# The Elasto-Plastic Contact Models of Cylinder-Based and Sphere-Based Fractal Rough Surfaces
## 1 Introduction  
For many engineering applications, knowledge about contact between surfaces is of fundamental importance to understand friction, wear, lubrication, friction-induced vibrations and noise, thermal and electrical contact resistance, etc. At the microscopic level, surfaces that include different asperities are rough. The contact between rough surfaces is characterized by the interaction of asperities. The surface topography results in an imperfect contact, which leads to the real area of contact being a fraction of the apparent or nominal area. The contact load on the per unit real area of contact is much greater than the design requirement, which leads to failure and wear in contact components. The investigation of mechanical properties about rough surfaces is continuous from the last century. A statistical contact model was firstly put forward to solve the rough surface contact problem by [PERSON] and [PERSON], called the GW model [1]. In this model, the rough surface is represented by a collection of asperities of a prescribed shape that are scattered over a reference plane. The height distribution of the asperities follows Gaussian, and each asperity owns an identical radius of curvature. The relations for the real contactarea and the total contact load as a function of the separation between the flat and the mean asperity level are derived. A 'plasticity index' is provided as a criterion, which indicates whether contact will be elastic or plastic. In the subsequence, some researchers extended the basic GW model. They used it to solve other problems of rough surface contact. These problems include the elastic contact of rough spheres [2], the properties of random surfaces [3], the contact of nominally flat rough surfaces [4], and the plastic contact of rough surfaces [5]. In order to obtain more accurate results, the contact mechanical properties of a single asperity with a rigid flat have been obtained [6; 7; 8; 9]. [PERSON] and [PERSON] developed an unloading contact model of rough surfaces based on the contact mechanical properties of a single asperity [10; 11]. These achievements make the statistical contact model more perfect [12; 13]. The variances in statistical parameters (such as asperity height, slop, and curvature) depend strongly on the resolution of the roughness-measuring instrument or any other form of filter and hence are not unique for a rough surface. To overcome the shortcoming of the statistical contact model, [PERSON], [PERSON] (1990), and [PERSON] (1990) used the Weierstrass-Mandelbrot function (WM function) to simulate a two-dimensional fractal rough surface profile [14; 15]. Then, [PERSON] and [PERSON] (1991) adopted the WM function to develop a fractal contact model, called the MB model [16]. The model predicts that a single asperity of a contact area smaller than a critical area is in plastic contact. When the load is increased, the asperity transfers from plasticity to elasticity. This unusual behavior contradicts classical contact mechanics, and the explanation is that some plastically deformed asperities join to form the elastic asperity. Then, the MB model was applied to the interfacial temperature distribution in the slow sliding regime [17; 18]. [PERSON] and [PERSON] (1998) extended the MB model from two-dimensional rough surface to three-dimensional rough surface [19]. [PERSON] and [PERSON] (2007) developed a revised elastic plastic contact model of a single fractal asperity [20]. The model shows that the critical area of asperity is scale-dependent; as the load and contact area increase, the deformation of asperity takes place from elastic to plastic. [PERSON] and [PERSON] (2010) developed an elastic plastic contact model of a single fractal asperity according to classical contact mechanics, and the total contact load and the real contact area were evaluated [21]. Then, the revised model was applied to sphere-based and cylinder-based fractal bodies contacting a rigid flat surface [22].