# Fractal Contact Mechanics Model for the Rough Surface of a Beveloid Gear with Elliptical Asperities
## 1 Introduction
[PERSON] suggested a normal contact model that took into account the contribution of elastically, plastically, and mixed elastic-plastically deformed asperities to the total normal load of rough surfaces [22]. [PERSON] created a novel elastoplastic asperity contact model that incorporated the continuity and smoothness of mean contact pressure and load values throughout a range of deformation regimes, from elastic to elastoplastic, and from elastoplastic to completely plastic [23]. [PERSON] suggested a new contact stiffness model for curved surfaces that incorporated friction and was based on the continuity of the asperity length scale [24]. In recent years, the contact behavior of elliptical asperity on rough surfaces has been widely studied. Asperities on the workpiece surface may have different curvature radii in various directions, according to [PERSON] and [PERSON] [25]. [PERSON] suggested an elastic-plastic contact model for elliptic contact spots between anisotropic rough surfaces [26]. [PERSON] presented a theoretical model for the elastic-plastic contact of ellipsoid bodies and introduced a new simple method to analyze the complexity of the elliptical integral by an accurate approximation [27]. [PERSON] and [PERSON] developed an elastic-plastic contact model that considered the elliptical contact of surface asperities [28]. To address the weaknesses of the previous elastic-plastic micro-contact model, [PERSON] proposed a new elliptical contact model that took into account elastic-plastic deformation [29]. [PERSON] and [PERSON] experimented to obtain data on the relationship between the normal deformation of asperity and contact area and contact load and then used the fitting approach to derive the formula [30]. [PERSON] devised an elastic-plastic contact model with ellipsoid surfaces taking friction factors into account, and he evaluated the contact load variation law for ellipsoid elastic bodies under low-velocity impact situations [31]. The research related to the contact analysis of rough curved surfaces such as tooth surfaces. [PERSON] described the fractal approach for evaluating contact stiffness for spheroidal contact bodies with friction [32]. [PERSON] proposed a contact model for determining the contact state of spherical pairs, taking into account the microscopic properties of the rough spherical surface and the friction factor [33]. [PERSON] developed the contact model of the loading-unloading process for cylindrical contact surfaces with friction in different deformation stages [34]. [PERSON] proposed the contact stiffness model of an involute arc cylindrical gear, considering the influence of sliding friction [35]. Some scholars developed a fractal contact model applicable to a gear pair contact, considering the effect of tooth surface roughness on normal contact stiffness [36; 37]. As demonstrated by the above literature review, while significant work has been conducted by domestic and foreign scholars to investigate the contact problems between rough surfaces, the current research on the fractal contact model between two rough curved surfaces has generally ignored the influence of surface texture on rough surface contact behavior. However, the topography of rough surfaces varies in the cutting direction and vertical direction, and the contact between two rough surfaces consists of a spectrum of elliptical asperities of varying diameters, with an anisotropic distribution of the elliptical asperities. Therefore, the contact area distribution function of elliptical asperity was proposed for the point contact of curved surfaces by transforming the elastic contact problem between gear meshing surfaces into the contact between elastic curved surfaces with an arbitrary radius of curvature. In addition, a fractal contact mechanics model for the rough surface of a beveloid gear with elliptical asperities was established. The influence of tooth surface topography on the contact load and contact stiffness under different fractal parameters was investigated.