# Crack Extension Analysis of Atmospheric Stress Corrosion Based on Peridynamics
## 2 Materials and Methods
### Atmospheric Stress Corrosion Coupling Model with Peridynamics  
#### 2.1.1 Peridynamics Fracture Model  
In the peridynamics theory proposed by [PERSON] et al. [30], it is assumed that the material is composed of material points that are uniformly distributed in space. At any time, each material point in the material represents the position of a material particle, and each particle has its particle volume and mass density. Particles move and deform when they are affected by a force and velocity. Peridynamics involves the study of the motion of objects due to the interaction between particles in objects. It is assumed that the interaction force between a particle \(x_{i}\) and a particle beyond a certain range (near field range \(\delta\)) is negligible, and the particles within its near field range are called family \(H_{x_{i}}\). PD theory studies the interaction between a particle and all particles within its near-field range. The existence of interaction between the mass point \(x_{i}\) and all the mass points in the family \(H_{x_{i}}\) produces a force density vector that acts on the mass point \(x_{i}\), which can be regarded as a force that is applied to the mass point \(x_{i}\) by the mass point \(x_{j}\). Similarly, the mass point \(x_{j}\) is acted upon by all the other mass points in the family \(H_{x_{j}}\). In the bonded peridynamics theory, a bond describes the interaction between two material locations, and a broken bond means that there is no longer an interaction force between the mass points. The mass points act in pairs, and the relative position state changes of two interacting material points when they are subjected to external forces are shown in Figure 1: \(x_{i}\) and \(x_{j}\) denote the initial positions of two material points, and \(x_{i}^{\prime}\) and \(x_{j}^{\prime}\) denote the positions of two mass points after a certain time t of external forces. The fundamental equations of motion for near-field dynamics are constructed based on [PERSON]'s second law. The mechanical parameters and characteristics of the materials are incorporated into the fundamental equations of motion through the instanton equations to achieve numerical simulations of damage to different materials. \(\mathbf{u}_{i}\) denotes the displacement of matter point \(x_{i}\), where its unit is \(m\), while \(\mathbf{u}_{j}\) denotes the displacement of matter point \(x_{j}\), \(m\). Let \(\mathbf{\eta}\) and \(\mathbf{\xi}\) represent the relative displacement and relative position of the material point \(\mathbf{x}_{i}\) and material point \(\mathbf{x}_{j}\), respectively, where the expressions are \(\mathbf{\xi}=\mathbf{x}_{j}-\mathbf{x}_{i}\) and \(\mathbf{\eta}=\mathbf{u}(\mathbf{x}_{j},t)-\mathbf{u}(\mathbf{x}_{i},t)=\left(\mathbf{x}_{j}^{\prime}- \mathbf{x}_{j}\right)-\left(\mathbf{x}_{i}^{\prime}-\mathbf{x}_{i}\right)\), \(m\). Within the theoretical framework of bonded peridynamics, the equation of motion of the material point \(x_{i}\) in time \(t\) can be formulated as [49]  
\[\rho_{i}(x_{i})\ddot{\mathbf{u}}_{\ i}(x_{i},t)=\int\limits_{H_{x_{i}}}\mathbf{f}(\bm {\eta},\mathbf{\xi},t)dV_{j}+\mathbf{b}_{i}(x_{i},t),\forall x_{j}\in H \tag{1}\]  
where \(\rho_{i}\) denotes the mass density of matter point \(x_{i}\), kg/m\({}^{3}\); \(\mathbf{b}_{i}\) denotes the external load in terms of the body force density (force per unit volume of matter), N/m\({}^{3}\); \(H_{x_{i}}\) is the near-field range of matter point \(x_{i}\); \(V_{j}\) is the volume of matter point \(x_{j}\), m\({}^{3}\); \(\rho_{i}(x_{i})\ddot{\mathbf{u}}_{\ i}(x_{i},t)\) represents the combined force that affects matter point \(x_{i}\); and \(\int_{H_{x_{i}}}\mathbf{f}(\mathbf{\eta},\mathbf{\xi},t)dV_{j}\) represents the sum of the bond forces acting with all matter points in the near-field region \(H_{x_{i}}\) of matter point \(x_{i}\).