## 5 Conclusions  
In this work, the compressive behaviors of CA mortar under uniaxial monotonic loading were experimentally investigated by taking into account the stochastic properties. In order to characterize the stochastic behaviors of CA mortar, an analytical stochastic constitutive model based on the statistical damage approach was proposed. Furthermore, the media process of transition from microscale to macroscale for CA mortar under compression and the comparisons with typical CRTS I CA mortar and concrete in terms of constitutive behaviors have also been demonstrated and conducted. Conclusions can be drawn as follows:  
1. The experimental results indicate that there are considerable variations in the mechanical responses including: compressive strength, critical strain, Young's modulus, and stress-strain curve for CA mortar specimens with the same mix proportions, even when identically prepared.
2. An analytical model based on the statistical damage approach was developed to describe the stochastic constitutive stress-strain relationship of CA mortar. The proposed model is capable of evaluating the mean and the standard deviation of stress-strain curves. The validation of the proposed model was performed by comparison between the predictions and the experimental results. It was shown that there is a relatively good coincidence between the predictions and the experimental results. In addition, a comparison of predictions in terms of the mean stress-strain curve by the proposed model and those obtained by the model in a previous study was also  
Figure 18: Comparisons of the mean constitutive relationships among CRTS II CA mortar, CRTS I CA mortar, and concrete under compression in terms of: **(a)** Probability density—\(\varepsilon\) curve; **(b)** Probability density—\(\varepsilon/\varepsilon_{critical}\) curve.  
conducted, which revealed that by adopting the proposed model, the accuracy of prediction is improved.
3. Further exploration of the behaviors of CA mortar was obtained by examining the media process in terms of transition from microscale (microstructure) to macroscale (macroseponse). It is found that the Lognormal distribution density function can well represent the damage probability density for CA mortar under compression when compared to the Weibull and Gauss distribution density functions. It is also summarized that the stochastic behaviors that emerged for CA mortar are attributed to the randomness of the damage probability density at the microscale. The media process of the transition during the loading process is represented by the stochastic evolution (SE) of specific variables at the macroscale.
4. Comparisons of the constitutive relationships among the CA mortar used in this work, typical CRTS-I type CA mortar, and concrete were conducted. It was demonstrated that the most ductile stress-strain response was obtained by CRTS-I CA mortar due to its lowest concentration in the probability density curve induced by the microstructure.  
Writing--original draft preparation, X.L.; investigation, P.L.; conceptualization, Z.Y. and Z.S.; methodology, Z.M. All authors have read and agree to the published version of the manuscript.  
This study was funded by the National Natural Science Foundation of China, grant numbers 51778632, U1434204, U1934217 and 51408614, the China Postdoctoral Science Foundation, grant numbers 2016M600675, 2017T100647 and 2018M642658, the Basic Research on Science and Technology Program of Shenzhen, grant numbers JCY20170818143541342 and [CYJ20180305123935198, the National Engineering Laboratory for High Speed Railway Construction of China, grant number HSR201903, and the Natural Science Foundation of Hunan Province of China, grant number 2017J3385.  
The authors declare no conflict of interest.