## 4 Discussion
### Identification of Circuit System Pattern  
Figure 8 shows that the excitation and response of the circuit system (\(H(t)\)) is a dynamic attenuation second-order system composed of LCR elements. To determine whether the system is composed of series or parallel LCR elements, a circuit system model must be established and theoretical analysis must be conducted, as shown in Figure 9. Figure 9 includes second-order circuit systems (\(H(t)\)) composed of LCR elements in series or parallel. When the circuit is in its initial state (\(t=0_{-}\)), the load \(P\) has \(U_{\sigma}(0)=E_{\sigma}\) at \(t=0\). The following mathematical model was established to solve for the impedance (\(Z_{\sigma}\)), current (\(I_{\sigma}(t)\)), and voltage (\(U_{C\sigma}(t)\)). We assumed that \"\(t=0_{-}\)\" denotes an instant just before \(t=0\). The component of impedance and its mathematical model of output voltage response when the specimen is not under external compressive loading, it is assumed that the initial impedance is \(Z_{0}\); the initial impedance (\(Z_{0}\)) is composed of the microelement total resistance (\(R_{0}\)) and the capacitive reactance (\(Z_{C\sigma}\)). However, when the specimen is subjected to external loading, its microstructural configuration changes with stress, and the impedance value of the specimen is no longer \(Z_{0}\), but a dynamic quantity composed of the microelement total resistance (\(R_{0}\)), the capacitive reactance (\(Z_{C\sigma}\)), and the inductive reactance (\(Z_{L\sigma}\)), etc. The microelement total resistance (\(R_{0}\)), the capacitive reactance (\(Z_{C\sigma}\)), and the inductive reactance (\(Z_{L\sigma}\)) are the components of the total impedance (\(Z_{\sigma}\)). The theoretical expression for the impedance (\(Z_{C\sigma}\)) in the frequency domain is:  
\[Z_{C\sigma}=1/\left(j\omega C_{\sigma}\right) \tag{5}\]  
In Equation (5), \(j\) represents the reactance, which is in the form of a complex number; \(\omega\) represents the angular frequency of the measurement instrument's signal source; and \(C_{\sigma}\) is the total capacitance of the specimen. When this component is subjected to the excitation shown in Figure 9, its voltage response model is as follows. \[U_{C\sigma}(t)=\frac{1}{C}\int_{0}^{t}I_{C\sigma}(t)dt \tag{6}\]  
Figure 9: A second-order circuit system composed of LCR components in series or parallel: (**a**) series connection of LCR components, (**b**) parallel connection of LCR components. The theoretical calculation expression of inductive impedance (\(Z_{L\sigma}\)) in the frequency domain is:  
\[Z_{L\sigma}=(j\omega L_{\sigma}) \tag{7}\]  
In Equation (7), \(L_{\sigma}\) is the dynamic inductance, which is expressed as  
\[L_{\sigma}=\mu^{\prime}\mu_{0}SN^{2}/l \tag{8}\]  
In Equation (8), \(\mu^{\prime}\) represents the magnetic permeability, \(\mu_{0}\) represents the permeability of the vacuum, \(S\) represents the cross-sectional area of the specimen, \(l\) represents the length of the specimen, and \(N\) represents the number of turns in the coil. Since the specimen does not have a coil structure, the inductance (\(L_{\sigma}\)) is zero when the load is zero; while when the load is applied, the inductance (\(L_{\sigma}\)) is solved jointly by the total impedance (\(Z_{\sigma}\)) and the capacitive impedance (\(Z_{C\sigma}\)). In the system shown in Figure 9, the voltage response model of the inductive reactance (\(Z_{L\sigma}\)) under the excitation of \(U_{\sigma}(t)\) is as follows:  
\[U_{L\sigma}(t)=LdI_{L\sigma}(t)/dt \tag{9}\]  
The last physical quantity that affects the total impedance (\(Z_{\sigma}\)) is the microelement total resistance (\(R_{\sigma}\)), and the formula for \(R_{\sigma}\) is:  
\[R_{\sigma}=\rho^{\prime}l/S \tag{10}\]  
In Equation (10), \(\rho^{\prime}\) represents the electrical conductivity, \(l\) represents the length of the specimen, and \(S\) represents the cross-sectional area of the specimen.