Hertzian contact

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Contact of Cylinders - the Hertz problem

Detailed discussions of this problem can be found in Hills and Nowells (1994) as well as Williams and Dwyer-Joyce (2001).

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:

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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 $b$.

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