# Bonded-Particle Model with Nonlinear Elastic Tensile Stiffness for Rock-Like Materials
## 2 Method Description
### Specim Preparation  
Uniaxial compression tests and direct tension tests are simulated in this study. The rock specimens used for simulations are created by the particle generating method proposed in [44], which can create dense and stable particle packings with disks once the geometric contour of the specimen and the minimum and maximum particle size are given. The effect of the size of particles and specimens on our method is not concerned in this study. Thus, square specimens with length \(L=50\) mm created by the method from [44] with \(R_{max}=0.9\) mm and minimum particle radius \(R_{min}=0.3\) mm are used for our simulations. Five specimens are created by using different seeds in the initiation of the particle generation, as shown in Figure 5. They are used to study the effects of the particle arrangement on the simulations. Some main parameters of the specimens are listed in Table 1, where \(R_{a}\) is the average radius of the particles and overlap ratio is the ratio of the overlap length to the sum of the radii of the two overlap particles. As presented in Figure 6, when the uniaxial compression tests are simulated, the specimens are placed between two parallel plates which are considered as rigid bodies. The bottom one is fixed and the top one moves towards the bottom one with a given constant velocity, \(V\). When the direct tension tests are simulated, the particles in the specimens near the top and bottom boundaries are fixed to the top and bottom plates respectively and have the same velocity with the plate. In order to eliminate the effect of the boundaries on the simulation results, a layer with a height of \(0.6L\) in tensile direction and a width of \(L\) perpendicular to the tensile direction is defined, which located in the center of the specimens. Only the bonds existing between the particles located in this layer are allowed to fail, similar to that in [12]. The bottom plate is fixed and the top plate moves away from the bottom one with a constant velocity \(V\). The lateral strain of the specimens is determined by the displacement of two particle groups. One particle group consists of four particles, which are closer to the middle point \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Specimens** & \(R_{a}\) **(mm)** & **Particle Number** & **Bonded Contacts** & **Porosity (\%)** & **Overlap Ratio (\(10^{-15}\))** \\ \hline Specim A & 0.544 & 2031 & 5100 & 13.77 & 5.4 \\ Specim B & 0.559 & 2006 & 5031 & 13.76 & 4.9 \\ Specim C & 0.557 & 2023 & 5089 & 13.65 & 4.7 \\ Specim D & 0.560 & 1996 & 5004 & 13.72 & 4.2 \\ Specim E & 0.564 & 1960 & 4912 & 14.02 & 6.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the specimens. Figure 5: Specimens with different particle arrangement. of the left side boundary of the specimen than other particles at the initial time. The other particle group consists of four particles, which are closer to the middle point of the right side boundary than other particles. The average distance of the two groups of particles in the direction perpendicular to the loading direction at the initial time and its variation during simulations are used to calculate the lateral strain. The axial strain is determined by the displacement of the two rigid plates. The loads are calculated by averaging the forces on the top and bottom plates. UCS and TS can be obtained by dividing the ultimate loads by the width of the specimen. The thickness of the specimen is assume to be 1 m in 2-D. \(E_{c}\), \(E_{t}\), \(v_{c}\) and \(v_{t}\) are determined at the state when the load is equal to half the ultimate load and the secant lines are used to estimate them as introduced in [8; 45; 46]. The NET-BPM model is compiled in Fortran 90 language for simulating uniaxial compression tests and direct tension tests. The values of the micro parameters in the model are presented in Table 2. The density of the particles is set to be 2900 kg/m\({}^{3}\).