## 4 Micro-Cracking Behaviors of Shale under Different Bedding Parameters
### Effects of Bedding Strength  
#### 4.1.1 Stress-Strain Responses and Micro-Crack Evolution  
This section mainly explores the impacts of bedding strength on the stress-strain responses and micro-cracking behaviors of shale under uniaxial compression loadings. The JSR in this section was taken as 0.8, 1.0 and 1.2 for both the tensile (\(\overline{\sigma}_{c}^{sj}\)) and shear strength (\(\overline{\tau}_{c}^{sj}\)) of the beddings. Further, the dip angle is taken as \(0^{\circ},30^{\circ},60^{\circ}\) and \(90^{\circ}\). The stress-strain and crack growth curves of shale are presented in Figure 7 when the JSR is 1.0. When the dip angle is \(0^{\circ}\), the axial stress increases linearly with axial strain at the pre-peak strength stage. Figure 7a shows that no micro-cracks are observed until the axial stress is at approximately 85% of the peak strength. After the peak strength of 91 MPa, the axial stress drops abruptly and shows obvious brittle failure. Simultaneously, different types of micro-cracks occur rapidly as the axial stress approaches the peak strength. This was observed by the acoustic emission of shale in the laboratory [47]. The growth rate and quantity of matrix cracks are dominant. Thus, micro-cracks occur mainly in the rock matrix. When the dip angle is \(30^{\circ}\), the stress-strain curve has no obvious change in shape (see Figure 7b), but the crack growth rate changes significantly. Particularly, the growth rate of shear crack in beddings increases significantly. At this time, shear cracks of beddings initiate just as the axial stress increases to approximately 60% of the peak strength. This is much earlier than that in the case of a dip angle of \(0^{\circ}\). When the dip angle is \(60^{\circ}\), the peak strength drops to 44 MPa (see Figure 7c). The shear slip of beddings initiates when the axial stress approaches about 80% of the peak strength and then increases considerably after the peak strength. In this circumstance, the damage of the rock matrix seems negligible. The stress-strain responses and the evolution of micro-cracks at the dip angle of \(90^{\circ}\) (see Figure 7d) are similar to those at the dip angle of \(0^{\circ}\). The matrix failure becomes dominant again. The stress-strain curve changes little in shape but the crack growth curve is different. The shear crack of beddings occurs first and then the matrix crack appears. Therefore, the dip angle is an important parameter to affect the local stress and the evolution of micro-cracking morphology. Figure 8 summarizes the variations of UCS with the dip angle under different JSR. When the dip angle is \(0^{\circ}\), \(30^{\circ}\) and \(90^{\circ}\), the UCS is almost the same. This implies that the dip angle has little impact. Further, the UCS is the lowest at the dip angle of \(60^{\circ}\) but this UCS also varies with JSR. The UCS increases from 35 MPa to 44 MPa and subsequently to 53 MPa when the JSR increases from 0.8 to 1.0 and then to 1.2, respectively. This growth rate is consistent with the increase of bedding strength. This implies that the UCS of the shale sample is the most sensitive to the bedding strength when the dip angle is \(60^{\circ}\). Similar behaviors were observed in uniaxial compression tests [3] and Brazilian splitting \begin{table}
\begin{tabular}{c c} \hline \hline
**Micro-Parameter** & **Value** \\ \hline Normal stiffness \(\overline{k}_{n}^{sj}\) (GPa/m) & 3000.0 \\ Shear stiffness \(\overline{k}_{s}^{sj}\) (GPa/m) & 2000.0 \\ Tensile strength \(\overline{\sigma}_{c}^{sj}\) (MPa) & 15.0 \\ Shear strength \(\overline{\tau}_{c}^{sj}\) (MPa) & 18.0 \\ Friction coefficient of beddings \(\mu_{f}\) & 0.6 \\ Dilation angle \(\varphi\) (degree) & 20.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Micro-constitutive parameters calibrated for beddings. tests [9]. This is because the failure pattern was changed. The micro-cracking morphology determines the UCS of shale formation. Figure 8: Variations of uniaxial compressive strength with dip angle at different JSR.