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Research Papers

# Numerical Analysis of Double Contacts of Similar Elastic Materials

[+] Author and Article Information
N. Sundaram

School of Aeronautics and Astronautics, Purdue University, 315 North Grant Street, West Lafayette, IN 47907-2023nsundara@purdue.edu

T. N. Farris

School of Aeronautics and Astronautics, Purdue University, 315 North Grant Street, West Lafayette, IN 47907-2023farrist@purdue.edu

This discussion only applies to smooth (incomplete) contacts, where the contact size is a function of the applied load; an example of a complete contact is the indentation of a half-plane by a flat rigid punch where the pressure is unbounded at the ends.

For $x$ only $∊$$L1$ or $x$ only $∊$$L2$, it may be readily recognized that this is a Cauchy SIE in which the integral kernel has a nondominant part. This nondominant part does not alter the singularity behavior at the ends.

Gladwell (3) used the term “interconnected parabolic punches;” “biquadratic” is used here to emphasize that the profile is not strictly parabolic.

In other words, of all admissible solutions that are bounded everywhere, the physically correct solution is the one that satisfies $V=0$.

J. Appl. Mech 75(6), 061017 (Aug 21, 2008) (9 pages) doi:10.1115/1.2967897 History: Received August 08, 2007; Revised July 09, 2008; Published August 21, 2008

## Abstract

A fast numerical method based on the Cauchy singular integral equations is presented to determine the contact pressure and extents for the contact of two-dimensional similar isotropic bodies when the contact area consists of two separate regions. The partial-slip problem is then solved to determine shear tractions using an equivalence principle. The extents of the contact are not all independent but related to a compatibility equation constraining the displacements of an elastic body in contact with an equivalent rigid body. A similar equation is found for the extents of the stick zones in partial-slip problems. The effects of load history are incorporated into the shear solution. The method is applicable to a wide range of profiles and it provides significant gains in computational efficiency over the finite element method (FEM) for both the pressure and partial-slip problems. The numerical results obtained are compared with that from the FEM for a biquadratic indenter with a single concavity and showed good agreement. Lastly, the transition behavior from double to single contacts in biquadratic profiles is investigated.

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## Figures

Figure 1

Double contact configuration

Figure 2

Analytical (solid lines) and SIE (markers) normalized pressure tractions p(x)/P* versus x/c for different loads

Figure 3

SIE (solid lines) and FEM (markers) normalized pressure tractions p(x)/P* versus x/c for P/F*=0.1677 and different moments M/M*

Figure 4

SIE (solid lines) and FEM (markers) normalized shear tractions q(x)/P* versus x/c for P/F*=0.1677, M=0, and different values of Q

Figure 5

SIE (solid lines) and FEM (markers) normalized shear tractions q(x)/P* versus x/c for P/F*=0.1677, M/M*=0.0314, and different values of Q

Figure 6

SIE (solid lines) and FEM (markers) normalized shear tractions q(x)/P* versus x/c for P/F*=0.1677, M/M*=0.0314, and a value of Q at which there is only one stick zone

Figure 7

SIE (solid lines) and FEM (markers) normalized shear tractions q(x)/P* versus x/c for normal loads P/F*=0.1887 and M=0. The shear traction reverses sign in the slip zones when Q is reduced from 0.81 μP to 0.41 μP.

Figure 8

SIE (solid lines) and FEM (markers) normalized shear tractions q(x)/P* versus x/c for P/F*=0.1887 and M/M*=−0.0314. Intermediate shear tractions when Q is reduced from 0.71 μP to −0.71 μP are shown.

Figure 9

(Left) SIE normalized pressure tractions p(x)/μpmax for the indentation of a half-space by a double-flat punch with rounded edges. The profile (not to scale) is also superposed. (Right) SIE normalized shear tractions q(x)/μpmax for a connected pair of cylinders with Q=0.455 μP applied in Step 2 (square markers) and partially reversed to Q=0 in load Step 3 (circular markers).

Figure 10

Transition behavior for a biquadratic punch; the solid lines represent a Type A transition boundary and the dashed line represents the Type B transition boundary. I and II represent the areas of single and double contacts, respectively.

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