**4. Dynamic analysis of laminated piezoelectric composite plates**

### **4.1 The electromechanical behavioral relations in the plate**

Consider laminated composite plates with general coordinate system (x, y, z), in which the x, y plane coincides with the neutral plane of the plate. The top and bottom surfaces of the plate are bonded to the piezoelectric patches or piezoelectric layers (actuator and sensor). The plate under the load acting on its neutral plane has any temporal variation rule (**Figure 5**).

Hypothesis: The piezoelectric composite plate corresponds with Reissner-Mindlin theory. The material layers are arranged symmetrically through the neutral plane of the plate, ideally adhesive with each other.

#### *4.1.1 Strain - displacement relations*

Based on the first-order shear deformation theory, the displacement fields at any point in the plate are [7, 8]:

$$\begin{aligned} u(\mathbf{x}, \mathbf{y}, \mathbf{z}, t) &= u\_0(\mathbf{x}, \mathbf{y}, t) + z \theta\_\mathbf{y}(\mathbf{x}, \mathbf{y}, t), \\ v(\mathbf{x}, \mathbf{y}, \mathbf{z}, t) &= v\_0(\mathbf{x}, \mathbf{y}, t) - z \theta\_\mathbf{x}(\mathbf{x}, \mathbf{y}, t), \\ w(\mathbf{x}, \mathbf{y}, \mathbf{z}, t) &= w\_0(\mathbf{x}, \mathbf{y}, t), \end{aligned} \tag{49}$$

where u, v and w are the displacements of a general point (x, y, z) in the laminate along x, y and z directions, respectively. u0, v0, w0, θ<sup>x</sup> and θ<sup>y</sup> are the displacements and rotations of a midplane transverse normal about the y-and xaxes respectively.

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

**Figure 5.**

**Case 2:** With structural damping, with piezoelectric damping (*Gv* = 0.5,

Consider laminated composite plates with general coordinate system (x, y, z), in which the x, y plane coincides with the neutral plane of the plate. The top and bottom surfaces of the plate are bonded to the piezoelectric patches or piezoelectric layers (actuator and sensor). The plate under the load acting on its neutral plane

Hypothesis: The piezoelectric composite plate corresponds with Reissner-Mindlin theory. The material layers are arranged symmetrically through the neutral

Based on the first-order shear deformation theory, the displacement fields at any

(49)

*u x*ð Þ¼ , *y*, *z*, *t u*0ð Þþ *x*, *y*, *t zθy*ð Þ *x*, *y*, *t* , *v x*ð Þ¼ , *y*, *z*, *t v*0ð Þ� *x*, *y*, *t zθx*ð Þ *x*, *y*, *t* ,

where u, v and w are the displacements of a general point (x, y, z) in the laminate along x, y and z directions, respectively. u0, v0, w0, θ<sup>x</sup> and θ<sup>y</sup> are the displacements and rotations of a midplane transverse normal about the y-and x-

*w x*ð Þ¼ , *y*, *z*, *t w*0ð Þ *x*, *y*, *t* ,

**4. Dynamic analysis of laminated piezoelectric composite plates**

**4.1 The electromechanical behavioral relations in the plate**

has any temporal variation rule (**Figure 5**).

*Vertical displacement response (Gv = 0.5, Gd = 30* � *Case 2).*

*Perovskite and Piezoelectric Materials*

*4.1.1 Strain - displacement relations*

point in the plate are [7, 8]:

axes respectively.

**202**

plane of the plate, ideally adhesive with each other.

*Gd* = 30).

**Figure 4.**

*Piezoelectric composite plate and coordinate system of the plate (a), and lamina details (b).*

The components of the strain vector corresponding to the displacement field (49) are defined as:

For the linear strain:

$$\begin{split} \boldsymbol{\varepsilon}\_{\mathbf{x}} &= \frac{\partial \boldsymbol{u}}{\partial \mathbf{x}} = \frac{\partial \boldsymbol{u}\_{0}}{\partial \mathbf{x}} + \boldsymbol{z} \frac{\partial \boldsymbol{\theta}\_{\mathbf{y}}}{\partial \mathbf{x}}, \boldsymbol{e}\_{\mathbf{y}} = \frac{\partial \boldsymbol{v}}{\partial \mathbf{y}} = \frac{\partial \boldsymbol{v}\_{0}}{\partial \mathbf{y}} - \boldsymbol{z} \frac{\partial \boldsymbol{\theta}\_{\mathbf{x}}}{\partial \mathbf{y}}, \\ \boldsymbol{\gamma}\_{\mathbf{xy}} &= \left( \frac{\partial \boldsymbol{u}}{\partial \mathbf{y}} + \frac{\partial \boldsymbol{v}}{\partial \mathbf{x}} \right) + \frac{\partial \mathbf{w}}{\partial \mathbf{x}} \cdot \frac{\partial \mathbf{w}}{\partial \mathbf{y}} = \left( \frac{\partial \boldsymbol{u}\_{0}}{\partial \mathbf{y}} + \frac{\partial \boldsymbol{v}\_{0}}{\partial \mathbf{x}} \right) + \boldsymbol{z} \left( \frac{\partial \boldsymbol{\theta}\_{\mathbf{y}}}{\partial \mathbf{x}} - \frac{\partial \boldsymbol{\theta}\_{\mathbf{x}}}{\partial \mathbf{y}} \right), \\ \boldsymbol{\gamma}\_{\mathbf{xx}} &= \frac{\partial \mathbf{u}}{\partial \mathbf{z}} + \frac{\partial \mathbf{w}}{\partial \mathbf{x}} = \frac{\partial \mathbf{w}\_{0}}{\partial \mathbf{x}} + \boldsymbol{\theta}\_{\mathbf{y}}, \boldsymbol{\gamma}\_{\mathbf{y}} = \frac{\partial \mathbf{v}}{\partial \mathbf{z}} + \frac{\partial \mathbf{w}}{\partial \mathbf{y}} = \frac{\partial \mathbf{w}\_{0}}{\partial \mathbf{y}} - \boldsymbol{\theta}\_{\mathbf{x}}, \end{split} \tag{50}$$

or in the vector form:

*εx εy γxy* 8 >>< >>: 9 >>= >>; ¼ *εo x εo y γo xy* 8 >>< >>: 9 >>= >>; þ *z κx κy κxy* 8 >>< >>: 9 >>= >>; ¼ *∂ ∂x* 0 <sup>0</sup> *<sup>∂</sup> ∂y ∂ ∂y ∂ ∂x* 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 *u*0 *v*0 ( ) þ *z* � *∂ ∂y* 0 <sup>0</sup> � *<sup>∂</sup> ∂x* � *∂ ∂y ∂ ∂x* 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 *θx θy* ( ) ¼ ¼ *D<sup>ε</sup>* ½ � *u*0 *v*0 ( ) þ *D<sup>κ</sup>* ½ � *θx θy* ( ) <sup>¼</sup> f g *<sup>ε</sup>*<sup>0</sup> <sup>þ</sup> *<sup>z</sup>*f g*<sup>κ</sup>* <sup>¼</sup> *<sup>ε</sup><sup>L</sup> b* � �, (51)

$$
\begin{Bmatrix} \mathcal{Y}\_{xx} \\ \mathcal{Y}\_{\mathcal{Y}^x} \end{Bmatrix} = \begin{bmatrix} \frac{\partial}{\partial \mathbf{x}} & \mathbf{0} & \mathbf{1} \\\\ \frac{\partial}{\partial \mathbf{y}} & -\mathbf{1} & \mathbf{0} \end{bmatrix} \begin{Bmatrix} w^\rho \\ \theta\_\mathbf{x} \\ \theta\_\mathbf{y} \end{Bmatrix} = \begin{bmatrix} \begin{Bmatrix} wD \end{Bmatrix} & -[I\_\circ] \end{Bmatrix} \begin{Bmatrix} w\_0 \\ \theta\_\mathbf{x} \\ \theta\_\mathbf{y} \end{Bmatrix} = \{\varepsilon\_\circ\}. \tag{52}
$$

and for the nonlinear strain:

$$\left\{ \begin{array}{c} \varepsilon\_{\pi} \\ \varepsilon\_{\mathcal{V}} \\ \chi\_{\mathfrak{N}} \end{array} \right\} = \left\{ \varepsilon\_{b}^{L} \right\} + \left\{ \varepsilon^{N} \right\} = \left\{ \varepsilon\_{b}^{N} \right\},\tag{53}$$

$$\left\{ \begin{matrix} \chi\_{\text{xx}} \\ \chi\_{\text{yx}} \end{matrix} \right\} = \{e\_i\},\tag{54}$$

$$\text{where } \{e^{N}\} = \frac{1}{2} \begin{bmatrix} \frac{\partial w\_{0}}{\partial x} & 0\\ 0 & \frac{\partial w\_{0}}{\partial y} \\ \frac{\partial w\_{0}}{\partial y} & \frac{\partial w\_{0}}{\partial x} \end{bmatrix} \begin{Bmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \end{Bmatrix} w\_{0} \text{ is the non-linear strain vector, } \{e^{L}\_{b}\} \text{ is} $$

the linear strain vector, {εs} is the shear strain vector.

#### *4.1.2 Stress-strain relations*

The equation system describing the stress-strain relations and mechanicalelectrical quantities is respectively written as [8, 14]:

$$\begin{aligned} \{\sigma\_b\} &= [Q]\{\varepsilon\_b^N\} - [e]\{E\}, \\ \{\pi\_b\} &= [Q\_s]\{\varepsilon\_t\}, \end{aligned} \tag{55}$$

<sup>Π</sup> <sup>¼</sup> <sup>1</sup> 2 ð

> þ ð

> > Ω

**layers**

where *q<sup>e</sup>*

node i.

**205**

*i*

*4.2.2 Dynamic equations*

damping) with in-plane loads is:

*4.2.1 Finite element models*

Ω f g *ε*<sup>0</sup>

*<sup>T</sup>*½ � *<sup>A</sup>* f g *<sup>ε</sup>*<sup>0</sup> *<sup>d</sup>*<sup>Ω</sup> <sup>þ</sup>

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

*<sup>ε</sup><sup>N</sup>* � �*<sup>T</sup>* ½ � *<sup>A</sup>* f g *<sup>ε</sup>*<sup>0</sup> � *Np*

1 2 ð

� � � � *<sup>d</sup>*<sup>Ω</sup> �

where Ω is the plane xy domain of the plate.

*<sup>ϕ</sup><sup>e</sup>* f g <sup>¼</sup> *: : <sup>ϕ</sup><sup>e</sup>*

*<sup>q</sup><sup>e</sup>* f g <sup>¼</sup> *<sup>q</sup><sup>e</sup>*

� � <sup>¼</sup> *ui vi wi <sup>θ</sup>xi <sup>θ</sup>yi*

½ � *Mbb* ½ � 0 ½ � 0 ½ � 0

<sup>¼</sup> f g*<sup>R</sup>* f g *Qel*

� � €*qbb* � �

<sup>þ</sup> ½ �þ *Kbb* ½ � *KG Kb<sup>ϕ</sup>*

*K<sup>ϕ</sup><sup>b</sup>*

The vector of degrees of freedom for the element {qe

1 � � *q<sup>e</sup>*

The equation of bending vibrations with out-of-plane loads is:

*<sup>j</sup> : :* � �*<sup>T</sup>*

in which NPL<sup>e</sup> is the number of piezoelectric layers in a given element.

2

� � … *q<sup>e</sup>*

The dynamic equations of piezoelectric composite plate can be derived by using Hamilton's principle, accordingly, the vibration equation of the membrane (without

*ϕ*€� � ( ) <sup>þ</sup> ½ � *CR* ½ � <sup>0</sup>

� �

� � � *<sup>K</sup>ϕϕ* � � " # *qbb* � �

where [Mss], [Kss] are the overall mass, membrane elastic stiffness matrix respectively, and *qss* � �, *<sup>q</sup>*\_ *ss* � �, €*qss* � � are respectively the membrane displacement, velocity, acceleration vector. [Mbb], [Kbb] and *qbb* � �, *<sup>q</sup>*\_ *bb* � �, €*qbb* � � are the overall mass, bending elastic stiffness matrix and the bending displacement, velocity, acceleration vector; [KG] is the overall geometric stiffness matrix; ([KG] is

� � *ϕ<sup>e</sup>* � �*<sup>T</sup>*

Ω f g*κ*

*<sup>T</sup>*½ � *<sup>D</sup>* f g*<sup>κ</sup> <sup>d</sup>*<sup>Ω</sup> <sup>þ</sup>

**4.2 Dynamic stability analysis of laminated composite plate with piezoelectric**

Nine-node Lagrangian finite elements are used with the displacement and strain fields represented by Eqs. (49), (53), and (54). In the developed models, there is one electric potential degree of freedom for each piezoelectric layer to represent the piezoelectric behavior and thus the vector of electrical degrees of freedom is [6, 14]:

ð

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

Ω f g *ε*<sup>0</sup> 1 2 ð

*<sup>T</sup> Np* � �*d*<sup>Ω</sup> �

Ω

ð

Ω f g*κ <sup>T</sup> Mp*

, *<sup>j</sup>* <sup>¼</sup> 1, … , *NPL<sup>e</sup>*

9

*Mss* ½ � €*qss* � � <sup>þ</sup> *Kss* ½ � *qss* � � <sup>¼</sup> f g *F t*ð Þ *:* (63)

� � *<sup>q</sup>*\_ *bb* � �

*<sup>ϕ</sup>*\_ � � ( )

½ � 0 ½ � 0

f g*ϕ* � �

� �, (64)

� � is the mechanical displacement vector for

} is:

f g *<sup>ε</sup><sup>s</sup> <sup>T</sup> As* ½ �f g *<sup>ε</sup><sup>s</sup> <sup>d</sup>*Ω<sup>þ</sup>

� �*d*<sup>Ω</sup> � *<sup>W</sup>*,

, (61)

, (62)

(60)

$$\{D\} = [\mathfrak{e}]\{\varepsilon\_b^N\} + [p]\{E\},\tag{56}$$

where f g *<sup>σ</sup><sup>b</sup>* <sup>¼</sup> *<sup>σ</sup><sup>x</sup> <sup>σ</sup><sup>y</sup> <sup>τ</sup>xy* � �*<sup>T</sup>* is the plane stress vector, f g *<sup>τ</sup><sup>b</sup>* <sup>¼</sup> *<sup>τ</sup>yz <sup>τ</sup>xz* � �*<sup>T</sup>* is the shear stress vector, [Q] is the ply in-plane stiffness coefficient matrix in the structural coordinate system, [Qs] is the ply out-of-plane shear stiffness coefficient matrix in the structural coordinate system. Notice that {τb} is free from piezoelectric effects.

The in-plane force vector at the state pre-buckling:

$$\left\{\boldsymbol{N}^{0}\right\} = \left\{\boldsymbol{N}\_{\rm x}^{0} \quad \boldsymbol{N}\_{\rm y}^{0} \quad \boldsymbol{N}\_{\rm xy}^{0}\right\}^{T} = \sum\_{k=1}^{n} \int\_{h\_{k-1}}^{h\_{k}} \begin{Bmatrix} \sigma\_{\rm x}^{0} \\ \sigma\_{\rm y}^{0} \\ \tau\_{\rm xy}^{0} \end{Bmatrix} d\boldsymbol{z}.\tag{57}$$

#### *4.1.3 Total potential energy*

The total potential energy of the system is given by:

$$\Pi = \frac{1}{2} \int\_{V\_p} \left\{ \varepsilon\_b^N \right\}^T \{\sigma\_b\} dV + \frac{1}{2} \int\_{V\_p} \{\varepsilon\_s\}^T \{\varepsilon\_b\} dV - \frac{1}{2} \int\_{V\_p} \{E\}^T \{D\} dV - W,\tag{58}$$

where W is the energy of external forces, Vp is the entire domain including composite and piezoelectric materials.

Introducing [A], [B], [D], [As], and vectors {Np}, {Mp} as [8]:

$$\begin{aligned} \left( [A], [B], [D] \right) &= \int\_{-h/2}^{h/2} (\mathbf{1}, z, z^2) [\mathbf{Q}] dz, \\ [A\_\ast] &= \int\_{-h/2}^{h/2} [\mathbf{Q}\_\ast] dz, \left( \{\mathbf{N}\_P\}, \{\mathbf{M}\_P\} \right) = \int\_{-h/2}^{h/2} (\mathbf{1}, z) [\mathbf{e}] \{\mathbf{E}\} dz, \end{aligned} \tag{59}$$

where h is the total laminated thickness and combining with (5), (6) the total potential energy equation (8) can be written

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

$$\begin{split} \Pi &= \frac{1}{2} \int\_{\Omega} \{\boldsymbol{\varepsilon}\_{0}\}^{T} [\boldsymbol{A}] \{\boldsymbol{\varepsilon}\_{0}\} d\Omega + \frac{1}{2} \int\_{\Omega} \{\boldsymbol{\kappa}\}^{T} [\boldsymbol{D}] \{\boldsymbol{\kappa}\} d\Omega + \frac{1}{2} \int\_{\Omega} \{\boldsymbol{\varepsilon}\_{\boldsymbol{\varepsilon}}\}^{T} [\boldsymbol{A}\_{\boldsymbol{s}}] \{\boldsymbol{\varepsilon}\_{\boldsymbol{\varepsilon}}\} d\Omega + \\ &+ \int\_{\Omega} \{\boldsymbol{\varepsilon}^{N}\}^{T} ([\boldsymbol{A}] \{\boldsymbol{\varepsilon}\_{0}\} - [\boldsymbol{N}\_{p}]) d\Omega - \int\_{\Omega} \{\boldsymbol{\varepsilon}\_{0}\}^{T} [\boldsymbol{N}\_{p}] d\Omega - \int\_{\Omega} \{\boldsymbol{\kappa}\}^{T} [\boldsymbol{M}\_{p}] d\Omega - \mathcal{W}, \end{split} \tag{60}$$

where Ω is the plane xy domain of the plate.

### **4.2 Dynamic stability analysis of laminated composite plate with piezoelectric layers**

#### *4.2.1 Finite element models*

where *<sup>ε</sup><sup>N</sup>* � � <sup>¼</sup> <sup>1</sup>

*4.1.2 Stress-strain relations*

where f g *σ<sup>b</sup>* ¼ *σ<sup>x</sup> σ<sup>y</sup> τxy*

*4.1.3 Total potential energy*

*Vp*

*εN b* � �*<sup>T</sup>*

composite and piezoelectric materials.

*As* ½ �¼

ð Þ¼ ½ � *A* , ½ � *B* , ½ � *D*

ð *h=*2

�*h=*2

potential energy equation (8) can be written

<sup>Π</sup> <sup>¼</sup> <sup>1</sup> 2 ð

**204**

tric effects.

2

*Perovskite and Piezoelectric Materials*

*∂w*<sup>0</sup> *∂x*

*∂w*<sup>0</sup> *∂y*

0

*∂ ∂x ∂ ∂y*

9 >>=

>>;

The equation system describing the stress-strain relations and mechanical-

*b* � � � ½ �*<sup>e</sup>* f g*<sup>E</sup>* ,

*b*

the shear stress vector, [Q] is the ply in-plane stiffness coefficient matrix in the structural coordinate system, [Qs] is the ply out-of-plane shear stiffness coefficient matrix in the structural coordinate system. Notice that {τb} is free from piezoelec-

� �*<sup>T</sup>* is the plane stress vector, f g *<sup>τ</sup><sup>b</sup>* <sup>¼</sup> *<sup>τ</sup>yz <sup>τ</sup>xz*

<sup>¼</sup> <sup>X</sup>*<sup>n</sup> k*¼1

> 2 ð

> > *Vp*

ð *h=*2

ð Þ 1, *z* ½ �*e* f g*E dz*,

�*h=*2

*h* ð*k*

*σ*0 *x σ*0 *y τ*0 *xy* 9 >=

>; *k*

f g*<sup>E</sup> <sup>T</sup>*f g *<sup>D</sup> dV* � *<sup>W</sup>*, (58)

8 ><

>:

*hk*�<sup>1</sup>

f g *<sup>σ</sup><sup>b</sup>* <sup>¼</sup> ½ � *<sup>Q</sup> <sup>ε</sup><sup>N</sup>*

f g¼ *<sup>D</sup>* ½ �*<sup>e</sup> <sup>ε</sup><sup>N</sup>*

*w*<sup>0</sup> is the non-linear strain vector, *ε<sup>L</sup>*

f g *<sup>τ</sup><sup>b</sup>* <sup>¼</sup> *Qs* ½ �f g *<sup>ε</sup><sup>s</sup>* , (55)

� � <sup>þ</sup> ½ � *<sup>p</sup>* f g*<sup>E</sup>* , (56)

*b* � � is

� �*<sup>T</sup>* is

*dz:* (57)

(59)

8 >><

>>:

*∂w*<sup>0</sup> *∂x*

<sup>0</sup> *<sup>∂</sup>w*<sup>0</sup> *∂y*

the linear strain vector, {εs} is the shear strain vector.

electrical quantities is respectively written as [8, 14]:

The in-plane force vector at the state pre-buckling:

The total potential energy of the system is given by:

1 2 ð

*Vp*

Introducing [A], [B], [D], [As], and vectors {Np}, {Mp} as [8]:

ð *h=*2

�*h=*2

*Qs* ½ �*dz*, *Np*

f g *σ<sup>b</sup> dV* þ

*<sup>x</sup> N*<sup>0</sup>

n o*<sup>T</sup>*

*<sup>y</sup> N*<sup>0</sup> *xy*

f g *<sup>ε</sup><sup>s</sup> <sup>T</sup>*f g *<sup>τ</sup><sup>b</sup> dV* � <sup>1</sup>

where W is the energy of external forces, Vp is the entire domain including

1, *<sup>z</sup>*, *<sup>z</sup>*<sup>2</sup> � �½ � *<sup>Q</sup> dz*,

where h is the total laminated thickness and combining with (5), (6) the total

� �, *Mp* � � � � <sup>¼</sup>

*<sup>N</sup>*<sup>0</sup> � � <sup>¼</sup> *<sup>N</sup>*<sup>0</sup>

Nine-node Lagrangian finite elements are used with the displacement and strain fields represented by Eqs. (49), (53), and (54). In the developed models, there is one electric potential degree of freedom for each piezoelectric layer to represent the piezoelectric behavior and thus the vector of electrical degrees of freedom is [6, 14]:

$$\{\phi^{\epsilon}\} = \begin{Bmatrix} \cdot & \cdot & \phi\_j^{\epsilon} & \cdot & \cdot \end{Bmatrix}^T, \quad j = 1, \ldots, NPL^{\epsilon},\tag{61}$$

in which NPL<sup>e</sup> is the number of piezoelectric layers in a given element. The vector of degrees of freedom for the element {qe } is:

$$\begin{Bmatrix} q^{\epsilon} \end{Bmatrix} = \begin{Bmatrix} \begin{Bmatrix} q\_1^{\epsilon} \end{Bmatrix} & \begin{Bmatrix} q\_2^{\epsilon} \end{Bmatrix} & \dots & \begin{Bmatrix} q\_9^{\epsilon} \end{Bmatrix} & \begin{Bmatrix} q^{\epsilon} \end{Bmatrix}^T,\tag{62}$$

where *q<sup>e</sup> i* � � <sup>¼</sup> *ui vi wi <sup>θ</sup>xi <sup>θ</sup>yi* � � is the mechanical displacement vector for node i.

#### *4.2.2 Dynamic equations*

The dynamic equations of piezoelectric composite plate can be derived by using Hamilton's principle, accordingly, the vibration equation of the membrane (without damping) with in-plane loads is:

$$[\mathbf{M}\_{\alpha}]\{\ddot{q}\_{\alpha}\} + [\mathbf{K}\_{\alpha}]\{q\_{\alpha}\} = \{F(t)\}.\tag{63}$$

The equation of bending vibrations with out-of-plane loads is:

$$
\begin{aligned}
\begin{aligned}
\begin{bmatrix}
[\mathcal{M}\_{bb}\big] & \mathbf{0}\big] \\
\hline
[\mathbf{0}] & \mathbf{0}
\end{bmatrix}
\end{bmatrix}
\begin{Bmatrix}
\begin{Bmatrix} \ddot{q}\_{bb}\end{Bmatrix} \\
\begin{Bmatrix} \ddot{\mathbf{0}}\end{Bmatrix}
\end{Bmatrix} + \begin{bmatrix}
[\mathbf{C}\_{R}\big] & \mathbf{0}\big] \\
\begin{Bmatrix} \ddot{\boldsymbol{\phi}}\end{Bmatrix}
\end{aligned}
\begin{aligned}
\begin{Bmatrix} \ddot{\boldsymbol{q}}\_{bb}\end{Bmatrix} &= \begin{bmatrix}
[\mathbf{C}\_{R}\big] & \mathbf{0}\big] \\
\begin{bmatrix} \dot{\boldsymbol{q}}\end{bmatrix}
\end{aligned}
\begin{aligned}
\begin{Bmatrix} \dot{\boldsymbol{q}}\_{bb}\mathbf{b}\end{Bmatrix} \\
\begin{aligned}
\begin{Bmatrix} [\mathbf{K}\_{bb}] + [\mathbf{K}\_{G}] & \begin{bmatrix} \mathbf{K}\_{bb} \end{bmatrix} \\
\begin{bmatrix} [\mathbf{K}\_{\phi b}] & -\begin{bmatrix} \mathbf{K}\_{\phi b} \end{bmatrix}
\end{Bmatrix}
\end{aligned}
\end{aligned}
\end{aligned}
\begin{aligned}
\begin{Bmatrix}
\begin{Bmatrix} \dot{\boldsymbol{q}}\_{bb}\mathbf{b}\end{Bmatrix} \\
\begin{Bmatrix} \begin{Bmatrix} \mathbf{K}\_{\phi b} \end{Bmatrix} \\
\begin{Bmatrix} \mathbf{K}\_{\phi b} \end{Bmatrix}
\end{bmatrix}
\end{aligned}
\end{aligned}
\end{aligned}
$$

$$
\begin{aligned}
\begin{Bmatrix}
\begin{Bmatrix} \mathbf{K}\_{\phi b} \end{Bmatrix} \\
=
\begin{Bmatrix}
\begin{Bmatrix} \mathbf{K} \end{Bmatrix} \\
\begin{Bmatrix} \mathbf{Q}\_{d\ell} \end{Bmatrix}
\end{Bmatrix}
\end{aligned}
\end{aligned}$$

where [Mss], [Kss] are the overall mass, membrane elastic stiffness matrix respectively, and *qss* � �, *<sup>q</sup>*\_ *ss* � �, €*qss* � � are respectively the membrane displacement, velocity, acceleration vector. [Mbb], [Kbb] and *qbb* � �, *<sup>q</sup>*\_ *bb* � �, €*qbb* � � are the overall mass, bending elastic stiffness matrix and the bending displacement, velocity, acceleration vector; [KG] is the overall geometric stiffness matrix; ([KG] is

a function of external in-plane loads); {F(t)} is the in-plane load vector, {R} is the normal load vector, {Qel} is the vector containing the nodal charges and in-balance charges.

The element coefficient matrices are:

$$
\left[K\_{\rm G}^{\epsilon}\right] = \left[K\_{\rm Gx}^{\epsilon}\right] + \left[K\_{\rm Gy}^{\epsilon}\right] + \left[K\_{\rm Gxy}^{\epsilon}\right], \tag{65}
$$

$$
\left[K\_{\rm Gx}^{\epsilon}\right] = \int\_{A\_{\epsilon}} N\_{\rm x}^{0} \left[N\_{\rm x}^{\prime}\right] \left[N\_{\rm x}^{\prime}\right]^{T} dA\_{\epsilon},
$$

$$
\text{where } \left[K\_{\rm Gy}^{\epsilon}\right] = \int\_{A\_{\epsilon}} N\_{\rm y}^{0} \left[N\_{\rm y}^{\prime}\right] \left[N\_{\rm y}^{\prime}\right]^{T} dA\_{\epsilon}, \tag{66}
$$

$$
\left[K\_{\rm Gxy}^{\epsilon}\right] = \int\_{A\_{\epsilon}} N\_{\rm xy}^{0} \left[N\_{\rm x}^{\prime}\right] \left[N\_{\rm y}^{\prime}\right]^{T} dA\_{\epsilon},
$$

where [Ab] and [Bs] are the stiffness coefficient matrix and strain-displacement

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

• In the case of plate subjected to periodic in-plane loads and without damping, the elastic stability problems become simple only by solving the linear

method can be proved effectively and the following dynamic stability criteria

◦ *Plate is considered to be stable if the maximum bending deflection is three times smaller than the plate's thickness: Eq. (71) has the solution (wi)max satisfying the condition* 0≤ wi j jmax <3*h, where wi is the deflection of the plate at node*

◦ *Plate is called to be in critical status if the maximum bending deflection of the plate is three times equal to the plate's thickness. Eq. (71) has the solution*

◦ *Plate is called to be at buckling if the maximum deflection of the plate is three times larger than the plate's thickness: Eq. (71) has the solution (wi)max*

The identification of critical forces is carried out by the iterative method.

Step 1. Defining the matrices, the external load vector and errors of load

stress vector is defined by (72), updating the geometric stiffness matrix [KG].

� If for all w*<sup>i</sup>* j j ¼ 0: increase load, recalculate from step 2;

stiffness matrix [KG]. Increase load, recalculate from step 2;

Step 2. Solving Eq. (71) to present unknown displacement vector, {qss} and the

Step 3. Solving Eq. (71) to present unknown bending displacement vector {qbb},

+ In case: 0 < w*<sup>i</sup>* j j*<sup>m</sup>*ax <3*h*: Define stress vector by Eq. (73), update the geometric

≤*εD*: Critical load p = pcr. End.

Stability analysis of piezoelectric composite plate with dimensions a � b � h,

composed of three layers, in which two layers of piezoelectric PZT-5A at its top and bottom are considered, each layer thickness hp = 0.00075 m; the middle layer material is Graphite/Epoxy material, with thickness h1 = 0.0005 m. The material properties for graphite/epoxy and PZT-5A are shown in Section 5.1 above. One short edge of the plate is clamped, the other three edges are free. The in-plane half-

where a = 0.25 m, b = 0.30 m, h = 0.002 m. Piezoelectric composite plate is

• In case of the plate under any in-plane dynamic load and with damping, the elastic stability problems become very complex. This iterative

matrix of the plane bending problem.

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

equations to determine the eigenvalues.

*(wi)max satisfying the condition* wi j jmax ¼ 3*h*.

*satisfying the condition* wi j jmax>3*h*.

Stability criteria [14]:

are used:

*number i*.

*4.2.4 Iterative algorithm*

and then testing stability conditions.

+ In case: 0 ≤ <sup>w</sup>*<sup>i</sup>* j j j jmax�3*<sup>h</sup>*

*4.2.5 Numerical analysis*

� If at least one value w*<sup>i</sup>* j j 6¼ 0:

w*<sup>i</sup>* j jmax

iterations.

**207**

$$\text{in which } \left[N\_x'\right] = \frac{\partial}{\partial \mathbf{x}} [N(\mathbf{x}, \boldsymbol{\mathcal{y}})], \left[N\_{\boldsymbol{\mathcal{y}}}'\right] = \frac{\partial}{\partial \boldsymbol{\mathcal{y}}} [N(\mathbf{x}, \boldsymbol{\mathcal{y}})],\tag{67}$$

$$\frac{\partial \mathbf{w}}{\partial \mathbf{x}} = \begin{bmatrix} \partial \mathbf{N} \\ \partial \mathbf{x} \end{bmatrix} \left\{ q\_{bb}^{\epsilon} \right\} = \begin{bmatrix} N\_{\mathbf{x}}^{\prime} \end{bmatrix} \left\{ q\_{bb}^{\epsilon} \right\}, \\ \frac{\partial \mathbf{w}}{\partial \mathbf{y}} = \begin{bmatrix} \partial \mathbf{N} \\ \partial \mathbf{y} \end{bmatrix} \left\{ q\_{bb}^{\epsilon} \right\} = \begin{bmatrix} N\_{\mathbf{y}}^{\prime} \end{bmatrix} \left\{ q\_{bb}^{\epsilon} \right\} \tag{68}$$

$$\left[K\_{\rm G}\right] = \sum\_{\rm ue} \left[K\_{\rm G}^{\epsilon}\right] \tag{69}$$

#### *4.2.3 Dynamic stability analysis*

When the plate is subjected to in-plane loads only ({R} = {0}), the in-plane stresses can lead to buckling, from Eqs. (63) and (64) the governing differential equations of motion of the damped system may be written as:

$$\begin{aligned} \left[\mathcal{M}\_{\rm s}\right]\left\{\ddot{q}\_{\rm s}\right\} + \left[\mathcal{K}\_{\rm s}\right]\{q\_{\rm s}\} &= \left\{F(t)\right\},\\ \left[\mathcal{M}\_{\rm b}\right]\left\{\ddot{q}\_{\rm b}\right\} + \left[\mathcal{C}\_{\rm R}\right]\{\dot{q}\_{\rm b}\} + \left(\left[\mathcal{K}\_{\rm b}\right] + \left[\mathcal{K}\_{\rm G}\right]\right)\{q\_{\rm b}\} + \left[\mathcal{K}\_{\rm b\phi}\right]\{\phi\} &= \{0\},\\ \left[\mathcal{K}\_{\phi\boldsymbol{b}}\right]\left\{q\_{\rm b}\right\} - \left[\mathcal{K}\_{\phi\boldsymbol{b}}\right]\{\phi\} &= \{Q\_{\rm d}\}. \end{aligned} \tag{70}$$

Eq. (70) is rewritten as:

$$\begin{aligned} \left[\mathcal{M}\_{\rm s}\right] \left\{ \ddot{q}\_{\rm s} \right\} + \left[\mathcal{K}\_{\rm s}\right] \left\{ q\_{\rm s} \right\} &= \left\{ F(t) \right\}, \\ \left[\mathcal{M}\_{bb}\right] \left\{ \ddot{q}\_{bb} \right\} + \left( \left[\mathcal{C}\_{\rm A}\right] + \left[\mathcal{C}\_{\rm R}\right] \right) \left\{ \dot{q}\_{bb} \right\} + \left( \left[\mathcal{K}^{\*}\right] + \left[\mathcal{K}\_{\rm G}\right] \right) \left\{ q\_{bb} \right\} &= \{ 0 \}. \end{aligned} \tag{71}$$

The overall geometric stiffness matrix [KG] is defined as follows:

• In the case of only tensile or compression plates (w = 0): Solving Eq. (71) helps us to present unknown displacement vector {qss}, and then stress vector:

$$\{\sigma\_{\alpha}\} = [A\_{\mathfrak{s}}][B\_{\mathfrak{s}}] \{q\_{\alpha}\},\tag{72}$$

where [As] and [Bs] are the stiffness coefficient matrix and strain-displacement matrix of the plane problem.

• In the case of bending plate (w 6¼ 0), the stress vector is:

$$\begin{aligned} \{\sigma\_{sb}\} &= \{\sigma\_{ts}\} + \{\sigma\_{bb}\},\\ \{\sigma\_{bb}\} &= [A\_b][B\_b] \{q\_{bb}\}, \end{aligned} \tag{73}$$

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

where [Ab] and [Bs] are the stiffness coefficient matrix and strain-displacement matrix of the plane bending problem.

Stability criteria [14]:

a function of external in-plane loads); {F(t)} is the in-plane load vector, {R} is the normal load vector, {Qel} is the vector containing the nodal charges and

> *Gx* � � <sup>þ</sup> *<sup>K</sup><sup>e</sup>*

> > ¼ ð

> > > ¼ ð

ð

*Ae N*<sup>0</sup> *<sup>x</sup> N*<sup>0</sup> *x* � � *N*<sup>0</sup> *x* � �*<sup>T</sup>*

*Ae N*<sup>0</sup> *<sup>y</sup> N*<sup>0</sup> *y* h i

*Ae N*<sup>0</sup> *xy N*<sup>0</sup> *x* � � *N*<sup>0</sup> *y* h i*<sup>T</sup>*

∂w *<sup>∂</sup><sup>y</sup>* <sup>¼</sup> <sup>∂</sup><sup>N</sup> *∂y* � �

When the plate is subjected to in-plane loads only ({R} = {0}), the in-plane stresses can lead to buckling, from Eqs. (63) and (64) the governing differential

½ �¼ *KG*

� � <sup>þ</sup> ð Þ ½ �þ *Kbb* ½ � *KG qbb*

� �f g*<sup>ϕ</sup>* <sup>¼</sup> f g *Qel :*

The overall geometric stiffness matrix [KG] is defined as follows:

*<sup>∂</sup><sup>x</sup>* ½ � *N x*ð Þ , *<sup>y</sup>* , *<sup>N</sup>*<sup>0</sup>

X *ne Ke G*

*Gy* h i <sup>þ</sup> *<sup>K</sup><sup>e</sup> Gxy* h i

*N*0 *y* h i*<sup>T</sup>*

*y* h i ¼ *∂ ∂y*

*qe bb* � � <sup>¼</sup> *<sup>N</sup>*<sup>0</sup>

� � <sup>þ</sup> *Kb<sup>ϕ</sup>*

� � <sup>þ</sup> *<sup>K</sup>* <sup>∗</sup> ð Þ ½ �þ ½ � *KG qbb*

• In the case of only tensile or compression plates (w = 0): Solving Eq. (71) helps us to present unknown displacement vector {qss}, and then stress vector:

where [As] and [Bs] are the stiffness coefficient matrix and strain-displacement

f g *σss* ¼ *As* ½ � *Bs* ½ � *qss*

f g¼ *σsb* f gþ *σss* f g *σbb* , f g *σbb* ¼ ½ � *Ab* ½ � *Bb qbb*

*dAe*,

*dAe*,

*dAe*,

*y* h i

� � (69)

� �f g*<sup>ϕ</sup>* <sup>¼</sup> f g<sup>0</sup> ,

� �, (72)

� �, (73)

� � <sup>¼</sup> f g<sup>0</sup> *:* (71)

*qe bb*

, (65)

½ � *N x*ð Þ , *y* , (67)

� � (68)

(66)

(70)

in-balance charges.

∂w <sup>∂</sup><sup>x</sup> <sup>¼</sup> <sup>∂</sup><sup>N</sup> ∂x � �

*Mss* ½ � €*qss*

½ � *Mbb* €*qbb*

The element coefficient matrices are:

*Perovskite and Piezoelectric Materials*

*Ke G* � � <sup>¼</sup> *<sup>K</sup><sup>e</sup>*

> *Ke Gx* � � <sup>¼</sup>

*Ke Gy* h i

*Ke Gxy* h i

*x* � � <sup>¼</sup> *<sup>∂</sup>*

*x* � � *q<sup>e</sup> bb* � �,

equations of motion of the damped system may be written as:

� � <sup>¼</sup> f g *F t*ð Þ ,

• In the case of bending plate (w 6¼ 0), the stress vector is:

� � <sup>¼</sup> f g *F t*ð Þ ,

� � � *<sup>K</sup>ϕϕ*

� � <sup>þ</sup> ð Þ ½ �þ *CA* ½ � *CR <sup>q</sup>*\_ *bb*

where

in which *N*<sup>0</sup>

*qe bb* � � <sup>¼</sup> *<sup>N</sup>*<sup>0</sup>

*4.2.3 Dynamic stability analysis*

� � <sup>þ</sup> *Kss* ½ � *qss*

*Kϕ<sup>b</sup>* � � *qbb*

Eq. (70) is rewritten as:

*Mss* ½ � €*qss*

½ � *Mbb* €*qbb*

matrix of the plane problem.

**206**

� � <sup>þ</sup> ½ � *CR <sup>q</sup>*\_ *bb*

� � <sup>þ</sup> *Kss* ½ � *qss*

	- *Plate is considered to be stable if the maximum bending deflection is three times smaller than the plate's thickness: Eq. (71) has the solution (wi)max satisfying the condition* 0≤ wi j jmax <3*h, where wi is the deflection of the plate at node number i*.
	- *Plate is called to be in critical status if the maximum bending deflection of the plate is three times equal to the plate's thickness. Eq. (71) has the solution (wi)max satisfying the condition* wi j jmax ¼ 3*h*.
	- *Plate is called to be at buckling if the maximum deflection of the plate is three times larger than the plate's thickness: Eq. (71) has the solution (wi)max satisfying the condition* wi j jmax>3*h*.

The identification of critical forces is carried out by the iterative method.

#### *4.2.4 Iterative algorithm*

Step 1. Defining the matrices, the external load vector and errors of load iterations.

Step 2. Solving Eq. (71) to present unknown displacement vector, {qss} and the stress vector is defined by (72), updating the geometric stiffness matrix [KG].

Step 3. Solving Eq. (71) to present unknown bending displacement vector {qbb}, and then testing stability conditions.

� If for all w*<sup>i</sup>* j j ¼ 0: increase load, recalculate from step 2;

� If at least one value w*<sup>i</sup>* j j 6¼ 0:

+ In case: 0 < w*<sup>i</sup>* j j*<sup>m</sup>*ax <3*h*: Define stress vector by Eq. (73), update the geometric stiffness matrix [KG]. Increase load, recalculate from step 2;

+ In case: 0 ≤ <sup>w</sup>*<sup>i</sup>* j j j jmax�3*<sup>h</sup>* w*<sup>i</sup>* j jmax ≤*εD*: Critical load p = pcr. End.

#### *4.2.5 Numerical analysis*

Stability analysis of piezoelectric composite plate with dimensions a � b � h, where a = 0.25 m, b = 0.30 m, h = 0.002 m. Piezoelectric composite plate is composed of three layers, in which two layers of piezoelectric PZT-5A at its top and bottom are considered, each layer thickness hp = 0.00075 m; the middle layer material is Graphite/Epoxy material, with thickness h1 = 0.0005 m. The material properties for graphite/epoxy and PZT-5A are shown in Section 5.1 above. One short edge of the plate is clamped, the other three edges are free. The in-plane halfsine load is evenly distributed on the short edge of the plate: p(t) = p0sin(2πft), where p0 is the amplitude of load, f = 1/T = 1/0.01 = 100 Hz (0 ≤ t ≤ T/2 = 0.005 s) is the excitation frequency, voltage applied V = 50 V. The iterative error of the load ε<sup>D</sup> = 0.02% is chosen.

**Figure 7** shows the time history of the vertical displacement at the plate centroid over the plate thickness when a voltage of 0, 50, 100, 150 and 200 V is applied. The

The results show that the voltage applied to the piezoelectric layers affects the stability of the plate. As the voltage increases, the critical load of the plate also

When the amplitude of the load changes from 0.25pcr to 1.5pcr (where pcr is the amplitude of the critical load), a voltage of 50 V is applied to the actuator layer of

The results show the time history response of the vertical displacement at the

*Time history of the vertical displacement at the plate centroid over the plate thickness when p0 = 0.25pcr, 0.5pcr,*

relation between critical load and voltages is shown in **Figure 8**.

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

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

plate centroid over the plate thickness as seen in **Figure 9**.

increases.

the plate.

**Figure 8.**

**Figure 9.**

**209**

*0.75pcr, 1.0pcr, 1.25pcr, and 1.5pcr.*

*Critical load-voltage relation.*

Consider two cases: with damping (ξ = 0.05, Gv = 0.5, Gd = 15) and without damping (ξ = 0.0, Gv = 0.0, Gd = 15). The response of vertical displacement at the plate centroid over the plate thickness for the two cases is shown in **Figure 6**.

The results show that the critical load of the plate with damping is larger than that without damping. In the two cases above, the critical load rises by 6.8%.

Analyze the stability of the plate with damping when a voltage of 200, 150, 100, 50, 0, 50, 100, 150 and 200 V is applied to the actuator layer of the piezoelectric composite plate.

**Figure 6.** *Vertical displacement response at the plate centroid over the plate thickness.*

**Figure 7.** *Vertical displacement response at the plate centroid over the plate thickness.*

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

**Figure 7** shows the time history of the vertical displacement at the plate centroid over the plate thickness when a voltage of 0, 50, 100, 150 and 200 V is applied. The relation between critical load and voltages is shown in **Figure 8**.

The results show that the voltage applied to the piezoelectric layers affects the stability of the plate. As the voltage increases, the critical load of the plate also increases.

When the amplitude of the load changes from 0.25pcr to 1.5pcr (where pcr is the amplitude of the critical load), a voltage of 50 V is applied to the actuator layer of the plate.

The results show the time history response of the vertical displacement at the plate centroid over the plate thickness as seen in **Figure 9**.

**Figure 8.** *Critical load-voltage relation.*

sine load is evenly distributed on the short edge of the plate: p(t) = p0sin(2πft), where p0 is the amplitude of load, f = 1/T = 1/0.01 = 100 Hz (0 ≤ t ≤ T/2 = 0.005 s) is the excitation frequency, voltage applied V = 50 V. The iterative error of the load

Consider two cases: with damping (ξ = 0.05, Gv = 0.5, Gd = 15) and without damping (ξ = 0.0, Gv = 0.0, Gd = 15). The response of vertical displacement at the plate centroid over the plate thickness for the two cases is shown in **Figure 6**. The results show that the critical load of the plate with damping is larger than

Analyze the stability of the plate with damping when a voltage of 200, 150,

that without damping. In the two cases above, the critical load rises by 6.8%.

100, 50, 0, 50, 100, 150 and 200 V is applied to the actuator layer of the

*Vertical displacement response at the plate centroid over the plate thickness.*

*Vertical displacement response at the plate centroid over the plate thickness.*

ε<sup>D</sup> = 0.02% is chosen.

*Perovskite and Piezoelectric Materials*

piezoelectric composite plate.

**Figure 6.**

**Figure 7.**

**208**

#### **Figure 9.**

*Time history of the vertical displacement at the plate centroid over the plate thickness when p0 = 0.25pcr, 0.5pcr, 0.75pcr, 1.0pcr, 1.25pcr, and 1.5pcr.*
