## 3 Simulation Results and Analysis
### Mechanical Analysis Under Compression Conditions
Therefore, the failure mode of horizontally bedded shale under uniaxial compressive stress is closely linked to the bedding structure, with the presence of bedding planes substantially reducing the overall strength of the rock mass. When the axial compressive stress \(\sigma_{z}\) exceeds the minimum strength threshold within the shale, shear failure occurs. This manifests macroscopically as shear cracks propagating along the bedding plane in the direction of axial stress (Figure 9d). Figure 15 illustrates the stress state of horizontally layered shale under triaxial compression, where the horizontal strain relationships are consistent with those observed under uniaxial compression. From the schematic of the layered structure, it is evident that, in addition to the axial compressive stress \(\sigma_{z}\), triaxial compression is also influenced by the combined effects of horizontal compressive stresses \(\sigma_{x}\) and \(\sigma_{y}\). For clarity, the structure is divided into parts A (matric) and C (bedding plane) and parts B (matric) and C. Due to differences in horizontal strain, frictional constraint stresses are generated at the contact surfaces between parts A and C. Specifically, the matrix is subjected to compressive stress states (\(\sigma_{xA}^{\prime}\) and \(\sigma_{yA}^{\prime\prime}\)), while the bedding plane experiences tensile stress states (\(\sigma_{xC}^{\prime}\) and \(\sigma_{yC}^{\prime\prime}\)). Consequently, the actual stress state is the superposition of the original stress and the frictional constraint stresses. As depicted, the frictional constraint stress increases the  
Figure 14: Horizontal bedding plane shale specimens under uniaxial loading conditions. horizontal stress states (\(\sigma_{xA}^{+}\) and \(\sigma_{yA}^{+}\)) in the matrix beyond the original horizontal stresses \(\sigma_{x}\) and \(\sigma_{y}\), with increments of \(\sigma_{xA}^{\prime}\) and \(\sigma_{yA}^{\prime}\), respectively. Conversely, the horizontal stresses \(\sigma_{\mathrm{xC}}^{*}\) and \(\sigma_{\mathrm{yC}}^{*}\) in the bedding plane are reduced compared to the original stresses, with reductions of \(\sigma_{\mathrm{xC}}^{\prime}\) and \(\sigma_{\mathrm{yC}}^{\prime}\), respectively. Thus, at the contact surface between parts A and C, the stress state is influenced by the frictional constraint stress. In contrast, the area outside the contact surface remains unaffected by these constraints, with the triaxial stresses remaining at \(\sigma_{x}\), \(\sigma_{y}\), and \(\sigma_{z}\). Similarly, at the contact surface between parts B and C, the frictional stresses on the matrix are compressive stress states (\(\sigma_{xB}^{\prime\prime}\) and \(\sigma_{yB}^{\prime\prime}\)), while those on the bedding plane are tensile stress states (\(\sigma_{xC}^{\prime\prime}\) and \(\sigma_{yC}^{\prime\prime}\)). As a result, the horizontal stress in the matrix increases from \(\sigma_{x}\) and \(\sigma_{y}\) to \(\sigma_{xB}^{*\ast}\) and \(\sigma_{yB}^{\prime\prime}\), with increments of \(\sigma_{xB}^{\prime\prime}\) and \(\sigma_{yB}^{\prime\prime}\), respectively. The horizontal stress in the bedding plane decreases from \(\sigma_{x}\) and \(\sigma_{y}\) to \(\sigma_{\mathrm{xC}}^{*\ast}\) and \(\sigma_{yC}^{*\ast}\), with reductions of \(\sigma_{xC}^{\prime\prime}\) and \(\sigma_{yC}^{\prime\prime}\), respectively. Therefore, the frictional constraint stress causes the stress state at the contact surface between B and C to differ from that outside the contact surface. From this analysis, it is found that the confining pressure greatly complicates the stress state of shale under triaxial compression. However, a comparative study of the uniaxial and triaxial stress states reveals certain similarities: the failure characteristics of horizontally bedded shale are closely linked to its layered structure, particularly at the contact areas between the bedding and the matrix.