## 3 Results and Discussion
### Results and Analysis of the Coefficient \(K\) at the Crack Front  
Many parameters influence the coefficient \(K\) of the SIF, such as long semi-axial length, short semi-axial length and component thickness [34; 35]. To analyze the relationship between the influence of various factors on the SIF, the effects of different crack ellipticity (\(a/c\)) and different crack relative depth (\(a/t\)) on the SIF were calculated in this paper, where \(t\) is the thickness of ampoule bottle body (\(t=R-r\)). Thus, the corrected shape coefficients of components under different relative thickness (\(R/r\)) were obtained. Supposing that the long half-axis length \(c\) was a fixed value, the values of \(a/c\) were set to 0.2, 0.4, 0.6, 0.8, respectively, the \(a/t\) values were set to 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8, respectively. The effects of these parameters on the coefficients \(K\) of the SIFs are shown in Figures 6-8. The abscissa shown in the figures represents the position ratio of different depth of the semi-elliptical crack front edge, i.e., \(\zeta=\theta/\theta_{max}\), and the origin is at the deepest point of the crack, where \(\theta\) and \(\theta_{max}\) are shown in Figure 2.  
Figure 6 shows that the type I SIF (\(K_{\rm I}\)) at the symmetrical position on both sides of the deepest semi-elliptical crack follows a symmetrical distribution. When the relative ellipticity ratio \(a/c\) is constant, but the relative crack depth \(a/t\) varies, that is, the ampoule bottle thickness gradually thickened, the curve changed from a concave and convex M-shape to a slightly protruding shape in the middle and a relatively smooth downward shape on both sides, and the SIF value at its corresponding position gradually decreased. When \(a/c=0.2\) and \(a/c=0.4\), the maximum value of SIF is located on both sides of the deepest crack point near the crack tip. When \(a/c=0.6\) and \(a/c=0.8\), the coefficient \(K_{\rm I}\) of the SIF is the minimum at the deepest crack point, and the SIF increases as it gets closer to the crack tip. In addition, when \(a/c=0.8\), the SIF of the deepest crack point at any \(a/t\) is smaller than those at \(a/t=0.7\) and at \(a/t=0.6\). It means that when the crack ellipticity rate  
Figure 5: The influence of mesh sizes on the coefficient \(K\) of the SIF. It can be inferred from Figure 6 that the depth of the crack has a great impact on the type I SIF at the deepest point, whereas it has a small impact on the type I SIF at the front of the cracks on both sides. The thickness of the ampoule bottle directly affects the SIF of the entire crack front. It suggests that the thicker the bottle thickness is, the smoother the corresponding SIF curve becomes and the smaller the change is in the corresponding coefficient \(K_{\rm I}\) of the SIF. When both the crack depth and the bottle thickness are changed, the effect of the crack depth on the crack SIF is greater than that of the thickness on the SIF. The distribution curves of type II SIF (\(K_{\rm II}\)) are shown in Figure 7, under both tensile stress and bending moment loads, which have intersection points in the region near the tip of both sides of the semi-elliptical crack. The \(K_{\rm II}\) of near intersection points are approximately equal. The \(K_{\rm II}\) is greatest at the deepest part of the crack and symmetrical at both sides of the deepest point. When \(a/c=0.2\), the \(K_{\rm II}\) fluctuates greatly. It gradually becomes smaller and smoother as the ampoule bottle thickness increases. It can be seen from Figure 8 that the \(K_{\rm III}\) at the deepest part of the crack is approximately 0, and the negative value indicates the opposite direction of the symmetric shear force on both sides of the semi-elliptic crack center. When \(a/c\) is constant, the larger the \(a/t\) value is, and the greater the \(K_{\rm III}\) value becomes. It indicates that when the thickness of the ampoule bottle becomes thinner, the larger the \(K_{\rm III}\) is, and the more likely the ampoule bottle is to break.