## 3 Results and Discussions
### Three-Point Bending of a Beam with Three Circular Holes
#### 3.1.2 Case II  
This specimen has the same outer dimensions as the one in case I, but it does not include any internal holes. Figure 8 illustrates the crack growth trajectory predicted by ANSYS, which closely aligns with the numerical path estimated by [PERSON] and [PERSON] [43], indicating a strong resemblance between the two predictions. The total deformation displayed in Figure 9 provides a visual representation of the crack growth in the hole-less specimen. It is evident that the crack propagated at a faster rate, eventually causing the specimen to fracture into two separate parts. This is in contrast to the previous specimen with holes, where the crack was effectively arrested at the middle hole, preventing further propagation. This highlights the value of using holes as crack arrestors to enhance structural integrity and prevent catastrophic failure. By redistributing stress, deflecting crack propagation, and inducing crack arrest, this technique significantly improves the reliability of various engineering structures and materials.  
In Figure 10, the von Mises stress values for the current case are displayed, along with the numerical results obtained using ABAQUS software with a mesh-independent computational algorithm through the D'Irik and Yalcinkaya [43] analysis. Notably, there is a remarkable agreement between the computed values. As depicted in the figure, it is evident that the von Mises stress for the hole-less specimen is considerably higher than that of the previous specimen with holes, as previously shown in Figure 7. This discrepancy in stress levels highlights the significance of hole presence or absence in influencing the distribution of stress  
Figure 6: Von Mises stress distribution for case I: **(a)** present study; **(b)** ABAQUS with a mesh-independent computational algorithm [43].  
Figure 7: Predicted crack growth path using ABAQUS without a mesh-independent computational algorithm [43].  
and, consequently, the von Mises stress values in the material. This disparity can be attributed to the absence of stress concentration points associated with the holes in the hole-less specimen. Without the presence of holes to redistribute and alleviate stress, the crack propagation in the hole-less specimen is less impeded, leading to higher stress concentrations and, consequently, higher von Mises stress values throughout the specimen.  
Figure 8: Crack growth path for case II: (**a**) present study and (**b**) ABAQUS with a mesh-independent computational algorithm [43].  
Figure 9: Total deformation of case II specimen.  
Figure 11 illustrates the calculated results of the first mode of the stress intensity factor (KI) for both case I and case II, taking into account the presence or absence of holes. The figure clearly illustrates that in case I, the stress intensity factor exhibits higher values compared to case II, even when accounting for the same crack length. As the crack continues to propagate in case II, the values of KI demonstrate a rapid and continuous increase until complete fracture occurs. The significance of holes on the stress intensity factor (SIF) is underscored by this observation, revealing distinct crack behavior in different cases. These findings highlight the importance of considering such factors in fracture mechanics and structural integrity assessments.