Difference between revisions of "Hertzian contact"
(→Normal contact) |
(→Contact of two cylinders with axes parallel) |
||
| Line 1: | Line 1: | ||
Click on [[Solid Mechanics]] to go back. | Click on [[Solid Mechanics]] to go back. | ||
| − | = Contact of | + | = Contact of cylinders - the Hertz problem = |
| − | |||
| − | |||
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 | ||
| Line 40: | Line 38: | ||
[[File:Screenshot_2016-11-16_16-13-26.png|300px]] [[File:Screenshot_2016-11-16_16-12-32.png|300px]] | [[File:Screenshot_2016-11-16_16-13-26.png|300px]] [[File:Screenshot_2016-11-16_16-12-32.png|300px]] | ||
| + | = Contact of cylinders under partial slip = | ||
| + | |||
| + | == The effect of bulk stress == | ||
= References = | = References = | ||
= FreeFem++ numerical solution = | = FreeFem++ numerical solution = | ||
Revision as of 16:24, 16 November 2016
Click on Solid Mechanics to go back.
Contents
Contact of cylinders - the Hertz problem
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 $b$.
