# Contact-Patch-Size Distribution and Limits of Self-Affinity in Contacts between Randomly Rough Surfaces

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model, Methods, and Theory

#### 2.1. Model

#### 2.2. Methods and Observables

#### 2.3. Theory

#### 2.3.1. Rough Estimate of ${A}_{\mathrm{min}}$

#### 2.3.2. Rough Estimate of ${A}_{\mathrm{max}}$

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The dimensionless coefficient $\kappa $ as a function of the discretization obtained with and without continuum corrections for $H=0.3$ and $H=0.8$. A value of $c=1/8$ was used in Equation (6) for the corrected contact area. $\kappa \equiv {a}_{\mathrm{r}}/{p}_{0}^{*}$, where ${p}_{0}^{*}\equiv {p}_{0}\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}{E}^{*}\overline{g}$ is the reduced pressure. System specification: $\mathcal{L}/{\lambda}_{\mathrm{r}}=1$, ${\lambda}_{\mathrm{r}}/{\lambda}_{\mathrm{s}}=256$, and ${p}^{*}=0.04$.

**Figure 2.**Characteristic contact area with and without continuum corrections as a function of ${\epsilon}_{\mathrm{c}}=a/{\lambda}_{\mathrm{s}}$ for $H=0.3$ (

**top**) and $H=0.8$ (

**bottom**). System specifications: $\mathcal{L}/{\lambda}_{\mathrm{r}}=1$, ${\lambda}_{\mathrm{r}}/{\lambda}_{\mathrm{s}}=512$, and ${p}_{0}^{*}=0.04$.

**Figure 3.**(

**Top left**,

**top right**,

**bottom right**) Visualization of contact stresses in a randomly rough contact at different magnifications. Stresses in the color bar are given in units of ${E}^{*}\overline{g}$; and (

**Lower left**) stress autocorrelation function. System specification: $\mathcal{L}/{\lambda}_{\mathrm{r}}=8$, ${\lambda}_{\mathrm{r}}/{\lambda}_{\mathrm{s}}=250$, and ${p}_{0}=0.01\phantom{\rule{3.33333pt}{0ex}}{E}^{*}/\overline{g}$, leading to a relative contact area of ${a}_{\mathrm{r}}\approx 0.02$.

**Figure 4.**Individual patch size ${A}_{\mathrm{n}}$ versus load carried by each patch ${L}_{n}$ (dots): (

**Left**) $H=0.3$; and (

**Right**) $H=0.8$. Stripes in the scatter plot at small A result from the spatial discretization. The open circles indicate running averages of the numerical data. Solid lines represents the theoretical prediction. System specifications: $\mathcal{L}/{\lambda}_{\mathrm{r}}=4$, ${\lambda}_{\mathrm{r}}/{\lambda}_{\mathrm{s}}=128$, and ${p}^{*}=0.04$. ${A}_{\mathrm{min}}$, which is used to undimensionalize the contact area, is determined from Equation (11) and ${L}_{\mathrm{min}}$ estimated as $2{A}_{\mathrm{min}}{E}^{*}\overline{g}$.

**Figure 5.**Distribution of contact-patch sizes $n\left(A\right)$ divided by the scaling law $n\left(A\right)\propto {A}^{2-H/2}$ at a reduced pressure of ${p}_{0}^{*}=0.04$: (

**Left**) $H=0.3$; and (

**Right**) $H=0.8$. System specification: $\mathcal{L}/{\lambda}_{\mathrm{r}}=4$ and ${\lambda}_{\mathrm{r}}/{\lambda}_{\mathrm{s}}=512$.

**Figure 6.**Characteristic contact-patch size ${A}_{\mathrm{c}}$ as a function of load. Closed symbols indicate the cluster size of percolated patches. Broken lines are based on fits of the form ${A}_{\mathrm{c}}\propto 1/{({a}_{\mathrm{cp}}-{a}_{\mathrm{r}})}^{\gamma}$. The exponents turned out to lie within $\gamma =2.2\pm 0.2$, while ${a}_{\mathrm{cp}}(H=0.8)=0.42$ and ${a}_{\mathrm{cp}}(H=0.3)=0.39$ were used. Solid lines represent fits to the small-contact-area domain. For the $H=0.3$ system, an exponential law of the form ${A}_{\mathrm{p}}\propto exp({a}_{\mathrm{r}}/{a}_{1})$ appeared best, while $H=0.8$ data were best represented at small ${a}_{\mathrm{cp}}$ with a power law ${A}_{\mathrm{cp}}\propto {a}_{r}^{\beta}$ with $\beta =0.55$. System specification: $L/{\lambda}_{\mathrm{r}}=4$ and ${\lambda}_{\mathrm{r}}/{\lambda}_{\mathrm{s}}=512$.

**Figure 7.**Characteristic contact-patch size as a function of the ratio ${\epsilon}_{\mathrm{f}}={\lambda}_{\mathrm{s}}/{\lambda}_{\mathrm{r}}$ at a fixed load: for a $H=0.3$ surface (

**top**); and for a $H=0.8$ surface (

**bottom**). The dashed lines are drawn to guide the eye. For the $H=0.8$, the dashed line reflects an ${\epsilon}_{\mathrm{f}}^{-1.5}$ power law. Note that the ordinate is linear for $H=0.3$ but logarithmic for $H=0.8$. System specification: $\mathcal{L}/{\lambda}_{\mathrm{r}}=1$ and ${p}^{*}=0.04$.

**Figure 8.**Stress ACF as a function of distance r for various values of the ratio ${\epsilon}_{\mathrm{f}}={\lambda}_{\mathrm{s}}/{\lambda}_{\mathrm{r}}$ at a fixed load for $H=0.3$ (

**left**) and $H=0.8$ (

**right**). System specifications: $\mathcal{L}/{\lambda}_{\mathrm{r}}=1$ and ${p}^{*}=0.04$.

**Figure 9.**Value of the stress ACF, ${G}_{\sigma \sigma}\left({r}_{\mathrm{c}}\right)$, at the characteristic contact radius ${r}_{\mathrm{c}}\equiv \sqrt{{A}_{\mathrm{c}}/\pi}$ for various values of ${\epsilon}_{\mathrm{f}}$ for $H=0.3$ (

**top**) and $H=0.8$ (

**bottom**). System specification: $\mathcal{L}/{\lambda}_{\mathrm{r}}=1$ and ${p}^{*}=0.04$.

**Figure 10.**Stress ACF as a function of distance r at selected values of the reduced pressure ${p}^{*}=p/{E}^{*}\overline{g}$ for: $H=0.3$ (

**left**); and $H=0.8$ (

**right**). System specification in both cases: $\mathcal{L}/{\lambda}_{\mathrm{r}}=4$ and ${\lambda}_{\mathrm{r}}/{\lambda}_{\mathrm{s}}=512$.

**Figure 11.**Value of the stress ACF evaluated at the characteristic patch radius ${r}_{\mathrm{c}}\equiv \sqrt{{A}_{\mathrm{c}}/\pi}$ as a function of reduced pressure for two different Hurst exponents. Open and closed symbols refer to data representing non-percolated and percolated contacts, respectively. System specification: $\mathcal{L}/{\lambda}_{\mathrm{r}}=4$ and ${\lambda}_{\mathrm{r}}/{\lambda}_{\mathrm{s}}=512$.

$\beta $ | exponent in the ${A}_{\mathrm{c}}\propto {a}_{\mathrm{r}}^{\beta}$ relation, valid at low-pressures for $H>0.5$ |

$\gamma $ | exponent in the ${A}_{\mathrm{c}}\propto 1/{({a}_{\mathrm{cp}}-{a}_{\mathrm{r}})}^{\gamma}$ relation, valid for large systems just below ${p}_{\mathrm{cp}}$ |

${\epsilon}_{\mathrm{c}},\phantom{\rule{0.277778em}{0ex}}{\epsilon}_{\mathrm{f}}$ | ${\epsilon}_{\mathrm{c}}=a/{\lambda}_{\mathrm{s}}$, ${\epsilon}_{\mathrm{f}}={\lambda}_{\mathrm{s}}/{\lambda}_{\mathrm{r}}$ |

$\kappa $ | dimensionless proportionality coefficient ${a}_{\mathrm{r}}/{p}^{*}$ |

${\lambda}_{\mathrm{r}},{\lambda}_{\mathrm{s}}$ | roll-off wavelength and short-wavelength cutoff |

$\sigma $ | stress |

$\tau $ | exponent in the $n\left(A\right)\propto {A}^{\tau}$ relation |

A | area of an individual contact patch |

${A}_{\mathrm{c}}$ | characteristic contact-patch area |

${A}_{\mathrm{min}}$ | crossover area from Hertz to self-affine scaling |

ACF | autocorrelation function |

$C\left(q\right)$ | height spectrum |

${E}^{*}$ | contact modulus |

${G}_{\sigma \sigma}\left(r\right)$ | stress ACF |

GFMD | Green’s function molecular dynamics |

H | Hurst exponent |

L | load or normal force |

a | discretization length used in the simulation |

${a}_{\mathrm{cp}}$ | relative contact area at percolation threshold |

${a}_{\mathrm{r}}$ | relative contact area |

$\overline{g}$ | rms height gradient |

$\overline{h}$ | rms height |

$h\left(\mathbf{r}\right),\tilde{h}\left(\mathbf{q}\right)$ | height in real-space and Fourier representation |

$n\left(A\right)$ | number density of contact-patch areas |

${p}_{0}$ | nominal contact pressure |

${p}^{*}$ | dimensionless contact pressure ${p}_{0}/{E}^{*}\overline{g}$ |

${p}_{\mathrm{cp}}$ | pressure at contact-percolation transition |

$\mathbf{q},q$ | wave vector and its magnitude |

${q}_{\mathrm{r}},{q}_{\mathrm{s}}$ | ${q}_{\mathrm{r}}=2\pi /{\lambda}_{\mathrm{r}}$, ${q}_{\mathrm{s}}=2\pi /{\lambda}_{\mathrm{s}}$ |

$\mathbf{r},r$ | in-plane vector and its magnitude |

${r}_{\mathrm{c}}$ | characteristic patch radius $\sqrt{{A}_{\mathrm{c}}/\pi}$ |

rms | root-mean square |

$u\left(\mathbf{r}\right),\tilde{u}\left(\mathbf{q}\right)$ | displacement in real-space and Fourier representation |

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**MDPI and ACS Style**

Müser, M.H.; Wang, A. Contact-Patch-Size Distribution and Limits of Self-Affinity in Contacts between Randomly Rough Surfaces. *Lubricants* **2018**, *6*, 85.
https://doi.org/10.3390/lubricants6040085

**AMA Style**

Müser MH, Wang A. Contact-Patch-Size Distribution and Limits of Self-Affinity in Contacts between Randomly Rough Surfaces. *Lubricants*. 2018; 6(4):85.
https://doi.org/10.3390/lubricants6040085

**Chicago/Turabian Style**

Müser, Martin H., and Anle Wang. 2018. "Contact-Patch-Size Distribution and Limits of Self-Affinity in Contacts between Randomly Rough Surfaces" *Lubricants* 6, no. 4: 85.
https://doi.org/10.3390/lubricants6040085