Technical Briefs

Determining a Surrogate Contact Pair in a Hertzian Contact Problem

[+] Author and Article Information
Anthony P. Sanders

 Ortho Development Corp., 12187 S. Business Park Dr., Draper, UT 84020tsanders@odev.com

Rebecca M. Brannon

Department of Mechanical Engineering, University of Utah, 50 S. Central Campus Dr., 2134 Merrill Engineering Bldg., Salt Lake City, UT 84112brannon@mech.utah.edu

J. Tribol 133(2), 024502 (Mar 17, 2011) (6 pages) doi:10.1115/1.4003492 History: Received March 21, 2010; Revised January 12, 2011; Published March 17, 2011; Online March 17, 2011

Laboratory testing of contact phenomena can be prohibitively expensive if the interacting bodies are geometrically complicated. This work demonstrates means to mitigate such problems by exploiting the established observation that two geometrically dissimilar contact pairs may exhibit the same contact mechanics. Specific formulas are derived that allow a complicated Hertzian contact pair to be replaced with an inexpensively manufactured and more easily fixtured surrogate pair, consisting of a plane and a spheroid, which has the same (to second-order accuracy) contact area and pressure distribution as the original complicated geometry. This observation is elucidated by using direct tensor notation to review a key assertion in Hertzian theory; namely, geometrically complicated contacting surfaces can be described to second-order accuracy as contacting ellipsoids. The surrogate spheroid geometry is found via spectral decomposition of the original pair’s combined Hessian tensor. Some numerical examples using free-form surfaces illustrate the theory, and a laboratory test validates the theory under a common scenario of normally compressed convex surfaces. This theory for a Hertzian contact substitution may be useful in simplifying the contact, wear, or impact testing of complicated components or of their constituent materials.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

(a) Section view through a general surface. (b) Section view through a surface with the laboratory basis positioned at the point of contact.

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Figure 2

Contacting curved surfaces (separated for clarity). Section planes contain principal curvatures. For clarity, points k and p are shown at an exaggerated distance from the origin.

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Figure 3

(a) Separation curves, joined by a transparent surface for visual reference. (b) In plan view, separation curves overlie the tangent ellipse; also, a c-basis is aligned with the principal axes of the tangent ellipse.

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Figure 4

(a) Free-form virtual surface; region at C extracted and transformed to contact the original surface at D. (b) Surfaces at D (viewed along tangent plane) initially touch at a point.

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Figure 5

Major semiaxis lengths from Sims 1 and 2, compared with predictions. There were two values for each semiaxis (i.e., c1 and c2) since the quasi-ellipses’ complementary semiaxes were not identical.

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Figure 6

Hertzian substitution concept: An arbitrary contact pair (a), with given principal curvatures and orientation, is substituted with a simpler contact pair (b) consisting of a spheroid and a plane

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Figure 7

Simulation results. (a) Sim A tangent ellipses: results show zero difference. (b) Sim B: Contact analysis: results for semiaxis lengths and pressure were identical for the pairs.

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Figure 8

Results of one validation test: comparison of contact patch major and minor axes. Legend: “OP” is the original pair, and “RP” is its replacing pair. Results for minor axes overlie one another.




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