Hertz equations for elliptical, spherical and cylindrical contacts

Hertz Contact Stresses
Tribology Wikipedia > Hertz equations for elliptical, spherical and cylindrical contacts

A theoretical background to the Hertz contact theory can be found here.

Line Contact (Cylindrical contact)

Cylinders in contact
Fig. 1. Contact of two cylinders

In case of two cylinders in contact (with radii  R_1, R_2 ), as shown in Fig. 1, the Hertzian radius of contact  b under applied normal load  F is given by the following equation:

(1)    \begin{eqnarray*} b = 2 \sqrt{\frac{2FR}{\pi BE'} }\\ \end{eqnarray*}

where  B is the length of the cylinders and  2/{E'}= {1 - {\nu_1}^2}/{E_1} + {{\nu_2}^2}/{E_2} is the reduced elastic modulus. Equivalent radius  R is given by the following relation:

(2)     \begin{eqnarray*} \frac{1}{R}=  \frac{1}{R_1} + \frac{1}{R_2} \\ \end{eqnarray*}

The mean and maximum pressures are given by:

(3)     \begin{eqnarray*} P_{mean}=  \frac{F}{2Bb}, P_{max}=\frac{4}{\pi P_{mean}}\\ \end{eqnarray*}

Corresponding Matlab code for Hertz solution can be found here. The online Hertz contact calculator can be found here.

Hertz contact radius
Fig. 2. Hertz radius of contact

Point Contact (Spherical contact)

Hertz contact of two spheres
Fig. 3. Contact of two spheres

For the case of two spheres in contact as shown in Fig.3, the Hertzian contact radius  a is given by the following equation:

(4)     \begin{eqnarray*} a=  \sqrt[\frac{1}{3}]{\frac{3FR}{E'}}\\ \end{eqnarray*}

, with  R is given by the following relation:

(5)     \begin{eqnarray*} \frac{1}{R}=  \frac{2}{R_1} + \frac{2}{R_2} \\ \end{eqnarray*}

The elastic approach  \delta (also know as rigid body approach) is given by the following expression:

(6)     \begin{eqnarray*} \delta= \sqrt[\frac{1}{3}]{\frac{9F^2}{8R{E'}^2}}\\ \end{eqnarray*}

The mean and maximum pressures are given by:

(7)     \begin{eqnarray*} P_{mean}=  \frac{F}{\pi a^2}, P_{max}=\frac{4}{\pi P_{mean}}\\ \end{eqnarray*}

Corresponding Matlab code for Hertz solution can be found here. The online Hertz contact calculator can be found here.

Elliptical Point Contact

Elliptical Hertzian contact
Fig. 4 Elliptical point contact

For the case of two spheres in contact as shown in Fig.4. In this case the Hertzian contact is an ellipse and is described by major ( a ) and minor  b axes of the contact ellipse:

(8)     \begin{eqnarray*} a= \alpha \sqrt[\frac{1}{3}]{\frac{3FR}{E'}} \\ b =\beta \sqrt[\frac{1}{3}]{\frac{3FR}{E'}}\\ \delta= \gamma \sqrt[\frac{1}{3}]{\frac{9F^2}{8R{E'}^2}}\\ \end{eqnarray*}

where  \frac{1}{R}=  \frac{1}{R_{1x}} + \frac{1}{R_{1y}} + \frac{1}{R_{2x}} + \frac{1}{R_{2y}} .

(9)     \begin{eqnarray*} \alpha \approx {\kappa}^{\frac{1}{3}} \sqrt[\frac{1}{3}]{\frac{2E(m)}{\pi}}\\ \beta \approx {\kappa}^{-\frac{2}{3}} \sqrt[\frac{1}{3}]{\frac{2E(m)}{\pi}}\\ \gamma \approx {\kappa}^{\frac{2}{3}} \sqrt[-\frac{1}{3}]{\frac{2E(m)}{\pi}}\frac{2}{\pi}K(m)\\ \end{eqnarray*}

The functions in the previous equations are approximated as follows:

(10)     \begin{eqnarray*} E(m) \approx \frac{\pi}{2}(1-m)[1+\frac{2m}{\pi (1-m)} -\frac{1}{8}ln(1-m)]\\ K(m) \approx \frac{\pi}{2}(1-m)[1+\frac{2m}{\pi (1-m)}ln(\frac{4}{\sqrt{1-m}}) -\frac{3}{8}ln(1-m)]\\ \kappa \approx {1+\sqrt{\frac{ln(16/\lambda)}{2\lambda}} -\sqrt{ln(4)} +0.16ln(\lambda) }^{-1} \\ m = 1 - {\kappa}^2 \\ \lambda = \frac{R_x}{R_y}, 0<\lambda<1 \\ \end{eqnarray*}

The mean and maximum pressures are given by:

(11)     \begin{eqnarray*} P_{mean}=  \frac{F}{\pi ab}, P_{max}=\frac{3}{2 P_{mean}}\\ \end{eqnarray*}

Corresponding Matlab code for Hertz solution can be found here. The online Hertz contact calculator can be found here.

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