Development of friction force during presliding displacement of an elastic contact

DEVELOPMENT OF THE FRICTION FORCE IN THE PROCESS OF PRELIMINARY DISPLACEMENT OF AN ELASTIC CONTACT

V. I. Maksak

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

The study of the interaction of contacting rough surfaces under mutual tangential displacement is the main subject of numerous works devoted to the problem of friction. The study of preliminary displacement provides a physical basis for the concept of an incomplete friction force and makes it possible to draw certain conclusions about its nature [1].

This paper considers such a preliminary displacement which, after the removal of the shearing and compressive forces, proves to be completely reversible and occurs under repeated loading.

The general character of the relationship between the shear force and preliminary displacement for different compressive forces is shown in Fig. 1. It is clear from the figure that an increase in the compressive force, on the one hand, increases the stiffness of the connection and, on the other hand, increases the value of the limiting displacement. The dashed line determines the values of the limiting displacement for different compressive forces.

Since the character of the development of deformation in the contact determines the limiting shear force, it is of interest to establish the relationship between the displacement and the ratio of this force to the compressive force, which is customarily called the coefficient of static friction. Such a relationship is shown in Fig. 2. Line 1 corresponds to displacement at maximum shear forces \(P = fN\). At a constant compressive force \(N\) and a given incomplete shear force \(P\), a decrease in the coefficient of friction \(f\) corresponds to an increase in displacement (lines 2–13). However, there exists a limit to the decrease in the coefficient of friction at a constant shear force, which is determined by the equality \(P = fN\). Further reduction of the coefficient of friction requires a reduction of the shearing force. In this case the magnitude of the elastic preliminary displacement decreases along line 1 (Fig. 2). At small coefficients of friction this relationship deviates from linearity toward a reduction of displacement, which may be explained by the increased influence of the adhesion component at small shear forces. Thus, a larger coefficient of static friction, as well as a greater compressive force, corresponds, on the one hand, to a greater stiffness of the connection and, on the other hand, to a greater limiting displacement.

In addition, the magnitude of the displacement is also influenced by the geometry of the rough surface and its mechanical properties. Carrying out such an experiment is very difficult, since it requires [[unclear: continuation cut off at bottom of page]]

…to exclude the different influence of other parameters on the displacement. It is not possible to obtain contacts with identical coefficients of friction.

Fig. 1. 1, 2, 3, 4, 5—experimental plots of preliminary displacement versus shear force for compressive forces having, respectively, the following values: 5, 15, 25, 35, 45 kg. 6—plot of limiting preliminary displacements (according to the data of I. R. Konyakhin [2])

Fig. 2. Experimental plots of the relationship between preliminary displacement and the coefficient of static friction for shear forces having, respectively, the following values in kg: 1/\(N\), 2/24, 3/22, 4/20, 5/18, 6/16, 7/14, 8/12, 9/10, 10/8, 11/6, 12/4, 13/2. Steel 3, \(N = 45\) kg, surface finish grade \(\nabla 8\)

However, despite these difficulties, an attempt to carry out such an experiment was nevertheless made. From Fig. 3a it is seen that

Fig. 3. a) Elastic preliminary displacements for specimens with surface-finish grades: 1/\(\nabla 5\), 2/\(\nabla 9\), 3/\(\nabla 4\), 4/\(\nabla 9\), in contact with a specimen having a surface finish of grade 8. The coefficients of friction of the corresponding pairs are equal to: 1/0.63, 2/0.47, 3/0.6, 4/0.2. Steel 3, \(N = 45\) kg.

b) Elastic preliminary displacements: 1—for contact of steel specimens with coefficient of friction 0.58; 2—for contact of copper specimens with coefficient of friction 0.45. \(N = 45\) kg, \(\nabla 8\)

for rougher surfaces the magnitude of displacement is somewhat greater. Thus the displacement of a specimen with surface-finish grade \(\nabla 4\) is greater than with surface-

of grade $\nabla 5$ (the coefficients of friction are approximately the same). The displacement for the specimen with machining finish $\nabla 9$ proved to be somewhat greater than with finish $\nabla 5$; however, this can be explained by the fact that this pair had a significantly lower coefficient of friction (the displacements are shown at identical shear forces). The fact that the influence of the coefficient of friction is large is evident from curve 4, for which the contact is the same as for curve 2. The only difference is that the coefficient of friction here is still lower and is equal to 0.2.

Experiments carried out on materials having different shear moduli showed only their quantitative influence on displacement. From Fig. 3b it is evident that, for one and the same shear force less than the limiting value, the displacement is smaller for the material with the lower shear modulus, and greater for the material with the higher shear modulus. Although the contact of the copper specimens had a lower coefficient of friction than the contact of the steel specimens, the difference in displacements clearly shows the influence of the shear modulus.

For an analytical expression of the relationship between the friction force and the preliminary displacement in the process of shear, one may use the formula obtained in [3] using the solutions of R. Mindlin [4] and N. B. Demkin [5]:

\[ \Delta = \frac{(2-\mu) f h_{\max}}{1-\mu} \left\{ \frac{0.75 + (1-\mu) r^{1/2} N}{b K_2 h_{\max}^{1/2} G A_c} \right\}^{\frac{2}{2\nu+1}} \left[ 1-\left(1-\frac{P}{fN}\right)^{\frac{2}{2\nu-1}} \right], \]

where

$\Delta$ — elastic preliminary displacement;
$\mu$ — Poisson’s ratio;
$G$ — shear modulus;
$h_{\max}$ — maximum height of asperities;
$r$ — radius of protrusions;
$A_c$ — contour area;
$b, \nu$ — parameters of the bearing-surface curve;
$K_2$ — coefficient depending on $\nu$ [5].

The formula is in satisfactory agreement with experiment.

LITERATURE

1. I. V. Kragel’skii, V. S. Shchedrov. Development of the Science of Friction. Publishing House of the Academy of Sciences of the USSR, 1956.

2. I. R. Konyakhin. Theory of Preliminary Displacements as Applied to Questions of the Contacting of Parts. Tomsk University Press, Tomsk, 1965.

3. V. I. Maksak. Quantitative evaluation of elastic preliminary displacement. Izv. vuzov, “Mashinostroenie”, No. 6, 1969.

4. R. D. Mindlin. Compliance of Elastic Bodies in Contact. J. Appl. Mech., No. 16, 1949.

5. N. B. Demkin. Actual Area of Contact of Solid Surfaces. Publishing House of the Academy of Sciences of the USSR, 1962.