## 3 Validation of the Proposed Approach
### Verification of Crack Growth Path Prediction  
Two configurations of the PMMA beam specimen (PMMA with and PMMA without holes) were chosen to validate the ABQUS-FRANC3D co-simulation approach for predicting the crack growth path. The specimen dimensions were as follows: width = 203.2 mm, length = 508 mm, and thickness = 12.7 mm. The PMMA beam specimens exhibited diverse configurations, showcasing variations in both the lengths and positions of initial crack offsets, as illustrated in Figures 3a and 4a. To provide a clearer description, a numerical identification system in Table 1 assigned specific numbers to the PMMA beam samples with distinct configurations. The parameters \"\(d\)\" and \"\(a_{0}\)\" represent the distance between the initial offset crack and the middle of the beam and the length of the initial crack, respectively. It is essential to underline that this section focuses on the crack growth path, allowing for continuous crack propagation without imposing specific load magnitude requirements. The stainless-steel material properties governing the specimens are characterized by a [PERSON]'s ratio \(v\) = 0.3 and Young's modulus \(E\) = 205 GPa. Figure 3a shows the finite element model. The element size is set to 2 mm, and a total of seven mesh layers are established along the thickness direction. For the FEM of the PMMA specimen with holes shown in Figure 4a, the mesh on the section enveloping the holes is refined to reduce the calculation time and improve accuracy. Comparisons between the propagation paths of the experiment obtained in ref. [24] and the proposed approach for the PMMA beam specimens without holes are shown in Figure 5. The propagation path derived from the combined simulation of the FRANC3D and ABAQUS and the propagation path of the observed experimental data matches well. This provides confirmation of FRANC3D's accuracy in simulating crack propagation. However, the ABAQUS results show convergency problems, and in many alternatives, the calculation cannot be finished. Figure 2: Comparison of the theoretical and numerical results of stress intensity factors: KI and KII with different initial crack lengths: (**a**) for \(\varphi\) = 30\({}^{\circ}\), and (**b**) for \(\varphi\) = 45\({}^{\circ}\). Figure 4: The PMMA specimen: (**a**) the finite element model with holes, (**b**) the crack growth path for specimen D, (**c**) the crack growth path for specimen E, (**d**) the crack growth path for specimen F.  
Figure 3: The polymethyl methacrylate (PMMA) beam specimens: (**a**) the finite element model without holes, (**b**) the crack growth path for specimen A, (**c**) the crack growth path for specimen B, (**d**) the crack growth path for specimen C.  
Figure 4b-d display the crack growth trajectories of the PMMA beam specimens with holes (specimens D, E, and F), while Figure 6 displays the obtained results in comparison with the experimental results and ABAQUS self-results. For specimens with holes, the crack growth ends at the edge of the hole. After this length of growth, the computation is no longer possible. As seen in Figure 6, the locations of the crack growth terminations for specimens D, E, and F change notably between the two approaches. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline  & \multicolumn{3}{c}{**Without Holes**} & \multicolumn{3}{c}{**With Holes**} \\ \hline
**Specimen No.** & **A** & **B** & **C** & **D** & **E** & **F** \\ \hline d (mm) & 152.4 & 152.4 & 127 & 152.4 & 152.4 & 127 \\ a\({}_{0}\) (mm) & 63.5 & 25.4 & 38.1 & 63.5 & 25.4 & 38.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Configurations and identification of different PMMA beam samples used in this study. Figure 5: Comparison of the crack propagation paths obtained using the proposed method with the experimental [24] and Abaqus self-method for the first three PMMA beam specimens (A, B, C). Figure 6: Comparison of the crack propagation paths obtained using the proposed method with the experimental [24] and the Abaqus self-method for the last three PMMA beam specimens (D, E, F).