Extending the analytical elastoplastic contact model of longitudinal-vibration friction reduction to real engineering surfaces (asperity interactions, measured height distributions, high-load accuracy, and experimental validation)

Liu, Ni, Wang and Zhao (Tribology International 146:106237, 2020) derived closed-form analytical predictions of the friction-reduction performance of a longitudinal vibration superimposed on sliding, using a Greenwood-Williamson-type statistical contact model whose asperities deform elastoplastically according to the Zhao-Maietta-Chang transition. The model rests on the standard idealizations of this framework: identical hemispherical asperities, neglected asperity interactions, a fitted modified exponential height distribution, and an interpolated elastic-to-plastic transition whose deviation from numerical reference results grows at high normal loads; validation was against numerical benchmarks and published Dahl-model results rather than broad experiments. The open problem is to build and experimentally validate a predictive model of longitudinal-vibration friction reduction for real engineering surfaces that removes these idealizations — incorporating measured topographies and asperity interactions, remaining accurate in strongly plastic regimes, and delineating its applicability bounds across materials, surface finishes, vibration amplitudes, frequencies, and sliding speeds. The required components exist separately in static contact modeling, but their coupling into the vibration-friction problem remains undone.

Background

When two nominally flat engineering surfaces are pressed together, they touch only where their highest microscopic peaks — called asperities — meet, so the real contact area is a small, load-dependent fraction of the nominal area. The classic Greenwood-Williamson (GW) model (1966) treats all asperity summits as identical hemispheres whose heights follow a Gaussian distribution and sums the individual elastic (Hertzian) contacts statistically, yielding closed-form relations among load, real contact area, and surface separation. Real contacts, however, quickly exceed the elastic limit of the asperity tips. The Zhao-Maietta-Chang (ZMC) model (2000) extends GW to elastoplasticity: below a critical interference $\omega_1$ (the geometric overlap at which an asperity tip first yields) the contact is elastic, above roughly $\omega_2 \approx 54\omega_1$ it is fully plastic with hardness-limited pressure, and in between an interpolation function $f(\omega)$ blends the two regimes. You and Chen (2011) added tangential (frictional) contact parameters to this statistical elastoplastic picture, and finite-element studies such as Jackson and Green (2005) later mapped the single-asperity elastoplastic transition more accurately than the ZMC interpolation.

Since the 1960s it has been known that superimposing high-frequency vibration on sliding reduces the average friction force. The effect is exploited in piezoelectric ultrasonic motors (Wallaschek 1998), ultrasonic-vibration-assisted machining and metal forming, and drill-string axial-oscillation friction-reduction tools (Gutowski and Leus 2012). The standard dynamic-friction description in this context is Dahl's model (1976), in which friction force depends on tangential displacement rather than speed, producing a pre-sliding hysteresis loop.

Liu, Ni, Wang and Zhao (2020) combined these threads into an analytical model predicting the friction-reduction performance of a longitudinal (in-plane, along the sliding direction) vibration of amplitude $V_d$ and frequency $f$ superimposed on a mean sliding speed $U_a$: a GW-type statistical contact whose asperities deform elastoplastically per the ZMC/You-Chen relations, with the tangential friction force coupled to the asperity tangential deformation through the vibration cycle. They derived closed-form solutions for the instantaneous and cycle-averaged friction force and the friction-reduction ratio (the relative change of the cycle-averaged friction force caused by the superimposed vibration), showed that the reduction stems mainly from the periodic reversal of the asperity tangential deformation, and validated the model against the ZMC numerical model and published Dahl-model results. The model, however, rests on the standard idealizations of the GW/ZMC framework: (i) all asperity summits are identical hemispheres; (ii) asperity interactions — the elastic coupling of neighboring asperities through the deforming bulk material between them — are neglected; and (iii) asperity heights follow a fitted "modified exponential" distribution rather than a measured topography. The elastoplastic transition uses the ZMC interpolation with $\omega_2 \approx 54\omega_1$, whose deviation from accurate numerical (finite-element) reference results grows at large interference, i.e. in strongly plastic regimes at high normal loads. Validation was numerical and against published friction-model results, not a broad experimental campaign across materials, topographies, vibration amplitudes $V_d$, frequencies $f$, or sliding speeds $U_a$.

Problem Statement

The problem, as left open by Liu et al. (2020), is to replace their idealized analytical model of longitudinal-vibration friction reduction with one that is predictive for real engineering surfaces and to establish its validity experimentally. The common objective is a model that predicts measured friction-reduction performance from independently measured surface topography and material properties, rather than from calibration. Concretely, the open objective is to construct an analytical or semi-analytical model of the friction-reduction performance of longitudinal vibration on rough, elastoplastically contacting surfaces that: (a) takes asperity geometry and height statistics from measured surface topographies — non-hemispherical asperity geometry and measured height distributions in place of the fitted modified exponential distribution — and includes asperity interactions; (b) represents the elastic-to-fully-plastic transition accurately enough that its predictions no longer drift from numerical (ZMC / finite-element) reference results as the normal load enters strongly plastic regimes; and (c) is validated by comparison with experimental friction-reduction measurements across a range of material pairs, surface topographies, vibration amplitudes $V_d$, frequencies $f$, and sliding speeds $U_a$, so that the model's applicability bounds — and how its height-distribution parameters must be calibrated or adapted for diverse real surfaces — are delineated. The three parts are jointly required; the material, topographic and operating parameters listed in (c) define the range the validation must span rather than an exhaustive set of combinations.

References

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