## 4 Numerical Simulation of the Adhesive-Rebar Interface
### Establishment of FE Model
The selection of each feature point is determined as follows:  
Figure 10: Schematic diagram of constitutive relationship. The average bond stress under the ultimate drawing bearing capacity of the planting bar was calculated by ignoring the influence of depth, and the formula is as follows:  
\[\tau_{u}=\frac{P_{u}}{\pi DL} \tag{1}\]  
where \(P_{u}\) is the ultimate drawing load of the planting bar, \(D\) is the diameter of the steel bar, and \(L\) is the anchor depth. (2) Elastic stage ultimate bond stress  
\[\tau_{e}=p\tau_{u} \tag{2}\]  
where \(p\) is the proportional coefficient of the elastic stage ultimate bond stress and the average ultimate bond stress; the data obtained by the test were taken as 0.71. (3) Ultimate load slip displacement  
The ultimate load slip displacement was the corresponding displacement when the specimen reached its ultimate drawing bearing capacity. The test conclusion and analysis were obtained as follows: the size of the ultimate load slip displacement was related to the fatigue loading times, the anchor depth and the steel bar diameter. Assuming that the ultimate load slip displacement satisfied the multivariate equation, and the calculation coefficients and constant terms were set, the expression of the ultimate load slip displacement is as follows:  
\[S_{u}=t_{1}L^{2}+t_{2}N^{2}+t_{3}D^{2}+t_{4}LN+t_{5}LD+t_{6}ND+t_{7}L+t_{8}N+t _{9}D+a \tag{3}\]  
where \(N\) is the fatigue loading times, \(t_{i}\) is the calculation coefficient, and \(a\) is the constant term. Using MATLAB to complete the calculation coefficients and the constant terms in Equation (3), the expression of the ultimate load slip displacement is as follows:  
\[ \begin{array}{ll}S_{u}&=2.34\times 10^{-4}L^{2}+1.3341\times 10^{-2}N^{2}+3. 804\times 10^{-2}D^{2}-2.906\times 10^{-3}LN\\ &+1.098\times 10^{-3}LD+1.21127\times 10^{-1}ND-4.7863\times 10^{-2}L-4.55822 \times 10^{-1}N\\ &+1.65341\times 10^{-1}D+0.345365\end{array}  \tag{4}\]  
Substituting the parameter information of the test specimen into Equation (4) for inverse calculation, the fitting function R\({}^{2}\) = 0.9637 of the ultimate load slip displacement was obtained. The calculated value of the formula is similar to the measured value of the test, and the fitting result is better. (4) Elastic stage ultimate slip displacement  
The ultimate slip displacement of the elastic stage is the corresponding displacement when the specimen reaches the peak of the drawing bearing capacity of the elastic bonding section, and the calculation formula is as follows:  
\[S_{e}=qS_{u} \tag{5}\]  
where \(q\) is the proportional coefficient of the elastic stage ultimate slip displacement and the ultimate load slip displacement; the data obtained by the test were taken as 0.13. (5) Residual slip displacement  
\[S_{r}=8(mm) \tag{6}\]  
(6) Residual bond strength  
\[\tau_{r}=0.5\tau_{u} \tag{7}\]According to the Equations (1)-(7) and the geometric characteristics of each stage's curve, the expression about \(\tau\)-\(S\) was obtained as follows:  
(1) Elastic bond stage  
According to the geometric characteristics of the curve shape, it can be assumed as a directly proportional function, and the expression of this section can be obtained by using the extreme point \((S_{e},\tau_{e})\) of the elastic stage:  
\[\tau=\frac{p\tau_{u}}{qS_{u}}S\ \ S\in[0,S_{e}] \tag{8}\]  
(2) Elastic-plastic slip stage  
According to the geometric characteristics of the curve shape, it can be assumed as a vertex form of quadratic function. Using the extreme points \((S_{e},\tau_{e})\) of the elastic stage and the extreme points \((S_{u},\tau_{u})\) of the whole curve, the expression of this section can be obtained:  
\[\left\{ \begin{array}{l}\tau=\lambda\frac{\tau_{u}}{S_{u}^{2}}S^{2}-2\lambda \frac{\tau_{u}}{S_{u}}S+(1+\lambda)\tau_{u}\ \ S\in[S_{e},S_{u}]\\ \lambda=(p-1)/(q-1)^{2}\end{array} \right.