# Dynamic and Stability Analysis of Multibolt Plane Joints under Normal Forces
## 2 Surface Fractal Contact Model and Verification  
The contact interface is an important nonlinearfactor of the bolt joint system. A three-dimensional fractal load-separation contact model modified with the domain extension factor was introduced and the contact stiffness can be obtained byderivationof the curve. To characterize features of the rough surface topography assuming isotropy, such as continuity, nondifferentiability, and self-affinity, a modified W-M function is given by [23]  
\[z(x)=L\Big{(}\frac{G}{L}\Big{)}^{D-1}(\ln\gamma)^{\frac{1}{2}}\sum_{n=0}^{n_{ max}}\gamma^{(D-2)n}\Bigg{[}\cos\varphi_{1n}-\cos\Bigg{(}\frac{2\pi\gamma^{n}x}{L}- \varphi_{1n}\Bigg{)}\Bigg{]} \tag{1}\]where \(z(x)\) is the fractal profile, \(L\) is the sample length, \(G\) is the fractal roughness parameter, \(D\left(1<D<2\right)\) is the fractal dimension, \(\gamma\) (\(\gamma>\)1) is the scaling parameter controls the density of the frequencies in the surface profile, and \(\varphi_{1n}\) is a random phase. In consideration of surface flatness and frequency distribution density, \(\gamma=1.5\) is suitable [21]. The approximate root-mean-square height is obtained as [23]  
\[\sigma=\frac{G^{D-1}L^{2-D}}{2\sqrt{\left(2-D\right)\ln\gamma}}. \tag{2}\]  
In previous studies, when two flexible bodies with nominally flat surfaces contact with each other, the system is usually replaced by a rigid smooth surface contact with an equivalent elastic rough surface with reduced elastic modulus \(E=\left[\left(1-v_{1}^{2}\right)/E_{1}+\left(1-v_{2}^{2}\right)/E_{2}\right]^ {-1}\) to make calculation convenient, where \(E_{1}\), \(E_{2}\) and \(\
u_{1}\), \(\
u_{2}\) are elastic moduli and Poisson ratios of original surfaces, respectively [23; 25]. The stress is derived from the pressure of contact asperities. To simplify the problem, it is normal and reasonable to assume that all asperities have spherical summits with different radiuses [17; 21]. For the simple case of elastic-fully plastic material, the normal elastic force \(p_{e}\) at an asperity can be written as [23]  
\[p_{e}(a^{\prime})=\frac{2^{\frac{9-2D}{2}}}{3\pi^{\frac{3-D}{2}}}(\ln\gamma)^ {\frac{1}{2}}G^{D-1}Ea^{\prime\frac{3-D}{2}} \tag{3}\]  
where \(a^{\prime}=2a_{e}\) is the truncated microcontact area belonging to a hypothetical section of the undeformed asperity at the contact position doubles the real microcontact area in elastic deformations and \(a^{\prime}=a_{p}\) in fully plastic deformations. The critical truncated area \(a_{c}^{\prime}\) is given by [23]  
\[a_{c}^{\prime}=\left[2^{7-2D}\pi^{D-1}b^{-1}G^{2\left(D-1\right)}\left(\frac{ E}{H}\right)^{2}\ln\gamma\right]^{\frac{1}{D-1}} \tag{4}\]  
in which \(b=\left(\pi q/2\right)^{2}\) and \(q=0.454+0.41\
u\), where \(\
u\) and \(H\) are the Poisson ratio and the hardness of the softer material, respectively. It is determined that asperities deform elastically when \(a^{\prime}>a_{c}^{\prime}\). In addition, the distribution function of the truncated microcontact area is expressed as [20]  
\[n(a^{\prime})=\frac{D}{2}\psi^{\frac{3-D}{2}}a_{l}^{\prime\frac{D}{2}}a^{ \prime-\frac{D+2}{2}} \tag{5}\]  
where \(a_{l}^{\prime}\) is the largest truncated microcontact area and \(\psi\) (\(\psi>1\)) is the domain extension factor. The value of \(\psi\) can be obtained by dichotomy method depends on a transcendental equation \(\psi^{(2-D)/2}-\left(1+\psi^{-D/2}\right)^{(D-2)/D}=(2-D)/D\). For light pressure of the interface, asperities are far enough to ignore the effect of their interactions, which leads to a real contact area much smaller than the nominal contact area.