*3.1. Two Internal Non-Colinear Cracks*

For this geometry, there were two internal, parallel, non-colinear, and non-angled cracks in a rectangular specimen with dimensions (90 mm/180 mm). The initial crack length was a = 10 mm for both cracks. As seen in Figure 4a, this geometry was subjected to acyclic tension (*<sup>σ</sup>*max = 160 N/mm, *σ*min = 0) at the upper end and restricted at the bottom side. The distance between the two tips was 15 mm in the horizontal direction and 5 mm in the vertical direction. The adaptive dense mesh is shown in Figure 4b. The selected material was aluminum, which has the material properties shown in Table 1.

**Figure 4.** (**a**) Problem statement for two internal non-colinear cracks (all dimensions in mm) and (**b**) adaptive mesh for the specimen.

**Table 1.** Material properties of aluminum.


This specimen contained four crack tips, which made it interesting to observe the interaction between cracks and to further explore the performance of the developed software in the simulation of multiple cracks.

The predicted crack growth is shown in Figure 5a, which closely resembled the experimental result of Tu and Cai (1993), as illustrated in Figure 5c. These predicted crack growth trajectories were also in agreemen<sup>t</sup> with the numerical results obtained by [29] using the linear smoothed extended finite element method, which was compared to the numerical results reported by [5] using a meshless method with enriched weight functions, as shown in Figure 5b.

**Figure 5.** (**a**) Crack propagation simulation for the two internal non-colinear cracks specimen, (**b**) the numerical results of [29], with permission from Elsevier 2019, and (**c**) the experimental results [30].

Figure 6 compares stress intensity factors in tips A and B along the crack length with the result from the meshless finite element [5]. Actually, the crack length values were the cumulative crack increment in tips A and B, starting at 10 mm in each stage, which was the original crack length. Only the upper right slip result was selected in the graph. The figure shows good agreemen<sup>t</sup> for the comparison results. The deviation of *KAI* at a crack length of 27 mm above was attributable to the contact with the opposite crack trajectories.

**Figure 6.** SIFs versus crack length comparison between the present study and the results of [5] for the two internal non-colinear crack specimens.

Both cracks demonstrated in the beginning a pure mode I of approximately the same SIF values. After that, the mode II of the SIF increased at tip A above that of tip B while the second mode of SIFs became negative at A, thus making the crack path curve towards the other break. Eventually the second mode of the SIFs at A tended to decrease as crack tip A moved closer when the first mode at B increased continually. Finally, the equivalent mode I of the SIF at B exceeded the fracture toughness and unstable fracture occurred at crack tip B. The fatigue life of the structure was evaluated as 6840 cycles, which was in good agreemen<sup>t</sup> with the results obtained by [5] using a meshless method, as shown in Figure 7, as well as with the numerical results obtained by [29].

**Figure 7.** Comparison for the fatigue lifecycles for the two internal non-colinear cracks.
