Research Papers

An Analytical Solution for Shear Stress Distributions During Oblique Elastic Impact of Similar Spheres

[+] Author and Article Information
Philip P. Garland1

Department of Mechanical Engineering, University of New Brunswick, Canada Fredericton, New Brunswick E3B 5A3, Canadaphil.garland@unb.ca

Robert J. Rogers

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

Under an assumption of Coulomb friction with equal static and kinetic friction coefficients, the maximum frictional force would be given by the product of the normal force and the friction coefficient. For impacts, we would expect the tangential force between the bodies to be equal to this value in the limiting case of full sliding of all coincident points in the contact zone. The shear stress distribution associated with the full sliding case is defined as the friction envelope.

A scalar, not vector, addition scheme is utilized.


Corresponding author.

J. Comput. Nonlinear Dynam 3(1), 011002 (Nov 02, 2007) (9 pages) doi:10.1115/1.2802112 History: Received July 27, 2006; Revised July 23, 2007; Published November 02, 2007

A new solution method for the oblique elastic impact of similar spheres with Coulomb friction is presented. The solution uses approximations of the shear stress distributions at each time step during impact. These distributions are solved from analytical formulations and are able to account for both full sliding and partial-slip scenarios that may both be present for this problem due to inclusion of tangential compliance and friction effects. Comparison to previous continuum models in the literature shows very good agreement for the contact force wave forms obtained. The major advantage of this method is the drastic reduction in computation time required compared to previous solutions.

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

Relationship between rigid body and elastic displacements for normal direction

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Figure 2

Configuration of spheres at initial contact

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Figure 3

Hertzian stress distribution

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Figure 4

Free body diagram for normal direction

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Figure 5

Cattaneo–Mindlin stress distribution

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Figure 6

Normalized tangential force versus normalized time using the algorithm of Maw (6)

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Figure 7

Typical shear stress distributions during impact (a) full sliding, (b) partial-slip loading, and (c) reversed partial slip. The outer curves are the friction envelopes.

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Figure 8

Relationship between rigid body, elastic, and relative displacements for (a) sticking of coincident points and (b) sliding of coincident points

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Figure 9

Free body diagram for tangential direction

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Figure 10

Normalized tangential force versus normalized time (Maw (6), solid; analytical shear stress, x)

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Figure 11

Normalized rebound angle versus normalized incidence angle (Maw (6), solid; analytical shear stress, +)

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Figure 12

Shear stress distributions for ψ=0.5. (a) During loading, (b) at maximum tangential force, (c) at maximum compression, and (d) at minimum tangential force. (Maw (6), solid; approximated shear stress, dash-dot; friction envelope, dash)



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