QUANTITATIVE ASSESSMENT OF ELASTIC PRESLIDING DISPLACEMENT UNDER DIRECTIONAL SURFACE ROUGHNESS

V. I. Maksak

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

The relative displacement of contacting bodies corresponding to an incomplete friction force is called presliding displacement [1].

In [2] it was noted that, for identical shearing forces, elastic presliding displacements (which disappear after the removal of the shearing and compressive forces) are greater along the machining traces than across them.

The displacement of contacting surfaces under conditions of presliding displacement occurs as a result of deformation of the asperities of the rough surface. The actual shape of the asperities of most rough surfaces is well described by an ellipsoid. For each of such asperities one can use P. Mindlin’s solutions for the displacement of elastic bodies with an elliptical contact area [3], or A. I. Lur’e’s solutions for the displacement of an elliptical punch [4].

The displacement of ellipsoidal asperities along the principal axes of the elliptical contact zone may be determined from the formula

\[ \Delta = \left\{ \begin{array}{ll} \dfrac{3 f N_i}{4 G a} \left[ 1-\left(1-\dfrac{P}{fN_i}\right)^{2/3} \right]\chi_{(e)}, & a>b, \\[1.2em] \dfrac{3 f N_i}{4 G b} \left[ 1-\left(1-\dfrac{P}{fN_i}\right)^{2/3} \right]\chi_{(e)}, & b>a, \end{array} \right. \tag{1} \]

where \(a\) and \(b\) are the principal semiaxes of the elliptical contact zone,

\[ \chi_{(e)} \]

is a quantity determined from the diagram [3, 4] (Fig. 1),

\(f\) is the coefficient of friction,

\(G\) is the shear modulus,

\(N_i\) is the compressive force on an asperity,

\(P\) is the shearing force.

Determining the displacement of a rough surface while taking into account the deformation of each asperity separately is rather complicated, and to simplify the solution the following is assumed:

a) all asperities are modeled by ellipsoids having equal corresponding dimensions;

b) the law of distribution of asperities by height satisfies the bearing-area curve;

c) the greatest compression is experienced by the highest asperities;

d) the coefficient of friction corresponding to a definite direction is the same over the entire contact.

Because of the nonuniform distribution of microirregularities by height, when the contact is loaded by a normal force they deform in compression differently: the highest protrusions are compressed more strongly, the less high ones less; of course, there are also some that come into contact only at the end of pressing or do not come into contact at all. As a result, all protrusions have different magnitudes of compressive force.

Fig. 1

During preliminary displacement, the less compressed protrusions will begin to enter sliding earlier. At this time the most compressed ones will be displaced elastically. Ever new protrusions will begin to enter sliding, each after its complete elastic preliminary displacement. Thus, sliding of the entire contact will not begin until the most compressed protrusion has entered sliding. Consequently, the magnitude of the total contact displacement must be determined by the magnitude of the total displacement of the most compressed protrusion, which is determined by formula (1) under the condition \(P=fN\), i.e.

\[ \Delta = \left\{ \begin{array}{ll} \dfrac{3fN_i}{4Ga}\,\chi(e), & a>b,\\[6pt] \dfrac{3fN_i}{4Gb}\,\chi(e), & b>a \end{array} \right\} \tag{2} \]

The major semiaxis \(a\) or \(b\) is determined by the Hertz formula [5]

\[ \left. \begin{array}{l} a\;(\text{for } a>b)\\ b\;(\text{for } b>a) \end{array} \right\} = n_a \sqrt[3]{\dfrac{3}{2}\left(\dfrac{1-\mu^2}{E}\right) \dfrac{r_{\mathrm{pr}}r_{\mathrm{pop}}}{r_{\mathrm{pr}}+r_{\mathrm{pop}}}\,N_i}, \]

and formula (2) is rewritten in the form

\[ \Delta = \dfrac{3fN_i^{2/3}\chi(e)} {4Gn_a \sqrt[3]{\dfrac{3}{2}\left(\dfrac{1-\mu^2}{E}\right) \dfrac{r_{\mathrm{pr}}r_{\mathrm{pop}}}{r_{\mathrm{pr}}+r_{\mathrm{pop}}}}} \tag{3} \]

The compressive force \(N_i\), according to the Hertz formula, is equal to

\[ N_i= \left( \dfrac{r_{\mathrm{pr}}r_{\mathrm{pop}}}{r_{\mathrm{pr}}+r_{\mathrm{pop}}} \right)^{1/2} \dfrac{h_{\max}^{3/2}\varepsilon^{3/2}E} {(1-\mu^2)n_b^{3/2}\cdot 0.653^{2}} \tag{4} \]

Here \(r_{\mathrm{pr}}, r_{\mathrm{pop}}\) are the radii of the asperity in the longitudinal and transverse directions, respectively; \(n_a, n_b\) are coefficients depending on the principal curvatures of the contacting bodies at the point of contact and on the angle between the planes of the principal curvatures [5]; \(\mu\) is Poisson’s ratio; \(E\) is the modulus of elasticity; \(h_{\max}\) is the maximum height of the irregularities; \(\varepsilon\) is the relative approach of the rough surfaces. The latter is determined from the condition

\[ N=\int_{0}^{n_r} N_i\,dn_r, \]

where \(n_r\) is the number of contacting asperities;
\(N\) is the compressive force of the rough surfaces.

Following the solution of N. B. Demkin [6], taking into account the shape of an ellipsoidal asperity, we obtain

\[ \varepsilon = \left[ \frac{\pi r_b^{3/2}(1-\mu)(r_{\mathrm{pr}}+r_{\mathrm{pop}})^{1/2}N} {1.88\,bK_2 h_{\max}^{1/2} E A_c} \right]^{\frac{2}{2\nu+1}}, \tag{5} \]

where \(b, \nu\) are parameters of the bearing-surface curve;
\(A_c\) is the contour area;
\(K_2\) is a coefficient depending on \(\nu\) and has the following values:

\[ \begin{aligned} &\text{for } \nu=1 \quad K_2=1,\\ &\text{for } \nu=2 \quad K_2=0.8,\\ &\text{for } \nu=3 \quad K_2=0.68. \end{aligned} \]

Further, assuming equality of the ratio of the shear and compression forces for the most highly compressed asperity and for the entire contact throughout the whole period of its displacement (such an assumption does not cause a discrepancy between the theoretical and experimental curves of more than 7%), taking into account (1), (3), (4), and (5), we obtain the magnitude of the elastic presliding displacement of a discrete contact along or across the machining marks, with allowance for the corresponding coefficient of friction:

\[ \Delta = \frac{2f h_{\max}\varepsilon\chi(e)} {(1-\psi)n_a n_b} \left[ 1-\left(1-\frac{P}{fN}\right)^{2/3} \right]. \tag{6} \]

For the contact of a rough surface with an absolutely rigid smooth surface, the magnitude of the displacement is reduced by a factor of two.

The formula obtained agrees satisfactorily with experimental data, at least in order of magnitude. Thus, for contacting hardened, ground specimens of steel 45 with surface finishes \(\Delta 5\) and \(\Delta 12\), with coefficients of friction along the machining marks of 0.23 and across them of 0.25, compressed by a force of 30 kgf and displaced by a force of 4 kgf, the magnitudes of the elastic presliding displacement along and across the machining marks, obtained experimentally, are respectively 0.22 and 0.18 μm, while those calculated by formula (6) are 0.19 and 0.14 μm.

REFERENCES

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

2. B. P. Mitrofanov, V. I. Maksak. Anisotropy of elastic presliding displacement. Vestnik of the Academy of Sciences of the BSSR. Series of Physical and Technical Sciences, Minsk, No. 1, 1968.

3. R. D. Mindlin. Compliance of Elastic Bodies in Contact. J. Appl. Mech., v. 16, 1949.

4. A. I. Lur’e. Spatial Problems of the Theory of Elasticity. Gostekhizdat, Moscow, 1955.

5. S. D. Ponomarev, V. L. Biderman, K. K. Likharev, V. M. Makushin, N. N. Malinin, V. I. Feodos’ev. Strength Calculations in Machine Building. Vol. II, Mashgiz, Moscow, 1958.

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