## 3 Results and Discussions  
### Three-Point Bending of a Beam with Three Circular Holes  
This case study seeks to verify the capability and accuracy of the ANSYS Smart Crack Growth feature in assessing crack propagation paths under mixed-mode loading conditions. To accomplish the validation, the process entails utilizing three-point bending beams with varying initial crack lengths and locations. The beam configurations displayed in Figures 1 and 2 represent two distinct scenarios: one without any holes and the other with three holes. These configurations are specifically designed to analyze and evaluate the crack behavior exhibited under different conditions. Table 1 presents a comprehensive overview of the material properties associated with the three-point bending beam utilized in this study. The material is characterized by linear elastic properties, which means that it exhibits a linear relationship between stress and strain within its elastic range. In several studies [42; 43; 44], there appears to be an inconsistency in referring to the beam as polymethyl methacrylate (PMMA) despite the use of different materials. It is important to note that this misnomer should be corrected to accurately reflect the material actually utilized in each study. The initial experimental work conducted by [PERSON] et al. [45] on PMMA specimens, which exhibited a linear elastic behavior, served as the basis for naming the beam as PMMA. However, it should be acknowledged that the authors of these studies expanded their investigations to include other linear elastic materials alongside PMMA. Therefore, the beam material designation should be adjusted to align with the specific material used in each study. To confirm the accuracy of the simulated crack paths, experimental results from [PERSON] et al. [45] are employed as a point of comparison. The specimens in this study are subjected to constraints that restrict all degrees of freedom, beginning from the location of the left pin. However, the right pin is permitted to move solely in one direction along the \(X\) axis.  
The geometry under consideration has specific dimensions: a length of 2 L = 508 mm, a width of 203.2 mm, and a thickness of 12.7 mm. All specimens share the same dimensions, hole positions, and hole sizes. However, primary variations occur in the initial crack length and its positions, as specified in Table 2. The specimens are subjected to a point load of 4448 kN, which is applied at the midpoint of the top section. Meanwhile, the bottom section of the specimens features two simply supported points, with one point fixed and the other point movable in the x-direction. This configuration allows for controlled loading conditions and facilitates the examination of the specimens' mechanical response under specific loading scenarios. Figure 3 displays the initial finite element meshes for specimens 1 and 2. In specimen 1, there are 253,462 nodes and 155,476 elements, while in specimen 2, there are 269,563 nodes and 165,231 elements.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Property** & **Value in Metric Unit** \\ \hline Modulus of elasticity, \(E\) & 205 GPa \\ Poisson’s ratio, \(v\) & 0.3 \\ Yield strength, \(\sigma_{y}\) & 285 MPa \\ Fracture toughness, \(K_{IC}\) & 730 MPa mm\({}^{0.5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mechanical characteristics of cold-rolled SAE 1020 steel [43].  
Figure 1: Geometrical representation of the three-point bending beam without holes (dimensions in mm).  
Figure 2: Geometrical representation of the three-point bending beam with three circular holes (dimensions in mm).