## 3 Experimental Results and Discussion
### Damage Model of Stress-Strain Relationship
#### 3.6.3 Model Verification  
In order to verify the effectiveness of the proposed model in this work, it has been validated by means of comparison with the predictions and experimental results. Figures 11 and 12 show the comparison of the results in terms of mean and STD. of the damage variable \(d\) and stress-strain curves for CA mortar specimens, respectively. It is observed that a very good matching between the predicted and experimental results in both of the two figures is obtained. However, in Figure 11a, the predicted  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{b}_{0}\) & \(\mathbf{b}_{1}\) & \(\mathbf{b}_{2}\) \\ \hline
0.001 & 0.237 & 42.592 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Identification results of parameters of STD. of the stress–strain curve \(STD_{\sigma}(\varepsilon)\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{A}_{1}\) & \(\mathbf{A}_{2}\) & \(\varepsilon_{0}\) & \(\mathbf{p}\) \\ \hline \(-0.013\) & 0.939 & 0.0081 & 3.2578 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Identification results of parameters of mean of the damage variable \(\mu_{d}(\varepsilon)\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{A}_{3}\) & \(\mathbf{A}_{4}\) & \(\mathbf{w}\) \\ \hline
0.12125 & 0.00814 & 0.00393 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Identification results of parameters of standard variation (STD) of the damage variable \(STD_{d}(\varepsilon)\).  
In summary, by examining the comparison results between the predictions and the experimental results in this work, it is verified that the proposed model is able to characterize the stochastic constitutive relationship (nonlinear and random stress-strain behaviors) of CA mortar under uniaxial monotonic compression with clear physical meaning based on a statistical damage approach, while some researchers used empirical relationships to predict certain mechanical behaviors of materials [11, 22, 24, 25, 48, 59, 60].  
In addition, from a practical point of view, although the proposed model can capture the nonlinearity and randomness of the stress-strain behavior, a number of models based on empirical and theoretical approaches proposed in foregoing studies [8, 21, 22, 23, 24, 25, 26, 27, 28, 29, 35, 46, 47, 48, 55, 60, 63] also provide lucid and acceptable results to predict relevant mechanical properties of certain materials. Thus, in order to further verify the applicability and accuracy for the proposed model, a comparison between the predicted mean stress-strain curve by using the proposed model and the model developed by [PERSON] et al. mentioned in the literature [35] were conducted. In detail, three stress-strain curves of CA mortar under uniaxial monotonic compressive test with strain rates of \(0.0033/\mathrm{ms^{-1}}\), \(0.17/\mathrm{ms^{-1}}\), and \(1.7/\mathrm{ms^{-1}}\) were selected for the verification process, and the results are illustrated diagrammatically in Figure 13. The observation shows that the predicted stress-strain curve generated by the proposed model is closer to the experimental results. Specifically, in Figure 13a,b, a certain degree of underestimations of the stresses in the ascending branch and overestimations in the descending branch are found for the predictions by [PERSON] et al. In Figure 13c, it seems that the differences in the predicted curves by these two models are relatively small. Furthermore, by considering the lack of capability for characterizing the stochastic behaviors (variance), the applicability of the proposed model is wider, to some extent, compared with the presented model mentioned in this work.  
Figure 12: Comparison between experimental and theoretical stress–strain curves for CA mortar: (a) Mean stress–strain curve; (b) STD. of stress–strain curve.  
Figure 13: _Cont._