# Fractal Contact Mechanics Model for the Rough Surface of a Beveloid Gear with Elliptical Asperities
## 1 Introduction  
The contact tooth surfaces are microscopically rough due to problems in machining accuracy and tooth surface wear. The contact between two rough tooth surfaces is a range of elliptical asperities of different sizes, and the distribution law of the elliptical asperities has a certain degree of anisotropy. Understanding the contact mechanics of rough tooth surfaces is critical for understanding phenomena such as tooth surface flash temperature, tooth surface wear, and gear vibration. Therefore, establishing a contact mechanics model for rough surfaces accurately and investigating the relationship between the real contact area and the contact load are particularly important. Numerous studies on the contact analysis of rough surfaces have been conducted by domestic and foreign scholars. [PERSON] and [PERSON] introduced the famous GW contact model and investigated the elastic contact between two rough surfaces by assuming a flat surface in contact with a rough surface [1]. [PERSON] and [PERSON] produced the MB contact model that considered isotropic rough surfaces to eliminate the scale-dependency of statistical theories [2]. Moreover, the normal contact load and contact area in the MB contact model were investigated using scale-independent parameters [3]. [PERSON] established the CEB contact model considering the elastic-plastic deformation of the asperities [4]. [PERSON] suggested an elastic-plastic asperity contact model, namely, the ZMC model with the continuity of variables across different deformation regimes [5]. [PERSON] and [PERSON] constructed the KE contact model between an elastic hemisphere and a rigid flat surface utilizing finite element methods and analyzed the contact behavior by quantitatively separating the rangesof elastic deformation, elastic-plastic deformation, and plastic deformation [6; 7]. [PERSON] presented a multi-scale contact model by employing the Fourier series coefficients obtained from an FFT of the rough surfaces to describe the surface geometry at multiple scales [8; 9]. [PERSON] proposed a revised contact mechanics model for rough surfaces based on the premise that the initial profile of the rough surface should be indifferent to the sample duration and contact area. Additionally, the effects of fractal dimension, fractal roughness, and contact pressure on contact stiffness were addressed in detail [10]. [PERSON] provided a novel elastoplastic contact model for rough surfaces that addressed a critical drawback of the MB model, which assumed that the asperities must be deformed [11]. [PERSON] introduced a new MB model of elastic-plastic contact between rough surfaces, and the revised model revealed that asperity levels have an effect on the mechanical characteristics of rough surfaces [12; 13]. [PERSON] established the contact mechanics model for rough surfaces based on shoulder-shoulder contact and fractal characteristics [14]. [PERSON] constructed an analytical model for contact stiffness that incorporated the friction factor between rough contact surfaces, and the findings indicated that the friction factor has a significant effect on the contact stiffness of the whole structure [15; 16]. [PERSON] suggested a new fractal contact model that took asperity interactions into account by representing the impacts of asperity interactions using the displacement of the mean of asperity heights [17]. [PERSON] suggested a contact stiffness model that took asperity interactions into account, and the proposed model's correctness was proved by experiments [18]. [PERSON] developed a novel contact stiffness model for the mechanical joint surface that accounted for the asperity's continuous smooth contact features while also taking the asperity interaction into account [19]. [PERSON] introduced an elastic-plastic spherical contact model for combined normal and tangential loadings, which integrated previously accurate finite element analyses for contact and static friction on rough surfaces [20; 21].