New Approximations for Elastic Spheres Under an Oscillating Torsional Couple

[+] Author and Article Information
Daniel J. Segalman

 Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0847djsegal@sandia.gov

Michael J. Starr, Martin W. Heinstein

 Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0847

J. Appl. Mech 72(5), 705-710 (Dec 02, 2004) (6 pages) doi:10.1115/1.1985430 History: Received July 23, 2004; Revised December 02, 2004

The Lubkin solution for two spheres pressed together and then subjected to a monotonically increasing axial couple is examined numerically. The Deresiewicz asymptotic solution is compared to the full solution and its utility is evaluated. Alternative approximations for the Lubkin solution are suggested and compared. One approximation is a Padé rational function which matches the analytic solution over all rotations. The other is an exponential approximation that reproduces the asymptotic values of the analytic solution at infinitesimal and infinite rotations. Finally, finite element solutions for the Lubkin problem are compared with the exact and approximate solutions.

Copyright © 2005 by American Society of Mechanical Engineers
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Figure 1

Lubkin considered two spheres pressed together by forces N and then subjected to a monotonically applied torsion M. The resulting relative rotation (one half the difference in absolute rotation) is indicated by β. Indicated in the inset is the radius a of the contact patch and the radius c of the stuck region.

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Figure 2

The Deresiewicz approximation for T(θ) for monotonic rotation appears to be adequate only for the very small values of rotation. A rational function (Padé) approximation overlies the values calculated from Lubkin’s integral equation and appears to be adequate over the whole range of values computed.

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Figure 3

The Cuttino and Dow approximation has the correct general form, but approaches incorrect asymptotic values. Another approximation of the same general form, but constructed to have the correct asymptotic values is a substantially better approximation though it does have some small visible error in the region 0.1<θ<0.5.

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Figure 4

The Padé approximation for the numerically obtained backbone curve is used to draw the hysteresis plots for θ=0.04 and θ=0.5. At large rotation angles, the slope of the hysteresis curve just before reversal becomes very small.

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Figure 5

The Deresiewicz approximation for dissipation per cycle appears to be adequate for only small values of applied torque. The dissipation at large torques calculated from hysteresis appears to go to infinity.

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Figure 6

The asymptotic behavior of the torque associated with large angles is shown in the plot of log(3π∕16−T(θ)), suggesting that the difference of the torque and its limiting value goes as θ−12.3. Note that the Padé approximation overlies the data points obtained through numerical evaluation of Lubkin’s equation.

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Figure 7

Once the peaks of the hysteresis curve approach the limiting torque, additional rotations result in additional dissipation approximately equal to the area of the rhombus shown

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Figure 8

A very fine mesh is employed to capture the Lubkin result with good precision. This mesh employs approximately 67 000 8-node hex elements and 70 000 nodes.

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Figure 9

The hysteresis curve calculated by the finite element code JAS for a normalized angle of θ=0.3. This hysteresis curve manifests the symmetries associated with Masing models.

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Figure 10

The ramp-up torque calculated by the finite element code is reasonably consistent with the numerical evaluation of Eqs. 1,2, as represented by the Padé approximation

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Figure 11

The dissipation per cycle calculated from the finite element analysis is consistent with the logarithmically singular behavior predicted by Eq. 14




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