## 5 Numerical Applications and Discussion
### Qualitative Failure Analysis  
PD theory can correctly analyze the crack propagation in quasi-brittle materials [37; 41], and the crack initiation and propagation are sensitive to the influence function [33]. In this section, a simulation on a typical mode-I fracture is carried out to investigate the impact of different influence functions on the crack extension morphology. The results are compared with experimental observation [42]. Then, the more effective influence function is selected by considering the simulation accuracy and stability. The geometrical condition of a precast notched brittle plate is revealed in Figure 13. The purpose of the precast notch is to clearly track the crack path. The upper and lower edges of the plate are subjected to a tensile stress load of 12 MPa. Material properties and PD model parameters are illustrated in Table 6.  
Figure 12: Eigenvalues of relative error. Figure 13: Schematic diagram of the plate. To assess the stability of crack growth, it is essential to determine if the crack growth velocity is within the theoretical limit of steady-state type-I fracture. The theoretical limit value is the Rayleigh wave speed, as calculated in Equation (26) [43]. The crack-tip is confirmed by tracing the rightmost point of \(D(x,t)=0.35\). The crack propagation velocity is determined in Equation (27). \[V_{R}=\frac{0.862+1.14v}{1+v}V_{s} \tag{26}\]  
where \(V_{s}\) is the transverse wave velocity, denoted as \(V_{s}=\sqrt{\frac{u}{\rho}}=\sqrt{\frac{E}{2\rho(1+v)}}\)  
\[v_{crack}=\left\|\frac{x_{c}-x_{c-1}}{t_{c}-t_{c-1}}\right\| \tag{27}\]  
in which \(x_{c}\) and \(x_{c-1}\) denote the crack-tip positions at time \(t_{c}\) and \(t_{c-1}\). Figure 14 presents the average and maximum crack propagation velocities under various influence functions. These eigenvalues are all less than \(V_{R}=3076\) m/s, which satisfies the stable propagation condition. This indicates that the cracks exhibit steady-state expansion under different influence functions. It is predictable that the damage will initially appear at the precast notch, where the maximum stress is located. As seen in Figure 15, the crack morphology is affected by the type of influence function, which is consistent with the findings of previous study [38]. However, the initial damage position and the main crack propagation paths observed in the simulation are similar to these discovered in experiment, that is, linear propagation first, followed by branching propagation, which has proved the ability to simulate material failure. In particular, the crack growth patterns obtained by influence functions \(g_{2}\), \(g_{3}\), \(g_{5}\) and \(g_{8}\) appear remarkably well in line with the experimental phenomenon. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**E (GPa)** & \(v\) & \(\rho\) **(kg/m\({}^{3}\))** & \(\Delta x\) **(m)** & \(\delta\) **(m)** & \(\Delta t\) **(m)** & **\(\infty\)** & **Influence Function** \\ \hline
65 & \(1/3\) & 2235 & \(0.25\times 10^{-3}\) & \(3\Delta x\) & 25\(ns\) & \(2.0934\times 10^{-3}\) & \(g_{1}-gs\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Material properties and PD model parameters. Figure 14: Crack propagation velocity eigenvalues under various influence functions. Combined with the research results in Section 5.2, the influence function \(g_{2}\) is the more effective influence function by a comprehensive trade-off between simulation accuracy and stability. To further illustrate the rationality of \(g_{2}\), Figure 16 depicts the change of PD force density and the crack evolution process. The PD force density is mainly concentrated at the pre-notch tip, where the damage initially occurs. As the PD force density increases near the tip, the accumulated wave promotes horizontal crack propagation, similar to the typical mode-I crack. At 850 steps, the horizontal propagation distance is 0.017 m. Subsequently, the main crack branches spontaneously and symmetrically, and the maximum PD force density appeared at the branches.