# Study on Effects of Energy Deposition and Thermal Shock Wave by Electron Beam Irradiation
## 5 Comparison and Discussion of Experimental and Numerical Simulation Results  
Since the PVDF sensor is placed on the back of the target plate, the stress measured by PVDF actually represents the transmitted shock wave stress after transmission and reflection action from the target to the PVDF interface. To obtain the peak stress of the shock wave in the target, that is, the incident wave stress, calculations must be done according to the transmission and reflection relationship of the wave at different medium interfaces. Assume that the stress recorded by the PVDF stress sensor is \(\sigma_{\rm q}\) and \(\sigma_{\rm a}\) is the stress of the component to be tested, then:  
\[\sigma_{\rm a}=\sigma_{\rm p}/T \tag{20}\]  
where \(T\) is the transmission coefficient, and  
\[\left\{ \begin{array}{l}T=T_{0}=\frac{2\rho_{\rm 0c}C_{\rm 0c}}{\rho_{\rm 0c}C_{ \rm 0c}+\rho_{\rm 0a}C_{\rm a}},\;\sigma_{\rm a}\leq\sigma_{\rm aH}\\ T=\frac{\rho_{\rm 0c}C_{\rm 0c}}{\rho_{\rm 0c}C_{\rm 0c}+\rho_{\rm 0a}D_{\rm a}},\; \sigma_{\rm a}>\sigma_{\rm aH}\end{array} \right. \tag{21}\]  
where \(\rho_{\rm 0c}C_{\rm 0c}\) and \(\rho_{\rm 0a}C_{\rm 0a}\) are the acoustic impedance of PVDF and target material, respectively, \(D_{\rm a}\) is the velocity of the shock wave, \(\sigma_{\rm aH}\) is the Hugoniot coefficient of target material, and \(T_{0}\) is the transmission coefficient of elastic waves. When \(\sigma_{\rm a}>\sigma_{\rm aH}\), use  
\[D_{\rm s}=C_{\rm os}+Su_{\rm s} \tag{22}\]  
and the shockwave front jumping conditions are:  
\[u_{\rm s}=\frac{C_{\rm os}}{2S}\Bigg{(}\sqrt{\frac{4S}{\rho_{\rm 0a}C_{\rm 0a} }}\Bigg{)}=\frac{\sigma_{\rm a}}{\rho_{\rm 0a}C_{\rm 0a}} \tag{23}\]  
Figure 15: Attenuation of thermal shock waves at different energy densities. \[\sigma_{\rm a}=\frac{\sigma_{\rm p}\big{(}\rho_{0\rm c}C_{0\rm c}+\rho_{0\rm a}C_{0 \rm a}\big{)}C_{0\rm a}}{\rho_{0\rm c}C_{0\rm c}C_{0\rm a}}=\frac{\sigma_{\rm p} }{T_{0}}\bigg{(}1+\frac{2\sigma_{\rm p}}{\rho_{0\rm c}C_{0\rm c}C_{0\rm a}} \bigg{)} \tag{24}\]  
In this experiment, the thermal shock time is 0.25-0.32 \(\upmu\)s, the half-width is 0.12-0.14 \(\upmu\)s, the rise time is 0.1-0.2 \(\upmu\)s, the electron beam energy injection range is 174\(\sim\)195 J/cm\({}^{2}\), the target material is Ly-12 aluminum and carbon phenolic, and the target thickness is 2-5 mm. Table 4 and Figure 16 respectively show the measurement and calculation results of the thermal shock wave (the energy deposition is calculated by the Monte Carlo method). In the table, D is the target thickness, \(\tau_{1/2}\) is the half-width of diode voltage, \(\Phi_{0}\) is the injected amount of electron beam energy at the target surface, and \(\sigma_{\rm q}\), \(\sigma_{\rm a}\), and \(\sigma_{\rm s}\) are the values of stress in the PVDF piezoelectric crystal, the stress in the target, and the numerical simulation of stress in the target, respectively. It can be seen that because the target material is placed in front of the PVDF sensor, the stress in the target is slightly greater than that in the sensor. When the electron incident depth is less than 3 mm, the calculated results are small, and when it is greater than 3 mm, the calculated results are large. By comparing the numerical simulation results with the measurement results, the calculation error is less than 10%. The stress decreases with the increase of target depth. Within the above energy injection range, there is a little difference in stress between aluminum and carbon phenolic at the same depth. Figure 16 shows the measured and calculated results of thermal shock waves for aluminum and carbon phenolic materials. Figure 17 shows the thermal shock peak value and measured voltage signal waveform of the No. 21040 shot. At each position, two waveforms of opposite polarity appear successively. The first waveform is the incident wave at the front surface, and the second is caused by the reflection of the first one at the back surface of the crystal.