DSpace at VNU: Nonlinear stability analysis of double-curved shallow fgm panels on elastic foundations in thermal environments

DSpace at VNU: Nonlinear stability analysis of double-curved shallow fgm panels on elastic foundations in thermal enviro...

NONLINEAR STABILITY ANALYSIS OF DOUBLE-CURVED

SHALLOW FGM PANELS ON ELASTIC FOUNDATIONS

IN THERMAL ENVIRONMENTS

Nguyen Dinh Duc * and Tran Quoc Quan

Keywords: functionally graded material, double-curved panels, imperfection, elastic foundation, thermal

environments

An analytical investigation into the nonlinear response of thick functionally graded double-curved shallow panels resting on elastic foundations and subjected to thermal and thermomechanical loads is presented Young’s modulus and Poisson’s ratio are both graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of constituents All formulations are based on the classical shell theory with account of geometrical nonlinearity and initial geometrical imperfection in the cases of Pasternak-type elastic foundations By applying the Galerkin method, explicit relations for the thermal load–deflection curves

of simply supported curved panels are found The effects of material and geometrical properties and foundation stiffness on the buckling and postbuckling load-carrying capacity of the panels in thermal environments are analyzed and discussed.

1 Introduction

Functionally graded materials (FGMs) made of a mixture of metal and ceramic constituents have received consider-able attention in recent years due to their high heat resistance and excellent mechanical characteristics in comparison with those of conventional composites By continuously and gradually varying the volume fraction of constituent materials in a specific direction, FGMs able to withstand ultrahigh temperature environments and extremely large thermal gradients can be obtained Therefore, these novel materials are being used in temperature-shielding structural components of aircraft, aerospace

Mechanics of Composite Materials, Vol 48, No 4, September, 2012 (Russian Original Vol 48, No 4, July-August, 2012)

University of Engineering and Technology - Vietnam National University,144 Xuan Thuy-Cau Giay- Hanoi-Viet Nam

*Corresponding author; tel: 84-4-37547565; fax: 84-4-37547460; e-mail: ducnd@vnu.edu.vn

Russian translation published in Mekhanika Kompozitnykh Materialov, Vol 48, No 4, pp 635-652 , July-August,

2012 Original article submitted April 13, 2012

vehicles, nuclear plants, and engineering structures As a result, the buckling and postbuckling behavior of FGM plate and shell structures under different types of loading become practically important problems in securing safe and optimal designs

The linear buckling behavior of simply supported perfect and imperfect FGM plates under thermal loads was inves-tigated in [1-5] by using an analytical approach and the classical and shear-deformation theories of plates Zhao and Liew [6] analyzed the mechanical and thermal buckling of FGM plates by using the element-free Ritz method The postbuckling behavior

of pure and hybrid FGM plates under various conditions of mechanical, thermal and electric loadings were investigated by Liew

et al [7, 8], who used the differential quadrate method, and Shen [9, 10], making use of the asymptotic perturbation technique, and Lee et al [11], who employed the element-free Ritz method Some investigations into the postbuckling of cylindrical FGM panels and cylindrical FGM shells subjected to a pressure loading in thermal environments was presented by Shen and Noda [12, 13] The postbuckling of cylindrical FGM panels under various loading types was treated in [14-16] by using numerical methods and different theories of shells The problem on the structural stability of functionally graded panels subjected to aerothermal loads was considered by Sohn and Kim [17] The thermomechanical postbuckling of cylindrical FGM panels with temperature-dependent properties was investigated by Yang et al [18] A geometrically nonlinear analysis of functionally graded shells was performed by Zhao and Liew [19] N D Duc and H V Tung carried out analytical investigations into the nonlinear response of thin and moderately thick cylindrical FGM panels [20, 21] and plates [22, 23] subjected to mechanical and thermomechanical loads They presented an analytical approach to obtain explicit expressions for the buckling load and postbuckling load–deflection curves in the case of constant Poisson’s ratio Huang and Han [24-26], investigated the case where Poisson’s ratio depended on plate thickness, but they studied only cylindrical shells, while in the present paper, we consider the general case of double-curved panels resting on elastic foundations

The structural components widely used in aircraft, reusable space transportation vehicles, and civil engineering are usually supported by an elastic foundation Therefore, it is also necessary to include the effects of elastic foundation for a better understanding of the buckling behavior and load-carrying capacity of such plates and shells In this connection, Librescu and his co-workers extended their analytical studies [27-29] to investigating the postbuckling behavior of flat and curved laminated composite panels resting on Winkler elastic foundations [30, 31] In spite of the practical importance and increasing use of FGM structures, explorations into the effects of elastic media on the response of FGM plates and shells are comparatively scarce The bending behavior of FGM plates on Pasternak-type foundations was studied by Huang et al [32] and Zenkour [33] using analytical methods and by Shen and Wang [34] employing the asymptotic perturbation technique Recently, Shen [35] and Shen et al [36] investigated the postbuckling behavior of FGM cylindrical shells surrounded by an elastic medium

of tensionless elastic foundation of Pasternak type and subjected to axial compressive loads and internal pressure

This paper presents an analytical approach to investigating the nonlinear response of double-curved shallow FGM panels (with Poisson’s ratio depending on plate thickness) resting on elastic foundations in thermal environments The formulations are based on the classical theory of shells with account of geometrical nonlinearity, initial geometrical imperfections, thermal loads, and elastic foundations The Pasternak model is used to describe the panel–foundation interaction Explicit expressions for the buckling loads and load–deflection curves of simply supported curved shallow FGM panels are found by the Galerkin method The effects of geometrical and material properties, in-plane restraints, foundation stiffness, and imperfections on the nonlinear response of the panels are analyzed and discussed

2 Double-Curved Shallow FGM Panels on Elastic Foundations

Consider a shallow double-curved ceramic-metal FGM panel with radii of curvature R x and R y , edges a and b, and uniform thickness h resting on an elastic foundation.

The panel is related to a coordinate system ( , , )x y z with the x and y axes in its middle surface and z is the thickness

direction ( /−h 2≤ ≤z h/ )2 , as shown in Fig 1

The volume fractions of constituents are assumed to vary through the thickness according to the power-law distribution

V z z h

N

m( )= + , c( ) m( ),





2

where N is the volume fraction index ( 0 ≤ < ∞ N ) The effective properties Preff of the FGM panel are determined by the

linear rule of mixtures

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

constituents, respectively

Expressions for the modulus of elasticity E, Poisson’s ratio ν ν= ( )z , the coefficient of thermal expansion α , and

the thermal conductivity coefficient K are obtained by substituting Eqs (1) into Eq (2):

E z v z( ), ( ), ( ), ( )α z K z E , , ,ν α K E ,ν ,α ,K

2

z h h

N

+









where

Emc =Em−Ec, νmc =νm−νc, αmc=αm−αc, Kmc=Km−Kc (4)

It is evident from Eqs (3) and (4) that the upper surface of the panel ( z= −h/ 2 ) is ceramic-rich, while the lower one

( z h= / 2 ) is metal-rich, and the percentage of the ceramic constituent in the panel grows when N increases.

The panel–foundation interaction is described by the Pasternak model as

q e=k w k1 − ∇2 2w, where ∇ = ∂ ∂ + ∂ ∂2 2/ x2 2/ y , w is the deflection of the panel, k2 1 is the modulus of the Winkler foundation, and k2 is the

stiffness of the shear layer of the Pasternak model

3 Theoretical Formulation

In this study, the classical theory of shells is used to establish the governing equations and to determine the nonlinear

response of curved FGM panels:

ε ε γ

ε ε γ

x y xy

x y xy

x y xy

z

k k k



















=























 +













0 0













x

y

z

h

R x

shear layer

R y

Fig.1 Geometry and coordinate system of a double-curved FGM panel on an elastic foundation

ε ε γ

x y xy

0 0 0

2 2

2 2

























=

,

x w y

























k k k

w w w

x y xy

x x

y y xy

























=

−

−

−





















, , ,

Here, u and v are the displacement components along the x- and y-directions, respectively.

Hooke’s law for the FGM panel is defined as

σ

ν γ

+

2 1( ) ,

where DT is the temperature rise from the stress-free initial state The force and moment resultants of the FGM panel are

determined by

/

/

h

h

=

−∫ σ 1 2

2

Insertion of Eqs (5) and (6) into Eq (7) gives the constitutive relations

N N M M x, y, x, y ( ,I I I I, , ) x ( , , , )I I I I y

20 10 21 11 0

+( , , ,I I I I k11 21 12 22) x+( , ,I I I I k21 11 22 12, ) y+( , ,Φ Φ Φ Φ1 1 2, 2) ,∆T (8)

N M xy, xy I I, xy I I, k xy,

31 32 2 γ where

h

h

2

2 1

=

−

−∫ ( )( ) , /

/ ν

h

h

2

2 1

=

−

−∫ ( ) ( )( ) , /

h

h

3 2

2

2 1

1 2

=

+

/

/

ν

/

/

Φ Φ1 2

2

2

= −

−∫ E z z z T z z dz

h

ν The nonlinear equilibrium equations of a perfect double-curved shallow FGM panel in the classical theory of shells are [37]

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

R

N R

x xx xy xy y yy x

x

y y

, +2 , + , + + + N w x ,xx+2N w xy ,xy+N w y ,yy+ −q k w k1 + ∇2 2w=0

It can be obtained from Eqs (8) that

ε0x=D I N0 10( x−I N20 y+D w1 ,xx+D w2 ,yy −D3 1Φ ∆T),

ε0y =D I N0 10( y−I N20 x+D w1 ,yy+D w2 ,xx−D3 1Φ ∆T), (11)

where

D

0

102 202 1 10 11 20 21

1

=

− , = − , D2 =I I10 21−I I20 11, D3 =I10−I20

Inserting Eqs (11) into the expression of M ij in (8) and then M ij into Eq (10) leads to

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

P1 4f P w N w x xx N w xy xy

R

N

x

y y

, + + + − 1 + ∇2 2 =0, where

Here, f(x, y) is the stress function, which is defined by

For an imperfect curved FGM panel, Eq (12) is transformed into the form

, , ,* , ( , , yyy* )+ f

R

f

yy x

xx y

where w x y*( , ) is a known function representing an initial small imperfection of the panel The compatibility equation for the imperfect double-curved shallow panel is written as

εx yy, εy xx, γxy xy, w,xy w w,xx ,yy w w,xy ,*xy w w xx yy

, ,*

R

w R

yy xx yy

x

xx y

From constitutive relations (11), in conjunction with Eqs (13), we have

ε0x=D I f0 10( ,yy−I f20 ,xx+D w1 ,xx+D w2 ,yy−D3 1Φ∆T),

ε0y =D I f0 10( ,xx−I f20 ,yy+D w1 ,yy+D w2 ,xx−D3 1Φ ∆T), (16)

γ0

xy = I (−f,xy+ I w,xy).

Inserting of Eqs (16) into Eq (15) gives the compatibility equation of the imperfect double-curved FGM panel as

f P w P w,xy w w,xx ,yy w w,xy ,*xy w w xx yy w w yy xx

, ,* , ,* w

R

w R

yy x

xx y





where P D

I

10

D I

4

0 10

1

Relations (14) and (17) are nonlinear equations in the variables w and f , and they are used to investigate the

stabil-ity of thick double-curved FGM panels on elastic foundations subjected to mechanical, thermal, and thermomechanical loads

In the present study, the edges of curved panels are assumed to be simply supported Depending on an in-plane restraint

at the edges, three cases of boundary conditions, labeled as Cases 1, 2, and 3, will be considered [26-30]

Case 1 Four edges of the panel are simply supported and freely movable (FM) The associated boundary conditions are

w N= xy =M x =0, N x=N x0 at x=0, ,a

Case 2 Four edges of the panel are simply supported and immovable (IM) In this case, the boundary conditions are

w u M= = x=0, N x =N x0 at x=0,a,

Case 3 All edges are simply supported The edges x=0, are freely movable, whereas the remaining ones y a =0, b

are immovable In this case, the boundary conditions are defined as

w N= xy =M x=0, N x=N x0 at x=0, ,a

where N x0 and N y0 are the in-plane compressive loads at the movable edges (i.e., Case 1 and the first of Case 3) or fictitious compressive edge loads at the immovable edges (i.e., Case 2 and the second of Case 3)

The approximate solutions w and f satisfying boundary conditions (18)-(20) are assumed in the form [27-31]

w w, * W h, sin m xsin n y,

f =A1 m x A+ 2 n y A+ 3 m x n y+ N y x0 2+ N x y

0 2

2

1 2

where λm=m aπ/ , δn =n bπ/ , W is the deflection amplitude, and m is the imperfection parameter The coefficients A i (i

= 1-3)are determined by inserting Eqs (21) and (22) into Eq (17):

m

2 2

n

2 2

n x

m y

3

= +





Introducing Eqs (21) and (22) into Eq (14) and applying the Galerkin procedure to the resulting equation, we obtain

m n

n x

m

π

λ δ

2

3 1 4





n x

m

4

2 2 2

+





+

+





 −



















 +

8

3

λ δ

m n

n x

m y

P

m n

m x

n

1 4

6

4

λ δ

+





 −















−P ab( + )W W( + h W) ( + h)−



y

N R

N R

q , (23)

where m and n are odd numbers This is the basic equation governing the nonlinear response of thick double-curved shallow

FGM panels under mechanical, thermal, and thermomechanical loading conditions In what follows, some thermal loading conditions will be considered

4 Nonlinear Stability Analysis of Double-Curved Shallow FGM Panels in Thermal Environments

4.1 Nonlinear thermal and thermomechanical response.

Let us consider a simply supported curved FGM panel on an elastic foundation, with all its edges considered

immov-able The panel is subjected to a uniform external pressure q and simultaneously exposed to a thermal environment or sub-jected to a through-the-thickness temperature gradient The in-plane condition of immovability at all edges, i.e., u = 0 at

x=0, and v = 0 at y a =0, , is fulfilled in the average sense as [27, 28]b

∂

∫ u x dxdy

a

0

0, ∫b∂∂v y dydx= 0

From Eqs (5) and (11), one can obtain the following expressions, in which Eq (13) and imperfection have been included:

∂

u

x D I f0 10 yy I f20 xx D1 xx D2 yy D3 1 w2x w w x x

1 2

R x,

∂

v

y D I f0 10 xx I f20 yy D1 yy D2 xx D3 1 w2y w w y y

1 2

R y.

(25)

Insertion of Eqs (21) and (22) into Eqs (25) and the result into Eqs (24) gives the fictitious compressive edge loads

N

x0= 1+ 4 2 21 n2+ 11 m2 −





Φ

mn

I R

I R

P

n x

m y

+









 − PP3 mb n2 W



























2

20 2

N

y0 = 1+ 4 2 11 n2+ 21 m2 −





Φ

mn

I R

I R

P

n x

m y

+









 − PP3 na m2 W



























8(I20λm2 I10δn2) (W W 2µh).

Let us determine expressions for the parameter Φ1 for the two cases of thermal loading mentioned

4.1.1 Uniform temperature rise

The curved FGM panel is exposed to a thermal environment whose temperature is uniformly raised from T i in the

stress-free initial state to its final value T f , with the temperature difference ∆T T= f −T i considered independent of the

thick-ness variable z The thermal parameter Φ1 is obtained from Eqs (9) as

N

E N

c cα c mcα mc cα mc mcα

4.1.2 Through-the-thickness temperature gradient

The temperature Tm of the metal-rich surface is maintained at the stress-free initial value, while the temperature Tc

of the ceramic-rich surface is elevated, and the nonlinear steady temperature conduction is governed by the one-dimensional Fourier equation

d

dz K z dT( )dz ,







=0 T z( = −h/ )2 =Tc, T z h( = / 2)=Tm (28)

Using K z( ) in Eqs (3), the solution of Eq (28) can be found in terms of a polynomial series The first eight terms

of this series gives the following approximation [21]:

jN

jN

j

j j

( )

/

−

+

−

+

=

=

∑

∑

m

mc c

mc c

1 0

5

0

where r=(2z h+ ) / ,2 and ∆T T T h = c− m is defined as the temperature change between two surfaces of the FGM panel

Introduction of Eq (29) into Eqs (9) gives the thermal parameter Φ1 as

Φ1=(L H h T,− ) ∆ where

H

jN

E jN

E

j

=

−

+

22 1

0 5

0 5









−

+

=

=

∑

∑

j

j j

jNmc c

Insertion of Eq (27) into Eqs (26) and the result into Eq (23) gives

q b W b W W= 14 + ( + )+b W W( + )+

54 2

where

B

a h

14

2 4

2 1 4

2 2

2 2 2

+

22 2

a

π4 2

2 2

a

3 1 4

6

2 1 3

aa2+n2 2













+

3

mn

B

h

4 3 3

2

P B

nB R m

mB R n

h





















4P B n R4 22 22 2 m B R2 22 2

nR m

mR B n

a

24

2 2 4 3

2 2 4 2 2

3

2

+

B h

3 4

π4

B h (m I B a + m n I B a +n I )

−  + + +

π2

4B h (I m B a I n B R a ) ax (I m B a I n R) by

+

B

a h

2 2 2 3

2 2 4 2

3 4

,,

B

B

h

a h

34

2

3

2 2 4 2

1 4 4

+



mn

44

6 4

6

4 10 4 4 20 2 2 2

a

54

2 16

1

Relation (30) is an explicit equation of pressure–deflection curves for curved FGM panels resting on elastic founda-tions and subjected to a combined action of a uniformly raised temperature field and a uniform external pressure A similar expression for curved FGM panels simultaneously subjected to a uniform external pressure and a temperature gradient across

the thickness can also be obtained in form (30) if L is replaced with L H−

4.2 Results and discussion

This section presents illustrative results for curved ceramic-metal panels made from aluminum and alumina with following properties [2-5]:

Em = 70 GPa, am = 23 · 10–6 °C–1, Km = 204 W/mK,

Ec = 380 GPa, a c = 7.4 · 10–6 °C–1, Kc = 10.4 W/mK

To characterize the behavior of the panels, deformations at which the central region of a panel occur towards its con-cave side are referred to as inward (or positive) deflections Deformations in the opposite direction are named outward (or negative) deflections [29] In addition, the results given in this section correspond to the deformation mode with numbers of

half-waves m n= =1, and unless otherwise stated, the FGM panel–foundation interaction is ignored

0 8

0 6

0 4

0 2

.

.

.

.

q, GPa

W, h

0

3

1 2 4 5

Fig 2 Effects of initial imperfection on the postbuckling curves of spherical FGM panels under a

uniform temperature rise at m = –0.5 (1), –0.2 (2), 0 (3), 0.2 (4), and 0.5 (5) a/b = 1, b/h – 30, N = 1,

m = n = 1 ΔT = 200, K1 = 0, K2 = 0 a/R x = 0.5, and b/R y = 0.5

As part of the effects of imperfections, the postbuckling load–deflection curves for spherical FGM panels are shown

in Fig 2 at m = –0.5, –0.2, 0, 0.2, and 0.5 Curved FGM panels subjected to thermal loads are not sensitive to initial

imperfec-tion In the case K1=K2 =0 (without an elastic foundation) and n = const, for cylindrical panels, the same results as in [12] were obtained (Fig 3)

Figures 4 and 5 show the effects of elastic foundations and temperature on the nonlinear behavior of spherical FGM panels under a uniform temperature rise A beneficial influence of elastic media on the nonlinear response of the panels is seen Specifically, their load-carrying ability is enhanced, and the intensity of their snap-through behavior is reduced due to the

0 4 0 8 1 2 1 6 2 0

0 035

0 030

0 025

0 020

0 015

0 010

.

.

.

.

.

.

q, GPa

W, h

0

1 2

3 4

Fig 3 Effects of initial imperfection on the postbuckling curves of cylindrical FGM panels under a

uniform temperature rise at m = 0.05 (1, 2) and 0.1 (3, 4) according to the present theory (1, 3) and

Shen [12] (2, 4) b/a = 1 Remaining parameters are the same as in Fig 2.

Fig 4 Effect of elastic foundations on the nonlinear response of spherical FGM panels at

(K1, K2) = (0, 0) (1), (20, 100) (2), (50, 200) (3), and (100, 250) (4) m = 0 and n = n(z) Remaining

parameters are the same as in Fig 2

0 5 1 0 1 5 2 0 2 5

q, GPa

W, h

0

0 07

0 05

0 03

0 01

.

.

.

.

4 3

1 2

5

0 5 1 5 2 5 3 5 4 5

q, GPa

W, h

0

0 18

0 14

0 10

0 06

0 02

.

.

.

.

.

1 2

3 4

Fig 5 Effects of temperature and elastic foundations on the nonlinear response of spherical FGM

panel at (DT , K1, K2) = (400, 100, 30) (1), (400, 200, 20) (2), (200, 200, 0) (3), (200, 0, 0) (4), and

(0, 0, 0) (5) m = 0 and n = n(z) Remaining parameters are the same as in Fig 2.

Fig 6 Effects of the index N and Poisson’s ratio on the pressure–deflection curves of cylindrical

FGM panels at N = 0 (1, 2) and 1 (2, 3) according to the present theory (1, 3) and Shen [12] (2, 4)

b/a = 1 and m = 0 The remaining parameters are the same as in Fig 2.

q, GPa

W, h

0

0 35

0 25

0 15

0 05

.

.

.

2 3 4