Difference between revisions of "Hertzian contact"
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| − | = Contact of | + | = Contact of Cylinders - the Hertz problem = |
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| + | Detailed discussions of this problem can be found in Hills and Nowells (1994) <ref> Hills, D. A. and Nowell, D. (1994). ''Mechanics of Fretting Fatique'', p. 20-25. Springer Science+Business Media, Dordrecht.</ref>, Williams and Dwyer-Joyce (2001). <ref>Williams, John A. and Dwyer-Joyce, Rob S. (2001). ''Contact Between Solid Surfaces'', p. 121 in '''Modern Tribology Handbook: Volume 1, Principles of Tribology''', editor: Bushan, Bharat. CRC Press LLC, Boca Raton.</ref> | ||
If two circular cylinders with radii $R_1$ and $R_2$ are pressed together by a force per unit length of magnitude $P$ with their axes parallel, then the contact patch will be of half-width $b$ such that | If two circular cylinders with radii $R_1$ and $R_2$ are pressed together by a force per unit length of magnitude $P$ with their axes parallel, then the contact patch will be of half-width $b$ such that | ||
Revision as of 16:38, 16 November 2016
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Contents
Contact of Cylinders - the Hertz problem
Detailed discussions of this problem can be found in Hills and Nowells (1994) [1], Williams and Dwyer-Joyce (2001). [2]
If two circular cylinders with radii $R_1$ and $R_2$ are pressed together by a force per unit length of magnitude $P$ with their axes parallel, then the contact patch will be of half-width $b$ such that \begin{equation} b = \sqrt{\frac{2PR}{\pi E^*}} \end{equation} where $R$ and $E^*$ are the reduced radius of contact and the contact modulus defined by \begin{equation} \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}, \end{equation} \begin{equation} \frac{1}{E^*} = \frac{1-{\nu_1}^2}{E_1} + \frac{1-{\nu_2}^2}{E_2}. \end{equation}
The resulting pressure distribution $p(x)$ is semielliptical, i.e., of the form \begin{equation} p(x) = p_0 \sqrt{1-\frac{x^2}{b^2}} \end{equation} where the peak pressure \begin{equation} p_0 = \sqrt{\frac{PE^*}{\pi R}}. \end{equation}
The coordinate $x$ is measured perpendicular to that of the cylinder axes. For the case of nominal contact between cylinders closed form analytical solutions are available.
The surfaces stresses are given by the equations:
The surface stresses and stresses along the line of symmetry are shown in the following two graphs. The $x$ and $z$ coordinates are normalized with the contact width $b$.
Contact of cylinders under partial slip
The effect of bulk stress
FreeFem++ numerical solution
References
- ↑ Hills, D. A. and Nowell, D. (1994). Mechanics of Fretting Fatique, p. 20-25. Springer Science+Business Media, Dordrecht.
- ↑ Williams, John A. and Dwyer-Joyce, Rob S. (2001). Contact Between Solid Surfaces, p. 121 in Modern Tribology Handbook: Volume 1, Principles of Tribology, editor: Bushan, Bharat. CRC Press LLC, Boca Raton.
