Influence of Rough Surface Geometry on Elastic Pre-Displacement and Energy Dissipated in Contact
Abstract Generated abstract
This paper examines how rough surface geometry affects elastic preliminary tangential displacement and energy dissipation in frictional contact under a shearing force below the limiting friction force. Modeling the surface as a set of spherical asperities, the authors apply Mindlin’s solution for elastic spheres and introduce averaged contact quantities to express both displacement and dissipated energy through a common geometric factor. Using Demkin’s relations for the number and real area of contacting asperities, they derive this factor in terms of roughness height, asperity radius, contour contact area, and bearing curve parameters. The analysis indicates that the geometric factor, and therefore the predicted displacement and dissipation, may vary by about a factor of two with roughness and increases for rougher surfaces.
Full Text
Volume 188
1974
INFLUENCE OF THE GEOMETRY OF A ROUGH SURFACE ON THE MAGNITUDES OF ELASTIC PRELIMINARY DISPLACEMENT AND THE ENERGY DISSIPATED IN CONTACT
B. P. Mitrofanov, V. I. Maksak
(Presented by the scientific seminar of the Department of Strength of Materials)
If a shearing force, equal in magnitude to an incomplete friction force, is applied to a body located on a hard surface, then a tangential displacement arises between the contacting bodies. This phenomenon is called preliminary displacement, and it is accompanied by such irreversible processes as energy dissipation.
By modeling a rough surface by a set of spherical asperities, for the preliminary displacement of each of which one may apply R. Mindlin’s solution on the displacement of elastic spheres [1], and using the concept of average quantities, it is possible to estimate the influence of the geometry of the rough surface.
Thus, the displacement for a single spherical asperity is
\[ \Delta=\frac{3(2-\mu)}{8Ga}\, fN\left[1-\left(1-\frac{P}{fN}\right)^{2/3}\right], \tag{1} \]
where
\[ \begin{aligned} \mu &\text{— Poisson’s ratio;}\\ G &\text{— shear modulus;}\\ a &\text{— radius of the contact spot;}\\ f &\text{— coefficient of friction;}\\ N &\text{— compressive force;}\\ P &\text{— shearing force.} \end{aligned} \]
Dividing the compressive force and the actual contact area \(A_r\) by the number of contacting asperities \(n_r\), we obtain average values, respectively, for the compressive force \(N\) and the actual contact area of a single asperity \(A_r\).
Thus,
\[ \overline{N}=\frac{N}{n_r},\qquad \overline{a}=\sqrt{\frac{A_r}{\pi n_r}}, \tag{2} \]
where
\[ \overline{a}\text{— average radius of contact of an asperity.} \]
Substituting the obtained values \(\overline{N}\) and \(\overline{a}\) into formula (1), we have:
\[ \overline{\Delta}=\frac{3(2-\mu)fN}{8G}\,\varphi \left[1-\left(1-\frac{P}{fN}\right)^{2/3}\right], \tag{3} \]
where
\[ \varphi=\left(\frac{\pi}{n_r A_r}\right)^{1/2}. \tag{4} \]
The influence of the contact-surface geometry on the elastic displacement \(\overline{\Delta}\) is incorporated in the term \(\varphi\). Thus, by calculating it, one can compare the average displacements of contacts with different geometries.
In an analogous way, one can determine the influence of the surface geometry on the magnitude of the dissipated energy. Thus, for a single spherical asperity the dissipation is
\[ \omega=\frac{(2-\mu)P^3}{36GafN}, \]
or, taking (2) into account, the average value is
\[ \overline{\omega}=\frac{2-\mu}{36Gf}\cdot\frac{P^3}{N}\cdot\varphi . \tag{5} \]
Here \(\varphi\) has the same value as in function (4). Consequently, the influence of the rough-surface geometry on the displacement and on the magnitude of the dissipated energy can be evaluated by the quantity \(\varphi\), determining \(n_r\) and \(A\) according to N. B. Demkin [2].
\[ n_r=\frac{A_c b\nu \varepsilon^{\nu-1}}{2\pi r h_{\max}}, \]
\[ A_r=A_c b\varepsilon^\nu, \]
\[ \varepsilon=\left\{\frac{3\pi(1-\mu^2)r^{1/2}N}{bK_2h_{\max}^{1/2}EA_c}\right\}^{\frac{2}{2\nu+1}} . \]
Then
\[ \varphi=\frac{4.44}{\nu^{1/2}} \left\{ \left[\frac{K_2}{3\pi(1-\mu^2)}\right]^{\frac{2\nu-1}{2\nu+1}} \left[\frac{h_{\max}^{2\nu}r}{b^2A_c^2}\right]^{\frac{1}{2\nu+1}} \right\}, \tag{6} \]
where
\(h_{\max}\) is the maximum height of the irregularities;
\(r\) is the radius of a spherical asperity;
\(A_c\) is the contour area of contact;
\(b\) and \(\nu\) are parameters of the bearing-surface curve.
The coefficient \(K_2\) has the following values:
\[ \begin{aligned} &\text{for } \nu=1 \qquad K_2=1;\\ &\text{for } \nu=2 \qquad K_2=0.8;\\ &\text{for } \nu=3 \qquad K_2=0.68. \end{aligned} \]
In numerical calculations it is convenient to compute \(\frac{\varphi_1}{\varphi_2}\), where \(\varphi_1\) is the function \(\varphi\) for one surface-finishing grade, and \(\varphi_2\) for another.
According to (6), the value of \(\varphi\) varies as a function of roughness by approximately a factor of 2, increasing for rough surfaces.
References
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R. D. Mindlin. Compliance of elastic bodies in contact. J. Appl. Mech., 16, 1949.
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N. B. Demkin. Actual Area of Contact of Solid Surfaces. Publishing House of the Academy of Sciences of the USSR, Moscow, 1962.