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Technical Briefs

# Perfect Mechanical Sealing in Rough Elastic Contacts: Is It Possible?

[+] Author and Article Information
IIya I. Kudish

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

Donald K. Cohen

Owner
Michigan Metrology
LLC, 17199 N. Laurel Park Drive
Suite 51 Livonia, MI 48152
e-mail: doncohen@michmet.com

Brenda Vyletel

Electron Microbeam Analysis Laboratory
University of Michigan
Ann Arbor, MI 48109
e-mail: bvyletel@umich.edu

According to the accepted mathematical definition, a singly connected region is such a region that any two points of it can be connected by a continuous curve, all points of which belong to the region. Otherwise, the region is multiply connected.

Chebyshev orthogonal polynomials of the first kind $Tk(x)$ and of the second kind $Uk(x)$ [13] are defined as follows $Tk(cosθ)=coskθ$ and $Uk(cosθ)=sin(k+1)θ/sinθ$, respectively. These polynomials satisfy the following properties $∫-11Tk(t)Tm(t)dt/1-t2=0$ if $k≠m$, $∫-11Tk2(t)dt/1-t2=π/2$ if $k≠$ 0 and the integral is equal to $π$ if k$=$ 0; $∫-111-t2Uk(t)Um(t)dt=$ 0 if k$≠$m, $∫-111-t2Uk2(t)dt=π/2$ if k$≥$ 0, and $∫-11Uk-1(t)1-t2dt/(t-x)=-πTk(x),k=1,2,…$

A region is convex when any of its two points can be connected by a segment of a straight line all points of which belong to the region.

Manuscript received December 1, 2011; final manuscript received June 4, 2012; accepted manuscript posted July 6, 2012; published online November 19, 2012. Assoc. Editor: Anand Jagota.

J. Appl. Mech 80(1), 014504 (Nov 19, 2012) (6 pages) Paper No: JAM-11-1458; doi: 10.1115/1.4007085 History: Received December 01, 2011; Revised June 04, 2012; Accepted July 06, 2012

## Abstract

Generally, it is assumed that under any applied force there will always be some gap between the surfaces in a contact of rough elastic surfaces, resulting in a discontinuous (i.e., multiply connected) contact. The presence of gaps along the line contact relates to the ability to form an adequate mechanical seal across an interface. This paper will demonstrate that for a twice continuously differentiable rough surface with sufficiently small asperity amplitude and/or sufficiently large applied load and/or sufficiently low material elastic modulus, singly connected contacts exist. The solution of a contact problem for a rough elastic half-plane and a perfectly smooth rigid indenter with sharp edges is considered. First, a problem with artificially created surface irregularity is considered and it is shown that, for such a surface, the contact region is always multiply connected. An exact solution of the problem for an indenter with sharp edges resulting in a singly connected contact region is considered and it is conveniently expressed in the form of a series in Chebyshev polynomials. A sufficient (not necessary) condition for a contact of an indenter with sharp edges and a rough elastic surface to be singly connected is derived. The singly connected contact condition depends on the surface microtopography, material effective elastic modulus, and applied load. It is determined that, in most cases, a normal contact of a twice continuously differentiable rough surface with sufficiently small asperity amplitude, sufficiently low material elastic modulus, and/or sufficiently large applied load is singly connected.

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## References

Greenwood, J. A., and Williamson, J. B. P., 1966, “Contact of Nominally Flat Surface,” Proc. R. Soc. London, Ser. A, 295, pp. 300–312.
Ciavarella, M., Delfine, V., and Demelio, V. G., 2006, “A “Re-Vitalized“ Greenwood and Williamson Model of Elastic Contact Between Fractal Surfaces,” J. Mech. Phys. Solids, 54(12), pp. 2569–2591.
Johnson, K. L., Kendall, K., and Roberts, A. D., 1971, “Surface Energy and the Contact of Elastic Solids,” Proc. R. Soc. London, Ser. A, 324, pp. 301–313.
Derjaguin, B. V., Muller, V. M., and Toporov, Y. P., 1975, “Effect of Contact Deformations on the Adhesion of Particles,” J. Colloid Interface Sci., 53, pp. 314–326.
Kalker, J. J., 1990, Three-Dimensional Elastic Bodies in Rolling Contact, Solid Mechanics and Its Applications, Vol. 2, Kluwer, Dordrecht, The Netherlands.
Kudish, I. I., and Covitch, M. J., 2010, Modeling and Analytical Methods in Tribology, Chapman & Hall/CRC, Boca Raton, FL.
Kudish, I. I., 2011, “Contact Fatigue of Elastic Surfaces With Small Roughness,” ASME J. of Tribol., 133, July, p. 031405.
Kudish, I. I., 1987, “Contact Problem of the Theory of Elasticity for Prestressed Bodies With Cracks,” J. Appl. Mech. Tech. Phys., 28, pp. 144–152.
Ciavarella, M., Demelio, G., Barber, J. R., and Jang, Y. H., 2000, “Linear Elastic Contact of the Weierstrass Profile,” Proc. Roy. Soc. London, Ser. A, 456, pp. 387–405.
Snidle, R. W., and Evans, H. P., 1994, “A Simple Method of Elastic Contact Simulation,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 208, pp. 291–293.
Lebeck, A. O., 1991, Principles and Design of Mechanical Face Seals, John Wiley & Sons, New York.
Galin, L. A., 1980, Contact Problems in Elasticity and Visco-Elasticity, Nauka, Moscow.
M.Abramowitz and I.A.Stegun, eds., 1964, Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, National Bureau of Standards, Vol. 55, Washington, DC.
Szegö, G.1959, Orthogonal Polynomials, Vol. XXIII, American Mathematical Society, Colloquim, New York.
Wong, R., 2001, Asymptotic Approximations of Integrals, Academic, New York.

## Figures

Fig. 1

Basic geometry for the problem of an indenter with sharp edges making contact with a surface with texture

Fig. 2

Measurement of a ground surface using an optical profiler over a field of view of ~100 μm×100 μm, with a diffraction limited lateral resolution of smaller than 1 μm and a height resolution of ~6 nm

Fig. 3

Measurement of a ground surface using an atomic force microscope over a field of view of ~2 μm × 2 μm, with a lateral resolution of ~16 nm, and a height resolution of ~0.1 nm. Note: the image is displayed on the same gray scale as for optical profiler measurements.

Fig. 6

Measurement of a ground surface using an atomic force microscope over a field of view of ~2 μm×2 μm, with a lateral resolution of ~4 nm, and a height resolution of ~0.1 nm. Note that the finer spaced parallel line features are scanning related artifacts.

Fig. 5

Measurement of a ground surface using an atomic force microscope over a field of view of ~2 μm×2 μm, with a lateral resolution of ~8 nm, and a height resolution of ~0.1 nm. Note that the finer spaced parallel line features are scanning related artifacts.

Fig. 4

Measurement of a ground surface using an atomic force microscope over a field of view of ~2 μm×2 μm, with a lateral resolution of ~16 nm, and a height resolution of ~0.1 nm. Note: the image is displayed on a different gray scale as for the optical profiler measurements to demonstrate the surface features. Note that the finer spaced parallel line features are scanning related artifacts.

## Errata

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