## 3 UDEC-Based Numerical Simulation on Discrete Element  
### Rheological Model Based on Strength Attenuation of Contact Face  
In the discrete element software Universal Distinct Element Code (UDEC), the deformation and damage of a rock block are jointly controlled by the block and the contact surface. UDEC merely provides a rheological model for the block, yet fails to offer a rheological model for the contact surface [22]. Based on the GBM-TtoF creep model developed by [PERSON] et al. [23], in this study, a rheological model based on the strength attenuation of the contact surface was established. The ideas of these two models are basically consistent. That is, the relationship between the block strength variation with time is established first and then converted into the strength attenuation of the contact surface through the coefficient. A schematic diagram of the block strength attenuation of the two models is illustrated in Figure 4. The models assume that the crack damage threshold is equal to the rheological damage threshold. By lowering the position of the strength line, rock damage under a static load that is lower than the ultimate strength is described. The ultimate strength of rock is the distance of AB:  
\[\sigma_{AB}=\left[c\cot\varphi+\frac{1}{2}(\sigma_{1}+\sigma_{3})\right]\sin\varphi \tag{1}\]  
where \(c\) is the cohesion of rock; \(\varphi\) is the internal friction angle of rock; and \(\sigma_{1}\) and \(\sigma_{3}\) are the maximum and minimum principal stresses, respectively. Figure 3: Surrounding rock loose circle test. (**a**) CT2 mine ultrasonic surrounding rock crack detector; (**b**) test borehole arrangement; (**c**) test results. Through rheological tests, [PERSON] and [PERSON] disclosed that the time dependence of the cohesion is much stronger than that of the internal friction angle [24]. Therefore, when the internal friction angle is ignored, the time-dependent change in the strength line position can be transformed into the time-dependent attenuation of the cohesion. According to the Mohr-Coulomb criterion, the maximum principal stress is:  
\[\sigma_{1}=\frac{1+\sin\varphi}{1-\sin\varphi}\sigma_{3}+\frac{2c\times\cos \varphi}{1-\sin\varphi} \tag{2}\]  
For uniaxial compression, \(\sigma_{3}\) is 0, and the uniaxial compressive strength of rock can be expressed as:  
\[\sigma_{ucs}=\frac{2c\times\cos\varphi}{1-\sin\varphi} \tag{3}\]  
where \(\sigma_{ucs}\) is the short-term strength of rock. Since the time-dependent change in the strength depends on the time-dependent change in cohesion, the long-term strength of rock \(\sigma_{l}\) can be expressed as:  
\[\sigma_{l}=\frac{2c_{l}\times\cos\varphi}{1-\sin\varphi} \tag{4}\]  
where \(c_{l}\) is the rock cohesion under long-term strength. [PERSON] proposed a relationship between the long-term strength \(\sigma_{l}\) and the short-term strength \(\sigma_{ucs}\) for rocks [25]:  
\[\sigma_{l}=\sigma_{ucs}(1-F\ln(t/t_{0})) \tag{5}\]  
where \(F\) is an empirical constant; \(t\) is the rheological time; and \(t_{0}\) is the test time for short-term strength. Equation (4) gives the relationship between the strength and the time. By substituting Equations (3) and (4) into Equation (5), the relationship between the cohesion and the time can be obtained:  
\[c_{l}=c(1-F\ln(t/t_{0})) \tag{6}\]  
Hence, the cohesion can be regarded as a function of time, and Equation (6) can be rewritten as [26]:  
\[c(t)=c(1-F\ln(t/t_{0})) \tag{7}\]  
The law of the time-dependent attenuation of the block strength can be obtained according to Equation (7) and the rheological damage threshold:  
\[c(t)=\left\{ \begin{array}{ll}c(1-F\ln(t/t_{0}))&\sigma_{1}{>}\sigma_{\rm d} \\ c&\sigma_{1}\leq\sigma_{\rm d}\end{array} \right. \tag{8}\]  
Figure 4: Schematic diagram of block strength attenuation. \[c_{j}(t)=\left\{ \begin{array}{ll}c_{j}(1-F\ln(t/t_{0}))\times K&\sigma_{1}{>} \sigma_{\rm d}\\ c_{j}&\sigma_{1}\leq\sigma_{\rm d}\end{array} \right.