# Hertzian contact equations for elliptical, spherical and cylindrical contacts

### Revision for “Hertzian contact equations for elliptical, spherical and cylindrical contacts” created on October 19, 2021 @ 04:41:18

Title Hertzian contact equations for elliptical, spherical and cylindrical contacts A theoretical background to the Hertz contact theory can be found here.

Line Contact (Cylindrical contact)

[caption id="attachment_10061" align="aligncenter" width="354"] Fig. 1. Contact of two cylinders[/caption] 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: $\begin{eqnarray} \label{complete_sys1} b = 2 \sqrt{\frac{2FR}{\pi BE'} }\\ \end{eqnarray}$ where $B$ is the length of the cylinders. It is important to note here that the reduced elastic modulus is defined as follows: $2/{E'}= (1 - {\nu_1}^2)/{E_1} + ({1-{\nu_2}^2})/{E_2}$. This definition is historically used in the field of hydrodynamic lubrication and it is different from the typical contact mechanics definition, where the reduced elastic modulus is given as $1/{E'}= ({1 - {\nu_1}^2})/{E_1} + ({1-{\nu_2}^2})/{E_2}$. Therefore, the given equations may slightly differ from the classical Hertzian equations given in contact mechanics textbooks (but they are equivalent). Equivalent radius $R$ is given by the following relation: $\begin{eqnarray} \label{complete_sys1} \frac{1}{R}= \frac{1}{R_1} + \frac{1}{R_2} \\ \end{eqnarray}$ The mean and maximum pressures are given by: $\begin{eqnarray} \label{complete_sys1} P_{mean}= \frac{F}{2Bb}, P_{max}=\frac{4 P_{mean}}{\pi}\\ \end{eqnarray}$ Corresponding Matlab code for Hertz solution can be found here. The online Hertz contact calculator can be found here. Further details of the contact theory can be found in Contact Mechanics by James Barber. [caption id="attachment_12859" align="aligncenter" width="464"] Fig. 2. Hertz radius of contact[/caption]

Point Contact (Spherical contact)

[caption id="attachment_10066" align="aligncenter" width="520"] Fig. 3. Contact of two spheres[/caption] For the case of two spheres in contact as shown in Fig.3, the Hertzian contact radius $a$ is given by the following equation: $\begin{eqnarray} \label{complete_sys1} a= \sqrt{\frac{3FR}{E'}}\\ \end{eqnarray}$, with $R$ is given by the following relation: $\begin{eqnarray} \label{complete_sys1} \frac{1}{R}= \frac{2}{R_1} + \frac{2}{R_2} \\ \end{eqnarray}$ The elastic approach $\delta$ (also know as rigid body approach) is given by the following expression: $\begin{eqnarray} \label{complete_sys1} \delta= \sqrt{\frac{9F^2}{8R{E'}^2}}\\ \end{eqnarray}$ The mean and maximum pressures are given by: $\begin{eqnarray} \label{complete_sys1} P_{mean}= \frac{F}{\pi a^2}, P_{max}=\frac{4}{\pi P_{mean}}\\ \end{eqnarray}$ Corresponding Matlab code for Hertz solution can be found here. The online Hertz contact calculator can be found here. Further details of the contact theory can be found in Contact Mechanics by James Barber.

Elliptical Point Contact

[caption id="attachment_10073" align="aligncenter" width="520"] Fig. 4 Elliptical point contact[/caption] 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: $\begin{eqnarray} \label{complete_sys1} a= \alpha \sqrt{\frac{3FR}{E'}} \\ b =\beta \sqrt{\frac{3FR}{E'}}\\ \delta= \gamma \sqrt{\frac{9F^2}{8R{E'}^2}}\\ \end{eqnarray}$ where $\frac{1}{R}= \frac{1}{R_{1x}} + \frac{1}{R_{1y}} + \frac{1}{R_{2x}} + \frac{1}{R_{2y}}$. $\begin{eqnarray} \label{complete_sys1} \alpha \approx {\kappa}^{\frac{1}{3}} \sqrt{\frac{2E(m)}{\pi}}\\ \beta \approx {\kappa}^{-\frac{2}{3}} \sqrt{\frac{2E(m)}{\pi}}\\ \gamma \approx {\kappa}^{\frac{2}{3}} ({\frac{2E(m)}{\pi}})^{-1/3} \frac{2}{\pi}K(m)\\ \end{eqnarray}$ The functions in the previous equations are approximated as follows: $\begin{eqnarray} \label{complete_sys1} E(m) \approx \frac{\pi}{2}(1-m)[1+\frac{2m}{\pi (1-m)} -\frac{1}{8}ln(1-m)]\\ K(m) \approx \frac{\pi}{2}(1-m)[1+\frac{2m}{\pi (1-m)}ln(\frac{4}{\sqrt{1-m}}) -\frac{3}{8}ln(1-m)]\\ \kappa \approx {1+\sqrt{\frac{ln(16/\lambda)}{2\lambda}} -\sqrt{ln(4)} +0.16ln(\lambda) }^{-1} \\ m = 1 - {\kappa}^2 \\ \lambda = \frac{R_x}{R_y}, 0<\lambda<1 \\ \end{eqnarray}$ The mean and maximum pressures are given by: $\begin{eqnarray} \label{complete_sys1} P_{mean}= \frac{F}{\pi ab}, P_{max}=\frac{3}{2 P_{mean}}\\ \end{eqnarray}$ Load as a function of rigid body approach can be calculated as follows: $\begin{eqnarray} \label{complete_sys1} F = \sqrt{\frac{8}{9} \frac{{E'}^2R}{{\gamma}^3}}{\delta}^{3/2}, \\ \end{eqnarray}$ Stiffness of the contact is defined as follows: $\begin{eqnarray} \label{complete_sys1} k = \frac{F}{\delta}, \\ \end{eqnarray}$ Hence, for an elliptical contact, stiffness can be found from the following expression: $\begin{eqnarray} \label{complete_sys1} k = \sqrt{\frac{8}{9} \frac{{E'}^2R}{{\gamma}^3}}{\delta}^{1/2}, \\ \end{eqnarray}$ Corresponding Matlab code for Hertz solution can be found here. The online Hertz contact calculator can be found here. Further details of the contact theory can be found in Contact Mechanics by James Barber. Here is an online calculator for an elliptical point contact.

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October 19, 2021 @ 04:41:18 [email protected]
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1. Owen says:

I believe the formula for the b (width of contact) is incorrect in two ways, firstly if b is the whole width it contradicts the diagram of the two cylinders in contact which shows 2b. So if it is the full width it should be b=2*(sqrt((4*F*R)/(pi*L*E’))) with a 4 instead of a 2 in the formula, OR if b is the half width as implied by the diagram it should be b=(sqrt((4*F*R)/(pi*L*E’))

2. Aydar Akchurin says:

Hi Owen,

Thank you for a comment. So b is the half-width, as shown in Figure 2. The equation for b that you posted in your comment assumes a different definition of the reduced elastic modulus (1/E’=…, while the equations posted in the wiki, assume 2/E’=…, see its definition after equation 1). If you substitute this formula to the equation in the article, you will get your equation.

3. geardyn.1 says:

Hi,
if we assume Steel material ,its ultimate strength is ~500MPa. if we calculate the contact pressure for a two sphere(0.1m) under contact in both elastic & plastic regime , the hertz contact pressure is reaching around 11.66E3 MPa. I have analytical & numerically(using abaqus) validated it. my concern is if the obtained contact pressure is so huge and crossing the ultimate strength of the material ,can i consider it for my design ?
Should Contact pressure or hertz contact stress be LESS than Ultimate stress ?

4. tribonet says:

I believe, if the Hertzian stress exceeds the ultimate strength of the material in your design, it cannot be a good sign. You will get a lot of plastic deformation and probably a failure in a short term. So it is good to rethink the design.

5. Jamie says:

Hi,
Are there any places that the derivation for the elliptical point contact equations are published? I’ve looked through some of the references and couldn’t find the exact equations.
Many thanks.