## 3 Validation of the Proposed Approach
### Verification of Mixed-Mode SIFs  
Predicting fatigue life and simulating the propagation of structural cracks relies on the precise computation of SIFs. An overwhelming number of cracks in practical engineering structures are mixed modes. In this phase, as illustrated in Figure 1, the center double slant crack with a hole specimen is chosen to validate the calculation process of the three-dimensional crack stress intensity factor using a coupling FRANC3D and Abaqus under a mixed-mode fracture.  
The finite element model (FEM) of this specimen without crack with length \(2l\), width \(2w\), thickness \(T\), and diameter of the circular hole \(d\) is first established using ABAQUS 2025 software. A uniform tensile stress \(\sigma\) is applied at both ends of the specimen. This specimen's length, width, height, and hole diameter are taken as 80 mm, 40 mm, 1 mm, and 10 mm, respectively, and the tensile stress is 1 MPa. The stainless-steel material properties are as follows: [PERSON]'s modulus \(E=210\) GPa, [PERSON]'s ratio \(\upsilon=0.3\). In this finite element model, three layers of mesh are arranged along the thickness direction using the C3D8 element with a size of 1 mm. It is important to note that the mesh sensitivity study indicates that this method can yield reliable results when the element size is approximately 1 mm and at least three layers of mesh are split along the thickness direction. After establishingthe finite element model, FRANC3D is used to implant the crack according to an initial crack length of \(2a\) and an initial crack slant angle of \(\varphi\). The different initial crack lengths are determined by referring to [24], considering two situations where the first crack slant angles are 30 and 45 degrees. The crack tip's region mesh has been refined to reduce the computation error. The template radius of the crack tip is \(r=0.6\) mm. Figure 1 shows an illustration of the FEM.  
The center double slant fracture with a hole specimen and analytical expressions of SIFs used in this work are provided as follows [25]:  
\[K_{I}=\sigma\sqrt{\sigma\tau a}F_{I}\Big{(}\frac{a}{w}\Big{)} K_{II}=\sigma\sqrt{\sigma\tau a}F_{II}\Big{(}\frac{a}{w}\Big{)} \tag{10}\]  
In where the structural form factors for mode I and mode II cracks are, respectively, \(F_{I}(a/w)\) and \(F_{II}(a/w)\). The precise values of \(F_{I}\) and \(F_{II}\) for various initial fracture lengths and slant angles are provided in [24; 25].  
For both angles, the SIFs for various beginning crack lengths are computed in turn. Figure 2 displays the computation results with comparisons to the theoretical answers. The outcomes demonstrate the accuracy of FRANC3D by showing a strong agreement between the numerical and analytical solutions. Furthermore, it can be ensured that the estimated fatigue crack development life is rather conservative because the computational results of SIFs are slightly higher than the analytical solutions. The given instance demonstrates that the suggested approach can accurately calculate the mixed-mode crack's SIFs.  
Figure 1: Finite element model illustration of the specimen: (**a**) without crack, (**b**) with center double slant crack.