## 4 Numerical Examples and Discussion
### Analysis of the Mechanical Behaviors of Double-Notched Concrete Beams under Three-Point Bending  
In actual concrete structures, there is usually more than one crack, and its distribution is often stochastic. It is costly to conduct systematic experimental studies. Even for simple double-notched concrete beams, the location of the notch is diverse. In this section, the mechanical response and fracture features of a typical double-notched concrete beam under three-point bending are analyzed, and the results are compared qualitatively with the experimental results.  
The geometry and boundary conditions are shown in Figure 11, where the primary notch depth \(a_{0}\) is located in the middle of the concrete beam, the secondary notch depth is noted as \(a_{1}\), and the distance between the primary and secondary notches is noted as \(d\). The following material parameters are considered: \(E_{0}\) = 33.5 GPa, \(r_{0}\) = 0.15, \(f_{t}\) = 2.96 MPa, with the linear softening curve considered. A prescribed displacement, \(u\) = 1 mm, is applied in 100 incremental steps, with each incremental step \(\Delta u\) fixed at 0.01 mm. For convenience, the primary notch depth \(a_{0}\) is fixed at 60 mm with the corresponding fracture energy \(G_{c}\) = 64 N/m. Only two groups of notch distance \(d\) = 30 and 60 mm are considered in this work; for each notch distance, three different secondary notch depths are selected (i.e., \(a_{1}\) = 30, 45 and 60 mm). The secondary notch is not higher than the primary notch.  
Figure 12 gives the numerical results for two typical specimens, and it is evident that the crack initiates and propagates at the primary notch of the specimens and that damage exists at the secondary notch as well. The closer the secondary notch is to the primary notch, the higher the notch depth and the more obvious it is. This is qualitatively in agreement with the experimental results of [PERSON] [54]. In particular, the specimen exhibits a typical mode-I failure in Figure 12a; the effect of the secondary notch is negligible compared to Figure 12b. It can be clearly observed in Figure 12b that as the secondary notch depth increases and the notch distance decreases, the primary notch crack path exhibits a pronounced reverse perturbation by the secondary notch, even without considering the nonuniformity at the mesoscale. The phenomenon is consistent with the experimental observation in Figure 13, and it cannot be objectively explained using the CZM. It can be assumed that the crack path perturbation caused by local stress concentrations would be more obvious as the distance between the primary and secondary notches becomes  
Figure 10: Relative error \(\delta\) of predicted peak loads with different \(\gamma\) values from PF-CZM and IH-PD.  
smaller. However, the crack would turn back to mode-I failure (without considering the heterogeneity at the mesoscale explicitly), as shown in Figure 12b.  
Furthermore, the predicted results of the peak load are shown in Table 3, and it can be found that when the secondary notch depth is the same, the limit load capacity of the beam is increased accordingly as the notch distance decreases. When the notch distance is fixed, the higher the secondary notch depth, the higher the limit load capacity of the beam. This phenomenon would be more obvious with an increase in the secondary notch depth. However, it should be noted that the increments of peak load are not obvious. On the other hand, this also shows the prospect of the applicability of the PF-CZM on the accurate failure modeling of multi-cracked structures.  
Figure 11: A concrete specimen: geometry, loading and boundary conditions.  
Figure 13: Experimental fracture feature of a double-notched concrete beam from [54].