# Crack Extension Analysis of Atmospheric Stress Corrosion Based on Peridynamics
## 2 Materials and Methods
### Atmospheric Stress Corrosion Coupling Model with Peridynamics
#### 2.1.3. Peridynamics Anodic Dissolution Model  
In the research model of PD, corrosion is considered a process of dissolution diffusion. [PERSON] et al. [52; 53] studied heat and mass transfer in discontinuous objects with evolving capabilities and proposed the following PD equation for mass transfer:  
\[\frac{\partial C(x_{i},t)}{\partial t}=\int_{H_{x_{i}}}k\big{(}x_{i},x_{j} \big{)}\frac{C\big{(}x_{j},t\big{)}-C(x_{i},t)}{||x_{j}-x_{i}||^{2}}dV_{x_{j}} \tag{20}\]  
Equation (20) simulates mass transfer by anodic dissolution at the metal interface and diffusion of metal ions in the electrolyte, where \(C(x,t)\) is the concentration of plasmas \(x_{i}\) at time \(t\), mol/m\({}^{3}\). The concentration-dependent damage model describes the relationship between the damage and concentration change during corrosion; \(k(x_{i},x_{j})\) is the diffusion coefficient of the bond between plasmas \(x_{i}\) and \(x_{j}\), which is defined as a function of mechanical damage \(d(x,t)\); and \(k\) is calculated as follows:  
\[k\big{(}x_{i},x_{j}\big{)}=\left\{ \begin{array}{ll}K_{L}(D),&d(x_{i},t)=1 \cup d(x_{j},t)=1\\ 0\,&d(x_{i},t)<1\cup d\big{(}x_{j},t)<1\\ K_{diss},&\big{[}d(x_{i},t)ord\big{(}x_{j},t\big{)}\big{]}<1\cup\big{[}d(x_{i}, t)ord\big{(}x_{j},t\big{)}\big{]}=1\end{array} \right. \tag{21}\]  
Equation (21) indicates that if both masses belong to the solid phase, no diffusion occurs, and if one point is in the solid phase and the other in the liquid phase, the bond is an interfacial bond that carries the anodic dissolution microflux for which \(K=K_{diss}\) and is called microdiffusivity. \(K_{L}(D)\) is the microdiffusivity of the liquid, which can be calculated based on the diffusivity \(D\) of the electrolyte:  
\[K_{L}(D)=\frac{9D}{2\pi\delta^{3}} \tag{22}\]  
The damage function \(d(x,t)\) in Equation (21) is  
\[d(x,t)=\left\{ \begin{array}{ll}1\,&C(x_{i},t)\leq C_{sat}\\ \frac{C_{solit}-C(x_{i},t)}{C_{solid}-C_{sat}},&C_{sat}<C(x_{i},t)<C_{solid} \\ 0\ \ \ \,&C(x_{i},t)=C_{solid}\end{array} \right. \tag{23}\]  
Equation (23) indicates that when the concentration of metal ions in the solid undergoes dissolution below the saturation concentration of metal ions in the electrolyte solution \(C_{sat}\), the solid becomes a liquid, i.e., the damage is 1, and the damage to the intact solid is 0. The damage to the region can be determined from the concentration of metal ions in Equation (23), and this damage is expressed by removing the mechanical bonds in the corroded region. Hydrogen is able to enter the specimen and promote anodic dissolution, and there is a synergistic effect of hydrogen and stress in promoting corrosion. Under the combined action of hydrogen and external stress, the metal anodic dissolution current density is  
\[i_{(\sigma-H)}=k_{(H)}k_{(\sigma)}k_{(\sigma,H)}i_{0} \tag{24}\]  
Under external stress alone, the metal anode dissolution current density is  
\[i_{(\sigma)}=k_{(\sigma)}i_{0} \tag{25}\]In atmospheric corrosion, Equations (24) and (25) can be combined and rewritten as  
\[i= \begin{cases}k_{(H)}k_{(\sigma)}k_{(\sigma,H)}i_{0},&\lambda<\lambda_{0}\\ k_{(\sigma)}i_{0},&\lambda\geq\lambda_{0}\end{cases}  \tag{26}\]  
where \(i_{(\sigma-H)}\) is the dissolution current (proportional to the dissolution rate) generated by adding a constant load after hydrogen charging, \(A\); \(i_{0}\) is the corrosion current when no hydrogen is loaded, \(A\); \(k_{(H)}\) and \(k_{(\sigma)}\) are the factors that increase the dissolution when hydrogen and stress are each present alone, respectively, and \(k_{(\sigma,H)}\) is the hydrogen and stress interaction factor.