# A Wear Simulation Method for Mechanical Face Seals under Friction Instability Conditions
## 2 Mathematical Model of Mechanical Face Seals under Friction Instability Conditions
### Mechanics Analysis of the Asperity Contact  
If boundary lubrication and mixed lubrication conditions are present, asperity contact will occur, and the influence of the asperity contact pressure on the mechanical face seals cannot be neglected. [PERSON] and [PERSON] proposed a model (G-W model) to calculate the asperity contact pressure [26]. In the G-W contact model, the contact between two rough surfaces is reduced to a contact between a smooth rigid surface and a rough surface. Independent Hertz contact is assumed to occur, as shown in Figure 3. It is assumed that the asperities have spherical summits, with a constant radius \(R_{e}\). Furthermore, the asperity deformation is assumed to be a pure elastic deformation. According to the G-W model, the asperity contact pressure \(p_{ac}\) is defined as:  
\[p_{ac}(z,t)=\frac{W_{a}}{A_{n}}=\frac{4}{3}\eta_{e}R_{e}^{\frac{1}{2}}E_{e} \int\limits_{\widetilde{h}(z,t)/\sigma_{e}}^{\infty}\phi(z)(z-\widetilde{h}( z,t)/\sigma_{e})^{\frac{3}{2}}dz, \tag{4}\]  
where \(\varphi(z)\) is the probability density function of the surface height distribution, \(\widetilde{h}(z,t)\) is the average value of the film thickness, \(\eta_{e}\) is the equivalent surface density of the asperities, \(W_{a}\) is the total asperity contact load, \(A_{n}\) is the nominal contact area, and \(E_{e}\) is the equivalent elastic modulus. The surface equivalent roughness \(\sigma_{e}\) is defined as the root mean square roughness of the heights of the stator and rotor [27]. However, the height distribution of the sealing surface cannot be approximated as a Gaussian distribution [28]. The surfaces are non-Gaussian because of the asperity contact and the running-in of the seals. Typically, the Johnson transformation is used for the transformation between Gaussian and non-Gaussian distributed series [29]. The four most important parameters of the Johnson  
Figure 3: Contact between a smooth rigid surface and a rough surface. Figure 2: Tapered film. transformation system (\(\gamma\), \(\varepsilon\), \(\lambda\), and \(\delta\)) can be obtained based on the first four moments, i.e., the mean, variance, skewness, and kurtosis of the non-Gaussian series. The frequency curve of the Johnson transformation system has three forms, namely: lognormal, unbounded, and bounded. After selecting the appropriate form of the frequency curve according to the Johnson transformation, the Johnson series \(Z_{\text{n-G}}\) is obtained from the transformation as:  
\[Z_{n-G}=\varepsilon+\lambda f^{-1}(\frac{Z_{G}-\gamma}{\delta}), \tag{5}\]  
where \(Z_{G}\) is the Gaussian distributed series, and \(Z_{\text{n-G}}\) is the non-Gaussian distributed series, and:  
\[f^{-1}(z)=\left\{ \begin{array}{cc}e^{z}&\log normal\\ \frac{1}{2}(e^{z}-e^{-z})&unbounded\\ 1/(1+e^{-z})&bounded\end{array} \right.. \tag{6}\]  
Therefore, the height distribution of the sealing surface, i.e., the non-Gaussian distributed surface, can be transformed from a Gaussian distributed surface with a specific skewness and kurtosis using the Johnson transformation. The Gaussian surfaces have a skewness coefficient of zero, indicating that the material is symmetrically distributed above and below the mean plane. The kurtosis coefficient of the Gaussian surface is three. This is not the case for mechanical seal faces, as demonstrated in [27]. Mechanical seals are assumed to operate in laminar flow and without a phase change. [PERSON] [30] suggested the compressive yield stress \(S_{C}\) as a plausible value for the mechanical contact pressure \(p_{c}\). The contact portion \(b_{\text{m}}\) of the mechanical seal can be estimated by a non-Gaussian cumulative distribution function:  
\[b_{m}(r)=\int_{h}^{\infty}\phi(z)dz. \tag{7}\]  
The interfacial contact pressure is expressed as [4]:  
\[p_{c}(r)=b_{m}S_{C}. \tag{8}\]  
Thus, the asperity contact load \(F_{c}\) is given as:  
\[F_{c}=\int_{r_{i}}^{r_{o}}2\pi rp_{c}(r)dr. \tag{9}\]  
The first contact between the smooth rotor and rough stator occurs at the highest asperity point.