## 6 Conclusions  
The deformation damage characteristic was evaluated using the finite element program RFPA2D Basic V2.0, based on three-point bending tests of semicircular beam (SCB) specimens, and considering the layer inclination and strength. The following are the key findings of this study:  
Under the influence of the weak surface and dip angle of laminae, the mechanical properties of sandstone have significant anisotropy. When the lamina strength is 0.3 and 0.6 times that of the rock matrix, the peak stress tends to decrease and then increase with the increase in the lamina dip angle, the overall shape is similar to a \"U\", and \(\theta=60^{\circ}\) is the cutoff point, i.e., the peak stress is the smallest at this time; under the same lamina dip angle, the peak stress gradually increases with the increase in the lamina. The failure mode of sandstone is controlled by the joint control of the weak surface of the lamina and the dip angle. When \(\theta=0^{\circ}\) and \(15^{\circ}\), the weak plane will show advanced fracture along the bedding direction, and the secondary cracks will lead the main crack to migrate along the bedding direction and then hinder the main crack from expanding in the direction of the loading. When \(\theta=30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), \(75^{\circ}\), or \(90^{\circ}\), several cracks appear on the weak surface along the bedding direction, and one of the cracks finally competes successfully and begins to expand. When \(\theta=30^{\circ}\), the cracks partially break through the bedding, while at other angles the cracks continue to expand along the bedding direction until the end of failure. The number of AE events corresponds to the failure mode, and the distribution range of AE events is the widest when the sample has a mixed fracture of cross-layer fracture and the fracture along the bedding plane. The distribution range of AE events is the smallest when the fracture is completely along the bedding plane. When pure translaminar rupture occurs, AE events have a small distribution range. When the strength of the bedding is weaker than that of the rock matrix, the cumulative number of AE events at \(\theta=0^{\circ}\) and \(15^{\circ}\) gradually decreases with the strengthening of the bedding. The cumulative number of AE events at \(\theta=30^{\circ}\) first increases and then decreases, while the cumulative number of AE events at the remaining dip angles gradually increases, and when the strength of the laminae reaches that of the rock matrix, the cumulative number of AE events at all dip angles tends to be equal. There was no preset crack in this study, and the failure type was not affected by the bedding strength and dip angle. According to the AE location maps of each sample, it can be seen that the failure type was a tensile failure. By observing the whole process of crack propagation, the failure mode of the samples can be divided into the following four types: pure lamellar tensile failure; dominant translaminar tensile fracture, accompanied by the migration of the main crack along the bedding plane; dominant tensile fractures along the bedding, with transbedding tensile fractures occurring locally; and complete tensile fracture along the bedding. Conceptualization, G.W. and T.M. ; methodology, T.M., G.W. and G.F.; investigation, G.W. and T.M. ; resources, L.M. ; software, W.Y. ; formal analysis, Z.Z. ; writing--original draft preparation, Z.Z. ; writing--review and editing, Y.H. ; project administration, X.L. All authors have read and agreed to the published version of the manuscript.