# Wave Electromechanical Coupling Factor for the Guided Waves in Piezoelectric Composites

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## Abstract

**:**

## 1. Introduction

## 2. Definitions of WEMCF

## 3. Wave and Finite Element Method (WFEM)

#### 3.1. Basic Procedure

#### 3.2. Reduced Model for the Piezoelectric Composites

## 4. Energy Formulas of WEMCF

#### 4.1. Two Energy Formulas

#### 4.2. Demonstrations

#### 4.2.1. Relationship between the Wavenumber and Wave Shape

**K**OC = HOC

**K**OC

**M**OC = HOC

**M**OC leading to

#### 4.2.2. Demonstration of Energy Formula ${\mathcal{K}}_{1\mathrm{b}}^{2}$

#### 4.2.3. Demonstration of Energy Formula ${\mathcal{K}}_{1\mathrm{f}}^{2}$

#### 4.3. Implementations

## 5. Validations

#### 5.1. Validation of the Implementations

#### 5.2. Validation against a Thin-Wall Piezoelectric Structure

## 6. Application: Designing the Resistive PZT Waveguide

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

EMCF | electromechanical coupling factor |

WEMCF | wave electromechanical coupling factor |

MEMCF | modal electromechanical coupling factor |

WFEM | wave and finite element method |

FRF | frequency response function |

TL | transmission loss |

OC | open circuit |

SC | short circuit |

DOF | Degree-of-freedom |

## Appendix A. Material Properties of the Piezoelectric Material (PZT4)

## References

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**Figure 4.**Unit cells of the piezoelectric waveguides: (

**a**) unit cell A; (

**b**) unit cell B which is a non-symmetric way of choosing the unit cell for the infinite periodic structure with unit cell A; (

**c**) unit cell C which has longer PZT patches. The Piezoelectric materials are polarized along the z axis, and the electrodes fully cover the surfaces parallel to the x-y plane (orthogonal to the polarized direction). The program code used to generate the Finite Element models in ANSYS can be found in the Supplementary Files.

**Figure 5.**Comparison of WEMCF calculated by using: full WFEM with ${\mathcal{K}}_{\mathrm{W}}$ (labelled by ‘Reference’), full WFEM with ${\mathcal{K}}_{1\mathrm{b}}$ (labelled by ‘using ${C}_{\mathrm{block}}$’), full WFEM with ${\mathcal{K}}_{1\mathrm{f}}$ (labelled by ‘using ${C}_{\mathrm{free}}$’) and reduced waveshape with ${\mathcal{K}}_{1\mathrm{f}}$ (labelled by ‘using ${C}_{\mathrm{free}}$ (reduced)’).

**Figure 6.**Dispersion curves and WEMCF for the piezoelectric waveguide with unit cell A: only wave 0 (z transverse) and 3 (longitudinal) have significant values.

**Figure 7.**Waveshape of wave 0 of unit cell A at the border frequencies of the band gaps: (

**a**) $11.634$ $\mathrm{k}$$\mathrm{Hz}$, low WEMCF; (

**b**) $12.556$ $\mathrm{k}$$\mathrm{Hz}$, high WEMCF; (

**c**) $46.475$ $\mathrm{k}$ $\mathrm{Hz}$, low WEMCF; (

**d**) $50.077$ $\mathrm{k}$$\mathrm{Hz}$, high WEMCF.

**Figure 8.**CPU time for the full dispersion characteristics with WFEM of unit cell A by the energy formula ${\mathcal{K}}_{1\mathrm{f}}^{2}$ where the energies are calculated by: (1) using full shapes obtained from full WFEM; (2) using full shapes obtained from reduced WFEM; (3) using reduced shapes. The label ’Wave selection and matching’ refers to the standard post-processing of the WFEM results in order to identify the same type of wave among different frequencies [36], not the matching between the results between the OC and SC status as required for computing the frequency formula of WEMCF.

**Figure 10.**Waveshape of wave 0 of unit cell C: (

**a**) at the first propagating zone; (

**b**) at the second propagating zone.

**Figure 11.**The unit cell of the thin-wall piezoelectric structure (

**a**) and its dispersion curves (

**b**). The program code used to generate the Finite Element model in ANSYS can be found in the Supplementary Files.

**Figure 12.**The comparison of WEMCF calculated by the frequency formula ${\mathcal{K}}_{\mathrm{W}}^{2}$ and the energy formula ${\mathcal{K}}_{1\mathrm{f}}^{2}$. Waves 0, 4 and 5 are selected from the full dispersion curves in Figure 11b. The ${\mathcal{K}}_{1\mathrm{f}}^{2}$ is calculated by the reduced wave shapes as recommended in the last section.

**Figure 15.**Parametric study of power transmission loss with respect to frequency and resistance, when the piezoelectric waveguides use: (

**a**) unit cell A; (

**b**) unit cell C.

**Figure 16.**The best TL and WEMCF with respect to the frequency for PZT waveguide with unit cell A and unit cell C.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Fan, Y.; Collet, M.; Ichchou, M.; Bareille, O.; Li, L.
Wave Electromechanical Coupling Factor for the Guided Waves in Piezoelectric Composites. *Materials* **2018**, *11*, 1406.
https://doi.org/10.3390/ma11081406

**AMA Style**

Fan Y, Collet M, Ichchou M, Bareille O, Li L.
Wave Electromechanical Coupling Factor for the Guided Waves in Piezoelectric Composites. *Materials*. 2018; 11(8):1406.
https://doi.org/10.3390/ma11081406

**Chicago/Turabian Style**

Fan, Yu, Manuel Collet, Mohamed Ichchou, Olivier Bareille, and Lin Li.
2018. "Wave Electromechanical Coupling Factor for the Guided Waves in Piezoelectric Composites" *Materials* 11, no. 8: 1406.
https://doi.org/10.3390/ma11081406