# Durability of Subsea Tunnels under the Coupled Action of Stress and Chloride Ions
## 3 Effect of Stress on Chloride Ions Transfer
### The Coefficient under the Influence of Stress  
The stress in the subsea tunnels is relatively low; therefore, the influence of external loads on the diffusibility of chloride ion in concrete is equivalent to the effect of modified concrete porosity [25; 30; 31; 32]. From a microcosmic perspective, the concrete is composed of cement paste and aggregate, and may further be subdivided into pores, solid phase of cement pastes, and aggregate. Figure 1 contrasts stressed and unstressed concrete. In the low stress state, the concrete displays small amounts of linear elastic deformation. According to the material mechanics, the volume of strain should be calculated by Equation (6):  
\[\theta=\frac{V^{\prime}-V}{V}=-\frac{1-2\mu}{E_{c}}(\sigma_{x}+\sigma_{y}+\sigma _{z}) \tag{6}\]  
where \(V\) is the volume of concrete in the unstressed state, \(V=V_{k}+V_{m}+V_{s}\); \(V^{\prime}\) is the volume of concrete under stress, \(V^{\prime}=V^{\prime}_{k}+V_{m}+V_{s}\); \(\mu\) is the Poisson ratio of concrete; \(E_{c}\) is the elastic modulus of concrete; and \(\sigma_{x}\), \(\sigma_{y}\), and \(\sigma_{z}\) are the stress of concrete in three directions; the compressive stress is positive and the tensile stress is negative. The relationship between the porosity of concrete and the porosity of cement paste in the unstressed state is shown in Equation (7):  
\[\varphi_{c0}=\frac{V_{k}}{V}=\frac{\varphi_{0}(V_{k}+V_{m})}{V}=\frac{\varphi _{0}(V-V_{s})}{V}=\varphi_{0}(1-\frac{V_{s}}{V}) \tag{7}\]  
where \(\varphi_{c0}\) is the porosity of concrete in the unstressed state, \(\varphi_{0}\) is the porosity of cement paste in the unstressed state, \(V_{k}\) is the pore volume inside the concrete, and \(V_{s}\) is the aggregate volume inside the concrete. Since the deformation is small, suppose \(V^{\prime}\approx V\). Therefore, the porosity of concrete under stress is:  
\[\varphi_{c\alpha}=\frac{V^{\prime}_{k}}{V^{\prime}}=\frac{V_{k}-(V-V^{\prime} )}{V^{\prime}}\approx\frac{V_{k}}{V}+\frac{(V^{\prime}-V)}{V}=\varphi_{c0}+\theta \tag{8}\]  
Through the combination of Equations (7) and (8), the porosity of cement paste under stress is obtained as shown in Equation (9):  
\[ \begin{array}{l}\varphi_{o}=\varphi_{c0}\frac{1}{1-V_{s}/V^{\prime}}\approx \varphi_{c0}\frac{1}{1-V_{s}/V}\\ =\varphi_{0}-\frac{1-2\mu}{E_{c}(1-V_{s}/V)}(\sigma_{x}+\sigma_{y}+\sigma_{z} )\end{array}  \tag{9}\]  
According to the Garboczi model and the Bentz model [33], the relationship between the diffusion coefficient of chloride ions and the porosity of cement paste is expressed by Equation (10):  
\[D_{m}=D_{1}[0.001+0.07\varphi^{2}+1.8H(\varphi-\varphi_{c})(\varphi-\varphi_{ c})^{2}] \tag{10}\]  
where \(\varphi_{c}\) is critical porosity, \(\varphi_{c}=0.18\); \(D_{1}\) is the diffusion coefficient of chloride ions in pore solution, \(D_{1}=2.032\times 10^{-9}\) m/s; and \(H(x)\) is a step function. If \(x>0\), \(H=1\); otherwise, \(H=0\). The concrete consists of cement paste and aggregate and it is assumed that aggregate and cement paste are impermeable to each other [34]. Therefore, we assumed that the diffusion coefficient of chloride ions in aggregate is zero. The diffusion coefficient of chloride ion in concrete is calculated by:  
Figure 1: Composition of concrete. \[ \begin{array}{ll}D_{\rm c}&=\frac{D_{m}A_{m}+D_{s}A_{s}}{A_{m}+A_{s}}=\frac{D_{m}A_ {m}}{A_{m}+A_{s}}\\ &=D_{m}\frac{V-V_{s}}{V}=D_{m}(1-\frac{V_{s}}{V})\end{array}  \tag{11}\]  
The coefficient under the influence of stress is defined as the ratio of the diffusion coefficient under stress to the diffusion coefficient in the unstressed state.