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u and l denote the material properties of the upper (material 1) and lower surfaces

(material 2), respectively, and α is the porosity volume fraction.

Type of uneven distribution, FGPMP-II, the porosities are concentrated

around the cross-section middle-surface and the amount of porosity discharges at the

top and bottom of the cross-section. In this case, the effective material properties are

computed by:

g

2z

( ciju − cijl ) hz + 12 + cijl − α2 ( ciju + cijl ) 1 − h ;

( i, j ) = {(1,1) , (1, 2 ) , (1,3) , ( 3,3) , ( 5,5) , ( 6,6 )}

cij ( z ) =

g

2z

α

z 1

eij ( z ) = ( e − e ) + + eijl − ( eiju + eijl ) 1 −

2

h

h 2

u

ij

l

ij

; ( i, j ) =

g

2z

α

z 1

kij ( z ) = ( k − k ) + + kijl − ( kiju + kijl ) 1 −

; ( i, j ) =

h

2

h 2

u

ij

l

ij

{( 3,1) , ( 3,3) , ( 3,5)}

(3. 38)

{(1,1) , ( 3,3)}

g

2z

α

z 1

ρ ( z ) = ( ρ − ρ ) + + ρ l − ( ρ u + ρ l ) 1 −

2

h

h 2

u

l

Figure 3.9. Geometry and cross sections of a FGPMP plate made of PZT-4/PZT-5H.

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In order to show the influence of porosity volume fraction on material

properties, the variation of elastic coefficient c11 of porous FGPM plate made of PZT4/PZT-5H versus the thickness is studied with various power index values as depicted

in Figure 3.10. It can be seen that the elastic coefficient of perfect FGPM, α = 0 , is

continuous through the top surface (PZT-4 rich) to the bottom surface (PZT-5H rich)

as shown in Figure 3.10a. As g = 0, the elastic coefficient is constant through the plate

thickness. The profiles of c11 are also plotted in Figure 3.10b and Figure 3.10c for

porous FGPMP-I and FGPMP-II, respectively. As seen, there has the same profile for

the perfect FGPM and FGPMP-I type with porosities. However, the magnitude of the

elastic coefficient of porous FGPMP-I is lower than that of perfect FGPM. Therefore,

the stiffness of the FGPMP is decreased with the presence of the porous parameter.

Moreover, when the porosities are distributed around the cross section mid-zone and

the amount of porosity diminishes on the top and bottom of the cross-section,

FGPMP-II type, the elastic coefficient is maximum on the bottom and top surface and

decreases towards middle zone direction as indicated in Figure 3.10c. Figure 3.10d

displays the influence of porosities on the elastic coefficient. It is found that the elastic

coefficient’s amplitude of FGPMP-II plate is equal to that of perfect FGPM on the

bottom and top surface, and equal to that of FGPMP-I plate at the mid-surface.

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a) Perfect FGPM

b) FGPMP-I

d) FGPMP, g=0.1

c) FGPMP-II

Figure 3.10. Variation of elastic coefficient c11 of FGPMP plate made of PZT4/PZT-5H with α = 0.2 .

3.7

Concluding remarks

In this chapter, an overview of plate theories used in all the next chapters are

given. In addition, the fundamentals of several materials are provided such as

laminated composite plates, piezoelectric laminated composite plates, piezoelectric

functionally graded porous plates reinforced by graphene platelets and functionally

graded piezoelectric material porous plates.

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Chapter 4

ANALYZE AND CONTROL THE LINEAR

RESPONSES OF THE PIEZOELECTRIC

LAMINATED COMPOSITE PLATES

4.1

Overview

In this chapter, an isogeometric finite element formulation based on Bézier

extraction for the non-uniform rational B-splines (NURBS) in combination with a

generalized unconstrained higher-order shear deformation theory (UHSDT) is

presented for analysis of static, free vibration and transient responses of plates. This

chapter based on two papers in refs. [98, 148]. Two types of plates such as the

laminated composite plates and the piezoelectric laminated composite plates are

studied. In addition, for the piezoelectric laminated composite plates, the active

response control of structures is also investigated. The displacement field is

approximated according to the proposed model and the linear transient formulation

for plates is solved by Newmark time integration. The presented method relaxes zeroshear stresses at the top and bottom surfaces of the plates and no shear correction

factors are used. NURBS can be written in terms of Bernstein polynomials and the

Bézier extraction operator as section 2.9. Through the thickness of each piezoelectric

layer, the electric potential variation is assumed to be linear. A closed-loop system is

used for active control of the piezoelectric laminated composite plates. The accuracy

and reliability of the proposed method are verified by comparing its numerical

predictions with those of other available numerical approaches.

4.2 Laminated composite plate formulation based on Bézier extraction for

NURBS

4.2.1 The weak form for laminated composite plates

The unconstrained theory based on HSDT (UHSDT) which is presented in 3.2.2

section is used for model 1 and model 2, in which f ( z ) = arctan( z ) and f(z) = sin( z ) ,

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respectively. Therefore, the UHSDT can be called the unconstrained inverse

trigonometric shear deformation theory (UITSDT) and the unconstrained sinusoidal

shear deformation theory (USSDT), respectively. By the assumption that a linear

constitutive relationship is employed for the analysis of the laminated composite plate

embedded in piezoelectric layers, the formulation for each field is approximated

separately.

The in-plane strain vector ε p is thus expressed by the following equation

[ε xx ε yy γ xy ]T= ε 0 + zε1 + f ( z )ε 2

ε=

p

(4. 1)

and the transverse shear strain vector γ has the following form

=

γ γ xz

T

γ yz =

ε 0s + f ' ( z )ε1s

(4. 2)

where f ' ( z ) is derivative of f(z) function and

u2, x

u0, x

u1, x

u1 + w, x

ε 0 = v0, y , ε1 = v1, y , ε 2 = v2, y , ε 0s =

,

+

v

w

y

1

,

v2, x + u2, y

v0, x + u0, y

v1, x + u1, y

(4. 3)

u

ε1s = 2

v2

A weak form of the static model for the plates under transverse loading q0 can be

written as

∫

Ω

δεTp Dε p dΩ + ∫ δγ T =

D s γ dΩ

Ω

∫

Ω

δ wq0dΩ

(4. 4)

where q0 is the transverse loading per unit area.

From Hooke’s law and the linear strains given by Eqs.(4. 1) and (4. 2), the stress

is computed by

σ p

=

σ =

τ

D 0 ε p

= cε

Ds γ

0

ε

c

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(4. 5)

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