## 2 The Improvement Considering the Vibration Deterioration Effect of the Structural Plane in FDEM
### Vibration Deterioration Effect of the Structural Plane  
A large number of studies related to rock dynamics have illustrated that the structural plane of rock mass will display a strength deterioration phenomenon under the continuous action of dynamic load: [PERSON] and [PERSON] (1982) explored the friction characteristics of the sliding surface during the motion of blocks through vibration tests and pointed out that the friction coefficient will vary dynamically with the cumulative displacement and the relative velocity [27]. Based on the above phenomenon, the differential equation of block motion was established and applied to the stability analysis of slopes. [PERSON] et al. (2007) summarized the research results of [PERSON] (1982) and proposed a seismic permanent displacement calculation method for the rock slope considering the vibration deterioration effect of structural planes based on the traditional Newmark method [28; 29]. [PERSON] et al. (2013) fitted the test results of [PERSON] and established a dynamic strength reduction method considering the vibration deterioration effect of the structural plane [30]. [PERSON] et al. (2001) carried out a laboratory shear cyclic test to investigate the deterioration law of the joint roughness and illustrated that it was greatly affected by the shear cyclic number [31]. Combining the studies of [PERSON] (1982) and [PERSON] et al. (2001), [PERSON] (2014) simultaneously considered the effects of the shear cyclic number, shear amplitude and relative velocity, and established a mathematical model of the vibration deterioration effect of the structural plane through a series of laboratory tests [27; 31; 32]. It is inferred that the dynamic load will cause the vibration deterioration effect of the structural plane in slope during the earthquake, and the static structural plane parameter design is no longer applicable. Therefore, the dynamic strength reduction method considering the vibration deterioration effect of the structural plane is proposed in the present study. The deterioration of the rock structural plane under the cyclic effect of seismic load is mainly reflected in the dynamic variation of structural plane parameters, which is an extremely complex process. Here, a time-dependent factor \(\eta\)(t), called the dynamic vibration deterioration factor of the structural plane, is introduced to portray the degree of deterioration of the structural plane. The joint element is calculated first in FDEM, and when the joint element breaks, the contact detection takes effect, and then the contact force is calculated. Therefore, this factor is applied to both failure initiation and post-failure stages of the joint element and can be expressed by the following equation  
\[\tau(t)=\tau_{0}\eta(t)=\left\{ \begin{array}{l}\sigma(t)\tan\varphi_{0}\eta( t)+c_{0}\eta(t)\\ \sigma(t)\tan\varphi_{0}\eta(t)\end{array} \right. \tag{1}\]  
where \(\tau(t)\) is the tangential stress of the structural plane at time \(t\), \(\tau_{0}\) is the initial shear strength of the structural plane, \(\eta(t)\) is the vibration deterioration factor of the structural plane at time \(t\), \(\sigma(t)\) is the normal stress of the structural plane at time \(t\), and \(\varphi_{0}\) and \(c_{0}\) are the initial friction angle and cohesion of the structural plane, respectively [33]. The friction angle, \(\varphi(t)\), and cohesion, \(c(t)\), of the structural plane at any time can be expressed as  
\[\left\{ \begin{array}{l}\varphi(t)=\arctan[\tan\varphi_{0}\eta(t)]\\ c(t)=c_{0}\eta(t)\end{array} \right. \tag{2}\]  
In summary, it can be seen that the cyclic effects of seismic load on the structural plane of rock mass are mainly reflected in two aspects: (1) The influence factor of vibration wear: Seismic loads are both dynamic and cyclic loads, the structural plane will be gradually worn out and blunted under the reciprocating motion, which is reflected in the parameters as the friction angle and cohesion will be gradually reduced. [PERSON] et al.