# Shear Properties of the Interface Between Polyurethane Concrete and Normal Concrete
## 4 Constitutive Model for PUC-NC Interface Bonding Performance  
The constitutive model for bonding performance plays a crucial role in the microstructural analysis of structures or components. The current interface bond-slip constitutive relationship can be obtained through two methods: placing strain gauges on the bonding interface to calculate the bond stress and slip at the interface through axial strain, which then allows the derivation of the bond-slip constitutive relationship; or by obtaining the load-slip curve through testing equipment or auxiliary instruments, from which the bond-slip curve can be further deduced. Table 8 presents four bonding constitutive models for analyzing the bond-slip relationship between PUC and NC in this study. The models by [PERSON] et al., [PERSON] et al., and [PERSON] et al. all assume that the failure of the bonding interface is brittle in these models, meaning the interface fails suddenly after reaching the maximum load and can no longer transmit stress. The [PERSON] et al. model effectively characterizes this process using a simplified bond-slip relationship, while the [PERSON] et al. model improves upon the Nakaba model by introducing empirical constants to enhance its applicability, particularly under varying materials and geometries. In contrast, the bilinear model assumes plastic failure dividing the bond-slip relationship into two stages: an initial linear increase, followed by a plastic stage, where the interface retains some load-bearing capacity after failure, exhibiting characteristics of partial plastic failure. Overall, the first two models focus on simulating brittle failure mechanisms, while the bilinear model considers the post-failure plastic behavior. Figure 11: SEM micrograph of PUC mixed with steel fiber: (**a**) between steel fiber and matrix; (**b**) high rate between steel fiber and matrix. Figure 12 presents the simulation results of shear strength between PUC and NC using four bond constitutive models, along with a comparison to experimental data. The error plot assesses the agreement between the simulation results and experimental data based on the diagonal line. The results indicate that the models by [PERSON] et al. and the bilinear model underestimate the shear strength of the PUC-NC interface, while the models by [PERSON] et al. and [PERSON] et al. exhibit significant scatter in their predictions. In this study, the failure mode observed in the experimental load-slip curve was brittle, which closely matches the model by [PERSON] et al. This model is fundamentally based on the assumption of brittle failure and effectively captures the failure mechanism of the bond interface after reaching the failure load. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & \\ \hline [PERSON] et al. [30] & \(\begin{cases}\tau=\tau_{max}\left(\frac{\delta}{s_{0}}\right)&if\;s<s_{0}\\ \tau=0&if\;s>s_{0}\end{cases}\) & \(\tau_{max}=1.8\beta_{\mu\
u}\times 0.202\) \\ \hline [PERSON] et al. [31] & \(\tau=\tau_{max}\left[\frac{\delta}{s_{0}}\frac{3}{2+(s/s_{0})^{2.8}}\right]\) & \(\tau_{max}=3.5f_{c}^{0.19}\) \\ \hline [PERSON] et al. [32] & \(\tau=\tau_{max}\left[\frac{\delta}{s_{0}}\frac{2.86}{1.86+(s/s_{0})^{2.86}}\right]\) & \(\tau_{max}=3.5f_{c}^{0.19}\) \\ \hline \hline \end{tabular}
\begin{tabular}{c c} \hline \hline Bilinear constitutive model [33] & \(\begin{cases}\tau=\tau_{max}\left(\frac{\delta}{s_{0}}\right)^{\alpha}&if\;s<s_{ 1}\\ \tau=\tau_{max}\frac{2\beta-s_{1}}{2s_{0}-s_{1}}&if\;s_{1}<s<s_{2}\\ \tau=0&if\;s>s_{2}\end{cases}\) & \(\tau_{max}=1.5\beta_{\mu\
u}f_{t}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Existing constitutive models for interface bonding behavior. Figure 12: Prediction of shear strength at the PUC–NC interface: (**a**) [PERSON] and Rostasy [30]; (**b**) [PERSON] et al. [31]; (**c**) [PERSON] et al. [32]; (**d**) bilinear model [33]. The proposed model is still developed within the framework of [PERSON] et al.