DISCRETE CONTACT AND ITS MECHANICAL PROPERTIES*

I. R. Konyakhin

1. Preliminary Displacement

Under the action of shearing forces on bodies in contact, before the onset of their mutual sliding, a certain relative displacement is observed, which has been called preliminary displacement [1]. Experiments have established that the magnitude of the greatest displacement \(S_k\) depends on the coefficient of static friction \(\mu_s\). It is known [2] that if, after shear of the contact (by a force smaller than the friction force), the shearing force is removed, then a reverse displacement \(S_y\)—elastic recovery—is observed. \(S_y\) is not equal to the elastic deformation of the surface, owing to the presence of frictional forces over the contact areas. But after the removal of the compressive force, the indicated resistance disappears and the bodies are additionally displaced (in the direction opposite to that of the shearing force) by a certain amount \(S_t\).

If the specimens are now again loaded with a compressive force and shear is produced, without allowing microslip at its end, then the magnitude of the displacement will be equal to

\[ S_y + S_t . \]

Thus, from an experiment it is always possible to obtain the magnitude of the limiting displacement

\[ S_k = S_y + S_t, \tag{1} \]

the value of which, as experimental investigations show, does not depend on the loading rate. Below, Table 1 gives the results of these experiments.

Table 1

* The present article contains fragments of a monograph being prepared by the author on the results of many years of research into the mechanical properties of contact.
The article was prepared by an editorial commission consisting of: G. A. Doshchinskii—Assoc. Prof., Cand. Tech. Sci.; B. P. Mitrofanov—Cand. Tech. Sci.; V. I. Maksak—postgraduate student; K. N. Tsukublina—assistant.

2. Dependence of the limiting displacement on the coefficient of static friction

After the preliminary displacement, the static friction force becomes known in the experiment, and consequently so does the value of the coefficient of friction

\[ \mu_s=\frac{P_s}{N}. \tag{2} \]

In the investigations carried out, the magnitude of the limiting displacement was measured for various coefficients of static friction \(\mu_s\). The increase in \(\mu_s\) was achieved by running the specimens in against one another under a pressure \(N=45\) kg. It was observed that the greater the value of \(\mu_s\), the greater also the value of the limiting displacement \(S_k\).

In Fig. 1 a graph is given of the dependence of \(S_k\) on \(\mu_s\), plotted from the data of an experiment involving displacement of a hardened-steel specimen on steel 3.

The dependence graph is a straight line. Its slope to the \(\mu_s\) axis is

\[ \frac{S_k}{\mu_s}=S_n, \tag{3} \]

where \(S_n\) is the reduced displacement that would be observed at \(\mu_s=1\).

Fig. 1

The indicated reduced displacement corresponds to the reduced shear force \(P_n\).

\[ P_n=\frac{P_s}{\mu_s}. \tag{4} \]

The dependence between the reduced force \(P_n\) and the reduced displacement \(S_n\) was determined experimentally. For the contact hardened steel on hardened steel, the values of the shear force \(P_s\) and the limiting displacement \(S_k\) were measured at a coefficient of friction \(\mu_s=0.53\). The results of the experiment are shown in the graph of Fig. 2, from which one may write that

\[ S_n=\sqrt{\frac{P_n}{R}}, \tag{5} \]

where \(R\) is the angular coefficient of the straight line of the graph. Since \(P_n=N\), then

\[ S_n=\sqrt{\frac{N}{R}}. \tag{5′} \]

An analogous formula was obtained by I. A. Karpov [3] in his study of the preliminary displacement of a wheel along a rail.

According to I. A. Karpov’s data, the coefficient \(R\) for steel is equal to \(8.3\cdot10^8\) kg/cm\(^2\). In the present investigations it proved to be considerably larger. Using (3) and (5), we obtain

\[ S_k=\mu_s\sqrt{\frac{P_n}{R}}, \]

but since, according to (4),

\[ P_n=\frac{P_s}{\mu_s}. \]

then

\[ S_k=\sqrt{\frac{\mu_s \cdot P_s}{R}}. \tag{6} \]

Fig. 2

Dependence (6) represents the law of limiting preliminary displacement.

3. Mechanical characteristic of the contact

Experiments were carried out with contacts in which the upper specimen was made of hardened steel, and the material of the lower specimens varied. In each of the experiments, several measurements of \(S_k\) and \(P_s\) were made at different pressures. From the obtained values, \(R\) was calculated.

Table 2

It turned out that the dependence between \(R\) and the corresponding shear modulus is linear. In Fig. 3, the ordinate axis shows the quantity \(R\), divided by \(G_1 = 0.9 \cdot 10^6\) kg/cm\(^2\), while the abscissa axis shows the shear modulus of the lower specimen.

Then

\[ \frac{G_2}{\dfrac{R}{G_1}}=\omega,\quad \text{or}\quad \omega R=G_1 \cdot G_2. \tag{7} \]

The quantity \(\omega\) turned out to depend on the boundary layer located between the contacting surfaces. Calculation showed that in the presence of oxide films \(\omega = 350\ \text{kg}/\text{cm}^2\), and in the presence of an adsorbed layer \(\omega = 1000\ \text{kg}/\text{cm}^2\).

4. Law of Preliminary Displacement

Let us consider the determination of the displacement \(S\) caused by a force \(P < P_{\mathrm{n}}\). For this purpose we shall use the results of experiments carried out on specimens: hardened steel on hardened steel. The displacements were produced under normal loads \(N = 43,\ 35,\ 25,\ 15\), and \(5\ \text{kg}\).

The limiting displacements observed in this case will be

\[ S_{\kappa} = \sqrt{\frac{\mu_s \cdot P_s}{R}}, \]

whence the intensity of growth of the tangential force during displacement can be represented in the form

\[ q = \frac{dP_s}{dS_{\kappa}} = \frac{2R \cdot S_{\kappa}}{\mu_s}. \]

Fig. 3

If the displacement curves are analyzed, it turns out that they are also parabolas with the same parameter as that of the parabola of limiting displacement, with the only difference that the sign of this parameter is negative, i.e., instead of the parameter

\[ \frac{2R}{\mu_s} \]

they will have

\[ -\frac{2R}{\mu_s}. \]

Thus,

\[ \frac{dq}{dS} = -\frac{2R}{\mu_s}, \]

or

\[ dq = -\frac{2R}{\mu_s}\,dS, \]

whence

\[ q = -\frac{2R}{\mu_s}S + C, \]

where \(C\) is a constant of integration, which can be determined from the initial conditions.

For \(S = 0\), \(q = q_0\), where \(q_0\) represents the contact stiffness at the beginning of displacement.

Consequently,

\[ C = q_0 \quad \text{and} \quad q = q_0 - \frac{2R}{\mu_s}S. \]

Or, taking into account that \(q = \dfrac{dP}{dS}\), we find

\[ \frac{dP}{dS} = q_0 - \frac{2R}{\mu_s}S. \]

Integrating this expression, we shall have

\[ P=q_0 S-\frac{R S^2}{\mu_s}+D. \]

The constant \(D\), representing the initial shear force, may be taken equal to 0. Thus, finally we obtain

\[ P=q_0 \cdot S-\frac{R S^2}{\mu_s}. \tag{8} \]

Expression (8) is the law of preliminary displacement.

At the point where the displacement curve meets the limiting parabola,

\[ S=S_{\mathrm{k}}, \quad P=P_s . \]

In this case equation (8) takes the form

\[ P_s=q_0 S_{\mathrm{k}}-\frac{R}{\mu_s} S_{\mathrm{k}}^2, \]

but since

\[ P_s=\frac{R}{\mu_s} S_{\mathrm{k}}^2, \]

then

\[ \frac{R S_{\mathrm{k}}^2}{\mu_s}=q_0 S_{\mathrm{k}}-\frac{R}{\mu_s} S_{\mathrm{k}}^2. \]

From this the initial contact stiffness is determined:

\[ q_0=\frac{2R \cdot S_{\mathrm{k}}}{\mu_s}. \tag{9} \]

Solving equations (8) and (9) simultaneously, we obtain

\[ P=\frac{R}{\mu_s}(2S_{\mathrm{k}}-S)\cdot S. \tag{10} \]

Formula (10) makes it possible to compute the shear force from the known displacement. However, for practical purposes the solution of the inverse problem is also needed. For this, we transform expression (10) to the form

\[ S^2-2S_{\mathrm{k}}\cdot S+\frac{\mu P}{R}=0, \]

whence

\[ S=S_{\mathrm{k}}-\sqrt{S_{\mathrm{k}}^2-\frac{\mu_s P}{R}}. \tag{11} \]

Introducing into (11) the substitution in the form

\[ S_{\mathrm{k}}=\sqrt{\frac{\mu_s \cdot P_s}{R}}, \]

we obtain

\[ S=\sqrt{\frac{\mu_s \cdot P_s}{R}} \left(1-\sqrt{1-\frac{P}{P_s}}\right). \tag{12} \]

It was obtained earlier that

\[ \omega R=G_1 G_2 . \]

In addition, we use Amontons’ law

\[ P_s=\mu_s \cdot N, \]

then (12) can be written as:

\[ S = \psi_s \sqrt{\frac{\omega N}{G_1 G_2}}\left(1-\sqrt{1-\frac{P}{\psi_s \cdot N}}\right). \tag{13} \]

Formula (13) makes it possible to calculate the magnitude of displacement under the action of a shear force smaller than, or equal to, the friction force, and for a known value of the normal load.

In what follows, displacements with simultaneous variation of both the shearing force and the normal load are considered.

On the basis of an analysis of models of discrete contact, an approach to the continuous case is possible, which would make it possible to create a certain mathematical apparatus for calculations.

Discrete Contact

Academician E. A. Chudakov [4] advanced the proposition that the relative displacement of asperities of a rough surface occurs non-simultaneously. This is explained by the nonuniformity of the load distribution over the contact area. The author indicates that, for a small value of the shear force, the asperities undergo elastic deformation. With a further increase of the shear force, when a force equal to the force of static friction acts on the microelements, a displacement will occur, accompanied by plastic deformation and destruction of the surface.

The experiments carried out in the present work with various materials made it possible to give a certain characterization of the frictional properties of discrete contact.

During displacement, a microelement passes through three stages: elastic deformation, plastic deformation, and microslip. In the last case, the material of the contact is not strengthened.

Owing to the difference in dimensions, the asperities of the surface have different stiffness both when they are compressed and when they are sheared. Let us represent the magnitude of displacement of an asperity as

\[ c = \frac{P_s}{q}. \tag{14} \]

Here \(C\) is the maximum displacement of the summit of the asperity;
\(P_s\) is the greatest friction force at its summit;
\(q\) is the stiffness.

Because of the difference in displacement of the protrusions, there are always two groups of microelements in the contact: deforming and sliding. During displacement, elements from the deforming group pass into the sliding group and, when all elements become sliding, the shear force \(P\) ceases to increase; microslip begins under a constant shear force, which may continue for a very long time without transition to ordinary sliding. The displacement that occurs for the first time after the formation of contact is accompanied by plastic deformation, whereas subsequent displacement proves to be elastic. However, the presence of microslip on the protrusions leads to a nonlinear dependence between the shear force and the displacement.

Removal of the shearing force after displacement is always accompanied by elastic return. During such return, the elements that have returned to the initial state begin to deform in the reverse direction, since elastic return continues at the expense of the elastic recovery of the remaining elements. Finally, a moment arrives when the resistance of the asperities deformed

in the opposite direction, becomes equal to the force of elements that have not yet recovered. At this time the external shear force is zero. Thus, after displacement and the subsequent elastic return, there is a certain reserve of elastic energy in the contact.

After this, with a gradual decrease now of the normal load, the formerly rubbing elements will begin to slip, while the recovering elements will move the specimen in the direction opposite to that of the preliminary displacement. This displacement is an elastic return of the second kind. At this time the contact is freed from stored energy; the elastic energy, owing to microfriction, is converted into heat.

Elastic-frictional model of discrete contact

In a discrete contact there is a large number of elements, owing to which in an experiment the averaged effect of deformation of asperities is observed. The basic qualities of a microprotrusion are preserved if it is replaced by a rectangular parallelepiped with the deformation properties of a real body.

If the adopted parallelepiped is divided by vertical planes into several parts, then the resulting group of elements will have the same stiffness as the whole element.

Contact between microelements in the contact occurs chaotically, as shown in Fig. 4.

Fig. 4

Fig. 5

In the contact there are different regions: horizontal and inclined ones. A microarea can be divided into a multitude of horizontal ones, and the real contact replaced by an identical one that will have only horizontal areas (Fig. 5).

Fig. 6

Fig. 7

After this, let us represent one of the contacting bodies as smooth and the other as rough (Fig. 6). In addition, let us rearrange the elements, placing them as shown in Fig. 7. Since there are many elements, by shifting them together we obtain along the tops a continuous line, which may be different. Experiments with the approach of bodies under compression of their nor-

showed that the approach law is expressed in most cases by an equation of the form

\[ N = bh^2, \tag{15} \]

where \(N\) is the normal load,
\(h\) is the magnitude of approach,
\(b\) is an experimental constant.

The contact stiffness in compression, determined from (15), is a linear function. Consequently, the peaks of the surface asperities in Fig. 7 are assumed to lie on a straight line. In addition, let us recall that the preliminary displacement is carried out according to a parabolic law.

Consequently, the contacting bodies may be represented in the form of a smooth triangular wedge.

Using this model, one can obtain relations for calculating approach and displacement.

Regularities of Various Types of Displacement

Earlier, the regularities of limiting and elastic-frictional displacements were established experimentally. Let us obtain these regularities on the model.

Elastic-Frictional Displacement

We shall carry out the investigation in reduced quantities, i.e., with \(\mu_s = 1\). If necessary, the actual coefficient of friction can always be taken into account. Thus,

\[ \frac{P}{\mu_s} = P_n,\qquad \frac{S}{\mu_s} = S_n. \]

Using (5′), we find the intensity of the conditional distributed load with respect to displacement

\[ q = \frac{dN}{dS_n} = 2R \cdot S_n. \]

This means that along the contact length, conditionally equal to the reduced displacement, the load is distributed according to a linear law. Then

\[ q = \frac{2N}{S_n}. \tag{16} \]

Fig. 8

Figure 8 shows the model for elastic-frictional displacement. The displacement on the model is represented by the upward motion of the line \(AB\). At a given moment of displacement, the shear force is represented by the area of the figure \(OABK\). The triangle \(OAq\) represents the resistance force of the rubbing elements, and the rectangle \(ABKq\) the elastic force of the deformable asperities. The displacement is equal to the segment \(S\).

The total shear force is

\[ P=\frac{qS}{2}+q_1(S_{\mathrm{p}}-S), \]

or, after simple transformations,

\[ P=q_1\left(S_{\mathrm{p}}-\frac{S}{2}\right). \]

Taking into account that \(q_1=\dfrac{q}{S_{\mathrm{p}}}S\), we obtain

\[ P=\frac{q}{S_{\mathrm{p}}}\left(S_{\mathrm{p}}-\frac{S}{2}\right)S. \]

Since

\[ q=\frac{2N}{S_{\mathrm{p}}}, \]

then

\[ P=R(2S_{\mathrm{p}}-S)S. \tag{17} \]

Formula (17) is identical to (10).

Digital model of contact

Sometimes the construction of the model considered above proves to be complicated. In this case one may apply a digital model, consisting of a table of the change in forces on each element.

The contact is divided into such a number of elements that the stiffness of each of them is equal to unity; then its displacement will be expressed by the same number as the normal load, since the coefficient of friction is taken as \(\mu_s=1\). Thus, the digital model will be represented in the form of a table.

Table 3

This table expresses the state before displacement begins. After a displacement by \(S=1\), the first element exhausts its friction reserve and becomes sliding; the remaining elements will deform.

Table 4

Let \(S=2\); then

Table 5

and so on.

Using the digital model, it is possible to construct the complete displacement cycle, which is shown in Fig. 9. In the graph, the separate parts of the curves are marked with numbers: 1 — elastic frictional displacement considered by us above; 2 — secondary displacement without removal of the normal load; 3 — elastic unloading; 4, 5 — displacement in the opposite direction; 6, 7, 8 — repetition in reverse order of branches 3, 4, 5.

Let us compile a table for branches 1, 2, 3 (Fig. 9). We divide the contact into 10 elements. Then the table of shear forces and displacements will have the following form:

Table 6

Losses of Mechanical Energy in a Discrete Contact

If an alternating tangential force acts on the contacting bodies, then displacement occurs alternately first in one direction, then in the other. Owing to microfriction, mechanical energy is dissipated in this process. This phenomenon can be used to damp vibrations of various devices containing joints produced by the contact of their surfaces.

Let us consider the displacement represented by branch 1 of the graph in Fig. 9. The formula for the displacement under consideration will be

\[ P = R(2S_{\mathrm{p}} - S)S. \tag{18} \]

According to this formula, the branch \(OA\) in Fig. 10 has been constructed. If expression (18) is solved for \(S\), then one obtains

\[ S=S_{\mathrm{n}}\left(1-\sqrt{1-\frac{P}{N}}\right). \tag{19} \]

Fig. 9

Fig. 10

The displacement energy is determined as the area of the graph, and its differential is

\[ d\varepsilon=P\,dS. \]

Substituting here the value of \(P\) from (18), we obtain

\[ d\varepsilon=R(2S_{\mathrm{n}}-S)S\cdot dS. \]

The total energy is represented as

\[ \varepsilon = R \int (2S_{\mathrm{p}} - S) S \cdot dS \]

or

\[ \varepsilon = R \left( S_{\mathrm{p}} S^{2} - \frac{S^{3}}{3} \right) + C. \]

Obviously, for \(S = 0,\ R = 0,\ \varepsilon = 0\), we obtain \(C = 0\). Thus,

\[ \varepsilon = \frac{R}{3} (3S_{\mathrm{p}} - S) S^{2}. \tag{20} \]

Substituting in (20) the quantity \(S\) in terms of the shear force \(P\) according to (18), we obtain

\[ \varepsilon = \frac{R \cdot S_{\mathrm{p}}^{3}}{3} \left( 2 + \sqrt{1 - \frac{P}{N}} \right) \left( 1 - \sqrt{1 - \frac{P}{N}} \right)^{2}. \tag{21} \]

The calculation of the energy of elastic recovery, which in Fig. 10 is represented by the area \(CAB\), presents no difficulty.

REFERENCES

1. A. V. Verkhovskii. The phenomenon of preliminary displacement in the friction of unlubricated surfaces from rest. El. prikl. fiz., vol. 3, No. 3, 1926.

2. I. R. Konyakhin. Theory of preliminary displacements as applied to questions of contact between parts. Publishing House of TSU, 1965.

3. I. V. Kragelskii, I. E. Vinogradova. Coefficient of Friction. Moscow, 1962.

4. E. A. Chudakov. On the question of friction of bodies possessing high elasticity. In: Friction and Wear in Machines. Publishing House of the Academy of Sciences of the USSR, 1939.