# Static Electromechanical Characteristic of a Three-Layer Circular Piezoelectric Transducer

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Electromechanical Characteristic of a Three-Layer Circular Piezoelectric Transducer

#### 2.1. Basic Assumptions

_{1}and t

_{3}, are made of non-piezoelectric material. One of these layers is the executive element of the transducer, while the other one can act as a protective layer/coating. The middle layer consists of two components (of the same thickness t

_{2}= t

_{4})—a piezoelectric disk and a ring made of non-piezoelectric material. In this case, the material of the ring is foam. Its task is to stabilize the electric wires supplying current to the electrodes of the piezoelectric disk. The inner radius of the ring is equal to the outer radius of the piezoelectric disk and is denoted as R

_{o}. The outer radius of all layers is the same and equals R. All disks are fixed to and supported by outer cylindrical surfaces. The transducer deformation occurs because of the transverse piezoelectric effect (caused by the action of a V voltage) occurring inside the piezoelectric disk. In order to simplify the mathematical model, the following assumptions were made:

- the total thickness of all layers is much smaller than their radius, therefore the Plate Theory [29] was used to determine transducer deflection;
- the thickness of the adhesive layers and electrodes is very small and has no effect on the transducer deflection;
- between individual transducer layers there are no slips and the cross-sections remain plane after deformation.
- in the piezoelectric disk, only transverse piezoelectric effect occurs.

#### 2.2. Analytical Description of Transducer Deformation

_{r}, M

_{r}—total force and bending moment in the radial direction; N

_{φ}, M

_{φ}—total force and bending moment in the circumferential direction.

- for the interval r < R
_{o}:$${u}_{2}={u}_{1}-\frac{{t}_{1}+{t}_{2}}{2}\frac{dw}{dr},\text{}{u}_{3}={u}_{1}-\frac{{t}_{1}+{t}_{3}+2{t}_{2}}{2}\frac{dw}{dr}$$ - for the interval R
_{o}< r < R$${u}_{4}={u}_{1}-\frac{{t}_{1}+{t}_{4}}{2}\frac{dw}{dr},\text{}{u}_{3}={u}_{1}-\frac{{t}_{1}+{t}_{3}+2{t}_{4}}{2}\frac{dw}{dr}.$$

- for the interval r < R
_{o}:$$\begin{array}{l}{\sigma}_{r1}=\frac{{E}_{1}\left({\epsilon}_{\phi}{}_{1}{\nu}_{1}+{\epsilon}_{r1}\right)}{1-{\nu}_{1}{}^{2}},\text{}{\sigma}_{\phi 1}=\frac{{E}_{1}\left({\epsilon}_{\phi}{}_{1}+{\epsilon}_{r1}{\nu}_{1}\right)}{1-{\nu}_{1}{}^{2}}\\ {\sigma}_{r2}=\frac{{E}_{2}\left({\epsilon}_{\phi}{}_{2}{\nu}_{2}+{\epsilon}_{r2}+{d}_{31}\frac{V}{{t}_{2}}\left(1+{\nu}_{2}\right)\right)}{1-{\nu}_{2}{}^{2}},\text{}{\sigma}_{\phi 2}=\frac{{E}_{2}\left({\epsilon}_{\phi}{}_{2}+{\epsilon}_{r2}{\nu}_{2}+{d}_{31}\frac{V}{{t}_{2}}\left(1+{\nu}_{2}\right)\right)}{1-{\nu}_{2}{}^{2}}\\ {\sigma}_{r3}=\frac{{E}_{3}\left({\epsilon}_{\phi}{}_{3}{\nu}_{3}+{\epsilon}_{r3}\right)}{1-{\nu}_{3}{}^{2}},\text{}{\sigma}_{\phi 3}=\frac{{E}_{3}\left({\epsilon}_{\phi}{}_{3}+{\epsilon}_{r3}{\nu}_{3}\right)}{1-{\nu}_{3}{}^{2}}\end{array}\}$$ - for the interval R
_{o}< r < R$$\begin{array}{l}{\sigma}_{r1}=\frac{{E}_{1}\left({\epsilon}_{\phi}{}_{1}{\nu}_{1}+{\epsilon}_{r1}\right)}{1-{\nu}_{1}{}^{2}},\text{}{\sigma}_{\phi 1}=\frac{{E}_{1}\left({\epsilon}_{\phi}{}_{1}+{\epsilon}_{r1}{\nu}_{1}\right)}{1-{\nu}_{1}{}^{2}}\\ {\sigma}_{r4}=\frac{{E}_{4}\left({\epsilon}_{\phi}{}_{4}{\nu}_{4}+{\epsilon}_{r4}\right)}{1-{\nu}_{4}{}^{2}},\text{}{\sigma}_{\phi 4}=\frac{{E}_{4}\left({\epsilon}_{\phi}{}_{4}+{\epsilon}_{r4}{\nu}_{4}\right)}{1-{\nu}_{4}{}^{2}}\\ {\sigma}_{r3}=\frac{{E}_{3}\left({\epsilon}_{\phi}{}_{3}{\nu}_{3}+{\epsilon}_{r3}\right)}{1-{\nu}_{3}{}^{2}},\text{}{\sigma}_{\phi 3}=\frac{{E}_{3}\left({\epsilon}_{\phi}{}_{3}+{\epsilon}_{r3}{\nu}_{3}\right)}{1-{\nu}_{3}{}^{2}}\end{array}\}$$

_{i}—Young’s modules; ν

_{i}—Poisson’s ratios; d

_{31}—piezoelectric constant.

- for the interval r < R
_{o}:$${N}_{r1}={\displaystyle \underset{-{t}_{1}/2}{\overset{{t}_{1}/2}{\int}}}{\sigma}_{r1}\mathrm{d}z,{N}_{r2}={\displaystyle \underset{-{t}_{2}/2}{\overset{{t}_{2}/2}{\int}}}{\sigma}_{r2}\mathrm{d}z,{N}_{r3}={\displaystyle \underset{-{t}_{3}/2}{\overset{{t}_{3}/2}{\int}}}{\sigma}_{r3}\mathrm{d}z$$$${N}_{\phi 1}={\displaystyle \underset{-{t}_{1}/2}{\overset{{t}_{1}/2}{\int}}}{\sigma}_{\phi 1}\mathrm{d}z,{N}_{\phi 2}={\displaystyle \underset{-{t}_{2}/2}{\overset{{t}_{2}/2}{\int}}}{\sigma}_{\phi 2}\mathrm{d}z,{N}_{\phi 3}={\displaystyle \underset{-{t}_{3}/2}{\overset{{t}_{3}/2}{\int}}}{\sigma}_{\phi 3}\mathrm{d}z$$$${M}_{r1}={\displaystyle \underset{-{t}_{1}/2}{\overset{{t}_{1}/2}{\int}}}\left({\sigma}_{r1}\mathrm{z}\right)\mathrm{d}z,{M}_{r2}={\displaystyle \underset{-{t}_{2}/2}{\overset{{t}_{2}/2}{\int}}}\left({\sigma}_{r2}\mathrm{z}\right)\mathrm{d}z,{M}_{r3}={\displaystyle \underset{-{t}_{3}/2}{\overset{{t}_{3}/2}{\int}}}\left({\sigma}_{r3}\mathrm{z}\right)\mathrm{d}z$$$${M}_{\phi 1}={\displaystyle \underset{-{t}_{1}/2}{\overset{{t}_{1}/2}{\int}}}\left({\sigma}_{\phi 1}\mathrm{z}\right)\mathrm{d}z,{M}_{\phi 2}={\displaystyle \underset{-{t}_{2}/2}{\overset{{t}_{2}/2}{\int}}}\left({\sigma}_{\phi 2}\mathrm{z}\right)\mathrm{d}z,{M}_{\phi 3}={\displaystyle \underset{-{t}_{3}/2}{\overset{{t}_{3}/2}{\int}}}\left({\sigma}_{\phi 3}\mathrm{z}\right)\mathrm{d}z$$ - for the interval R
_{o}< r < R$${N}_{r1}={\displaystyle \underset{-{t}_{1}/2}{\overset{{t}_{1}/2}{\int}}}{\sigma}_{r1}\mathrm{d}z,{N}_{r4}={\displaystyle \underset{-{t}_{4}/2}{\overset{{t}_{4}/2}{\int}}}{\sigma}_{r4}\mathrm{d}z,{N}_{r3}={\displaystyle \underset{-{t}_{3}/2}{\overset{{t}_{3}/2}{\int}}}{\sigma}_{r3}\mathrm{d}z$$$${N}_{\phi 1}={\displaystyle \underset{-{t}_{1}/2}{\overset{{t}_{1}/2}{\int}}}{\sigma}_{\phi 1}\mathrm{d}z,{N}_{\phi 4}={\displaystyle \underset{-{t}_{4}/2}{\overset{{t}_{4}/2}{\int}}}{\sigma}_{\phi 4}\mathrm{d}z,{N}_{\phi 3}={\displaystyle \underset{-{t}_{3}/2}{\overset{{t}_{3}/2}{\int}}}{\sigma}_{\phi 3}\mathrm{d}z$$$${M}_{r1}={\displaystyle \underset{-{t}_{1}/2}{\overset{{t}_{1}/2}{\int}}}\left({\sigma}_{r1}\mathrm{z}\right)\mathrm{d}z,{M}_{r4}={\displaystyle \underset{-{t}_{4}/2}{\overset{{t}_{4}/2}{\int}}}\left({\sigma}_{r4}\mathrm{z}\right)\mathrm{d}z,{M}_{r3}={\displaystyle \underset{-{t}_{3}/2}{\overset{{t}_{3}/2}{\int}}}\left({\sigma}_{r3}\mathrm{z}\right)\mathrm{d}z$$$${M}_{\phi 1}={\displaystyle \underset{-{t}_{1}/2}{\overset{{t}_{1}/2}{\int}}}\left({\sigma}_{\phi 1}\mathrm{z}\right)\mathrm{d}z,{M}_{\phi 2}={\displaystyle \underset{-{t}_{2}/2}{\overset{{t}_{2}/2}{\int}}}\left({\sigma}_{\phi 2}\mathrm{z}\right)\mathrm{d}z,{M}_{\phi 3}={\displaystyle \underset{-{t}_{3}/2}{\overset{{t}_{3}/2}{\int}}}\left({\sigma}_{\phi 3}\mathrm{z}\right)\mathrm{d}z$$

_{o}; i = 1, 2, 4 for the interval R

_{o}< r < R.

_{o}; j = II for the interval R

_{o}< r < R.

_{o}(20):

- for the interval r < R
_{o}:$${M}_{rI}({R}_{o})={M}_{r1}({R}_{o})+{M}_{r2}({R}_{o})+{M}_{r3}({R}_{o})+{N}_{r3}({R}_{o})\frac{\left({t}_{1}+{t}_{3}+2{t}_{2}\right)}{2}+{N}_{r2}({R}_{o})\frac{\left({t}_{1}+{t}_{2}\right)}{2}$$ - for the interval R
_{o}< r < R:$${M}_{rII}({R}_{o})={M}_{r1}({R}_{o})+{M}_{r4}({R}_{o})+{M}_{r3}({R}_{o})+{N}_{r3}({R}_{o})\frac{\left({t}_{1}+{t}_{3}+2{t}_{4}\right)}{2}+{N}_{r4}({R}_{o})\frac{\left({t}_{1}+{t}_{4}\right)}{2},$$

_{ri}(R

_{o}) and N

_{ri}(R

_{o}) are determined using the Equations (7) and (9) (in Formula (21)), and Equations (11) and (13) (in Formula (22)).

## 3. Verification of the Analytical Solution

_{3}= E

_{4}= 0, t

_{3}= t

_{4}= 0.

^{−3}m, R

_{o}= 5 × 10

^{−3}m, t

_{1}= 2 × 10

^{−4}m, t

_{2}= 1.5 × 10

^{−4}m. In addition, the driving voltage was set to 200 V.

_{2}= 1/s

_{11}= 7.69 × 10

^{10}Pa, ν

_{2}= −s

_{12}/s

_{11}= 0.334, d

_{31}= −1.3 × 10

^{−10}m/V, E

_{1}= 13 × 10

^{10}Pa, ν

_{1}= 0.34. Comparing the results obtained with the literature data (Figure 4), it can be concluded that both solutions are approximately equal—the maximum error is 1.3%. This slight difference in results could be due to the fact that comparative data (dashed line, Figure 4) was obtained by digitizing the graph from [25].

- (a)
- geometrical dimensions: R = 2.54 × 10
^{−2}m, R_{o}= 1.27 × 10^{−2}m, t_{1}= 5.08 × 10^{−4}m, t_{2}= 1.127 × 10^{−4}m - (b)
- material data:
- bottom layer made of aluminium; E
_{1}= 70 × 10^{9}Pa, ν_{1}= 0.33; - piezoelectric disk made of PZT-5H (Piezo Material Lead Zirconate Titanate); E
_{2}= 1/s_{11}= 6.06 × 10^{10}Pa, ν_{2}= −s_{12}/s_{11}= 0.289, d_{31}= −2.74 × 10^{−10}m/V;

- (a)
- geometrical dimensions: R = 6 × 10
^{−2}m, R_{o}= 5.5 × 10^{−2}m, t_{1}= 2.5 × 10^{−4}m, t_{2}= t_{4}= 2.5 × 10^{−4}m, t_{3}= 3 × 10^{−4}m. - (b)
- material data:
- bottom layer made of copper; E
_{1}= 13 × 10^{10}Pa, ν_{1}= 0.34; - upper layer made of PTFE (Polytetrafluoroethylene); E
_{3}= 0.4 × 10^{9}Pa, ν_{3}= 0.46; - piezoelectric disk made of PZT-5H (Piezo Material Lead Zirconate Titanate); E
_{2}= 1/s_{11}= 6.06 × 10^{10}Pa, ν_{2}= −s_{12}/s_{11}= 0.289, d_{31}= −2.74 × 10^{−10}m/V; - the middle ring made of foam; E
_{4}= 35.8 × 10^{6}Pa, ν_{4}= 0.383.

- (c)
- applied electrical load: V = −100 V.

## 4. Influence of Geometrical-Material Parameters on the Electromechanical Characteristics of a Three-Layer Transducer

- the relative thickness of piezo and non-piezo elements: t
_{g}= t_{2}/(t_{1}+ t_{3}); - the elastic moduli ratio of piezo and non-piezoelectric components: E
_{g}= E_{2}/(E_{1}+ E_{3}); - the relative radius of piezoelectric disk and non-piezoelectric layers: R
_{g}= R_{o}/R;

- the relative thickness of the top and bottom layers: t
_{np}= t_{3}/t_{1}; - the ratio of elastic moduli of non-electrical layers: E
_{np}= E_{3}/E_{1}.

_{4}= t

_{2}, E

_{4}= 35.8 × 10

^{6}Pa, ν

_{4}= 0.383—were adopted for this material. In addition, it was assumed that the reference material is piezoelectric material PZT-5H, for which, E

_{2}= 1/s

_{11}= 6.06 × 10

^{10}Pa, ν

_{2}= −s

_{12}/s

_{11}= 0.289, d

_{31}= −2.74 × 10

^{−10}m/V. The dimensions and material constants of the other elements were variable and depended on the values of factors (t

_{g}, E

_{g}, R

_{g}, t

_{np}, E

_{np}), which were taken into account when testing the transducer deformation conditions.

_{g}) and stiffness (E

_{g}) of piezo and non-piezoelectric materials. However, the most important role is played by the relative radius of the piezoelectric disk and non-piezoelectric layers—R

_{g}. If the values of this parameter are too small or too large, the transducer’s deflection is very small. In the case where R

_{g}tends to 0 (a piezoelectric element with a small radius R

_{o}), there is a small elongation of the piezoelectric element, and therefore a small deflection of the transducer. In the opposite situation, when the radius of the piezoelectric element is similar to the radius on which the transducer is mounted (R

_{g}tends to 1), the elongation of the piezoelectric element is blocked by forces occurring in the mounting place, which results in a decrease in transducer deflection.

- the transducer deflection increases as the rigidity of one of the non-electrical components decreases (E
_{np}decrease); - an increase in the relative thickness t
_{np}, depending on the ratio of elastic moduli E_{np}, may cause an increase or decrease in the transducer deflection value.

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Fragment of the piezoelectric transducer structure: (

**a**) deformed radial section, (

**b**) system of forces acting on a small element of a transducer.

**Figure 4.**Comparison of deflection curves: solid line—analytical solution (23); dashed line—analytical solution taken from the work [25] (p. 36), V = 200 V, R = 6 × 10

^{−3}m, R

_{o}= 5 × 10

^{−3}m, t

_{1}= 2 × 10

^{−4}m, t

_{2}= 1.5 × 10

^{−4}m, Pz26/Cu.

**Figure 5.**Comparison of the analytical solution (23) obtained for the unimorph actuator with experimental data [24] (Adapted from Mo, C.; Wright, R.; Slaughter, W.S.; Clark, W.W Behaviour of a unimorph circular piezoelectric actuator. Smart Mater. Struct. 2006, 15, p. 1102).

**Figure 6.**Comparison of deflection curves for the three-layer transducer: solid line—analytical solution (20); dashed line—FEM (Finite Element Method) solution; V = −100 V.

**Figure 7.**Influence of the ratio of elastic moduli, the relative thickness and the radius of piezo and non-piezoelectric materials on transducer deflection: (

**a**) E

_{g}= 0.1, (

**b**) E

_{g}= 1, (

**c**) E

_{g}= 10; ν

_{1}= 0.34, ν

_{3}= 0.46, R = 0.06 m, t

_{3}= 0.0001 m, V = 100 V.

**Figure 8.**Influence of relative thickness and the ratio of elastic moduli of non-electrical materials on transducer deflection (

**a**) E

_{g}= 10, (

**b**) E

_{g}= 1; ν

_{1}= 0.34, ν

_{3}= 0.46, R = 0.06m, R

_{o}/R = 0.9, V = 100 V.

© 2019 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**

Mieczkowski, G.; Borawski, A.; Szpica, D.
Static Electromechanical Characteristic of a Three-Layer Circular Piezoelectric Transducer. *Sensors* **2020**, *20*, 222.
https://doi.org/10.3390/s20010222

**AMA Style**

Mieczkowski G, Borawski A, Szpica D.
Static Electromechanical Characteristic of a Three-Layer Circular Piezoelectric Transducer. *Sensors*. 2020; 20(1):222.
https://doi.org/10.3390/s20010222

**Chicago/Turabian Style**

Mieczkowski, Grzegorz, Andrzej Borawski, and Dariusz Szpica.
2020. "Static Electromechanical Characteristic of a Three-Layer Circular Piezoelectric Transducer" *Sensors* 20, no. 1: 222.
https://doi.org/10.3390/s20010222