## 3 Results  
Potential candidate or alternative state-of-the-art piezoelectric materials for the material selection analysis were selected from each piezoelectric material category discussed in the material section on the basis of their performance for actuator applications, i.e., based on their \(d_{33}\) values. The selected alternative state-of-the-art piezoelectric materials and their properties are presented in Tables 1 and 2. Various physical/electrical properties affect the performance of a piezoelectric actuator. However, the piezoelectric strain coefficient \(d\) is the most important piezoelectric property in determining the effectiveness of piezoelectric materials in actuator applications [6]. A high value of \(d\) implies a high strain for a given applied voltage. In particular, in this work, the piezoelectric strain coefficient in the longitudinal mode \(d_{33}\) is treated as the figure of merit (FOM) for piezoelectric materials for actuator application. In addition to the piezoelectric strain coefficient \(d\), other properties considered in this article for the TOPSIS approach were: coupling coefficient \(K\), elastic compliance \(S^{E}\) at constant electric field, and dielectric constant \(\varepsilon^{T}\) at constant stress. From these properties, the desirable properties of piezoelectric materials for actuator applications are high values of \(K\), \(\varepsilon\), \(e\), and a low value of \(S\)[14]. The TOPSIS indices of the alternative materials are determined by considering the relative weights of \(d_{33}\), \(k_{33}\), \(S^{E}_{33}\), and \(T_{33}\). For piezoelectric actuators, the relative importance \begin{table}
\begin{tabular}{c c} \hline \hline
**Material** & **Category** \\ \hline PZT-5H & Soft ceramics \\ PZT-5J & Soft ceramics \\ PZT-5A & Soft ceramics \\ PZT-4 & Hard ceramics \\ PZT-8 & Hard ceramics \\ PVDF & Polymer \\ (PVDF-TrFE)75/25 & Polymer \\ PZT film (4 \(\mu\)m on Si) & Thin film \\ ZnO film & Thin film \\ AlN film & Thin film \\ PMN-28\% PT & Single crystal binary composite \\ PIN24\%-PMN-PT & Single crystal ternary composite \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Potential piezoelectric materials for the material selection analysis and their material categories. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Materials** & \(k_{33}\) & \(S^{E}_{33}\left(p\frac{\pi^{2}}{N}\right)\) & \(\varepsilon^{T}_{33}\left(\frac{\mu E}{m}\right)\) & \(d_{33}\left(p\frac{\xi}{N}\right)\) & **Reference** \\ \hline PZT-5H & 0.75 & 21.0 & 33.63 & 650 [19] & [19] \\ PZT-5J & 0.74 & 18.8 & 18.59 & 500 [19] & [19] \\ PZT-5A & 0.72 & 18.6 & 15.93 & 390 [19] & [19] \\ PZT-4 & 0.71 & 17.0 & 14.60 & 372 & [38] \\ PZT-8 & 0.64 & 13.5 & 9.8 & 275 & [38] \\ PVDF & 0.18 & 472 & \(6.73\times 10^{-2}\) & 32.5 & [21; 39] \\ (PVDF-TrFE)75/25 & 0.23 & 300 & \(7.0\times 10^{-2}\) & 33.5 & [21; 39] \\ PZT film (4 \(\mu\)m on Si) & 0.69 & 14.7 & 12.39 & 246 & [40] \\ ZnO film & 0.33 & 4.74 & \(7.2\times 10^{-2}\) & 12 & [30; 40] \\ AlN film & 0.24 & 2.53 & \(8.85\times 10^{-2}\) & 7 & [30; 41] \\ PMN-28\% PT & 0.9 & 36.15 & 48.68 & 1190 & [39] \\ PIN24\%-PMN-PT & 0.89 & 49.04 & 42.06 & 1285 & [39] \\ \hline \end{tabular}
\end{table} 
Table 2: Potential piezoelectric materials and their properties selected for the analysis. of the piezoelectric strain coefficient is higher than that of the coupling coefficient and, in turn, higher than that of the dielectric constant [7]. However, the relative importance of the dielectric constant and compliance are assumed to be equal [7]. Thus, based on these assertations, relative weights of 30%, 25%, 22.5%, and 22.5% were assumed for \(d_{33}\), \(k_{33}\), \(S^{E}{}_{33}\), and \(T_{33}\), respectively, in this work. The ranking of the above state-of-the art piezoelectric materials according to the FOM and TOPSIS approaches is given in Table 3.  
Therefore, the best materials for the piezoelectric plate actuator, according to the results of the FOM and TOPSIS methods, are PIN24%-PMN-PT and PMN-28% PT, respectively.