### 2.1 Generation of Rough Surface Profile Samples

Rough surfaces with Gaussian height distribution and exponential ACF are expressed as

$$\left\{ {\begin{array}{*{20}l} {\phi \left( z \right) = \frac{1}{{\sqrt {2{\uppi }} \sigma }}\exp \left( { - \frac{{\left( {z - \mu } \right)^{2} }}{{2\sigma^{2} }}} \right),} \hfill \\ {R\left( \tau \right) = \sigma^{2} \exp \left[ {\vartheta \cdot \left( {{\tau \mathord{\left/ {\vphantom {\tau \beta }} \right. \kern-\nulldelimiterspace} \beta }} \right)} \right],} \hfill \\ \end{array} } \right.$$

(1)

where *z* is surface height; *μ*, *σ* and \(\phi\) are the mean, the standard deviation and the probability density function (PDF) of [*z*], respectively; *R* is auto-correlation, *τ* is lag length, and *β* is correlation length. It should be noted that [*z*] is actually Gaussian white noise when *β* = 0. *ϑ* is high pass filtering constant, representing high pass filtering with different cut-off lengths. Obviously, correlation decays faster with a larger absolute value of *ϑ*. Whitehouse and Archard [19] defined correlation length as the distance where *R* decayed to 1/10 of the origin value. This corresponds to *ϑ* = − 2.3. Besides, *ϑ* = − 0.844 and *ϑ* = − 1 were used by Hirst and Hollander [20] and Aramaki et al. [21] respectively. These definitions will all be chosen for investigating the effect of *ϑ*.

In order for the generation of [*z*] with given statistical distributions, a pseudorandom number generator is used to generate Gaussian white noise sequence [*ε*] with a specified standard deviation *σ*. To go further, a convolution operation is performed on the white noise sequence and on the filter coefficient [*h*] decided by ACF. That is

$$z\left( i \right) = \sum\limits_{{k = - \frac{T}{2} + 1}}^{{\frac{T}{2} - 1}} {h\left( k \right)} \varepsilon \left( {i + k} \right),$$

(2)

where *T* is ACF truncation length and the filter coefficient *h*(*k*) is

$$h\left( k \right) = \frac{1}{T}\sum\limits_{{\omega = - \frac{T}{2} + 1}}^{{\frac{T}{2} - 1}} {H\left( \omega \right)} \exp ( - jk\omega ).$$

(3)

To determine the frequency response *H*, the Fourier transform of ACF is first calculated:

$$S\left( \omega \right) = \frac{1}{T}\sum\limits_{{k = - \frac{T}{2} + 1}}^{{\frac{T}{2} - 1}} {R\left( k \right)} \exp ( - jk\omega ),\;{\kern 1pt} {\kern 1pt} \omega = {{ - T} \mathord{\left/ {\vphantom {{ - T} 2}} \right. \kern-\nulldelimiterspace} 2} + 1, \cdots ,{T \mathord{\left/ {\vphantom {T 2}} \right. \kern-\nulldelimiterspace} 2} - 1,$$

(4)

where *S*(*ω*) is the power spectral density.

According to the definition of ACF and Eq. (2), it gives

$$H\left( \omega \right) = \sqrt {{{S\left( \omega \right)} \mathord{\left/ {\vphantom {{S\left( \omega \right)} C}} \right. \kern-\nulldelimiterspace} C}} ,$$

(5)

where *C* is a constant, the Fourier transform of the sequence [*ε*]. Thereupon, any desired height sequence [*z*] with prescribed *σ* and *β* can be generated.

The detailed theory and methodology for rough surface simulation based on FFT can be found in Ref. [22]. It was suggested by He et al. [23] that the ratio of ACF truncation length *T* to correlation length *β* should be greater than 6 so that the asperity distribution parameters could converge to their true values.

### 2.2 Modeling of Asperity Peak Distribution

According to three-point definition of asperity peak, any point *z*_{i} from [*z*] becomes a peak when

$$z_{i} > z_{i - 1} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \& {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} z_{i} > z_{i + 1} .$$

(6)

All the heights of *z* satisfying Eq. (6) form the height distribution of asperity peaks [*ξ*]. The mean *ξ*_{avg}, the root mean square (RMS) *σ*_{ξ}, the skewness *ξ*_{sk} and the kurtosis *ξ*_{ku} of asperity height distribution can be calculated according to their definitions, i.e.,

$$\left\{ {\begin{array}{*{20}l} {\xi_{{{\text{avg}}}} = \frac{1}{{n_{{\text{p}}} }}\sum\limits_{i = 1}^{{n_{{\text{p}}} }} {\xi_{i} } ,} \hfill \\ {\sigma_{{\upxi }} = \sqrt {\frac{1}{{n_{{\text{p}}} }}\sum\limits_{i = 1}^{{n_{{\text{p}}} }} {\left( {\xi_{i} - \xi_{{{\text{avg}}}} } \right)^{2} } } ,} \hfill \\ {\xi_{{{\text{sk}}}} = \frac{1}{{n_{{\text{p}}} \sigma_{{\upxi }}^{3} }}\sum\limits_{i = 1}^{{n_{{\text{p}}} }} {\left( {\xi_{i} - \xi_{{{\text{avg}}}} } \right)^{3} ,} } \hfill \\ {\xi_{{{\text{ku}}}} = \frac{1}{{n_{{\text{p}}} \sigma_{{\upxi }}^{4} }}\sum\limits_{i = 1}^{{n_{{\text{p}}} }} {\left( {\xi_{i} - \xi_{{{\text{avg}}}} } \right)^{4} } ,} \hfill \\ \end{array} } \right.$$

(7)

where *n*_{p} is the number of asperity peaks.

The curvature of asperity peak is

$$\kappa = {{\left( { - z_{i + 1} + 2z_{i} - z_{i - 1} } \right)} \mathord{\left/ {\vphantom {{\left( { - z_{i + 1} + 2z_{i} - z_{i - 1} } \right)} {d^{2} }}} \right. \kern-\nulldelimiterspace} {d^{2} }},$$

(8)

where *d* is sampling interval. The curvature radius of asperity peak is

$$r_{{\text{p}}} = {1 \mathord{\left/ {\vphantom {1 \kappa }} \right. \kern-\nulldelimiterspace} \kappa },$$

(9)

The density of asperity peak is

$$\eta = \frac{{n_{{\text{p}}} }}{L} = \frac{{n_{{\text{p}}} }}{N \cdot d},$$

(10)

where *L* is sampling length and *N* is the number of sampling points. Unit sampling interval will be used to calculate *r*_{p} and *η* in the following, and non-dimensional results will be obtained.

In addition to the peak identification scheme based method, a mathematical model of asperity peak distribution was established by Nayak [9] based on central moments of surface power spectral density (PSD) using random process theory, namely spectral moment approach. The assumption that the joint distribution of surface height, slope and curvature was normal was made during the derivation. From the deduced model, the height distribution of asperity peaks is

$$\begin{gathered} \phi_{{{\text{peak}}}} \left( {\xi^{*} } \right) = \frac{\delta }{\sqrt 2 \uppi }\left\{ {\exp \left[ { - \left( {\xi^{*2} /2\delta^{2} } \right)} \right]} \right. + \hfill \\ \left. {\sqrt {\uppi } \chi \exp \left( { - 0.5\xi^{*2} } \right)\left( {1 + {\text{erf}} \chi } \right)} \right\}, \hfill \\ \end{gathered}$$

(11)

where \(\xi^{*} = {\xi \mathord{\left/ {\vphantom {\xi \sigma }} \right. \kern-\nulldelimiterspace} \sigma }\), *α* is band width coefficient and \(\alpha = {{m_{0} m_{4} } \mathord{\left/ {\vphantom {{m_{0} m_{4} } {m_{2}^{2} }}} \right. \kern-\nulldelimiterspace} {m_{2}^{2} }}\), \(\delta = \left[ {{{\left( {\alpha - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\alpha - 1} \right)} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}\), \(\chi = \left( {\frac{1 - \delta }{{2\delta^{2} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \xi^{*}\), and the central moments of order zero to order four of PSD are

$$\begin{gathered} m_{0} = R\left( 0 \right) = \sigma^{2} , \hfill \\ m_{2} = \frac{{{\text{d}}^{2} R\left( 0 \right)}}{{{\text{d}}\tau^{2} }}, \hfill \\ m_{4} = \frac{{{\text{d}}^{4} R\left( 0 \right)}}{{{\text{d}}\tau^{4} }}, \hfill \\ \end{gathered}$$

(12)

where *m*_{2} and *m*_{4} are RMS slope and RMS curvature of rough surface profile, respectively.

The curvature distribution of asperity peaks is

$$\overline{\kappa } \left( {\xi^{*} } \right) = \sqrt 2 m_{4}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \delta \frac{{\left[ {\chi + \sqrt {\uppi } \exp (\chi^{2} )\left( {1 + {\text{erf}} \chi } \right)\left( {\chi^{2} + 0.5} \right)} \right]}}{{\left[ {1 + \chi \exp (\chi^{2} )\sqrt {\uppi } \left( {1 + {\text{erf}}\chi } \right)} \right]}}.$$

(13)

Substituting the mean height of asperity peaks into Eq. (13), the average curvature radius of asperity peaks can be calculated approximately as

$$r_{{\text{p}}} = \frac{1}{{\overline{\kappa } \left( {{{\xi_{{{\text{avg}}}} } \mathord{\left/ {\vphantom {{\xi_{{{\text{avg}}}} } \sigma }} \right. \kern-\nulldelimiterspace} \sigma }} \right)}}.$$

(14)

The density of asperity peak is

$$\eta = \frac{1}{{2{\uppi }}}\left( {{{m_{4} } \mathord{\left/ {\vphantom {{m_{4} } {m_{2} }}} \right. \kern-\nulldelimiterspace} {m_{2} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} .$$

(15)

On the basis of the work by Nayak, and on the assumption that the height distribution of asperities was Gaussian, the mean and the RMS of asperity height distribution were further obtained by McCool [10] as follows:

$$\xi_{{{\text{avg}}}} = \frac{4\sigma }{{\sqrt {{\uppi }\alpha } }},$$

(16)

$$\sigma_{{\upxi }} = \left( {1 - {\raise0.7ex\hbox{${0.8968}$} \!\mathord{\left/ {\vphantom {{0.8968} \alpha }}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$\alpha $}}} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \sigma .$$

(17)