## 4 Proposed Method for Notch Fatigue Analysis  
### Theoretical Background of the Proposed Model  
As mentioned in the introduction, in many engineering applications, the crack initiation stage does not exist, and some initial microcracks generated by the manufacturing or processing process pre-exist in the engineering component, causing the fatigue limit of materials in practical structural engineering, such as steel [33; 34], to be mainly determined by non-propagating fatigue crack behavior. It is well known that the formation of non-propagating cracks is attributed to the crack closure effect [42; 43; 44]. After the initial microcrack propagates for a short distance, when the applied load is insufficient to overcome the crack closure resistance caused by the plastic wake at the crack tip, the crack stops growing and forms a non-propagating crack, as shown in Figure 9. Therefore, the critical state is reached when the elastic stress of the non-propagating crack wake (point B) is equal to the opening stress (\(\sigma_{op}\)) of the crack closure effect. Which can be expressed as:  
\[\sigma_{\max}\sqrt{\frac{a}{2S}}=\sigma_{op} \tag{6}\]  
where \(S\) is the length of the non-propagating crack; the crack length of the notch (\(a\)) can be expressed as [45; 46; 47]:  
\[a=\frac{1}{\pi}\bigg{(}\frac{K_{th,eff}}{\sigma_{0}}\bigg{)}^{2}=\frac{1}{\pi} \bigg{(}\frac{K_{th}}{\sigma_{0}}\bigg{)}^{2}\bigg{(}\frac{1{-}\lambda}{1{-}R} \bigg{)}^{2} \tag{7}\]  
where crack opening stress coefficient \(\lambda\)= \(\sigma_{op}/\sigma_{\max}\); stress ratio \(R=\sigma_{\min}/\sigma_{\max}\).  
Figure 8: Results from finite element analysis (FEA). (**a**) Stress distribution near the root of the notch along the direction perpendicular to the applied load (Y-axis); (**b**) relationship between the element size and stress at different locations.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Specimen No.**} & \multicolumn{2}{c}{**Experimental Value**} & \multicolumn{1}{c}{**PM**} & \multirow{2}{*}{**Deviation**} & \multicolumn{1}{c}{**LM**} & \multirow{2}{*}{**Deviation**} \\  & **(MPa)** & & **(MPa)** & & **(MPa)** & \\ \hline V1 & 180 & 139 & \(-\)22.65\% & 163 & \(-\)9.45\% \\ V2 & 200 & 149 & \(-\)25.38\% & 177 & \(-\)11.70\% \\ V3 & 215 & 153 & \(-\)28.84\% & 175 & \(-\)18.60\% \\ V4 & 235 & 169 & \(-\)28.18\% & 192 & \(-\)18.14\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparing the predicted values of PM and LM with the fatigue experimental results.  
Substituting Equation (7) into Equation (6), the length of the non-propagating crack can be obtained:  
\[S=\frac{1}{\pi}\left(\frac{K_{th}}{\sigma_{0}}\right)^{2}\left(\frac{1{-}\lambda} {\sqrt{2}\lambda(1{-}R)}\right)^{2} \tag{8}\]  
The formation of the non-propagating crack changes the position of the crack and causes the crack tip to extend from the initial position (point A) to the non-propagating crack tail (point B) as shown in Figure 9. Since the conventional critical distance method is based on the theory of linear elastic fracture mechanics, the change of the crack position will affect the calculation of the critical length, thereby affecting the accuracy of the fatigue strength prediction. Therefore, the present study attempts to combine the length of the non-propagating crack and the conventional critical distance as a new critical distance.  
\[\sigma^{*}{}_{eff}(PM)=\sigma(L/2+S)=\sigma_{0} \tag{9}\]