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# Frictional energy dissipation in contact of nominally flat rough surfaces under harmonically varying loads

**Frictional energy dissipation in contact of nominally flat rough surfaces under harmonically varying loads** Original Research Article

*Journal of the Mechanics and Physics of Solids, , Available online 1 October 2011, *

C. Putignano, M. Ciavarella, J.R. Barber View Abstract

If the nominal contact tractions at an interface are everywhere below the Coulomb friction limit throughout a cycle of oscillatory loading, the introduction of surface roughness will generally cause local microslip between the contacting asperities and hence some frictional dissipation. This dissipation is important both as a source of structural damping and as an indicator of potential fretting damage. Here we use a strategy based on the Ciavarella-Jäger superposition and a recent solution of the general problem of the contact of two half spaces under oscillatory loading to derive expressions for the dissipation per cycle which depend only on the normal incremental stiffness of the contact, the external forces and the local coefficient of friction. The results show that the dissipation depends significantly on the relative phase between the oscillations in normal and tangential load—a factor which has been largely ignored in previous investigations. In particular, for given load amplitudes, the dissipation is significantly larger when the loads are out of phase. We also establish that for small amplitudes the dissipation varies with the cube of the load amplitude and is linearly proportional to the second derivative of the elastic compliance function for all contact geometries, including those involving surface roughness. It follows that experimental observations of less than cubic dependence on load amplitude cannot be explained by reference to roughness alone, or by any other geometric effect in the contact of half spaces.

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## also in a seminar at Caltech by Jim Barber October 18

October 18, 2011, Tuesday at 11:00 a.m.

James R. Barber, University of Michigan, Ann Arbor, Michigan

Roughness, Fractality, Contact and FrictionThe multiscale features of rough surfaces lead to approximately linear relations between macroscale physical quantities, such as normal force, electrical contact conductance, friction force, etc., even when the microscale relations between these quantities are highly non-linear. Surfaces are often found to be quasi-fractal at fine scales, in which case the resulting contact problem is best approached by considering the incremental change in the statistical distribution of (for example) contact pressure when the high frequency cutoff is slightly extended, and then using inductive arguments. The parameters of macroscale phenomena are generally determined by deviations from fractality, or by the occurrence of a length scale in the governing physical laws such as the constitutive equation. Some important quantities, including incremental stiffness and electrical contact resistance, are determined by the coarse scale ‘roll-off’ of the power spectral density, and rigorous bounds can then be established using relatively unsophisticated numerical models. By contrast, although multiscale arguments provide a plausible ‘explanation’ of Coulomb’s law of friction, the friction coefficient is determined at the fine scale

and is therefore difficult to predict.

If two conforming bodies are in nominally static contact but subjected to vibration, ‘microslip’ will occur between some of the contacting asperities, leading to fretting fatigue and hysteretic damping. The energy dissipation per unit nominal area can be related to the tangential contact stiffness, using an extension of a general result for elastic contact problems due to Ciavarella and J¨ager. Since this stiffness is determined at the coarse scale, it proves unecessary to investigate the morphology of the fine scale microslip process. A general sinusoidal loading cycle shows that energy dissipation (and hence potential fretting damage) is significantly higher when the oscillations in normal and tangential forces are out of phase with each other.

## but the enigma remains! Any suggestions?

Although the analytical solution for halfspace contact problem under harmonic loading (normal and tangential), is simplifying considerably the Mindlin solution, the result is also negative.

Indeed, we also found that any halfspace geometry including roughness or any other geometrical effect, and not just Hertzian geometry studied by Johnson in 1961, would show cubic dependence of the energy dissipation on tangential load amplitude (even when normal load is changing), so the "challenge" to justify the non-cubic dependence seen in all experiments (power law is usually between 2 and 3) cannot be justified within this model.

Hence, the enigma originally noted by Johnson in 1961, and later observed by others and never explained, remains.... Room for brigth and energetic young people to think about it!

Johnson,K.L.,1961. Energy dissipation at spherical surfaces in contact transmitting oscillating forces. J. Mech. Eng.Sci.3,362–368.