## 4 Trans-Scale Peridynamic Fatigue Model
### Trans-Scale Fatigue Model  
As shown in Figure 11a,b, The peridynamic solid undergoes cycle loading acting on the body boundary. For a given bond \(\xi_{kj}\) within the horizon of point \(\mathbf{x}_{(k)}\), the force density acting on the spring-like bond varies in a noncyclic way. The spectrum loading of the bond strain \(\varepsilon_{kj}\), shown in Figure 11d, varies irregularly with every loading cycle acting on the body boundary. Defines the maximum and minimum bond strains in the current cycle \(s_{\max}\) and \(s_{\min}\). Figure 10: The progressive failure in the peridynamic body. As shown in Equation (5), to express the damage accumulation rate of a given bond explicitly, the concise representation of the critical active energy \(Q(\sigma_{a},\sigma_{m},D)\) is obtained by using inverse analysis through the damage fatigue failure process and damage fatigue law. As shown in Figure 12, the normalized fatigue damage quantity can be expressed by fictitious time \(t_{f}\) as follows:  
\[\left\{ \begin{array}{c}\lambda\big{(}\textbf{x},\xi,t_{f}\big{)}=NC\exp(m \varepsilon)\\ t_{f}=\frac{N}{N_{f}}\end{array} \right. \tag{23}\]  
where \(N_{f}\) is the cycle number to failure, \(N\) is the current cycle number, \(\varepsilon=|\textsl{s}_{\text{max}}-\textsl{s}_{\text{min}}|\) is the current cycle bond strain in the bond, \(C\) and \(m\) are positive parameters corresponding to the material property. The normalized fatigue damage quantity \(\lambda\big{(}\textbf{x},\xi,t_{f}\big{)}\) is defined as the consumption life of the bond. It involves the loading cycle \(N\) increases. \[\left\{ \begin{array}{c}\frac{d\lambda}{dN}(N)=C\exp(m\varepsilon)\\ \lambda(0)=0\end{array} \right. \tag{24}\]  
The bond breaks at the loading cycle \(N\) that  
\[\lambda(N)\geqslant 1 \tag{25}\]  
Figure 11: Fatigue model for a given bond.\(\xi_{kj}\). (**a**) Tensile and compress cycle loading; (**b**) Peridynamic solid; (**c**) The bond \(\xi_{kj}\); (**d**) Spectrum loading. Figure 12: Scheme of fatigue damage for each bond. For the fatigue crack initiation phase, parameter \(C\) and \(m\) are calibrated by the S-N curve expressed in exponent form, as shown in Figure 13. Equation (25) means that the cycle \(N\) is the first bond breaks. Parameter \(C\) and \(m\) are set to  
\[\left\{ \begin{array}{l}C=C_{1}\\ m=m_{1}\end{array} \right. \tag{26}\]  
Combining Equations (23), (25) and (26), the first bond in the entire domain \(\mathcal{R}\) breaks with \(\varepsilon_{1}\) is the cyclic strain range. \[N_{1}C_{1}\exp(m_{1}\varepsilon_{1})=1 \tag{27}\]  
where \(N_{1}\) is the smallest cycle at which this bond breaks and \(\lambda(N_{1})=1\). The first bond breaks in the fatigue crack initiation phase when its fatigue cycle number \(N\) becomes larger than \(N_{1}\). \[N_{1}\geqslant\frac{1}{C_{1}\exp(m_{1}\varepsilon_{1})} \tag{28}\]  
When the first bond breaks with the cyclic bond \(\varepsilon_{1}\) in fatigue crack initiation, new static solutions for other bonds are calculated in the same cycle. If those bonds strain ranges are larger than \(\varepsilon_{1}\), then the bonds break and new static solutions continue until no more bonds break at the same cycle. The largest bond strain is calculated for the next cycle number by using the static solution in the deformed peridynamic material configuration. For the fatigue crack growth phase, parameter \(C\) and \(m\) are calibrated by the well-known Paris law. Parameter \(C\) and \(m\) are set to  
\[\left\{ \begin{array}{l}C=C_{2}\\ m=m_{2}\end{array} \right. \tag{29}\]  
As shown in Figure 14, a fixed bond \(\xi\) normal to the axis of the growing model-I fatigue crack. During the cycle loading, the deformation in the vicinity of the crack tip is constant and the crack growth rate is defined as \(da/dN\) in each loading cycle. The relative distance between the bond and the crack tip is expressed as follows. \[z=x-\frac{da}{dN}N \tag{30}\]  
Figure 13: Calibration parameters for the initiation phase.