## 2 Mesoscopic Parameter Calibration and Basic Theory  
### Specim Preparation and Mesoscopic Parameter Calibration  
The cement mortar, as a typical rock-like material for laboratory uniaxial compression tests, is made from a mixture of 42.5R ordinary Portland cement, quartz sand, and water, with a mass ratio of 1:2.34:1.35, respectively. In order to promote the fluidity of cement mortar, a small amount of water reducing agent was added during the preparation of specimens. The physico-mechanical parameters of this rock-like material are listed in Table 1. The physico-mechanical properties are similar to those of typical rock materials such as sandstone, and the ratio of tensile strength to compressive strength is close to 1:10, indicating that it is a comparatively ideal rock-like material with high brittleness. Therefore, it can be used as a rock-like material to study the evolution of secondary crack propagation in brittle rocks. The specimens used in uniaxial test series are cuboid blocks with dimensions of 140 mm in height, 70 mm in width, and 40 mm in thickness. Prior to casting the cement mortar specimens, the iron piece, fixed in the mold, is smeared with a little epoxy resin on the surface. After meticulous maintenance in the mold for 24 h, two flaws, created by pulling out the thin iron pieces, are formed through the thickness of the specimens during casting in such a way that the plane of the flaws is perpendicular to the faces of the specimens. Prefabricated cracks are open cracks with a certain degree of openness, so the internal faces do not touch each other during fabrication and loading. Two flaws are always parallel to each other, and have a constant length of 12 mm. The thickness of the flaws is 1.2 mm approximately. To study the influence of the crack inclination angle on the mechanical properties and failure process of the rock mass, three flaw inclination angles of 30\({}^{\circ}\), 45\({}^{\circ}\), and 60\({}^{\circ}\) are used, and the spacing of pre-existing parallel flaws is 15 mm. Two flaws are located at the center of the specimen. Six cement mortar specimens (a total of 18 specimens) were prepared with the same flaw inclination angle, and the average values of the test data are used for analysis. The calculations performed in PFC2D (Particle Flow Code in 2 dimensions) is based on [PERSON]'s second law and a force-displacement law at the contacts. [PERSON]'s second law is used to determine the motion of each particle arising from the contact and body forces acting upon it, while the force-displacement law is used to update the contact forces arising from the relative motion at each contact [36]. Particle Flow Code (PFC) has great advantage in simulating the micro-mechanical behavior and investigating the mechanism of crack propagation in brittle materials. However, the straightforward adoption of circular (or spherical) particles cannot fully capture the behavior of complex shaped and highly interlocked grain structures [37]. Further, PFC fails to simulate the mechanical properties of brittle rock with higher internal friction angle. A parallel bond model is adopted as the numerical model of specimens, and the model size and crack layout are shown in Figure 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Compressive Strength \(\sigma_{c}\) (MPa)** & **Tensile Strength \(\sigma_{t}\)** & **Young’s Modulus E** & **Poisson’s Ratio \(\
u\)** & **Density \(\rho\)** \\  & **(MPa)** & **(MPa)** & **(GPa)** & **Ratio \(\
u\)** & **(g/cm\({}^{3}\))** \\ \hline Cement mortar & 58.25 & 5.62 & 11.63 & 0.20 & 2.38 \\ Sandstone & 20–170 & 4–25 & 3–35 & 0.02–0.25 & 2.10–2.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physico-mechanical parameters of cement mortar and Sandstone. The loading stops when the axial stress drops to 50% of the peak strength. [PERSON] et al. [38] studied the effect of loading rate on the crack propagation and failure modes of the specimen under uniaxial loading.