## 2 Probabilistic Approach for Cracked Concrete under Carbonation  
### Carbonation Model of Sound Concrete  
Carbonation relates to both the material properties and environmental conditions of concrete. The quantity of carbonatable materials in concrete, for example, calcium hydroxide (CH) and calcium silica hydrate (CSH), relies on the cement content in the concrete mixing proportions and reaction amount of cement. In line with the hydration model within our previous studies [1; 9], the CH contents can be established by the following:  
\[CH(t)=RCH_{CE}*C_{0}*\alpha \tag{1}\]  
where \(RCH_{CE}\) is the mass of CH from the hydration of 1-unit mass cement; \(C_{0}\) is cement content; and \(\alpha\) is the degree of cement hydration. \(RCH_{CE}\) can be determined by using the mineral compositions of cement [9] and \(\alpha\) can be determined using the integral method in the time domain as \(\alpha=\int_{0}^{t}\frac{da}{dt}dt\) and the detailed equation for \(\frac{da}{dt}\) can be found in [9]. The kinetic cement hydration model \(\frac{da}{dt}\) can be used to calculate the age-dependent properties of concrete such as the carbonatable substance contents and porosity. The calculation results of the kinetic cement hydration model can be used for the carbonation model. The input parameters of the kinetic cement hydration model include the binder compound compositions and Blaine surfaces, the concrete mixing proportions and curing conditions. The output parameters of this kinetic cement hydration model are time dependent properties such as the reaction degree of the binder, strength and heat evolution [9].  
By using the cement content and reaction degree of cement, CSH contents can be calculated as follows [9]:  
\[CSH(t)=2.85f_{S,C}*C_{0}*\alpha \tag{2}\]  
where \(f_{S,C}\) is the silica weight fraction in cement. The coefficient 2.85 denotes the mass ratio between the CSH molar weight and SiO\({}_{2}\) weight in CSH [9].  
The hydration products from the cement hydration deposits in the capillary pore spaces of concrete and the porosity of concrete will be reduced due to cement hydration. The porosity of concrete, \(\varepsilon\), can be determined as follows [2; 9]:  
\[\varepsilon(t)=W/\rho_{W}-0.25*C_{0}*\alpha-\Delta\varepsilon_{C} \tag{3}\]  
where \(W\) is the water content in concrete; \(\rho_{W}\) is the density of water; and \(\Delta\varepsilon_{C}\) is the porosity reduction due to the carbonation of concrete [2; 9].  
[PERSON] [2] proposed that when relative humidity in the environment is higher than 0.55, the carbonation of concrete is controlled by the diffusion of CO\({}_{2}\). The carbonation depth of concrete can be determined as follows:  
\[x_{c}=\sqrt{\frac{2D[CO_{2}]_{0}t}{[\text{CH}]+3[\text{CSH}]}} \tag{4}\]  
\[D=A\left(\frac{\varepsilon}{\frac{C_{0}}{\rho_{c}}+\frac{W_{0}}{\rho_{w}}} \right)^{a}\left(1-\frac{RH}{100}\right)^{2.2} \tag{5}\]  
where \(x_{c}\) is the carbonation depth of concrete; \(D\) is the CO\({}_{2}\) diffusivity; \(\left[CO_{2}\right]_{0}\) is the CO\({}_{2}\) molar concentration at the concrete surface; \([\text{CH}]\) is the molar concentration of calcium hydroxide; \([\text{CSH}]\) is the molar concentration of calcium silicate hydrate; \(\rho_{c}\) is the density of cement; \(A\) and \(a\) are the CO\({}_{2}\) diffusivity parameters; and RH is the environmental relative humidity. \([\text{CH}]+3[\text{CSH}]\) in the denominator of Equation (4) is the content of carbonatable material. The dependence of CO\({}_{2}\) diffusivity on temperature can be considered by using Arrhenius's Law [7].