## 1 Introduction  
At a microscopic level, the rough surface consists of numerous asperities of different sizes which may be caused by form errors, waviness and roughness. Although the waviness affects the contact of surfaces [1], the influence of surface roughness is more significant for rough machined fractal sealing surfaces. When two contacting surfaces are under load, these asperities will deform, affecting the pressure bearing and sealing performance of the surfaces. Mechanical analysis of rough surface is the main content of surface contact theory, which involves real contact area, contact pressure and the relationship between them. A variety of contact mechanics models were established by predecessors, promoting the continuous development of the surface contact theory. [PERSON] studied the contact mechanics of two spherical elastomers without friction and laid the theoretical foundation of modern contact mechanics [2]. [PERSON] and [PERSON] combined the simulated surface with [PERSON] contact theory and established the GW model [3] based on statistical characterization parameters. Based on GW model, [PERSON] [4] and [PERSON] [5] further modified and improved the height distribution, shape and curvature radius of asperities. Assumed that all the asperities are purely plastic deformed, [PERSON] [6] established the corresponding plastic contact model of rough surfaces. On the principle of constant volume of asperities during plastic deformation, [PERSON] [7]proposed the CEB model to analyze the elastic and plastic contact characteristics of rough surfaces. However, the CEB model did not consider the elastic-plastic deformation, so the results are discontinuous. [PERSON] et al. re-analyzed the deformation process of asperities under load [8; 9] and put forward a contact model including the elastic, elastic-plastic and plastic deformation of asperities. [PERSON] [10], [PERSON] [11], [PERSON] [12] and [PERSON] [13] took the asperity base deformation caused by the asperity deformation into consideration and established various statistical contact models for the interaction of asperities. In addition, many researchers established the loading-unloading contact model [14; 15], rough surface contact stiffness model [16; 17] and viscoelastic contact model [18; 19] to meet various working requirements. Based on the GW statistical model and probability density function of height distribution for asperity, [PERSON] [20] established the total stiffness model of rough surface contact by considering the contact state of elastic, elastic-plastic and purely plastic deformation. [PERSON] proposed an unlubricated contact model for random rough surfaces [21] and analyzed the contact between equivalent rough surfaces and rigid planes based on statistical theory. Contact mechanical models of rough surface with statistical characterization were established under the premise that \"The contact area is less than the cross-sectional area of the profile opening area\", however, this assumption is not consistent with the principle of volume conservation during deformation. Moreover, the statistical parameters in these models are greatly influenced by the sample length and resolution of the instrument, which is highly scale-dependent and not universal. [PERSON] and [PERSON] [22] established the rough surface contact model (M-B model) based on fractal theory for the first time. They deduced the elastic and plastic contact deformation of the single asperity, but they did not consider the elastic-plastic contact problem. [PERSON] and [PERSON] modified the M-B model by considering the effect of the expansion factor of the size distribution domain of the micro contact on the deformation of asperities [23]. [PERSON] and [PERSON] initially put forward a three-dimensional rough surface contact model with fractal characteristics [24] and obtained the total contact load and contact area of the rough surface by derivation.