# The Elasto-Plastic Contact Models of Cylinder-Based and Sphere-Based Fractal Rough Surfaces
## 2 Rough Surface Characterization  
### The Cylinder-Based Rough Surface Characterized by the Weierstrass-Mandelbrot Function  
The characteristics of fractal geometry have been found in different disciplinary fields, such as geography, turbulence, precipitation, and surface topography. Continuity, non-differentiability, and self-affinity are major features of fractal geometry [28]. [PERSON] extended a function devised by [PERSON] in 1872, which is called the WM function [29]. The WM function is continuous but non-differentiable and self-affine and is used to characterize and simulate surface profiles. In order to characterize a cylinder-based fractal rough surface, the cylindrical coordinates \((W_{C},\theta,z)\) are introduced. In this system, the cylinder-based fractal rough surface profiles can be written as  
\[W_{C}(\theta)=\rho+\left(\frac{\ln\gamma}{M}\right)^{1/2}\sum_{m=1}^{M}A_{m} \sum_{n=n_{\min}}^{n_{\max}}\left(k\gamma^{n}\right)^{(D-3)}\{\cos(\Psi_{m,n})- \cos[k\gamma^{n}\rho\cos(\theta-\alpha_{m})+\Psi_{m,n}]\}\]  
\[z=z_{0}(2<D<3,\gamma>1) \tag{1}\]  
where \(W_{C}\) illustrates the distance between any point on the fractal surface and the axis of the cylinder body as \(z=z_{0}\). \(\rho\) is the base circle radius, which denotes the mean values of the \(W_{C}\) at all \(\theta\) angles in a given \(z\) plane. The parameter \(n\) is a frequency index, and \(\gamma(\gamma>1)\) is a parameter that determines the spectral density of frequencies in the profile. In order to satisfy the requirements for high spectral density and phase randomization, we set \(\gamma=1.5\). The parameter \(M\) denotes the number of superposed ridges used to construct the surface. \(D\) is the fractal dimension of the surface profile, and its value ranges from 2 to 3 for a continuous surface. For isotropic surfaces, the parameter \(A_{m}=2\pi(2\pi/G)^{(2-D)}\) for all \(m\) values, where \(G\) is the length parameter. The arbitrary angle \(\alpha_{m}\) is used to offset the ridges in the azimuthal direction. In the paper, the ridges are equally offset, and thus, \(\alpha_{m}=\pi m/M\). A random number generator is used to uniformly distribute the values of random phase \(\Psi_{m,n}\) in the interval \([0,2\pi]\). The parameter \(k\) is the wave number related to the sample size, \(k=2\pi/L_{C}\), where \(L_{C}\) is the sample length. Figure 1 illustrates an isotropic cylinder-based fractal surface obtained from Equation (1) with \(D=2.5\), \(G=2.1\times 10^{-3}\), and \(\rho=0.1\) m. The fractal rough surfaces appear to be random, multiscale, and disordered. The fractal rough surfaces are continuous and non-differentiable at each point of them. The section profile of the cylinder-based rough surface is wiggly and jagged in Figure 1b. This section is at an arbitrary position perpendicular to the axis of the cylinder.  
As a cylinder-based fractal rough surface contacts another cylinder-based fractal rough surface, the contact of two cylinder-based fractal rough surfaces can be replaced by an equivalent cylinder-based fractal rough surface in contact with a rigid flat surface. The equivalent cylinder-based fractal rough surface is generated by Equation (1). Figure 2 shows the cylinder-based fractal rough surface contacts with a rigid flat surface. The base circle radius is denoted by \(\rho\) (the base circle is not deformable). The parameter \(\omega\) is contact interference and \(d\) is the distance between the bottom of the base circle and rigid flat surface. The parameter \(\omega\) is the total vertical deformation when a rough surface contacts a flat one. It starts at 0 and can reach the height of the rough surface's asperities.