## 4 Numerical Examples and Discussion
When the notch is located in the middle of the span (\(\gamma\) = 0), the specimen has a typical mode-I failure. Within a certain offset range, the crack growths will gradually change from mode-I to a mixed mode fracture as the notch offset increases. When the offset is greater than the critical value of offset distance (\(\gamma\) = 0.7), the fracture will occur anywhere in the mid-span of the beam instead of extending along the cut tip; this feature can be well reproduced by the PF-CZM. It can also be found that when the offset is 50 mm (\(\gamma\) = 0.5), the results obtained using the PF-CZM exhibit a tendency to change to mode-I in the late stage of the fracture. In the experimental results, a critical state for the second transition of the fracture mode exists at \(\gamma\) = 0.7. The PF-CZM can naturally capture this process of fracture mode transition states. Figure 8 gives the damage profile of the three groups of specimens at \(\gamma\) = 0.70, 0.72 and 0.77, when the crack initiates in the span for the first time. It can be seen that when \(\gamma\) gradually approaches 0.7, the damage zone at the tip of the notch gradually  
Figure 6: Notched concrete beam (\(\gamma\) = 0) under three-point bending: finite element mesh. Figure 7: Ultimate damage profile of six specimens with the rising notch offset (\(\gamma\) = 0, 0.5, 0.7, 0.72, 0.77 and 0.875 from top to bottom): (**a**) PF-CZM; (**b**) experimental results [53]; (**c**) intermediately homogenized peridynamic (IH-PD) model; (**d**) fully homogenized peridynamic (FH-PD) model [44]. expands. Compared with \(\gamma=0.50\) in Figure 7a, the damage continues to extend directly along the notch tip and does not initiate in the span, which can prove that, at this time, there is indeed a critical state. To further validate the model, we quantitatively compared the peak loads of concrete beams at different offsets using different methods, and the results are shown in Figure 9. The peak loads obtained from the experimental and theoretical analyses in [53], the PF-CZM, and the peridynamic model [44] are shown in Figure 9. It can be noted that the peak loads of the PF-CZM do not differ much from the experimental results as well as the analytical solution, and the evolution tendency with \(\gamma\) is consistent with the test, i.e., the effect of the notch offset on the limit peak capacity of the specimen gradually decreases with an increase in the notch offset, which effectively verifies the accuracy of the model. Figure 8: Damage profile when crack initiates at the mid-span for the first time: (**a**) \(\gamma=0.70\); (**b**) \(\gamma=0.72\); (**c**) \(\gamma=0.77\). (The phase field values are rounded to one decimal place). Figure 9: Comparison of peak loads with different \(\gamma\) values from PF-CZM, IH-PD, FH-PD [44], the experimental results and the analytical solution [53]. Where \(\delta=\frac{|\frac{P_{1}-P_{2}}{P_{2}}|}{P_{2}}\times 100\%\), \(P_{1}\) is the predicted peak load using the PF-CZM and IH-PD, and \(P_{2}\) is the analytical results from [53]. It can be found in Figure 10 that the relative error of the PF-CZM is smaller than IH-PD for the same group materials with different notch features. Thus, we could assume that once the optimal softening law is determined, it is suitable for the same group of materials. However, it should be noted that whether the method is applicable to large-scale structures depends on further research.