**3. Numerical Results and Discussion**

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*3.1. Modified Four-Point Bending Beam*

This case corresponds to a single cracked beam with a hole, loaded in the upper two points and constrained in the lower two points, i.e., a modified four-point bending specimen as shown in Figure 1. This refers to a problem of plane strain that was solved numerically in [23,24]. The geometry is 125 × 30 × 10 mm<sup>3</sup> in size, and the hole radius, *R* = 5.2 mm, was located 9.3 mm from the left of the original crack and 14.8 mm above it. This specimen was simulated under fatigue loading with a constant amplitude load ratio, R= 0.1, and the quantity of the applied loads were *P* = 100 N. The initial mesh of this geometry is shown in Figure 2. The material for this specimen was cold-rolled SAE 1020 steel with the following properties as shown in Table 1:

**Figure 1.** Geometry of the four-point bending beam (dimensions in mm).

**Figure 2.** Initial mesh of the four-point bending beam.


**Table 1.** Materials properties for cold-rolled SAE 1020 steel.

The predicted crack growth trajectory was smooth and identical to the experimental path predicted by [23] and can be further compared to the predicted trajectories obtained using other numerical methods, such as the finite element method based on local Lepp– Delaunay mesh refinement used in [24], the finite element with configurational forces used in [25], and the coupled extended meshfree–smoothed meshfree method used in [26], as shown in Figure 3a–e, respectively. In the initial period, the crack grew with a small increment when the crack tip was relatively far from the hole. The crack growth direction changed with a large angle and gradually affected the hole with the crack proceeding. Figure 4 illustrates six different steps of the crack growth represented in the von Mises stress distribution, whereas the three-dimensional distribution of the von Mises stress distribution with and without deformation is shown with a legend in Figure 5.

**Figure 3.** Comparison of the crack growth trajectory for the four-point bending beam; (**a**) present study; (**b**) experimental observation reproduced from [23] with permission from Elsevier 2003; (**c**) numerical reproduced from [24] with permission from Elsevier 2010; (**d**) numerical reproduced from [25] with permission from Elsevier 2017; (**e**) numerical reproduced from [26] with permission from Elsevier 2020.

**Figure 4.** From top to bottom, six different steps of crack growth for the four-point bending beam.

**Figure 5.** Von Mises stress distribution for the last step of the crack growth with and without deformation.

> The distribution of the maximum principal stress is shown in Figure 6 with enlargement of the crack tip area.

**Figure 6.** Maximum principal stress distribution.

For fatigue life evaluation, the SIFs are the important criterion. For a normal four-point bending beam, various handbooks may include analytical calculations of the SIFs. For the regular four-point bending beam without a hole the SIFs solution is formulated as follows [27]:

$$K\_I = f(a/\mathcal{W})\frac{6P(\mathbf{s} - r)\sqrt{\pi a}}{W^2 t} \tag{10}$$

where *KI* is the first mode of SIFs, *f*(*a*/*W*) refers to the dimensionless SIF, *W* is the beam width, *t* is beam thickness, *P* is load applied, *s* and *r* are the distances defined in Figure 1, and *a* is the length of the crack. The dimensionless regular stress intensity factor for the point bending beam without holes was formulated as [27]:

$$f(a/W) = \frac{1.1215}{\left(1 - \frac{a}{W}\right)^{(3/2)}} \left[ \frac{5}{8} - \frac{5}{12} (a/W) + \frac{1}{8} (a/W)^2 + 5(a/W)^2 (1 - \frac{a}{W})^6 + \frac{3}{8} \exp(-6.1342(a/W)/(1 - \frac{a}{W})) \right. \tag{11}$$

The presence of a hole created a curved crack trajectory in this modified geometry, hence, Equation (11) was no longer valid as a consequence of the curved crack direction. ANSYS can obtain accurate expected *f*(*a*/*W*) values rather than manual solutions for the regular four-point bending beam specimen. In order to achieve the dimensionless stress factor *f*(*a*/*W*), mode I SIFs (*KI*) were obtained from ANSYS and substituted into Equation (11). Fitting the fifth-degree polynomial into the stress intensity factors for the modified four-point bending beam gave the following equation:

$$f(a/\text{W}) = 12.116(a/\text{W}) - 88.937(a/\text{W})^2 + 336.46(a/\text{W})^3 - 595.59(a/\text{W})^4 + 417.66(a/\text{W})^3 + 0.4287 \tag{12}$$

A generalized linear regression method facilitates usage of the formula, which displays SIFs as a function of both the relevant crack and contact parameters, easing assessment of crack growth behavior. For the modified four-point beam specimen used in the above analysis, the numerical dimensionless SIFs were compared with the analytical solution in Equation (11) for the standard beam without a hole, as well as with the dimensionless SIF values calculated by [14] applying the boundary element method (BEM) with BemCracker2D (BC2D) software as shown in Figure 7.

**Figure 7.** Dimensionless stress intensity factors for the standard and modified four-point bending beams.

The predicted values of both modes of stress intensity factors, i.e., *KI* and *KII* are shown below in Figure 8. As seen in this figure, the crack started to grow in a straight line as the first mode of stress intensity factors dominated the crack growth direction. When the crack direction was influenced by the presence of the hole, the crack grew toward the hole and changed its direction, increasing the values of the second mode of stress intensity factors. The predicted fatigue life according to the number of cycles was compared, as shown in Figure 9, to the experimental results performed by [14] alongside the numerical results for the same authors with two software programs: Vida and BemCracker2D. According to this figure, there was a strong correlation between the present study's result and the Vida software compared to that of the BemCracker2D. According to Figure 8, the bimodality ratio (*KII/KI*) was not zero. The direction of the crack was dominated by *KI* at the beginning

of the crack growth since the *KII* values were small compared to the *KI* values. After that, as the second mode of stress intensity factors, *KII* was increased gradually up to a maximum value of 21 MPa(mm)1/2, leading to a change in the direction of the crack toward the hole.

**Figure 8.** Predicted values of the stress intensity factors.

**Figure 9.** Comparison for fatigue life of the modified four-point bending beam.
