## 2 The Improvement Considering the Vibration Deterioration Effect of the Structural Plane in FDEM
### Mathematical Model of the Structural Plane Considering the Vibration Deterioration Effect  
As mentioned above, it can be seen that the vibration deterioration effect of seismic load on the structural plane is mainly affected by three influence factors: cumulative displacement, shear cycle number, and relative velocity. Therefore, the present study introduces a vibration deterioration model of the structural plane considering the above three factors based on the previous studies [29; 31; 33]. Here, it is assumed that these three factors are opposite to each other during the entire duration of the earthquake, and the vibration deterioration factor of the structural plane at any time can be expressed as  
\[\eta(t)=\alpha(t)\beta(t)\gamma(t) \tag{3}\]  
where \(\alpha(t)\) is the influence factor of cumulative displacement (IFOCD), \(\beta(t)\) is the influence factor of relative velocity (IFORV), and \(\gamma(t)\) is the influence factor of shear cycle number (IFOSCN).  
The IFOCD and IFOSCN can be obtained from the curves of shear strength with displacement and shear cycle number, respectively, while the IFORV can be derived from the block dynamic friction test. The test curves of each factor are shown in Figure 1.  
As shown in Figure 1a, the influence factor of cumulative displacement \(\alpha=1\) when the cumulative displacement \(u=0\). As the cumulative displacement increases, its influence factor gradually decreases, and then the decrease slows down, finally approaching a  
Figure 1: The laboratory test results [27; 32] and fitting curves with test data. (**a**) The variation law of IFOCD with cumulative displacement. (**b**) The variation law of IFORV with relative velocity. (**c**) The variation law of IFOSCN with shear cycle number.  
convergent value. As a whole, it is consistent with the form of the negative exponential function, and the IFORV and IFOSCN are similar to it. Therefore, the present study adopts Equation (4) to describe the variation of the influence factors.  
\[\left\{\begin{array}{l}\alpha(t)=A+(1-A)e^{-mu(t)}\\ \beta(t)=B+(1-B)e^{-\imath\pi(t)}\\ \gamma(t)=C+(1-C)e^{-\eta m(t)}\end{array}\right. \tag{4}\]  
where \(A\), \(B\), and \(C\) are the convergence values of the IFOCD, IFORV, and IFOSCN, respectively, and \(m\), \(n\), and \(p\) are the undetermined coefficients.  
The fitting results are also shown in Figure 1. It can be seen from the figure that the fitting curve has high fitting accuracy on the test data points, and the coefficient of determination, \(R^{2}\), is above 0.96. It illustrates that it is reasonable to adopt Equation (4) to describe the variation law of each vibration deterioration influence factor of the structural plane. Parameters to be determined are as follows: \(A\) = 0.862, \(m\) = 1.529; \(B\) = 0.816, \(n\) = 13.336; \(C\) = 0.83, \(p\) = 0.631. After substituting the above parameters into Equations (3) and (4), the vibration deterioration factor of the structural plane at any time can be simplified as follows:  
\[\eta(t)=(0.862+0.138e^{-1.529\mu(t)})(0.816+0.184e^{-13.33\sigma(t)})(0.83+0.17 e^{-0.631\mu(t)}) \tag{5}\]  
It should be noted here that the present study assumes that the deterioration factor will not increase with time after decreasing, so the deterioration factor in FDEM is set as monotonically decreasing.