## 3 Verification of Proposed Model
### Carbonation Depth of Cracked Concrete
By using the experimental results on carbonation depth for sound concrete, the CO\({}_{2}\) diffusivity parameters A and a in Equation (5) were set as 2.66\(\times\)10\({}^{-6}\) and 4.75\(\times\)10\({}^{-6}\), respectively. These CO\({}_{2}\) diffusivity parameters did not change with different water to cement ratios. The calculated results regarding the carbonation depth of sound concrete are shown in Figure 3. Concrete with lower water-to-cement ratios showed lower carbonation depths as concrete with a lower water to cement ratio has higher carbonatable material contents and lower porosity (shown in Figure 2). According to [PERSON] et al. [5], cracks were induced into concrete through splitting tests. A Linear Variable Differential Transformer (LVDT) was used for measuring crack width in a fully unloaded state. It is very difficult to obtain the required crack width, so [PERSON] et al. [5] tried to determine the crack width by 0.1 mm-division after the splitting test. In this study, 0.05 mm cracks ranged from 0 to 0.1 mm; 0.15 mm cracks ranged from 0.1 mm to 0.2 mm; and 0.25 mm crack ranged from 0.2 mm to 0.3 mm. In Equation (7), the value of \(\frac{\omega}{T}\) equals the area ratio between the crack zone and total surface of concrete. The value of \(\frac{D_{M}}{D}\) was regressed based on the experimental results of the carbonation depth of cracked concrete. First, for each water to cement ratio such as 0.65, 0.55 and 0.45, based on the comparison of carbonation depths of sound concrete and cracked concrete by using the equivalent CO\({}_{2}\) diffusivity (Equation (8)), the value of \(\frac{D_{M}}{D}\) was determined. It was found that \(\frac{D_{M}}{D}\) was independent on crack width and was only dependent on the water to cement ratio of concrete. Second, linear  
Figure 3: Carbonation depth of sound concrete. regression was used to find the relationship between \(\frac{D_{cr}}{D}\) and the water to cement ratio. We proposed that the relationship between \(\frac{D_{cr}}{D}\) and the water-to-cement ratio was set as follows:  
\[\frac{D_{cr}}{D_{28}}=79223-102299\frac{W}{C} \tag{14}\]  
where \(D_{28}\) is the CO\({}_{2}\) diffusivity after 28 days of curing. Figure 4 shows the ratio of \(\frac{D_{\alpha 1}}{D}\) versus the crack width for different water-to-cement ratios. One hundred mm and 150 mm were used as the representative values of crack interspace. As shown in this figure, the ratio of \(\frac{D_{\alpha 1}}{D}\) increased with crack width. With the decrease of water-to-cement ratios, the ratio of \(\frac{D_{\alpha 1}}{D}\) increased. Hence, high strength concrete is more vulnerable due to cracking than low strength concrete. With the increase of crack interspaces, the ratio of \(\frac{D_{\alpha 1}}{D}\) decreased. It should be noted that in Equation (14), the curing period of concrete before the start of the carbonation tests was 28 days. When the concrete curing periods before the carbonation tests changed, the porosity and CO\({}_{2}\) diffusivity of concrete will vary correspondingly. By dividing time dependent CO\({}_{2}\) diffusivity \(D(t)\) at the left- and right-hand sides of Equation (14), Equation (14) can be transformed as follows:  
\[\frac{D_{cr}}{D(t)}=\left[79223-102299\frac{W}{C}\right]*\frac{D_{28}}{D(t)} \tag{15}\]  
After this transformation, the effect of the curing period on CO\({}_{2}\) diffusivity was considered. By using Equation (15) and the carbonation reaction (Equation (8)), the carbonation depth of cracked concrete with different curing periods could be calculated. In this study, we used equivalent CO\({}_{2}\) diffusivity to calculate the carbonation depth of cracked concrete. As we considered the fundamental mechanisms involved in carbonation such as hydration, carbonation and equivalent CO\({}_{2}\) diffusivity, the calculation could be carried out not only within the experimental period but also in the extension period beyond the carbonation experiment.