# Numerical Modeling of Crack Growth under Mixed-Mode Loading
## 3 Numerical Results and Discussions
### Three-Point Bending Beam with Different Configurations
#### 3.1.1 Specimen 1  
The simulated crack propagation in this specimen moves between the lower and middle hole and approaches the middle hole from its right side. This indicates that the shear stress intensity factor (\(K_{ll}\)) around the holes is dramatically increased, which forces the step-sizes of crack to be decreased. During propagation, the results of the crack trajectory agree excellently with the experimental data given by [45; 46] and with  
\begin{table}
\begin{tabular}{c c c c} \hline
**Specimen No.** & **Length of Crack, a (mm)** & **Crack Position, b (mm)** & **Hole Layout** \\ \hline
1 & 25.4 & 152.4 & With hole \\
2 & 25.4 & 152.4 & Without hole \\
3 & 38.1 & 127 & With hole \\
4 & 38.1 & 127 & Without hole \\
5 & 63.5 & 152.4 & With hole \\
6 & 63.5 & 152.4 & Without hole \\ \hline \end{tabular}
\end{table}
Table 1: Configurations of the three-point bending beam specimen.  
Figure 1: Geometry of the three-point bending beams: with holes (**left**), without holes (**right**).  
the numerical results when using the coupled extended meshfree-smoothed meshfree method presented by [PERSON] et al. (2020) [47], XFEM results using ABAQUS software obtained by [PERSON] and [PERSON] [36], and numerical results obtained by [PERSON] et al. (2019) [5] using A polygonal XFEM with numerical integration for linear elastic fracture mechanics as presented in Figure 3a-d, and e, respectively. As shown in this figure, the predicted crack growth path in the present study is closer to the experimental trajectory compared to other numerical results.  
Figure 3: Specimen 1, crack growth trajectory (**a**) Ansys simulation, (**b**) experimental [45; 46], (**c**) numerical results [47], (**d**) numerical results [36], (**e**) numerical results [5].  
Figure 2: Initial mesh for the three-point bending beam.  
The crack propagation trajectory predicted in the present study was also matched to those results for the same geometry and boundary condition using the Dual Boundary Element Method (DBEM) introduced by [PERSON] and [PERSON] (2018) and to the numerical result reported by ([PERSON] and [PERSON], 2018) using the Boundary element method (BEM), as shown in Figure 4 (left). The predicted crack growth trajectory in the present study as shown in Figure 4 (right) was more closely matching the experimental crack growth path predicted by [PERSON] et al. (1996) ([PERSON] et al., 1996) than to those predicted in Figure 4 (left).  
The findings of this simulation for dimensionless stress factors were compared with those from XFEM using the smooth nodal stress technique by ([PERSON] and [PERSON], 2018), as shown in Figure 5 with identical results. It is observed that as the crack approaches the hole, SIFs tend to change at a greater amplitude. Figure 6 shows the equivalent Von-Mises stress distribution for specimen 1 with enlargement of the area around the crack tip.  
Figure 4: Crack growth path using the Dual Boundary Element Method (DBEM) ([PERSON] and [PERSON], 2018) and the Boundary element method (BEM) ([PERSON] and [PERSON], 2018) (**left**), Ansys and experimental crack trajectory for the crack tip coordinate of specimen 1 (**right**).  
Figure 5: Comparison for the dimensionless stress intensity factors for specimen 1.