# Simulation of the Fracturing Process of Inclusions Embedded in Rock Matrix under Compression
## 2 Fracturing Behaviors of Single Inclusion Embedded in Rock Matrix  
### Stress Distribution Inside an Inclusion  
Before investigating the fracture spacing behavior in an isolated inclusion, the stress distribution is firstly illustrated by using the RFPA code. Figure 2 is the employed numerical model in this study, in which an isolated inclusion is embedded in the rock matrix. The geometry of the whole model is 1200 mm \(\times\) 1200 mm and the diameter of the inclusion is 200 mm. The specimen is meshed into 300 \(\times\) 300 = 90,000 mesoscopic elements. The vertical compressive far-field stresses are applied to the top boundary. The model bottom is fixed in the Y-direction. The mechanical parameters for the rock sample are listed in Table 1. [PERSON] and [PERSON] (1992) [1] have mentioned that the Poisson ratio \(v_{i}\) of the inclusion has a small effect on the stress solution, whereas the Poisson ratio \(v_{r}\) of the matrix has a profound effect. Thus, the Poisson ratio \(v_{i}\) of the inclusion is constant, i.e., 0.25 in the study. In this section, the homogeneity index, \(m\), is chosen to be 100.0, representing the assumption that the rock sample is a homogeneous material. The influence of material heterogeneity on fracturing behavior will be discussed in detail in the following sections. [PERSON] and [PERSON] (1979) [7] proposed that the analytical solution of stress inside an inclusion under the constraints can be expressed as follows:  
\[\sigma_{(\max)i}=\sigma_{(\max)r}\frac{[K(X_{r}+2)+X_{i}]K[X_{r}+1]}{2[2K+X_{i}-1 ][XX_{r}+1]}+\sigma_{(\min)r}\frac{[K(X_{r}-2)-X_{i}+2]K[X_{r}+1]}{2[2K+X_{i}-1 ][XX_{r}+1]} \tag{1}\]  
\[\sigma_{(\min)i}=\sigma_{(\max)r}\frac{[K(X_{r}-2)-X_{i}+2]K[X_{r}+1]}{2[2K+X_{ i}-1][KX_{r}+1]}+\sigma_{(\min)r}\frac{[K(X_{r}+2)+X_{i}]K[X_{r}+1]}{2[2K+X_{i}-1 ][KX_{r}+1]} \tag{2}\]  
\[K=\mu_{i}/\mu_{r},\ X_{r}=3-4\upsilon_{r},\ X_{i}=3-4\upsilon_{i}, \tag{3}\]  
where \(\sigma_{(\max)r}\) and \(\sigma_{(\min)r}\) are the maximum and minimum compressive remote stress, respectively. \(\sigma_{(\max)i}\) and \(\sigma_{(\min)i}\) are the maximum and minimum compressive stress inside the inclusion, as shown in Figure 2. \(\mu_{i}\) and \(\mu_{r}\) are the shear moduli, and \(\upsilon_{i}\) and \(\upsilon_{r}\) are the Poisson ratios of the inclusion and the matrix, respectively. Figure 3 shows the analytical result and numerical result of stress varying with the Poisson ratio of the matrix. Figure 3a is the stress variation for the case under uniaxial \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Inclusion** & **Matrix** \\ \hline Homogeneity index (\(m\)) & 100 & 100 \\ Young’s modulus (E)I/GPa & 100 & 5 \\ Uniaxial compressive strength(\(\ell_{0}\))/MPa & 50 & 100 \\ Tensile strength (\(\ell_{10}\))/MPa & 2.5 & 10 \\ Poisson ratio (\(\upsilon\)) & 0.25 & 0.25, 0.35, 0.4, 0.45, 0.5 \\ Friction angle (\(\varphi\))/\({}^{\circ}\) & 30 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physico-mechanical parameters employed in the model with single inclusion. Figure 2: Numerical model with a single circular inclusion embedded in matrix under compression conditions. \(\sigma_{(\max)r}\) is the load of 0.02 MPa/step applied in Y-direction of specimen. The yellow dotted rectangle (1/3 area of entire model) refers to the zone for making analysis of fracturing mode and stress distribution. The dotted red line A–A’ is the cross-section of the inclusion for the stress analysis in the following sections. compressive loading, i.e., \(\sigma_{\text{(min)}r}=0\). Figure 3b is the stress variation for the case under biaxial compressive loading, i.e., \(\sigma_{\text{(min)}r}=0.002\) MPa/step, \(\sigma_{\text{(max)}r}=0.02\) MPa/step. It is shown that the numerical solutions agree well with the analytical solutions in both cases. The value of \(\sigma_{\text{(min)}}/\sigma_{\text{(max)}r}\) inside the inclusion is changed from positive to negative with the increase in the Poisson ratio \(v_{r}\) of the matrix.