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Buckling analysis of higher order graded smart piezoelectric plates with porosities resting on elastic foundation
Author’s Accepted Manuscript Buckling analysis of higher order graded smart piezoelectric plates with porosities resting on elastic foundation M.R. Barati, M.H. Sadr, A.M. Zenkour www.elsevier.com/locate/ijmecsci
PII: DOI: Reference:
S0020-7403(16)30218-1 http://dx.doi.org/10.1016/j.ijmecsci.2016.09.012 MS3417
To appear in: International Journal of Mechanical Sciences Received date: 31 March 2016 Revised date: 28 July 2016 Accepted date: 8 September 2016 Cite this article as: M.R. Barati, M.H. Sadr and A.M. Zenkour, Buckling analysis of higher order graded smart piezoelectric plates with porosities resting on elastic f o u n d a t i o n , International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2016.09.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Buckling analysis of higher order graded smart piezoelectric plates with porosities resting on elastic foundation M.R. Barati*1, M.H. Sadr1, A.M. Zenkour2,3 1
Department of Aerospace Engineering and Center of Excellence in Computational Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran
2
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 3
Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh 33516, Egypt *
Corresponding author: [email protected]
Abstract In this study, examination of buckling behavior of functionally graded piezoelectric (FGP) plates with porosities is conducted employing a refined four-variable plate theory. By capturing shear deformation effects, the present inverse trigonometric theory is needless of shear correction factor. Electro-elastic material properties of porous FGP plate vary across the thickness based on modified power-law model. Implementing an analytical approach which satisfies different boundary conditions, governing equations derived from Hamilton’s principle are solved. The obtained results are compared with those provided in literature. It is indicated that the buckling behavior of piezoelectric plates is significantly influenced by elastic foundation parameters, external voltage, porosity distribution, power-law index, boundary conditions and aspect ratio Keywords: Electro-mechanical buckling, Four-variable plate theory, Functionally graded piezoelectric plate, Porosities, Analytical method.
1. Introduction Multi-phase composite materials, which are known as functionally graded materials (FGMs) possess spatially graded microstructure to achieve particular mechanical properties to suit the functionality of structures. The gradually composition variations of the material constituents from one surface to another provide a proper solution to the problem of induced transverse shear stresses due to the two bonded dissimilar materials with high difference in material properties. Also, due to containing outstanding mechanical properties, FGMs are appropriate for design of engineering structures. Moreover, piezoelectric materials are known as a type of smart structures
which can create electricity when exposed to mechanical stresses. Also, they will work in reverse, producing a strain by the application of an electric field. Until now, several articles have investigated mechanical behavior of functionally graded piezoelectric (FGP) structures. Among them, Kargarnovin et al. (2007) performed vibration analysis of piezo-patched FGM plates under an electric field. Ebrahimi and Rastgoo (2008) researched vibrational characteristics of FG piezoelectric annular plates. Free and forced vibration response of thermo-piezo-electrically actuated higher order graded beams is investigated by Doroushi et al. (2011). Also, stability analysis of FGPM beams under thermo-electrical field is conducted by Kiani et al. (2011). The effects of temperature and electric fields on vibrational characteristics of a post-buckled FGPM beam are studied by Komijani et al. (2013). Jadhav and Bajoria (2013) investigated free and forced vibration control of FG piezoelectric plate under electro-mechanical loading. Also, Rouzegar and Abad (2015) studied vibration of a FGM plate with piezoelectric patches using four-variable refined plate theory. Barati et al. (2016) examined electro-mechanical vibration of smart piezoelectric FG plates with porosities according to a refined four-variable theory. During manufacturing process, porosities are found in FG materials. Porous FG materials are made of two components which are solid and liquid such as woods and stones. Porosity effect on vibration of FGM beams is examined by Wattanasakulpong and Chaikittiratana (2015) via Timoshenko beam theory applying Chebyshev collocation method. Ebrahimi and Zia (2015) studied large amplitude nonlinear vibration analysis of functionally graded Timoshenko beams with porosities. Ebrahimi et al. (2016) investigated thermal vibration response of FGM beams with porosities. They stated that the volume fraction of porosity has a notable effect on natural frequencies for every material graduation. Mechab et al. (2016) investigated the effects of porosities in functionally graded small scale plates resting on Winkler–Pasternak elastic foundations. They applied a modified rule of mixture to approximate material distribution of the functionally graded plates considering the porosity volume fraction. As one can see, some papers have studied mechanical behaviors of FGM structures considering porosities. But, examination of porosity effect on buckling of FG piezoelectric plates is a novel topic which not reported till to now. In this context, the classical plate theory model (CPL) or Kirchoff theory, which ignores the shear deformation effect, only give acceptable results for thin plates. Then the first order shear deformation (FSDT) based in Reissner and Mindlin was utilized but this theory needs a shear correction factor, which is difficult to calculate. Therefore higher-order shear deformation theories (HSDTs) were introduced to avoid the shear correction factors. Further, The HSDTs can be developed using polynomial or non-polynomial thickness functions. Zenkour (2009, 2015) employed a refined sinusoidal theory for functionally graded and composite plates embedded in elastic medium. Moreover, Grover et al. (2015) suggested a secant function based theory for analysis of composite plates. Recently, inverse trigonometric shear deformation theory is recommended by Sahoo and Singh (2013) and Thai et al. (2014). Also, an inverse hyperbolic theory for analysis of plates is proposed by Nguyen et al. (2014). Kulkarni et al. (2015) investigated bending and buckling of FGM plates using refined inverse cotangential theory.
Zenkour et al. (2015a, b) studied thermo-Mechanical bending response of exponentially graded and layered plates resting on elastic foundations using the sinusoidal theory. Most recently, Barati et al. (2016) presented buckling analysis of embedded small scale FG plates under temperature distributions employing an inverse cotangential shear strain function. Also, Barati and Shahverdi (2016) presented an analytical solution for thermal vibration of compositionally graded nanoplates with arbitrary boundary conditions based on physical neutral surface position. This paper investigates buckling response of higher-order shear deformable plates made of functionally graded piezoelectric (FGP) materials with porosities embedded in an elastic foundation. Material properties of FGP plate change continuously in thickness direction based on power-law model. Employing Hamilton’s principle, the governing equations of FGP plate embedded in elastic foundation are obtained. To predict buckling behavior of embedded FGP plates, an analytical solution is applied to solve the governing equations. Numerical results demonstrate the influences of various parameters such as elastic foundation, external electric voltage, power-law index, porosity coefficient and aspect ratio on the buckling loads of porous FGP plates.
2. Theoretical formulations 2.1. The material properties of porous FG piezoelectric plates Consider a functionally graded plate composed of PZT-4 and PZT-5H piezoelectric materials exposed to an electric potential (x , z ,t ) , as shown in Figure 1. The effective material properties of the porous FGM-I piezoelectric plate are variable across the thickness direction based on the modified power-law model as:
hz 12 cijl ciju cijl 2 (i, j ) {(1,1), (1, 2), (1,3), (3,3), (5,5), (6, 6)} p
cij ( z ) ciju cijl
eij ( z ) e
u ij
hz 12 eijl eiju eijl 2
(1a)
p
eijl
z 1 kij ( z ) kiju kijl kijl kiju kijl 2 h 2
(i, j ) {(3,1), (3,3), (3,5)}
(1b)
(i, j ) {(1,1), (3,3)}
(1c)
p
where
p
is an exponent and evaluates the material distribution across the plate thickness and u
and l are correspond to the material properties of the top and bottom surfaces, respectively; is the porosity volume fraction. Also, C ijkl , e kij and k ik are elastic, piezoelectric and dielectric constant, respectively. It must be noted that, the top surface at fully PZT-4 , whereas the bottom surface ( z
h / 2 )
z h / 2
of porous FGP plate is
is fully PZT-5H . Moreover, when the
porosities are distributed around the cross section mid-zone and the amount of porosity
diminishes at the top and bottom of the cross-section. In this case (FGM-II), the effective material properties can be defined by: p 2z z 1 cij ( z ) c c cijl ciju cijl (1 ) 2 h h 2
u ij
l ij
u ij
eijl
eij ( z ) e
(2a)
p 2z z 1 h 2 eijl 2 eiju eijl (1 h )
(2b)
p 2z z 1 kij ( z ) kiju kijl kijl kiju kijl (1 ) 2 h h 2
(2c)
2.2 Basic assumptions of four-variable theory The assumptions of the present theory are as follows: i. The displacements are small in comparison with the plate thickness and, therefore, strains involved are infinitesimal. ii. The transverse normal stress z is negligible in comparison with in-plane stresses x and y . iii. The transverse displacement u3 includes two components of bending wb and shear ws . These
components
are
functions u3 ( x, y, z ) wb ( x, y) ws ( x, y) .
of
coordinates
x
and
y
only.
That
is
iv. The in-plane displacements u1 and u2 consist of extension, bending, and shear components. That is u1 u ub us and u2 v vb vs . - The bending components ub and vb are assumed to be similar to the displacements given by the classical plate theory. Therefore, the expressions for ub and vb are given, respectively, by
ub z
w wb and vb z b . y x
- The shear components us and vs give rise, in conjunction with ws , to the sinusoidal variations of shear strains xz , yz and hence to shear stresses xz , yz through the thickness h of the plate in such a way that shear stresses xz , yz are zero at the top and bottom surfaces of the plate. Consequently, the expressions for us
vs f ( z )
ws . y
2.3. Kinematic relations
and vs can be given as us f ( z )
ws x
and
Here, a non-polynomial shear deformation theory with four unknowns is employed to model the plates composed from porous functionally graded piezoelectric materials. The displacement field of present theory can be expressed by:
u1 x, y, z u x, y z
wb w f ( z) s x x
(3)
u 2 x, y , z v x , y z
wb w f ( z) s y y
(4)
u3 ( x, y, z ) wb ( x, y) ws ( x, y)
(5)
where the theory possesses an inverse cotangential function stated by (Kulkarni et al. 2015):
f ( z) g ( z) z ; g ( z ) cot 1 (rh / z )
(6)
in which
4r / [h(4r 2 1)] with r=0.46 Also,
u
(7)
and v are displacements of the mid-surface and wb and ws are the bending and shear
transverse displacement, respectively. The criteria of continuity and differentiability (Reddy, 1984) are satisfied for the choice of shear strain shape function. The parameter is a constant evaluated by implementing the transverse shear stress boundary conditions so that the transverse shear stresses at the boundary vanish. The parameter r is optimized in the post-processing technique. By comparing the results of newly developed theories for various values of shape parameter r with the exact solution and for various side-to-thickness ratios, the optimized value of r is ascertained. Grover et al. (2013) obtained non-dimensional deflection of four-layer symmetric cross-ply plate for a wide range of r parameter, and compared with the exact solution for the side-to-thickness ratio as 4 and 10. So, they selected an optimized value of r=0.46 for the present theory. The distribution of electric potential across the thickness is expressed by the following relation: ( x, y, z ) cos ( z ) ( x, y)
where / h and
V
2z V h
(8)
is the applied electric voltage applied; and ( x,, y, t ) denotes the function of
the electrical potential in spatial coordinate. Nonzero strains of the present theory are expressed by:
b s x0 x x s yz x 0 b s yz z f , g y y y y s 0 b s xz xz xy xy xy xy
(9)
Where 2w 2 b u ws 2 2 b x s x 0 x x s ws x x y 2 2 wb ws yz 0 v b , sy , y , y 2 2 s ws y y y x z 0 b s xy xy u v x 2 w xy 2 w b s y x 2 xy 2 xy
(10)
Using Eq. (8), the electric field ( Ex , Ey , Ez ) can be stated by:
, x E y , y cos ( z ) , y
(11)
Ex , x cos ( z )
Ez , z sin ( z )
(12)
2V h
(13)
Employing Hamilton’s principle in the following form, the governing equations can be obtained:
t
( 0
where
S
W ) dt 0
S
(14)
and W denote strain energy and work of external forces. The strain energy of present
model is obtained as:
S ij ij dV v
( v
x
x
y y xy xy yz yz xz xz Dx Ex Dy E y Dz Ez ) dV
where D i is electric displacement. Inserting Eqs.(9) and (10) into Eq.(15) gives:
(15)
S
a
0
N xy (
a
0
b
0
[Nx
2 2 2 2 wb u v s ws b wb s ws M xb M N M M x y y y x x 2 x 2 y y 2 y 2
2 2 ws ws u v b wb s ws ) 2 M xy 2 M xy Qyz Qxz ]dxdy y x xy xy y x
b
h/2
0
h/2
(16)
Dx cos( z ) Dz sin( z ) dzdxdy Dy cos( z ) x y
Where the variables existing in above equation are defined by:
( Ni , M ib , M is ) (1, z, f ) i dA, i ( x, y, xy ) A
Qi g i dA, i ( xz , yz )
(17)
A
The work done by applied forces can be expressed by: ( wb ws ) ( wb ws ) ( wb ws ) ( wb ws ) N y0 0 0 x x y y ( wb ws ) ( wb ws ) ( wb ws ) ( wb ws ) 2 N xy0 kw ( wb ws ) k p )dxdy x y x y
W
a
b
( N x0
(18)
where N x0 , N y0 , N xy0 are in-plane applied loads and kw , k p are elastic foundation parameters. For a piezo-electrically actuated porous FGPM plate, the constitutive relations may be expressed as:
ij Cijkl kl ekij Ek
(19)
Di eikl kl kik Ek
(20)
where ij , ij and E i denote the stress, strain and electric field components, respectively. Finally, the equivalent form of Eqs. (19) and (20) are:
xx c11 xx c12 yy e31 Ez
(21)
yy c12 xx c11 yy e31 Ez
(22)
xy c66 xy xz c55 xz e15 Ex
(23)
yz c55 yz e15 Ex
(25)
Dx e15 xz k11Ex
(26)
Dy e15 yz k11E y
(27)
Dz e31 xx e31 yy k33 Ez
(28)
(24)
where cij , eij and kij are reduced constants for the FGP plate under the plane stress state and are given as:
c132 c132 c11 c11 , c12 c12 , c66 c66 c33 c33
(29)
2 c13e33 e33 e31 e31 , k11 k11 , k33 k33 c33 c33
The following governing equations are obtained by inserting Eqs. (16)-(18) in Eq. (14) when the coefficients of u, v, wb and ws are equal to zero: N x N xy 0 x y N xy x
N y y
(30)
0
(31)
b 2 M xy 2 M yb 2 M xb 2 ( N E N b ) 2 ( wb ws ) kw ( wb ws ) k p 2 ( wb ws ) 0 2 2 x xy y
s 2 M xy 2 M ys Qxz Qyz 2 M xs 2 ( N E N b ) 2 ( wb ws ) x 2 xy y 2 x y
(32)
(33)
kw ( wb ws ) k p ( wb ws ) 0 2
Dy D cos( z ) x cos( z ) sin( z) Dz h /2 x y
h /2
dz 0
(34)
Where N x0 N y0 N E , N xy0 0 and electric load can be expressed as (Doroushi et al. 2011, Jadhav and Bajoria 2013):
N E
h/ 2
h/ 2
e31
2V dz h
(35)
By integrating Eq. (21)-(28) over the area of plate cross-section, the following relations for the FGP plate can be obtained:
N x A11 N y A12 0 N xy
A12 A22 0
u x B11 0 v 0 B12 y 0 A66 u v y x
B12 B22 0
2 wb x 2 s B 11 0 2 w s b 0 B12 y 2 B66 0 2 wb 2 xy
s B12 s B22 0
2 ws x 2 e A 0 31 2 ws e 0 A31 2 y s 0 B66 2w s 2 x y
(36)
b M x B11 b M y B12 b 0 M xy
s Bs M x 11 s s M y B12 s 0 M xy
Qxz Q yz
u x D11 0 v 0 D12 y B66 u v 0 y x
B12 B22 0
u Ds 0 x 11 s v 0 D12 y s 0 B66 u v y x
s B12 s B22 0
ws 0 x s w s A55 y
A 0
s 44
D12 D22 0
s D12 s D22 0
A15e
2 wb x 2 s D 11 0 2 w s b 0 D12 y 2 D66 0 2 wb 2 xy
2 wb 2 Hs 0 x 11 2w s b 0 H 2 12 y s 0 D66 2w b 2 xy
s D12 s D22 0
s H12 s H 22 0
2 ws 2 Ee 0 x 31 2 w e s 0 E31 2 y s 0 D66 2w s 2 xy
2 ws 2 F e 0 x 31 2 ws e 0 F31 2 y s 0 H 66 2w s 2 xy
x y
(37)
(38)
(39)
ws
y
y
Dx e x e x cos( z ) dz E F 15 w 11 h/2 Dy s
(40)
h /2
D sin( z)dz A h /2
h /2
e 31
z
(
u v e ) E31 2 wb F31e 2 ws F33e x y
(41)
in which the cross-sectional rigidities are defined as follows: c A11 , B11 , B11s , D11 , D11s , H11s 11 h /2 s s s 2 2 A12 , B12 , B12 , D12 , D12 , H12 h /2 c12 (1, z , f , z , zf , f )dz s s s c66 A66 , B66 , B66 , D66 , D66 , H 66
(42)
A
, E31e , F31e
(43)
A
, E15e
e 31
e 15
h /2
h /2
h /2
h /2
e15 cos( z ) 1, gdz
F , F k e 11
e 33
h /2
h /2
s s A44 A55
e31 sin( z ) 1, z, f dz
h /2
11
(44)
cos2 ( z ), k33 2 sin 2 ( z) dz
c g 2 dz
h /2 55
(45) (46)
The governing equations for a refined four-variable porous FGP plate in terms of the displacement can be derived by substituting Eqs. (36)-(41), into Eqs. (30)-(34) as follows: A11
3 3 wb 3 wb 2u 2u 2v s ws A ( A A ) B ( B 2 B ) B 66 12 66 11 12 66 11 x 2 y 2 xy x 3 xy 2 x 3
s ( B12s 2 B66 )
A66
3 3 wb 3 wb 2v 2v 2u s ws A ( A A ) B ( B 2 B ) B 22 12 66 22 12 66 22 x 2 y 2 xy y 3 x 2y y 3
s ( B12s 2 B66 )
B11
3 ws e A31 0 2 xy x
2( D12 2 D66 )
B11s
(48)
3 ws e A31 0 2 x y y
4 wb 3u 3u 3v 3v e ( B 2 B ) ( B 2 B ) B D E31 2 12 66 12 66 22 11 x 3 xy 2 x 2 y y 3 x 4
s D22
(47)
(49)
4 4 wb 4 wb 4 ws s ws s s D D 2( D 2 D ) 22 11 12 66 x 4 y 2 y 4 x 4 x 4 y 2
4 ws ( N E N b ) 2 ( wb ws ) kw ( wb ws ) k p 2 ( wb ws ) 0 y 4
4 2 3 3u 3u 3v s s s s s v s wb s ws ( B 2 B ) ( B 2 B ) B D A 12 66 12 66 22 11 55 x 3 xy 2 x 2 y y 3 x 4 x 2 4 4 2 ws 4 wb 4 ws s s s wb s ws s s 2( D 2 D ) D H 2( H 2 H ) 12 66 22 11 12 66 y 2 x 4 y 2 y 4 x 4 x 4 y 2
s A44
s H 22
e A31 (
(50)
4 ws F31e 2 ( N E N b ) 2 ( wb ws ) k w ( wb ws ) k p 2 ( wb ws ) A15e 2 0 y 4
u v ) E31e 2 wb F31e 2 ws E15e 2 ws F11e 2 F33e 0 x y
(51)
3. Solution procedure Here, an exact solution of the governing equations for buckling of a FG piezoelectric plate with simply-supported (S), clamped (C) or free (F) edges or combinations of these boundary conditions is presented which they are given as (Sobhy, 2013):
Simply-supported (S):
wb ws N x M x 0 at x=0,a wb ws N y M y 0 at y=0,b
Clamped (C):
u v wb ws 0 at x=0,a and y=0,b Free (F):
M x M xy Qxz 0 at x=0,a M y M xy Qyz 0 at y=0,b
The following expansions have been assumed to satisfy various boundary conditions:
u Amn mn ( x, y )
(52)
m 1 n 1
v Bmn mn ( x, y )
(53)
m 1 n 1
wb Cmn mn ( x, y)
(54)
m 1 n 1
ws Dmn mn ( x, y)
(55)
m 1 n 1
Emn mn ( x, y)
(56)
m 1 n 1
where ( Amn , Bmn , Cmn , Dmn , Emn ) are the unknown coefficients and ( mn , mn , mn , mn , mn ) are admissible functions which are expressed as (Sobhy, 2014):
Fm ( x) Fn ( y) x F ( y) Tmn ( x, y) Fm ( x) n y mn ( x, y)
(57)
(58)
mn ( x, y) Fm ( x) Fn ( y)
(59)
mn ( x, y) Fm ( x) Fn ( y)
(60)
mn ( x, y) Fm ( x) Fn ( y)
(61)
Where the function Fm ( x) is presented for considered boundary conditions as ( m m / a ): SS: Fm ( x) sin(m x) CS: Fm ( x) sin m x [cos m x 1] CC: Fm ( x) sin (m x ) 2
(62)
FF: Fm ( x) cos (m x)[sin (m x) 1] 2
2
However, one can obtain the function Fn ( y ) by replacing x, m , m and a in the above equations by y, n ,n and b respectively as ( n n / b ): SS: Fn ( y) sin(n y) CS: Fn ( y) sin n y [cos n y 1] CC: Fn ( y) sin (n y) 2
(63)
FF: Fn ( y) cos (n y)[sin (n y) 1] 2
2
Inserting Eqs. (52) - (56) into Eqs. (47) -(51) respectively, leads to: k1,1 k2,1 k3,1 k 4,1 k5,1
Where
k1,2 k2,2 k3,2 k4,2 k5,2
k1,3 k2,3 k3,3 k4,3 k5,3
k1,4 k2,4 k3,4 k4,4 k5,4
k1,5 Amn k2,5 Bmn k3,5 Cbmn 0 k4,5 Dsmn k5,5 E mn
(64)
k1,1 A11 12 A66 8 , k1,2 A12 A66 8 , k1,3 B11 12 B12 2 B66 8 , k1,4 B11s 12 B12s 2 B66s 8 ,
k1,5 A31e 9 , k2,2 A22 4 A66 10 , k2,3 B22 4 B12 2 B66 10 , k2,4 B22s 4 B12s 2 B66s 10 ,
k2,5 A31e 3 , k3,5 ( E31e ) 3 9 k3,3 D11 13 2 D12 2 D66 11 D22 5 k w 1 ( N E N b k p ) 3 9 , k3,4 D11s 13 2 D12s 2 D66s 11 D22s 5 k w 1 ( N E N b k p ) 3 9 , k4,4 H11s 13 2 H12s 2 H 66s 11 H 22s 5 A44s 9 A55s 3 k w 1 ( N E N b k p ) 3 9 , k4,5 ( E31e E15e ) 3 9 , k5,5 F11e 3 9 F33e 1.
(65) In which
( 1 , 3 , 5 )
a
0
b
0
( 9 , 11 , 13 )
a
0
( 6 , 8 , 12 )
a
( 2 , 4 , 10 )
a
0
0
( Fm Fn , Fm Fn'' , Fm Fn'''' ) Fm Fn dxdx
b
0
b
b
0
0
( Fm'' Fn , Fm'' Fn'' , Fm'''' Fn ) Fm Fn dxdy
( Fm' Fn , Fm' Fn'' , Fm''' Fn ) Fm' Fn dxdy ( Fm Fn' , Fm Fn''' , Fm'' Fn' ) Fm Fn' dxdy
(66)
Also, the dimensionless form of buckling load and foundation parameters are adopted as: 2 k pa2 kw a 4 b a u 3 ˆ NN , Kw , Kp , Dc C11 h Dc Dc Dc
(67)
4. Numerical results and discussions 4.1 Validation In this section, numerical and illustrative examples are provided to explore electro-mechanical buckling response of FGP plates on elastic foundation with porosities employing a higher order trigonometric plate theory. The electro-elastic material properties of the porous FGP plate vary across the thickness according to the modified power-law model. The influences of electric
voltage, various boundary conditions, porosity coefficient, power-law index and aspect ratio on the buckling behavior of the porous FGP plate are explored. The correctness of the presented buckling results are compared with those of first and third order plate theories presented respectively by Mohammadi et al. (2010) and Thai and Choi (2012) for a perfect FGM plate and the results are tabulated in Table 2. It is indicated that the present plate model and solution procedure can accurately predict buckling loads of FGM plates. For comparison study, the material properties are selected as:
= 420 GPa
= 70 GPa and 0.3 .
4.2 Effect of porosity distributions Dimensionless buckling load of FGPM-I and FGPM-II plates under different boundary conditions, namely SSSS, CSSS, CSCS, CCCC and CCFF, electric voltages and power-law exponents at a/h=100 is tabulated in Tables 3 and 4, respectively. It can be deduced that even porosity distribution provides lower buckling loads than uneven porosity distribution at a constant external voltage, regardless the type of boundary conditions. Hence, type of porosity distribution plays an important role in buckling behavior of FGP plates. Table 5 presents dimensionless buckling load of porous FGPM-I and FGPM-II plates resting on elastic foundation for various boundary conditions at V=+500 , 𝛼=0.2 and a=b=100h. It is found that an embedded porous FGP plate has larger buckling loads than a foundationless FGP plate which indicates hardening effect of elastic foundation on the plate structure. Moreover, it is deduced that Pasternak parameter has more considerable influence on buckling loads of porous FGP plate than Winkler parameter. The dimensionless buckling load of FGPM plates with porosities (α) for various values of sideto-thickness ratio (a/h) is presented in Table 6. It is deduced that side-to-thickness ratio (a/h) has an increasing effect on the value of dimensionless buckling loads. So, it can be concluded that thinner FGP plates possesses larger buckling loads.
4.3 Effect of power-law exponent Examination of the effect of power-law exponent on the buckling loads of FGP plates for different porosities at voltage V=0 with and without elastic foundation is indicated in Fig. 2. It is observable that as the power-law exponent increases, the buckling load decreases for every value
of porosity coefficient. Also, increasing porosity coefficient shows a reducing impact on buckling loads. This reduction in buckling load is less sensible for uneven model.
4.4 Effect of boundary condition A study on boundary condition effect on natural frequency of FGM-II piezoelectric plates is presented in Fig. 3 versus porosity coefficient when a=b=100h, V=200 and p=5. It can be mentioned that increasing the number constraints at the edges leads to increment in the flexural rigidity of the plate and yields to a larger buckling loads. Therefore, it is found that a CCFF porous FGP plate provides largest natural frequency, and SSSS plate provide lowest one. Also, for all boundary conditions increasing porosity coefficient has reduced buckling loads of FGP plate.
4.5 Effect of electric voltage Influence of external electric voltage on buckling load of FGP plates for different porosities when a=b=100h and p=5 is depicted in Fig.4. Increasing porosity coefficient based on both even and un-even models decreases buckling loads of porous FGP plate at a fixed external voltage. Also, the electric field possesses a decreasing effect on buckling loads of FGP plates. The variation of non-dimensional buckling load of FGM-II piezoelectric plate versus applied electric voltage for various boundary conditions (SSSS, CSSS, CSCS, CCCC and CCFF) when a=b=100h, 𝛼=0.2 and p=5 is plotted in Fig.5. It is observed that increasing the magnitude of negative voltage enlarges the buckling load, while increasing the magnitude of positive voltages reduces the buckling load. Another observation from this figure is that at a prescribed voltage, the CCFF porous FGPM plate has the highest buckling load results followed by CCCC, CSCS, CSCS and SSSS respectively. So, for more accurate design of smart porous FGP plates, it is essential to consider various boundary conditions. 4.6 Effect of geometrical parameters In Fig. 6 the influence of side-to-thickness ratio (a/h) on the dimensionless buckling load with various external electric voltages is presented for a simply supported (SSSS) square perfect and porous (FGM-I) piezoelectric plate at p=5 and 𝛼=0.2. It is clear that a perfect FGP plate has higher buckling loads than porous FGM-I piezoelectric plate at a constant electric voltage. Also, it is observable from this figure that the effect of applied voltage is significant when the side-to-
thickness ratio value is large, but it is ignorable at smaller side-to-thickness ratios. Also, it is found that with the rise of the side-to-thickness ratio, the negative values of applied voltage increase the buckling load of porous FGP plate, while positive voltages reduce the buckling load. Moreover, when V=0 the buckling response of porous FGP plate is independent of side-tothickness ratio. Also, Fig. 7 indicates the effect of plate aspect ratio (a/b) on the non-dimensional buckling load of porous FG piezoelectric plates with un-even porosity distribution and different boundary conditions at a/h=100, p=5, V=200 and 𝛼=0.2. It is seen that an increase in plate aspect ratio (a/b) results in a remarkable increment in the buckling loads. Furthermore, for larger values of aspect ratio the difference between buckling loads according to various boundary conditions rises. 4.7 Effect of elastic medium Figs. 8 and 9 indicate the variation of the dimensionless buckling as a function of the Winkler and Pasternak parameters for both even and un-even porosity distributions and various values of porosity coefficient (𝛼). It is seen from these figures that as the Winkler or Pasternak foundation parameters rise, the buckling load becomes higher, irrespective of the value of the porosity coefficient. Moreover, by increasing the porosity coefficient the buckling load becomes smaller for both even and uneven models. At a constant foundation parameters, FGPM-I porosity coefficient has more sensible effect on buckling loads than FGPM-II porosity coefficient. Fig. 10 investigates the effects of foundation parameters on buckling load of FGPM-II plates with various boundary conditions at a/h=100, p=5, V=200 and 𝛼=0.2. It is evident that the Pasternak constant has more prominent influence on buckling load than Winkler constant for all type of boundary conditions. Also, at a prescribed contact condition, CCFF boundary edges provides higher buckling loads than CCCC and the later provides larger buckling followed by CSCS, CSSS and SSSS.
5. Conclusions This article presents a higher-order refined plate model for buckling analysis of piezoelectric FG plates embedded on two-parameter elastic foundation. Governing equations are obtaine from Hamilton’s principle and are solved applying an analytical solution method. Electro-mechanical properties of FGP plate are supposed to vary continuously through the thickness direction according to power-law model. A detailed parametric study is conducted to investigate the influences of the elastic foundation, porosity, external electric voltage, material composition and aspect ratio on the buckling characteristics of FGP plates. It is found that for all values of elastic foundation parameters, increasing power-law exponent lead to reduction in both rigidity of the
plate structure and buckling loads. But with the rise in magnitude of Winkler or Pasternak constants the rigidity of the FGP plate as well as the buckling load results increase. A change in the external electric voltage from a negative value to a positive value leads to the decrease of the buckling loads due to the fact that the axial compressive and tensile forces will be produced in FGP plate due to applied positive and negative voltages, respectively. The most important observation is that even porosity distribution gives smaller buckling loads than uneven porosity distribution.
References 1.
2. 3.
4. 5. 6. 7. 8.
9. 10. 11. 12.
13.
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Tables: Table 1. Electro-mechanical coefficients of material properties for PZT-4 and PZT-5H (Ramirez et al., 2006). PZT-5H Properties PZT-4 c11 c 22 (GPa)
138.499
99.201
c12
77.371
54.016
c13
73.643
50.778
c 33
114.745
86.856
c 55
25.6
21.1
30.6
22.6
e 31 (Cm )
-5.2
-7.209
e 33
15.08
15.118
12.72
12.322
k 11 (C m N )
1.306e-9
1.53e-9
k 33
1.115e-9
1.5e-9
c 66 -2
e15 2
-2
-1
Table 2. Comparison of non-dimensional buckling load of perfect FGM plates for various power-law exponents (a=b=10). FSDT (Mohammadi et al. 2010)
SSSS TSDT (Thai et al. 2012)
Present ICPT
FSDT(Mohammadi et al. 2010)
CSCS TSDT (Thai et al. 2012)
Present ICPT
p=0 p=1
18.6854 18.8566
18.6861 18.8572
18.7054 18.8735
33.3206 33.9966
34.1195 34.6939
34.4056 34.978
p=2
18.8545
18.8021
18.8123
33.9881
34.5084
34.7691
Table 3. Variation of dimensionless buckling load of FGPM-I plate for various boundary conditions and electric voltages. (a=b=100h). B.C.
Perfect FGPM (𝛼=0) p=0.2 p=1
p=5
Porous FGPM-I (𝛼=0.2) p=0.2 p=1
p=5
SSSS
V=-500 V=0 V=+500
1.95762 1.84860 1.73957
1.95762 1.73195 1.62001
1.76521 1.65061 1.53600
1.58506 1.49850 1.41193
1.46926 1.37977 1.29029
1.39068 1.29859 1.20650
CSSS
V=-500 V=0 V=+500
2.79610 2.68707 2.57805
2.62507 2.51313 2.40118
2.50735 2.39274 2.27814
2.26577 2.17920 2.09264
2.09170 2.00222 1.91273
1.97425 1.88216 1.79007
CSCS
V=-500 V=0 V=+500
3.52735 3.41833 3.30930
3.30744 3.19550 3.08356
3.15616 3.04155 2.92694
2.85918 2.77261 2.68604
2.6354 2.54591 2.45642
2.48450 2.39241 2.30032
CCCC
V=-500 V=0 V=+500
3.94192 3.83289 3.72386
3.68181 3.56987 3.45793
3.50556 3.39096 3.27635
3.19847 3.11190 3.02534
2.93395 2.84446 2.75497
2.75843 2.66634 2.57425
CCFF
V=-500 V=0 V=+500
4.42855 4.31952 4.21049
4.12819 4.01624 3.90430
3.92597 3.81136 3.69676
3.59513 3.50857 3.42200
3.28975 3.20026 3.11077
3.08852 2.99643 2.90434
Table 4. Variation of dimensionless buckling of FGPM-II plate for various boundary conditions and electric voltages. (a=b=100h). B.C.
Perfect FGPM (𝛼=0) p=0.2 p=1
p=5
Porous FGPM-II (𝛼=0.2) p=0.2 p=1
p=5
SSSS
V=-500 V=0 V=+500
1.95762 1.84860 1.73957
1.95762 1.73195 1.62001
1.76521 1.65061 1.53600
1.8593 1.7615 1.6637
1.74458 1.64386 1.54313
1.66529 1.56194 1.45859
CSSS
V=-500 V=0 V=+500
2.79610 2.68707 2.57805
2.62507 2.51313 2.40118
2.50735 2.39274 2.27814
2.65829 2.56049 2.46269
2.48583 2.38510 2.28438
2.36736 2.26402 2.16067
CSCS
V=-500 V=0 V=+500
3.52735 3.41833 3.30930
3.30744 3.19550 3.08356
3.15616 3.04155 2.92694
3.3550 3.2572 3.1594
3.13328 3.03255 2.93183
2.98109 2.87775 2.77440
CCCC
V=-500 V=0 V=+500
3.94192 3.83289 3.72386
3.68181 3.56987 3.45793
3.50556 3.39096 3.27635
3.75035 3.65256 3.55476
3.48825 3.38753 3.28680
3.31142 3.20807 3.10473
CCFF
V=-500 V=0 V=+500
4.42855 4.31952 4.21049
4.12819 4.01624 3.90430
3.92597 3.81136 3.69676
4.21419 4.11639 4.01859
3.91159 3.81086 3.71014
3.70894 3.60559 3.50224
Table 5. Variation of dimensionless buckling load of porous FGPM plate on elastic foundation for various boundary conditions (a=b=100h,V=+500, 𝛼=0.2). B.C.
Kw=10, Kp=0 p=0.2 p=1
p=5
Kw=10, Kp=10 p=0.2 p=1
p=5
SSSS
FGPM-I FGPM-II
1.91854 2.17031
1.79689 2.04974
1.71311 1.9652
11.9185 12.1703
11.7969 12.0497
11.7131 11.9652
CSSS
FGPM-I FGPM-II
2.48233 2.85239
2.30242 2.67408
2.17977 2.55036
12.4823 12.8524
12.3024 12.6741
12.1798 12.5504
CSCS
FGPM-I FGPM-II
3.00267 3.47603
2.77305 3.24846
2.61695 3.09103
13.0027 13.476
12.7731 13.2485
12.617 13.091
CCCC
FGPM-I FGPM-II
3.40529 3.93471
3.13493 3.66676
2.95421 3.48468
13.4053 13.9347
13.1349 13.6668
12.9542 13.4847
CCFF
FGPM-I FGPM-II
3.8229 4.41949
3.51167 4.11103
3.30524 3.90314
13.8229 14.4195
13.5117 14.111
13.3052 13.9031
Table 6. Variation of dimensionless buckling of FGPM plate for various boundary conditions and side-tothickness ratios. B.C.
Porous FGPM-I (𝛼=0.2) a/h=5 a/h=10
a/h=20
Porous FGPM-II (𝛼=0.2) a/h=5 a/h=10
a/h=20
SSSS
0.60184
1.32566
1.37202
1.22878
1.56543
1.63095
CSSS
1.09349
1.49123
1.94592
1.24125
1.74223
2.30855
CSCS
1.62136
2.32703
2.49777
1.83336
2.72691
2.96210
CCCC
1.38138
2.09484
2.71967
1.55643
2.43765
3.22018
CCFF
1.74687
2.74526
3.08080
1.96483
3.19999
3.64661
Figures:
(a)
(b) Fig. 1. Geometry of porous functionally graded piezoelectric plate (a) and examples of plate cross sections (b).
2
Dimensionless buckling load
Dimensionless buckling load
2.2 𝛼=0
FGPM-I Kw=Kp=0
𝛼=0.1
1.8
𝛼=0.2
1.6 1.4 1.2 1
12.7 𝛼=0
FGPM-I Kw=Kp=10
12.5
𝛼=0.1 𝛼=0.2
12.3 12.1 11.9 11.7 11.5
0
2
4
6
8
10
0
Power-law exponent (p)
2
4
6
8
10
Power-law exponent (p)
Dimensionless buckling load
Dimensionless buckling load
2.4 𝛼=0
2.2
FGPM-II Kw=Kp=0
2
𝛼=0.1 𝛼=0.2
1.8 1.6 1.4 1.2 1
12.7 𝛼=0
FGPM-II Kw=Kp=10
12.5
𝛼=0.1 𝛼=0.2
12.3 12.1 11.9 11.7 11.5
0
2
4
6
Power-law exponent (p)
8
10
0
2
4
6
8
10
Power-law exponent (p)
Fig 2. Variation of dimensionless buckling load of SSSS FG piezoelectric plates versus power-law exponent for various porosity coefficient (a=b=100h, V=0).
Dimensionless buckling load
CCFF CCCC CSCS CSSS SSSS
4.5
3.5
2.5
1.5
0.5 0
0.1
0.2
0.3
0.4
0.5
Porosity coefficient (𝛼)
Fig 3. Variation of dimensionless buckling load of FGM-II piezoelectric plates versus porosity coefficient for various boundary conditions (a=b=100h, V=200, p=5).
12.8 𝛼=0
2 1.8
FGPM-I Kw=Kp=0
Dimensionless buckling load
Dimensionless buckling load
2.2 𝛼=0.1 𝛼=0.2
1.6 1.4 1.2 1 0.8 -500
-300
-100
100
Applied voltage (V)
300
500
12.6 12.4
𝛼=0
FGPM-II Kw=Kp=0
𝛼=0.1 𝛼=0.2
12.2 12 11.8 11.6 11.4 11.2 -500
-300
-100
100
Applied voltage (V)
300
500
2
13 𝛼=0
FGPM-I Kw=Kp=10
𝛼=0.1
1.8
𝛼=0
12.8
Dimensionless buckling load
Dimensionless buckling load
2.2
12.6
𝛼=0.2
12.4
1.6
𝛼=0.1
FGPM-II Kw=Kp=10
𝛼=0.2
12.2
1.4
12
11.8
1.2
11.6
1 0.8 -500
11.4
-300
-100
100
300
11.2 -500
500
-300
Applied voltage (V)
-100
100
300
500
Applied voltage (V)
Fig 4. Variation of dimensionless buckling load of simply-supported FGM piezoelectric plates versus applied voltage for various porosity coefficient (a=b=100h, p=5).
Dimensionless buckling load
5 CCFF CSCS SSSS
4
CCCC CSSS
3
2
1 -500
-300
-100
100
300
500
Applied voltage (V)
Fig 5. Variation of dimensionless buckling load of FGM-II piezoelectric plates versus applied voltage for various boundary conditions (a=b=100h, p=5, 𝛼=0.2).
Dimensionless buckling load
2 V=-500 V=0 V=+500
1.8
1.6
perfect FGPM
1.4
1.2
porous FGPM
1 20
40
60
80
100
Side-to-thickness ratio (a/h)
Fig 6. Variation of dimensionless buckling load of perfect (𝛼=0) and porous (𝛼=0.2) FGPM-I plates versus side-to-thickness ratio for various electric voltages (a/b=1, p=5).
Dimensionless buckling load
10 CCFF CCCC CSCS CSSS SSSS
8
6
4
2
0 0.5
1
1.5
2
Aspect ratio (a/b)
Fig 7. Variation of dimensionless buckling load of FGM-II piezoelectric plates versus aspect ratio for various boundary conditions (a=100h, p=5,V==200, 𝛼=0.2).
5 𝛼=0
Dimensionless buckling load
Dimensionless buckling load
5
𝛼=0.1
4
𝛼=0.2 3
FGPM-I
2
1
𝛼=0 𝛼=0.1
4
𝛼=0.2 3
FGPM-II
2
1 0
10
20
30
40
50
0
10
Winkler parameter, Kw
20
30
40
50
Winkler parameter, Kw
Fig 8. Variation of dimensionless buckling load of simply-supported FGPM plates versus Winkler parameter for various porosity coefficient (a=b=100h, p=5, V=200).
14
Dimensionless buckling load
Dimensionless buckling load
14 𝛼=0
12
𝛼=0.1
10
𝛼=0.2 8 6 4
FGPM-I
2 0
12
𝛼=0
10
𝛼=0.1 𝛼=0.2
8 6 4
FGPM-II
2 0
0
2
4
6
Pasternak parameter, Kp
8
10
0
2
4
6
8
10
Pasternak parameter, Kp
Fig 9. Variation of dimensionless buckling load of simply-supported FGPM plates versus Pasternak parameter for various porosity coefficient (a=b=100h, p=5, V=200).
16
Dimensionless buckling load
Dimensionless buckling load
6
4
CCFF CCCC CSCS CSSS SSSS
2
0
CCFF CCCC CSCS CSSS SSSS
14 12 10 8 6 4 2 0
0
10
20
30
Winkler parameter, Kw
40
50
0
2
4
6
8
10
Pasternak parameter, Kp
Fig 10. Variation of dimensionless frequency of FGPM-II plates versus Pasternak parameter for various boundary conditions (a=b=100h, p=5, V=200, 𝛼=0.2).
HIGHLIGHTS
Buckling analysis a porous FGM piezoelectric plate resting on elastic foundation is conducted. Two type of porosity distribution are considered: even and uneven. FG piezoelectric plate is modeled via a four-variable refined plate theory. Various boundary conditions (SSSS, CSSS, CCSS, CSCS, CCCC and CCFF) are considered.
PII: DOI: Reference:
S0020-7403(16)30218-1 http://dx.doi.org/10.1016/j.ijmecsci.2016.09.012 MS3417
To appear in: International Journal of Mechanical Sciences Received date: 31 March 2016 Revised date: 28 July 2016 Accepted date: 8 September 2016 Cite this article as: M.R. Barati, M.H. Sadr and A.M. Zenkour, Buckling analysis of higher order graded smart piezoelectric plates with porosities resting on elastic f o u n d a t i o n , International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2016.09.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Buckling analysis of higher order graded smart piezoelectric plates with porosities resting on elastic foundation M.R. Barati*1, M.H. Sadr1, A.M. Zenkour2,3 1
Department of Aerospace Engineering and Center of Excellence in Computational Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran
2
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 3
Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh 33516, Egypt *
Corresponding author: [email protected]
Abstract In this study, examination of buckling behavior of functionally graded piezoelectric (FGP) plates with porosities is conducted employing a refined four-variable plate theory. By capturing shear deformation effects, the present inverse trigonometric theory is needless of shear correction factor. Electro-elastic material properties of porous FGP plate vary across the thickness based on modified power-law model. Implementing an analytical approach which satisfies different boundary conditions, governing equations derived from Hamilton’s principle are solved. The obtained results are compared with those provided in literature. It is indicated that the buckling behavior of piezoelectric plates is significantly influenced by elastic foundation parameters, external voltage, porosity distribution, power-law index, boundary conditions and aspect ratio Keywords: Electro-mechanical buckling, Four-variable plate theory, Functionally graded piezoelectric plate, Porosities, Analytical method.
1. Introduction Multi-phase composite materials, which are known as functionally graded materials (FGMs) possess spatially graded microstructure to achieve particular mechanical properties to suit the functionality of structures. The gradually composition variations of the material constituents from one surface to another provide a proper solution to the problem of induced transverse shear stresses due to the two bonded dissimilar materials with high difference in material properties. Also, due to containing outstanding mechanical properties, FGMs are appropriate for design of engineering structures. Moreover, piezoelectric materials are known as a type of smart structures
which can create electricity when exposed to mechanical stresses. Also, they will work in reverse, producing a strain by the application of an electric field. Until now, several articles have investigated mechanical behavior of functionally graded piezoelectric (FGP) structures. Among them, Kargarnovin et al. (2007) performed vibration analysis of piezo-patched FGM plates under an electric field. Ebrahimi and Rastgoo (2008) researched vibrational characteristics of FG piezoelectric annular plates. Free and forced vibration response of thermo-piezo-electrically actuated higher order graded beams is investigated by Doroushi et al. (2011). Also, stability analysis of FGPM beams under thermo-electrical field is conducted by Kiani et al. (2011). The effects of temperature and electric fields on vibrational characteristics of a post-buckled FGPM beam are studied by Komijani et al. (2013). Jadhav and Bajoria (2013) investigated free and forced vibration control of FG piezoelectric plate under electro-mechanical loading. Also, Rouzegar and Abad (2015) studied vibration of a FGM plate with piezoelectric patches using four-variable refined plate theory. Barati et al. (2016) examined electro-mechanical vibration of smart piezoelectric FG plates with porosities according to a refined four-variable theory. During manufacturing process, porosities are found in FG materials. Porous FG materials are made of two components which are solid and liquid such as woods and stones. Porosity effect on vibration of FGM beams is examined by Wattanasakulpong and Chaikittiratana (2015) via Timoshenko beam theory applying Chebyshev collocation method. Ebrahimi and Zia (2015) studied large amplitude nonlinear vibration analysis of functionally graded Timoshenko beams with porosities. Ebrahimi et al. (2016) investigated thermal vibration response of FGM beams with porosities. They stated that the volume fraction of porosity has a notable effect on natural frequencies for every material graduation. Mechab et al. (2016) investigated the effects of porosities in functionally graded small scale plates resting on Winkler–Pasternak elastic foundations. They applied a modified rule of mixture to approximate material distribution of the functionally graded plates considering the porosity volume fraction. As one can see, some papers have studied mechanical behaviors of FGM structures considering porosities. But, examination of porosity effect on buckling of FG piezoelectric plates is a novel topic which not reported till to now. In this context, the classical plate theory model (CPL) or Kirchoff theory, which ignores the shear deformation effect, only give acceptable results for thin plates. Then the first order shear deformation (FSDT) based in Reissner and Mindlin was utilized but this theory needs a shear correction factor, which is difficult to calculate. Therefore higher-order shear deformation theories (HSDTs) were introduced to avoid the shear correction factors. Further, The HSDTs can be developed using polynomial or non-polynomial thickness functions. Zenkour (2009, 2015) employed a refined sinusoidal theory for functionally graded and composite plates embedded in elastic medium. Moreover, Grover et al. (2015) suggested a secant function based theory for analysis of composite plates. Recently, inverse trigonometric shear deformation theory is recommended by Sahoo and Singh (2013) and Thai et al. (2014). Also, an inverse hyperbolic theory for analysis of plates is proposed by Nguyen et al. (2014). Kulkarni et al. (2015) investigated bending and buckling of FGM plates using refined inverse cotangential theory.
Zenkour et al. (2015a, b) studied thermo-Mechanical bending response of exponentially graded and layered plates resting on elastic foundations using the sinusoidal theory. Most recently, Barati et al. (2016) presented buckling analysis of embedded small scale FG plates under temperature distributions employing an inverse cotangential shear strain function. Also, Barati and Shahverdi (2016) presented an analytical solution for thermal vibration of compositionally graded nanoplates with arbitrary boundary conditions based on physical neutral surface position. This paper investigates buckling response of higher-order shear deformable plates made of functionally graded piezoelectric (FGP) materials with porosities embedded in an elastic foundation. Material properties of FGP plate change continuously in thickness direction based on power-law model. Employing Hamilton’s principle, the governing equations of FGP plate embedded in elastic foundation are obtained. To predict buckling behavior of embedded FGP plates, an analytical solution is applied to solve the governing equations. Numerical results demonstrate the influences of various parameters such as elastic foundation, external electric voltage, power-law index, porosity coefficient and aspect ratio on the buckling loads of porous FGP plates.
2. Theoretical formulations 2.1. The material properties of porous FG piezoelectric plates Consider a functionally graded plate composed of PZT-4 and PZT-5H piezoelectric materials exposed to an electric potential (x , z ,t ) , as shown in Figure 1. The effective material properties of the porous FGM-I piezoelectric plate are variable across the thickness direction based on the modified power-law model as:
hz 12 cijl ciju cijl 2 (i, j ) {(1,1), (1, 2), (1,3), (3,3), (5,5), (6, 6)} p
cij ( z ) ciju cijl
eij ( z ) e
u ij
hz 12 eijl eiju eijl 2
(1a)
p
eijl
z 1 kij ( z ) kiju kijl kijl kiju kijl 2 h 2
(i, j ) {(3,1), (3,3), (3,5)}
(1b)
(i, j ) {(1,1), (3,3)}
(1c)
p
where
p
is an exponent and evaluates the material distribution across the plate thickness and u
and l are correspond to the material properties of the top and bottom surfaces, respectively; is the porosity volume fraction. Also, C ijkl , e kij and k ik are elastic, piezoelectric and dielectric constant, respectively. It must be noted that, the top surface at fully PZT-4 , whereas the bottom surface ( z
h / 2 )
z h / 2
of porous FGP plate is
is fully PZT-5H . Moreover, when the
porosities are distributed around the cross section mid-zone and the amount of porosity
diminishes at the top and bottom of the cross-section. In this case (FGM-II), the effective material properties can be defined by: p 2z z 1 cij ( z ) c c cijl ciju cijl (1 ) 2 h h 2
u ij
l ij
u ij
eijl
eij ( z ) e
(2a)
p 2z z 1 h 2 eijl 2 eiju eijl (1 h )
(2b)
p 2z z 1 kij ( z ) kiju kijl kijl kiju kijl (1 ) 2 h h 2
(2c)
2.2 Basic assumptions of four-variable theory The assumptions of the present theory are as follows: i. The displacements are small in comparison with the plate thickness and, therefore, strains involved are infinitesimal. ii. The transverse normal stress z is negligible in comparison with in-plane stresses x and y . iii. The transverse displacement u3 includes two components of bending wb and shear ws . These
components
are
functions u3 ( x, y, z ) wb ( x, y) ws ( x, y) .
of
coordinates
x
and
y
only.
That
is
iv. The in-plane displacements u1 and u2 consist of extension, bending, and shear components. That is u1 u ub us and u2 v vb vs . - The bending components ub and vb are assumed to be similar to the displacements given by the classical plate theory. Therefore, the expressions for ub and vb are given, respectively, by
ub z
w wb and vb z b . y x
- The shear components us and vs give rise, in conjunction with ws , to the sinusoidal variations of shear strains xz , yz and hence to shear stresses xz , yz through the thickness h of the plate in such a way that shear stresses xz , yz are zero at the top and bottom surfaces of the plate. Consequently, the expressions for us
vs f ( z )
ws . y
2.3. Kinematic relations
and vs can be given as us f ( z )
ws x
and
Here, a non-polynomial shear deformation theory with four unknowns is employed to model the plates composed from porous functionally graded piezoelectric materials. The displacement field of present theory can be expressed by:
u1 x, y, z u x, y z
wb w f ( z) s x x
(3)
u 2 x, y , z v x , y z
wb w f ( z) s y y
(4)
u3 ( x, y, z ) wb ( x, y) ws ( x, y)
(5)
where the theory possesses an inverse cotangential function stated by (Kulkarni et al. 2015):
f ( z) g ( z) z ; g ( z ) cot 1 (rh / z )
(6)
in which
4r / [h(4r 2 1)] with r=0.46 Also,
u
(7)
and v are displacements of the mid-surface and wb and ws are the bending and shear
transverse displacement, respectively. The criteria of continuity and differentiability (Reddy, 1984) are satisfied for the choice of shear strain shape function. The parameter is a constant evaluated by implementing the transverse shear stress boundary conditions so that the transverse shear stresses at the boundary vanish. The parameter r is optimized in the post-processing technique. By comparing the results of newly developed theories for various values of shape parameter r with the exact solution and for various side-to-thickness ratios, the optimized value of r is ascertained. Grover et al. (2013) obtained non-dimensional deflection of four-layer symmetric cross-ply plate for a wide range of r parameter, and compared with the exact solution for the side-to-thickness ratio as 4 and 10. So, they selected an optimized value of r=0.46 for the present theory. The distribution of electric potential across the thickness is expressed by the following relation: ( x, y, z ) cos ( z ) ( x, y)
where / h and
V
2z V h
(8)
is the applied electric voltage applied; and ( x,, y, t ) denotes the function of
the electrical potential in spatial coordinate. Nonzero strains of the present theory are expressed by:
b s x0 x x s yz x 0 b s yz z f , g y y y y s 0 b s xz xz xy xy xy xy
(9)
Where 2w 2 b u ws 2 2 b x s x 0 x x s ws x x y 2 2 wb ws yz 0 v b , sy , y , y 2 2 s ws y y y x z 0 b s xy xy u v x 2 w xy 2 w b s y x 2 xy 2 xy
(10)
Using Eq. (8), the electric field ( Ex , Ey , Ez ) can be stated by:
, x E y , y cos ( z ) , y
(11)
Ex , x cos ( z )
Ez , z sin ( z )
(12)
2V h
(13)
Employing Hamilton’s principle in the following form, the governing equations can be obtained:
t
( 0
where
S
W ) dt 0
S
(14)
and W denote strain energy and work of external forces. The strain energy of present
model is obtained as:
S ij ij dV v
( v
x
x
y y xy xy yz yz xz xz Dx Ex Dy E y Dz Ez ) dV
where D i is electric displacement. Inserting Eqs.(9) and (10) into Eq.(15) gives:
(15)
S
a
0
N xy (
a
0
b
0
[Nx
2 2 2 2 wb u v s ws b wb s ws M xb M N M M x y y y x x 2 x 2 y y 2 y 2
2 2 ws ws u v b wb s ws ) 2 M xy 2 M xy Qyz Qxz ]dxdy y x xy xy y x
b
h/2
0
h/2
(16)
Dx cos( z ) Dz sin( z ) dzdxdy Dy cos( z ) x y
Where the variables existing in above equation are defined by:
( Ni , M ib , M is ) (1, z, f ) i dA, i ( x, y, xy ) A
Qi g i dA, i ( xz , yz )
(17)
A
The work done by applied forces can be expressed by: ( wb ws ) ( wb ws ) ( wb ws ) ( wb ws ) N y0 0 0 x x y y ( wb ws ) ( wb ws ) ( wb ws ) ( wb ws ) 2 N xy0 kw ( wb ws ) k p )dxdy x y x y
W
a
b
( N x0
(18)
where N x0 , N y0 , N xy0 are in-plane applied loads and kw , k p are elastic foundation parameters. For a piezo-electrically actuated porous FGPM plate, the constitutive relations may be expressed as:
ij Cijkl kl ekij Ek
(19)
Di eikl kl kik Ek
(20)
where ij , ij and E i denote the stress, strain and electric field components, respectively. Finally, the equivalent form of Eqs. (19) and (20) are:
xx c11 xx c12 yy e31 Ez
(21)
yy c12 xx c11 yy e31 Ez
(22)
xy c66 xy xz c55 xz e15 Ex
(23)
yz c55 yz e15 Ex
(25)
Dx e15 xz k11Ex
(26)
Dy e15 yz k11E y
(27)
Dz e31 xx e31 yy k33 Ez
(28)
(24)
where cij , eij and kij are reduced constants for the FGP plate under the plane stress state and are given as:
c132 c132 c11 c11 , c12 c12 , c66 c66 c33 c33
(29)
2 c13e33 e33 e31 e31 , k11 k11 , k33 k33 c33 c33
The following governing equations are obtained by inserting Eqs. (16)-(18) in Eq. (14) when the coefficients of u, v, wb and ws are equal to zero: N x N xy 0 x y N xy x
N y y
(30)
0
(31)
b 2 M xy 2 M yb 2 M xb 2 ( N E N b ) 2 ( wb ws ) kw ( wb ws ) k p 2 ( wb ws ) 0 2 2 x xy y
s 2 M xy 2 M ys Qxz Qyz 2 M xs 2 ( N E N b ) 2 ( wb ws ) x 2 xy y 2 x y
(32)
(33)
kw ( wb ws ) k p ( wb ws ) 0 2
Dy D cos( z ) x cos( z ) sin( z) Dz h /2 x y
h /2
dz 0
(34)
Where N x0 N y0 N E , N xy0 0 and electric load can be expressed as (Doroushi et al. 2011, Jadhav and Bajoria 2013):
N E
h/ 2
h/ 2
e31
2V dz h
(35)
By integrating Eq. (21)-(28) over the area of plate cross-section, the following relations for the FGP plate can be obtained:
N x A11 N y A12 0 N xy
A12 A22 0
u x B11 0 v 0 B12 y 0 A66 u v y x
B12 B22 0
2 wb x 2 s B 11 0 2 w s b 0 B12 y 2 B66 0 2 wb 2 xy
s B12 s B22 0
2 ws x 2 e A 0 31 2 ws e 0 A31 2 y s 0 B66 2w s 2 x y
(36)
b M x B11 b M y B12 b 0 M xy
s Bs M x 11 s s M y B12 s 0 M xy
Qxz Q yz
u x D11 0 v 0 D12 y B66 u v 0 y x
B12 B22 0
u Ds 0 x 11 s v 0 D12 y s 0 B66 u v y x
s B12 s B22 0
ws 0 x s w s A55 y
A 0
s 44
D12 D22 0
s D12 s D22 0
A15e
2 wb x 2 s D 11 0 2 w s b 0 D12 y 2 D66 0 2 wb 2 xy
2 wb 2 Hs 0 x 11 2w s b 0 H 2 12 y s 0 D66 2w b 2 xy
s D12 s D22 0
s H12 s H 22 0
2 ws 2 Ee 0 x 31 2 w e s 0 E31 2 y s 0 D66 2w s 2 xy
2 ws 2 F e 0 x 31 2 ws e 0 F31 2 y s 0 H 66 2w s 2 xy
x y
(37)
(38)
(39)
ws
y
y
Dx e x e x cos( z ) dz E F 15 w 11 h/2 Dy s
(40)
h /2
D sin( z)dz A h /2
h /2
e 31
z
(
u v e ) E31 2 wb F31e 2 ws F33e x y
(41)
in which the cross-sectional rigidities are defined as follows: c A11 , B11 , B11s , D11 , D11s , H11s 11 h /2 s s s 2 2 A12 , B12 , B12 , D12 , D12 , H12 h /2 c12 (1, z , f , z , zf , f )dz s s s c66 A66 , B66 , B66 , D66 , D66 , H 66
(42)
A
, E31e , F31e
(43)
A
, E15e
e 31
e 15
h /2
h /2
h /2
h /2
e15 cos( z ) 1, gdz
F , F k e 11
e 33
h /2
h /2
s s A44 A55
e31 sin( z ) 1, z, f dz
h /2
11
(44)
cos2 ( z ), k33 2 sin 2 ( z) dz
c g 2 dz
h /2 55
(45) (46)
The governing equations for a refined four-variable porous FGP plate in terms of the displacement can be derived by substituting Eqs. (36)-(41), into Eqs. (30)-(34) as follows: A11
3 3 wb 3 wb 2u 2u 2v s ws A ( A A ) B ( B 2 B ) B 66 12 66 11 12 66 11 x 2 y 2 xy x 3 xy 2 x 3
s ( B12s 2 B66 )
A66
3 3 wb 3 wb 2v 2v 2u s ws A ( A A ) B ( B 2 B ) B 22 12 66 22 12 66 22 x 2 y 2 xy y 3 x 2y y 3
s ( B12s 2 B66 )
B11
3 ws e A31 0 2 xy x
2( D12 2 D66 )
B11s
(48)
3 ws e A31 0 2 x y y
4 wb 3u 3u 3v 3v e ( B 2 B ) ( B 2 B ) B D E31 2 12 66 12 66 22 11 x 3 xy 2 x 2 y y 3 x 4
s D22
(47)
(49)
4 4 wb 4 wb 4 ws s ws s s D D 2( D 2 D ) 22 11 12 66 x 4 y 2 y 4 x 4 x 4 y 2
4 ws ( N E N b ) 2 ( wb ws ) kw ( wb ws ) k p 2 ( wb ws ) 0 y 4
4 2 3 3u 3u 3v s s s s s v s wb s ws ( B 2 B ) ( B 2 B ) B D A 12 66 12 66 22 11 55 x 3 xy 2 x 2 y y 3 x 4 x 2 4 4 2 ws 4 wb 4 ws s s s wb s ws s s 2( D 2 D ) D H 2( H 2 H ) 12 66 22 11 12 66 y 2 x 4 y 2 y 4 x 4 x 4 y 2
s A44
s H 22
e A31 (
(50)
4 ws F31e 2 ( N E N b ) 2 ( wb ws ) k w ( wb ws ) k p 2 ( wb ws ) A15e 2 0 y 4
u v ) E31e 2 wb F31e 2 ws E15e 2 ws F11e 2 F33e 0 x y
(51)
3. Solution procedure Here, an exact solution of the governing equations for buckling of a FG piezoelectric plate with simply-supported (S), clamped (C) or free (F) edges or combinations of these boundary conditions is presented which they are given as (Sobhy, 2013):
Simply-supported (S):
wb ws N x M x 0 at x=0,a wb ws N y M y 0 at y=0,b
Clamped (C):
u v wb ws 0 at x=0,a and y=0,b Free (F):
M x M xy Qxz 0 at x=0,a M y M xy Qyz 0 at y=0,b
The following expansions have been assumed to satisfy various boundary conditions:
u Amn mn ( x, y )
(52)
m 1 n 1
v Bmn mn ( x, y )
(53)
m 1 n 1
wb Cmn mn ( x, y)
(54)
m 1 n 1
ws Dmn mn ( x, y)
(55)
m 1 n 1
Emn mn ( x, y)
(56)
m 1 n 1
where ( Amn , Bmn , Cmn , Dmn , Emn ) are the unknown coefficients and ( mn , mn , mn , mn , mn ) are admissible functions which are expressed as (Sobhy, 2014):
Fm ( x) Fn ( y) x F ( y) Tmn ( x, y) Fm ( x) n y mn ( x, y)
(57)
(58)
mn ( x, y) Fm ( x) Fn ( y)
(59)
mn ( x, y) Fm ( x) Fn ( y)
(60)
mn ( x, y) Fm ( x) Fn ( y)
(61)
Where the function Fm ( x) is presented for considered boundary conditions as ( m m / a ): SS: Fm ( x) sin(m x) CS: Fm ( x) sin m x [cos m x 1] CC: Fm ( x) sin (m x ) 2
(62)
FF: Fm ( x) cos (m x)[sin (m x) 1] 2
2
However, one can obtain the function Fn ( y ) by replacing x, m , m and a in the above equations by y, n ,n and b respectively as ( n n / b ): SS: Fn ( y) sin(n y) CS: Fn ( y) sin n y [cos n y 1] CC: Fn ( y) sin (n y) 2
(63)
FF: Fn ( y) cos (n y)[sin (n y) 1] 2
2
Inserting Eqs. (52) - (56) into Eqs. (47) -(51) respectively, leads to: k1,1 k2,1 k3,1 k 4,1 k5,1
Where
k1,2 k2,2 k3,2 k4,2 k5,2
k1,3 k2,3 k3,3 k4,3 k5,3
k1,4 k2,4 k3,4 k4,4 k5,4
k1,5 Amn k2,5 Bmn k3,5 Cbmn 0 k4,5 Dsmn k5,5 E mn
(64)
k1,1 A11 12 A66 8 , k1,2 A12 A66 8 , k1,3 B11 12 B12 2 B66 8 , k1,4 B11s 12 B12s 2 B66s 8 ,
k1,5 A31e 9 , k2,2 A22 4 A66 10 , k2,3 B22 4 B12 2 B66 10 , k2,4 B22s 4 B12s 2 B66s 10 ,
k2,5 A31e 3 , k3,5 ( E31e ) 3 9 k3,3 D11 13 2 D12 2 D66 11 D22 5 k w 1 ( N E N b k p ) 3 9 , k3,4 D11s 13 2 D12s 2 D66s 11 D22s 5 k w 1 ( N E N b k p ) 3 9 , k4,4 H11s 13 2 H12s 2 H 66s 11 H 22s 5 A44s 9 A55s 3 k w 1 ( N E N b k p ) 3 9 , k4,5 ( E31e E15e ) 3 9 , k5,5 F11e 3 9 F33e 1.
(65) In which
( 1 , 3 , 5 )
a
0
b
0
( 9 , 11 , 13 )
a
0
( 6 , 8 , 12 )
a
( 2 , 4 , 10 )
a
0
0
( Fm Fn , Fm Fn'' , Fm Fn'''' ) Fm Fn dxdx
b
0
b
b
0
0
( Fm'' Fn , Fm'' Fn'' , Fm'''' Fn ) Fm Fn dxdy
( Fm' Fn , Fm' Fn'' , Fm''' Fn ) Fm' Fn dxdy ( Fm Fn' , Fm Fn''' , Fm'' Fn' ) Fm Fn' dxdy
(66)
Also, the dimensionless form of buckling load and foundation parameters are adopted as: 2 k pa2 kw a 4 b a u 3 ˆ NN , Kw , Kp , Dc C11 h Dc Dc Dc
(67)
4. Numerical results and discussions 4.1 Validation In this section, numerical and illustrative examples are provided to explore electro-mechanical buckling response of FGP plates on elastic foundation with porosities employing a higher order trigonometric plate theory. The electro-elastic material properties of the porous FGP plate vary across the thickness according to the modified power-law model. The influences of electric
voltage, various boundary conditions, porosity coefficient, power-law index and aspect ratio on the buckling behavior of the porous FGP plate are explored. The correctness of the presented buckling results are compared with those of first and third order plate theories presented respectively by Mohammadi et al. (2010) and Thai and Choi (2012) for a perfect FGM plate and the results are tabulated in Table 2. It is indicated that the present plate model and solution procedure can accurately predict buckling loads of FGM plates. For comparison study, the material properties are selected as:
= 420 GPa
= 70 GPa and 0.3 .
4.2 Effect of porosity distributions Dimensionless buckling load of FGPM-I and FGPM-II plates under different boundary conditions, namely SSSS, CSSS, CSCS, CCCC and CCFF, electric voltages and power-law exponents at a/h=100 is tabulated in Tables 3 and 4, respectively. It can be deduced that even porosity distribution provides lower buckling loads than uneven porosity distribution at a constant external voltage, regardless the type of boundary conditions. Hence, type of porosity distribution plays an important role in buckling behavior of FGP plates. Table 5 presents dimensionless buckling load of porous FGPM-I and FGPM-II plates resting on elastic foundation for various boundary conditions at V=+500 , 𝛼=0.2 and a=b=100h. It is found that an embedded porous FGP plate has larger buckling loads than a foundationless FGP plate which indicates hardening effect of elastic foundation on the plate structure. Moreover, it is deduced that Pasternak parameter has more considerable influence on buckling loads of porous FGP plate than Winkler parameter. The dimensionless buckling load of FGPM plates with porosities (α) for various values of sideto-thickness ratio (a/h) is presented in Table 6. It is deduced that side-to-thickness ratio (a/h) has an increasing effect on the value of dimensionless buckling loads. So, it can be concluded that thinner FGP plates possesses larger buckling loads.
4.3 Effect of power-law exponent Examination of the effect of power-law exponent on the buckling loads of FGP plates for different porosities at voltage V=0 with and without elastic foundation is indicated in Fig. 2. It is observable that as the power-law exponent increases, the buckling load decreases for every value
of porosity coefficient. Also, increasing porosity coefficient shows a reducing impact on buckling loads. This reduction in buckling load is less sensible for uneven model.
4.4 Effect of boundary condition A study on boundary condition effect on natural frequency of FGM-II piezoelectric plates is presented in Fig. 3 versus porosity coefficient when a=b=100h, V=200 and p=5. It can be mentioned that increasing the number constraints at the edges leads to increment in the flexural rigidity of the plate and yields to a larger buckling loads. Therefore, it is found that a CCFF porous FGP plate provides largest natural frequency, and SSSS plate provide lowest one. Also, for all boundary conditions increasing porosity coefficient has reduced buckling loads of FGP plate.
4.5 Effect of electric voltage Influence of external electric voltage on buckling load of FGP plates for different porosities when a=b=100h and p=5 is depicted in Fig.4. Increasing porosity coefficient based on both even and un-even models decreases buckling loads of porous FGP plate at a fixed external voltage. Also, the electric field possesses a decreasing effect on buckling loads of FGP plates. The variation of non-dimensional buckling load of FGM-II piezoelectric plate versus applied electric voltage for various boundary conditions (SSSS, CSSS, CSCS, CCCC and CCFF) when a=b=100h, 𝛼=0.2 and p=5 is plotted in Fig.5. It is observed that increasing the magnitude of negative voltage enlarges the buckling load, while increasing the magnitude of positive voltages reduces the buckling load. Another observation from this figure is that at a prescribed voltage, the CCFF porous FGPM plate has the highest buckling load results followed by CCCC, CSCS, CSCS and SSSS respectively. So, for more accurate design of smart porous FGP plates, it is essential to consider various boundary conditions. 4.6 Effect of geometrical parameters In Fig. 6 the influence of side-to-thickness ratio (a/h) on the dimensionless buckling load with various external electric voltages is presented for a simply supported (SSSS) square perfect and porous (FGM-I) piezoelectric plate at p=5 and 𝛼=0.2. It is clear that a perfect FGP plate has higher buckling loads than porous FGM-I piezoelectric plate at a constant electric voltage. Also, it is observable from this figure that the effect of applied voltage is significant when the side-to-
thickness ratio value is large, but it is ignorable at smaller side-to-thickness ratios. Also, it is found that with the rise of the side-to-thickness ratio, the negative values of applied voltage increase the buckling load of porous FGP plate, while positive voltages reduce the buckling load. Moreover, when V=0 the buckling response of porous FGP plate is independent of side-tothickness ratio. Also, Fig. 7 indicates the effect of plate aspect ratio (a/b) on the non-dimensional buckling load of porous FG piezoelectric plates with un-even porosity distribution and different boundary conditions at a/h=100, p=5, V=200 and 𝛼=0.2. It is seen that an increase in plate aspect ratio (a/b) results in a remarkable increment in the buckling loads. Furthermore, for larger values of aspect ratio the difference between buckling loads according to various boundary conditions rises. 4.7 Effect of elastic medium Figs. 8 and 9 indicate the variation of the dimensionless buckling as a function of the Winkler and Pasternak parameters for both even and un-even porosity distributions and various values of porosity coefficient (𝛼). It is seen from these figures that as the Winkler or Pasternak foundation parameters rise, the buckling load becomes higher, irrespective of the value of the porosity coefficient. Moreover, by increasing the porosity coefficient the buckling load becomes smaller for both even and uneven models. At a constant foundation parameters, FGPM-I porosity coefficient has more sensible effect on buckling loads than FGPM-II porosity coefficient. Fig. 10 investigates the effects of foundation parameters on buckling load of FGPM-II plates with various boundary conditions at a/h=100, p=5, V=200 and 𝛼=0.2. It is evident that the Pasternak constant has more prominent influence on buckling load than Winkler constant for all type of boundary conditions. Also, at a prescribed contact condition, CCFF boundary edges provides higher buckling loads than CCCC and the later provides larger buckling followed by CSCS, CSSS and SSSS.
5. Conclusions This article presents a higher-order refined plate model for buckling analysis of piezoelectric FG plates embedded on two-parameter elastic foundation. Governing equations are obtaine from Hamilton’s principle and are solved applying an analytical solution method. Electro-mechanical properties of FGP plate are supposed to vary continuously through the thickness direction according to power-law model. A detailed parametric study is conducted to investigate the influences of the elastic foundation, porosity, external electric voltage, material composition and aspect ratio on the buckling characteristics of FGP plates. It is found that for all values of elastic foundation parameters, increasing power-law exponent lead to reduction in both rigidity of the
plate structure and buckling loads. But with the rise in magnitude of Winkler or Pasternak constants the rigidity of the FGP plate as well as the buckling load results increase. A change in the external electric voltage from a negative value to a positive value leads to the decrease of the buckling loads due to the fact that the axial compressive and tensile forces will be produced in FGP plate due to applied positive and negative voltages, respectively. The most important observation is that even porosity distribution gives smaller buckling loads than uneven porosity distribution.
References 1.
2. 3.
4. 5. 6. 7. 8.
9. 10. 11. 12.
13.
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Tables: Table 1. Electro-mechanical coefficients of material properties for PZT-4 and PZT-5H (Ramirez et al., 2006). PZT-5H Properties PZT-4 c11 c 22 (GPa)
138.499
99.201
c12
77.371
54.016
c13
73.643
50.778
c 33
114.745
86.856
c 55
25.6
21.1
30.6
22.6
e 31 (Cm )
-5.2
-7.209
e 33
15.08
15.118
12.72
12.322
k 11 (C m N )
1.306e-9
1.53e-9
k 33
1.115e-9
1.5e-9
c 66 -2
e15 2
-2
-1
Table 2. Comparison of non-dimensional buckling load of perfect FGM plates for various power-law exponents (a=b=10). FSDT (Mohammadi et al. 2010)
SSSS TSDT (Thai et al. 2012)
Present ICPT
FSDT(Mohammadi et al. 2010)
CSCS TSDT (Thai et al. 2012)
Present ICPT
p=0 p=1
18.6854 18.8566
18.6861 18.8572
18.7054 18.8735
33.3206 33.9966
34.1195 34.6939
34.4056 34.978
p=2
18.8545
18.8021
18.8123
33.9881
34.5084
34.7691
Table 3. Variation of dimensionless buckling load of FGPM-I plate for various boundary conditions and electric voltages. (a=b=100h). B.C.
Perfect FGPM (𝛼=0) p=0.2 p=1
p=5
Porous FGPM-I (𝛼=0.2) p=0.2 p=1
p=5
SSSS
V=-500 V=0 V=+500
1.95762 1.84860 1.73957
1.95762 1.73195 1.62001
1.76521 1.65061 1.53600
1.58506 1.49850 1.41193
1.46926 1.37977 1.29029
1.39068 1.29859 1.20650
CSSS
V=-500 V=0 V=+500
2.79610 2.68707 2.57805
2.62507 2.51313 2.40118
2.50735 2.39274 2.27814
2.26577 2.17920 2.09264
2.09170 2.00222 1.91273
1.97425 1.88216 1.79007
CSCS
V=-500 V=0 V=+500
3.52735 3.41833 3.30930
3.30744 3.19550 3.08356
3.15616 3.04155 2.92694
2.85918 2.77261 2.68604
2.6354 2.54591 2.45642
2.48450 2.39241 2.30032
CCCC
V=-500 V=0 V=+500
3.94192 3.83289 3.72386
3.68181 3.56987 3.45793
3.50556 3.39096 3.27635
3.19847 3.11190 3.02534
2.93395 2.84446 2.75497
2.75843 2.66634 2.57425
CCFF
V=-500 V=0 V=+500
4.42855 4.31952 4.21049
4.12819 4.01624 3.90430
3.92597 3.81136 3.69676
3.59513 3.50857 3.42200
3.28975 3.20026 3.11077
3.08852 2.99643 2.90434
Table 4. Variation of dimensionless buckling of FGPM-II plate for various boundary conditions and electric voltages. (a=b=100h). B.C.
Perfect FGPM (𝛼=0) p=0.2 p=1
p=5
Porous FGPM-II (𝛼=0.2) p=0.2 p=1
p=5
SSSS
V=-500 V=0 V=+500
1.95762 1.84860 1.73957
1.95762 1.73195 1.62001
1.76521 1.65061 1.53600
1.8593 1.7615 1.6637
1.74458 1.64386 1.54313
1.66529 1.56194 1.45859
CSSS
V=-500 V=0 V=+500
2.79610 2.68707 2.57805
2.62507 2.51313 2.40118
2.50735 2.39274 2.27814
2.65829 2.56049 2.46269
2.48583 2.38510 2.28438
2.36736 2.26402 2.16067
CSCS
V=-500 V=0 V=+500
3.52735 3.41833 3.30930
3.30744 3.19550 3.08356
3.15616 3.04155 2.92694
3.3550 3.2572 3.1594
3.13328 3.03255 2.93183
2.98109 2.87775 2.77440
CCCC
V=-500 V=0 V=+500
3.94192 3.83289 3.72386
3.68181 3.56987 3.45793
3.50556 3.39096 3.27635
3.75035 3.65256 3.55476
3.48825 3.38753 3.28680
3.31142 3.20807 3.10473
CCFF
V=-500 V=0 V=+500
4.42855 4.31952 4.21049
4.12819 4.01624 3.90430
3.92597 3.81136 3.69676
4.21419 4.11639 4.01859
3.91159 3.81086 3.71014
3.70894 3.60559 3.50224
Table 5. Variation of dimensionless buckling load of porous FGPM plate on elastic foundation for various boundary conditions (a=b=100h,V=+500, 𝛼=0.2). B.C.
Kw=10, Kp=0 p=0.2 p=1
p=5
Kw=10, Kp=10 p=0.2 p=1
p=5
SSSS
FGPM-I FGPM-II
1.91854 2.17031
1.79689 2.04974
1.71311 1.9652
11.9185 12.1703
11.7969 12.0497
11.7131 11.9652
CSSS
FGPM-I FGPM-II
2.48233 2.85239
2.30242 2.67408
2.17977 2.55036
12.4823 12.8524
12.3024 12.6741
12.1798 12.5504
CSCS
FGPM-I FGPM-II
3.00267 3.47603
2.77305 3.24846
2.61695 3.09103
13.0027 13.476
12.7731 13.2485
12.617 13.091
CCCC
FGPM-I FGPM-II
3.40529 3.93471
3.13493 3.66676
2.95421 3.48468
13.4053 13.9347
13.1349 13.6668
12.9542 13.4847
CCFF
FGPM-I FGPM-II
3.8229 4.41949
3.51167 4.11103
3.30524 3.90314
13.8229 14.4195
13.5117 14.111
13.3052 13.9031
Table 6. Variation of dimensionless buckling of FGPM plate for various boundary conditions and side-tothickness ratios. B.C.
Porous FGPM-I (𝛼=0.2) a/h=5 a/h=10
a/h=20
Porous FGPM-II (𝛼=0.2) a/h=5 a/h=10
a/h=20
SSSS
0.60184
1.32566
1.37202
1.22878
1.56543
1.63095
CSSS
1.09349
1.49123
1.94592
1.24125
1.74223
2.30855
CSCS
1.62136
2.32703
2.49777
1.83336
2.72691
2.96210
CCCC
1.38138
2.09484
2.71967
1.55643
2.43765
3.22018
CCFF
1.74687
2.74526
3.08080
1.96483
3.19999
3.64661
Figures:
(a)
(b) Fig. 1. Geometry of porous functionally graded piezoelectric plate (a) and examples of plate cross sections (b).
2
Dimensionless buckling load
Dimensionless buckling load
2.2 𝛼=0
FGPM-I Kw=Kp=0
𝛼=0.1
1.8
𝛼=0.2
1.6 1.4 1.2 1
12.7 𝛼=0
FGPM-I Kw=Kp=10
12.5
𝛼=0.1 𝛼=0.2
12.3 12.1 11.9 11.7 11.5
0
2
4
6
8
10
0
Power-law exponent (p)
2
4
6
8
10
Power-law exponent (p)
Dimensionless buckling load
Dimensionless buckling load
2.4 𝛼=0
2.2
FGPM-II Kw=Kp=0
2
𝛼=0.1 𝛼=0.2
1.8 1.6 1.4 1.2 1
12.7 𝛼=0
FGPM-II Kw=Kp=10
12.5
𝛼=0.1 𝛼=0.2
12.3 12.1 11.9 11.7 11.5
0
2
4
6
Power-law exponent (p)
8
10
0
2
4
6
8
10
Power-law exponent (p)
Fig 2. Variation of dimensionless buckling load of SSSS FG piezoelectric plates versus power-law exponent for various porosity coefficient (a=b=100h, V=0).
Dimensionless buckling load
CCFF CCCC CSCS CSSS SSSS
4.5
3.5
2.5
1.5
0.5 0
0.1
0.2
0.3
0.4
0.5
Porosity coefficient (𝛼)
Fig 3. Variation of dimensionless buckling load of FGM-II piezoelectric plates versus porosity coefficient for various boundary conditions (a=b=100h, V=200, p=5).
12.8 𝛼=0
2 1.8
FGPM-I Kw=Kp=0
Dimensionless buckling load
Dimensionless buckling load
2.2 𝛼=0.1 𝛼=0.2
1.6 1.4 1.2 1 0.8 -500
-300
-100
100
Applied voltage (V)
300
500
12.6 12.4
𝛼=0
FGPM-II Kw=Kp=0
𝛼=0.1 𝛼=0.2
12.2 12 11.8 11.6 11.4 11.2 -500
-300
-100
100
Applied voltage (V)
300
500
2
13 𝛼=0
FGPM-I Kw=Kp=10
𝛼=0.1
1.8
𝛼=0
12.8
Dimensionless buckling load
Dimensionless buckling load
2.2
12.6
𝛼=0.2
12.4
1.6
𝛼=0.1
FGPM-II Kw=Kp=10
𝛼=0.2
12.2
1.4
12
11.8
1.2
11.6
1 0.8 -500
11.4
-300
-100
100
300
11.2 -500
500
-300
Applied voltage (V)
-100
100
300
500
Applied voltage (V)
Fig 4. Variation of dimensionless buckling load of simply-supported FGM piezoelectric plates versus applied voltage for various porosity coefficient (a=b=100h, p=5).
Dimensionless buckling load
5 CCFF CSCS SSSS
4
CCCC CSSS
3
2
1 -500
-300
-100
100
300
500
Applied voltage (V)
Fig 5. Variation of dimensionless buckling load of FGM-II piezoelectric plates versus applied voltage for various boundary conditions (a=b=100h, p=5, 𝛼=0.2).
Dimensionless buckling load
2 V=-500 V=0 V=+500
1.8
1.6
perfect FGPM
1.4
1.2
porous FGPM
1 20
40
60
80
100
Side-to-thickness ratio (a/h)
Fig 6. Variation of dimensionless buckling load of perfect (𝛼=0) and porous (𝛼=0.2) FGPM-I plates versus side-to-thickness ratio for various electric voltages (a/b=1, p=5).
Dimensionless buckling load
10 CCFF CCCC CSCS CSSS SSSS
8
6
4
2
0 0.5
1
1.5
2
Aspect ratio (a/b)
Fig 7. Variation of dimensionless buckling load of FGM-II piezoelectric plates versus aspect ratio for various boundary conditions (a=100h, p=5,V==200, 𝛼=0.2).
5 𝛼=0
Dimensionless buckling load
Dimensionless buckling load
5
𝛼=0.1
4
𝛼=0.2 3
FGPM-I
2
1
𝛼=0 𝛼=0.1
4
𝛼=0.2 3
FGPM-II
2
1 0
10
20
30
40
50
0
10
Winkler parameter, Kw
20
30
40
50
Winkler parameter, Kw
Fig 8. Variation of dimensionless buckling load of simply-supported FGPM plates versus Winkler parameter for various porosity coefficient (a=b=100h, p=5, V=200).
14
Dimensionless buckling load
Dimensionless buckling load
14 𝛼=0
12
𝛼=0.1
10
𝛼=0.2 8 6 4
FGPM-I
2 0
12
𝛼=0
10
𝛼=0.1 𝛼=0.2
8 6 4
FGPM-II
2 0
0
2
4
6
Pasternak parameter, Kp
8
10
0
2
4
6
8
10
Pasternak parameter, Kp
Fig 9. Variation of dimensionless buckling load of simply-supported FGPM plates versus Pasternak parameter for various porosity coefficient (a=b=100h, p=5, V=200).
16
Dimensionless buckling load
Dimensionless buckling load
6
4
CCFF CCCC CSCS CSSS SSSS
2
0
CCFF CCCC CSCS CSSS SSSS
14 12 10 8 6 4 2 0
0
10
20
30
Winkler parameter, Kw
40
50
0
2
4
6
8
10
Pasternak parameter, Kp
Fig 10. Variation of dimensionless frequency of FGPM-II plates versus Pasternak parameter for various boundary conditions (a=b=100h, p=5, V=200, 𝛼=0.2).
HIGHLIGHTS
Buckling analysis a porous FGM piezoelectric plate resting on elastic foundation is conducted. Two type of porosity distribution are considered: even and uneven. FG piezoelectric plate is modeled via a four-variable refined plate theory. Various boundary conditions (SSSS, CSSS, CCSS, CSCS, CCCC and CCFF) are considered.