Solution of the Contact Zone Orientation for Normal Elliptical Hertzian Contact

Many mechanical designs have parts that come into, or lose, contact with each other. When elastic bodies with second order surface geometries come into contact, the contact patch is expected to be approximately flat and to have an elliptical boundary. Classic Hertzian contact mechanics can be used to model such contacts, but since there is no closed-form analytical solution to predict the major and minor axes of the contact zone ellipse, approximate numerical methods have been developed, some of which are very accurate. Predictions of the mutual approach of the bodies and the contact pressure distribution can then be made. Although the shape of the contact ellipse has been modeled and solved for, to date there has been no solution for the orientation of the contact ellipse with respect to either of the contacting bodies. The contact ellipse orientation is needed in order to model the shear stress distributions that occur when sticking friction forces are developed and separate contact zones of sticking and slipping are expected. Using the results of a numerical solution for the conventional contact parameters, this paper presents an analytical solution of the orientation of the contact ellipse, which is shown to depend only on the curvatures and the relative orientation of the contacting bodies. In order to validate the analytical solution, the results are compared with those from ABAQUS^™ finite element simulations for cases of identical bodies and bodies with dissimilar curvatures. The predictions of the contact ellipse orientation angles and the major and minor semi-axes agree very well for all cases considered.