The Effect of Asperity Flattening During Cyclic Normal Loading of a Rough Spherical Contact
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Abstract
The effect of asperity flattening of a rough spherical contact during cyclic loading is investigated experimentally. Two types of surfaces are examined; the first is an “as-manufactured” isotropic surface and the second a smooth “laser-polished” surface. Both the surfaces exhibit a large amount of hysteresis of the load–displacement curve during the first load–unload cycles. This hysteresis is found to decrease as a function of the number of load cycles. A comparison of the experimental results with results obtained from a numerical model for a rough spherical contact shows good correlation. The model shows that for rough surfaces the total displacement is a function of the contacting asperities while for smooth surfaces the main contribution comes from the bulk displacement.
Keywords
Contact mechanics Spherical contact Rough surface Cyclic load–unloadAbbreviations
- a
Radius of contact
- E
Young’s modulus
- H
Hardness
- h_{0}
Minimum separation
- \( h_{0}^{*} \)
Dimensionless minimum separation \( h_{0}/\sigma \)
- \( h_{1}^{*} \)
Dimensionless displacement of the contacting asperities\( h_{1} /\sigma \)
- L
Spherical shell height
- P
Normal load
- P_{c}
Critical normal load
- P^{*}
Dimensionless normal load P/P _{c}
- R
Radius of the spherical shell
- Y
Yield strength
- t
Spherical shell thickness
- β
Roughness parameter ηρσ
- γ
Total displacement of the rough spherical contact
- η
Area density of asperities
- ρ
Asperity tip radius of curvature
- σ
Standard deviation of surface heights
- γ^{*}
Dimensionless total displacement γ/σ
- γ_{max}
Maximum displacement
- \( \gamma_{\text{res}} \)
Residual displacement
- λ
Shell parameter
- v
Poison’s ratio
- σ_{s}
Standard deviation of asperity heights
- Ψ
Plasticity index
- ω
Displacement of the sphere bulk
- ω^{*}
Dimensionless displacement ω/σ
- ω_{c}
Critical interference
- \( \omega_{\text{c}}^{*} \)
Dimensionless critical interference ω _{c}/σ
1 Introduction
Elastic–plastic contacts of rough spherical surfaces can occur in many engineering applications such as MEMS micro-switches [1], AFM systems [2], or magnetic storage devices [3, 4]. In these applications, loading and unloading of the contacting surfaces occur in a repetitive, cyclic manner. Cyclic loading of contacts can generate wear particles which can significantly shorten the lifetime of the devices.
Several numerical models can be found in the literature for the contact of a smooth sphere and a rigid flat (see references [6, 7, 8, 9]). These models, which can represent a single asperity of a rough surface, are used in statistical descriptions of spherical contact of rough surfaces with either Gaussian distribution [10] or other statistical distribution of asperity heights [11, 12].
A number of experimental studies of a single spherical contact also exist in the literature [13, 14, 15, 16, 17, 18]. Parker and Hatch [13] investigated hemispherical specimens of indium and lead pressed against a glass flat under elastic–plastic or purely plastic deformation. Chaudhri and Yoffe [14] investigated the contact area of steel and WC balls pressed against fused silica, soda-lime glass and sapphire flats in purely elastic deformation. The measured contact radii were compared with the Hertz theory and good agreement was obtained for a steel ball in contact with a glass flat. Ovcharenko et al. [15] investigated the contact of copper and stainless steel spheres of different diameters that were pressed against a sapphire flat during loading, unloading, and cyclic load–unload in the elastic–plastic regime of deformations. An in situ, optical technique was used to observe the evolution of the contact area. Good agreement was found between experimental [15] and theoretical [6, 8, 9] results for the contact area and mean contact pressure.
Several experimental investigations have been conducted that were specifically aimed at studying contact of rough spherical surfaces [19] or flattening of the asperities of rough surfaces [20]. Jamari and Schipper [19] performed an experimental investigation into the deformation behavior of a smooth sphere and a real rough surface. Their experimental results were in good agreement with theoretical prediction. They also observed that the total deformation consisted of asperity and bulk contributions. Lo et al. [20] conducted a series of experiments using aluminum strips to investigate the evolvement of surface roughness and the resistance of surface roughness to flattening in forming processes.
Several theoretical models have been published concerning unloading and cyclic load–unload of a deformable single sphere or a rough spherical contact [21, 22, 23, 24, 25]. Only two published experimental investigations [15, 26] seem to have been performed concerning the contact area during unloading and cyclic load–unload. Qian et al. [26] used a nano-indenter to perform cyclic load–unload tests (which they termed radial nano-fretting) of a Berkovich diamond tip against typical structural materials used in MEMS technology. Their experimental results exhibit hysteretic behavior of force versus displacement curve. However, the residual deformation decreases quickly to zero, and both the contact stiffness and the projected area of the indents approach a constant value, after the first cycle.
As can be seen from the above literature review, only a few experimental studies exist in the literature on surface roughness effects of rough spherical contacts. Furthermore, no experimental studies are present for asperity flattening during multiple loading–unloading cycles of rough spherical contact. It is, therefore, the main goal of the present investigation to focus on this missing information by studying experimentally the effect of surface roughness on asperity flattening during cyclic loading of a rough spherical contact. Such studies could be useful, for example, in understanding the generation of wear particles in the dimple/gimbal interface of a hard disk drive, or other similar applications.
2 Experimental Details
Mechanical properties of typical “rough” and “laser-polished” dimples, and of the sapphire flat
Specimen | Material | H [GPa] | Y [GPa] | ν | E [GPa] | P_{c} [mN] | ω_{c} [nm] |
---|---|---|---|---|---|---|---|
Rough dimple | Stainless steel 304 | 4.31 | 1.68 | 0.31^{a} | 181 | 212 | 186 |
Laser-polished dimple | Stainless steel 304 | 3.7 | 1.42 | 0.31^{a} | 181 | 128 | 133 |
Rigid flat | Sapphire | 19^{a} | 2.95^{a} | 0.27^{a} | 435^{a} | – | – |
All experiments were carried out at room temperature of 20–25 °C and relative humidity of 40–60%. Each experiment was performed on a new dimple. Both the dimple and the sapphire surfaces were cleaned by acetone prior to testing using an Ultrasonic cleaner (Cole-Parmer, USA). All experiments were performed under dry condition. Each load–unload cycle was completed in 10 s (5 s for loading and 5 s for unloading).
The surface roughness of each tested dimple was measured both before and after the test to identify asperity flattening.
3 Experimental Results
4 Comparison Between Experimental and Theoretical Results
The values of ω _{c} for the rough and laser-polished dimples are given in Table 1.
A dimensionless transition load \( P_{\text{t}}^{*} \) was found in reference [31] as a function of \( \omega_{\text{c}}^{*} \) and ψ. At this transition load, the displacement \( h_{1}^{*} \) of the asperities equals the displacement \( \omega^{*} \)of the bulk of the sphere. If the value of P ^{*} is much smaller than \( P_{\text{t}}^{*} \), the contribution of the asperities to the total displacement is dominant. On the other hand, if P ^{*} is much larger than \( P_{\text{t}}^{*} \) roughness effect is negligible.
To compare the present experimental measurements with the theoretical results of reference [31], the roughness parameters ρ, σ, and σ _{s} were calculated from the surface roughness measurements of the dimples using the procedure described in [32]. It should be noted here that the measured roughness parameters and, hence, the plasticity index ψ depend on the scan size and the sampling interval of the measurement [33]. For this reason, various scan sizes ranging from 2.5 × 2.5 to 50 × 50 μm and various sampling intervals from 0.039 to 1.17 μm were used. The resulting calculated plasticity indices before testing varied from 3.2 to 30 for the rough dimple and from 0.4 to 12 for the laser-polished dimple. We observed that the best correlation with the theoretical prediction, for both the rough and the laser-polished dimple corresponds to ψ values that resulted from a scan size of 10 × 10 μm and a sampling interval of 0.69 μm. The measured roughness parameters for the rough dimple under this scanning condition were: ρ = 8.30 μm, and σ ≈ σ _{s} = 90 nm. These roughness parameters result in: ψ = 3, \( \omega_{\text{c}}^{*} \) = 2.07, and \( P_{\text{t}}^{*} \) = 3.03. For the laser-polished dimple, the corresponding values were: ρ = 12.50 μm, σ ≈ σ _{s} = 30 nm, ψ = 2.5, \( \omega_{\text{c}}^{*} \) = 4.43, and \( P_{\text{t}}^{*} \) = 0.25.
5 Conclusion
The effect of asperity flattening on the displacement of a rough spherical contact during repetitive load-cycling was investigated experimentally. Tests were performed with dimples of a hard disk drive suspension that were pressed against a sapphire flat using a modified nano-indenter test rig. Two different types of dimples, a rough dimple and a smoother laser-polished dimple were studied.
Both the dimple types exhibit substantial hysteresis of the load–displacement curve during the first load–unload cycles and a tendency toward reduced plastic deformation with subsequent load–unload cycles (elastic shakedown). The values of the maximum displacement, the residual displacement, and the energy dissipation of the smoother laser-polished dimple were smaller than their corresponding values for a rough dimple. Surface roughness measurements, before and after tests, showed substantial flattening of contacting asperities. This flattening, which results in a smoother contact area seems to be the main cause for the observed tendency to elastic shakedown.
A good correlation between the experimental results and the theoretical prediction of a recent model for rough spherical contact was found. The model clearly shows that for rougher surfaces the total displacement is mainly contributed by the contacting asperities while for smoother surfaces the main contribution comes from the bulk displacement.
Notes
Acknowledgments
We would like to thank Mr. Edward B. Fanslau, and Hanya-san of NHK International Corporation for their interest and support of this research. We would also like to thank Mr. J. McGuire and M. Berg from Hysitron Inc. for providing the plastic holders for the nano-indenter test rig, and help with providing photographs of the test equipment. L. Li thanks the China Scholarship Council (CSC) and Prof. G. Zhang from Harbin Institute of Technology, for supporting his Ph.D. studies at UCSD.
Open Access
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