Investigation of Deformation of the Surface Layer under Contact Loading

I. R. Konyakhin

Static Friction

According to the experiments of Coulomb—Amontons, the friction force is proportional to the normal pressure:

\[ P=\mu N, \]

where \(\mu\) is the coefficient of static friction;

hence

\[ \mu=\frac{P}{N}. \]

For a given pair of surfaces, the coefficient of friction is a constant quantity.

There is an opinion that the coefficient of friction depends directly on the magnitude of the protrusions of the asperities. We have established that the magnitude of these protrusions, or irregularities, plays a significant role in static friction, but it cannot be said that the irregularities have a direct influence on the magnitude of the friction force: the greater the roughness, the greater the friction force. True, sometimes this is indeed the case, but there are also cases when the matter stands otherwise.

We carried out the following experiment. First, the coefficient of friction of ground surfaces was determined. After this, the surfaces were treated with coarse emery paper. The coefficient of static friction did not change in this case. From this it may be concluded that, of course, asperities are of importance, but not directly; rather, their influence is indirect.

Influence of Specific Pressure on the Coefficient of Static Friction

According to the Coulomb—Amontons law, the friction force does not depend on the specific pressure. This proposition will be correct only under certain conditions, namely, if the specific pressure is small.

In calculating the specific pressure one may take the nominal area; then

\[ P=\frac{N}{F_{\text{nom}}}. \]

By this formula a greatly underestimated specific pressure is obtained. The actual contact area is considerably smaller. The contact area is formed as a result of deformation of the irregularities. Deforming plastically, they increase the contact area until an equilibrium state is reached. When their plastic deformation ceases under the given normal pressure, then the greatest contact area will be obtained, which is evidently limited by the yield point

\[ F_k=\frac{N}{\sigma_s}. \]

In our experiments \(F_k = 1.4\ \text{cm}^2\). If the yield strength for steel is taken as \(\sigma_s = 2000\ \text{kg}/\text{cm}^2\), then at \(N = 24.4\ \text{kg}\)

\[ F_k=\frac{24.4}{2000}=0.012\ \text{cm}^2=1.2\ \text{mm}^2. \]

After an experiment with friction, under a microscope one can see smoothed areas, that is, those points at which actual contact occurred. The area of these smoothed spots is of the same order of magnitude as that calculated by the formula, but only in the case when the surfaces have been run in under load; without running-in, the contact areas obtained are smaller.

The contact irregularities are deformed in the normal direction in different ways: some within the elastic range, others beyond it, so that the true contact area consists, as it were, of two areas: the area of elastically deformed irregularities and the area of plastically deformed irregularities

\[ F_k = F_y + F_p. \]

If the nominal specific pressure is small, the elastic contact area prevails, since only an insignificant number of irregularities have so far been plastically deformed.

At low specific pressures, indeed, as experiments show, the coefficient of static friction retains its value and friction follows Coulomb’s law.

It may thus be concluded that, for surfaces within the elastic range, the friction force is proportional to the normal pressure. With an increase in the plastic contact area, the coefficient of friction increases. In particular, this explains the increase in the coefficient of static friction as a function of the duration of preliminary contact; plastic deformation of irregularities in the normal direction occurs over time; consequently, the contact area likewise increases, which entails an increase in the friction force.

Dependence of the coefficient of static friction on the rate of increase of the shearing force

During our experiments with static friction it was found that the coefficient of friction depends on the shearing force itself, namely, on its increase over time.

If the force \(P\) is applied to the body being displaced not all at once, but by gradually increasing it from zero to a maximum, and very slowly at that, then the body will move when the force \(P\) reaches its greatest value \(P_{\max}\). In this case

\[ \mu=\frac{P_{\max}}{N} \]

Fig. 1

If, however, the force \(P\) is increased more rapidly, the body will move from its place earlier than \(P_{\max}\) is reached, that is, at a smaller coefficient of friction.

Shown below is the dependence of the coefficient of static friction on the rate of increase of the shearing force (Fig. 1).

The decrease in the coefficient of friction follows a curve resembling a hyperbola.

The Phenomenon of Temporary Rest

For displacement from rest, as we have seen, it is not necessary that the displacing force have the value \(P_{max}\). Displacement from rest can be produced by a force \(P<P_{max}\). In this case, if the force \(P\) differs little from \(P_{max}\), the displacement will occur, but not immediately; rather, after the lapse of some time. Consequently, the body may remain at rest temporarily. This will constitute temporary rest. The greater the difference \(P_{max}-P\), the greater the duration of temporary rest.

Fig. 2

With a force \(P=P_{max}\), the duration of temporary rest is zero. This phenomenon is manifested more sharply at high specific pressures.

Fig. 2 shows the dependence of the duration of temporary rest on the difference of forces \(P_{max}-P\).

At a certain value \(\Delta P=\Delta P_0\), displacement does not occur. \(\Delta P=P_{max}-P\) can be chosen close to the greatest force of permanent rest. In this case the surfaces still do not slide. If, under such conditions, the specimens are held for a long interval of time, the coefficient of friction increases.

In our experiments the rubbing surface was a flat ring with outside diameter \(D=22\) mm and inside diameter \(d=18\) mm.

Increase in the Coefficient of Static Friction by Means of Repeated Breakaways

The maximum couple of forces produces a breakaway of the contact adhesion, that is, a rapid transition from rest to motion. After the breakaway, for example clockwise, and the subsequent stop, the surface may be displaced in the same direction with greater difficulty, and in the opposite direction more easily. In the latter case, the breakaway force is as great as before, that is, as in a clockwise breakaway. Each change in the direction of breakaway increases the coefficient of friction. The greater the number of such breakaways, the greater the coefficient of friction. But the coefficient of friction cannot be increased in this way to an arbitrarily large value; a saturation limit is reached.

We carried out an experiment with cast iron in contact with cast iron. Fresh surfaces, not yet run in, had a coefficient of friction \(\mu=0.17\). After 100 rapid breakaways the coefficient of friction increased to \(\mu=0.36\), and after an additional 30 breakaways—to \(\mu=0.42\). After this, additional breakaways no longer produced an increase in the coefficient of friction—saturation had occurred.

It should be noted that the coefficient of friction increased in this way is not stable.

In the experiment just described, the coefficient of friction was brought to the value \(\mu=0.42\), but after 20 hours it had already become \(\mu=0.28\),

Fig. 3

i.e., it decreased, but now it was possible to increase it still more: after 20 breakaways it rose to \(\mu = 0.57\). The results of this experiment are presented in the graph (Fig. 4).

It is evident from the graph that a fresh specimen increases the coefficient of friction less intensively than a specimen that has already been in service and has long been under the action of normal pressure.

In the second experiment, after 20 hours under load \(N\), the coefficient of friction decreased to \(\mu = 0.49\). The decrease of the coefficient of friction at \(N = 0\) proceeds considerably faster. In the last experiment, after removal of the normal pressure, the coefficient of friction decreased in 5 minutes from \(\mu = 0.49\) to \(\mu = 0.31\).

Fig. 4

The coefficient of friction achieved in the preceding experiment is not at all the greatest possible. Contact of cast iron on cast iron can give a considerably higher coefficient of friction. We carried out experiments on increasing the coefficient of friction of certain materials, and the saturation of \(\mu\) was different.

For example, contact of steel on glass gives no increase: \(\mu = 0.08\), \(\mu_n = 0.08\), that is, there is no increment of \(\mu\).

Steel—cast iron: \(\mu = 0.12\), \(\mu_n = 0.74\), an increase by a factor of 6.7. Steel on steel: \(\mu = 0.12\), \(\mu_n = 0.96\), that is, an increase by a factor of 8. Copper on copper: \(\mu = 0.12\), \(\mu_n = 1\), an increase by a factor of 9.

After such an experiment, a seizure sometimes forms on the contact surface. If the oscillations during breakaways are made of very small amplitude, then no seizure is detected. Evidently, the speed of motion during breakaway has an influence here. The surface layer is destroyed owing to fatigue, producing small particles in the form of scales, which are detected with the aid of a microscope. The sizes of the scales reach up to \(0.1\) mm in width and length. It has not yet been possible to determine their thickness, since a Brinell microscope with a magnification of 20 times was used.

Elastic properties of the contact

The elastic properties of the contact were investigated on specimens with an annular contact

Fig. 5

surface, while the circular displacement was caused by a pair of forces applied to the upper specimen. The experiments were carried out on a specially designed apparatus giving a visible magnification of the displacement by 50,000 and 250,000 times.

The magnitude of the displacement is of the order of fractions of a micron. The unit of force was taken to be 1 kg, and the unit of displacement \(10^{-6}\) cm. With such small displacements, obviously, there is no difference whether this is an annular surface or a straight strip; therefore, for simplicity of exposition, we shall consider the translational displacement of a body.

Elastic displacement

Under the action of a force \(P\), the specimen is displaced in the direction of the action of the force \(P\) by a certain microscopically small amount, and the greater the force, the greater the displacement. The dependence of the displacement on the shearing force was investigated at contacts of various materials. It was found that in the tangential direction the contact possesses elastic properties; if the specimen is displaced by a small amount measured in fractions of a micron, then after removal of the displacing force \(P\) it returns back. In this case, at the beginning of the motion the displacement occurs according to a linear law: \(P = KS\), where \(S\) is the magnitude of the displacement, \(K\) is the coefficient of proportionality, which henceforth we shall call the coefficient of elasticity.

Displacement by the force \(P\), generally speaking, occurs in two stages. Under the action of this force the specimen at once shifts forward by a certain amount, after which it continues a slow motion and stops.

Removal of the force \(P\) causes a reverse motion, which also occurs in two stages: first a rapid backward displacement and then a slow motion until stopping. In other words, the contact possesses the property of elastic aftereffect, which can be observed in mechanical testing of metals.

The magnitude of the displacement also depends on whether the specimen is displaced from a state of rest or from a state of unfinished motion during elastic aftereffect; in the second case the magnitude of the displacement will be greater.

Displacement by means of a slowly increasing moving force

A slowly increasing force \(P\) gives a displacement occurring along the curve \(OAB\). The shearing force increases not in proportion to the displacement. When the force \(P\) reaches the value \(P_{\max}\), its increase is stopped, and under this force, which maintains a constant value, the displacement proceeds uniformly with constant velocity. A further increase of the force causes accelerated motion and then breakaway, that is, the transition of micromotion into ordinary motion.

Fig. 6

A small decrease of this force, however, leads to retarded motion and stopping. Generally speaking, uniform motion cannot be achieved. By uniform motion here one should understand motion with very small deceleration. After stopping, the moving force may be removed; its removal is accompanied by reverse displacement, but now according to a linear law; in the graph (Fig. 6) this is shown by the straight line \(BB_1\), or, if the force \(P_{\max}\) has not been reached, by the straight line \(AA_1\).

In this case a purely elastic displacement takes place. If the force \(P\) is now increased again, the displacement proceeds along the same straight line \(AA_1\) or \(BB_1\) along which unloading occurred.

Coefficient of elasticity

The elastic properties of the contact can be characterized by the ratio

\[ K=\frac{BB_2}{B_1B_2} \]

(Fig. 6), but \(BB_2\) is the elastic force \(P_y\), while the quantity \(B_1B_2\) is the reverse elastic displacement.

In view of the linearity of the elastic displacement,

\[ K=\frac{P_y}{S_y}; \]

here \(S_y\) is the magnitude of the elastic displacement.

Such a displacement, as shown in the graph of Fig. 6, can be carried out in portions; that is, first one can produce a displacement along the curve \(OA_1\). Removal of the load \(P\) will cause a reverse motion along \(A_1A\), and \(S_y=A_1A_2\) will be obtained. After this, it is again possible to produce a forward displacement, which will now go first along the straight line \(A_1A\), then further along the curve \(AB\). After this we carry out a descent along the straight line \(BB_1\), which gives \(S_{y2}=B_1B_2\).

The coefficient of elasticity \(K\) determines the ratio of the elastically displacing force along the straight line \(A_1A\) to the magnitude of the elastic displacement. Thus we obtain:

\[ K_1=\frac{P_{y1}}{S_{y1}};\quad K_2=\frac{P_{y2}}{S_{y2}}. \]

It is possible in this way to obtain a whole series of values of the coefficient of elasticity for a given contact.

Fig. 7

We performed many experiments to determine the coefficients of elasticity with various contacts at various roughnesses, and in all cases a slight decrease of the coefficient of elasticity is observed as the displacement proceeds forward, that is, \(K_1>K_2>K_3\), etc.

Figure 7 shows a graph of the change in the coefficient of elasticity as a function of the magnitude of displacement.

Elastic zone of displacement

In the appended graph of Fig. 8, the displacement of the specimen along the curve \(OA\) is shown, and then a descent along the straight line \(AA_1\); the specimen returns backward incompletely, since the displacement was not only elastic but also plastic at the same time.

The area of the graph represents the work absorbed by the contact; in this case the area \(OAA_1\) is the work of plastic deformation, and \(A_1AA_2\) is elastic, so that the displacement consists of two zones: the elastic \(A_1A_2\) and the plastic \(OA_1\). As the specimen advances forward, the elastic zone increases to a certain magnitude. Beyond point \(A\) the curve becomes...

almost into a horizontal straight line; therefore the triangle \(A_1 A_2 A\) no longer increases in height, and the same occurs with the base of the triangle \(A_1 A_2\). Figure 9 shows the dependence of the magnitude of the elastic zone \(S\) on the magnitude of the total displacement.

The magnitude of the coefficient of static friction depends entirely on the magnitude of the elastic zone of displacement,

\[ \mu = \frac{P_y}{N}, \]

i.e., \(\mu\) depends on \(P_y\); in turn, the magnitude \(P_y\) is related to \(S_y\) by the coefficient of elasticity.

Fig. 8

Fig. 9

It may be assumed that \(P_y = K S_y\); consequently,

\[ \mu = \frac{K S_y}{N}. \]

We carried out an experiment with an increase in the coefficient of friction. During its increase, the elastic zone also increased invariably. The increase was produced by running-in. The results of this experiment are shown in Fig. 10.

Fig. 10

If the slight change in the coefficient of elasticity is not taken into account, it may be assumed that \(P_y\), and along with it also \(\mu\), are directly proportional to the magnitude of the elastic zone:

\[ \mu = \frac{K S_y}{N}, \]

where \(K S_y = P_y\).

Plastic Properties of Contact

Yielding

It was shown above that the displacement of the contact surfaces is not entirely elastic; together with elastic properties, plastic properties also appear. Yielding, albeit slight, takes place in a fresh contact under a force of any magnitude. The motion of the specimen may be regarded as consisting, as it were, of two motions: elastic and plastic.

At the beginning of the displacement, elastic displacement predominates, while the plastic part of the displacement is small. The farther the advance of the specimen proceeds, the more the plastic part increases. Finally, when the shearing force reaches its maximum value, a smooth, almost uniform motion occurs.

The displacement of the specimen may also be carried out in increments. The applied force \(P\) will cause an elastic displacement forward; then plastic motion will take place with decreasing velocity until complete stoppage. An increase in the shearing force again produces elastic displacement, after which the specimen again moves plastically with decreasing velocity, but over a path already greater than in the preceding motion. And the farther the displacement proceeds, the greater is the plastic travel produced by one and the same increment of force. At the same time the magnitude of the deceleration becomes ever smaller—the velocity approaches a constant value and, when this value is reached, the motion takes place at constant velocity under a constant force.

Relaxation

At a certain value of the driving force, as we have seen, purely plastic motion begins; in this case the elastic zone no longer grows and all the work is expended on plastic resistance. In this case a rapid decrease of the driving force will cause, as we know, elastic motion backward. If, however, the force is decreased according to a definite law, it is possible to achieve cessation of the forward motion, while at the same time there will be no motion backward; that is, the contact surfaces will be at rest relative to one another. But for this the force \(P\) must not remain constant, otherwise the motion will proceed forward; nor must it decrease very rapidly. The force \(P\) must be decreased according to some law \(P=f(t)\). Thus, if one seeks to hold the specimen in place while gradually reducing the force \(P\), one obtains the phenomenon of contact relaxation.

In our experiments the dependence of the decrease of the driving force on time was determined as follows. During the experiment we made sure that there was no displacement, so that the specimen remained in one place. At the same time, of course, the force was reduced and its decrease was observed with a stopwatch. Finally, when the force \(P\) had been reduced to the value \(P_y\), the specimen stopped. The fall of the force during relaxation will be \(P-P_y=q\).

The experiment was carried out with contacts: steel on steel, copper on lead, and copper on lead with a saturated coefficient of friction.

The normal pressure was \(N=24.4\) kg, the magnification 143,000 times. The results are shown in the graph in Fig. 11, where three curves, 1, 2, and 3, are plotted.

1 — steel on steel — fresh contact,
2 — copper on lead — fresh contact, and
3 — copper on lead — contact with a saturated coefficient of friction.

It is clear from the graph that relaxation of the steel-on-steel contact proceeds faster than that of the copper-on-copper contact. Saturation of the contact leads to relaxation proceeding more slowly.

All this difference is possibly explained by the fact that, for the copper–lead contact, the actual contact area, owing to the greater plasticity, proves to be considerably larger than for the steel-on-steel contact.

Fig. 11

For a saturated contact, of course, the contact area is still larger; and since this is so, consequently the number of asperities participating in relaxation is also larger. Here it is clear only that the relaxation time depends on the actual contact area or, in any case, is somehow connected with it.

Driving factor

The displacement of the contacting surfaces is caused by the driving force \(P\). If, by reversing the removal of the force \(P\), an elastic zone is created, then in this zone there is no plastic motion until the force \(P\) exceeds the value of the elastic force \(P_y\). For plastic motion an additional force \(q\) is necessary:

\[ P = P_y + q. \]

If the force \(q = 0\), then there is no motion. Consequently, during motion the work is performed by the force \(P\), but, owing to the presence of \(q\), the force \(q\) as it were permits the force \(P\) to perform work. We shall call the additional permitting force \(q\) the driving factor, for although this force is itself incapable of producing motion, it creates the condition for the possibility of motion by means of the force \(P\).

The driving factor \(q\) affects plastic motion in the following way: the greater \(q\) is, the greater the velocity of plastic motion.

The graph (Fig. 12) shows the dependence of the driving factor on the velocity of motion. The driving factor was determined by relaxation, which we began at different velocities of plastic motion.

The moving force \(\rho\) was reduced until rest occurred at contact under a constant, nondecreasing force, that is, in order to create rest it was no longer necessary to produce relaxation—the relaxation ended.

Fig. 12

Time-dependent elasticity

Figure 13 shows the displacement curve \(OAD\). Suppose that, by a slow increase of the force \(P\), we have produced displacement up to point \(A\). To cause further displacement, an additional force must be applied.

With a slow increase of the factor \(q\), the motion will proceed along the curve farther along the segment \(AD\); but if the factor \(q\) is increased very rapidly, then an elastic displacement will occur along the straight line \(AC\), which is a continuation of the straight line \(A_1A\). Having produced such a displacement, we can immediately remove back the moving factor \(q\), and we shall obtain a reverse elastic motion along \(CA\) to point \(A\).

If, however, the descent along \(CA\) is not carried out, that is, if the moving factor \(q\) is left without removing it, then the motion will proceed along the straight line \(CD\), and \(D\) will end up on the curve at point \(D\).

Fig. 13

After this, the descent can be carried out only along the straight line \(DD_1\); in this case the residual path \(OD_1\) will be obtained. The elastic energy of the moving factor, expressed by the triangle \(ABC\), has a time-dependent character. True, this energy still remains elastic energy (triangle \(GED\)), but this elastic energy will no longer be

represent the energy of the driving factor, for now it is no longer capable of permitting further displacement; its resolving capacity has become equal to zero, and the driving factor \(q = CB\) has turned into the increment of the elastic force \(P = DE\). Before plastic motion along the straight line \(CD\), the factor \(q = CB\) was temporary; consequently, the elasticity expressed by the triangle \(ACB\) was temporary. Thus the temporary elasticity \(ACB\) was transformed into the permanent \(DGE\). A qualitative change in the elastic energy occurred; the elastic factor \(q\) disappeared.

Motion along \(CD\) may be prevented by removing, through relaxation, the driving factor along the straight line \(CC_1\), and arriving at the same curve at point \(C\); then there will be no plastic motion. In this case the elastic force \(P_y\) will receive the increment \(q_4 = CB\). This means that the force \(q_2 = CC_1\) was expended on plastic motion. At the end of the plastic motion we had an increment equal to \(q = FE\); consequently, during the plastic motion a redistribution of the elastic factor \(q\) took place.

At any point of the plastic motion,

\[ q_2 = q_{2y} + q_{2n}. \]

At the beginning of the plastic motion \(q_{2y} = 0,\ q_{2n} = q_{2n\,max}\).

At its end \(q_{2y} = max,\ q_{2n} = 0\),

that is, during this motion there occurred a transition of \(q_{2n}\) into \(q_{2y}\).

If the segment of the curve \(C_1D\) is taken as a straight line, then it may be considered that the ratio

\[ \varepsilon = \frac{q_{2n}}{q_{2y}} \]

changes according to a hyperbolic law; since in this case the segment of the curve \(C_1H\) is taken as a straight line, it is therefore the diagonal of the rectangle \(CDC_1H\).

The ratio

\[ \varepsilon = \frac{q_{2y}}{q_{2y}} \]

can be expressed through the length of the plastic travel

\[ S_n \frac{q_{2y}}{S_n - x} = q_2 \frac{q_{2}}{S_n}, \quad \text{consequently,} \quad \frac{q_{2n}}{x} = \frac{q_{2y}}{S_n - x}; \quad \frac{q_{2y}}{q_{2n}} = \frac{S_n - x}{x} \]

or

\[ \varepsilon = \frac{S_n}{x} - 1, \]

that is, the change in the ratio of the driving factors occurs according to a hyperbolic law. Obviously, the velocity of plastic motion also changes according to this same law, since the velocity graph resembles precisely a hyperbola; therefore it may be concluded that the principal factor in plastic motion is the factor \(q_2\). The velocity of motion is proportional to the active remainder of the driving factor \(q_{2y}\).

After a complete qualitative change of the driving factor into the passive factor \(P\) has occurred, the motion will cease; in order to cause further motion, it is necessary once again to create an active driving factor \(q_{2y}\).

Contact energy during displacement

As is evident from the graph in Fig. 13, at the very beginning of displacement there is no longer any purely elastic or purely plastic motion. Purely elastic displacement is observed only during backward displacement, and also during repeated forward displacement. A newly established contact gives the first displacement in a mixed manner; in this case the elastic part of the displacement is reversible, that is, by whatever amount the specimen is displaced forward, by the same amount it moves back after removal of the displacing force. In a newly established contact there is no such complete reversibility, because immediately, at the beginning

displacement, plasticity comes into play, so that in the subsequent motion the supplied energy is distributed into two unequal parts: elastic and plastic.

As the displacement proceeds, the ratio between the amounts of elastic and plastic energies changes. The share of elastic energy decreases, while the plastic part increases, but their sum is equal to the total absorbed work,

\[ W_{\text{tot}} = W_{\text{el}} + W_{\text{pl}} . \]

An experiment with displacement of the specimens makes it possible to separate one energy from the other. To determine the ratio of these energies, many experiments were carried out. Fig. 14 shows the displacement of steel over celluloid at a normal pressure

\[ N = 17.6\ \text{kg}. \]

At various points of the curve, elastic unloadings were performed. Thus, in the course of the experiment we were able to obtain several triangles of the kind shown in Fig. 13; these are the triangles \(AA_1A_2\) and \(DD_1D_2\).

The total work is represented by the area \(OADD_2\), and the elastic energy by a triangle. Subtracting the area of the triangle from the total area, we obtain the magnitude of the plastic energy.

Fig. 14 shows the curves of the change in energy as a function of the displacement of one contact surface relative to the other.

Fig. 14

Since the elastic force tends toward a limit, the work also tends toward a certain limit. In the graph of Fig. 14, the curve \(W_{\text{el}}\) tends in this case to the value \(W_{\text{el}} = 500 \cdot 10^{-6}\ \text{kg}\cdot\text{cm}\); this is the limiting elastic saturation. The curve of plastic energy passes into an inclined straight line. This means that further motion must proceed under a constant force, and then \(W = PS\), which is expressed by a straight line. For the same reason, the total energy further on in purely plastic motion is represented by a straight line parallel to the graph of \(W_{\text{pl}}\).