# Transient Response of Dynamic Stress Concentration around a Circular Opening: Incident SH Wave
## 5 Numerical Examples and Discussion  
From the convergence of the Bessel function, it can be seen that when \(m\geq\mathrm{M}\), the space function \(\tau_{\theta_{2}}^{(n)}(r,\theta)\) can converge. After many calculations, the calculation accuracy can meet the requirements when \(m=20\). Using the common material 16 NiCr4 in industry as an example, its density is \(\rho=7.8\times 10^{3}\) kg\(\cdot\)m\({}^{-3}\) and its shear modulus is \(\mu=7.9\times 10^{10}\) N\(\cdot\)m\({}^{-2}\). The characteristics of SH wave incidence are as follows: (i) the impact time of incident wave is short, and the response time of circular opening is generally 5-20 ms and (ii) the frequency of the incident wave is high and the frequency band is wide, generally 50-1000 Hz. Considering that the distance from the wave source to the edge of the opening is \(s=0\), the spectral function of SH wave in the example \(\varphi^{(i)}(r,\theta,\omega)\) is given in Table 1, where \(n=10\), circular frequency \(\omega_{0}=100\pi\), and harmonic amplitude coefficient \(a_{n}\in(0,\,0.2)\) mm. The waveform diagram corresponding to the spectral function in the time domain is shown in Figure 2.  
Figure 2 shows the temporal distribution of the wavefield generated by the incident wave. At time T = 10 ms, the wavefield is fully established. The maximum displacement occurs at time \(t=0.71\) ms, with a value of \(a=0.86\) mm. Initially, the displacement undergoes significant variations, indicating a rapid change in the amplitude of the wavefield during this time interval. After approximately \(t=2\) ms, the displacement exhibits more minor variations. It gradually decreases to zero at time \(t=\mathrm{T}\). This signifies a gradual decrease in the amplitude of the wavefield until it eventually vanishes. This temporal behavior of the wavefield provides insights into the incident wave's dynamic response and propagation characteristics. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \(\omega\) & \(\omega_{0}\) & \(2\omega_{0}\) & \(3\omega_{0}\) & \(4\omega_{0}\) & \(5\omega_{0}\) & \(6\omega_{0}\) & \(7\omega_{0}\) & \(8\omega_{0}\) & \(9\omega_{0}\) & \(10\omega_{0}\) \\ \hline \(a_{n}/\)mm & 1.66 & 1.64 & 1.14 & 1.14 & 0.572 & 1.40 & 1.60 & 0.883 & 0.892 & 0.931 \\ \hline \end{tabular}
\end{table} 
Table 1: Spectrum function of periodic shockwave. Based on the data presented in Figure 3, we can observe the temporal variation of the time stress peak coefficient (TSPC) peak at different radii. The TSPC consists of the maximum value of the DSCC at each moment. For a radius of 1, the peak of TSPC reaches 2.27, corresponding to a time of 0.44 ms. Similarly, at a radius of 2, the peak of TSPC is 1.75, occurring at 1.11 ms. In the case of a radius of 3, the peak of TSPC is 1.41, with a corresponding time of 1.08 ms. From the figure, it can be observed that peaks are predominantly concentrated in the initial transient response, indicating that the maximum value of TSPC is achieved during the initial stages of the system. Subsequently, after exceeding 2 ms, the amplitude of TSPC significantly attenuates. This observation is consistent with the temporal waveform figure in Figure 2, further reinforcing this relationship. These findings are essential for understanding the system's dynamic behavior and identifying appropriate parameter configurations and optimization control strategies. The opening edge DSCC corresponding to the peak of each curve in Figure 3 is shown in Figure 4. It is evident from the figure that each curve is not complete, indicating that the  
Figure 3: Change of TSPC with time at different radii, \(r=1,2,3\). Figure 2: Time domain waveform of transient SH wave incident wave. TSPC only exists during the initial stage of shockwave impact. Specifically, when the radius of the opening is 1, the stress response at the opening edge is significantly greater than when the opening radii are 2 and 3. This suggests that, under a constant incident wave, the smaller the aperture, the higher the peak of DSCC generated at the opening edge.