## 4 Bonding-Sliding Constitutive Model
### Bond-Slip Constitutive Relationship  
#### 4.2.1 Ascending Segment  
The curve between the average bond stress and the amount of slip is called the bond-slip whole-process curve. According to its characteristics, the process curve is divided into four segments: ascending segment (0-\(s_{1}\)), peak segment (\(s_{1}\)-\(s_{2}\)), descending segment (\(s_{2}\)-\(s_{3}\)) and residual segment (\(s_{3}\)-). According to the research of [PERSON] and others [16], by combining the characteristics of the bond-slip curve in the rising section, the curve of the rising section is described by Equation (7):  
\[\tau=\tau_{u}\left(\frac{s}{s_{1}}\right)^{\alpha}(0\leq s\leq s_{1}) \tag{7}\]  
where \(\tau\) is the bonding strength; \(s\) is relative slip; and \(\alpha\) is the correlation coefficient between 0 and 1. The magnitude of the upward slope of the bond-slip curve is significantly influenced by the parameter \(\alpha\). When \(\alpha=0\), the bond stress reaches its maximum value instantaneously. In contrast, when \(\alpha=1\), a linear relationship exists between the bond stress and slip. Researchers have conducted extensive studies on the coefficient \(\alpha\), proposing various values based on specific experimental outcomes. The variation in the value of \(\alpha\) can be attributed to differences in concrete strength, the geometry and the type of reinforcing  
Figure 11: Fitting values and other prediction models. bars. According to the method suggested in Fib's recent report, the coefficient \(\alpha\) is derived through the process of nonlinear curve fitting. By combining previous research findings and experimental data, a value of \(\alpha=0.4\) is obtained. The peak slip s1 is significantly affected by numerous factors, including concrete strength, embedded length, protective layer thickness, stirrup constraints and loading rate, as corroborated by a wealth of previous test results. In previous studies, the test value of s1 demonstrates minimal variation. However, given the variegated nature of each situation, s1 must be determined based on specific test data. For steel bar shear pull-out failure, s1 is assumed to be inversely proportional to the square root of concrete strength, as proposed by [PERSON] et al. [17]. However, [PERSON] et al. [18] discovered no correlation between s1 and concrete strength or constraints, but rather a primary relationship with the spacing of the steel bars. For bars with a diameter less than 25 mm, s1 can be approximately equated to 0.2 times the spacing of the transverse ribs. [PERSON] et al. [19] also found that s1 is not particularly sensitive to changes in concrete strength but rather increases with the increase in the spacing of the cross ribs (\(C_{\text{clear}}\)). The results of a statistical regression analysis revealed that the value of s1 was approximately 0.1 \(C_{\text{clear}}\). [PERSON] et al. [20] conducted a bond test on large-diameter steel bars and observed that s1 was nearly 0.07 d, which is more aligned with the findings of [PERSON] et al. The adequacy of the concrete protective layer to provide restraint or equivalent functionality to stirrups determines the mode of bond failure, which commonly presents as pull-out failure. In this scenario, the concrete between the steel bars is completely severed or crushed. The bond stress is primarily determined by the shear strength of the concrete, rather than the splitting capacity of the protective layer. Consequently, the additional restraint provided by the concrete cover or stirrups has a limited impact on the bonding behavior. Thus, the variable s1 in the failure mode of steel bar shearing and pull-out is dictated by the spacing of the steel bars' transverse ribs. The modifications in peak slip s1 and the spacing \(C_{\text{clear}}\) of the steel bars' transverse ribs based on the previous test data are depicted in Figure 12.  
Figure 12: Fitting curve of \(C_{\text{clear}}\). Note: \(\tau\) is the bond stress between steel bars and concrete. S is the value of steel bar slip.