*3.2. Cracked Plate with Three Holes*

Consider a 120 mm × 65 mm × 16 mm rectangular plate with two 13 mm diameter holes near both ends and a 20 mm hole near the middle of the plate, as seen in Figure 10. At the middle of the plate is an initial edge crack of 10 mm. The plate was made from aluminum 7075-T6, with the material properties shown in Table 2, and the amount of the fatigue load was *P* = 20 kN with a stress ratio *R* = 0.1. Linear elastic material behavior was assumed. The initial XFEM ANSYS model with an eight-node tetrahedron mesh is shown in Figure 10b. The size of the mesh element was set as 1 mm, creating a mesh of 581,980 nodes and 398,566 elements.

**Figure 10.** (**a**) Description geometry of the cracked plate with three holes (dimensions in mm); (**b**) initial mesh.

**Table 2.** Materials properties for aluminum 7075-T6.


The crack path growth simulated with ANSYS software was compared, and had strong agreement, with both experimental and numerical results from ABAQUS software obtained by [28] as well as with numerical results performed by [29] using XFEM with a controllable :=

crack propagation strategy, as seen in Figure 11a–d, respectively. The distribution of the maximum principal stress, the von Mises stress, and the equivalent strain are shown in Figures 12–14, respectively.

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**Figure 11.** Comparison of crack growth trajectory; (**a**) present study; (**b**) experimental observation reproduced from [28] with permission from Elsevier 2009; (**c**) numerical reproduced from [28] with permission from Elsevier 2009; (**d**) numerical reproduced from [29] with permission from Elsevier 2018.

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**Figure 12.** The maximum principal stress distribution.

**Figure 14.** The equivalent elastic strain distribution.

The maximum value of the von Mises stress was in the last step of the crack growth, in which the area around the crack tip is known as a plastic zone. In this area, the behavior of the material is plastic. The plastic zone is created when the stress goes from minimum to maximum values and is called uploading. The plastic zone is plastically elongated in the loading direction. It becomes longer than it was before. As a consequence, the zone is

loaded in compression during unloading and reversed plasticity occurs. As pointed out by Rice [30], reversed plasticity requires a local stress increment in the reserved direction in the order of twice the yield stress.

The predicted values of the two modes of stress intensity factors, i.e., *KI* and *KII*, are shown in Figures 15 and 16, respectively. As shown in Figure 14, the crack starts to grow in a straight direction, indicating the domination of *KI* followed by a curved direction with an increasing negative value of the second mode, *KII*, that results in the crack growing toward the hole.

**Figure 15.** Predicted values of the first mode of stress intensity factors.

**Figure 16.** Predicted values of the second mode of stress intensity factors.
