# Computational Simulation of 3D Fatigue Crack Growth under Mixed-Mode Loading
## 3 Numerical Results and Discussion
### Compact Tension Shear Specim (CTS)  
Figure 8a shows CTS geometry, while Figure 8b shows its proposed loading angles. The distribution of forces according to the load angles considered in ANSYS is shown in Figure 9. The applied load F is compared with the corresponding loads on other holes of the six holes, based on the given formulations [39]:  
\[F_{1}=F_{6}=F(0.5\cos\alpha+\frac{c}{b}\sin\alpha) \tag{10}\]  
\[F_{2}=F_{5}=F\sin\alpha \tag{11}\]  
\[F_{3}=F_{4}=F(0.5\cos\alpha-(c/b)\sin\alpha) \tag{12}\]  
where \(c\) and \(b\) are length parameters (\(c\) = \(b\) = 54 mm) and \(\alpha\) is the loading angles as represented by Figure 2. Table 1 shows the results for all loading forces at different loading angles. Figure 8: (**a**) Geometrical details for CTS specimen and (**b**) loading angles. Figure 7: Fatigue life cycles versus crack extension for various specimen thicknesses. The considered material was Aluminum alloy having the following material properties as shown in Table 2:  
The simulation is conducted under fatigue loading ratio \(R=0.1\), applied load \(F=14\) kN, and the ratio of the initial crack length to the width of the specimen (\(a/W=0.5\)). This simulation considered three different thicknesses, which are 3 mm, 6 mm, and 12 mm, for three angles of loading \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\). The Ansys simulated model shown in Figure 7 is entirely compatible with the experimental work conducted by [40; 41; 42; 43]. Figure 10 shows the typical finite element mesh for each specimen with different thickness associated with the number of nodes and elements simulated by ANSYS. The element size in all simulation was 1 mm, and the total number of nodes and element for \begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties** & **Metric Units Value** \\ \hline Elasticity Modulus, \(E\) & 74 GPa \\ \hline [PERSON]’s ratio, \(\
u\) & 0.33 \\ \hline Yield strength, \(\sigma_{y}\) & 517 MPa \\ \hline Ultimate strength, \(\alpha u\) & 579 MPa \\ \hline Threshold SIF, \(K_{th}\) & 3.15 MPa \(\sqrt{m}\) \\ \hline Fracture toughness, \(K_{lC}\) & 32.95 MPa \(\sqrt{m}\) \\ \hline Paris’ law coefficient, \(C\) & 4.3378 \(\times\) 10\({}^{-7}\) \\ \hline Paris law exponent, \(m\) & 2.6183 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of the considered material. Figure 9: Initial mesh for CTS with different thicknesses: (**a**) t = 12 mm, (**b**) t = 6 mm, and (**c**) t = 3 mm. the three different thicknesses are shown in Table 3. The geometry is meshed using (Q8) eight-noded isoparametric quadrilateral elements. At each step of the crack propagation, the finite element meshes are completely re-meshed, and the mixed-mode SIFs, \(K_{I}\) and \(K_{II}\), are estimated. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **t = 12 mm** & **t = 6 mm** & **t = 3 mm** \\ \hline No. of nodes & 897,045 & 623,867 & 443,766 \\ No. of elements & 615,479 & 419,579 & 288,809 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Total number of nodes and element for each specimen with different thicknesses. Figure 10: Distribution of the Von-Mises stress for the predicted crack growth path with different loading angles and thicknesses. The predicted crack growth path for the three different angles and three different thicknesses are shown in Figure 9 through the Von-Mises stress distribution. The effects of loading angles on the crack growth trajectory were demonstrated clearly. Three different thicknesses, 3, 6, and 12, were used to investigate the effect of thickness on fatigue crack propagation for this geometry. The crack growth trajectory was unaffected by the geometrical thickness, whether the thickness increased or decreased. However, there was a significant impact on the stress and strain distribution, as seen in Figure 10 for the corresponding equivalent Von-Mises stress, which decreased with approximately the same ratio as the specimen thicknesses were increased.