## 3 A Case Study for Validating the Proposed Model and Process
### Results and Discussion  
Damage plots from this study, shown for all three mesh densities (\(\delta\) = 1.2, 0.6 mm and 0.5 mm) in Figures 7-9, demonstrate that for a mesh density which is sufficiently refined, crack path behavior is in agreement with experimental results. Figure 6: Crack Path for CT Specimens from Experimental Study by [PERSON] et al. [PERSON] et al. (2018). Figure 8: Crack Path as Determined from Damage Plot, \(\delta\) = 0.6 mm. Figure 7: Crack Path as Determined from Damage Plot, \(\delta\) = 1.2 mm. Figure 9: Crack Path as Determined from Damage Plot, \(\delta\) = 0.5 mm. Figure 10 compares the simulated crack path to the experimental crack path with the X-coordinate representing horizontal growth of the crack parallel to the initial precrack and the Y-coordinate representing growth in the vertical direction. The path for the coarsest mesh (\(\delta\) = 1.2 mm) propagates into the tertiary hole and does not converge to the experimental path over any portion of the advance. Despite this, it is obvious that the path results are very similar for horizon sizes of \(\delta\) = 0.6 mm and \(\delta\) = 0.5 mm providing confidence that the mesh density is sufficient to obtain converged results. The location at X = 22 is the initial crack location. Correlation of both refined models to the experimental crack path is achieved for crack tip spatial position of the X-coordinate in the range of 14 mm to 22 mm. Thus, the simulation accurately predicts the crack path over the first 200,000 cycles. At X values less than 14 mm the path flattens ahead of the \"knee\" in the experimental crack path. It was previously noted that for a mode I crack, the crack would grow purely in the horizontal direction. Deviation from this horizontal path occurs when mixed-mode loading occurs (i.e., introduction of \(K_{II}\)). When deviation occurs towards the end of the simulation the crack is growing purely in the horizontal direction suggesting the \(K_{II}\) component is under-represented. Results for X \(<\) 14 mm might be improved with further refinement of the grid size near the tertiary hole. As observed in Figure 11, the results of [PERSON] et al. [18] matched very well for horizon sizes of \(\delta\) = 1.2 mm and \(\delta\) = 0.6 mm up to X = 14 mm, where the X-coordinate is aligned to the horizontal direction. For X values less than 14 mm, the \(\delta\) = 1.2 mm mesh significantly over-predicted the Y-coordinate of the crack path while the \(\delta\) = 0.6 mm model under-predicted the Y-coordinate. This result indicates that the relative coarseness of the mesh at the tertiary hole has a significant effect on the strain field near the crack front as the crack path nears the hole. Figure 12 shows the comparisons of the simulations of this study with the study of [PERSON] et al. [18]. Results in the present study underestimated the Y-coordinate of the crack path for X \(<\) 14 mm for \(\delta\) = 0.5 mm and \(\delta\) = 0.6 mm similar to the study by [PERSON] et al. [18]. Figure 10: Spatial Position of Crack Tips of This Study With Experimental Data. It is important to note that differing convergence rates were achieved in the present study and the study by [PERSON] due to the choice of micromodulus function. A constant micromodulus function was utilized in the present study due to its ease of implementation in ABAQUS code, while the work by [PERSON] utilized a conical micromodulus function. This conical micromodulus has been shown to converge at a faster rate compared to classical elasticity solutions [29]. Further refinement of the mesh near the tertiary hole via a non-uniform mesh is left for future study. In the simulation as well as the experimental tests after which the simulation is modeled, a near constant stress intensity factor range of 20 Mpa (m)\({}^{1/2}\) was maintained at the crack tip. Through material testing [PERSON] et al. [23] showed that this stress intensity factor (SIF) range lies in the stage II regime of a Paris Law plot allowing for the use of Equation (26), stated below.