## 2 Theory Based on Resonance Method
### Radial Extension (RAD)  
In this mode, we have a thin round plate in which the polarization and vibrations are both oriented along the depth. Table 1 presents the geometric shape and material constants. The aspect ratios of the component must satisfy \(\varnothing>20t\), and we use five specimens with dimensions \(\varnothing=19.9\pm 0.05\) mm and \(t=0.81\) mm. In 1973, [PERSON], [PERSON], and [PERSON] [21] derived the constants of radial vibrations in round plates. The formula for \(k_{p}\), the electromechanical coupling coefficient of the radial vibrations in a piezoelectric round plate is  
\[\frac{k_{p}^{2}}{1-k_{p}^{2}}=\frac{\Big{(}1-\sigma^{E}\Big{)}J_{1}[\eta_{1}(1 +Hf/f_{r})]-\eta_{1}(1+\Delta f/f_{r})J_{0}[\eta_{1}(1+\Delta f/f_{r})]}{(1- \sigma^{E})J_{1}[\eta_{1}(1+\Delta f/f_{r})]} \tag{25}\]  
where \(k_{p}\) denotes the electromechanical coupling coefficient of the radial mode shape; \(\Delta f=f_{a}-f_{r}\) is the difference between the anti-resonant frequency and the resonant frequency; \(J_{0}\) is Bessel Function of First Kind and Zero Order; \(J_{1}\) is Bessel Function of First Kind and First Order; \(\sigma^{E}\) is Poisson's Ratio; and \(\eta_{1}\) is the first positive root from the function \(\Big{(}1-\sigma^{E}\Big{)}J_{1}(\eta)=\eta J_{0}(\eta)\). However, in 2005, [PERSON], [PERSON], and [PERSON] [22] modified the electromechanical coupling relation above using approximation into  
\[k_{p}^{2}=\frac{1}{P}\frac{f_{a}^{2}-f_{r}^{2}}{f_{a}^{2}} \tag{26}\]  
where  
\[P=\frac{2\Big{(}1+\sigma^{E}\Big{)}}{\Big{\{}\eta_{1}^{2}-\Big{[}1-\big{(} \sigma^{E}\big{)}^{2}\Big{]}\Big{\}}} \tag{27}\]  
Based on RM by IEEE standard [14; 15], several formula were simplified also by [PERSON] et al. [22] and we used them to calculate some parameters, i.e., \(\varepsilon_{33}^{T}\)\(\varepsilon_{33}^{S}\), \(\varepsilon_{11}^{T}\). RM uses the vibration characteristics corresponding to the five types of piezoelectric components above to derive the elastic, piezoelectric, and dielectric constants. Furthermore, Poisson's ratio is defined as the ratio of lateral strain to longitudinal strain:  
\[\sigma^{E}=-\frac{S_{12}^{E}}{S_{11}^{E}} \tag{28}\]  
where the superscript \(E\) indicates values measured in a fixed electric field. The negative indicates that the two vibrations are opposite in direction, which means that when the longitudinal vibration is extending, the lateral vibration is contracting, and when longitudinal vibration is contracting, the lateral vibration is extending. Obtaining the [PERSON]'s ratio generally requires the measurement of the fundamental frequency and one overtone frequency of the piezoelectric round plate. The formula used is \(\Big{(}1-\sigma^{E}\Big{)}J_{1}(\eta)=\eta J_{0}(\eta)\), the source and usage of which are explained in Ref [23; 24]. Then, \(\eta_{1}\) and are calculated, and we can look up the fundamental frequency \(f_{r}=f_{1}^{r}\) and the overtone frequency \(f_{2}^{r}\) in the table provided in the Appendix to determine the [PERSON]'s ratio \(\sigma^{E}\), which is substituted into Equation (25) to derive \(k_{p}\), the electromechanical coupling coefficient of the radial mode shape. Although RM can be used to obtain the material constants of piezoelectric materials, there are still some constants that cannot be measured via experiment, and some conversions between constants are required to derive the constants that cannot be directly measured before all of the elastic, piezoelectric, and dielectric constants are obtained. Below is the solved equation including the piezoelectric strain constant:  
\[d_{ij}^{2}=k_{ij}^{2}\varepsilon_{ii}^{T}s_{ij}^{E} \tag{29}\]  
where \(ij=31,33,15\). Based on the equation above, it \(k_{ij}\), \(\varepsilon_{ij}^{T}\), and \(s_{ij}^{E}\) (i.e., the electromechanical coupling coefficient, the dielectric constant in constant stress, and the elastic compliance constant in a constant electric field) are known, then \(d_{31}\), \(d_{33}\), and \(d_{15}\), (i.e., the piezoelectric strain constants) can be obtained.