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TECHNICAL PAPERS

# Rolling of an Elastic Ellipsoid Upon an Elastic-Plastic Flat

[+] Author and Article Information
Daniel Nélias, Eduard Antaluca, Vincent Boucly

LaMCoS, INSA-Lyon, CNRS UMR5259, Villeurbanne F69621, France

J. Tribol 129(4), 791-800 (Jun 07, 2007) (10 pages) doi:10.1115/1.2768078 History: Received March 09, 2007; Revised June 07, 2007

## Abstract

The paper presents a numerical analysis of the rolling contact between an elastic ellipsoid and an elastic-plastic flat. Numerical simulations have been performed with the help of a contact solver called Plast-Kid®, with an algorithm based on an integral formulation or semi-analytical method. The application of both the conjugate gradient method and the discrete convolution and fast Fourier transform technique allows keeping the computing time reasonable when performing transient 3D simulations while solving the contact problem and calculating the subsurface stress and strain states. The effects of the ellipticity ratio $k$—ranging from 1 to 16—and of the normal load—from 4.2 GPa to 8 GPa—are investigated. The reference simulation corresponds to the rolling of a ceramic ball on a steel plate made of an AISI 52100 bearing steel under a load of 5.7 GPa. The results that are presented are, first, the permanent deformation of the surface and, second, the contact pressure distribution, the von Mises stress field, the hydrostatic pressure, and the equivalent plastic strain state within the elastic-plastic body. A comparison with an experimental surface deformation profile is also given to validate the theoretical background and the numerical procedure.

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## Figures

Figure 1

Comparison of a vertical loading/unloading with and without rolling (k=1−PHertz=5.7GPa, ceramic ball (elastic)/flat 52100 (elastic-plastic) surface, after a single cycle, rolling distance of 4mm)

Figure 2

Surface displacement at the center of the rolling track after unloading (k=1−PHertz=5.7GPa, ceramic ball (elastic)/flat 52100 (elastic-plastic) surface, after one, two, and three cycles, rolling distance of 2mm)

Figure 3

Transverse surface profile after unloading (k=1−PHertz=5.7GPa, ceramic ball (elastic)/flat 52100 (elastic-plastic) surface, after one, two, and three cycles)

Figure 4

Cross-section view of the von Mises stress under load in the steady-state region (k=1−PHertz=5.7GPa, ceramic ball (elastic)/flat 52100 (elastic-plastic) surface, after the second loading cycle)

Figure 5

Figure 6

Distribution of the residual stress in the plane y=0 (k=1−PHertz=5.7GPa, ceramic ball (elastic)/flat 52100 (elastic-plastic) surface, after the second loading cycle)

Figure 7

Figure 8

Isovalues of the equivalent plastic strain (in %) in the plane y=0 (k=1−PHertz=5.7GPa, ceramic ball (elastic)/flat 52100 (elastic-plastic) surface, after the second loading cycle)

Figure 9

Surface displacement at the center of the rolling track after unloading for various ellipticity ratios and after one and two cycles (PHertz=5.7GPa, ceramic ellipsoid (elastic)/flat 52100 (elastic-plastic) surface). Loading, rolling, and unloading are represented by the thick arrows

Figure 10

Transverse surface profile after unloading for various ellipticity ratios and after one and two cycles. Profiles taken in the steady-state region (PHertz=5.7GPa, ceramic ellipsoid (elastic)/flat 52100 (elastic-plastic) surface)

Figure 11

Contact pressure distribution found in the steady-state region for various ellipticity ratios and after the second cycle (PHertz=5.7GPa, ceramic ellipsoid (elastic)/flat 52100 (elastic-plastic) surface)

Figure 12

Profile of the equivalent plastic strain found at the center of the track and in the steady-state region for various ellipticity ratios and after one and two cycles (PHertz=5.7GPa, ceramic ellipsoid (elastic)/flat 52100 (elastic-plastic) surface)

Figure 13

Profile of the von Mises stress found under loading at the center of the track and in the steady-state region for various ellipticity ratios and after one and two cycles (PHertz=5.7GPa, ceramic ellipsoid (elastic)/flat 52100 (elastic-plastic) surface)

Figure 14

Profile of the residual von Mises stress found after unloading at the center of the track and in the steady-state region for various ellipticity ratios and after one and two cycles (PHertz=5.7GPa, ceramic ellipsoid (elastic)/flat 52100 (elastic-plastic) surface)

Figure 15

Hydrostatic pressure found after unloading at the center of the track and in the steady-state region for various ellipticity ratios and after one and two cycles (PHertz=5.7GPa, ceramic ellipsoid (elastic)/flat 52100 (elastic-plastic) surface)

Figure 17

Profile of the von Mises stress found after unloading at the center of the track and in the steady-state region for various normal loads and after one and two cycles (k=8, ceramic ellipsoid (elastic)/flat 52100 (elastic-plastic) surface)

Figure 18

Profile of the hydrostatic pressure found after unloading at the center of the track and in the steady-state region for various normal loads and after one and two cycles (k=8, ceramic ellipsoid (elastic)/flat 52100 (elastic-plastic) surface)

Figure 19

Displacement at the center of the rolling track after unloading for various normal loads and after one and two cycles (k=8—ceramic ellipsoid (elastic)/flat 52100 (elastic-plastic) surface)

Figure 16

Profile of the equivalent plastic strain found at the center of the track and in the steady-state region for various normal loads and after one and two cycles (k=8, ceramic ellipsoid (elastic)/flat 52100 (elastic-plastic) surface)

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