DSpace at VNU: MECHANICAL AND THERMAL POSTBUCKLING OF SHEAR-DEFORMABLE FGM PLATES WITH TEMPERATURE-DEPENDENT PROPERTIES

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MECHANICAL AND THERMAL POSTBUCKLING

OF SHEAR-DEFORMABLE FGM PLATES

WITH TEMPERATURE-DEPENDENT PROPERTIES

N D Duc a, * and H V Tung b

Keywords: postbuckling, functionally graded materials, temperature-dependent properties, imperfection

An analytical approach to investigating the stability of simply supported rectangular functionally graded plates under in-plane compressive, thermal, and combined loads is presented The material properties are as-sumed to be temperature-dependent and graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of constituents The equilibrium and compatibility equations for the plates are derived by using the first-order shear deformation theory of plates, taking into account both the geo-metrical nonlinearity in the von Karman sense and initial geogeo-metrical imperfections The resulting equations are solved by employing the Galerkin procedure to obtain expressions from which the postbuckling load–de-flection curves can be traced by an iterative procedure A stability analysis performed for geometrically midplane-symmetric FGM plates shows the effects of material and geometric parameters, in-plane boundary conditions, temperature-dependent material properties, and imperfections on the postbuckling behavior of the plates.

1 Introduction

Functionally graded materials (FGMs) are microscopically inhomogeneous composites usually made of a mixture of metals and ceramics By gradually varying the volume fraction of their constituents, it can be achieved that the effective proper-ties of FGMs exhibit a smooth and continuous change from one surface to another, thus eliminating interface problems and mit-igating thermal stress concentrations Due to the high heat resistance, FGMs are used as structural components operating in ultrahigh-temperature environments and subjected to extremely high thermal gradients, such as aircraft, space vehicles, nuclear plants, and other engineering applications

These new materials pose interesting topics for structural mechanics, e.g., problems relating to the buckling and postbuckling behavior of FGM structures subjected to mechanical, thermal, and thermomechanical loads Eslami and his co-workers [1-7] have treated a series of problems relating to the linear buckling of simply supported rectangular FGM plates, with and without imperfections, under mechanical and thermal loads By using an analytical approach, they obtained

a

University of Engineering and Technology, Vietnam National University, Ha Noi, Viet Nam.bFaculty of Civil Engi-neering, Hanoi Architectural University, Ha Noi, Viet Nam Russian translation published in Mekhanika Kompozitnykh Materialov, Vol 46, No 5, pp 679-700, September-October, 2010 Original article submitted February 19, 2010

*Corresponding author; e-mail: ducnd@vnu.edu.vn

Mechanics of Composite Materials, Vol 46, No 5, 2010

closed-form expressions for buckling loads Following this direction, Lanhe [8] examined the thermal postbuckling of rectan-gular moderately thick FGM plates But the effects of prebuckling deformation, the temperature dependence of material prop-erties, and postbuckling behavior of FGM plates have not been considered in these works Shen [9-11] investigated the postbuckling behavior of FGM plates subjected to transverse and in-plane [9], thermoelectromechanical [10], and thermal [11] loads By using Reddy’s higher-order shear deformation theory in conjunction with a two-step perturbation technique, he suc-cessfully analyzed the postbuckling of FGM plates with both temperature-dependent material properties and initial imperfec-tions accounted for Liew et al analyzed the postbuckling of FGM plates under simultaneous acimperfec-tions of various loads [12] and uniform temperature changes [13] by using a higher-order shear deformation theory They also investigated the postbuckling

of cylindrical panels under combined thermomechanical loads within the framework of the classical shell theory [14] By em-ploying differential quadrature method and an iteration technique, the postbuckling curves for fully clamped plates were traced

in these studies Na and Kim [15] investigated the thermal postbuckling of FGM plates by using the three-dimensional finite-el-ement method Zhao et al [16] and Zhao and Liew [17] analyzed the buckling and postbuckling behavior of FGM plates by in-voking the element-free kp-Ritz method It is evident from the literature that investigations considering the temperature de-pendence of material properties are few in number However, in practice, the FGM structures are usually exposed to high-temperature environments, where significant changes in material properties are unavoidable Therefore, the temperature dependence of their properties should be considered for an accurate and reliable prediction of deformation behavior of the composites

This paper presents an analytical approach to investigating the buckling and postbuckling behavior of simply sup-ported rectangular FGM plates subjected to in-plane compressive, thermal, and thermomechanical loads Their material prop-erties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power-law distribu-tion in terms of volume fracdistribu-tions of constituents The governing equadistribu-tions are derived within the framework of the first-order shear deformation theory, with account of both the von Karman nonlinearity and initial imperfections The resulting equations are solved by the Galerkin method to obtain expressions from which the buckling loads and postbuckling curves are deter-mined by an iteration procedure A stability analysis carried out for geometrically in-plane symmetric FGM plates shows the effects of material and geometric parameters, in-plane boundary conditions, temperature-dependent material properties, and imperfections on the postbuckling behavior of the plates

2 Functionally Graded Plates

Consider a rectangular plate that consists of two layers made of functionally graded ceramic and metal materials and is midplane-symmetric The outer surface layers of the plate are ceramic-rich, but the midplane layer is purely metallic The

length, width, and total thickness of the plate are a, b, and h, respectively The plate is referred to a Cartesian coordinate system

x, y, z, where xy is the midplane of the plate and z is the thickness coordinate,-h 2£ £z h 2

By applying a simple power-law distribution, the volume fractions of metal and ceramic,Vm andVc, are assumed as

z h

z h

m

k

k

( )

=

+ æ è

ø

- + æ è

ø

ì í

2

2

ï ï î

ï

where the volume fraction index k is a nonnegative number that defines the material distribution and can be chosen to optimize

the structural response

It is assumed that the effective properties Peff of the functionally graded plate, such as the modulus of elasticity E and

the coefficient of thermal expansiona, vary in the thickness direction z and can be determined by the linear rule of mixture as [2, 3]

Peff =Prm V m( ) Prz + c V c( ),z (2)

where Pr denotes a temperature-dependent material property, and the subscripts m and c stand for the metal and ceramic

con-stituents, respectively

From Eqs (1) and (2), the effective properties of the FGM plate can be written as follows:

z h

z h h

k

+ æ è

ø

- +

2

2 æ è

ø

ì í

ï ï î

ï ï

k

z h

where

E mc =E m-E c, amc=am -ac, and the Poisson rationis assumed constant,n( )z =n

Obviously, due to the temperature-dependent properties of constituents, the effective properties E andaof the FGM plate are both temperature- and position-dependent

3 Governing Equations

In the present study, the first-order shear deformation theory is used to obtain the equilibrium and compatibility equa-tions, as well as expressions for determining the buckling loads and the postbuckling load–deflection curves of FGM plates

The strain–displacement relations taking into account the von Karman nonlinear terms are

e e g

e e g

x y xy

xm ym xym

x y xy

z

k k k

æ è

ç ç ç

ö ø

÷

÷

÷=

æ è

ç ç ç

ö ø

÷

÷

÷+

æ è

ç ç ç

ö ø

÷

÷

÷,

g g

f f

xz yz

x x

y y

w w

æ è

çç öø÷÷ =æèçç , ++ öø÷÷

with

e e g

xm ym xym

x x

æ è

ç ç ç

ö ø

÷

÷

÷=

+ +

2 2

2 2

w,y

æ è

ç ç ç ç

ö ø

÷

÷

÷

÷ ,

k k k

x y xy

x x

y y

x y y x

æ è

ç ç ç

ö ø

÷

÷

æ è

ç ç ç

ö ø

÷

÷

÷

f f

, ,

,

(5)

whereexm andeym are the normal strains,gxym is the shear strain on the midplane of the plate, andgxzandgyz are the

trans-verse shear strains; u, v, and w are the midplane displacement components along the x, y, and z axes;fxandfzare the rotation

angles in the xz and yz planes, respectively; (, ) indicates a partial derivative.

Hooke’s law for the plate, including the thermal effects, is

s

E

T

=

1

1

E

T

=

1

1

(6) s

n g

E

= +

2 1( ) , s

n g

E

= +

2 1( ) , s

n g

E

= +

2 1( ) . The force and moment resultants of the plate can be expressed in terms of stress components across the plate thickness as

(N i,M i) i( , )z dz

h

h

=

-òs 1

2

2

, i=x y xy, , , Q i j dz

h

h

=

-òs

2

2

, i=x y , ; j=xz yz,

(7)

Inserting Eqs (3), (4), and (6) into Eqs (7) gives the constitutive relations

(N x,M x)= [(E E, )( xm ym)

1

1 n2 1 2 e ne +(E2,E3)(k x+nk y) (- +1 n F F)( m, b)], (N y,M y)= [(E E, )( ym xm)

1

1 n2 1 2 e ne +(E2,E3)(k y+nk x) (- +1 n F F)( m, b)], (8)

1

+1

2 1 n g g , where

k

mc

1

1

+

3

(9)

(Fm,Fb) ( , ) ( , )D ( , )

h

h

=

2

2

Within the framework of the first-order shear deformation theory, with the assumption that the temperature field is

raised uniformly, the nonlinear equilibrium equations for a perfect plate can be written in terms of deflection w and force

resul-tants as [7, 8]

N x x, +N xy y, =0, N xy x, +N y y, =0,

(10)

E N w x xx N xy w xy N w y yy

1 2

2 1

2

(N w x ,xx 2N xy w,xy N w y ,yy) 0,

where

Ñ = ¶ + ¶2 2

2

2 2

¶x ¶y

and D E E E

E

-1 3 22

1(1 n2).

For an imperfect plate, let w*( , )denote a known small imperfection This parameter represents a small initial devi-x y

ation of the plate plane from a flat shape When the imperfection is considered, equilibrium equations (10) obtain the form [7]

N x x, +N xy y, =0, N xy x, +N y y, =0,

1 2

2 1

2

n

y(w,yy+w,*yy)] (11)

-[N x(w,xx+w*,xx)+2N xy(w,xy+w*,xy)+N y(w,yy+w*,yy)]=0

Considering the first two of Eqs (11), a stress function f may be defined as

N x = f,yy , N y = f,xx , N xy= -f,xy (12) Inserting Eq (12) into the third of Eqs (11) leads to

E f yy w xx w xx f xy w xy w xy

1 2

2 1

2

n

)+ f,xx(w,yy+w*,yy)]

-[f,yy(w,xx+w*,xx)-2f,xy(w,xy+w*,xy)+ f,xx(w,yy+w,*y y)]=0 (13)

Equation (13) includes two dependent unknowns, w and f To obtain a second equation, relating the unknowns, the

geometrical compatibility [18]

exm yy, +eym xx, -gxym xy, =w,2xy-w,xx w,yy

can be used

For an imperfect plate, this equation may be transformed to the form

exm yy, +eym xx, -gxym xy, =w,2xy-w,xx w,yy+ w,xy w*,xy-w,

2 xx w,*yy-w,yy w*,xx (15)

From constitutive relations (8), with E2 = 0, one can write

(exm,eym) [( x, y) n( y, x) m( , )]

1

=2 1+

1

Inserting the previous equations into Eq (15), with account of Eq (14), leads to the compatibility equation for an im-perfect FGM plate

f E w( ,xy w,xx w,yy w,xy w,xy w,xx w,yy w,yy w , x*x )=0 (16) Equations (13) and (16) are the basic relations used to investigate the buckling and postbuckling of FGM plates They

are nonlinear in the 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, ther-mal, and combined loads Three cases of boundary conditions, labeled Cases 1, 2, and 3, will be considered [19]

Case 1 Plate edges are simply supported and freely movable (FM) The associated boundary conditions are

w=fy =M xx =N xy=0, N x =N x0, x=0, ,a

(17)

w=fx =M yy =N xy =0, N y =N y0, y=0, b

Case 2 The edges are simply supported and immovable (IM) The associated boundary conditions are

w= =u fy =M xx=0, N x =N x0, x=0, ,a

(18)

w= =v fx =M yy=0, N y =N y0, y=0, b Case 3 The edges are simply supported, and uniaxial edge loads operate in the x-coordinate direction The edges

x=0, and x a =0, are considered freely movable, but the other two are load-free and immovable For this case, the boundaryb

conditions are

w=fy =M xx =N xy =0, N x =N x0, x=0, ,a (19)

w= =v fx =M yy=0, N y =N y0, y=0, ,b (19)

where N x0 and N y0 are the prebuckling force resultants in the x and y directions, 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 Eqs

(13) and (16) for the unknowns w and f , with consideration of boundary conditions (17)-(19), we assume that [19, 20]

w W= sinlm xsinmn y,

(20)

f =A1cos2lm x+A2cos2mn y+A3cos2lm xcos2mn y+A4 m x n y+1N x0y2+ N y0x2

2

1 2

wherelm =m ap andmn =n bp (m, n = 1, 2, ) are the numbers of half-waves in the x and y directions, respectively, and W is the deflection amplitude; A i (i = 1-4) are the coefficients to be determined.

Considering boundary conditions (17)-(19), the imperfections of the plate are assumed in the form [5-7, 18]

w* =mhsinlm xsinmn y ; m, n = 1, 2, , (21)

where the coefficientm, varying between 0 and 1, represents the size of the imperfections

After substituting Eqs (20) and (21) into Eq (16), the coefficients A i are found:

m

2 2

32

2

E

m n

2 2

32

2

Introducing Eqs (20)-(22) into Eq (13) and applying the Galerkin method to the resulting equation, we obtain

D(l2m mn2 2) W (1 v D) [5l m l2m n2( 2m mn2) l6m mn6]

8

ë

ê ê

û

ú

E

16 (l m ) ( m )( 2m )

ë

û

2 1

1

v D

This equation, derived for odd values of m and n, is used to determine the buckling loads and postbuckling curves for

rectangular FGM plates under mechanical, thermal, and combined loads

4.1 Analysis of mechanical stability A simply supported FGM plate with freely movable edges (Case 1) is assumed

to be under in-plane compressive loads P x and P y uniformly distributed along the edges x=0, and y a =0, , respectively Theb

material properties are assumed to be temperature-independent in this case

The prebuckling force resultants are [1, 18]

The introduction of Eq (24) into Eq (23) gives

W W

E m B

a

p

p

2

+

é ë

ê ê

n

B h m B a n L mn

4

16

)

+ + + + +

+

p

b

8

(

W W

mn

)

ù û

ú

where

a

a = , B b

h

h = , D D

h

h

1= 1, W W

h

= ,

E B

mn

h

a

2

1 2

p

, b =P

P

y x

Equation (25) may be used to trace the postbuckling load–deflection curves for FGM plates subjected to in-plane com-pressive loads For a perfect plate,m= 0, Eq (25) leads to an equation from which the buckling compressive load P xb may be obtained:

+

p

b

(26)

The critical buckling load P xcr is found for the values of m and n which make the preceding expression minimum In

contrast, whenm ¹0, the imperfection sensitivity of the plates may be predicted The counterparts of Eqs (25) and (26) corre-sponding to the classical theory can also be easily obtained, but the pertinent expressions are omitted for the sake of brevity

4.2 Analysis of thermal stability A simply supported FGM plate with temperature-dependent material properties

and immovable edges (Case 2) under a thermal load is considered The condition expressing the immovability of edges, u = 0 at

x=0, and v = 0 at y a =0, , is fulfilled on the average [11, 19]:b

¶

¶

u

x dxdy

a b

0 0

0

ò

¶

v

y dydx

b à

0 0

0

ò

(27)

From Eqs (5) and (8), the following relations can be obtained in which Eq (12) and the imperfection have been in-cluded:

¶

u

m

2

1

2

1

,

(28)

¶

v

m

2

1

2

1

Introducing Eqs (20) and (21) into Eqs (28) and then into Eqs (27) yields

2

=

F

(29)

2

=

F

Equations (29) express the compressive stresses making the edges immovable These stresses depend on the thermal parameterFm and the prebuckling deflection If the prebuckling deflection is ignored, Eq (29) leads to

N x0 N y0 m

1

-F

n.

This equation may also be derived by using the membrane form of equilibrium equations and the method proposed by Meyers and Hyer [21] Such an approach was employed by Eslami and his co-workers [2-4, 6, 7] and by Lanhe [8] in investi-gating pure FGM plates

Inserting Eqs (29) into Eq (23) gives the following expression for the thermal parameter:

mn

b L

W

+

p

m

2

1

+

8

(

a 2)L mn

é ë

ê ê

+

p2

1 16

a

ù û

ú ú

p2

2

8 1

W W

a

( +2mh ).

(30)

In this study, it is assumed that the temperature is uniformly raised from an initial value T i to a final one T f and the temperature differenceDT=T f -T i is constant

By using Eqs (9), the thermal parameterFm can be expressed in terms ofDT:

k

E k

cac camc mcac mcamc

Although DT is included in the expression for P due to the temperature dependence of material properties (T=T0+DT), one may formally expressDT from Eqs (30) and (31) as follows:

B PL

W W

a

h mn

+

p

m

2

1

8

(

B P m B

é ë

ê

+

p2

1 16

a

ù û

ú ú

p2

2

8 1

W

(W +2m),

(32)

which is a closed-form expression for thermal postbuckling in the case of temperature-independent material properties Con-versely, when material properties are temperature-dependent, an iteration procedure must be used to determine the thermal

postbuckling curves of FGM plates Specifically, for given material and geometric parameters and specific values of W h and

m, the temperature difference is gradually increased from an initial valueDT = 0 ( T =T0)to the final oneDT* (T=T0+DT*)

at which the difference in values between both sides of Eq (32) reaches a small value prescribed

When the initial imperfection is ignored, Eq (32) leads to an equation from which the buckling temperature difference

DT b may be derived as

DT D v m B n

B PL

h mn

2

1

(33)

An iterative process is adopted to determine the critical values of DT bwhen material properties are temperature-de-pendent Whenm ¹0, no bifurcation buckling point exists, and the imperfection sensitivity of the plate may be predicted More-over, the counterparts of Eqs (32) and (33) corresponding to the classical theory can readily be derived

4.3 Analysis of thermomechanical stability Let us consider a simply supported plate, with movable edgesx=0,a and immovable ones y=0, (Case 3), subjected to the simultaneous action of a thermal field and an in-plane compressive loadb

P x distributed uniformly along the edges x=0, a

From the first of Eqs (24) and the second of Eqs (27) and (28), we have

N x0 = -P h x , N y0 vN x0 m E1 n W W h

2

Employing these relations in Eq (23) yields

W W

p

m

+

8

(

h a n2)L mn

ì í

ï îï

+

ü ý ï þï +

p2

2 16

W W

2

m)

-+

n P T

m B a vn

D

(34)

Equation (34) is utilized to trace the postbuckling curves of FGM plates under combined mechanical and thermal loads Specifically, it is used to determine the in-plane compressive loads as functions of total deflection (for a given uniform temperature rise) or the variation in temperature as a function of total deflection (for a given compressive edge load) This

equa-tion shows that, when the temperature dependence of properties is accounted for, the funcequa-tion P W x( )is affected by the

temper-ature field in all its terms and is not merely shifted along the P x axis by the amount DP n P TD

x

a

=

-+

2

2 2 2 as its temperature-independent counterpart

From Eq (34),DT can be formally expressed in terms of the remaining members The resulting equation may be

im-mediately used to trace the thermal postbuckling paths, and theDT W( )curves can be displaced along theDT axis by the amount

-(m B +vn )P

n P

2 when the temperature dependence of material properties is ignored In contrast, when the properties are temperature-dependent, an iterative process must be used to determine the thermal postbuckling curves for given compressive loads

5 Results and Discussion

Here, several numerical examples will be presented for perfect and imperfect simply supported midplane-symmetric FGM plates The silicon nitride and stainless steel are regarded as constituents of the FGM plates A material property Pr, such

as the elastic modulus and the thermal expansion coefficient, can be expressed as a nonlinear function of temperature [22]:

Pr= P0(P T-1 -1+ +1 P T1 1+P T2 2+P T3 3), (35)

where T=T0+DT and T0= 300 Ê (room temperature); P0, P-1, P1, P2, and P3are temperature-dependent coefficients charac-terizing the constituent materials The typical values of the coefficients of the materials mentioned are listed in Table 1 In what follows, the temperature-dependent and temperature-independent properties will be referred to as T-D and T-ID, respectively Moreover, Poisson’s ratio is chosen to ben= 0.3, and the T-ID properties are those computed by formula (35) at T=T0in this study, unless stated otherwise

To validate the present formulation for buckling and postbuckling of FGM plates under mechanical, thermal, and combined loads, the postbuckling of a midplane-symmetric FGM plate with T-D properties under a uniform temperature rise is considered This problem was also analyzed by Shen [11], which used the asymptotic perturbation method and Reddy’s higher-order shear deformation theory The plate is simply supported and immovable at all edges (Case 2) The postbuckling load–deflection curves of FGM plates with and without an initial imperfection are compared with Shen’s results in Fig 1 It is evident that a good agreement has been achieved in this comparison study

Table 2 shows the effects of the volume fraction index k and the aspect ratio a b on the difference of buckling

tempera-ture of FGM plates in comparison with Shen’s results [11] It is seen that the plates buckle sooner when their material properties

are temperature-dependent, and the buckling temperature increases when k increases and a b decreases, as expected.

Table 3 shows the effects of temperature field on the buckling loads of FGM plates under uniaxial compression Obvi-ously, an increasing environment temperature lowers the buckling load

The effects of material and geometric parameters, as well as temperature field, on the postbuckling behavior of FGM plates are depicted in Figs 2-10

In the case of mechanical stability, a simply supported square FGM plate under uniaxial compression was considered

as an example Here, the critical buckling load of perfect plates corresponds to m = n =1, which is the first buckling mode.

Figure 2 shows the effects of the volume fraction index k on the postbuckling behavior of FGM plates with movable

edges and T-ID properties under a uniaxial compressive load As expected, the postbuckling strength of the plates increased

with k, i.e., with the volume content of silicon nitride in the FGM plates.

The effects of in-plane boundary conditions on the postbuckling behavior of FGM plates under a uniaxial

compres-sion are illustrated in Fig 3 Two types of in-plane conditions at the edges y=0, , referred to as freely movable (FM) and im-b

movable (IM) edges, were considered In this figure, the postbuckling curves for the cases of FM and IM edges were traced by using Eq (25) withb =0 and Eq (34) withDT = 0, respectively As seen, the postbuckling strength of the plates increases when the edges y=0, are made immovable and the deflection is sufficiently large.b

In the case of thermal stability, the perfect FGM plates buckled at m= n = 1 for any aspect ratio a b Figure 4 shows the

pronounced effect of T-D material properties on the thermal postbuckling behavior of the FGM plates It is evident that the postbuckling load-carrying capacity of the plates decreases drastically when the temperature dependence of material properties

is taken into consideration Figure 5 demonstrates the increased thermal postbuckling load-carrying capability of the FGM

plates with T-D material properties when k increases.

Figure 6 shows the effects of the aspect ratio a b on the thermal postbuckling behavior of the FGM plates with T-ID and T-D material properties As can be seen, the postbuckling load-deflection curves become lower when k increases and the

temperature dependence of material properties is taken into account

TABLE 1 Temperature-Dependent Coefficients of the Silicon Nitride and Stainless Steel [23]

E, Pà Silicon nitride 0 348.43×109 –3.070×10-4 2.160×10-7 –8.946×10-11

Stainless steel 0 201.04×109 3.079×10-4 –6.534×10-7 0

a, 1/K Silicon nitride 0 5.8723×10-6 9.095×10-4 0 0

Stainless steel 0 12.330×10-6 8.086×10-4 0 0