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Research Papers: Contact Mechanics

# Elastic Contact Between a Geometrically Anisotropic Bisinusoidal Surface and a Rigid Base

[+] Author and Article Information
Yang Xu, Robert L. Jackson

Mechanical Engineering Department,
Auburn University,
Auburn, AL 36849

Amir Rostami

G. W. Woodruff School
of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332

This is also true for the plane problem. However, the shape of the contact area in the plane problem is known. For the 3D problem, the shape of the contact area depends on the external average contact pressure [15].

In the rest of the article, the material is always linear, isotropic, and homogeneous. “Geometrically isotropic” is reserved for the geometrical property of the rough surface. Readers may find the terminology, “isotropic,” inappropriate here because it is usually applied to describe the rough surface not the smooth one. A rough surface is called isotropic, if every profile of the rough surface, measured in the arbitrary direction, is statistically the same. Axisymmetric is an equivalent terminology used for smooth surfaces. In this study, however, in order to be consistent with the terminology used by Johnson et al. [9], geometrically isotropic is used here to describe the bisinusoidal waviness with the same wavelengths on two principal axes. If the wavelengths are different, the bisinusoidal waviness is called geometrically anisotropic.

The linear, frictionless, elastic contact between the sinusoidal waviness, and the rigid flat can be extended to a broader case where the sinusoidal waviness is in contact with an elastic half-space. Then, the effective material modulus can be written as $1/E*=(1-ν12/E1+1-ν22/E2)$. Ei is the Young’s modulus of the sinusoidal waviness (i = 1) and the half-space (i = 2), respectively. νi is the Poisson’s ratio of the sinusoidal waviness (i = 1) and the half-space (i = 2), respectively.

Using the multiples of two, e.g., 512, 1024 instead of 600, can efficiently use the FFT libraries. We also test the FFT model with 512 × 512 which gives us almost the same answer for all five different $k'$.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received August 22, 2014; final manuscript received December 31, 2014; published online February 5, 2015. Assoc. Editor: Sinan Muftu.

J. Tribol 137(2), 021402 (Apr 01, 2015) (8 pages) Paper No: TRIB-14-1210; doi: 10.1115/1.4029537 History: Received August 22, 2014; Revised December 31, 2014; Online February 05, 2015

## Abstract

In the current study, a semi-analytical model for contact between a homogeneous, isotropic, linear elastic half-space with a geometrically anisotropic (wavelengths are different in the two principal directions) bisinusoidal surface on the boundary and a rigid base is developed. Two asymptotic loads to area relations for early and almost complete contact are derived. The Hertz elliptic contact theory is applied to approximate the load to area relation in the early contact. The noncontact regions occur in the almost complete contact are treated as mode-I cracks. Since those cracks are in compression, an approximate relation between the load and noncontact area can be obtained by setting the corresponding stress intensity factor (SIF) to zero. These two asymptotic solutions are validated by two different numerical models, namely, the fast Fourier transform (FFT) model and the finite element (FE) model. A piecewise equation is fit to the numerical solutions to bridge these two asymptotic solutions.

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## Figures

Fig. 1

Schematic representation of a half-plane with the sinusoidal waviness on the top boundary (only the part in one complete period is shown) and a rigid flat (a) before and (b) in contact

Fig. 2

Schematic representation of a typical elliptic contact (noncontact) area in the vicinity of the peak A (valley D') on a sinusoidal waviness shown in Fig. 3

Fig. 3

Contour plot of a 3D geometrically anisotropic bisinusoidal waviness within one period, λx > λy. Peaks of the waviness are located on points A, B, C, D, and E. Valleys are located on points B',C',D', and E'.

Fig. 4

Nonlinear relation between the elliptic modulus, k'1, and the wavelength ratio, k'

Fig. 5

Schematic representation of the decomposition of (a) almost complete contact under the contact pressure distribution p(x',y') into (b) complete contact under p1(x',y') and (c) a pressurized crack embedded inside an half-space with p2(x',y') acting on the crack surfaces (only half of the crack is shown due to the symmetry)

Fig. 6

Meshed sinusoidal surface for k' = 0.5

Fig. 7

Contact ratio to the dimensionless average pressure relation predicted by: (i) the FE model, (ii) the FFT model, and (iii) two asymptotic solutions (Eqs. (19) and (29)) when k' = λy/λx = 0.5

Fig. 8

Dimensionless contact area to dimensionless average pressure relation predicted by: (i) the FE model, (ii) the FFT model, and (iii) two asymptotic solutions (Eqs. (19) and (29)) when k' = λy/λx equals to (a) 1.0, (b) 0.7, (c) 0.3, and (d) 0.1

Fig. 9

Contact ratio to dimensionless average pressure relation predicted by curve-fit equation, Eq. (37), and the FFT model

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