## 4 Shear Instability Analysis  
### Shear Instability Mechanism  
Figure 9a-e show the damage evolution diagram of shear zone asperities at different stages, as well as the accumulation and release process of elastic energy. Figure 9b shows the accumulation of elastic energy in the elastic stage (\(o\)-\(a\)), and only elastic deformation (\(u_{e}\)) occurs on both sides of the joint surface. As shown in Figure 9c, the elastic energy and elastic deformation (\(u_{e}\)) continue to increase during the yield stage (\(a\)-\(b\)) while the plastic deformation (\(u_{s}\)) occurs. Due to the different shapes, sizes, and distributions of the asperities, the strength of different areas of the shear zone is different. With the increase in shear displacement, the low strength region weakens, while the actual shear stress  
Figure 8: Influence of CNL on shear mechanical properties of rock joints. (**a**) Peak shear displacement \(\tau_{p}\), (**b**) peak shear displacement \(u_{p}\), (**c**) post-peak modulus \(S\), and (**d**) stress drop \(\Delta\tau\). and elastic deformation increase in the high-strength region, resulting in the shear stress-displacement curve gradually deviating from a straight line (e.g., the yield stage). Thus, the damage of the joint surface only concentrates in the local area of the shear zone, that is, the shear failure process of rock joints is localized. As shown in Figure 9d, the shear stress decreases after the peak, the shear zone weakens locally, and the elastic deformation rebounds (i.e., \(u_{e}\) is converted into \(u_{s}\)), and elastic energy release occurs on both sides of the joint (i.e., the shear instability). Therefore, the local damage evolution of the shear zone and the elastic deformation rebound of the specimen jointly determine the shear instability characteristics of rock joints. The shear stress-displacement curve is an average reflection of the shear deformation and failure process of rock joints, which can be divided into four stages, as shown by the black curve in Figure 9f. However, the shear displacement (\(u\)) of rock joints is composed of elastic deformation (\(u_{e}\)) and plastic deformation (\(u_{s}\)), as shown in Figure 9a-e. In the elastic stage \(o\)-\(a\), the elastic deformation (\(u_{e}\)) increases linearly, and the shear displacement (\(u\)) is mainly composed of elastic deformation (\(u_{e}\)). In the yield stage \(a\)-\(b\), plastic deformation (\(u_{s}\)) appears, and the elastic deformation (\(u_{e}\)) reaches the maximum at the peak. In the soften stage \(b\)-\(c\), the elastic deformation (\(u_{e}\)) partially recovers, and the plastic deformation (\(u_{s}\)) continues to increase, that is, \(u_{e}\) transforms into \(u_{s}\). In the residual stage \(c\)-\(d\), the elastic deformation (\(u_{e}\)) remains unchanged, while the plastic deformation (\(u_{s}\)) increases continuously. In this respect, the rock joint shear weakening process should be described by the plastic shear displacement (\(u_{s}\)) of the damage area of the shear zone rather than the shear displacement (\(u\)). Therefore, the green shear stress-plastic shear displacement curve of rock joints in Figure 9f can be obtained by removing the elastic deformation part of the black shear stress-displacement curve according to Equation (1). \[u_{s}=u-\tau/E \tag{1}\]  
Figure 9: Schematic diagram of shear instability mechanism of rock joints. (**a**) Initial stage \(o\), (**b**) Elastic stage \(o\)-\(a\), (**c**) Yield stage \(a\)-\(b\), (**d**) Soften stage \(b\)-\(c\), (**e**) Residual stage \(c\)–\(d\), (**f**) Shear stress–plastic shear displacement. \(S_{p}\) in Figure 9f is the post-peak soft modulus of the shear stress-plastic shear displacement curve, which is used to describe the shear weakening process or shear instability behavior of rock joints.