# A Discussion on Present Theories of Rubber Friction, with Particular Reference to Different Possible Choices of Arbitrary Roughness Cutoff Parameters

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## Abstract

**:**

## 1. Introduction

_{max}= 2πv/λ

_{min}is introduced into the integral of their Equation (35), where λ

_{min}is the lower cutoff length of the excitation spectra and v is the velocity. They showed that the results strongly depend on λ

_{min}as v

_{2}in their Equation (36) does depend on λ

_{min}. On the other hand, differently from Persson’s theory, they use the elastic contact model of the Greenwood and Williamson (GW) asperity model for the contact mechanics, which has later found to have some limitations and to be inaccurate, especially for broad spectra of roughness. A profound analysis of their theory is not simple: GW seems to lead to reasonable predictions for the mean penetration depth, which appears to depend only on macroscopic wavelengths, as does Persson’s theory much later [6] (see also [7]). For the cutoff λ

_{min}, they introduce an energy balance [3] in the presence of adhesion, which turns out to be relevant only in the normal force regime, i.e., when the external load is dominant for the formation of elastic contacts, giving a cutoff λ

_{min}very sensible to the surface fractal dimension, especially in the limit of D ≅ 2, which in turns is the most common case for real surfaces. Contrary to Persson [4], Klüppel and Heinrich [3] suggest that adhesion-induced hysteretic losses may play a role only for extremely smooth surfaces, such as glasses with D = 2, but they do not seem to be relevant on rough surfaces such as road tracks with typical D = 2.2. Klüppel and Heinrich [3] try to apply the theory to the original cases of Grosch, namely friction on the silicon carbide paper and the glass surface, by changing the broadness of the spectrum, although only very qualitatively, and not really explaining the appearance of the two maxima in the case of Grosch data on silicon carbide: choices of cutoffs are not immediately clear to the reader.

_{1}= 2π/λ

_{min}: it is defined where the rms slope reaches

_{rms}(q

_{1}) = 1.3

_{s0}is a static shear stress for very low velocities and v

_{c}is a certain critical velocity to a plateau. There are explicit expressions for τ

_{s0}based on contact angle measurements; obviously E

_{∞}/E

_{0}is the ratio of dynamic modulus in a glassy and rubbery state, n can be estimated from the power law behavior of the relaxation time spectra H(τ) in the glass transition range. Notice that this term is then multiplied by the real contact area, and hence the dependence on roughness spectrum and its truncation is very important. Even the viscoelastic properties, which in Equation (2) are limited to two elastic moduli, with no reference to relaxation spectra, could have more effect on the contact area. The agreement with an extensive set of measurements is quite good over an extensive range of velocities, except perhaps at very low ones of the order of 10

^{−4}m/s, where the discrepancy is significant.

_{0}is the relative contact area, p

_{0}is the nominal contact pressure and v is the sliding velocity. This adhesive contribution on one hand, is also very simple in form, and basically as semi-empirical as is Klüppel’s one, but on the other, is symmetrical contrary to the Lang & Klüppel [12]. It maintains the complications (or the effects) of the full-contact mechanics theory with roughness, since the contact area A/A

_{0}depends in principle on the full viscoelastic spectrum, and on the full roughness spectrum. However, in the end, the choices of the factors τ

_{f0}, c, and v* are not from independent experiments such as contact angle measurements in Lang & Klüppel [12].

## 2. Experimental Data

#### 2.1. Surface PSD

_{rms}, the RMS slope h′

_{rms}, and the RMS curvature h″

_{rms}[16]. Referring to the Persson formulation, Figure 1 shows the 2D surface roughness power spectrum of the concrete surface. The significant range of interest of this self-affine fractal power spectrum can be assumed to be a power law:

_{0}= 0.001152 m^

^{(2−2H)}. The wavenumber q

_{0}considered in this study is q

_{0}= 10

^{2.7}[1/m] but fortunately the choice of this truncation is not very relevant for friction estimation, while the choice of the large wavenumber cutoff is extremely more sensible and arbitrary, as debated in the next section.

#### 2.2. Material Viscoelastic Properties

_{f}≅ 0.1, where S

_{f}is the strain softening factor [15].

#### 2.3. Friction

_{const}= 0.2 (an empirical choice, for the lack of better data), which the authors refer to the “scratching of the concrete surface by the hard filler particles”.

## 3. Discussion on the Viscoelastic and Adhesive Contributions

_{rms}is the rms slope of the surface that depends on the cutoff wavenumber q

_{1}and E(q

_{1}v) is the complex viscoelastic modulus of the rubber (for a given temperature), at the circular frequency 2πf = q

_{1}v. Adopting the power law approximation of the moduli at low frequencies, the coefficient K is calculated as:

_{const}= 0.2. The parameters are the same as those used for the calculation with the Persson theory and reported in Table 2.

- -
- the choice of the cutoff wavenumber q
_{1}influences strongly both viscoelastic and adhesive contributions; - -
- the term μ
_{const}= 0.2 which is attributed to scratching of the concrete surface by the hard filler particles is quite arbitrary; - -
- The reference velocity v* and the τ
_{f0}significantly influence the adhesive curve; - -
- The assumption that for sliding friction on the rough surfaces the deformation is ε ≅ 1 implies a reduction of the strain modulus E(ω) of a strain factor S
_{f}≅ 0.1. Due to the non-linear effects related to the viscoelastic modulus of the rubber, lower strain values would cause sensible variation of the S_{f}and therefore of the adhesive friction.

_{1}of several orders of magnitude obtaining almost equally good fits for the friction coefficient. This poses fundamental questions on the physical meaning of this quantity and on the role that it plays in modern theories of viscoelastic friction.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Real (solid blue) and imaginary (solid red) parts of the viscoelastic modulus of rubber Compound A (

**a**), B (

**b**), C (

**c**) in TP [14]. The dashed lines indicate the power law approximations at low frequencies.

**Figure 3.**The measured (symbols, from [14]) and calculated (lines) friction coefficient using the Persson formulation for Compound A (

**a**), B (

**b**), C (

**c**).

**Figure 4.**The measured (symbols, from [14]) and calculated (lines) friction coefficient on concrete as a function of the logarithm of the sliding speed, using the simplified formulation for the viscoelastic contribution, for Compound A (

**a**), B (

**b**), C (

**c**).

**Figure 5.**The measured (symbols, form [14]) and calculated (lines) friction coefficient on concrete as a function of the logarithm of the sliding speed considered five different sets of arbitrary parameters for each Compound A (

**a**), B (

**b**), C (

**c**).

Compound | α_{r} | β_{r} | α_{i} | β_{i} |
---|---|---|---|---|

Compound A | 1.4193 | 0.0820 | 0.5375 | 0.0939 |

Compound B | 1.3262 | 0.0501 | 0.2312 | 0.0507 |

Compound C | 1.4140 | 0.0639 | 0.4713 | 0.0737 |

Compound | q_{1} | h′_{rms} | μ_{const} | log_{10} v* (m/s) | τ_{f0} (MPa) | S_{f} |
---|---|---|---|---|---|---|

Compound A | 2·10^{6} | 1.3 | 0.2 | −2.47 | 3.6 | 0.1 |

Compound B | 2·10^{6} | 1.3 | 0.2 | −1.97 | 4.0 | 0.1 |

Compound C | 2·10^{6} | 1.3 | 0.2 | −2.53 | 4.1 | 0.1 |

**Table 3.**Summary of the different parameter sets adopted for the friction calculation of the Compound A.

SET | q_{1} | h′_{rms} | μ_{const} | log_{10} v* (m/s) | τ_{f0} (MPa) | S_{f} |
---|---|---|---|---|---|---|

A | 2·10^{9} | 3.6 | 0 | −1.5 | 8 | 0.1 |

B | 2·10^{6} | 1.3 | 0.3 | −2.2 | 4.1 | 0.1 |

C | 2·10^{7} | 1.85 | 0.3 | −1.9 | 7 | 0.1 |

D | 2·10^{5} | 0.9 | 0.35 | −2.1 | 7 | 0.3 |

E | 3·10^{4} | 0.63 | 0.35 | −2.1 | 7 | 0.5 |

**Table 4.**Summary of the different parameter sets adopted for the friction calculation of the Compound B.

SET | q_{1} | h′_{rms} | μ_{const} | log_{10} v* (m/s) | τ_{f0} (MPa) | S_{f} |
---|---|---|---|---|---|---|

A | 2·10^{10} | 4.5 | 0 | −1 | 4.0 | 0.1 |

B | 2·10^{6} | 1.3 | 0.3 | −1.80 | 8 | 0.19 |

C | 2·10^{5} | 0.9 | 0.3 | −1.8 | 8 | 0.3 |

D | 3·10^{4} | 0.63 | 0.35 | −1.8 | 8 | 0.5 |

E | 10^{7} | 1.66 | 0.25 | −2 | 5.5 | 0.1 |

**Table 5.**Summary of the different parameter sets adopted for the friction calculation of the Compound C.

SET | q_{1} | h′_{rms} | μ_{const} | log_{10} v* (m/s) | τ_{f0} (MPa) | S_{f} |
---|---|---|---|---|---|---|

A | 2·10^{6} | 1.3 | 0.25 | −2.0 | 5.25 | 0.1 |

B | 10^{7} | 1.66 | 0.25 | −2 | 6.9 | 0.1 |

C | 3·10^{7} | 1.96 | 0.25 | −2 | 8 | 0.1 |

D | 10^{5} | 0.8 | 0.3 | −2.1 | 7.1 | 0.3 |

E | 10^{4} | 0.5 | 0.3 | −2.1 | 6.4 | 0.5 |

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**MDPI and ACS Style**

Genovese, A.; Farroni, F.; Papangelo, A.; Ciavarella, M.
A Discussion on Present Theories of Rubber Friction, with Particular Reference to Different Possible Choices of Arbitrary Roughness Cutoff Parameters. *Lubricants* **2019**, *7*, 85.
https://doi.org/10.3390/lubricants7100085

**AMA Style**

Genovese A, Farroni F, Papangelo A, Ciavarella M.
A Discussion on Present Theories of Rubber Friction, with Particular Reference to Different Possible Choices of Arbitrary Roughness Cutoff Parameters. *Lubricants*. 2019; 7(10):85.
https://doi.org/10.3390/lubricants7100085

**Chicago/Turabian Style**

Genovese, Andrea, Flavio Farroni, Antonio Papangelo, and Michele Ciavarella.
2019. "A Discussion on Present Theories of Rubber Friction, with Particular Reference to Different Possible Choices of Arbitrary Roughness Cutoff Parameters" *Lubricants* 7, no. 10: 85.
https://doi.org/10.3390/lubricants7100085