## 4 Numerical Simulation
### Finite Element Model
#### 4.1.4 Plastic Damage Model  
The premise for studying the plastic damage model for RAC is reflected in the constitutive relationship. The constitutive relationship between the compression and tension in RAC is basically the same as that of ordinary concrete. The specific differences are reflected in several coefficients related to the replacement rate. The equation proposed by [PERSON] et al. (2018) was used for calculation in this study. The stress and strain at different strength levels are shown in Table 10.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Concrete Strength** & **Compressive Strength\(f_{\epsilon}\)/MPa** & \(\epsilon_{\epsilon}\) & **Tensile Strength\(f_{\epsilon}\)/MPa** & \(\epsilon_{t}\) & **Elastic Modulus/MPa** \\ \hline C20 & 16.408 & 0.00124 & 1.396 & 0.00008 & 18,480 \\ C30 & 24.647 & 0.00147 & 1.830 & 0.00010 & 23,420 \\ C40 & 34.291 & 0.00168 & 2.281 & 0.00011 & 28,510 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Characteristic parameters of RAC.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**No.** & **Section 1** & **Node 1** & **Section 2** & **Node 2** & **No.** & **Section 1** & **Node 1** & **Section 2** & **Node 2** \\ \hline  
\hline \hline \end{tabular} Note: Table cells are expressed in absolute form; C4, D4, and E4 are the columns where the x, y, and z coordinates of Section 1 are located. G4: G1143 represents the query columns, which correspond to the C4 column of Section 1; the rest can be deduced by analogy. F4: F1143 is the column for the output formula.  
\end{table}
Table 9: Relationship of node for SRRC-1.  
The plastic damage model of concrete is mainly used to provide a universal analysis model for analyzing the structure of concrete under cyclic and dynamic loads. This model is based on plastic and isotropic failure assumptions, and it can be used in unidirectional loading, cyclic loading, and other functions [39].  
The evolution of the yield or failure surface is controlled by \(\overline{\varepsilon}_{c}^{Int}\) and \(\overline{\varepsilon}_{t}^{Int}\), where \(\overline{\varepsilon}_{c}^{Int}\) represents a compressive inelastic strain and \(\overline{\varepsilon}_{t}^{Int}\) represents a tensile inelastic strain.  
\[\overline{\varepsilon}_{c}^{Int}=\varepsilon_{c}-\varepsilon_{0c}^{el} \tag{18}\]  
\[\varepsilon_{0c}^{el}=\sigma_{c}/E_{0} \tag{19}\]  
Let the plastic strain in the inelastic strain be \(\overline{\varepsilon}_{c}^{pl}\), then the proportion of the inelastic strain is \(\beta_{c}\), according to the Abaqus user manual:  
\[\overline{\varepsilon}_{c}^{pl}=\overline{\varepsilon}_{c}^{int}-\frac{d_{c}} {1-d_{c}}\times\frac{\sigma_{c}}{E_{0}} \tag{20}\]  
\[d_{c}=\frac{(1-\beta_{c})\overline{\varepsilon}_{c}^{Int}E_{0}}{\sigma_{c}+(1 -\beta_{c})\overline{\varepsilon}_{c}^{int}E_{0}} \tag{21}\]  
\[d_{t}=\frac{(1-\beta_{t})\overline{\varepsilon}_{t}^{Int}E_{0}}{\sigma_{t}+(1 -\beta_{t})\overline{\varepsilon}_{t}^{int}E_{0}} \tag{22}\]  
where \(d_{c}\) is the concrete compression damage factor; \(d_{t}\) is the concrete tensile damage factor; \(\sigma_{c}\) is the peak compressive stress of RAC; \(\sigma_{t}\) is the peak tensile stress of RAC; \(\beta_{c}\) is 0.6; \(\beta_{t}\) is 0.8; \(\overline{\varepsilon}_{c}^{int}\) is the inelastic compressive strain of RAC; \(\overline{\varepsilon}_{t}^{int}\) is the inelastic tensile strain of RAC.