# Failure of Rock Slope with Heterogeneous Locked Patches: Insights from Numerical Modelling
## 2 Brief Description of RFPA and Numerical Model  
We used the rock failure process analysis code (RFPA) to simulate the locked patch failure. The code has been widely adopted to simulate the mechanical behaviour of rock and rock mass under various loading conditions at both laboratory and field scales [24; 25; 26; 27; 28; 29; 30]. The elements of the rock are assumed to be linearly elastic, isotropic, and damage-free before loading. The mechanical properties of the elements are assigned statistically through a given Weibull distribution \(\varphi(\alpha)\)[31] as follows:  
\[\varphi(\alpha)=\frac{m}{\alpha_{0}}\cdot(\frac{\alpha}{\alpha_{0}})^{m-1} \cdot e^{-(\frac{\alpha}{\alpha_{0}})^{m}} \tag{1}\]  
where \(\alpha\) represents a given property of material; \(\alpha_{0}\) is a scaling parameter denoting the average value of the material property; \(m\) is a shape parameter representing material homogeneity. Figure 1: Schematic of the rock slope with locked patches (Reprinted with permission from [23], 2021, Chinese Journal of Geophysics). The cumulative probability function \(P(\alpha)\) is expressed as follows:  
\[P(\alpha)=1-e^{-(\frac{\alpha}{\epsilon_{0}})^{m}} \tag{2}\]  
The element damage obeys the one-dimensional linear elasticity law:  
\[\sigma=E\epsilon=E_{0}\epsilon(1-D) \tag{3}\]  
where \(\sigma\), \(\epsilon\) represent stress and elastic strain, respectively; \(E\) and \(E_{0}\) are elastic modulus of damaged and undamaged material, respectively; \(D\) is the damage variable. The constitutive relationship of an element under uniaxial tension is:  
\[D=\left\{ \begin{array}{ccc}0&&\epsilon\leq\epsilon_{t0}\\ 1-\frac{\sigma_{tr}}{E_{0}\epsilon}&&\epsilon_{t0}<\epsilon\leq\epsilon_{tu}\\ 1&&\epsilon_{tu}<\epsilon\end{array} \right. \tag{4}\]  
where \(\epsilon_{t0}\) is the elastic tensile strain limit; \(\epsilon_{tu}\) is the ultimate tensile strain of the element. \(\sigma_{tr}\) is the residual uniaxial tensile strength. When the element is under uniaxial compression, the damage variable \(D\) is expressed as:  
\[D=\left\{ \begin{array}{ccc}0&&\epsilon\leq\epsilon_{c0}\\ 1-\frac{\sigma_{tr}}{E_{0}\epsilon}&&\epsilon_{c0}<\epsilon\end{array} \right. \tag{5}\]  
where \(\epsilon_{c0}\) is the elastic compression strain limit; \(\sigma_{cr}\) is the residual uniaxial compression strength. Based on the stress and strain of the element and the chosen failure criterion (Mohr-Coulomb in this test), the damage state of element can be judged. The tensile criterion takes precedence over the compression and shear failure criterion in RFPA. Once the element is damaged, all the elastic energy stored in the element is released in the form of acoustic emission (AE). In the AE image, the element that reaches the tensile failure criterion is represented by a red circle while the shear failure element is represented by a white one and the circle diameter represents the relative size of AE energy. Specifically, there is an empirical relationship between the mean element uniaxial compression strength and macroscopic uniaxial compression strength of samples using the following formula [32]:  
\[\frac{\sigma_{c}}{\sigma_{c0}}=0.2602\ln m+0.0233(1.2\leq m\leq 50) \tag{6}\]  
where \(\sigma_{c0}\) is the mean element uniaxial compression strength. \(\sigma_{c}\) is the macroscopic uniaxial compression strength. The locked patches control the locked rockslide where the weak fillings have limited effect. To understand the fracture process of the locked patches better, a series of simplified numerical simulations with omitted fillings was performed. As presented in Figure 2, the model was simplified to a plane model dimension of 10 \(\times\) 5 m. After meshing, there were a total of 400 \(\times\) 200 = 80,000 quadrilateral elements. The upper load was fixed, and the shear force was applied through the displacement-controlled mode. For the real rock slope, the geological settings are complex and the geometry and number of the locked patch are different.