## 4 Bonding-Sliding Constitutive Model
### Comparison and Verification of Constitutive Models  
The predicted stress-slip curve was compared with the measured average bond stress-slip curve. It can be observed that the ascending section of all prediction models aligns well with the corresponding section of the measured curve, as illustrated in Figure 16a. According to the literature [12], bonding stress in the ascending section primarily relies on chemical bonding and friction forces. The strength of concrete cement-based structures can be enhanced by incorporating basalt and polypropylene fiber, without increasing the frictional resistance between the steel bars and the concrete. Furthermore, hybrid fibers improve bonding performance by hindering crack development, and reducing crack width to ensure the fiber is in a fully tensioned state. However, there is only limited sliding between the steel and concrete in the ascending section, resulting in most fibers being in an untensioned or marginally tensioned.  
Figure 15: Bond–slip constitutive model of hybrid fiber-reinforced concrete and steel bars.  
Consequently, during the initial ascending stage of the curve, the hybrid fiber makes a negligible contribution to the bond stress between the steel bars and the concrete. The descending section of the forecast model differs from the corresponding section of the measured curve. The falling portion of the measured curve is underestimated, as illustrated in Figure 16b. Cracks emerge in the concrete at this stage, and the mixed fiber at the crack is in tension. After concrete cracks, the hybrid fiber can provide a certain degree of lateral restraint, inhibiting the rapid expansion and extension of cracks. Consequently, the bonding stress does not plummet abruptly due to concrete cracking, resulting in a smoother curve. Therefore, in the descending phase of the curve, the contribution of hybrid fibers to the bond stress between steel and concrete is underestimated. As a result, the descending segment of the prediction model differed from the descending segment of the measured curve.  
Figure 17a compares the average bond stress-slip curve obtained from [PERSON]'s experiment with the prediction model [21]. This represents the average bond stress-slip curve in comparison with the predicted model. The model demonstrates substantial agreement with the ascending section of the test curve. However, the descending section exhibited a discrepancy from the test curve. Increasing the steel fiber content results in higher bond strength, peak slip and bond toughness. Polypropylene fiber has a low modulus of elasticity; thus, it has a minimal impact on bond strength and peak slip. The prediction model presented in this study is based on basalt and polypropylene hybrid fiber concrete, while the measured curve pertains to steel fiber and polypropylene hybrid fiber concrete. Steel fiber enhances the strength and toughness of concrete more significantly than basalt. The restraint effect of steel fiber and polypropylene hybrid fiber on concrete cracking is underestimated by the prediction model.  
Consequently, the prediction model differs from the measured curve. To further validate the accuracy of the model, the measured curves from Chu [8], Byung [22], and Xiao [23] are employed for comparison. As illustrated in Figure 17b, the quantity of steel fiber has a pronounced impact on bond strength and bond toughness. The inhibitory effect of steel fiber on concrete cracks is significantly underestimated by the prediction model. The descending segment of the predicted curve lies below the measured curve, as shown in Figure 17c,d. Despite some differences, the prediction model exhibited considerable agreement with the measured curve. Thus, the prediction model proposed in this article differs from the curve obtained from the experiment; however, the discrepancy is limited. The proposed prediction model demonstrates higher accuracy and is a valuable reference.  
Figure 16: Prediction model and measured curve. (**a**) \(d\) = 14 mm; (**b**) \(d\) = 16 mm.