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An eigenvalue problem for elastic contact with finite friction

Published online by Cambridge University Press:  24 October 2008

D. A. Spence
D. A. Spence
Affiliation:
Mathematics Research Center, University of Wisconsin
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Abstract

The two-dimensional indentation of an elastic half space by a rigid punch under a slowly applied normal load is considered, for the case in which there is a finite coefficient of friction μ between the surfaces. The contact area is then divided into an inner adhesive region −c < x < c in which the surface displacements are known, surrounded by regions c < |x| < 1 in which the friction is limiting and the lateral displacement (which must increase in proportion to the overall load) is not known in advance. The problem is formulated in terms of a coupled pair of singular integral equations for the normal and shear stresses σ and τ at the surface; these are combined to give a single homogeneous Fredholm equation with positive kernel for a quantity π proportional to the difference τ – μσ in the adhesive region. The largest eigenvalue of this equation, for which π > 0, gives the adhesive boundary c in terms of μ and Poisson's ratio ν. A similarity transformation shows that c has the same value for both flat-faced and power law punches.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 73 , Issue 1 , January 1973 , pp. 249 - 268
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

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