**1. Introduction**

638 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

for Intelligent Systems, 2008. MFI 2008. pp. 434-438.

[32] Cannata, G, Maggiali, M, Metta, G, Sandini, G (2008) An embedded artificial skin for humanoid robots. IEEE International Conference on Multisensor Fusion and Integration

> Lead zirconate titanate (PZT) ceramics in special electronic devices such as structural health monitoring systems of liquid rocket engines and microvalves for space applications are subjected to cryogenic temperatures. PZT ceramics are also used in active fuel injectors under severe environments (Senousy et al. 2009a; 2009b). In the application of the PZT stack actuators to hydrogen fuel injectors, the actuators are operated under electric fields at cryogenic temperatures. Hence, it is important to understand the cryogenic electromechanical response of the PZT actuators under electric fields.

> In this chapter, we address the present state of piezomechanics in PZT stack actuators for fuel injectors at cryogenic temperatures. First, we discuss the cryogenic response of PZT stack actuators under direct current (DC) electric fields (Shindo et al. 2011). A thermodynamic model is used to predict a monoclinic phase around a morphotropic phase boundary (MPB). A shift in the boundary between the tetragonal and rhombohedral/monoclinic phases with decreasing temperature is determined, and the temperature dependent piezoelectric coefficients are evaluated. Temperature dependent coercive electric field is also predicted based on the domain wall energy. A finite element analysis (FEA) is then performed, considering the shift in the MPB and polarization switching, to calculate the electromechanical fields of the PZT stack actuators from room to cryogenic temperatures. In addition, experimental results on the DC electric field induced strain, which verify the model, are presented. Next, we discuss the dynamic response of PZT stack actuators under alternating current (AC) electric fields at cryogenic temperatures (Shindo et al. 2012). Dynamic electromechanical fields of the PZT stack actuators from room to cryogenic temperatures are simulated by the FEA with MPB shift and domain wall motion effects. Dynamic strain measurements of the PZT stack actuators under AC electric fields are also presented, and a comparison is made between calculations and measurements to validate the predictions. Moreover, a parametric study using FEA is performed to

© 2012 Shindo and Narita, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

investigate the factors affecting the cryogenic response of PZT stack actuators and to provide a basis for selecting desirable design details.

## **2. Analysis**

#### **2.1. Basic equations**

*y*

Consider the orthogonal coordinate system with axes *x*, *y*, and *z*. The Newton's second law (the equations of motion) and Gauss' law for piezoelectric materials are given by

$$\begin{aligned} \sigma\_{xx,x} + \sigma\_{yx,y} + \sigma\_{zx,z} &= \rho u\_{x,tt} \\ \sigma\_{xy,x} + \sigma\_{yy,y} + \sigma\_{zy,z} &= \rho u\_{y,tt} \\ \sigma\_{xz,x} + \sigma\_{yz,y} + \sigma\_{zz,z} &= \rho u\_{z,tt} \\ \\ D\_{x,x} + D\_{y,y} + D\_{z,z} &= 0 \end{aligned} \tag{1}$$

where (σ*xx*, σ*yy*, σ*zz*, σ*xy* = σ*yx*, σ*yz* = σ*zy*, σ*zx* = σ*xz*) and (*Dx*, *Dy*, *Dz*) are the components of stress tensor and electric displacement vector, (*ux*, *uy*, *uz*) are the components of displacement vectors, ρ is the mass density, and a comma denotes partial differentiation with respect to the coordinates or the time *t*. Constitutive relations for PZT ceramics poled in the *z*-direction can be written as

/ 2 / 2 <sup>=</sup> + 11 12 13 12 11 13 13 13 33 44 44 66 31 31 33 15 15 000 000 000 0 0 0 /2 0 0 0 0 0 0 /2 0 0 0 0 0 0 /2 0 0 0 0 0 0 0 0 0 0 0 00 *xx xx yy yy zz zz yz yz zx zx xy xy ε σ sss ε σ sss ε σ sss ε σ s ε σ s ε σ s d d d d d* <sup>+</sup> *r xx r yy x r zz y r yz z r zx r xy ε ε E ε E ε E ε ε* (3) T 15 11 T 15 11 T 31 31 33 33 0000 0 0 0 0 0 0 00 0 0 0 00 0 0 *xx yy <sup>x</sup> zz yz <sup>z</sup> zx xy D d D d D ddd* σ σ σ σ σ σ <sup>∈</sup> = + <sup>∈</sup> <sup>+</sup> <sup>∈</sup> *r x x r y y <sup>r</sup> <sup>z</sup> <sup>z</sup> E P E P E P* (4)

where (ε*xx*, ε*yy*, ε*zz*, ε*xy* = ε*yx*, ε*yz* = ε*zy*, ε*zx* = ε*xz*) and (*Ex*, *Ey*, *Ez*) are the components of strain tensor and electric field intensity vector, ( *εεεεεε* ,,,,, *r r rr rr xx yy zz xy yz zx* ) and ( , , *rrr <sup>x</sup> <sup>y</sup> <sup>z</sup> PPP* ) are the remanent strain and polarization components, (*s*11, *s*12, *s*13, *s*33, *s*44, *s*66 = 2(*s*11- *s*12)) are the elastic compliances, ( *d*<sup>31</sup> , *d*<sup>33</sup> , *d*<sup>15</sup> ) are the temperature dependent piezoelectric coefficients, and ( <sup>T</sup> <sup>∈</sup><sup>11</sup> , <sup>T</sup> <sup>∈</sup><sup>33</sup> ) are the dielectric constants. The quantities ε *r ij* and *P<sup>r</sup> <sup>i</sup>* are taken to be due entirely to polarization switching. The strain components are

$$\begin{aligned} \varepsilon\_{xx} &= u\_{x,x'} & \varepsilon\_{yy} &= u\_{y,y'} & \varepsilon\_{zz} &= u\_{z,z'}\\ \varepsilon\_{xy} &= \frac{1}{2} (u\_{x,y} + u\_{y,x}) & \varepsilon\_{yz} &= \frac{1}{2} (u\_{y,z} + u\_{z,y}) & \varepsilon\_{zx} &= \frac{1}{2} (u\_{z,x} + u\_{x,z}) \end{aligned} \tag{5}$$

The electric field components are related to the electric potential φby

$$E\_x = -\phi\_{,x'} \qquad E\_y = -\phi\_{,y'} \qquad \quad E\_z = -\phi\_{,z} \tag{6}$$

#### **2.2. Temperature dependent piezoelectric coefficient**

640 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

provide a basis for selecting desirable design details.

**2. Analysis** 

where (σ*xx*, σ*yy*, σ*zz*, σ*xy* = σ*yx*, σ*yz* = σ*zy*, σ*zx* = σ

**2.1. Basic equations** 

investigate the factors affecting the cryogenic response of PZT stack actuators and to

Consider the orthogonal coordinate system with axes *x*, *y*, and *z*. The Newton's second law

++= ++= ++=

*σσσ ρu σσσ ρu σσσρu*

and electric displacement vector, (*ux*, *uy*, *uz*) are the components of displacement vectors,

time *t*. Constitutive relations for PZT ceramics poled in the *z*-direction can be written as

/ 2

0 0

15

*d*

0 0

0 00

*<sup>x</sup> zz*

*<sup>z</sup> zx*

11 12 13 12 11 13 13 13 33

/ 2

15

*D d D d D ddd*

*d*

+

*y*

mass density, and a comma denotes partial differentiation with respect to the coordinates or the

<sup>=</sup>

*xx xx yy yy zz zz yz yz zx zx xy xy*

*ε σ sss ε σ sss ε σ sss ε σ s ε σ s ε σ s*

0 0 0 /2 0 0 0 0 0 0 /2 0 0 0 0 0 0 /2

44

*r xx r yy x r zz y r yz z r zx r xy*

*ε ε*

*ε*

*ε*

*ε ε*

T

T

T

 <sup>+</sup> 

15 11

*xx yy*

 

σ

σ

σ

15 11

*yz*

*xy*

 

0 00 0 0

σ

σ

σ

31 31 33 33

0000 0 0 0 0 0 0 00 0 0

<sup>∈</sup> = + <sup>∈</sup> <sup>+</sup>

<sup>∈</sup>

000 000 000

66

44

*E*

*E*

*E*

31 31 33

*d d d*

*xx,x yx,y zx,z x,tt xy,x yy,y zy,z y,tt xz,x yz,y zz,z z,tt*

(1)

ρis the

(3)

*r x x r y y <sup>r</sup> <sup>z</sup> <sup>z</sup>*

(4)

*E P E P E P*

++= 0 *DDD x,x y,y z,z* (2)

*xz*) and (*Dx*, *Dy*, *Dz*) are the components of stress tensor

(the equations of motion) and Gauss' law for piezoelectric materials are given by

Temperature dependent piezoelectric coefficient is outlined here. Figure 1 shows the phase diagram of PZT established in Jaffe et al. (1971) and Noheda et al. (2000). As the temperature *T* is lowered, PZT undergoes a paraelectric-to-ferroelectric phase transition, and the cubic unit cell is distorted depending on the mole fraction *X* of PbTiO3. In the Zr-rich region, the paraelectric phase changes to orthorhombic phase. An intermediate monoclinic phase exists between the Zr-rich rhombohedral perovskite and Ti-rich tetragonal perovskite phases. Compositions between Zr/Ti ratios 90/10 and 65/35 reveals a ferroelectric-to-ferroelectric transition between rhombohedral space groups. This transition involves the oxygen octahedral tilt.

The MPB between the tetragonal and rhombohedral/monoclinic phases is the origin of the unusually high piezoelectric response of PZT, and this MPB is numerically predicted. For simplicity here, we ignore the octahedral tilt transition which differentiates the high temperature (HT) and low temperature (LT) rhombohedral phases, and the orthorhombic phase.

An energy function for the solid solution between the two end-members PbTiO3 and PbZrO3 is given by (Bell & Furman 2003)

$$
\Delta G\_{\rm PZT} = X G\_{\rm PT} \left( p\_i \right) + \left( 1 - X \right) G\_{\rm PZ} \left( q\_i \right) + G\_{\rm C} \left( p\_i, q\_i \right) \tag{7}
$$

where *pi* and *qi* (*i* = 1,2,3) denote the polarizations of the PbTiO3 and PbZrO3, respectively, and

$$\begin{aligned} G\_{PT}\left(p\_i\right) &= 3.74 \times 10^5 \left(T - T\_{p1}\right) \left(p\_1^2 + p\_2^2 + p\_3^2\right) - 7.9 \times 10^7 \left(p\_1^4 + p\_2^4 + p\_3^4\right) \\ &+ 7.5 \times 10^8 \left(p\_1^2 p\_2^2 + p\_2^2 p\_3^2 + p\_3^2 p\_1^2\right) + 2.61 \times 10^8 \left(p\_1^6 + p\_2^6 + p\_3^6\right) \\ &+ 6.3 \times 10^8 \left[p\_1^4 \left(p\_2^2 + p\_3^2\right) + p\_2^4 \left(p\_3^2 + p\_1^2\right) + p\_3^4 \left(p\_1^2 + p\_2^2\right)\right] - 3.66 \times 10^9 \, p\_1^2 p\_2^2 p\_3^2 \end{aligned} \tag{8}$$

$$\begin{aligned} \mathbf{G}\_{PZ}\left(q\_{i}\right) &= 2.82 \times 10^{5} \left(T - T\_{pZ}\right) \left(q\_{1}^{2} + q\_{2}^{2} + q\_{3}^{2}\right) + 5.12 \times 10^{8} \left(q\_{1}^{4} + q\_{2}^{4} + q\_{3}^{4}\right) \\ &- 6.5 \times 10^{8} \left(q\_{1}^{2}q\_{2}^{2} + q\_{2}^{2}q\_{3}^{2} + q\_{3}^{2}q\_{1}^{2}\right) + 5.93 \times 10^{8} \left(q\_{1}^{6} + q\_{2}^{6} + q\_{3}^{6}\right) \\ &+ 2 \times 10^{9} \left\{q\_{1}^{4}\left(q\_{2}^{2} + q\_{3}^{2}\right) + q\_{2}^{4}\left(q\_{3}^{2} + q\_{1}^{2}\right) + q\_{3}^{4}\left(q\_{1}^{2} + q\_{2}^{2}\right)\right\} - 9.5 \times 10^{9} q\_{1}^{2} q\_{2}^{2} q\_{3}^{2} \end{aligned} \tag{9}$$

$$\mathbf{G}\_{C}\left(p\_{i}, q\_{i}\right) = \mathcal{Y}\_{200}\left(p\_{1}^{2}q\_{1}^{2} + p\_{2}^{2}q\_{2}^{2} + p\_{3}^{2}q\_{3}^{2}\right) + \mathcal{Y}\_{220}\left[p\_{1}^{2}\left(q\_{2}^{2} + q\_{3}^{2}\right) + p\_{2}^{2}\left(q\_{3}^{2} + q\_{1}^{2}\right) + p\_{3}^{2}\left(q\_{1}^{2} + q\_{2}^{2}\right)\right] \tag{10}$$

In Eqs. (8) - (10), *GPT* and *GPZ* are identical to a Landau-Devonshire potential (free energy of a ferroelectric crystal) up to sixth order, *GC* represents the coupling energy, *TPT* = 766 K and *TPZ* = 503 K are the Curie temperatures of PbTiO3 and PbZrO3, respectively, and γ200 and γ220 are unknown coefficients.

**Figure 1.** PZT phase diagram

The thermodynamic equilibrium state can be determined via minimization of Δ*GPZT* with respect to *pi* and *qi*. For simplicity, only positive values of *pi* and *qi* are considered. For each temperature *T* and mole fraction *X*, the local minima in Δ*GPZT* are systematically obtained for the following phases:

Cubic (C)

$$p\_1 = p\_2 = p\_3 = 0, \qquad q\_1 = q\_2 = q\_3 = 0 \tag{11}$$

Tetragonal (T)

$$p\_1 = p\_2 = 0, p\_3 \neq 0, \qquad q\_1 = q\_2 = 0, q\_3 \neq 0 \tag{12}$$

Rhombohedral (R)

642 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

6

unknown coefficients.

γ

*T* (K)

**Figure 1.** PZT phase diagram

for the following phases:

Cubic (C)

Tetragonal (T)

750

500

0

250

() ( )() ()

10 .12 10

= × − ++ + × ++

*q q q q q q q q q qqq*

<sup>200</sup> ( ) ( )( )( ) <sup>220</sup> { } 22 22 22 22 2 22 2 22 2 *G p ,q p q p q p q p q q p q q p q q C ii* 11 22 33 12 3 23 1 31 2 (10)

*X* (Ti)

PbZrO3 PbTiO3

The thermodynamic equilibrium state can be determined via minimization of Δ*GPZT* with respect to *pi* and *qi*. For simplicity, only positive values of *pi* and *qi* are considered. For each temperature *T* and mole fraction *X*, the local minima in Δ*GPZT* are systematically obtained

0, == ≠ == ≠ 0, 12 3 12 3 *pp p qq q* 0, 0 (12)

0.5 1.0

=== === 12 3 123 *ppp qqq* 0, 0 (11)

Monoclinic

Tetragonal

5 222 84 4 4

123 123

(9)

γ200 and γ220 are

( ) ()

( ) = + + + ++ ++ +

In Eqs. (8) - (10), *GPT* and *GPZ* are identical to a Landau-Devonshire potential (free energy of a ferroelectric crystal) up to sixth order, *GC* represents the coupling energy, *TPT* = 766 K and *TPZ*

*G q TT q q q q q q PZ i PZ qq qq qq qqq*

8 22 22 22 86 6 6 12 23 31 123 942 2 42 2 42 2 9222 12 3 23 1 31 2 123

10 10

 γ

{ } ( )( )( )

2 9.5

= 503 K are the Curie temperatures of PbTiO3 and PbZrO3, respectively, and

Cubic

Rhombohedral

HT

LT

Rhombohedral

Orthorhombic

+× + + + + + − ×

−× + + + × ++

.5 10 5.93 10

2.82 5

$$p\_1 = p\_2 = p\_3 \neq 0, \qquad q\_1 = q\_2 = q\_3 \neq 0 \tag{13}$$

Monoclinic (M)

$$p\_1 = p\_2 \neq 0, p\_3 \neq 0, \qquad q\_1 = q\_2 \neq 0, q\_3 \neq 0 \tag{14}$$

The energies of the minima are then compared to define the stable state. In the simulation, γ200 = 6 × 108 and γ220 = 1.2 × 108 are assumed.

We now show a numerical example and comparison with experiments. Fig. 2 shows the

**Figure 2.** Calculated and experimental phase diagram of PZT

calculated PZT phase diagram. We also plot the experimental phase diagrams of PZT. The open square, open circle and solid circle are the results from Jeffe et al. (1971), Noheda et al. (2000) and Pandey et al. (2008). The simulation result reasonably agrees with the experimental data. It is noted that at room temperature, the MPB is located at about *X* = 0.44, and that there is an apparent shift of about 13 mol% in the MPB location on cooling to 0 K. In the work of Boucher et al. (2006), the piezoelectric coefficient *d*333 of PZT (Mn-doped) presented a significant decrease (about 85 %) due to a shift of 12 mol% of Zr at room temperature. Thus, from the consideration of Fig. 2, we propose the temperature dependent piezoelectric coefficient *dijk* ( *d d* 311 31 = , *d d* 333 33 = , *d d* 131 15 = / 2 here) for *X* = 0.44, i.e.,

$$\overline{d}\_{\vec{n}\vec{k}} = \begin{cases} (1.7 \times 10^{-4} T + 0.95) d\_{\vec{n}\vec{k}} & 192 \le T \\ (1.0 \times 10^{-5} T^2 + 2.4 \times 10^{-3} T + 0.15) d\_{\vec{n}\vec{k}} & 0 < T \le 192 \end{cases} \tag{15}$$

where *dijk* (*d*311=*d*31, *d*333=*d*33, *d*131=*d*15/2 here) is the piezoelectric coefficient at 298 K. In Eqs. (3) and (4), reduced indices for the full notations of elastic compliances *sijkl* and temperature dependent piezoelectric coefficients *dijk* are used, with the following correspondence between the one and two indices: 1 = 11, 2 =22, 3 = 33, 4 = 23, 5 = 31, 6 = 12. We can also predict the piezoelectric coefficient for other mole fractions. For example, the temperature dependent piezoelectric coefficient for *X* = 0.56 can be expressed as

$$\overline{d}\_{ijk} = \begin{cases} (-1.7 \times 10^{-4} T + 0.25) d\_{ijk} & 192 \le T \\ (-1.0 \times 10^{-5} T^2 - 2.4 \times 10^{-3} T + 1.1) d\_{ijk} & 20 < T \le 192 \\ (1.0 \times 10^{-5} T^2 + 2.4 \times 10^{-3} T + 0.95) d\_{ijk} & 0 < T \le 20 \end{cases} \tag{16}$$

#### **2.3. Polarization switching**

High electromechanical fields lead to the polarization switching. We assume that the direction of a spontaneous polarization *Ps* of each grain can change by 180o or 90o for ferroelectric switching induced by a sufficiently large electric field opposite to the poling direction. The 90o ferroelastic domain switching is also induced by a sufficiently large stress field. The criterion states that a polarization switches when the electrical and mechanical work exceeds a critical value (Hwang et al. 1995)

$$\begin{aligned} \sigma\_{xx}\Delta\varepsilon\_{xx} + \sigma\_{yy}\Delta\varepsilon\_{yy} + \sigma\_{zz}\Delta\varepsilon\_{zz} + 2\sigma\_{xy}\Delta\varepsilon\_{xy} + 2\sigma\_{yz}\Delta\varepsilon\_{yz} + 2\sigma\_{zx}\Delta\varepsilon\_{zx} \\ + \overline{E\_x}\Delta P\_x + \overline{E\_y}\Delta P\_y + \overline{E\_z}\Delta P\_z \geq 2P^s\overline{E\_c} \end{aligned} \tag{17}$$

where *<sup>c</sup> E* is a temperature dependent coercive electric field, and Δε*ij* and Δ*Pi* are the changes in the spontaneous strain and polarization during switching, respectively. The values of Δε*ij* = ε *r ij* and Δ*Pi*<sup>=</sup> *<sup>P</sup><sup>r</sup> <sup>i</sup>* for 180o switching can be expressed as

$$\begin{aligned} \Delta \varepsilon\_{xx} &= 0, & \Delta \varepsilon\_{yy} &= 0, & \Delta \varepsilon\_{zz} &= 0, & \Delta \varepsilon\_{xy} &= 0, & \Delta \varepsilon\_{yz} &= 0, & \Delta \varepsilon\_{zx} &= 0, \\ \Delta P\_x &= 0, & \Delta P\_y &= 0, & \Delta P\_z &= -\Delta P^s \end{aligned} \tag{18}$$

For 90o switching in the *zx* plane, the changes are

$$\begin{aligned} \Delta \varepsilon\_{xx} &= \gamma^s, \qquad \Delta \varepsilon\_{yy} = 0, \qquad \Delta \varepsilon\_{zz} = -\gamma^s, \qquad \Delta \varepsilon\_{xy} = 0, \qquad \Delta \varepsilon\_{yz} = 0, \qquad \Delta \varepsilon\_{zx} = 0, \\\Delta P\_x &= \pm P^s, \qquad \Delta P\_y = 0, \qquad \Delta P\_z = -P^s \end{aligned} \tag{19}$$

where *<sup>s</sup>* γis a spontaneous strain. For 90o switching in the *yz* plane, we have

$$\begin{aligned} \Delta \varepsilon\_{xx} &= 0, & \Delta \varepsilon\_{yy} &= \mathfrak{p}^s, & \Delta \varepsilon\_{zz} &= -\mathfrak{p}^s, & \Delta \varepsilon\_{xy} &= 0, & \Delta \varepsilon\_{yz} &= 0, & \Delta \varepsilon\_{zx} &= 0, \\ \Delta P\_x &= 0, & \Delta P\_y &= \pm P^s, & \Delta P\_z &= -P^s \end{aligned} \tag{20}$$

The constitutive equations after polarization switching are

$$
\begin{bmatrix}
\varepsilon\_{xx} \\
\varepsilon\_{yy} \\
\varepsilon\_{zz} \\
\varepsilon\_{xz} \\
\varepsilon\_{yz} \\
\varepsilon\_{zx} \\
\varepsilon\_{xy}
\end{bmatrix} = \begin{bmatrix}
s\_{11} & s\_{12} & s\_{13} & 0 & 0 & 0 \\
s\_{12} & s\_{11} & s\_{13} & 0 & 0 & 0 \\
s\_{13} & s\_{13} & s\_{33} & 0 & 0 & 0 \\
0 & 0 & 0 & s\_{44} / 2 & 0 & 0 \\
0 & 0 & 0 & 0 & s\_{44} / 2 & 0 \\
0 & 0 & 0 & 0 & 0 & s\_{66} / 2
\end{bmatrix} \begin{bmatrix}
\sigma\_{xx} \\
\sigma\_{yy} \\
\sigma\_{zz} \\
\sigma\_{zz} \\
\sigma\_{yz} \\
\sigma\_{zx} \\
\sigma\_{xy}
\end{bmatrix},
$$

$$
\begin{bmatrix}
\widetilde{\widetilde{d}}\_{111} & \widetilde{\widetilde{d}}\_{211} & \widetilde{\widetilde{d}}\_{311} \\
\widetilde{d}\_{112} & \widetilde{d}\_{222} & \widetilde{d}\_{322} \\
\widetilde{d}\_{123} & \widetilde{d}\_{223} & \widetilde{d}\_{333} \\
\widetilde{d}\_{133} & \widetilde{d}\_{233} & \widetilde{d}\_{333} \\
\widetilde{d}\_{212} & \widetilde{d}\_{223} & \widetilde{d}\_{332}
\end{bmatrix} \begin{bmatrix}
\varepsilon'\_{xx} \\
\varepsilon'\_{yy} \\
\varepsilon'\_{zz} \\
\varepsilon'\_{yz} \\
\varepsilon'\_{yz} \\
\varepsilon'\_{zx}
\end{bmatrix}.
\tag{21}
$$

\*\*\*\*\*\* 111 122 133 123 131 112 T <sup>11</sup> \*\*\*\*\*\* <sup>T</sup> 211 222 233 223 231 212 11 \*\*\*\*\*\* T 311 322 333 323 331 312 33 0 0 0 0 0 0 *xx yy <sup>x</sup> zz y yz <sup>z</sup> zx xy D D D* σ σ σ σ σ σ <sup>∈</sup> <sup>=</sup> + ∈ <sup>∈</sup> *x y z dddddd <sup>E</sup> dddddd E E dddddd* <sup>+</sup> *r x r y r z P P P* (22)

where

644 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

dependent piezoelectric coefficient for *X* = 0.56 can be expressed as

direction of a spontaneous polarization *Ps*

work exceeds a critical value (Hwang et al. 1995)

σσσ

00 2 *<sup>s</sup> P P PP*

0

*s s PP P PP*

*s s P PP PP*

Δ = Δ =± Δ =−

*xy z*

, ,

γ

Δ =± Δ = Δ =−

, ,

*x yz*

*s s*

Δ = Δ = Δ =−

*xyz*

For 90o switching in the *zx* plane, the changes are

, ,

+Δ+ Δ+ Δ≥

**2.3. Polarization switching** 

= ε

*r*

where *<sup>s</sup>* γ

*ij* and Δ*Pi*<sup>=</sup> *<sup>P</sup><sup>r</sup>*

γ

0

where *dijk* (*d*311=*d*31, *d*333=*d*33, *d*131=*d*15/2 here) is the piezoelectric coefficient at 298 K. In Eqs. (3) and (4), reduced indices for the full notations of elastic compliances *sijkl* and temperature dependent piezoelectric coefficients *dijk* are used, with the following correspondence between the one and two indices: 1 = 11, 2 =22, 3 = 33, 4 = 23, 5 = 31, 6 = 12. We can also predict the piezoelectric coefficient for other mole fractions. For example, the temperature

4

× + × + <≤

High electromechanical fields lead to the polarization switching. We assume that the

ferroelectric switching induced by a sufficiently large electric field opposite to the poling direction. The 90o ferroelastic domain switching is also induced by a sufficiently large stress field. The criterion states that a polarization switches when the electrical and mechanical

> σ

*E*

in the spontaneous strain and polarization during switching, respectively. The values of Δ

Δ= Δ = Δ= Δ= Δ= Δ=

*εε εε εε*

*xx yy zz xy yz zx*

Δ = Δ = Δ =− Δ = Δ = Δ =

*ε εε εεε*

*xx yy zz xy yz zx*

 γ

is a spontaneous strain. For 90o switching in the *yz* plane, we have

 γ

Δ = Δ = Δ =− Δ = Δ = Δ =

*εε ε εεε*

*xx yy zz xy yz zx*

*s s*

*εεε ε ε ε*

*xx yy zz xy yz zx*

000000

,,,,,,

,, ,,,,

0 000

, , ,,,,

Δ+ Δ + Δ+ Δ+ Δ+ Δ

2 *xx yy zz xy yz zx*

*<sup>s</sup> <sup>c</sup> xyz EP EP EP P*

where *<sup>c</sup> E* is a temperature dependent coercive electric field, and Δ

*<sup>i</sup>* for 180o switching can be expressed as

*xyz*

−

 −× + <sup>≤</sup> =− × − × + <≤

( 1.7 10 0.25) 192 ( 1.0 10 2.4 10 1.1) 20 192 (1.0 10 2.4 10 0.95) 0 20

*T Td T*

*ijk*

*Td T*

*ijk*

222

0 000

 σ

of each grain can change by 180o or 90o for

 σ

ε

(16)

(17)

ε*ij*

(18)

(19)

(20)

*ij* and Δ*Pi* are the changes

5 2 3 5 2 3

− − − −

*d T Td T*

*ijk ijk*

$$\overline{\overline{d}}\_{i|k}^\* = \{ \overline{d}\_{33} \boldsymbol{\eta}\_i \boldsymbol{\eta}\_k \boldsymbol{\eta}\_l + \overline{d}\_{31} (\boldsymbol{\eta}\_i \boldsymbol{\delta}\_{kl} - \boldsymbol{\eta}\_i \boldsymbol{\eta}\_k \boldsymbol{\eta}\_l) + \frac{1}{2} \overline{d}\_{15} (\boldsymbol{\delta}\_{ik} \boldsymbol{\eta}\_l - 2 \boldsymbol{\eta}\_i \boldsymbol{\eta}\_k \boldsymbol{\eta}\_l + \boldsymbol{\delta}\_{il} \boldsymbol{\eta}\_k) \} \tag{23}$$

In Eq. (23), *ni* is the unit vector in the poling direction and δ*ij* is the Kronecker delta.

#### **2.4. Domain wall motion**

A domain wall displacement causes changes of strain and polarization (Cao et al. 1999). For simplicity, here the applied AC electric field *Ez*=*E*0exp(iω*t*) is parallel to the direction of spontaneous polarization *Ps* in one of the domains (see Fig. 3); *E*0 is the AC electric field amplitude and ω is angular frequency(=2π*f* where *f* is frequency in Hertz). The changes of the strains and polarization due to the domain wall displacement Δ*l* (Arlt and Dederichs, 1980) can be written as

$$\begin{aligned} \Delta \boldsymbol{\varepsilon\_{xx}}^{\*} &= \Delta \mathbf{s}\_{11} \boldsymbol{\sigma}\_{xx} + \Delta \mathbf{s}\_{13} \boldsymbol{\sigma}\_{zz} + \Delta d\_{311} E\_z \\ \Delta \boldsymbol{\varepsilon\_{zz}}^{\*} &= \Delta \mathbf{s}\_{13} \boldsymbol{\sigma}\_{xx} + \Delta \mathbf{s}\_{33} \boldsymbol{\sigma}\_{zz} + \Delta d\_{333} E\_z \\ \Delta P\_z^{\*} &= \Delta d\_{311} \boldsymbol{\sigma}\_{xx} + \Delta d\_{333} \boldsymbol{\sigma}\_{zz} + \Delta \boldsymbol{\varepsilon\_{z3}^{T}} E\_z \end{aligned} \tag{24}$$

where all terms with Δ are the contributions from the domain wall motion, and

$$\begin{aligned} \Delta \mathbf{s}\_{11} &= \frac{\left(\mathbf{y}^{s}\right)^{2}}{2lf\_{\mathrm{D}}}, \qquad \Delta \mathbf{s}\_{13} = -\frac{\left(\mathbf{y}^{s}\right)^{2}}{2lf\_{\mathrm{D}}}, \qquad \Delta \mathbf{s}\_{33} = \frac{\left(\mathbf{y}^{s}\right)^{2}}{2lf\_{\mathrm{D}}},\\ \Delta d\_{311} &= -\frac{\mathbf{y}^{s}\mathbf{P}^{s}}{2lf\_{\mathrm{D}}}, \qquad \Delta d\_{333} = \frac{\mathbf{y}^{s}\mathbf{P}^{s}}{2lf\_{\mathrm{D}}}, \qquad \Delta \mathbf{e}\_{33} = \frac{\left(\mathbf{P}^{s}\right)^{2}}{2lf\_{\mathrm{D}}} \end{aligned} \tag{25}$$

In Eq. (25), *l* is the domain width and *fD* is the force constant for the domain wall motion process. The strains ε*xx*, ε*zz* and electric displacement *Dz* are given by

**Figure 3.** Schematic drawing of many grains which in turn consist of domains and basic unit of a piezoelectric crystallite with a domain wall.

$$\begin{aligned} \varepsilon\_{\text{xx}} &= \stackrel{\ast}{s}\_{11}^{\*} \sigma\_{\text{xx}} + \stackrel{\ast}{s}\_{12} \sigma\_{yy} + \stackrel{\ast}{s}\_{13}^{\*} \sigma\_{zz} + \stackrel{\ast}{d}\_{31}^{\*} E\_z \\ \varepsilon\_{zz} &= \stackrel{\ast}{s}\_{13}^{\*} \sigma\_{\text{xx}} + \stackrel{\ast}{s}\_{13}^{\*} \sigma\_{yy} + \stackrel{\ast}{s}\_{33}^{\*} \sigma\_{zz} + \stackrel{\ast}{d}\_{31}^{\*} E\_z \\ D\_z &= d\_{31}^{\*} \sigma\_{\text{xx}} + d\_{31}^{\*} \sigma\_{yy} + d\_{33}^{\*} \sigma\_{zz} + \stackrel{\ast}{e}\_{33}^{\*} E\_z \end{aligned} \tag{26}$$

where

$$\begin{aligned} \stackrel{\circ}{s}\_{11}^{\*} &= \mathbf{s}\_{11} + \Delta \mathbf{s}\_{11\prime} & \stackrel{\circ}{s}\_{13}^{\*} &= \mathbf{s}\_{13} + \Delta \mathbf{s}\_{13\prime} & \stackrel{\circ}{s}\_{33}^{\*} &= \mathbf{s}\_{33} + \Delta \mathbf{s}\_{33} \\ \stackrel{\circ}{d}\_{31}^{\*} &= \overline{d}\_{31} + \Delta d\_{311\prime} & \stackrel{\circ}{d}\_{33}^{\*} &= \overline{d}\_{33} + \Delta d\_{333\prime} \\ & \mathbf{e}\_{33}^{T\*} &= \mathbf{e}\_{33}^{T} + \Delta \mathbf{e}\_{33}^{T} \end{aligned} \tag{27}$$

Experimental studies on PZT ceramics have shown that 45-70% of dielectric and piezoelectric moduli values may originate from the extrinsic contributions (Luchaninov et al. 1989, Cao et al. 1991). The extrinsic dielectric constant <sup>Δ</sup> <sup>T</sup> <sup>∈</sup>33 is approximately estimated as the two thirds of the bulk properties (Li et al. 1993). Here, the following equation (Narita et al. 2005) is utilized to describe <sup>Δ</sup> <sup>T</sup> <sup>∈</sup>33 in terms of AC electric field amplitude *E*0 and temperature dependent coercive electric field *<sup>c</sup> E* :

$$
\Delta \in \overset{\text{T}}{\varepsilon\_{33}} = \in \overset{\text{T}}{\varepsilon\_{33}} \frac{2E\_0}{\mathfrak{3}\overline{E}\_c} \tag{28}
$$

By substituting Eq. (28) into the sixth of Eq. (25), *lfD* = 3(*P*<sup>s</sup> )2 *<sup>c</sup> <sup>E</sup>* /(4 <sup>T</sup> <sup>∈</sup><sup>33</sup> *<sup>E</sup>*0) is obtained. By eliminating *lfD*, the changes in the elastic compliances and piezoelectric coefficients in Eq. (25) can be rewritten in terms of AC electric field amplitude etc.

#### **2.5. Finite element model**

646 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

*\* xx \* zz \* z*

*ε ε P*

γ

process. The strains

where

ε*xx*, ε

piezoelectric crystallite with a domain wall.

*xx zz z*

γ

311 333

11 13 311 13 33 333

σ

Δ =Δ +Δ +Δ Δ =Δ +Δ +Δ Δ =Δ +Δ +Δ∈

σ

where all terms with Δ are the contributions from the domain wall motion, and

*ss s*

Δ = Δ =− Δ =

11 13 33

σ

*s s dE s s dE dd E*

 σ

 σ

*xx zz z xx zz z xx zz z*

311 333 33

 σ

2 22

*D DD s s s s s*

*2lf 2lf 2lf*

, ,,

*D DD*

*P*s

*z*

( ) () ()

*s ss*

 γ

*2lf 2lf 2lf*

*P PP d d*

In Eq. (25), *l* is the domain width and *fD* is the force constant for the domain wall motion

*zz* and electric displacement *Dz* are given by

Δ*l*

Domain

**Figure 3.** Schematic drawing of many grains which in turn consist of domains and basic unit of a

12

*Dd d d*

\*\* \* 11 13 33 \* \* 31 33 \* 33

*ss s*

*ε s sd ε sssd*

σσσ

\* \*\* 11 13 31 \*\*\*\* 13 13 33 31 \*\*\* \* 31 31 33 33

=+++ =+++ = + + +∈

σσσ

σσσ

*xx yy zz z xx yy zz z xx yy zz z*

11 11 13 13 33 33

*ss ss ss*

, ,

31 311 33 333 TT T 33 33

Experimental studies on PZT ceramics have shown that 45-70% of dielectric and piezoelectric moduli values may originate from the extrinsic contributions (Luchaninov et al. 1989, Cao et al. 1991). The extrinsic dielectric constant <sup>Δ</sup> <sup>T</sup> <sup>∈</sup>33 is approximately estimated as the two thirds of the bulk properties (Li et al. 1993). Here, the following equation (Narita

, ,

*dd dd* = +Δ = +Δ = +Δ

= +Δ = +Δ ∈ =∈ +Δ ∈

*s E*

, ,

Δ =− Δ = Δ∈ =

T

γγ

T 33 (24)

(25)

ω

(26)

2

*x*

*t*) *<sup>z</sup>* <sup>0</sup>

*E* = *E* exp(*i*

*l*

*l*

T

*d d* (27)

*E E*

γ<sup>s</sup> <sup>O</sup>

( )

Consider a PZT stack actuator with 300 PZT layers of width *Wp* = 5.2 mm and thickness *hp* = 0.1 mm, thin electrodes, and elastic coating layer of thickness *he* = 0.5 mm as shown in Fig. 4. A rectangular Cartesian coordinate system O-*xyz* is used and the origin of the coordinate system coincides with the center of the stack actuator. Each PZT layer is sandwiched between thin electrodes. An external electrode is attached on both sides of the actuator to address the internal electrodes.

In order to discuss the electromechanical fields near the internal electrode, the problem of the stack actuator is solved using the unit cell model (two layer piezoelectric composite with |*x*| ≤ *Wp*/2 + *he*, |*y*| ≤ *Wp*/2 + *he*, 0 ≤ *z* ≤ 2*hp*) shown in Fig. 5. Electrodes of length *a* and width *Wp* = 5.2 mm are attached to the PZT layer of thickness *hp* = 0.1 mm, and a *Wp* - *a* tab region exists on both sides of the layer. Because of the geometric and loading symmetry, only the half needs to be analyzed. The electrode layers are not incorporated into the model.

**Figure 4.** Schematic drawing of PZT stack actuator.

First, we consider the PZT stack actuator under DC electric fields. The electric potential on two electrode surfaces (-*Wp*/2 ≤ x ≤ -*Wp*/2+*a*, 0 ≤ *y* ≤ *Wp*/2, *z* = 0, 2*hp*) equals the applied voltage, φ = *V*0. The electrode surface (*Wp*/2-*a* ≤ *x* ≤ *Wp*/2, 0 ≤ *y* ≤ *Wp*/2, *z* = *hp*) is connected to the ground, so that φ = 0. The mechanical boundary conditions include the traction-free conditions on the coating layer surfaces at *x* = ± (*Wp*/2 + *he*), *y* = *Wp*/2 + *he* and the symmetry conditions on the *xz* plane at *y* = 0 and *xy* planes at *z* = 0, 2*hp*. In addition, the origin is constrained against the displacement in the *x*-direction, to avoid rigid body motion. We next consider the PZT stack actuator under AC electric fields. The electric potential on two electrode surfaces (-*Wp*/2 ≤ x ≤ -*Wp*/2+*a*, 0 ≤ *y* ≤ *Wp*/2, *z* = 0, 2*hp*) equals the applied voltage, φ = *V*0exp(*i*ω*t*), and the electrode surface (*Wp*/2-*a* ≤ *x* ≤ *Wp*/2, 0 ≤ *y* ≤ *Wp*/2, *z* = *hp*) is connected to the ground.

**Figure 5.** Unit cell of the PZT stack actuator.

Each element in ANSYS is defined by eight-node 3-D coupled field solid for the PZT layers and eight-node 3-D structural solid for the coating layer. *Ps* = 0.3 C/m2 and γ*s* = 0.004 are used. In order to calculate the electromechanical fields, we need the temperature dependent coercive electric field *<sup>c</sup> E* . First-principles free energy calculations for ferroelectric perovskites (Kumar and Waghmare 2010) have shown that the domain wall energy increases linearly about 50% as the temperature *T* decreases from room temperature to 260 K. Since higher domain wall energy leads to higher coercive electric field, the temperature dependent coercive electric field is assumed to be the following equation:

$$
\overline{E}\_{\varepsilon} = (4.84 - 0.0129T)E\_{\varepsilon} \tag{29}
$$

where *Ec* is a coercive electric field at 298 K.

#### **3. Experiment**

The stack actuator is fabricated using 300 soft PZT N-10 layers (NEC/Tokin Co. Ltd., Japan) of width *Wp* = 5.2 mm and thickness *hp* = 0.1 mm (see Fig. 6). A rectangular Cartesian

**Figure 6.** Experimental setup.

voltage,

*V*0exp(*i*ω

the ground.

*hp*

*Wp*/2

φ

**Figure 5.** Unit cell of the PZT stack actuator.

where *Ec* is a coercive electric field at 298 K.

**3. Experiment** 

φ

the ground, so that

φ

First, we consider the PZT stack actuator under DC electric fields. The electric potential on two electrode surfaces (-*Wp*/2 ≤ x ≤ -*Wp*/2+*a*, 0 ≤ *y* ≤ *Wp*/2, *z* = 0, 2*hp*) equals the applied

conditions on the coating layer surfaces at *x* = ± (*Wp*/2 + *he*), *y* = *Wp*/2 + *he* and the symmetry conditions on the *xz* plane at *y* = 0 and *xy* planes at *z* = 0, 2*hp*. In addition, the origin is constrained against the displacement in the *x*-direction, to avoid rigid body motion. We next consider the PZT stack actuator under AC electric fields. The electric potential on two electrode surfaces (-*Wp*/2 ≤ x ≤ -*Wp*/2+*a*, 0 ≤ *y* ≤ *Wp*/2, *z* = 0, 2*hp*) equals the applied voltage,

*a*

*a*

Poling

and eight-node 3-D structural solid for the coating layer. *Ps*

*Wp*

dependent coercive electric field is assumed to be the following equation:

O

*z*

φ

Each element in ANSYS is defined by eight-node 3-D coupled field solid for the PZT layers

used. In order to calculate the electromechanical fields, we need the temperature dependent coercive electric field *<sup>c</sup> E* . First-principles free energy calculations for ferroelectric perovskites (Kumar and Waghmare 2010) have shown that the domain wall energy increases linearly about 50% as the temperature *T* decreases from room temperature to 260 K. Since higher domain wall energy leads to higher coercive electric field, the temperature

The stack actuator is fabricated using 300 soft PZT N-10 layers (NEC/Tokin Co. Ltd., Japan)

of width *Wp* = 5.2 mm and thickness *hp* = 0.1 mm (see Fig. 6). A rectangular Cartesian

= *V*0. The electrode surface (*Wp*/2-*a* ≤ *x* ≤ *Wp*/2, 0 ≤ *y* ≤ *Wp*/2, *z* = *hp*) is connected to

*t*), and the electrode surface (*Wp*/2-*a* ≤ *x* ≤ *Wp*/2, 0 ≤ *y* ≤ *Wp*/2, *z* = *hp*) is connected to

Ground

= 0. The mechanical boundary conditions include the traction-free

*he*

= 0.3 C/m2 and

(4.84 0.0129 ) *<sup>c</sup>* = − *T E <sup>c</sup> E* (29)

*y*

*x*

γ*s*

= 0.004 are

φ=


**Table 1.** Material properties of N-10.

coordinate system O-*xyz* is used and the origin of the coordinate system coincides with the center of the actuator. The electrode length is *a* = *Wp* = 5.2 mm (full electrodes), and the actuator is coated with epoxy layer of thickness *he* = 0.5 mm. The total dimensions of the specimen are width of 6.2 mm and length of 40.5 mm. Table 1 lists the material properties of N-10. The coercive electric field of N-10 at room temperature is approximately *Ec* = 0.36 MV/m. Young's modulus, Poisson's ratio and mass density of epoxy layer are 3.35 GPa, 0.214 and 1100 kg/m3, respectively.

The actuator is bonded to the test rig of a SUS304 stainless steel plate using epoxy bond, and DC voltage (0 Hz) and AC voltage (400 Hz) are applied using a power supply. Two strain gages are attached at the center of the *y* = ± 3.1 mm planes, and the magnitude of strain is measured. To control the temperature of the actuator, an automated helium refill system (TRG-300, Taiyo Toyo Sanso Co. Ltd., Japan) is used. For the reliability of the test, two specimens are experimented, and four strain values are obtained.
