## 3 Results and Discussion
### The Constitutive Model of Bond-Slip between RBFC and Rebar
The fifth stage is the residual section (after D), in which only part of the friction and residual mechanical occlusion forces remain, and the main characteristic is that the load values are basically stabilized at a steady value. Due to the small value of \(s_{0}\) (between 0.05 and 0.2 mm), the artificially defined inflection point A has a large error. The empirical formulas fitted to \(s_{0}\) are lower due to the greater dispersion of the data. Therefore, the model is simplified by defining the 0.05 mm displacement (\(\tau\)) as the initial bond stress (\(\tau_{0}\)) in the micro-slip section, and no displacement occurs during the micro-slip stage (\(s_{0}=0\)). Therefore, the model is established by focusing on the micro-slip section. The simplified model is shown in Figure 14. Points A-D are the inflection points for each segment. The equations of the modified model are shown in Equations (2)-(6):  
\[\tau:0\rightarrow\tau_{0} \tag{2}\]  
rising section (\(0<s\leq s_{g}\)):  
\[\tau=\tau_{0}+s(\frac{\tau_{g}-\tau_{0}}{s_{g}}), \tag{3}\]  
Figure 14: Simplified bond stress–displacement relationship diagram. slipping section (\(s_{g}<s\leq s_{u}\)):  
\[\tau=\tau_{g}+\big{(}\tau_{u}-\tau_{g}\big{)}\bigg{(}\frac{s-s_{g}}{s_{u}-s_{g}} \bigg{)}^{a}, \tag{4}\]  
descending section (\(s_{u}<s\leq s_{c}\)):  
\[\tau=\tau_{u}-\bigg{(}\frac{\tau_{u}-\tau_{c}}{s_{c}-s_{u}}\bigg{)}(s-s_{u}), \tag{5}\]  
residual section (\(s>s_{c}\)):  
\[\tau=\tau_{c}. \tag{6}\]  
To avoid the problem of large scatter in the data set, the mean of their associated eigenvalues was taken. The statistics related to the RBFC were organized to obtain the empirical parameter statistics in Table 9. The value \(a\) was negatively correlated with bonding length and rebar diameter. This indicates that for specimens with larger bonding lengths and rebar diameters, the slipping section lasted longer and the concrete in front of the rib experienced a longer crushing section. Individual factors such as bonding length and rebar diameter had a greater effect on bond strength. Therefore, the bonding length (\(l\)) and the rebar diameter (\(d\)) were selected for this 2D surface fitting. Substituting the fitting results into the fitting formula, as shown in Equation (7), provided an empirical formula for each parameter:  
\[\left\{ \begin{array}{ll}s_{g}=-0.00036d^{2}+0.0580d+0.0042l^{2}-0.1367l-0.06 78\\ s_{u}=-0.00015d^{2}+0.0223d-0.0017l^{2}+0.0250l+1.2444\\ s_{c}=0.00038d^{2}-0.0787d-0.0125l^{2}+0.3033l+7.4267\\ \tau_{0}=0.00049d^{2}-0.0909d-0.0013l^{2}+0.0108l+6.8867\\ \tau_{g}=-0.00318d^{2}+0.3763d+0.0592l^{2}-1.8317l+19.5856\\ \tau_{u}=0.00130d^{2}-0.3618d+0.0125l^{2}-0.7867l+46.4067\\ \tau_{c}=0.00094d^{2}-0.2103d-0.0050l^{2}+0.1817l+16.3533\\ a=0.00026d^{2}-0.0553d+0.0025l^{2}-0.0283l+2.9933\\ \end{array} \right. \tag{7}\]  
The test values of all rebar and RBFC specimens were verified using the constitutive model, and the results are shown in Figure 15.  
As shown in Figure 15, the bond stress-slip curves of the simulated and test values fit well. In particular, there was a high degree of overlap for the rising and slipping sections of the stress-slip curves. It is shown that the bond stress-slip curve obtained by fitting the test data can characterize the bonding properties between the deformed rebar and the RBFC.