## 4 Further Discussion  
### Transition from Microscale to macroscale  
For the purpose of understanding the intrinsic random nature for materials in the aspect of varied scales, it is essential to focus on the issue of the media process in terms of the transition from microscale to macroscale. Figure 14 demonstrates the stochastic and the mean behaviors of the probability density for the damage variable \(d\) in the perspective of microscale. The probability density can be obtained by differentiating the relative damage evolution function, viz., Equation (23). Specifically, in Figure 13a, it is illustrated that different probability densities are endowed for CA mortar specimens induced by the randomness of the microstructure, consequently resulting in the varieties of the damage evolution processes during loading stage. In Figure 15, three commonly used density functions in the statistical damage approach are examined by fitting the mean curve shown in Figure 13b. It is indicated that the best agreement is obtained for CA mortar by adopting the Lognormal distribution function, which is represented in Figure 15a, when compared with the Weibull distribution function in Figure 15b and Gauss distribution function in Figure 15c. Moreover, the stochastic evolution (SE) illustrated in Figure 4 (in terms of elastic modulus degradation) and in Figure 10 (in terms of damage accumulation) are also indispensable for investigating the media process of CA mortar under loading. Precisely, the SE demonstrates the stochastic media process of transition from microscale (microstructure) to macroscale (macroresponse). It is concluded  
Figure 13: Comparison between experimental ([PERSON] et al., 2014) and theoretical curves: (**a**) Strain rate at \(0.0033/\mathrm{ms}^{-1}\); (**b**) Strain rate at \(0.17/\mathrm{ms}^{-1}\); (**c**) Strain rate at \(1.7/\mathrm{ms}^{-1}\). Comparison of Constitutive Relationships among CA Mortar for CRTS I, II and Concrete under Monotonic Uniaxial Compression  
In this section, the comparisons of the mean compressive constitutive relationships among the CA mortar studied in this work (CRTS-II type CA mortar), typical CRTS-I CA mortar, and concrete are obtained by examining on the strain-stress responses, evolutions of damage variables, and probability density. The relevant experimental data used in this sub-section are obtained from the literature [21; 51]. Figure 16a shows the compressive stress-strain responses for the three materials. It is found that the largest peak stress occurs in the stress-strain curve of concrete, which is approximately 3.5 times larger than CRTS-II type CA mortar and 10 times larger than CRTS-I type CA mortar. Figure 16b,c represents the stress-strain responses by examining the relationships of normalized strain (\(\varepsilon/\varepsilon_{critical}\)) versus stress \(\sigma\) and normalized strain (\(\varepsilon/\varepsilon_{critical}\)) versus normalized stress (\(\sigma/\sigma_{peak}\)). It is revealed that the flattest response appeared in the relevant curve for CRTS-I CA mortar. The reasons can be attributed to CRTS-I type CA mortar being more ductile when compared with concrete and CRTS-II type CA mortar, which results in the appearance of the lowest peak stress among the three materials. Figure 15: Fitting of the probability density for the thresholds: (**a**) Lognormal distribution; (**b**) Weibull distribution; (**c**) Gauss distribution. Figure 17a,b illustrated the evolution of the damage variable in terms of strain \(\varepsilon\) and normalized strain (\(\varepsilon/\varepsilon_{critical}\)). It is found that that concrete has the most intensive development of the damage, as shown in Figure 17a, when compared to CRTS-II and I type CA mortars. A more apparent appearance can be noticed in Figure 17b when the curve is constructed in terms of damage variable versus the normalized strain (\(\varepsilon/\varepsilon_{critical}\)).