6 Functionally graded piezoelectric material porous plates (FGPMP)
Tải bản đầy đủ - 0trang
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 ) ,
60
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)