## 5 Numerical Examples
### Kalthoff-Winkler Impact Test  
The Kalthoff-Winkler test, as a classic benchmark problem for studying dynamic fracture and peridynamics simulation results [44; 58], is considered here to verify the ability of the proposed model and the fracture criterion. It can be observed that a brittle fracture mode occurs with impact speed of 32 m/s, and the crack propagates around 70\({}^{\circ}\) with respect to the pre-existing cracks during the test. The specimen geometry and loading boundary are shown in Figure 6.  
Figure 5: Contour plots of plate at \(t=0.1\) s: (**a**) temperature and (**c**) displacement obtained by the FEM, (**b**) temperature and (**d**) displacement obtained by the proposed model.  
Figure 6: The geometry of the Kalthoff–Winkler with boundary conditions.  
Both the bond stretch criterion and the proposed criterion are used to simulate this test. The critical stretch \(s_{0}\) is set to be 0.01 in the bond stretch criterion. The yield stress is set to be the ultimate tensile 2000 MPa [58], and the fracture strain is considered to be very small in the proposed criterion. Other material properties are given in Table 2. The horizon \(\delta\) is set to be 0.75 mm, and \(l\) is set to be 0.25 mm. The time increment is chosen as \(\Delta t=0.087\) us. To save computational costs, a half of the model is taken.  
The results of crack propagation are shown in Figure 7. It can be seen that the cracks start to initiate at about 26.1 us, and then propagate at the angle of 81\({}^{\circ}\) and 69\({}^{\circ}\) for applying the different fracture criteria, respectively. In the case of the bond stretch criterion, the angle of crack propagation is far from the experimental result [59]. Moreover, the rate of crack propagation is slower compared with the case of the proposed criterion and exhibits an unreasonable crack path at t = 113.1 us. In contrast, the proposed criterion gives more stable results and is in good agreement with the pioneer's work [60; 61].  
Figure 8 further shows contour plots of von Mises stress obtained by the two criteria, which demonstrates why the application of the bond stretch criterion in the NOSB-PD framework to simulate the fracture problem often fails. As shown in Figure 8, the stress  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(E\) & \(v\) & \(\rho\) & \(\sigma_{\text{V}_{0}}\) & \(s_{0}\) & \(D_{\text{c}}\) & \(\bar{z}_{0}^{\text{p}}\) & \(\bar{z}_{t}^{\text{p}}\) \\ \hline GPa & & kg/m\({}^{3}\) & MPa & & & & \\ \hline
191 & 0.3 & 8000 & 2000 & 0.01 & 0.99 & 0.0 & 0.001 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mechanical properties of material [58].  
Figure 7: Comparisons of crack propagation obtained by (**a–d**) the bond stretch criterion, (**e–h**) the proposed criterion, and (**i**) the experimental observations [59].  
level is increasing at the bottom of the pre-existing crack and leads to crack initiation. However, using the bond to determine a crack cannot guarantee that the deformation at the crack tip will not produce the numerical oscillation, which leads to chaotic displacement and extreme stress [see Figure 8a]. Instead, using the proposed criterion gives a more stable displacement field and low effect of stress concentration [see Figure 8b].