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Technical Briefs

# Solution of the Contact Zone Orientation for Normal Elliptical Hertzian Contact

[+] Author and Article Information
Philip P. Garland1

School of Biomedical Engineering, Dalhousie University, Halifax, NS, B3H 2Y9, Canadaphil.garland@unb.ca

Robert J. Rogers

Department of Mechanical Engineering, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

The elastic deformation of body 2 is negative in the global $z$-direction.

The authors have found a slight error in Eqs. (7)–(9) in Ref. 8 and suggest using the equivalent equations found in Ref. 3 to solve the sought-after values.

1

Corresponding author.

J. Appl. Mech 78(3), 034501 (Feb 16, 2011) (5 pages) doi:10.1115/1.4003365 History: Received March 13, 2010; Revised November 21, 2010; Posted January 05, 2011; Published February 16, 2011; Online February 16, 2011

## Abstract

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.

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Copyright © 2011 by American Society of Mechanical Engineers
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## Figures

Figure 2

Normal direction deformation

Figure 3

Projection of disks’ surface profiles onto contact zone plane. (Note: ϕ2 is negative in this figure.)

Figure 1

Geometry of radiused disks in contact

Figure 4

Comparison of contact zone geometries and orientation for identically radiused disks of identical material: FEA solution (solid); analytical solution (dashed)

Figure 5

Comparison of contact zone geometries and orientation for dissimilarly radiused disks of identical material: FEA solution (solid); analytical solution (dashed)

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