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Research Papers: Contact Mechanics

# Contact Fatigue of Rough Elastic Surfaces in Two-Dimensional Formulation

[+] Author and Article Information
Ilya I. Kudish

Professor of Mathematics ASME Fellow Department of Mathematics,  Kettering University, Flint, MI, 48439ikudish@kettering.edu

Chebyshev orthogonal polynomials of the, first kind $Tk(υ)$ and of the second kind $Uk(υ)$ [20] are defined as follows $Tk(cosθ)=coskθ$ and $Uk(cosθ)=[sin(k+1)θ/sinθ]$, respectively. These polynomials satisfy the following properties $∫-11[Tk(υ)Tm(υ)dv/1-υ2]=$ 0 if k$≠$ m, $∫-11[Tk2(υ)dυ/1-υ2]=π2$ if k$≠0$ and the integral is equal to $π$ if $k=0$; $∫-111-υ2Uk(υ)Um(υ)dυ=$ 0 if k $≠$ m, $∫-111-υ2Uk2(υ)dυ=π/2$ if k $≥$ 0, $∫-11dt/[1-t2(t-υ)]=$ 0, and $∫-11Tk(t)dt/[1-t2(t-υ)]=πUk-1(υ),k≥1$. It is well known that almost any continuous function on interval $[-1,1]$ can be expanded in Chebyshev polynomials $Tk(υ)$ [21].

J. Tribol 133(3), 031405 (Jul 25, 2011) (9 pages) doi:10.1115/1.4004346 History: Received September 13, 2010; Revised May 31, 2011; Published July 25, 2011; Online July 25, 2011

## Abstract

Solution of a contact problem for a rough elastic half-plane is considered. Surface roughness is assumed to be small and stochastic. A perturbation solution of the problem for relatively small roughness with singly connected contact region is proposed and is conveniently expressed in terms of Chebyshev polynomials. Mean distribution of pressure and mean size of the contact are obtained analytically. A pitting model for rough surfaces is considered based on a generalization of an earlier proposed contact model with some stochastic parameters. An analytical formula relating subsurface originated fatigue is considered and fatigue life of rough and smooth surfaces is obtained which shows that fatigue life of rough solids is slightly shorter than of the smooth ones. In the general case of a contact region of rough surfaces with multiple connectivity subsurface originated fatigue possesses properties similar to the case of singly connected contact region. Surface roughness may have a significant effect only on surface and near surface originated fatigue such as wear, micropitting, and shallow flaking.

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

Figure 1

Distributions of the normal stress intensity factors k1± at the tips of the crack as a function of x in case of a subsurface crack with dimensionless half-length ℓ=0.05, angle of orientation α=1.047198, y=-0.3, q0=-0.001 for friction coefficient λ=0.04 (curve 1) and λ=0.08 (curve 2) (after Kudish and Covitch [23])

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