## 1 Introduction  
With the wide application of pressure vessels in the industry, the life assessment of pressure vessels has become more and more important. When using fracture mechanics or failure assessment diagrams to evaluate the life of pressure vessels, it is necessary to know the stress intensity factor at the crack tip in advance. [PERSON] and other scholars [1, 2] proposed the use of various aspect ratios (ratio of crack depth to crack length) and relative depth (ratio of crack depth to pipe thickness) to characterize different semi-elliptical defects, and evaluate the SIFs with surface defects. [PERSON] [3] modeled the low-cycle fatigue of crack propagation exploring J-integral. [PERSON] [4] modeled and explained the mechanism for the low-cycle fatigue crack propagation of the pressure vessel. [PERSON]. [5] combined the simulation approach of the initiation-growth-arrest algorithm with XFEM, and proposed a new simulation approach for crack initiation and propagation. [PERSON] [6, 7] considered the influence of the distribution form on the stress intensity factor, using the power series expansion method and the weight function method, and obtained the applicable range of the formula for calculating the stress intensity factor under uniform and non-uniform distribution pressures. [PERSON] [8] established a flexural crack model on a wave-shaped interface, and studied the influence of structural parameters on porosity, pore distribution and TC layer thickness, interface roughness, and crack tip on the wave-shaped interface. Scholars, such as [PERSON] [9, 10], considered the influence of the relative positions of multiple cracks on their stress intensity factors, analyzed the interaction mechanism of multiple cracks, and obtained the strengthened and weakened areas of the stress intensity factor. [PERSON] [11] considered the uneven distribution of stress and calculated the complex stress intensity factor solution to verify the closed form weight function derived from several nonlinear normal and shear stress distributions. [PERSON] [12, 13] proposed using the hot-spot stress method to calculate the stress intensity factor of a tubular T-joint, but its limitation is that it does not consider the effect of the crack angle. [PERSON] [14] computed the crack propagation of the pressure vessel using a hybrid of the coupled particle and Eulerian methods. [PERSON] [15] investigated the crack propagation for the composite plate.  
All the above models assume that the initial crack angle of the curved surface is \(0^{\circ}\), equivalent to the normal direction of the curved surface. The influence of the angle \(\alpha\) between the initial crack direction and the normal direction is not considered for the crack growth. In fact, the size of the initial crack angle affects the direction of the crack propagation and the stress distribution of the model. Therefore, we take the neck joint surface body of the pressure vessel bottle as the research object, considering the initial crack angle of the curved surface. To perform this, we modify the stress intensity factor formula and then apply the modified formula to optimize the crack propagation model of the pressure vessel.