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Linear elastic contact of the Weierstrass profile

Linear elastic contact of the Weierstrass profile
Linear elastic contact of the Weierstrass profile
A contact problem is considered in which an elastic half-plane is pressed against a rigid fractally rough surface, whose profile is defined by a Weierstrass series. It is shown that no applied mean pressure is sufficiently large to ensure full contact and indeed there are not even any contact areas of finite dimension - the contact area consists of a set of fractal character for all values of the geometric and loading parameters. A solution for the partial contact of a sinusoidal surface is used to develop a relation between the contact pressure distribution at scale n - 1 and that at scale n. Recursive numerical integration of this relation yields the contact area as a function of scale. An analytical solution to the same problem appropriate at large n is constructed following a technique due to Archard. This is found to give a very good approximation to the numerical results even at small n, except for cases where the dimensionless applied load is large. The contact area is found to decrease continuously with n, tending to a power-law behaviour at large n which corresponds to a limiting fractal dimension of (2 - D), where D is the fractal dimension of the surface profile. However, it is not a 'simple' fractal, in the sense that it deviates from the power-law form at low n, at which there is also a dependence on the applied load. Contact segment lengths become smaller at small scales, but an appropriately normalized size distribution tends to a limiting function at large n.
multifractals, Weierstrass, contact mechanics, rough surfaces, asperities, multiscale models
1364-5021
387-405
Ciavarella, M.
d5aa6350-b3d4-4a78-a670-9d78242f58c5
Demelio, G.
cb38aabe-9837-4ab2-aa87-60028fd7b82c
Barber, J.R.
af9edfeb-4ab0-4c9a-9974-12f1e993e10b
Jang, Y.H.
40a660a5-ec02-4ee8-aab2-adafd3be7e2a
Ciavarella, M.
d5aa6350-b3d4-4a78-a670-9d78242f58c5
Demelio, G.
cb38aabe-9837-4ab2-aa87-60028fd7b82c
Barber, J.R.
af9edfeb-4ab0-4c9a-9974-12f1e993e10b
Jang, Y.H.
40a660a5-ec02-4ee8-aab2-adafd3be7e2a

Ciavarella, M., Demelio, G., Barber, J.R. and Jang, Y.H. (2000) Linear elastic contact of the Weierstrass profile. Proceedings of the Royal Society of London A, 456 (1994), 387-405. (doi:10.1098/rspa.2000.0522).

Record type: Article

Abstract

A contact problem is considered in which an elastic half-plane is pressed against a rigid fractally rough surface, whose profile is defined by a Weierstrass series. It is shown that no applied mean pressure is sufficiently large to ensure full contact and indeed there are not even any contact areas of finite dimension - the contact area consists of a set of fractal character for all values of the geometric and loading parameters. A solution for the partial contact of a sinusoidal surface is used to develop a relation between the contact pressure distribution at scale n - 1 and that at scale n. Recursive numerical integration of this relation yields the contact area as a function of scale. An analytical solution to the same problem appropriate at large n is constructed following a technique due to Archard. This is found to give a very good approximation to the numerical results even at small n, except for cases where the dimensionless applied load is large. The contact area is found to decrease continuously with n, tending to a power-law behaviour at large n which corresponds to a limiting fractal dimension of (2 - D), where D is the fractal dimension of the surface profile. However, it is not a 'simple' fractal, in the sense that it deviates from the power-law form at low n, at which there is also a dependence on the applied load. Contact segment lengths become smaller at small scales, but an appropriately normalized size distribution tends to a limiting function at large n.

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Published date: 2000
Keywords: multifractals, Weierstrass, contact mechanics, rough surfaces, asperities, multiscale models

Identifiers

Local EPrints ID: 21414
URI: https://eprints.soton.ac.uk/id/eprint/21414
ISSN: 1364-5021
PURE UUID: ba4196fe-9507-4cb2-bb31-05d2dd961841

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Date deposited: 06 Feb 2007
Last modified: 17 Jul 2017 16:26

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Contributors

Author: M. Ciavarella
Author: G. Demelio
Author: J.R. Barber
Author: Y.H. Jang

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