# Determination of the PIC700 Ceramic’s Complex Piezo-Dielectric and Elastic Matrices from Manageable Aspect Ratio Resonators

^{1}

^{2}

^{*}

## Abstract

**:**

_{33}charge coefficient, is a good starting point in establishing the potential applicability of piezoceramics. However, piezoceramics are only completely characterized by consistent piezoelectric-elastic-dielectric material coefficient matrices in complex form, i.e., including all losses. These matrices, which define the various alternative forms of the constitutive equations of piezoelectricity, are required for reliable virtual prototyping in the design of new devices. To meet this need, ten precise and accurate piezoelectric dielectric and elastic coefficients of the material, including all losses, must be determined for each alternative. Due to the difficulties arising from the coupling of modes when using the resonance method, this complete set of parameters is scarcely reported. Bi

_{0.5}Na

_{0.5}TiO

_{3}-based solid solutions are already commercially available in Europe and Japan. Here, we report a case study of the determination of these sets of material coefficients (d

_{iα}, g

_{iα}, e

_{iα}and h

_{iα}; s

^{E}

^{,D}

_{αβ}and c

^{E}

^{,D}

_{αβ}; ε

^{T}

_{ik}and ε

^{S}

_{ik}; and β

^{T}

_{ik}and β

^{S}

_{ik}), including all losses, of the commercial PIC700 eco-piezoceramic. Plate, disk, and cylinder ceramic resonators of a manageable aspect ratio were used to obtain all the material coefficients. The validation procedure of the matrices is also given by FEA modeling of the considered resonators.

## 1. Introduction

_{0.5}Na

_{0.5}TiO

_{3}-based solid solutions are already commercially available in Europe under the denomination, PIC700 [1]. Moreover, there are commercial ceramics based on the same material in Japan [2]. Achieving good piezoelectric properties, such as the widely reported d

_{33}charge coefficient for sensors, is a good starting point in establishing their potential applicability. However, this knowledge is largely insufficient. Consistent piezoelectric-elastic-dielectric material coefficient matrices in complex form, i.e., including all losses, which define the various alternative forms of the constitutive equations of piezoelectricity, are required for reliable virtual prototyping in the design of new devices, which is a relatively inexpensive and time-saving procedure [3,4].

_{kij}, d

_{kij}, g

_{kij}, and h

_{kij}are the piezoelectric coefficient matrices; s

^{E}

^{,D}

_{ijkl}and c

^{E}

^{,D}

_{ijkl}are the elastic compliances and elastic stiffness tensors, respectively, at a constant electric field (E), closed circuit, or constant electric displacement (D), open circuit; ε

^{T}

_{ik}and ε

^{S}

_{ik}, β

^{T}

_{ik}, and β

^{S}

_{ik}are the free (at constant, zero, stress (T)) and clamped (at constant, zero, strain (S)) relative dielectric permittivity matrices and dielectric impermeabilities or impermittivities (β = 1/ε), respectively. These equations are currently being simplified using Einstein´s summation convention of repeated subscripts (i, k = 1, 2, 3; ij = α; and kl = β, where α, β = 1, 2, …, 6) and the subscript 3 to denote the polarization direction in the material. The mentioned matrices’ inconsistency derives from the use of resonator modes at a relatively wide range of frequencies and whose microstructural characteristics or poling levels may not be identical. For this reason, it is a good practice to provide one or more validity and consistency criteria of the set of parameters determined [8].

_{15}, s

^{E}

_{55}, and ε

^{S}

_{11}) from the electrically-induced shear mode cannot be excluded from this reduced resonator set. Two types of difficulties arise when using the standard thin shear plate, which is length-poled and excited in thickness. They cause the associated parameters to be scarcely reported and affect the consistency of the matrices. On the one hand, longitudinal poling is a problem when dealing with materials of a high coercive field [11] or low dielectric breakdown [12]. On the other hand, the coupling of shear modes and other lateral and contour modes, which takes place in the shear resonance of standard length-poled shear plates, is unavoidable [3,4,13,14]. Unfortunately, this leads to the stringent requirement of aspect ratios between the length for poling (L), the width (w), and the thickness for electrical excitation (t) of the plate of L, w > 20 t [15] and to an underestimation of the piezoelectric coefficients [13].

## 2. Materials and Methods

#### 2.1. Materials

_{0.5}Na

_{0.5}TiO

_{3}-based commercial lead-free (PIC700; PI Ceramic GmbH, Lederhose, Germany) poled ceramic resonators were studied in this work [19]. The ceramics were prepared from high-purity oxide precursors by the conventional ceramic method. Samples were cut and lapped to the desired dimensions. Afterwards, they were electroded and poled in an oil bath, according to the company standards [19], whose details are protected knowledge. All the resonators under study were poled to saturation in the same conditions to ensure the consistency of the results obtained from them. All measurements were taken for at least two samples—although statistical analysis is not an issue discussed in this work—to verify that the results are not affected by microstructure accidents. Changes from batch to batch could take place. The density of the considered samples was 5.49 g·cm

^{−3}.

#### 2.2. Material Coefficient Determination

_{iα}, g

_{iα}, e

_{iα}, and h

_{iα}; s

^{E}

^{,D}

_{αβ}and c

^{E}

^{,D}

_{αβ}; ε

^{T}

_{ik}and ε

^{S}

_{ik}; and β

^{T}

_{ik}and β

^{S}

_{ik}), accounting for all losses, were determined at room temperature from measurements of complex impedance curves. The data were automatically acquired from an HP-4192A LF precision impedance analyzer (Hewlett-Packard, Palo Alto, CA, USA) and controlled by a PC via a GPIB-PCIIA (National Instruments). An automatic iterative method explained elsewhere [3,16,17] was here used for the analysis of the curves for each resonance considered. Contrarily to standard methods, this method does not require additional measurements, besides those of the resonance. Four complex impedance spectra, Z*, of the electromechanical resonance modes of three poled ceramic resonators were analyzed. These modes are: (i) Shear mode of non-standard, thickness-poled, and longitudinally excited shear plates, (ii) Radial and (iii) Thickness extensional modes of thickness-poled thin disks; and (iv) Length extensional mode of a longitudinally poled cylinder. The software used is available to the scientific community [20].

_{s}and f

_{p}, respectively, are automatically determined by the software. For each of the considered modes, some material coefficients were directly determined by the automatic iterative solution of the impedance/admittance expression as a function of these coefficients and the resonator density (ρ) and dimensions (thickness between electrodes for electrical excitation (t); electroded surface area (S)) [3,16,17]. In each iteration of the analysis, the software solves a system of four equations for the values of Z* or Y* measured at four frequencies, including f

_{s}and f

_{p}, by a numerical method, until a convergence criterion of the last determined coefficients in comparison with the previous ones is met.

_{x}, where x = 33, 31, 15, p (radial mode of the disc) and t (thickness mode of the disc)) and the corresponding frequency numbers (${N}_{x}={f}_{s}^{x}\left(\mathrm{kHz}\right)\cdot D\left(mm\right)$, where ${f}_{s}^{x}$ is the series resonance of the x mode, and D is the leading dimension of the resonance), as well as the planar Poisson’s ratio.

^{2}). This parameter accounts for the validity of the analytical expression and these coefficients for 1D modeling of this mode of resonance. The higher the coupling of the resonance with other undesired resonances, the lower the ℜ

^{2}. The closer the experimental resonance to a monomodal resonance, the closer ℜ

^{2}is to 1.

#### 2.3. Finite Element Modelling

^{®}4.3) was used. The piezoelectric device module (PZD) was used to simulate the piezoelectric response of the four resonance modes of the three resonators considered here. The mesh used typically has five nodes per wavelength, which allowed for simulating the modes of resonance that are excited, together with the fundamental resonance under study. The number of frequencies analyzed was chosen to obtain the needed resolution of each calculated spectra. The shear plate was simulated as a full 3D item. This typically results in a calculation time of up to 5 h for each sweep of 1000 frequencies. The cylinder and disk resonators were simulated using their rotational symmetry, which results in faster calculations. The 3D harmonic analysis used here provides the complex impedance values in a given interval of frequencies. The intervals of the frequency of interest in the present work were those of the experimental measurements.

## 3. Results and Discussion

#### 3.1. Measurements and Calculation of the Piezoelectric, Elastic, and Dielectric Material Coefficients

#### 3.1.1. Shear Resonance of a, Non-Standard, Thickness-Poled and Longitudinally Excited Shear Plate

_{s}= 797.7 kHz and is marked as 3, four other modes are measured. The resonances marked as 1 and 2 correspond to modes taking place at lower frequencies, which are associated with the larger lateral dimensions of the plate. Their overtones appear at periodically distributed frequencies. Those marked with 4 and 5 are coupled with the fundamental shear resonance, which is marked 3. This invalidates the sample for the accurate determination of complex material coefficients.

^{2}). Figure 1a also shows the resonance spectra, measured again after the thickness of the plate was reduced in steps to a plate with t = 1.66 mm (L/t = 4.52), which has a virtually uncoupled fundamental shear mode taking place at f

_{s}= 892.0 kHz.

^{2}, from the initial 0.8420 to the intermediate value of 0.9003 and up to 0.9780 for the sample of t = 1.66 mm. Figure 1b shows also the data for a plate with L/t = 3.89, after a further thickness reduction to t = 1.61 mm, which again shows a coupled shear mode due to interference, with modes 4 and 5 marked in the spectra of the t = 1.66 mm sample in Figure 1a. For this sample, ℜ

^{2}is reduced to 0.9560. This indicates that the maximum decoupling of modes was surpassed, and the resonators evolve away from a monomodal resonance once again.

^{S}

_{11}, e

_{15}, and s

^{E}

_{55}, and, from these, ε

^{T}

_{11}, d

_{15}, h

_{15}, g

_{15}, s

^{D}

_{55}, c

^{E}

_{55}, and c

^{D}

_{55}are also calculated. Due to the symmetry d

_{24}= d

_{15}, e

_{24}= e

_{15}, h

_{24}= h

_{15}, g

_{24}= g

_{15}, ε

^{T}

^{,S}

_{22}= ε

^{T}

^{,S}

_{11}, s

^{D}

^{,E}

_{44}= s

^{D}

^{,E}

_{55}, and c

^{D}

^{,E}

_{44}= c

^{D}

^{,E}

_{55}are also determined. Additionally, the electromechanical coupling factor and frequency number, k

_{15}and N

_{15}, are obtained from this resonance mode. These are shown in Table 1, Table 2 and Table 3.

#### 3.1.2. Resonances of a Thickness-Poled and Excited Thin Disk

- (a)
- Extensional radial resonance of the disk

_{2s}) is used to calculate the first estimate of the planar Poissón´s ratio (σ

^{P}) from the ratio, f

_{2s}/f

_{s}[17].

^{2}= 0.9999.

^{T}

_{33}, d

_{31}, and s

^{E}

_{11}, which are the coefficients that can otherwise be obtained by the LTE of a thickness-poled bar. The coefficient s

^{E}

_{12}is also determined in the radial mode from the Poisson´s ratio and given the expression σ

^{P}= (−s

_{12}

^{E}⁄s

_{11}

^{E}). Besides, g

_{31}, s

^{D}

_{11}, and s

^{D}

_{12}are also calculated from the previous values, along with s

^{D}

^{,E}

_{66}= 2(s

^{D}

^{,E}

_{11}− s

^{D}

^{,E}

_{12}) and c

^{D}

^{,E}

_{66}= 1/s

^{D}

^{,E}

_{66}. Due to the symmetry d

_{32}= d

_{31,}g

_{32}= g

_{31,}s

_{21}= s

_{12}, and s

_{22}= s

_{11}are determined. Additionally, the electromechanical coupling factors and frequency numbers, k

_{p}, N

_{p}, and k

_{31}, were obtained. These are shown in Table 1, Table 2 and Table 3.

- (b)
- Extensional thickness resonance of the disk

^{2}= 0.8720. Undoubtedly, research must still be conducted with the aim of enhancing this experimental result, which compromises the overall determination of matrices. However, this type of spectra is what is commonly accepted as a manageable aspect ratio and impedance curve for analysis when working with the thickness modes of disks [9,23,24,25].

^{S}

_{33}, h

_{33}, and c

^{D}

_{33}. They can also be obtained from the LE mode of a thickness-poled thin plate. From these coefficients, e

_{33}and c

^{E}

_{33}are also calculated. At this stage of the process, the ε

^{S}

_{ij}matrix is complete, which allows for the calculation of the β

^{S}

_{ij}matrix. Additionally, the electromechanical coupling factor and frequency number, k

_{t}and N

_{t}, are obtained. These are shown in Table 1, Table 2 and Table 3.

#### 3.1.3. Longitudinal Resonance of a Longitudinally Poled and Excited Cylinder

^{2}= 0.9998). The directly calculated coefficients in this mode are ε

^{T}

_{33}, g

_{33}, and s

^{D}

_{33}. Due to the frequency dependence of the permittivity, the ε

^{T}

_{33}determined in this mode is higher than the one determined in the thickness mode taking place at a higher frequency (Figure 3). Consequently, at this point, there is a need for an expert selection of the value that is more amenable for the validity of the matrices for FEA modelling [10].

_{iα}and ε

^{T}

_{ij}matrices are complete, which allows the β

^{T}

_{ij}matrix to be calculated. From these coefficients, d

_{33,}which completes the d

_{iα}matrix as well, and s

^{E}

_{33}are also calculated. Due to the frequency dependence of the material coefficients, this d

_{33}value at resonance is lower than the quasi-static one measured at 100 Hz. Additionally, the electromechanical coupling factor and frequency number, k

_{t}and N

_{t}, are obtained. These are shown in Table 1, Table 2 and Table 3.

#### 3.1.4. Combined Determination of the Remaining Material Coefficients

- (a)
- from the inversion of the matrices, we have the following expression:

^{E}

_{13}. From s

^{E}

_{13}, the symmetry considerations provide s

^{E}

_{23}= s

^{E}

_{13}, s

^{E}

_{31}= s

^{E}

_{13}, and s

^{E}

_{32}= s

^{E}

_{13}. This completes the s

^{E}

_{αβ}matrix and allows the calculation of its inverse, c

^{E}

_{αβ}, which at this point was only lacking the values of c

^{E}

_{11,}c

^{E}

_{12}, and c

^{E}

_{13}, to be completed. This also ensures consistency between the stiffness and compliance matrices.

- (b)
- knowing c
^{E}_{11,}c^{E}_{12}, and c^{E}_{13}, we can make use of Equation (62) in [8] to obtain e_{31}from the following:

_{iα}matrix.

- (c)
- by the relationships between the coefficients, we can calculate h
_{31}using the following expression:

_{iα}matrix.

- (d)

^{D}

_{23}= s

^{D}

_{13}, s

^{D}

_{31}= s

^{D}

_{13}, and s

^{D}

_{32}= s

^{D}

_{13}. This completes the s

^{D}

_{αβ}matrix and allows the calculation of its inverse, c

^{D}

_{αβ}, which at this point was lacking the values of c

^{D}

_{11}, c

^{D}

_{12}, and c

^{D}

_{13}, to be completed. This completes all the matrix determinations, as shown in Table 1, Table 2 and Table 3, and determines the consistency between the c

^{D}

_{αβ}and s

^{D}

_{αβ}matrices.

^{2}values. Additionally, we have used a, non-standard, thickness-poled and longitudinally excited shear plate producing accurate complex shear coefficients, as the shear mode can be efficiently decoupled from other modes. The procedure reported here for the calculation of the coefficients by combining parameters that were obtained from previously analyzed spectra is not a universal solution, as there are options for calculating the indirectly determined coefficients [8,10].

#### 3.2. Validation of the Piezoelectric, Elastic, and Dielectric Material Coefficients

#### 3.2.1. Meaningful Losses

#### 3.2.2. Finite Element Analysis

## 4. Conclusions

^{2}), was accomplished. The selection of spectra with a high ℜ

^{2}allows for the use of resonators of a manageable aspect ratio. The matrices of the coefficients are consistent and show the validity of PIC700 material modeling using an FEA in a frequency interval from 100 kHz to a few MHz.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Shear resonance mode of a, non-standard, thickness-poled and longitudinally excited shear plate: (

**a**) Impedance spectra in (Z, θ) plots, and (

**b**) the same spectra in the equivalent (R, G) plots. In the (Z, θ) plots, the lines are only a “guide for the eye” in between the experimental points, and the frequency interval covers the lateral modes (marked as 1, 2, 4, and 5) at both sides of the fundamental shear mode (marked as 3). In the (R, G) plots, the symbols are the experimental data, and the lines are the reconstructed peaks after the calculation of the coefficients for the fundamental shear mode of the plates. The values of the regression factors for these calculations (ℜ

^{2}) are shown. Figure (

**a**) shows two spectra: the one corresponding to the initial plate with t = 2 mm (red circles; full symbols for Z values and open symbols for θ values) and the one for the reduced plate with t = 1.66 mm (black squares; full symbols for Z values and open symbols for θ values). The other two dimensions, L = 7.49 mm and w = 7.45 mm, remain unaltered in the process. Figure (

**b**) shows the two equivalent spectra for the same plates as in Figure (

**a**), (red circles for t = 2 mm and black squares for t = 1.66 mm). In Figure (

**b**) the full symbols are used for the G values, and open symbols are used for the R values. Additionally, in Figure (

**b**) the spectra of an intermediate plate with t = 1.93 mm and L/t = 3.89 (blue triangles) and of a plate after a further thickness reduction to t = 1.61 mm and L/t = 4.66 (blue rhombus) are shown to complete the description of the resonance decoupling process.

**Figure 2.**Fundamental radial extensional resonance mode of a thickness-poled and excited thin disk. Impedance spectra in an (R, G) plot of a disk with a diameter of 12 mm and t = 1.15 mm. The symbols are the experimental data, and the lines are the reconstructed peaks after the calculation of the coefficients. The value of the regression factor for this calculation (ℜ

^{2}) is shown. The black symbols and lines (G data) can be read in the left-y-axis, and the blue ones (R data) are in the right-y-axis.

**Figure 3.**Fundamental thickness extensional resonance mode of a thin disk. The impedance spectra in an (R, G) plot of a disk with a 12 mm diameter and 1.00 mm thickness. The symbols are the experimental data, and the lines are the reconstructed peaks after the calculation of the coefficients. The value of the regression factor for this calculation (ℜ

^{2}) is shown. The black symbols and lines (G data) can be read in the left-y-axis, and the blue ones (R data) are in the right-y-axis.

**Figure 4.**Fundamental length extensional resonance of a thickness-poled and excited cylinder. The impedance spectra in an (R, G) plot of a cylinder with a 6 mm diameter and 15 mm length. The symbols are the experimental data, and the lines are the reconstructed peaks after the calculation of the coefficients. The value of the regression factor for this calculation (ℜ

^{2}) is shown. The black symbols and lines (G data) can be read in the left-y-axis, and the blue ones (R data) are in the right-y-axis.

**Figure 5.**3D FEA modeled shear resonance of a, non-standard, thickness-poled and longitudinally excited shear plate (L): (

**a**) impedance spectra in (Z, θ) plots and (

**b**) spectra in the equivalent (R, G) plots. In the (Z, θ) plots, the lines are only a “guide for the eye” in between the modeled points. In Figure (

**a**), the spectra for plates with L = 7.49 mm, w= 7.45 mm, and with an initial t = 2 mm (red circles) and reduced t = 1.66 mm (black squares) are shown. Full symbols are used for the Z values and open symbols for the θ values. The corresponding experimental (Z, θ) plots are given in Figure 1a. In the equivalent (R, G) plots in Figure (

**b**), the 3D modeled spectrum of the plate with a reduced t = 1.66 mm is shown as a black thick line for the G peak and blue thick line for the R peak. For comparison, the experimental spectrum and the 1D modeled one (the spectra reconstructed using the iterative method) are also shown in Figure (

**b**). The symbols are the experimental data of G (closed black circles) and R (open blue circles), and the thin lines are the reconstructed peaks (1D model) (black line for G and blue line for R). In Figure (

**a**,

**b**), the frequency interval covers the lateral modes (marked 1, 2, 4 and 5) on both sides of the fundamental shear mode (marked as 3).

**Figure 6.**Fundamental radial extensional resonance mode of a thickness-poled and excited thin disk. The FEA-modeled (thick lines) impedance spectrum in an (R, G) plot of a disk with a diameter of 12 mm and a thickness of 1.15 mm. The experimental data (symbols) and the reconstructed spectrum after the calculation of the parameters in a 1D model (thin lines) are also shown for comparison. Black symbols and lines (G data) can be read in the left-y-axis, and the blue ones (R data) are in the right-y-axis.

**Figure 7.**Fundamental thickness extensional resonance of a thickness-poled and excited thin disk. 3D FEA-modeled impedance spectrum: (

**a**) (Z, θ) plot; and (

**b**) equivalent (R, G) plot of a disk with a diameter of 12 mm and a thickness of 1.00 mm, shown as thick lines. The two types of plot also show the experimental spectrum (symbols). Figure (

**b**), (R, G) plots showing the reconstructed spectrum after the calculation of the parameters (1D model) as thin lines. Black symbols and lines (Z and G data) can be read in the left-y-axis, and blue ones (θ and R data) are in the right-y-axis.

**Figure 8.**Fundamental length extensional resonance of a thickness-poled and excited cylinder. 3D FEA-modeled impedance spectrum in an (R, G) plot of a cylinder with a diameter of 6 mm and a length of 15 mm (shown as thick lines). The experimental and reconstructed spectra after the calculation of the parameters (1D model) are also shown as symbols and thin lines, respectively, for comparison. Black symbols and lines (G data) can be read in the left-y-axis, and blue ones (R data) are in the right-y-axis.

s^{E}^{,D}_{αβ}= s′ + is″/10^{−12}·m^{2}·N^{−1} | s^{E}_{11} | s^{E}_{12} | s^{E}_{13} | s^{E}_{33} | s^{E}_{55} | s^{E}_{66} | s^{D}_{11} | s^{D}_{12} | s^{D}_{13} | s^{D}_{33} | s^{D}_{55} | s^{D}_{66} |

s′ | 8.5998 | −2.0200 | 1.673 | 8.9803 | 20.7702 | 21.24 | 8.5219 | −2.0979 | 2.004 | 7.4639 | 18.6114 | 21.24 |

s″ | −0.0663i | +0.0154i | −0.012i | −0.114i | −0.2550i | −0.163i | −0.0577i | +0.024i | −0.03i | −0.0486i | −0.2322i | −0.163i |

c^{E}^{,D}_{αβ}= c′ + ic″/10^{10} N·m^{−2} | c^{E}_{11} | c^{E}_{12} | c^{E}_{13} | c^{E}_{33} | c^{E}_{55} | c^{E}_{66} | c^{D}_{11} | c^{D}_{12} | c^{D}_{13} | c^{D}_{33} | c^{D}_{55} | c^{D}_{66} |

c′ | 13.102 | 3.686 | −3.127 | 12.299 | 4.8139 | 4.708 | 14.056 | 4.641 | −5.019 | 16.092 | 5.3722 | 4.708 |

c″ | +0.107i | +0.034i | −0.044i | −0.164i | +0.0591i | +0036i | +0.043i | −0.029i | +0.049i | +0.033i | +0.067i | +0036i |

^{(}*

^{)}Directly calculated parameters are quoted in bold.

d_{iα} = d′ + id″/10^{−12}C·N^{−1} | d_{31} | d_{33} | d_{15} | e_{iα} = e′ + ie″/C·m^{−2} | e_{31} | e_{33} | e_{15} |

d′ | −21.1408 | 89.685 (**) | 102.9606 | e′ | −6.357 | 11.0862 | 4.9590 |

d″ | +1.4046i | −3.162i | −4.4218i | e″ | +0.265i | +0.5252i | −0.1520i |

h_{iα} = h′ + ih″/10^{8} V·m^{−2} | h_{31} | h_{33} | h_{15} | g_{iα} = g′ + ig″/mV·N^{−1} | g_{31} | g_{33} | g_{15} |

h′ | −15.173 | 26.4709 | 11.2442 | g′ | −3.6952 | 15.632 | 20.9383 |

h″ | −0.408i | +0.5634i | +0.5044i | g″ | +0.1393i | −0.0180i | +0.6777i |

^{(}*

^{)}Directly calculated parameters are quoted in bold. (**) The quasi-static value is 120 × 10

^{−12}C.N

^{−1}at a Berlincourt d(sub 33)-meter (100 Hz).

**Table 3.**Complex dielectric coefficients, Poisson´s ratios, electromechanical coupling factors, and frequency numbers of the considered resonances of the PIC700 eco-piezoceramic

^{(}*

^{)}.

ε^{S,T}_{ik} = ε′ + iε″/ε_{0} | ε^{S}_{11} | ε^{S}_{33} | ε^{T}_{11} | ε^{T}_{33} | β^{S,T}_{ik} = β′+iβ″/10^{−4}/ε_{0} | β^{S}_{11} | β^{S}_{33} | β^{T}_{11} | β^{T}_{33} | Poisson´s Ratio (σ^{P}) |

ε′ | 554.02 | 472.31 | 496.43 | 648.00 | β′ | 17.948 | 21.073 | 20.029 | 15.414 | 0.235 |

ε″ | −41.78i | −32.46i | −37.54i | −22.1i | β″ | +1.353i | +1.448i | +1.515i | +0.526i | +0.00002i |

kx = k′ + ik″ | k_{31} | k_{33} | k_{15} | k_{p} | k_{t} | N_{x}/kHz·mm | N_{33} | N_{15} | N_{p} | N_{t} |

k′ | 0.07632 | 0.41102 | 0.32239 | 0.14737 | 0.43482 | 2231.47 | 1480.72 | 3021.04 | 2381.00 | |

k″ | −0.00405i | −0.00504i | −0.00028i | −0.00783i | −0.00776i |

^{(}*

^{)}Directly calculated parameters are quoted in bold.

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## Share and Cite

**MDPI and ACS Style**

Pardo, L.; García, Á.; Schubert, F.; Kynast, A.; Scholehwar, T.; Jacas, A.; Bartolomé, J.F.
Determination of the PIC700 Ceramic’s Complex Piezo-Dielectric and Elastic Matrices from Manageable Aspect Ratio Resonators. *Materials* **2021**, *14*, 4076.
https://doi.org/10.3390/ma14154076

**AMA Style**

Pardo L, García Á, Schubert F, Kynast A, Scholehwar T, Jacas A, Bartolomé JF.
Determination of the PIC700 Ceramic’s Complex Piezo-Dielectric and Elastic Matrices from Manageable Aspect Ratio Resonators. *Materials*. 2021; 14(15):4076.
https://doi.org/10.3390/ma14154076

**Chicago/Turabian Style**

Pardo, Lorena, Álvaro García, Franz Schubert, Antje Kynast, Timo Scholehwar, Alfredo Jacas, and José F. Bartolomé.
2021. "Determination of the PIC700 Ceramic’s Complex Piezo-Dielectric and Elastic Matrices from Manageable Aspect Ratio Resonators" *Materials* 14, no. 15: 4076.
https://doi.org/10.3390/ma14154076