## 6 Bond-Slip Constitutive Model  
The fracture damage of the bolts occurred with an anchorage length of more than 300 mm in this study. Figure 11 shows the load-displacement curves for these specimens with different anchorage lengths. It can be seen that the curves can be divided into two stages: one stage extends from the start to the bolt yielding point, which is called the ascending stage, and another stage extends from the bolt yielding point to the bolt fracturing point, which is called the intensifying stage. The experimental data were compared with the ascending section of the bond-slip experimental model developed by [PERSON] [30] and the bond-slip experimental model by [PERSON] [31], as shown in Figure 15. As can be seen in Figure 15, the experimental data differ significantly from the two bond-slip curves; thus, neither of the curves can represent the bond-slip constitutive relationship in this experiment. Therefore, the bond-slip constitutive model of the fractured bolts was investigated by fitting the bond-slip curve.  
Figure 14: Schematic diagram of the bond strength fitting curve of the fractured bolt.  
Upon analyzing the ascending stage of the fitted curve, a power function can be used to describe the change law of these curves. Referring to CEB-FIP 2010 [32] in regard to the curve expression of the ascending stage in the bond-slip relationship under monotonic loading, the power function can be written as:  
\[\tau=\tau_{\text{u}}\left(\frac{s}{s_{1}}\right)^{\alpha} \tag{12}\]  
where \(s\) is the bolt displacement (mm), while \(s_{1}\) is the corresponding displacement of the bolt when it yields (mm).  
The power function was chosen to fit the ascending stage, and the fitting results are shown in Figure 16.  
Since the ascending stage of the bond stress-displacement curve is similar, the coefficients of each fitting formula can be averaged to obtain the bond stress-displacement constitutive model for the ascending stage of the fractured bolt, as well as the average values of six specimens, which is given by:  
\[\tau=\tau_{\text{u}}\left(\frac{s}{s_{1}}\right)^{0.6434};\quad(0\leq s\leq s_ {1}) \tag{13}\]  
The intensifying stage corresponds to the process that occurs between the bolt yielding and fracturing. By observing the curve-change pattern, the power function is selected to fit the intensifying stage, and the fitting results are shown in Figure 17.  
Figure 15: Comparison between different bond–slip constitutive curves.  
Figure 16: Fitting schematic diagram of the bond stress–displacement at the ascending stage. (**a**) Fitting ascending stage with an anchorage length of 300 mm. (**b**) Fitting ascending stage with an anchorage length of 400 mm (**c**) Fitting ascending stage with an anchorage length of 500 mm.  
The fitting results are treated in the same way as the ascending stage, and the bond stress-displacement constitutive model for the intensifying stage of the fractured bolt is obtained as follows:  
\[\tau=0.79905\tau_{\max}s^{0.09596};\quad(s_{1}<s\leq s_{2}) \tag{14}\]  
Figure 17: Fitting schematic diagram of the bond stress–displacement relationship at the intensifying stage. (**a**) Fitting intensifying stage with an anchorage length of 300 mm. (**b**) Fitting intensifying stage with an anchorage length of 400 mm. (**c**) Fitting intensifying stage with an anchorage length of 500 mm.  
where \(r_{\rm max}\) is the maximum value of bond stress (MPa), while \(s_{2}\) is the corresponding bolt displacement that occurs when the bolt fractures (mm).