Stability analysis of a simply supported rectangular functionally graded plate shows the effects of the volume frac-tion index, plate geometry, in-plane boundary condifrac-tions, and imp
Trang 1Nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads
a
Faculty of Civil Engineering, Hanoi Architectural University, Ha Noi, VietNam
b
College of Technology, Vietnam National University, Ha Noi, VietNam
a r t i c l e i n f o
Article history:
Available online 20 October 2009
Keywords:
Nonlinear analysis
Functionally graded materials
Postbuckling
Imperfection
a b s t r a c t
This paper presents a simple analytical approach to investigate the stability of functionally graded plates under in-plane compressive, thermal and combined loads Material properties are assumed to be temper-ature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents Equilibrium and compatibility equations for functionally graded plates are derived by using the classical plate theory taking into account both geometrical non-linearity in von Karman sense and initial geometrical imperfection The resulting equations are solved
by Galerkin procedure to obtain explicit expressions of postbuckling load–deflection curves Stability analysis of a simply supported rectangular functionally graded plate shows the effects of the volume frac-tion index, plate geometry, in-plane boundary condifrac-tions, and imperfecfrac-tion on postbuckling behavior of the plate
Ó 2009 Elsevier Ltd All rights reserved
1 Introduction
Functionally Graded Materials (FGMs) are microscopically
inho-mogeneous composites usually made from a mixture of metals and
ceramics By gradually varying the volume fraction of constituent
materials, their material properties exhibit a smooth and continuous
change from one surface to another, thus eliminating interface
prob-lems and mitigating thermal stress concentrations By high
perfor-mance heat resistance capacity, FGMs are now developed for
general use as structure components in ultrahigh temperature
envi-ronments and extremely large thermal gradients such as aircraft,
space vehicles, nuclear plants, and other engineering applications
Buckling and postbuckling behaviors are one of main interest in
design of structural components such as plates, shells and panels
for optimal and safe usage Therefore, it is important to study the
buckling and postbuckling behaviors of FGM plates under
mechan-ical, thermal and combined thermomechanical loads for accurate
and reliable design Some works about the stability of FGM
struc-tures relating to present study are introduced in the following
Javaheri and Eslami [2–4] and Shariat and Eslami [5] reported
mechanical and thermal buckling of rectangular functionally
graded plates by using the classical plate theory[2,3]and higher
order shear deformation plate theory [4,5] They used energy
method to derive governing equations that analytically solved to
obtain the closed-form solutions of critical loading The same
authors and Shariat[6–8]extended their these studies when influ-ences of initial geometrical imperfection on the critical buckling loading are taken into consideration Lanhe[9]used the first order shear deformation theory to derive closed-form relations for buck-ling temperature difference of simply supported moderately thick rectangular FGM plates Three dimensional thermal buckling anal-ysis of functionally graded composite plates, using finite element method, is reported by Na and Kim[10] The research on thermo-elastic stability of FGM cylindrical shells is introduced by Eslami and his co-workers[12–14] and Lanhe et al [15] Except [10], above mentioned works used analytical approach to study buck-ling of FGM plates and shells Furthermore, by linear buckbuck-ling anal-ysis effects of prebuckling deformation and postbuckling behavior have not been considered in these works Recently, Darabi et al [16]presented nonlinear analysis of dynamic stability for function-ally graded cylindrical shells under periodic axial loading by analytical approach Some investigations about postbuckling behavior of functionally graded plates are also reported by Liew
et al.[17,18]using differential quadrature method, Shen [19,20] using perturbation asymptotic method, and Zhao and Liew[21] using the element-free kp-Ritz method The influences of shear deformation, initial imperfection, piezoelectric actuators, and tem-perature-dependent properties on postbuckling behavior of FGM plates are also taken into consideration in these works
This paper presents a simple analytical approach to investigate buckling and postbuckling behaviors of functionally graded plates subjected to in-plane compressive, thermal, and combined loads The motivation of this study results from practical significance of
0263-8223/$ - see front matter Ó 2009 Elsevier Ltd All rights reserved.
* Corresponding author.
E-mail address: htung0105@gmail.com (H.V Tung).
Contents lists available atScienceDirect Composite Structures
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m p s t r u c t
Trang 2relatively simple closed-form expressions of buckling load and
postbuckling load–deflection curves in the design Formulation is
based on the classical plate theory with both von Karman type of
kinematic nonlinearity and initial geometrical imperfection are
ac-counted for By Galerkin procedure, the resulting equations are
solved to obtain closed-form expressions of postbuckling
equilib-rium paths Stability analysis is carried out for a rectangular FGM
plate simply supported on all edges and effects of material and
geometric parameters, in-plane boundary conditions, and
imper-fection on the postbuckling behavior are discussed
2 Functionally graded plates
Consider a rectangular functionally graded plate of length a,
width b, and thickness h, referred to the rectangular Cartesian
coordinates ðx; y; zÞ, where ðx; yÞ plane coincides with middle
sur-face of the plate and z is the thickness coordinate ðh=2 6 z 6 h=2Þ
By applying a simple power law distribution, the volume fractions
of metal and ceramic, Vmand Vc, are obtained as follows[3,4,9,11]:
VcðzÞ ¼ 2z þ h
2h
where volume fraction index k is a nonnegative number that defines
the material distribution and can be chosen to optimize the
struc-tural response
It is assumed that the effective properties Peff of functionally
graded plate, such as the modulus of elasticity E, the coefficient
of thermal expansiona, and the coefficient of thermal conduction
K, change in the thickness direction z and can be determined by the
linear rule of mixture as[3,4,9,11]
where P denotes a temperature-independent material property, and
subscripts m and c stand for the metal and ceramic constituents,
respectively
From Eqs.(1) and (2), the effective properties of FGM plate can
be written as follows in which Poisson’s ratiomis assumed to be
constant
½EðzÞ;aðzÞ; KðzÞ ¼ ½Em;am;Km þ ½Ecm;acm;Kcm 2z þ h
2h
mðzÞ ¼m
ð3Þ
where
Ecm¼ Ec Em; acm¼acam; Kcm¼ Kc Km ð4Þ
3 Governing equations
In the present study, the classical plate theory is used to obtain
the equilibrium and compatibility equations as well as expressions
of buckling loads and postbuckling equilibrium paths of FGM
plates
The strains across the plate thickness at a distance z from the
mid-plane are[1]
ex¼exmþ zkx; ey¼eymþ zky; cxy¼cxymþ 2zkxy ð5Þ
whereexmandeymare the normal strains,cxymis the shear strain at
the middle surface of the plate, and kijare the curvatures
In the framework of classical plate theory, the strains at the
middle surface and the curvatures are related to the displacement
components u;v;w in the coordinates as[1]
exm¼ u;xþ w2
;x=2; eym¼v;yþ w2
;y=2; cxym¼ u;yþv;xþ w;xw;y;
kx¼ w;xx; ky¼ w;yy; kxy¼ w;xy
ð6Þ
where geometrical nonlinearity in von Karman sense is accounted for and subscript (,) indicates the partial derivative Hooke law for
a plate is defined as
ðrx;ryÞ ¼ ½E=ð1 m2Þ½ðex;eyÞ þmðey;exÞ ð1 þmÞa DTð1; 1Þ;
The force and moment resultants of a plate are expressed in terms of the stress components through the thickness as
ðNij;MijÞ ¼
Z h=2
h=2
Substituting Eqs.(3), (5) and (7)into Eq.(8)gives the constitu-tive relations
Nx¼ E1
1 m2ðexmþmeymÞ þ E2
1 m2ðkxþmkyÞ Um
1 m
Ny¼ E1
1 m2ðeymþmexmÞ þ E2
1 m2ðkyþmkxÞ Um
1 m
Nxy¼ E1 2ð1 þmÞcxymþ E2
1 þmkxy
ð9Þ
Mx¼ E2
1 m2ðexmþmeymÞ þ E3
1 m2ðkxþmkyÞ Ub
1 m
My¼ E2
1 m2ðeymþmexmÞ þ E3
1 m2ðkyþmkxÞ Ub
1 m
Mxy¼ E2 2ð1 þmÞcxymþ E3
1 þmkxy
ð10Þ
where
E1¼ Emh þ Ecmh=ðk þ 1Þ; E2¼ Ecmh2½1=ðk þ 2Þ 1=ð2k þ 2Þ;
E3¼ Emh3=12 þ Ecmh3½1=ðk þ 3Þ 1=ðk þ 2Þ þ 1=ð4k þ 4Þ;
ðUm;UbÞ ¼
Z h=2
h=2
Emþ Ecm
2z þ h 2h
amþacm 2z þ h
2h
DTð1; zÞdz
ð11Þ
The nonlinear equilibrium equations of a perfect plate based on the classical plate theory are given by
Nx;xþ Nxy;y¼ 0
Nxy;xþ Ny;y¼ 0
Mx;xxþ 2Mxy;xyþ My;yyþ Nxw;xxþ 2Nxyw;xyþ Nyw;yy¼ 0
ð12Þ
If the temperature distributes uniformly in x and y directions and when Eqs.(9) and (10)are substituted into Eq.(12), the equi-librium equations can be written in terms of deflection variable w and force resultants as
Nx;xþ Nxy;y¼ 0
Nxy;xþ Ny;y¼ 0
Dr4w ðNxw;xxþ 2Nxyw;xyþ Nyw;yyÞ ¼ 0
ð13Þ
wherer2
¼ @2=@x2þ @2=@y2, and
D ¼ E1E3 E
2
For an imperfect plate, let wðx; yÞ denotes a known small imperfection This parameter represents a small initial deviation
of the plate plane from a flat shape When imperfection is consid-ered, the equilibrium Eq.(13)is modified into form as[6–8]
Trang 3Nx;xþ Nxy;y¼ 0
Nxy;xþ Ny;y¼ 0
Dr4w Nxðw;xxþ w
;xxÞ þ 2Nxyðw;xyþ w
;xyÞ þ Nyðw;yyþ w
;yyÞ
¼ 0 ð15Þ
Considering the first two of Eqs.(15), a stress function f may be
defined as
Substituting Eq.(16)in the third of Eqs.(15)leads to
Dr4w f;yyðw;xxþ w
;xxÞ 2f;xyðw;xyþ w
;xyÞ þ f;xxðw;yyþ w
;yyÞ
¼ 0 ð17Þ
The Eq.(17)includes two dependent unknowns, w and f To
ob-tain a second equation relating these two unknowns, the
compat-ibility equation may be used
The geometrical compatibility equation is written as[1]
exm;yyþeym;xxcxym;xy¼ w2;xy w;xxw;yy ð18Þ
For a imperfect plate, the above equation may be modified into
form as
exm;yyþeym;xxcxym;xy¼ w2
;xy w;xxw;yyþ 2w;xyw
;xy w;xxw
;yy
w;yyw
From the constitutive relations(9), one can write
ðexm;eymÞ ¼ 1
E1
½ðNx;NyÞ mðNy;NxÞ E2ðkx;kyÞ þUmð1; 1Þ;
cxym¼ 2
E1½ð1 þmÞNxy E2kxy
ð20Þ
Substituting the above equations in Eq.(19), with the aid of Eqs
(6) and (16), leads to the compatibility equation of an imperfect
FGM plate as
r4f E1 w2
;xy w;xxw;yyþ 2w;xyw
;xy w;xxw
;yy w;yyw
;xx
¼ 0 ð21Þ
Eqs.(17) and (21)are the basic equations used to investigate
the stability of functionally graded plates They are nonlinear
equa-tions in terms of two dependent unknowns w and f
4 Stability analysis
In this section, an analytical approach is used to investigate the
stability of FGM plates subjected to mechanical, thermal, and
com-bined loads Depending on the in-plane behavior at the edges,
three cases of boundary conditions, labelled Cases (1), (2) and (3)
will be considered[22]
Case (1) The edges are simply supported and freely movable
(FM) The associated boundary conditions are
w ¼ Mxx¼ Nxy¼ 0; Nx¼ Nx0; on x ¼ 0; a
Case (2) The edges are simply supported and immovable (IM)
The associated boundary conditions are
w ¼ u ¼ Mxx¼ 0; Nx¼ Nx0 on x ¼ 0; a
Case (3) The edges are simply supported Uniaxial edge loads
are applied in the direction of the x-coordinate The edges
x ¼ 0; a are considered freely movable, the remaining two edges
being unloaded and immovable For this case, the boundary
condi-tions are
w ¼ Mxx¼ Nxy¼ 0; Nx¼ Nx0 on x ¼ 0; a
where Nx0;Ny0are prebuckling force resultants in directions x and y, respectively, for Case (1) and the first of Case (3), and are fictitious compressive edge loads rendering the edges immovable for Case (2) and the second of Case (3) To solve two Eqs.(17) and (21)for un-knowns w and f, and with the consideration of the boundary condi-tions(22)–(24), we assume the following approximate solutions [22–24]
w ¼ W sin kmx sinlny
f ¼ A1cos 2kmx þ A2cos 2lny þ A3cos 2kmx cos 2lny
þ A4sin kmx sinlny þ1
2Nx0y
2þ1
2Ny0x
2
ð25Þ
where km¼ mp=a;ln¼ np=b; m; n ¼ 1; 2; are number of half waves in x and y directions, respectively, and W is amplitude of deflection Also, Aiði ¼ 1 4Þ are coefficients to be determined Considering the boundary conditions(22)–(24), the imperfec-tions of the plate are assumed as[1,6–8]
where the coefficientlvarying between 0 and 1 represents imper-fection size By substituting Eqs.(25) and (26) into Eq.(21), the coefficients Aiare determined as
A1¼E1l2
32k2 m
WðW þ 2lhÞ; A2¼E1k
2 m
32l2WðW þ 2lhÞ;
Introduction of Eqs.(25) and (26)into Eq (17) and applying Galerkin method for the resulting equation yield
Dðk2
mþl2Þ2W þ 2k2
ml2ðA1þ A2Þ þ k2mNx0þl2Ny0
ðW þlhÞ ¼ 0
ð28Þ
Eq.(28), derived for odd values of m; n, is used to determine buckling loads and postbuckling curves of rectangular FGM plates under mechanical, thermal, and combined loads
4.1 Mechanical stability analysis The simply supported FGM plate with freely movable edges (that is, Case (1)) is assumed to be under in-plane compressive loads Pxand Py(in Pascals), uniformly distributed along the edges
x ¼ 0; a and y ¼ 0; b, respectively
The prebuckling force resultants are[1,2]
Introduction of Eqs.(27) and (29)into Eq.(28)gives
Px¼
p2D m 2B2aþ n22
B2hm2B2aþ bn2 W
p2E1m4B4aþ n4 16B2hm2B2aþ bn2 WðW þ 2lÞ
ð30Þ
where
Ba¼ b=a; Bh¼ b=h; D ¼ D=h3; E1¼ E1=h; W ¼ W=h;
For a perfect plate,l¼ 0, Eq.(30)leads to equation from which buckling compressive load Pxbmay be obtained as
Pxb¼p2D m 2B2aþ n22
Trang 4The above equation has been reported by Javaheri and Eslami
[2]when they analyze linear buckling of perfect FGM plates under
in-plane compressive loadings The critical buckling loads Pxcr is
obtained for values of m and n that make the preceding expression
a minimum In contrast, whenl–0, imperfection sensitivity of
the plates may be predicted Specifically, no bifurcation-type
buck-ling occurs, and the plates start to deflect at the onset of
compression
Eq (30) may be used to trace postbuckling load–deflection
curves of FGM plates subjected to in-plane compressive loads
4.2 Thermal stability analysis
A simply supported FGM plate with immovable edges (that is,
Case (2)) under thermal loads is considered The condition
express-ing the immovability on the edges, u ¼ 0 (on x ¼ 0; a) andv¼ 0 (on
y ¼ 0; b), is fulfilled on the average sense as[20,22]
Z b
0
Z a
0
@u
@xdxdy ¼ 0;
Z a 0
Z b 0
@v
From Eqs.(6) and (9)one can obtain the following relations in
which Eq.(16)and imperfection have been accounted for
@u
@x¼
1
E1
ðf;yymf;xxÞ þE2
E1
w;xx1
2w
2
;x w;xw
;xþUm
E1
@v
@y¼
1
E1
ðf;xxmf;yyÞ þE2
E1
w;yy1
2w
2
;y w;yw
;yþUm
E1
ð34Þ
Substituting Eqs.(25) and (26)into Eq.(34)and then into Eq
(33)yield
Nx0¼ Um
1 mþ
4E2
mnp2ð1 m2Þ k
2
mþml2
W
8ð1 m2Þðk
2
mþml2ÞWðW þ 2lhÞ
Ny0¼ Um
1 mþ
4E2
mnp2ð1 m2Þl2þmk2m
W
8ð1 m2Þl2þmk2m
WðW þ 2lhÞ
ð35Þ
Eq.(35)represents the compressive stresses making the edges
immovable and depending on thermal parameter and prebuckling
deflection It should be noted that when prebuckling deflection is
ignored Eq.(35)leads to
Nx0¼ Ny0¼ Um
which is derived by Javaheri and Eslami[3]by solving the
mem-brane form of equilibrium equations and using the method
pro-posed by Meyers and Hyer[25]
By substituting Eqs.(27) and (35)into Eq.(28)we obtain the
following expression for thermal parameter
Um¼
p2Dð1 mÞ m2B2
aþ n2
b2
W
W þlh
þ
4E2 m4B4
aþ 2mm2n2B2
aþ n4
mnð1 þmÞb2 m2B2
aþ n2
þp2E1hð3 m2Þðm4B4aþ n4Þ þ 4mm2n2B2ai
16ð1 þmÞb2ðm2B2
By Eq.(37)the postbuckling behavior of rectangular FGM plates
under two types of thermal loads will be analyzed
4.2.1 Uniform temperature rise Under mentioned boundary conditions, temperature can be uniformly raised from initial value Tito final one Tf and tempera-ture differenceDT ¼ Tf Tiis a constant
The thermal parameterUmcan be expressed in terms of theDT from Eq.(11)and then introduction of the result into Eq.(37)one obtains
DT ¼p2Dð1 mÞðm2B2aþ n2Þ
B2P
W
þ 4E2 m4B4
aþ 2mm2n2B2
aþ n4
mnð1 þmÞB2P m2B2
aþ n2
þ
p2E1 ð3 m2Þ m4B4
aþ n4
þ 4mm2n2B2
a
16ð1 þmÞB2hPðm2B2aþ n2Þ WðW þ 2lÞ ð38Þ
where
P ¼ EmamþEmacmþ Ecmam
Ecmacm
2k þ 1; E2¼ E2=h
2
ð39Þ
When imperfection is not taken into consideration, Eq (38) leads to expression from which bifurcation-type buckling temper-ature differenceDTbmay be obtained as
DTb¼p2Dð1 mÞ m 2B2aþ n2
B2
This equation has been derived by Javaheri and Eslami[3]when they analyze linear buckling of perfect FGM plates under uniform temperature rise When minimization methods are carried, the critical buckling temperature difference of perfect plates is ob-tained for m ¼ n ¼ 1 In addition, with this buckling mode Eq (38)may be used to trace postbuckling curves of FGM plates sub-jected to thermal load under consideration
4.2.2 Nonlinear temperature change across the thickness
In this case, the temperature through thickness is governed by the one-dimensional Fourier equation of steady-state heat conduction
d
dz KðzÞ
dT dz
¼ 0; Tðz ¼ h=2Þ ¼ Tc; Tðz ¼ h=2Þ ¼ Tm ð41Þ
where Tcand Tmare temperatures at ceramic-rich and metal-rich surfaces, respectively The solution of Eq.(41)can be obtained by means of polynomial series Taking the first seven terms of the ser-ies, the solution for temperature distribution across the plate thick-ness becomes[3,7,9]
TðzÞ ¼ TmþDTr
P5 n¼0 ðr k K cm =K m Þn nkþ1
P5 n¼0 ðK cm =K m Þ n nkþ1
ð42Þ
where r ¼ ð2z þ hÞ=2h andDT ¼ Tc Tmis defined as the temperature difference between ceramic-rich and metal-rich surfaces of the plate
By following the same procedure as the preceding loading case, and assuming the metal surface temperature as reference temper-ature, yields
DT ¼p2Dð1 mÞ m2B2
aþ n2
B2H
W
þ 4E2 m4B4
aþ 2mm2n2B2
aþ n4
mnð1 þmÞB2H m2B2
aþ n2
þ
p2E1 ð3 m2Þ m4B4
aþ n4
þ 4mm2n2B2
a
16ð1 þmÞB2H m2B2
aþ n2
Trang 5H ¼
P5
n¼0
ðK cm =K m Þ n
nkþ1
E m a m nkþ2þE m a cm þE cm a m ðnþ1Þkþ2 þ E cm a cm
ðnþ2Þkþ2
P5 n¼0 ðK cm =K m Þ n nkþ1
ð44Þ
It is similar to preceding loading case, when initial imperfection
is ignored Eq.(43)is reduced to expression from which buckling
temperature change may be obtained as Eq.(40), provided P is
re-placed by H Such a result has been reported by Javaheri and Eslami
[3]by linear buckling analysis of the perfect FGM plates under
non-linear temperature gradient The imperfection sensitivity of the
plates under thermal loads may be predicted from Eqs.(38) and
(43), that is, postbuckling curves originate from coordinate origin
because no bifurcation buckling point exists whenl–0
4.3 Thermomechanical stability analysis
A simply supported plate with movable edges x ¼ 0; a and
immovable edges y ¼ 0; b (that is, Case (3)) and subjected to the
simultaneous action of a thermal field and an uniaxial compressive
loading Px, uniformly distributed along the edges x = 0, a is
considered
Employing Nx0¼ Pxh, Eq.(27)and the second of Eqs.(33), (34)
in Eq.(28)yields
Px¼
p2D m 2B2aþ n22
B2hm2B2aþmn2 W
4E2n3
B2hm m 2B2aþmn2 W þ
p2E1 m4B4
aþ 3n4
16B2 m2B2
aþmn2
2PDT
m2B2
aþmn2 ð45Þ
Eq.(45) is employed to trace postbuckling curves of the FGM
plates under combined mechanical and thermal loads Specifically,
it is used to determine the dependence of the in-plane compressive
edge loads vs total deflection (for given uniform temperature rise)
and conversely, the variation of the temperature rise vs total
deflec-tion (for given compressive edge load) Obviously, temperature
changes can shift PxðWÞ curves along the Px- axis by an amount
DPx¼ n2PDT=ðm2B2aþmn2Þ and conversely,DTðWÞ curves can be
displaced along theDT- axis an amount ðm2B2
aþmn2ÞPx=ðn2PÞ due
to the presence of axial compressive load
5 Results and discussion
To validate the present formulation in buckling and postbuck-ling of plates under mechanical, thermal and combined loads, the postbuckling of a homogeneous isotropic plate under uniaxial compression is considered, which was also analyzed by Shen[19] using the perturbation asymptotic method and Reddy’s higher-or-der shear deformation theory The plate is simply supported on all edges (Case (1)) The postbuckling load–deflection curves of an iso-tropic plate ðm¼ 0:326Þ with and without initial imperfection are compared in Fig 1 with Shen’s results It is evident that good agreement is achieved in this comparison study As a second com-parison study, postbuckling of a simply supported isotropic plate with all immovable edges (Case (2)) under uniform temperature rise is considered, which was also analyzed by Bhimaraddi and Chandrashekhara[26]using the single mode approach and the par-abolic shear deformation theory The postbuckling paths of perfect
0
0.5
1
1.5
2
2.5
W/h
Shen [19], μ = 0
Shen [19], μ = 0.1
Present, μ = 0
Present, μ = 0.1
Px/Pxcr
isotropic thin plate (ν = 0.326) b/a = 1.0, (m,n) = (1,1)
Fig 1 Comparisons of postbuckling curves for isotropic thin plates under uniaxial
0 0.5 1 1.5 2 2.5 3
W/h
Ref [26], μ = 0 Ref [26], μ = 0.1 Present, μ = 0 Present, μ = 0.1 ΔT/ΔTcr
isotropic thick plate b/a = 1.0, b/h = 10
Fig 2 Comparisons of postbuckling curves for isotropic plates under uniform temperature rise.
0 0.2 0.4 0.6 0.8 1 1.2 1.4
1.6
perfect imperfect (μ=0.1)
W/h
Px (GPa)
b/a = 1.0, b/h = 40, β = 0 k = 0
k = 1
k = 5
Trang 6and imperfect isotropic plates are compared inFig 2with results
in Ref.[26] As can be observed, a good agreement is obtained in
this comparison study
To illustrate the present approach, we consider a ceramic–metal
functionally graded plate that consist of aluminum and alumina
with the following properties[3,5,9]
Em¼ 70 GPa; am¼ 23:106 C1
Ec¼ 380 GPa; ac¼ 7; 4:106 C1; Kc¼ 10; 4 W=mK ð46Þ
In the case of mechanical stability, a simply supported square
FGM plate under uniaxial compression is considered as a example
In this case, the critical buckling load of perfect plates corresponds
to m ¼ n ¼ 1, which is the first buckling mode
Fig 3shows variation of postbuckling equilibrium paths of a
FGM plate with side-to-thickness ratio b=h ¼ 40 under uniaxial
compressive load vs three different values of volume fraction
in-dex k (=0, 1, 5) As can be seen, the postbuckling curves become lower as the k increases as expected, and postbuckling curves of imperfect plates are lower than those of perfect plates when deflection is small Effects of in-plane boundary conditions on post-buckling behavior of FGM plates under uniaxial compression are illustrated in Fig 4 Two types of in-plane conditions on edges
y ¼ 0; b, referred to as freely movable (FM) and immovable (IM) edges, are considered In Fig 4, postbuckling curves of the FM and IM cases are traced byEq (30)with b ¼ 0 and Eq.(45)with
DT ¼ 0, respectively It is shown that postbuckling strength of the plate is increased when the edges y ¼ 0; b are immovable and the deflection is sufficiently large
In the case of thermal stability, the perfect FGM plates buckle when m ¼ n ¼ 1 for arbitrary aspect ratio b=a.Figs 5 and 6give postbuckling temperature–deflection curves of a square FGM plate with three various values of k and under two types of thermal load-ings As can be seen, postbuckling curves to be lower with
0
0.2
0.4
0.6
0.8
1
1.2
1.4
W/h
1: FM, μ=0.0
2: FM, μ=0.1
3: IM, μ=0.0
4: IM, μ=0.1
Px (GPa)
1
2 3 4
b/a = 1.0, b/h = 40, k = 1.0
Fig 4 Effect of in-plane boundary conditions on postbuckling behavior of FGM
plates under uniaxial compression.
0
100
200
300
400
500
W/h
perfect
imperfect (μ = 0.1)
ΔT (oC)
b/a = 1.0, b/h = 40 k = 0
k = 1
k = 5
0 200 400 600 800 1000
W/h
perfect imperfect (μ = 0.1)
ΔT (oC)
k = 0
k = 1
k = 5 b/a = 1.0, b/h = 40
Fig 6 Postbuckling curves of FGM plates under nonlinear temperature change
vs k.
0 200 400 600 800 1000
W/h
perfect imperfect (μ = 0.1)
ΔT (oC)
1 2
3
b/h = 40, k = 1.0 1: b/a = 1.0 2: b/a = 1.5 3: b/a = 2.0
Trang 7ing values of k as above, and postbuckling loading carrying
capabil-ity of the plate under nonlinear temperature gradient is higher
than that of plate under uniform temperature rise Furthermore,
postbuckling strength of imperfect plates is higher than that of
perfect plates when the deflection is sufficiently large Effects of
as-pect ratio b=a on thermal postbuckling behavior of FGM plates are
depicted inFig 7 It is seen that the postbuckling strength of the
plates under uniform temperature rise is considerably increased
when b=a ratio increases.Fig 8shows effects of temperature field
on postbuckling behavior of FGM plates under uniaxial
compres-sion Conversely, the effects of in-plane compressive load on
post-buckling behavior of FGM plates under uniform temperature rise
are depicted in Fig 9 It is shown in these Figs that the
(pre-stressed) preheated FGM plates exhibit a decreasing tendency in
postbuckling loading carrying capacity when they are subjected
to action of (thermal) compressive loads as mentioned Finally,
the effects of initial imperfection on postbuckling behavior of
FGM plates with all FM edges subjected to uniaxial compressive loads are depicted inFig 10 It is shown that postbuckling loading capacity of the plates is reduced with increasing values of imper-fection size l when the deflection is small However, a inverse trend occurs when the deflection is sufficiently large Similarly, variation of postbuckling curves of FGM plates under uniform tem-perature rise vs different values oflis plotted inFig 11 As can be observed, when the deflection exceeds a specific value, the curves become higher whenlis increased In other words, initial imper-fection makes FGM plates more stable under temperature field
6 Concluding remarks The paper presents a simple analytical approach to investigate buckling and postbuckling behaviors of functionally graded plates under in-plane edge compressive, thermal, and combined loads The formulation is based the classical plate theory with both von Karman nonlinear terms and initial imperfection are incorporated
By using Galerkin method, closed-form expressions of
0
0.5
1
1.5
2
2.5
W/h
perfect
imperfect (μ = 0.1)
Px (GPa)
b/a = 1.0, b/h = 30, k = 0.5
1: ΔT = 0 2: ΔT = 100 (oC) 3: ΔT = 200 (oC)
1 2 3
Fig 8 Effects of temperature rise on postbuckling behavior of FGM plates under
uniaxial compression.
0
200
400
600
800
1000
W/h
perfect imperfect (μ = 0.1)
ΔT (oC)
b/a = 1.0, b/h = 30, k = 0.5
1: Px = 0 2: Px = 0.2 GPa 3: Px = 0.4 GPa
1 2 3
Fig 9 Effects of compressive load on postbuckling behavior of FGM plates under
uniform temperature rise.
0 0.2 0.4 0.6 0.8 1
W/h
1: μ = 0 2: μ = 0.1 3: μ = 0.2 4: μ = 0.3
Px (GPa)
b/a = 1.0, b/h = 40, k = 1.0
1 2
Fig 10 Postbuckling curves of FGM plates under uniaxial compression vs.l.
0 50 100 150 200 250 300 350
W/h
1: μ = 0 2: μ = 0.1 3: μ = 0.2 4: μ = 0.3
ΔT (oC)
1
2 3 4
b/a = 1.0, b/h = 40, k = 1.0
Fig 11 Postbuckling curves of FGM plates under uniform temperature rise vs.l.
Trang 8ling load–deflection curves of a simply supported FGM plate are
determined for all mentioned types of load with and without
imperfection From these explicit expressions, closed-form
rela-tions of buckling loads of perfect plates, obtained in foregoing
works by linear buckling analysis, may be derived as particular
cases The results show that postbuckling behavior of FGM plates
are greatly influenced by material and geometric parameters, and
in-plane boundary conditions Furthermore, it is also shown that
initial imperfection has significant effects on postbuckling
behav-ior of FGM plates
Acknowledgement
The authors would like to express their science thank to
Profes-sor Dao Huy Bich for offering many valuable suggestions The
financial support by the research project of Vietnam National
Uni-versity – Ha Noi, coded QGTD.09.01 is gratefully acknowledged
References
[1] Brush DO, Almroth BO Buckling of bars, plates and shells New York:
McGraw-Hill; 1975.
[2] Javaheri R, Eslami MR Buckling of functionally graded plates under in-plane
compressive loading ZAMM 2002;82(4):277–83.
[3] Javaheri R, Eslami MR Thermal buckling of functionally graded plates AIAA J
2002;40(1):162–9.
[4] Javaheri R, Eslami MR Thermal buckling of functionally graded plates based on
higher order theory J Therm Stress 2002;25(1):603–25.
[5] Samsam Shariat BA, Eslami MR Buckling of thick functionally graded plates
under mechanical and thermal loads Compos Struct 2007;78:433–9.
[6] Samsam Shariat BA, Javaheri R, Eslami MR Buckling of imperfect functionally
graded plates under in-plane compressive loading Thin-Wall Struct
2005;43:1020–36.
[7] Samsam Shariat BA, Eslami MR Thermal buckling of imperfect functionally
graded plates Int J Solids Struct 2006;43:4082–96.
[8] Samsam Shariat BA, Eslami MR Effect of initial imperfection on thermal
buckling of functionally graded plates J Therm Stress 2005;28:1183–98.
[9] Lanhe W Thermal buckling of a simply supported moderately thick rectangular FGM plate Compos Struct 2004;64(2):211–8.
[10] Na H-S, Kim J-H Three-dimensional thermomechanical buckling analysis for functionally graded composite plates Compos Struct 2006;73:413–22 [11] Reddy JN, Chin CD Thermomechanical analysis of functionally graded cylinders and plates J Therm Stress 1998;21:593–626.
[12] Shahsiah R, Eslami MR Thermal buckling of functionally graded cylindrical shells J Therm Stress 2003;26(3):277–94.
[13] Shahsiah R, Eslami MR Functionally graded cylindrical shell thermal instability based on improved Donnell equations AIAA J 2003;41:1819–24 [14] Mirzavand B, Eslami MR, Shahsiah R Effect of imperfections on thermal buckling of functionally graded cylindrical shells AIAA J 2005;43:2073–6 [15] Lanhe W, Jiang Z, Liu J Thermoelastic stability of functionally graded cylindrical shells Compos Struct 2005;70:60–8.
[16] Darabi M, Darvizeh M, Darvizeh A Non-linear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading Compos Struct 2008;82:201–11.
[17] Liew KM, Yang J, Kitipornchai S Postbuckling of piezoelectric FGM plates subjected to thermo-electro-mechanical loading Int J Solids Struct 2003;40:3869–92.
[18] Liew KM, Yang J, Kitipornchai S Thermal post-buckling of laminated plates comprising functionally graded materials J Appl Mech ASME 2004;71:839–50 [19] Shen H-S Postbuckling of FGM plates with piezoelectric actuators under thermo-electro-mechanical loadings Int J Solids Struct 2005;42:6101–21 [20] Shen H-S Thermal postbuckling behavior of shear deformable FGM plates with temperature-dependent properties Int J Mech Sci 2007;49:466–78 [21] Zhao X, Liew KM Geometrically nonlinear analysis of functionally graded plates using the element-free kp-Ritz method Comput Methods Appl Mech Eng 2009;198:2796–811.
[22] Librescu L, Stein M A geometrically nonlinear theory of transversely isotropic laminated composite plates and its use in the post-buckling analysis Thin-Wall Struct 1991;11:177–201.
[23] Librescu L, Chang MY Imperfection sensitivity and postbuckling behavior of shear-deformable composite doubly-curved shallow panels Int J Solids Struct 1992;29(9):1065–83.
[24] Librescu L, Souza MA Post-buckling of geometrically imperfect shear-deformable flat panels under combined thermal and compressive edge loadings J Appl Mech ASME 1993;60:526–33.
[25] Meyers CA, Hyer MW Thermal buckling and postbuckling of symmetrically laminated composite plates J Therm Stress 1991;14:5247–66.
[26] Bhimaraddi A, Chandrashekhara K Nonlinear vibrations of heated antisymmetric angle-ply laminated plates Int J Solids Struct 1993;30(9):1255–68.