**2. Electromechanical interaction of piezoelectric materials**

#### **2.1 Mechanical-electrical behavior relations**

Let us consider a block of elastic material in an environment with an electric field of zero, the relationship between stress and strain is followed Hooke's law, and written as follows [1, 2]:

$$\{\sigma\} = [\mathbf{c}]\{\mathbf{c}\},\tag{1}$$

where {σ} is the mechanical stress vector, {ε} is the mechanical strain vector, and [c] is the material stiffness matrix of beam.

Mechanical-electrical relations in piezoelectric materials have an interactive relationship, strain {ε} will produce eε - polarization, where e is the voltage stress factor when there is no mechanical strain. The imposed electric field E produces the -eE stress in the piezoelectric material according to the reverse voltage effect. Therefore, we have a mathematical model that describes the mechanical-electrical interaction relationship in piezoelectric materials as follows [3–6]:

$$\{\boldsymbol{\sigma}\} = [\mathbf{c}] \{\mathbf{c}\} - [\mathbf{e}] \{\mathbf{E}\},\tag{2}$$

Finally, Eq. (6) becomes:

*DOI: http://dx.doi.org/10.5772/intechopen.89303*

s11 s12 s13 000 s12 s22 s23 000 s13 s23 s33 000 0 0 0s44 0 0 0 0 0 0s55 0 0 0 0 0 0s66

*Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates*

Eq. (4) can be written in the matrix form as [4, 6, 7]:

d11 d12 d13 d14 d15 d16 d21 d22 d23 d24 d25 d26 d31 d32 d33 d34 d35 d36

0 0 0 0d15 0 0 0 0d24 0 0 d31 d32 d33 0 00

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

σ<sup>11</sup> σ<sup>22</sup> σ<sup>33</sup> τ<sup>23</sup> τ<sup>13</sup> τ<sup>12</sup> 9

>>>>>>>>>>>>=

þ

p11 p12 p13 p21 p22 p23 p31 p32 p33

p11 0 0 0 p22 0 0 0p33

8 >><

>>:

E1 E2 E3 9 >>=

>>; ,

(9)

8 >><

>>:

E1 E2 E3

(8)

9 >>=

>>; ,

> >>>>>>>>>>>;

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

σ<sup>11</sup> σ<sup>22</sup> σ<sup>33</sup> τ<sup>23</sup> τ<sup>13</sup> τ<sup>12</sup>

9

>>>>>>>>>>>>=

>>>>>>>>>>>>;

þ

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

where E1, E2, and E3 are electric fields in the 1, 2, and 3 directions, respectively.

The induction charge equation of piezoelectric sensor layers is derived from

The non-zero piezoelectric strain constants are d31, d32, d15, d24, and d33, in which d31 = d32, d15 = d24. And the non-zero dielectric coefficients are p11, p22, and

where D1, D2, D3, p11, p22, and p33 are the displacement charge, dielectric

Normally, the voltage is transmitted through the thickness of the

σ<sup>11</sup> σ<sup>22</sup> σ<sup>33</sup> τ<sup>23</sup> τ<sup>13</sup> τ<sup>12</sup>

9

>>>>>>>>>>>>=

>>>>>>>>>>>>;

þ

E1 E2 E3 9 >>=

>>; ,

(7)

8 >><

>>:

ε<sup>11</sup> ε<sup>22</sup> ε<sup>33</sup> γ<sup>23</sup> γ13 γ12 9

>>>>>>>>>>>>=

>>>>>>>>>>>>;

*2.2.2 Piezoelectric sensors*

D1 D2 D3 9 >>=

p33, where p11 = p22. Eq. (8) becomes:

constant in the 1, 2, and 3 directions, respectively.

>>; ¼

8 >><

>>:

¼

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

f gε ¼

f g D ¼

f g D ¼

D1 D2 D3 9 >>=

>>; ¼

8 >><

>>:

actuator layers.

**195**

$$\{\mathbf{D}\} = \left[\mathbf{e}\right]^{\mathrm{T}} \{\mathbf{e}\} + \left[\mathbf{p}\right] \{\mathbf{E}\},\tag{3}$$

$$\text{for } \{\mathbf{D}\} = \left[\mathbf{d}\right]^{\text{T}} \{\mathbf{o}\} + \left[\mathbf{p}\right] \{\mathbf{E}\},\tag{4}$$

where [e] is the piezoelectric stress coefficient matrix, [p] is the dielectric constant matrix, {E} is the vector of applied electric field (V/m), and {D} is the vector of electric displacement (C/m<sup>2</sup> ).

For the linear problem and small strain, strain vector in the piezoelectric structures can be defined as follows:

$$\{\mathbf{e}\} = [\mathbf{s}]\{\mathbf{e}\} + [\mathbf{d}]\{\mathbf{E}\},\tag{5}$$

in which [s] is the matrix of compliance coefficients (m<sup>2</sup> /N), [d] is the matrix of piezoelectric strain constants (m/V).

In the field of engineering, piezoelectric materials are used by two types. The first type, the piezoelectric layers or the piezoelectric patches act as actuators, called the piezoelectric actuators. In this case, the piezoelectric layers are strained when imposing an electric field on it. The second type, the piezoelectric layers or piezoelectric patches act as sensors, called piezoelectric sensors. In this case, the voltage is generated in piezoelectric layers when there is mechanical strain.

#### **2.2 Piezoelectric actuators and sensors**

#### *2.2.1 Piezoelectric actuators*

Eq. (5) can be written in the matrix form as follows [4, 6]:

f gε ¼ ε<sup>11</sup> ε<sup>22</sup> ε<sup>33</sup> γ<sup>23</sup> γ13 γ12 8 >>>>>>>>>>>< >>>>>>>>>>>: 9 >>>>>>>>>>>= >>>>>>>>>>>; ¼ s11 s12 s13 s14 s15 s16 s21 s22 s23 s24 s25 s26 s31 s32 s33 s34 s35 s36 s41 s42 s43 s44 s45 s46 s51 s52 s53 s54 s55 s56 s61 s62 s63 s64 s65 s66 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 σ<sup>11</sup> σ<sup>22</sup> σ<sup>33</sup> τ<sup>23</sup> τ<sup>13</sup> τ<sup>12</sup> 8 >>>>>>>>>>>< >>>>>>>>>>>: 9 >>>>>>>>>>>= >>>>>>>>>>>; þ d11 d21 d31 d12 d22 d32 d13 d23 d33 d14 d24 d34 d15 d25 d35 d16 d26 d36 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 E1 E2 E3 8 >>< >>: 9 >>= >>; , (6)

Assuming that the device is pulled along the axis 3, and viewing the piezoelectric material as a transversely isotropic material, which is true for piezoelectric ceramics, many of the parameters in the above matrices will be either zero, or can be expressed through each other. In particular, the non-zero compliance coefficients are s11, s12, s13, s21, s22, s23, s31, s32, s33, s44, s55, s66, in which s12 = s21, s13 = s31, s23 = s32, s44 = s55, s66 = 2(s11 � s12).

*Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates DOI: http://dx.doi.org/10.5772/intechopen.89303*

Finally, Eq. (6) becomes:

factor when there is no mechanical strain. The imposed electric field E produces the -eE stress in the piezoelectric material according to the reverse voltage effect. Therefore, we have a mathematical model that describes the mechanical-electrical

where [e] is the piezoelectric stress coefficient matrix, [p] is the dielectric constant matrix, {E} is the vector of applied electric field (V/m), and {D} is the

For the linear problem and small strain, strain vector in the piezoelectric struc-

).

In the field of engineering, piezoelectric materials are used by two types.

actuators, called the piezoelectric actuators. In this case, the piezoelectric layers

8

>>>>>>>>>>><

>>>>>>>>>>>:

Assuming that the device is pulled along the axis 3, and viewing the piezoelectric

ceramics, many of the parameters in the above matrices will be either zero, or can be expressed through each other. In particular, the non-zero compliance coefficients are s11, s12, s13, s21, s22, s23, s31, s32, s33, s44, s55, s66, in which s12 = s21, s13 = s31,

material as a transversely isotropic material, which is true for piezoelectric

σ<sup>11</sup> σ<sup>22</sup> σ<sup>33</sup> τ<sup>23</sup> τ<sup>13</sup> τ<sup>12</sup>

9

>>>>>>>>>>>=

>>>>>>>>>>>;

þ

d11 d21 d31 d12 d22 d32 d13 d23 d33 d14 d24 d34 d15 d25 d35 d16 d26 d36

E1 E2 E3

(6)

9 >>=

>>; ,

8 >><

>>:

The first type, the piezoelectric layers or the piezoelectric patches act as

are strained when imposing an electric field on it. The second type, the piezoelectric layers or piezoelectric patches act as sensors, called piezoelectric sensors. In this case, the voltage is generated in piezoelectric layers when there is

Eq. (5) can be written in the matrix form as follows [4, 6]:

s11 s12 s13 s14 s15 s16 s21 s22 s23 s24 s25 s26 s31 s32 s33 s34 s35 s36 s41 s42 s43 s44 s45 s46 s51 s52 s53 s54 s55 s56 s61 s62 s63 s64 s65 s66

f g¼ σ ½ � c f g� ε ½ � e f gE , (2)

or Df g <sup>¼</sup> ½ � <sup>d</sup> <sup>T</sup>f g<sup>σ</sup> <sup>þ</sup> ½ � <sup>p</sup> f g<sup>E</sup> , (4)

f gε ¼ ½ �s f gσ þ ½ � d f gE , (5)

/N), [d] is the matrix of

<sup>T</sup>f g<sup>ε</sup> <sup>þ</sup> ½ � <sup>p</sup> f g<sup>E</sup> , (3)

interaction relationship in piezoelectric materials as follows [3–6]:

f g D ¼ ½ � e

in which [s] is the matrix of compliance coefficients (m<sup>2</sup>

vector of electric displacement (C/m<sup>2</sup>

piezoelectric strain constants (m/V).

**2.2 Piezoelectric actuators and sensors**

tures can be defined as follows:

*Perovskite and Piezoelectric Materials*

mechanical strain.

f gε ¼

**194**

*2.2.1 Piezoelectric actuators*

9

>>>>>>>>>>>=

>>>>>>>>>>>;

¼

s23 = s32, s44 = s55, s66 = 2(s11 � s12).

ε<sup>11</sup> ε<sup>22</sup> ε<sup>33</sup> γ<sup>23</sup> γ13 γ12

8

>>>>>>>>>>><

>>>>>>>>>>>:

$$
\begin{Bmatrix} \mathbf{e} \end{Bmatrix} = \begin{Bmatrix} \mathbf{e}\_{11} \\ \mathbf{e}\_{22} \\ \mathbf{e}\_{33} \\ \mathbf{e}\_{34} \\ \mathbf{e}\_{13} \\ \mathbf{e}\_{13} \\ \mathbf{e}\_{12} \end{Bmatrix} = \begin{Bmatrix} \mathbf{s}\_{11} & \mathbf{s}\_{12} & \mathbf{s}\_{13} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{s}\_{12} & \mathbf{s}\_{22} & \mathbf{s}\_{23} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{s}\_{13} & \mathbf{s}\_{23} & \mathbf{s}\_{33} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{s}\_{55} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{s}\_{66} \end{Bmatrix} \begin{Bmatrix} \sigma\_{11} \\ \sigma\_{22} \\ \sigma\_{33} \\ \tau\_{33} \\ \tau\_{33} \\ \tau\_{33} \\ \tau\_{12} \end{Bmatrix} + \begin{Bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{d}\_{31} \\ \mathbf{0} & \mathbf{0} & \mathbf{d}\_{32} \\ \mathbf{0} & \mathbf{d}\_{24} & \mathbf{0} \\ \mathbf{0} & \mathbf{d}\_{24} & \mathbf{0} \\ \mathbf{d}\_{15} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \end{Bmatrix} \begin{Bmatrix} \mathbf{E}\_{1} \\ \mathbf{E}\_{2} \\ \mathbf{E}\_{3} \\ \$$

where E1, E2, and E3 are electric fields in the 1, 2, and 3 directions, respectively.

#### *2.2.2 Piezoelectric sensors*

The induction charge equation of piezoelectric sensor layers is derived from Eq. (4) can be written in the matrix form as [4, 6, 7]:

$$
\begin{Bmatrix} \mathbf{D} \end{Bmatrix} = \begin{Bmatrix} \mathbf{D}\_1 \\\\ \mathbf{D}\_2 \\\\ \mathbf{D}\_3 \end{Bmatrix} = \begin{bmatrix} \mathbf{d}\_{11} & \mathbf{d}\_{12} & \mathbf{d}\_{13} & \mathbf{d}\_{14} & \mathbf{d}\_{15} & \mathbf{d}\_{16} \\\\ \mathbf{d}\_{21} & \mathbf{d}\_{22} & \mathbf{d}\_{23} & \mathbf{d}\_{24} & \mathbf{d}\_{25} & \mathbf{d}\_{26} \\\\ \mathbf{d}\_{31} & \mathbf{d}\_{32} & \mathbf{d}\_{33} & \mathbf{d}\_{34} & \mathbf{d}\_{35} & \mathbf{d}\_{36} \end{Bmatrix} \begin{Bmatrix} \sigma\_{11} \\\\ \sigma\_{22} \\\\ \sigma\_{33} \\\\ \tau\_{33} \\\\ \tau\_{31} \\\\ \tau\_{12} \end{Bmatrix} + \begin{Bmatrix} \mathbf{p}\_{11} & \mathbf{p}\_{12} & \mathbf{p}\_{13} \\\\ \mathbf{p}\_{21} & \mathbf{p}\_{22} & \mathbf{p}\_{23} \\\\ \mathbf{p}\_{31} & \mathbf{p}\_{32} & \mathbf{p}\_{33} \end{Bmatrix} \begin{Bmatrix} \mathbf{E}\_1 \\\\ \mathbf{E}\_2 \\\\ \mathbf{E}\_3 \end{Bmatrix}, \tag{8}$$

The non-zero piezoelectric strain constants are d31, d32, d15, d24, and d33, in which d31 = d32, d15 = d24. And the non-zero dielectric coefficients are p11, p22, and p33, where p11 = p22. Eq. (8) becomes:

$$
\begin{Bmatrix} \mathbf{D} \end{Bmatrix} = \begin{Bmatrix} \mathbf{D}\_1 \\ \mathbf{D}\_2 \\ \mathbf{D}\_3 \end{Bmatrix} = \begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{d}\_{15} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{d}\_{24} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{d}\_{31} & \mathbf{d}\_{32} & \mathbf{d}\_{33} & \mathbf{0} & \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{Bmatrix} \sigma\_{11} \\ \sigma\_{22} \\ \sigma\_{33} \\ \tau\_{23} \\ \tau\_{31} \\ \tau\_{13} \\ \tau\_{12} \end{Bmatrix} + \begin{Bmatrix} \mathbf{p}\_{11} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{p}\_{22} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{p}\_{33} \end{Bmatrix} \begin{Bmatrix} \mathbf{E}\_1 \\ \mathbf{E}\_2 \\ \mathbf{E}\_3 \end{Bmatrix}, \tag{9}
$$

where D1, D2, D3, p11, p22, and p33 are the displacement charge, dielectric constant in the 1, 2, and 3 directions, respectively.

Normally, the voltage is transmitted through the thickness of the actuator layers.
