# Fatique Crack Growth Studies under Mixed-Mode Loading in AISI 316 Stainless Steel
## 3 Numerical Results and Discussions
### Compact Tensile Shear (CTS) Specim
By performing this comparison, the present study aimed to gain important insights into the behavior of cracks under different loading conditions, which can aid in the development of more accurate predictive models for designing materials and structures that are less susceptible to failure. The three figures demonstrated that the predicted paths of crack extension were similar to the experimental trajectories. Figure 5: Predicted crack growth paths compared to experimental data by [36]. Figure 8: Comparison of predicted and experimental crack propagation trajectories for a 60\({}^{\circ}\) loading angle: (**a**) present study and (**b**) experimental [15]. Figure 6: Comparison of predicted and experimental crack propagation trajectories for a 30\({}^{\circ}\) loading angle: (**a**) present study and (**b**) experimental [15]. Figure 7: Comparison of predicted and experimental crack propagation trajectories for a 45\({}^{\circ}\) loading angle: (**a**) present study and (**b**) experimental [15]. The comparison of the estimated fatigue life cycles for mixed-mode loading (I/II) at different loading angles (30\({}^{\circ}\), 45\({}^{\circ}\), and 60\({}^{\circ}\)) between the present study and the experimental results provided by [PERSON] et al. [36] is illustrated in Figures 9-11. The results obtained from these figures provide an insight into the accuracy of the present study's predictions for the fatigue life cycles of materials under mixed-mode loading conditions. These figures illustrate the remarkable consistency between the fatigue life cycles estimated by the present and the experimental data obtained in mixed-mode loading conditions with an average error of less than 0.5%. It is worth noting that the relationship between the fatigue lifetime and loading angle can depend on various factors, such as the material properties, loading conditions, and crack morphology. The R-squared value, also known as the coefficient of determination, is a widely used statistical measure to assess the quality of the fit between the experimental data and the results obtained by modeling. In the current study, the R-squared values for the crack growth path and fatigue life were determined to be 0.996 and 0.993, respectively. Figure 10: A comparison of fatigue life for a loading angle 45\({}^{\circ}\), experimental [36] versus simulated results. Figure 9: A comparison of fatigue life for a loading angle 30\({}^{\circ}\), experimental [36] versus simulated results. [PERSON] et al. [49] proposed an analytical solution for determining the mixed-mode stress-intensity factors in the compact tension shear specimen. This solution involves a set of equations that can be used to calculate the stress-intensity factors for both mode \(I\) and mode _II_ loading conditions as follows:  
\[K_{I}=\frac{F}{Wt}\sqrt{\pi a}\frac{\cos\alpha}{\left(1-a/W\right)}\sqrt{\frac {0.26+2.65a/\left(W-a\right)}{1+0.55a/\left(W-a\right)-0.08\left(a/\left(W-a \right)\right)^{2}}} \tag{8}\]  
Figure 4:0  
where \(F\) represents the uniaxial load, while \(a\) denotes the length of a crack, and \(W\) and \(t\) denote the width and thickness, respectively. The approach outlined in this solution is based on the assumption of linear elastic fracture mechanics and is applicable to CTS specimens with different loading configurations and material properties. Figures 12 and 13 depict an excellent agreement between the predicted values and analytical solutions given by Equations (8) and (9) for the opening and in-plane shear mode SIFs at loading angles of \(30^{\circ}\), \(45^{\circ}\), and \(60^{\circ}\). The behavior of crack growth in the compact tension shear specimens is influenced by the SIFs, _K\({}_{I}\)_ and _K\({}_{II}\)_, which are related to the magnitude and direction of the applied load. The SIFs are used to quantify the intensity of the stress fields near the crack tip and are critical in predicting the onset and propagation of cracks in materials. At a loading angle of \(30^{\circ}\), _K\({}_{I}\)_ is the dominant factor, and _K\({}_{II}\)_ is relatively small.