## 3 Experimental Results and Discussion
### Energy Dissipation  
Energy dissipation reveals the damage evolution process for material under external load. In a closed system, the total input energy of certain material in monotonic compression case could be obtained by the following equation [8; 61; 62]:  
\[U=U_{d}+U_{e} \tag{6}\]  
where \(U,~{}U_{d},~{}U_{e}\) denote the total input energy, dissipated strain energy, and elastic strain energy, respectively. The total input energy \(U\) can be calculated as:  
\[U=Ah\int\sigma d\varepsilon \tag{7}\]  
where \(A\) and \(h\) denote the cross-section area and the height of the specimen, respectively. The elastic strain energy is calculated as:  
\[U_{e}=\frac{Ah\alpha^{2}}{2E_{0}} \tag{8}\]  
where \(E_{0}\) denotes the Young's modulus of the material. It is worth stressing that when a material is subjected to an external load less than its yield strength, the difference between unloading modulus \(E_{u}\) and [PERSON]'s modulus \(E_{0}\) is negligible. However, by adopting the above equation, after the external load exceeded its yield strength, an underestimation will have occurred in the results of the elastic strain energy \(U_{e}\). Nevertheless, due to the fact of lacking experimental results for determining the unloading modulus \(E_{u}\) and for simplicity purposes, the modulus adopted for relative calculations is selected as the [PERSON]'s modulus \(E_{0}\) in this work with reference to the literature [8; 62]. Therefore, the dissipated strain energy \(U_{e}\) can be calculated as:  
\[ \begin{array}{ll}U_{d}&=U-U_{e}\\ &=Ah(\int\sigma d\varepsilon-\frac{\alpha^{2}}{2E_{0}})\end{array}  \tag{9}\]  
Since the specimens for this test were of the same size, the input strain energy, dissipated strain energy, and elastic strain energy can be further expressed as:  
\[AhW=AhW_{d}+AhW_{e} \tag{10}\]  
where \(W,~{}W_{d},~{}W_{e}\) denote the total input strain energy density, dissipated strain energy density, and elastic strain energy density, respectively. Hence, the relationship among the input strain energy density, dissipated strain energy density, and elastic strain energy density can be expressed as:  
\[W=W_{d}+W_{e} \tag{11}\]  
where \(W,~{}W_{d},~{}W_{e}\) denote the total input strain energy density, dissipated strain energy density, and elastic strain energy density, respectively. A schematic diagram is illustrated in Figure 6.  
Specifically, in Figure 6, the boundary of strain energy density is defined as a straight line with slope equals to the Young's modulus of material. By constructing a line perpendicular to the horizontal axis from the point of intersection between the boundary of strain energy density and the stress-strain curve, the accumulative elastic strain energy density \(W_{e}\) can be addressed. Moreover, the total input strain energy density \(W\) could be represented as the enclosed area by stress-strain curve, horizontal axis, and the line perpendicular to the x-axis correspond to current strain. The total input strain energy density can be calculated by following equation:  
\[W=\int_{0}^{e}\sigma d\varepsilon \tag{12}\]  
By assuming the strain energy boundary intersects with the stress-strain curve at point \((\varepsilon_{1},\sigma_{1})\), it can be expressed as:  
\[\sigma=E_{0}\varepsilon+q \tag{13}\]  
where \(q\) denotes the intercept between strain energy boundary and vertical axis. Then the elastic strain energy density can be calculated by Equation (14):  
\[W_{e}=\frac{1}{2}\bigg{(}\varepsilon_{1}-\frac{E_{0}\varepsilon_{1}-\sigma_{1 }}{E_{0}}\bigg{)}\sigma_{1} \tag{14}\]  
The dissipated strain energy density is determined by the following equation:  
\[W_{d}=W-W_{e} \tag{15}\]  
The calculated results of relative energy density with increased strain are shown in Figure 7. In Figure 7a, the total input strain energy density of CA mortar specimens under monotonic compression tests are given, with the mean and STD. curves illustrated in Figure 7b.