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Research Papers: Contact Mechanics

# A Simplified Version of Persson's Multiscale Theory for Rubber Friction Due to Viscoelastic Losses

[+] Author and Article Information
M. Ciavarella

Center of Excellence in
Computational Mechanics,
Politecnico di BARI,
Viale Gentile 182,
Bari 70126, Italy
e-mail: Mciava@poliba.it

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received February 1, 2017; final manuscript received April 27, 2017; published online August 2, 2017. Assoc. Editor: James R. Barber.

J. Tribol 140(1), 011403 (Aug 02, 2017) (6 pages) Paper No: TRIB-17-1042; doi: 10.1115/1.4036917 History: Received February 01, 2017; Revised April 27, 2017

## Abstract

We show that the full multiscale Persson's theory for rubber friction due to viscoelastic losses can be approximated extremely closely to simpler models, like that suggested by Persson in 1998 and similarly by Popov in his 2010 book (but notice that we do not make any use of the so-called “Method of Dimensionality Reduction” (MDR)), so it is essentially a single scale model at the so-called large wavevector cutoff. The dependence on the entire spectrum of roughness is therefore only confusing, at least for range of fractal dimensions of interest $D≃2.2$, and we confirm this with actual exact calculations and reference to recent data of Lorenz et al. Moreover, we discuss the critical assumption of the choice of the “free parameter” best fit truncation cutoff.

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## References

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## Figures

Fig. 1

Adapted from Persson's [8] Fig. 3 in turn using Grosch [10] results on silicon carbide paper (solid curve) and a smooth glass surface (dashed curve)

Fig. 2

Real (solid blue) and imaginary (solid black) parts of the viscoelastic modulus in Ref. [6] rubber compound C, together with power-law approximations (dashed lines) at low frequencies. ImE=100.075f0.07 and ReE=101.075f0.05.

Fig. 3

A comparison of our approximate solution (12)R1appr/π≃ (ImE(q1v))/|E(q1v)| (solid lines) for the viscoelastic moduli ratio (10)R1/π (dashed lines). The data are for decreasing velocities going from left to right (black, blue, and red line), v=1,0.1,0.01 m/s.

Fig. 4

Increase of friction coefficient with cutoff wavevector q1 in Ref. [6] (red dots), against our approximation (blue solid line) (16)

Fig. 5

Increase of friction coefficient with speed with different choices of wavevector q1. The black line is the “best fit” with pure hysteresis loss contribution, which corresponds to a cutoff of hrms′≃3.5, whereas the solid blue line is the “theoretical limit” at atomic scale suggested by Lorenz et al. [6].

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