## 2 Numerical Simulations  
According to the laboratory test results in the paper by [PERSON] et al. [40], a set of numerical models were generated by Particle Flow Code (PFC). As a discrete element method, the particles and bonds determine the behavior of the macro material, and it can indicate a similar simulation result in the macro-phenomenon of specimen's failure [41]. The particle discrete element model provides support to comprehend the failure process of rock-like material from macroscopic-scale phenomena and statistics. In this paper, the parallel bond was used to generate the bonded-particle models (BPMs).  
In the laboratory test, cement, sand, and water were used to make samples with a size of 50 mm (length) \(\times\) 100 mm (height) \(\times\) 30 mm (width) [40]. The ratio of the sample length to height complies with the ISRM standards. Following the block size in the laboratory test, the simulated specimens in the PFC were 100 mm high and 50 mm long. We adopted the two-dimensional models to investigate the surface deformation characteristics. Therefore, the results and conclusions only involve two-dimensional situations and might be slightly different from actual situations or three-dimensional models.  
The intact BPM has a uniform particle-size distribution bounded by \(R_{\text{min}}\) (0.25 mm) and \(R_{\text{max}}\) (0.415 mm). The micro-parameters of the parallel bonds were determined by the calibration process of the simulated uniaxial compression test of the intact model. The parameters of the parallel bonds were adopted when the simulated results reproduced the laboratory test values, including ball-ball contact modulus (\(E_{c}\)) of 2.5 GPa, normal strength (\(\sigma_{n}\)) of 19 MPa, shear strength (\(\sigma_{s}\)) of 27.1 MPa, and ball stiffness ratio (\(k_{n}\)/\(k_{s}\)) of 2.5. The macro properties of the intact specimens in the physical experiment and numerical simulation are listed in Table 1, which proves the rationality of the numerical model. The uniaxial compressive strength of the cement specimens corresponds to some kinds of homogeneous and isotropic sandstone and argillite shale. After the reproduction of the intact sample, the specimen with 45\({}^{\circ}\) flaw was created by deleting some particles in a specific area. The defined flaw in the model had the same geometry as that in the experimental samples.  
Figure 1 shows the stress-strain curves of intact and flawed samples in the laboratory test and numerical simulation. It is apparent from Figure 1 that the stress-strain curve of the BPM has an obvious similarity with those of the physical test, both in the intact sample and the flawed sample, indicating the models are feasible.  
The numerical models and their failure modes are shown in Figure 2. The regularly arranged small circles on the model surface are measurement circles (MCs), which are used to obtain a further investigation about the process of the deformation localization of the specimen. The final collapses of the intact specimens in Figure 2a show a shear failure with shear cracks (blue dot) and a few tensile cracks (red dot). At 45\({}^{\circ}\) tips, small slippages emerged, and tensile cracks interpenetrated each other, while shear cracks quickly developed to lead to failure, as shown in Figure 2b.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Intact Specim**} & \multicolumn{2}{c}{**45\({}^{\circ}\)****Flaw Specim**} \\ \hline  & **BPM** & \
\begin{tabular}{c} **Physical** \\ **Experiment** \\ \end{tabular} & **BPM** &
\begin{tabular}{c} **Physical** \\ **Experiment** \\ \end{tabular} \\ \hline Uniaxial compressive strength & 27.58 & 26.96 & 18.9 & 19.12 \\ (UCS)/MPa & 3.2 & 3.0 & 2.9 & 3.0 \\ (P Poisson’s ratio & 0.24 & 0.17 & 0.28 & **–** \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties of physical experiment and numerical study.  
Figure 1: Stress–strain curves of PFC models and tests.