## 2 Existing Durability Model in Chlorine Environment
### Initial Corrosion Model of Steel Reinforcement  
When the chloride concentration on the surface of the steel bar reaches a certain value \(C_{cr}\) (critical chloride concentration), the steel bar loses its passivation layer and corrode, which is defined as the initial corrosion durability limit state [18]. The code 'Code for design of concrete structures' [19] specifies that for marine concrete structures with a design service life of 50 years, the maximum chloride ion content is 0.1% (of the total amount of cementitious materials), and the maximum chloride ion content in prestressed concrete components is 0.06% (of the total amount of cementitious materials). The American Concrete Institute's ACI 318 standard [20] has clear regulations on the maximum allowable chloride ion content in concrete to ensure that the reinforcement is not corroded. For prestressed concrete, the chloride ion content must not exceed 0.06% of the weight of the total amount of cementitious materials. Therefore, to simplify the calculations, taking the initial chloride ion content inside the concrete equal to 0, as \(C_{0}=0\), the durability limit state equation is expressed as follows:  
\[G=C_{\sigma}-\varepsilon_{1}\cdot C_{s}\Big{[}1-erf\Big{(}x_{d}/\Big{(}2\sqrt{ D_{Cl}t}\Big{)}\Big{)}\Big{]} \tag{6}\]  
Here, \(x_{d}\) is the thickness of the concrete protective layer, with the unit in mm. The diffusion coefficient of chloride ions \(D_{Cl}\) in the model is a function of time [12], expressed as follows:  
\[D_{Cl}(t)=D_{Cl}^{0}(t_{0}/t)^{a}=D_{Cl}^{0}\cdot\eta(t_{0},t,a) \tag{7}\]  
where \(D_{Cl}^{0}\) is the instantaneous chloride diffusion coefficient at the \(t_{0}\) moment, \(D_{Cl}(t)\) is the chloride diffusion coefficient at the \(t\) moment, and \(a\) is the age attenuation index. Generally, it takes \(t_{0}\) as 28 days and \(D_{Cl}^{0}\) as the chloride ion diffusion coefficient of concrete at 28 days (m\({}^{2}\)/s). Usually, the chloride ion diffusion coefficient in concrete materials does not decay exponentially over time. Life-365 [21] defines the diffusion coefficient of chloride ions as only increasing within 25 years, assuming a constant after 25 years. In the durability design of the Hong Kong-Zhuhai-Macau offshore passage project, the decay time of the chloride ion diffusion coefficient over time is set to 30 years [22]. That is to say, after 30 years, the chloride ion diffusion coefficient of concrete is no longer attenuated:  
\[\eta(t_{0},t,a)\mid_{t>t_{D}}\;=\eta(t_{0},t_{D},a)\;\text{with}\;t_{D}=30\; \text{years} \tag{8}\]  
The effect of the fly ash and slag incorporation on the age decay index is the following:  
\[a=0.2+0.4(\%FA/50+\%SG/70) \tag{9}\]  
Life-365 [20] also specifies the relationship between the diffusion coefficient of chloride ions and temperature:  
\[D_{Cl}(T)=D_{Cl}^{0}\cdot\exp[U/R\cdot(1/T_{0}-1/T)] \tag{10}\]  
where \(D_{Cl}(T)\) is the diffusion coefficient at time \(t\) and temperature \(T\); \(D_{Cl}^{0}\) is the chloride diffusion coefficient over a 28-day period; \(T_{0}\) is the temperature, set at 293 K (20 \({}^{\circ}\)C); \(U\) is the activation energy of the diffusion process (35,000 J/mol); \(R\) is the gas constant; and \(T\) is the temperature.  
When \(G=0\) in Equation (6), we can calculate the initial corrosion time \(t_{ini}\):  
\[t_{ini}={x_{d}}^{2}\cdot(4D_{Cl})^{-1}\cdot\Big{[}erf^{-1}(1-C_{\sigma}/( \varepsilon_{1}C_{s}))\Big{]}^{-2} \tag{11}\]  
According to the limit equation of the state of initial corrosion durability of reinforcement, considering the statistical characteristics of each parameter in the equation, the probability of reinforcement initial corrosion at any time \(t\) can be calculated as follows:  
\[\begin{array}{l}p_{\text{cor}}(t)=\Pr\{G<0\}=\Pr\{C_{\sigma}-\varepsilon_{ 1}\cdot C_{s}\big{[}1-erf\big{(}x_{d}/\big{(}2\sqrt{D_{Cl}t}\big{)}\big{)}\big{]} <0\}\\ 0<p_{\text{cor}}(t)<1\end{array} \tag{12}\]