## 5 Conclusions  
In this study, the influence of grain size (\(d\)) and constant normal load (\(CNL\)) on the shear mechanical properties of rock joints was investigated through experiments and theoretical method, which further revealed the shear instability mechanism of rock joints. The following conclusions can be drawn:  
1. The grain size (\(d\)) and normal load (\(CNL\)) have significant effects on the shear mechanical properties of rock joints, such as the peak shear strength (\(\tau_{p}\)), peak shear displacement (\(u_{p}\)), post-peak modulus (\(S\)), and stress drop (\(\Delta\tau\)). On the one hand, the \(\tau_{p}\), \(u_{p}\), \(S\), and \(\Delta\tau\) of rock joints generally increase first and then decrease with  
Figure 14: Shear instability model derivation and discriminant logical diagram.  
the increase in grain size (\(d\)). On the other hand, the \(\tau_{p}\), \(u_{p}\), \(S\), and \(\Delta\tau\) of rock joints generally increase with the increase in normal load (_CNL_).
* Marble specimens with a smaller grain size have a compact structure, fewer inherent cracks, and a higher proportion of dolomite with large Mohs hardness, which improves \(\tau_{p}\) and \(u_{p}\). Marble specimens with a larger grain size have a loose structure, fewer grain boundaries, and a higher proportion of calcite with lower Mohs hardness, resulting in decreases in shear stress dispersion, \(\tau_{p}\), and \(u_{p}\). The mineral composition and microstructure have certain influences on the shear mechanical properties of rock joints.
* The established mechanical model of the shear instability of rock joints and the proposed shear instability parameter of the post-peak soften modulus (\(S_{p}^{\prime}\)) can be used to predict the possibility of the shear instability of marble joints. The shear instability behavior of marble joints is verified by comparing the empirical result (\(S_{p}^{\prime}\)) with the experimental result (\(S_{p}\)), that is, the rock joint specimens B4, B5, B6, C4, C5, and C6 are determined as shear instability.
* The proposed shear instability parameter (\(S_{p}\)) can describe the weakening behavior of the local damage shear zone, while \(S\) describes the average weakening behavior of the whole shear zone. The shear instability analysis proves that \(S_{p}\) is more suitable than \(S\) to describe the shear instability behavior of rock joints.
* This paper only studied the shear instability characteristics and influencing factors of marble joints (metamorphic rock), and the research on other types of rock is still insufficient. In future research, the shear instability characteristics and influencing factors of sandstone (sedimentary rock) and granite (igneous rock) joints will be studied to improve our understanding of the shear instability mechanism of different types of rock joints.  
Conceptualization, H.D.; methodology, H.D.; validation, B.G.; formal analysis, H.D.; investigation, H.D.; data curation, H.D.; writing--original draft preparation, H.D.; writing--review and editing, H.D. and B.G.; visualization, K.S., Q.M. and H.W.; supervision, B.G. All authors have read and agreed to the published version of the manuscript.  
This work was financially supported by the Youth Fund of the Natural Science Foundation of Henan Province (Grant No. 242300421457), Anhui Province Joint Key Laboratory of Intelligent and Green Mining of Deep Coal Resources (Grant No. IGMDCR202404), North China University of Water Resources and Electric Power Launch Fund for High-level Talents Research (Grant No. 40937), and Henan Province Science and Technology Research Project (Grant No. 242102320364).  
Not applicable.  
The data are contained within the article.  
The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.