ELASTIC PRE-DISPLACEMENT AND STIFFNESS OF MECHANICAL CONTACT

V. I. Maksak

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

This work considers the displacement of contacting surfaces caused by a tangential force smaller than the friction force, and referred to as pre-displacement [1]. It is known [2] that under the first loading of a contact the displacement is elastoplastic and, in magnitude, greater than the displacements caused by subsequent loading in the same direction by forces not exceeding the first. Usually, in steady-state processes the second case occurs. Such displacements are called elastic [3], since after removal of the tangential and compressive forces there are no residual displacements, and in general form their dependence on the tangential force is shown by one of the curves in Fig. 1, a. It is evident from the figure that a larger coefficient of friction corresponds, on the one hand, to greater stiffness of the joint and, on the other hand, to greater pre-displacement. Accordingly, the elastic [2] and second (BO) returns are larger. For the limiting elastic pre-displacement, experiment shows a linear relationship with the coefficient of friction; however, at small values of the latter the displacement is somewhat lower, which is possibly explained by the increased influence of the adhesive component at small tangential forces.

The relationship between elastic pre-displacement and the coefficient of static friction is shown in Fig. 1, b. Line 1 corresponds to displacement at maximum tangential forces \(P=fN\). At a constant compressive force \(N\) and a definite tangential force \(P\), a reduction in the coefficient of friction corresponds to an increase in displacement (lines 2—13). However, for a reduction in the coefficient of friction at a constant tangential force there exists a limit determined by the equality \(P=fN\). Further reduction of the coefficient of friction requires a reduction of the tangential force. In this case, the magnitude of the elastic pre-displacement decreases along line 1 (Fig. 1, b).

The influence of the compressive force on pre-displacement is shown in Figs. 2, a and 2, b. An increase in the compressive force, like an increase in the coefficient of friction, on the one hand increases the stiffness of the joint and, on the other hand increases the value of the limiting displacement.

Since pre-displacement is determined by deformation of the rough surface, it is natural to assume an influence of the dimensions of the asperities on the magnitude of the displacement. Carrying out such an experiment is very difficult, since it requires eliminating various ...

Fig. 1

a) Plots of total elastic pre-displacements for coefficients of friction, respectively: 1) 0.22; 2) 0.29; 3) 0.4; 4) 0.51; 5) 0.56. Steel 3. Surface finish $\triangle 8$, $N = 45$ kg.

b) Plots of the relationship between elastic pre-displacement and the coefficient of static friction for shear forces having, respectively, the following values in kg:
1) $fN$; 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. $N = 45$, $\triangledown 8$

of the influence of other parameters on displacement. It does not appear possible to obtain contacts with identical coefficients of friction. Nevertheless, despite these difficulties, an attempt was made to conduct such an experiment. From Fig. 2,b it is evident that for rougher surfaces the magnitude of the displacement is somewhat greater. Thus, the displacement of a specimen with surface finish $\triangledown 4$ is greater than with finish $\triangledown 5$ (the coefficients of friction are approximately the same). The displacement for the specimen with finish $\triangledown 9$ proved to be somewhat greater than with finish $\triangledown 5$; however, this can be explained by the fact that this pair had a significantly lower coefficient of friction (the displacements are shown for 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 here the coefficient of friction is even 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. 2,c it is evident that the displacement at the same shear force is less before the limiting value for the material with the smaller shear modulus. Although the contact of copper specimens had a coefficient of friction lower than that of the contact of steel specimens, the difference in displacements demonstrates the evident influence of the shear modulus.

For selecting a contact of optimal stiffness, it is often of interest to know its tangential compliance as a function of the parameters investigated above. By compliance is understood here the magnitude of displacement per unit shearing force. The change in contact compliance as a function of shearing force for different coefficients of static friction is shown in Fig. 3,a. When the shearing force reaches its limiting value, the compliance of the contact increases without bound. It is characteristic that at small shear forces the compliance is constant and does not depend on the coefficient of friction [4]. The latter is also evident from Fig. 1,a, where the displacements coincide at the beginning of shear. In contrast to the coefficient of friction, the compressive force has a noticeable effect on the initial compliance, when the shearing force is equal

Fig. 2

a) 1, 2, 3, 4, 5 — graphs of preliminary displacement due to shear force, for compressive forces equal respectively to 5, 15, 25, 35, and 45 kg. 6 — graph of preliminary displacement due to shear force equal to \(fN\) (according to data of I. R. Konyakhin).

b) Dependence of the magnitude of displacement on the normal load for the case \(P=fN\) (according to data of I. R. Konyakhin).

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

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

zero (Fig. 2, a and line 1 in Fig. 3, b). For different shearing forces the compliance curves are obtained by equidistant displacement of them along the \(N\) axis. With an increase in the compressive force, the compliance of the contact decreases, which is in good agreement with I. V. Kragel’skii’s indication [1] of the increasing stiffness of the bond. The influence of the coefficient of friction is shown in Fig. 3, c.

The theoretical description of the process may be based on representations of the displacement of a unit asperity in accordance with P. Mindlin’s solution for the displacement of elastic spheres and on the adoption of a continuous law of distribution of asperities by height according to the bearing-surface curve [5, 6]. According to these representations, the magnitude of the shearing force in the contact of rough surfaces is written in the form

Fig. 3

Graphs of the tangential compliance of the contact
a) along the path of its loading by a shearing force for coefficients of static friction:
1) 0.68; 2) 0.55; 3) 0.46; 4) 0.26; 5) 0.22; 6) 0.16; steel 3, \(N=45\) kg.
b) as a function of the compressive force at the following shearing forces:
1) \(P=0\); 2) \(P=1\) kg; 3) \(P=2\) kg, copper.
c) as a function of the coefficient of static friction for shearing forces in kg: 1) 24; 2) 20;
3) 16; 4) 12; 5) 10; 6) 8; 7) 6; 8) 4; 9) 2; steel 3, \(N=45\) kg.

\[ P=\int_{0}^{n_{rm}} P_i\,dn_r+\int_{n_{rm}}^{n_r} P_i\,dn_r , \tag{1} \]

where \(P_i\) is the shearing force on a unit asperity; \(n_{rm}\) is the number of displaced asperities; \(n_r\) is the number of contacting asperities;

\[ dn_r=\frac{A_c b\nu(\nu-1)x^{\nu-2}\,dx}{2\pi h_{\max}^{\nu}\sqrt{r_{\mathrm{pr}}r_{\mathrm{pop}}}} . \tag{2} \]

Taking into account Mindlin’s solution [7] and the Hertz problem, the magnitude of the shearing force of each displaced asperity may be represented in the form

\[ P_i=fN_i\left\{1-\left[1-\frac{\Delta(1-\mu)n_a n_b}{2f h_{\max}(\varepsilon-x)}\right]^{3/2}\right\}, \tag{3} \]

and for a slipping asperity

\[ P_i=fN_i, \tag{4} \]

where \(N_i\) is the compressive force on a unit asperity.

\[ N_i=\frac{1.88}{n_a^{3/2}} \left(\frac{r_{\mathrm{pr}}r_{\mathrm{pop}}}{r_{\mathrm{pr}}+r_{\mathrm{pop}}}\right)^{1/2} \frac{E h_{\max}^{3/2}(\varepsilon-x)^{3/2}}{1-\mu^2}; \tag{5} \]

\(\varepsilon\) is the relative approach of the surfaces,

\[ \varepsilon= \left[ \frac{\pi n_a^{3/2}(1-\mu^2)(r_{\mathrm{pr}}+r_{\mathrm{pop}})^{1/2}N} {0.94\,bK_{\nu}h_{\max}^{1/2}EA_c} \right]^{\frac{2}{2\nu+1}} . \tag{6} \]

Here and above \(A_c\) is the contour area; \(b,\nu\) are parameters of the bearing-surface curve, and \(K_2\) is the integration constant according to N. B. Demkin [6]; \(h_{\max}\) is the maximum height of the irregularities; \(r_{\mathrm{pr}}, r_{\mathrm{pop}}\) are the radii of the asperities in the longitudinal and transverse directions of machining; \(N\) is the contact compressive force; \(f\) is the coefficient of sliding friction; \(\Delta\) is the displacement magnitude; \(n_a, n_{\mathrm{v}}, n_i\) are the coefficients of the principal curvatures of the asperities; \(E\) is the modulus of elasticity; \(\mu\) is Poisson’s ratio; \(\chi\) is a coefficient accounting for the ellipticity of the unit contact spot [7].

The level \(m\) of the asperity peaks, separating the displaced and the slipping asperities, is determined from the condition of limiting displacement at \(P_i=fN_i\)

\[ m=\varepsilon-\frac{\Delta(1-\mu)n_a n_{\mathrm{v}}}{2\chi f h_{\max}}. \tag{7} \]

From (1), taking into account (2)–(7), after integration and some transformation we obtain

\[ \tau=fq\left\{1-\left[1-\frac{\Delta(1-\mu)n_a n_{\mathrm{v}}}{2\chi f \varepsilon h_{\max}}\right]^{\frac{2\nu+1}{2}}\right\}. \tag{8} \]

Whence the displacement

\[ \Delta=\frac{2\chi f \varepsilon h_{\max}}{(1-\mu)n_a n_{\mathrm{v}}} \left[1-\left(1-\frac{\tau}{fq}\right)^{\frac{2}{2\nu+1}}\right] \tag{9} \]

and the compliance

\[ S=\frac{d\Delta}{d\tau}= \frac{4\chi\varepsilon h_{\max}}{(2\nu+1)(1-\mu)n_a n_{\mathrm{v}}q} \left(1-\frac{\tau}{fq}\right)^{\frac{1-2\nu}{1+2\nu}}. \tag{10} \]

Here \(\tau\) and \(q\) are the specific shear and compressive forces.

Analysis of formulas (9) and (10) confirms their good agreement with experimental data.

The results of the experimental and theoretical investigation make it possible to say that the greatest tangential contact stiffness corresponds to: 1) maximum pressing of the surfaces; 2) the greatest modulus of elasticity; 3) a larger value of the coefficient of friction; 4) better cleanliness of machining of the contacting surface.

REFERENCES

1. I. V. Kragel’skii. Friction and Wear. Mashinostroenie Publishing House, Moscow, 1968.

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

3. B. P. Mitrofanov. The Nature of Elastic Preliminary Displacement. Theory of Friction and Wear. Nauka Publishing House, Moscow, 1965.

4. B. P. Mitrofanov, V. I. Maksak. Elastic preliminary displacement in the presence of an intermediate layer. Izv. TPI, vol. 183, 1968.

5. V. I. Maksak. Quantitative assessment of elastic preliminary displacement. Izv. Vuzov, “Mashinostroenie,” Moscow, No. 8, 1969.

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

7. R. D. Mindlin. Compliance of Elastic Bodies in Contact. J. of Applied Mechanics, 16, No. 3, 1949.