## 4 Numerical Examples and Discussion  
Analysis of the Fracture Mode and Mechanical Response of Concrete Beams under Three-Point Bending with Different Notch Offsets  
The fracture mode would be changed with a different notch, and the corresponding mechanical response would also be varied. [PERSON] [44] modeled the fracture mode transitions of notched concrete beams with different offsets under three-point bending based on the modified PDs. The author focused mainly on the fracture mode transitions of the concrete beams with a large offset and critical value when the transition happened. Since the PF-CZM does not need to consider complex crack tracking, it can be presumed that it can capture the mode transitions in concrete beams as the notch offset increases. Consider a set of quasi-static tests carried out by [PERSON] [53]; these tests contained six different notch offsets of the specimens. The geometry and loading boundary conditions of the specimen are shown in Figure 5. Since the material parameters are not provided in the literature and the material parameters taken in different studies vary, the material parameters we adopted here are: [PERSON]'s modulus \(E_{0}\) = 31.5 GPa, [PERSON]'s ratio \(v_{0}\) = 0.17, failure strength \(f_{\text{f}}\) = 3.19 MPa, fracture energy \(G_{\text{c}}\) = 99.67 N/m. The maximum tensile stress failure criterion and the [PERSON] [49] softening curve are adopted with the method proposed in Section 3. To improve computational efficiency, we used a subdomain implementation scheme. The mesh size of the element used to discretize the damage subdomain is taken as \(h=\ell_{0}/5\), while a larger mesh size is used for the elastic domain, as shown in Figure 6. In addition, the length scale parameter \(\ell_{0}\) is 2.5 mm, the plane stress assumption is considered, and the CPS4T and CPS4R elements are adopted for the damage subdomain and the elastic domain, respectively, in Abaqus. Displacement-controlled is used for the acquisition of response curves, and the prescribed displacement \(u\) = 0.6 mm is applied in 60 incremental steps, with each incremental step \(\Delta u\) fixed at 0.01 mm. Figure 7 gives a comparison of the crack growth with the experimental results using the PF-CZM and the peridynamic model for six different notch offset specimens, with \(\gamma\) = 0, 0.5, 0.7, 0.72, 0.77 and 0.875 from top to bottom. The two groups of specimens with \(\gamma\) = 0.5 and 0.7 in Figure 7a are directly calculated using the span length L. As can be seen from Figure 7, the PF-CZM (Figure 7a) and IH-PD models (Figure 7c) are in good agreement with the experimental results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Strength Level** & **Experimental Mean** & **Max Predicted** & **Min Predicted** & **Optimal Predicted** & **Optimal** \\  & **Peak Load (kN)** & **Peak Load (kN)** & **Peak Load (kN)** & **Peak Load (kN)** & **Softening Law** \\ \hline C60 & 3.501 & 4.055 & 3.618 & 3.618 & Exponential \\ C100 & 4.761 & 4.967 & 4.418 & 4.553 & Petersson \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Predicted peak loads using the PF-CZM with different softening laws. Figure 5: Notched concrete beam: geometry, loading and boundary conditions. agreement with the experimental results, including the crack growth angle and fracture mode, except for a slight difference with the experimental results at \(\gamma\) = 0.7. However, the IH-PD model has taken into account the factor of mesoscale inhomogeneity and the predicted results of the FH-PD model (Figure 7d) without this factor differing greatly from the experimental results; the crack initiates and grows from the notch tip, no matter where the notch is located. For the PF-CZM, without considering the mesoscale inhomogeneity, it is still possible to capture the twice transitions of the fracture mode of the specimen, and the damage zone is clearer and more explicit compared to the peridynamic modeling. However, it should be noted that PF-CZM still has the problem of computational efficiency.