## 2 Theory and Method of Fracture Mechanics Based on the FEM
### Verification Method of Shape Coefficient Correction  
In Equation (5), the coefficient \(F\) is the corrected shape coefficient containing the V-shaped notch. To verify its effectiveness, the corrected shape coefficients not containing the V-shaped notch are obtained by converting Equation (5), and a comparison is made to the corrected shape coefficients without V-shaped notches in the literature. The relationship between the shape coefficient for the cracks with the V-shaped notch and without the V-shaped notch can be approximately expressed as:  
\[F_{\rm N}=\left\{ \begin{array}{l}F/K_{t},\\ F,\end{array} \right.\qquad\qquad \begin{array}{l}\rho\\ R\end{array} \to 0\\ \begin{array}{l}\rho\\ R\end{array} \gg 0\end{array} \tag{6}\]  
where \(F_{\rm N}\) is the corrected shape coefficient of the crack without the V-shaped notch, \(K_{t}\) is the elastic stress concentration coefficient with the V-shaped notch during stretching or bending. [PERSON] et al. [31] investigated the variation law of the SIF of semi-elliptical cracks on the cylinder surface under bending and torsion loads with the influence of relative depth of ellipse and relative thickness of cylinder and relative depth of crack and obtained the curve of the corrected shape coefficient at the deepest point of the crack. [PERSON] et al. [32] calculated the three-dimensional stress field at the crack front at the root of the V-shaped notch using a three-dimensional FEM, analyzed the effect of notch restraint on the stress state at the crack front and established an empirical solution for the SIF by considering the effect of notch restraint. However, for special structures such as an ampoule bottle, whose neck is a special V-shaped cut, cracks exist on its surface. As the ampoule bottle is subjected to the loads, fatigue cracks are most likely to occur at the notch, resulting in an unstable fracture. The schematic of the V-shaped notch area is shown in Figure 4, where \(r_{0}\) represents the depth of the arc at the notch, \(\rho\) is the radius of curvature of the arc at the root of the notch, \(\beta\) is the notch angle and \(p\) and \(q\) are auxiliary parameters to determine the relationship between \(\rho\) and \(r_{0}\). [PERSON] et al. [33] obtained an accurate stress field equation for calculating notch stress by compensating the influence of finite size on the stress field. In this paper, the stress field equation of the Filippi notch under tensile stress load can be expressed as:  
\[\left( \begin{array}{c}\sigma_{\theta}\\ \sigma_{r}\\ \tau_{\theta}\end{array} \right)= \lambda_{1}r^{\lambda_{1}-1}a_{1}\left(\left\{ \begin{array}{c}(1 +\lambda_{1})\cos(1-\lambda_{1})\theta\\ (3-\lambda_{1})\cos(1-\lambda_{1})\theta\\ (1-\lambda_{1})\sin(1-\lambda_{1})\theta\end{array} \right\}+\chi_{b_{1}}(1- \lambda_{1})\left\{ \begin{array}{c}\cos(1+\lambda_{1})\theta\\ -\cos(1+\lambda_{1})\\ \sin(1+\lambda_{1})\theta\end{array} \right\}\right. \tag{7}\] \[+\frac{q}{4(q-1)}\left(\frac{r}{r_{0}}\right)^{\mu_{1}-\lambda_{1 }}\left[\chi_{d_{1}}\left\{ \begin{array}{c}(1+\mu_{1})\cos(1-\mu_{1}) \theta\\ (3-\mu_{1})\cos(1-\mu_{1})\theta\\ (1-\mu_{1})\sin(1-\mu_{1})\theta\end{array} \right\}\right]\] \[+\chi_{c_{1}}\left\{ \begin{array}{c}\cos(1+\mu_{1})\theta\\ -\cos(1+\mu_{1})\theta\\ \sin(1+\mu_{1})\theta\end{array} \right\}\right)\]  
Under shear load, the stress field equation of the Filippi notch can be written as:  
\[\left( \begin{array}{c}\sigma_{\theta}\\ \sigma_{r}\\ \tau_{\theta}\end{array} \right)= \lambda_{2}r^{\lambda_{2}-1}a_{2}\left(\left\{ \begin{array}{c}(1 +\lambda_{2})\sin(1-\lambda_{2})\theta\\ (3-\lambda_{1})\sin(1-\lambda_{2})\theta\\ (1-\lambda_{1})\cos(1-\lambda_{2})\theta\end{array} \right\}+\chi_{b_{1}}(1+ \lambda_{1})\left\{ \begin{array}{c}\sin(1+\lambda_{2})\theta\\ -\sin(1+\lambda_{2})\\ \cos(1+\lambda_{2})\theta\end{array} \right\}\right. \tag{8}\] \[+\frac{1}{4(\mu_{2}-1)}\left(\frac{r}{r_{0}}\right)^{\mu_{1}- \lambda_{1}}\left[\chi_{d_{2}}\left\{