after another, as the bodies approach each other during the increase of the normal load. As a result, the degree of compressive deformation in them is different, and therefore each compressed element of contact experiences a pressure different from ...

pressures on other asperities. Therefore the force of static friction on each element also has its own magnitude. In addition, microasperities, having different dimensions, possess different stiffnesses when deformed by their shear. In view of these circumstances, different contact elements under shear have different degrees of deformation, determined, on the one hand, by stiffness and, on the other, by the force of static microfriction. The greater the stiffness, the smaller the deformation, and conversely, the greater the force of static friction, the later microslip of the contact elements occurs and the greater their shear deformation.

Thus, the magnitude of the shear deformation of a microasperity is proportional to the force of static friction and inversely proportional to the stiffness of the given contact element.

\[ C=\frac{F}{K}, \]

where \(C\) is the displacement of the element, \(F\) is the friction force, \(K\) is the stiffness in \(kg/\mu\), and \(C\) is the magnitude of the displacement of the tip of a microasperity relative to its base before the onset of slip.

Naturally, very many microasperities have different displaceability \(C\).

Elements having a greater value of \(C\) under microshear are capable of deformation of greater magnitude than elements with small \(C\). The former may be called active elements, the latter—weak ones.

During displacement the weakest elements begin to slip first, and the most active ones last.

The displacement that occurs for the first time after the specimens have been set up for the experiment is accompanied by plastic deformation of the contact elements; repeated displacement is accompanied by elastic deformation, since during the primary shear the microasperities are strengthened by plastic deformation.

Under repeated displacement, a nonlinear dependence is observed between the displacing force and the magnitude of microshear, due to the fact that the microasperities do not begin to slide simultaneously.

Removal of the shearing force is accompanied by elastic return of the upper specimen over the lower one. At this time elastic recovery of the deformed elements takes place. During this return, the weaker elements, having fully recovered, begin to deform in the opposite direction, since the return continues at the expense of recovery of the active elements. Finally, the resistance of all weak elements becomes equal to the force of those not yet recovered; equilibrium of forces sets in, and at this time the tractive force is reduced to zero.

Thus, as a result of displacement and return, accumulated elastic energy of deformed elements—weak and active, shifted in different directions—is present in the contact.

With a gradual decrease now of the normal load, the weak elements, having lost adhesion, will begin to slip in the direction of the shear that occurred, and the active elements, possessing sufficient adhesion to the upper specimen, will carry it in the direction of the elastic return that had occurred before this. A return of the second kind is obtained. At this time the contact is freed from the elastic energy of the deformed elements.

2. Types of Static Displacements

Static displacement of one body over another is effected by a slow increase in the shearing force acting on the upper

specimen. In this process, the magnitude of the force is increased so slowly that plastic shear deformations of the microasperities have time to occur in the contact material. During such displacement, stopping the increase of the tractive force leads to an immediate stop of the specimen. If the tractive force is increased faster than is required by the static displacement, then after its increase is stopped the body does not stop, but continues its micromotion with decreasing velocity. Such displacement is no longer static, as in the preceding case.

Static displacement, if it is performed for the first time after the specimens have been set up and loaded by a normal force, occurs plastically.

In the course of this shear, the material at the points of contact of the bodies is hardened. As a result, after removal of the shearing force a residual displacement is obtained.

A repeated microshear occurs elastically, but because the weaker elements of the contact slip, the shear becomes similar to a plastic one. Since the nature of such a microdisplacement is determined by the elasticity and microfriction of the contact elements, it may be called elastic-frictional.

Let us describe one of the experiments on displacement of a specimen made of hardened steel over bronze. In this case, only the microasperities of the rough surface of the lower body undergo plastic shear deformation.

Figure 1 shows the graph of the initial displacement in the form of the curve \(OA\). The latter represents a static displacement caused by the tractive force \(P\), increasing from zero.

Fig. 1.

Another curve—\(AB\)—represents the elastic return that occurred during removal of the displacing force. The horizontal straight line \(BC\) corresponds to the second return, which took place during gradual removal of the normal load. The magnitude \(CO\) shows the residual deformation.

After this experiment the specimens were again loaded by normal pressure. Then, by increasing the tractive force from zero, a repeated displacement was produced, which occurred according to the curve \(CA\) (Fig. 1). This curve represents an elastic-frictional displacement, which occurred in material hardened by the preceding displacement. As the curve \(CA\) shows, the displacement occurred with more intense resistance than in the preceding case.

After the displacement, a gradual decrease of the tractive force to zero was accompanied by elastic return of the upper specimen. The return occurred according to the curve \(AB\) of the graph, that is, in the same way as in the preceding case—

case. After removal of the shearing force, a residual displacement \(CB\) was obtained. With the subsequent gradual decrease of the normal pressure, the specimen moved opposite to the initial displacement, making a second elastic return according to the graph. Thus, the cycle of repeated displacement proved to be closed—the figure \(CABC\). The area of this figure corresponds to the frictional energy losses in the contact.

Displacement along the curve \(CA\) occurred after the normal load had been removed and then applied again. This operation eliminated the frictional bonds of the elements and enabled them to free themselves of elastic energy by recovery. It may be said that in this case the contact was discharged of the elastic energy contained in it up to that time owing to the frictional bonds caused by the normal pressure. If, after displacement, such a discharge is not carried out, then the subsequent displacement does not occur as shown by the curve \(CA\) of the graph, but along the steeper line \(BA\), shown by a dashed line and forming, with the elastic-return curve \(AB\), a hysteresis loop. It should be noted that the hysteresis loop is also due to the frictional bonds of the contact elements; with a certain method of displacement it is absent.

As a result of the experiment described, three different curves of static displacement were obtained—the curves of the graph \(OA\), \(CA\), \(BA\).

3. Residual contact forces caused by frictional bonds

As was described earlier, after preliminary displacement and the subsequent elastic return upon removal of the shearing force, residual forces will exist in the contact, caused by the braking of the contact elements by frictional bonds as they tend to recover. The magnitude of such an elastic force can be determined in the following way.

As we have already seen, during a gradual decrease of the normal load the specimen moves backward under the action of the elastic force of the elements under consideration. If the specimen is not allowed to execute this motion, then the elastic contact elements, while being freed from the frictional bonds during the decrease of pressure, will nevertheless be unable to undergo elastic recovery because they are held back by the supporting force. When the entire normal load has been removed while the specimen is immobile, the supporting force will be equal to the residual contact force.

The residual contact forces described have a noticeable influence on the character of static displacement. We shall show this in an experiment with specimens—hardened steel on steel.

First, static displacement was carried out at a normal load \(N_c = 45\) kg with the aid of a tensile force increased from zero to \(P_c = 25\) kg. As a result, curve 1 of the graph in Fig. 2 was obtained. Then the same displacement was carried out, with only the difference that some preliminary preparation had first been performed, consisting of the following.

The specimens, loaded by a normal force \(N_c = 20\) kg, were displaced by a tensile force \(P_c = 14\) kg, after which, without removing this force, the normal pressure was increased to \(N_c = 45\) kg. After removal of the shearing force, displacement was carried out in the same way as in the preceding case, by a force \(P_c = 25\) kg.

The graph curves of both displacements are shown in Fig. 2.

Curve 2 on the graph occupies a higher position than curve 1. Consequently, during the second displacement the resistances to shear at the corresponding moments were greater than in the first case.

Before the second microdisplacement, the residual forces in the contact were increased by increasing the normal load. This also affected the character of the displacement in the second experiment, despite the fact that the displacements in both cases were carried out in exactly the same way. It is also possible to show a case in which partial relief of the contact from residual forces lowers its resistance to microdisplacement.

Fig. 2.

The experiment was carried out on specimens: the upper one was made of hardened steel, the lower one of copper.

First the upper specimen was displaced along the lower one by a force \(P_c = 13\) kg at a normal pressure \(N_c = 45\) kg. As a result, in the contact,

Fig. 3.

after removal of the displacing force, residual forces are present. After this, the specimens were displaced with the residual elastic forces already present in the contact. As a result, curve 1 in Fig. 3 was obtained.

After the displacing force was removed, the normal load was reduced to \(N_c = 20\) kg and immediately increased again to its former value, \(=45\) kg. This operation released some elements of the contact from the stressed state—namely those in which the frictional bonds were small; when the pressure was reduced, these elements, being only weakly pressed, slipped, as a result of which the elastic force of the entire contact decreased.

The displacement after this proceeded along curve 2, which shows that the resistance to shear had decreased.

The experiment was then repeated twice more, with the normal load reduced to 10 kg and to 3 kg. After discharging the contact by the indicated reductions in pressure, the specimens were displaced, as before, by the same tensile forces of 13 kg at normal loads of 45 kg. As a result of the last experiments, curves 3 and 4 were obtained.

The graphs in Fig. 3 show that the greater the discharge of the contact from residual forces, the smaller the resistance to displacement obtained.

The resistance of the contact to static microdisplacement is obtained as the smallest if the preliminary displacement is made in the opposite direction, without changing the magnitude of the normal load. In the present case, however, it matters whether or not the specimens were relieved of the normal load before their subsequent forward displacement. If they were relieved, then the displacement proceeds as shown by curve 4 of the graph in Fig. 3. If, after the preliminary backward displacement, no relief from normal pressure is carried out and the forward displacement is produced, then it proceeds as shown by curve 5, which occupies the lowest position on the graph.

Since in the last experiment (curve 5), before the static displacement, the contact was not temporarily relieved of the normal load, there was no discharge in it, and residual forces remained in it; but those of them that were due to active elements were directed toward the subsequent displacement, and not opposite to it, as had been the case in the preceding experiments. Consequently, the residual force in the contact in the last case assisted the tensile force in producing displacement, rather than opposing it, as in the previous cases. All that has been said applies to the active elements of the contact; the weak ones, having small frictional bonds, are subject to slipping, and therefore their influence on the displacement proves to be smaller, since, once they begin to slip, they no longer participate in further elastic deformation, and therefore their resistance during further microshear does not increase but remains constant, whereas the tensile force increases continuously.

Microshear of one body over another can be produced not only by increasing the tensile force from zero, but also by decreasing the normal load while a small tensile force of constant magnitude is acting. Being applied to the body being displaced, it advances it slightly, after which the specimen remains immobile. A subsequent slow decrease in the normal pressure causes a microdisplacement in the direction of the applied small tangential force. In this case the displacement continues until almost the entire normal load has been removed.

Similar experiments were carried out on materials consisting of hardened steel on copper. A tensile force \(P_c = 3.3\) kg was applied to the upper specimen at a normal pressure \(N_c = 45\) kg. The latter was then smoothly reduced. In the course of this, the upper body moved in the direction of the force \(P_c = 3.3\) kg.

An analogous experiment was carried out with a displacing force of 6.6 kg. In the second case the displacement of the specimen was more intensive.

The graphs of both experiments are shown in Fig. 4.

An increase in the tractive force during displacement helps overcome the resistance to microdisplacement of the contact elements. During slip, their resistance, owing to friction, remains constant, and the variability of the tractive force is due to the elastic resistance of the remaining, non-slipping elements. In displacement by means of a decrease in normal pressure, the weak elements, being released from frictional bonds, no longer participate in shear deformation and therefore offer neither elastic resistance to their deformation nor frictional resistance. Therefore, in the present case microdisplacement proceeds differently than in displacement by means of an increase in the displacing force.

Fig. 4.

The latter proposition is further confirmed by the following. The curves of static displacements obtained from experiments with shear by an increasing tractive force under a constant normal load are, in their form, close to parabolas of the second degree, whereas the curves obtained from displacement by decreasing the normal pressure at a constant value of the tractive force are, in their form, close to parabolas of the fourth degree.

All the presented static curves of preliminary displacements, obtained by different methods, testify to a certain diversity among them, and also to the fact that the energy losses in the contact are to a significant extent due to microfriction and plastic deformation of the microasperities of the rough surface.

LITERATURE

1. V. D. Kuznetsov, Physics of the Solid State, vol. 4, Tomsk, 1947.