## 2 Methods
### State-Based Peridynamic Theory
\right.,s_{ij}=\frac{\Big{|}\mathbf{y}_{j}-\mathbf{ y}_{i}\Big{|}-\big{|}\mathbf{x}_{j}-\mathbf{x}_{i}\Big{|}}{\big{|}\mathbf{x}_{j}-\mathbf{x}_{i} \Big{|}}=\frac{\big{|}\mathbf{Y}\big{<}\mathbf{\tilde{\xi}}_{ij}\big{>}\big{|}- \big{|}\mathbf{\tilde{\xi}}_{ij}\big{|}}{\big{|}\mathbf{\tilde{\xi}}_{ij}\big{|}}, \tag{5}\]  
where \(s_{ij}\) is the bond stretch. Then, the damage at a node can be defined as a ratio between the broken and the initial number of bonds [31]:  
\[\phi(\mathbf{x}_{i})=1-\frac{\int_{H_{\mathbf{x}_{i}}}\omega(\mathbf{x}_{i})dV_{\mathbf{x}_{ j}}}{\int_{H_{\mathbf{x}_{i}}}\ dV_{\mathbf{x}_{j}}}. \tag{6}\]  
The PD fatigue damage model used in this study was introduced in [32] and used in [33; 34; 35]. Other researchers have also developed fatigue damage models [36; 37]; however, these models use bond-based peridynamic theory and simulate only the crack growth phase. A small overview of the model is given here for completeness; Equations (7) through (11) were first presented in [32]. Figure 2: The most conservative two-dimensional (2D) case when it could be thought that a crack has appeared. Some bonds are drawn curved to avoid overlapping. A body undergoes some cyclic deformation between two extremes \(+\) and \(-\), then bond strains at each extreme are \(s_{ij}^{+}\), \(s_{ij}^{-}\) and the cyclic bond strain \(\varepsilon_{ij}\) is:  
\[s_{ij}^{+}=\frac{\left|\mathbf{Y}^{+}\big{\langle}\mathbf{\tilde{\varepsilon} }_{ij}\big{\rangle}\right|-\left|\mathbf{\tilde{\varepsilon}}_{ij}\right|}{ \left|\mathbf{\tilde{\varepsilon}}_{ij}\right|},s_{ij}^{-}=\frac{\left| \mathbf{Y}^{-}\big{\langle}\mathbf{\tilde{\varepsilon}}_{ij}\big{\rangle} \right|-\left|\mathbf{\tilde{\varepsilon}}_{ij}\right|}{\left|\mathbf{ \tilde{\varepsilon}}_{ij}\right|},\varepsilon_{ij}=\left|s_{ij}^{+}-s_{ij}^{- }\right|. \tag{7}\]  
For each bond, a variable called the \"remaining life\" \(\lambda_{ij}\big{(}\mathbf{x}_{i},\ \mathbf{\tilde{\varepsilon}}_{ij},N\big{)}\) is defined. It degrades at each loading cycle \(N\), and a bond breaks when the remaining life is reduced to zero:  
\[\lambda_{ij}(N)\leq 0. \tag{8}\]  
At the beginning, when \(N=0\):  
\[\lambda_{ij}(0)=1, \tag{9}\]  
at each cycle in the crack nucleation phase (phase I), the change of \(\lambda\) is given by  
\[\frac{d\lambda_{ij}}{dN}(N)=\left\{ \begin{array}{cc}-A_{I}\big{(}\varepsilon _{ij}-\varepsilon_{\infty}\big{)}^{m_{I}}&,\ if\ \varepsilon_{ij}> \varepsilon_{\infty}\\ 0&,\ if\ \varepsilon_{ij}\leq\varepsilon_{\infty}\end{array} \right., \tag{10}\]  
where \(\varepsilon_{\infty}\) is the fatigue limit under which no fatigue damage occurs, and \(A_{I},\ m_{I}\) are parameters for phase I. In phase II, the remaining life changes according to:  
\[\frac{d\lambda_{ij}}{dN}(N)=\ -A_{II}\varepsilon_{ij}^{m_{II}}, \tag{11}\]  
where \(A_{II},\ m_{II}\) are parameters for phase II. The transition from phase I to phase II is handled by applying the phase I model to bonds connected to \(\mathbf{x}_{i}\) until there is a node \(\mathbf{x}_{j}\) in \(H_{\mathbf{x}_{i}}\) with damage  
\[\phi\big{(}\mathbf{x}_{j}\big{)}\geq\phi_{c}, \tag{12}\]  
where \(\phi_{c}\) is the damage at which phase II begins. Then, reset the remaining life of bonds connected to \(\mathbf{x}_{i}\) to 1 and switch to the phase II model.