## 2 Numerical Analysis Procedure  
Among the prominent software tools available for FCG analysis, ANSYS Workbench has emerged as a versatile and powerful platform. Its capabilities extend beyond traditional finite element analysis, encompassing a wide range of functionalities specifically designed for fatigue analysis. To streamline the analysis process, an unstructured mesh strategy was utilized, reducing preprocessing time and enabling faster meshing. To accommodate crack propagation, the analysis employed Separating Morphing Adaptive and Remeshing Technology (SMART) meshing, utilizing higher-order SOLID187 tetrahedral elements. At each step of the analysis, automatic mesh updates were performed to accurately capture the changes in the crack front. This approach ensured that the mesh effectively adapted to the evolving crack geometry, enabling precise simulations of crack propagation behavior. The SIFs were determined using the interaction integral evaluation technique, which is a well-established and widely accepted approach in fracture mechanics. By utilizing the interaction integral evaluation method, this study ensures precise and robust analysis of the SIFs. In this study, the determination of the FCG angle was based on the widely used maximum tangential stress criterion. According to this theory, when isotropic materials are subjected to mixed-mode loading, the crack tends to propagate along a path perpendicular to the direction of the maximum tangential tensile stress. The tangential stresses in polar coordinates can be mathematically represented by the following expressions [47; 48; 49; 50]:  
\[\begin{array}{l}\sigma_{r}=\frac{1}{\sqrt{2\pi r}}\cos(\theta/2)\left(K_{I} [1+\sin^{2}(\theta/2)]+\frac{3}{2}K_{II}\sin\theta-2K_{II}\tan(\theta/2) \right)\\ \sigma_{\theta}=\frac{1}{\sqrt{2\pi r}}\cos(\theta/2)\left[K_{I}\cos^{2}( \theta/2)-\frac{3}{2}K_{II}\sin\theta\right]\\ \tau_{r\theta}=\frac{1}{\sqrt{2\pi r}}\frac{\cos(\theta/2)}{2}[K_{I}\sin\theta +K_{II}(3\cos\theta-1)]\end{array} \tag{1}\]  
where \(\sigma_{r}\) and \(\sigma_{\theta}\) denote the normal stress in the radial direction and tangential direction, respectively, and \(\tau_{r\theta}\) denotes the shear stress.  
By solving \(d\sigma_{\theta}/d\theta=0\) for \(\theta\), the result is expressed as follows:  
\[K_{I}\sin\theta+K_{II}(3\cos\theta-1)=0 \tag{2}\]  
From which the FCG angle in ANSYS is determined using the following formula:  
\[\theta=\cos^{-1}\left(\frac{3K_{II}^{2}+K_{I}\sqrt{K_{I}^{2}+8K_{II}^{2}}}{K_{ I}^{2}+9K_{II}^{2}}\right) \tag{3}\]  
where symbols \(K_{I}\) and \(K_{II}\) represent the SIFs for the opening mode and in-plane shear mode, respectively.  
The SMART crack growth tool in ANSYS incorporates the use of the equivalent SIF range (\(\Delta K_{eq}\)) to predict fatigue life under mixed-mode loading. For correlating FCG with \(\Delta K_{eq}\), a power law relationship was proposed by [PERSON] [51] in the form of a modified Paris law formula as follows:  
\[\frac{da}{dN}=C(\Delta K_{eq})^{m} \tag{4}\]  
where \(a\) is the crack length, \(N\) represents the number of cycles, and \(C\) and \(m\) are the Paris constant and Paris exponent, respectively. The equivalent stress intensity formula can be expressed as follows [52; 53]:  
\[\Delta K_{eq}=\frac{1}{2}\cos(\frac{\theta}{2})[\Delta K_{I}(1+\cos\theta)-3 \Delta K_{II}\sin\theta] \tag{5}\]where  
\[\begin{array}{l}\Delta K_{I}=K_{I}^{\max}-K_{I}^{\min}=(1-\mathrm{R})K_{I}^{\max} \\ \Delta K_{II}=K_{II}^{\max}-K_{II}^{\min}=(1-\mathrm{R})K_{II}^{\max}\end{array} \tag{6}\]  
and \(R\) is the stress ratio.