DSpace at VNU: Nonlinear postbuckling of imperfect doubly curved thin shallow FGM shells resting on elastic foundations and subjected to mechanical loads

DSpace at VNU: Nonlinear postbuckling of imperfect doubly curved thin shallow FGM shells resting on elastic foundations...

NoNliNear postbuckliNg of imperfect doubly

curved thiN shallow fgm shells restiNg

oN elastic fouNdatioNs aNd subjected

to mechaNical loads

Nguyen dinh duc * and tran Quoc Quan

Keywords: nonlinear postbuckling, doubly curved thin shallow FGM shell, classical theory of shells,

imperfection, elastic foundation

The nonlinear response of buckling and posbuckling of imperfect thin functionally graded doubly curved thin shallow shells resting on elastic foundations and subjected to some mechanical loads is investigated analytically The elastic moduli of materials, Young’s modulus, and Poisson ratio are all graded in the shell thickness direction according to a simple power-law in terms of volume fractions of constituents All formulations are based on the classical theory of shells with account of geometrical nonlinearity, an initial geometrical imperfection, and a Pasternak-type elastic foundation By employing the Galerkin method, explicit relations for the load–deflection curves of simply supported doubly curved shallow FGM shells are determined The effects of material and geometrical properties, foundation stiffness, and imperfection of shells on the buckling and postbuckling load-carrying capacity of spherical and cylindrical shallow FGM shells are analyzed and discussed.

1 introduction

Functionally graded materials (FGMs), which are microscopically laminated composites made from a mixture of metal and ceramic constituents, have received considerable attention in recent years due to their high performance, good heat resistance capacity, and excellent mechanical characteristics in comparison with those of conventional composites Therefore, the buckling and postbuckling behavior of FGM plate and shell structures under different types of loading attracts the attention

Mechanics of Composite Materials, Vol 49, No 5, November, 2013 (Russian Original Vol 49, No 5, September-October, 2013)

Vietnam National University,Hanoi, Vietnam

*Corresponding author; tel.: +84-4-37547978; fax: +84-4-37547724; e-mail: ducnd@vnu.edu.vn

Russian translation published in Mekhanika Kompozitnykh Materialov, Vol 49, No 5, pp 737-756, September-October, 2013 Original article submitted May 2, 2012; revision submitted February 27, 2013

of many researchers all over the world The postbuckling of cylindrical FGM panels under axial compression were studied

in [1] and under external pressure in [2] by using a higher-order theory of shells in conjunction with the boundary-layer the-ory of buckling of shells The postbuckling of cylindrical FGM panels with piezoelectric actuators in thermal environments was investigated in [3] Cylindrical FGM shells in a thermal environment under various types of loading were treated in [4-7]

by similar methods and different shell theories In these investigations, a semianalytical approach was used to expand the deflection and stress functions in the form of power functions of small parameters, and then iteration was employed to deter-mine the buckling loads and postbuckling curves In [8, 9], the nonlinear buckling of cylindrical FGM shells under axial and external pressure was examined utilizing a semianalytical approach The thermomechanical postbuckling of cylindrical FGM panels with temperature-dependent properties was addressed in [10] The stability of compositionally graded ceramic-metal cylindrical shells under a periodic axial impulsive loading was studied in [11] A geometrically nonlinear analysis of function-ally graded shells was considered in [12] The structural stability of FGM shells subjected to aerothermal loads was investi-gated in [13], but the stability of compositionally graded cylindrical ceramic-metal shells under periodic axial impulsive loadings was explored in [11] In [14], the linear buckling of shallow spherical FGM shells under two types of thermal loads was analyzed An analytical method and an equilibrium criterion, with the assumption of small deflections, was used in [15]

to determine the critical buckling loads of truncated conical FGM shells subjected to mechanical and thermal loadings The stability of truncated conical FGM shells under compression, external pressure, and impulsive and thermal loads was also treated in [16-18] by using an analytical method Analytical investigations into the nonlinear response of thin and moderately thick cylindrical FGM panels [19-21] subjected to mechanical and thermomechanical loads can be found in [19-21], where

an analytical approach to obtaining explicit expressions for the buckling load and postbuckling load-deflection curves in the case of a constant Poisson ratio (ν = const was employed The case where the Poisson ratio n depends on the shell thickness )

coordinate z (ν ν= ( ))z was considered in [21-23] A nonlinear stability analysis of imperfect functionally graded plates and

cylindrical panels with ν ν= ( )z subjected to mechanical and thermal loads can be found in [24] and [25], respectively In

recent years, there have been several works dealing with more complicated FGM shells, for example, spherical, conical, and doubly curved shallow FGM shells In [26], the linear buckling of shallow spherical FGM shells under two types of thermal loads was examined An analytical method and an equilibrium criterion, with the assumption of small deflections, was used

in [27] to determine the critical buckling loads of truncated conical FGM shells subjected to mechanical and thermal loadings The stability of truncated conical FGM shells under compression, external pressure, and impulsive and thermal loads was also treated in [16-18] by utilizing an analytical method An analytical approach to investigating the nonlinear buckling of a shal-low spherical FGM shell under a uniform external pressure, including temperature effects, was used in [28] and of conical panels under mechanical loads in [29] The nonlinear postbuckling of functionally graded plates and shells with stiffeners was explored in [30]

The components of structures widely used in aircraft, reusable space transportation vehicles, and civil engineering are usually supported by an elastic foundation Therefore, it is necessary to include the effects of elastic foundation to better understand the buckling behavior and the load-carrying capacity of plates and shells The postbuckling behavior of flat and curved laminated composite panels resting on Winkler elastic foundations was investigated in [31, 32] In spite of the practical importance and increasing use of FGM structures, investigations into the effects of elastic media on the response of FGM plates and shells are rather scarce The bending behavior of FGM plates on Pasternak-type foundations were studied in [33, 34] by using analytical methods and in [35] with the help of the asymptotic perturbation technique In [36, 37], the postbuck-ling behavior of cylindrical FGM shells subjected to axial compressive loads and internal pressure, surrounded by an elastic medium, and resting on a tensionless elastic foundation of Pasternak type were explored All above-mentioned papers consid-ered FGM plates and shells on an elastic foundation only in the case of a constant Poisson ratio (ν = const across the shell ) thickness

More recently, the nonlinear response of thin doubly curved shallow FGM panels resting on an elastic foundation with

E E z= ( ) and ν ν= ( )z under thermal loads was examined in [38]

Our paper presents an analytical approach to investigating the nonlinear response of buckling and postbuckling of doubly curved thin shallow FGM shells (with elastic moduli of materials, Young’s modulus, and Poisson ratio depending on

the shell thickness coordinate z) resting on elastic foundations and subjected to some mechanical loads All formulations are

based on the classical theory of shells with account of geometrical nonlinearity, an initial geometrical imperfection, and elas-tic foundations The Pasternak model is used to represent the shallow shell – foundation interaction Explicit expressions for the buckling loads and load–deflection curves of simply supported curved shallow FGM shells 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 shells are analyzed and discussed

2 doubly curved fgm shells on elastic foundations

Consider a ceramic-metal doubly curved FGM shell, with the radii of curvature R x and R y , edges lengths a and b, and uniform thickness h , resting on an elastic foundation.

A curvilinear coordinate system ( , , )x y z is introduced whose ( , ) x y surface coincides with the middle surface of the shell and z is the thickness coordinate (−h2£ £z h 2), as shown in Fig 1

The volume fractions of constituents of the shell are assumed to vary across the thickness according to the power law

N

m( )= + , c( ) m( ),





 = −

2

where N is the volume fraction index ( 0£N < ∞ ) The effective properties Preff of the FGM shell are determined by the linear rule of mixture

Pr ( ) Preff z = m mV z( ) Pr+ c cV z( ), (2) where Pr denotes a temperature-independent material property, and the subscripts m and c denite the metal and ceramic constituents, respectively

Expressions for the elastic modulus E z( ) and Poisson ratio ν ν= ( )z are obtained by inserting Eq (1) into Eq (2)

h

N

( ), ( )ν ,ν ,ν ,

 +   + 

where

Ecm =Ec−E m, νcm=νc−νm, (4) and, unlike in other papers, the Poisson ratio is assumed to vary smoothly across the shell thickness, ν ν= ( ).z It is evident

from Eqs (3) and (4) that the upper surface of the shell (z= −h2 is ceramic-rich, while the lower surface () z h= 2 is ) metal-rich, and the percentage of te ceramic constituent in the shallow shell grows when N increases.

x

y

z h

R x

1

R y

Fig 1 Geometry and the coordinate system of a doubly curved FGM shell on an elastic foundation,

where 1 is the shear layer

The shell–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 shell, k2 1 is the modulus of Winkler foundation, and k2 is the modu-lus of the shear layer of foundation 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 doubly curved thin shallow FGM shells [43]:

ε ε γ

ε ε γ

x y xy

x y xy

x y xy z

k k k



















=























 +













0 0













Here, the nonlinear strain–displacement relationships, which are based upon the von Karman theory for moderately large deflections and small strains, are

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

xx yy xy

























=

−

−

−





















, , , ,

where u and n are displacement components along the x and y axes, respectively.

Hooke’s law for an FGM shell is defined as

σ σ

ν ε ε ν ε ε

x, y E ( , )x y ( , ) ,y x

−  + 

ν γ

+

2 1( ) . (6) The force and moment resultants of the FGM shell are determined by

( , ) ( , ) ,

/

/

N M i i i z dz

h

h

=

−∫ σ 1 2

2

The introduction 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

ε ε ( , , ,I I I I k11 21 12 22) x+( , ,I I I I k21 11 22 12, ) ,y

(8)

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

31 32 2

where I i ij( =1 2 3, , ;j=0 1 2 are , , )

z z dz

h

h

2

2 1

=

−

−∫ ( )( )

/

/

ν , I E z z

z z dz

h

h

2

2 1

=

−

−∫ ( ) ( )( )

/

ν ,

h

h

3 2

2

2 1

1 2

=

+



 

= −

/ /

The nonlinear equilibrium equations of the doubly curved thin shallow FGM shell, based on the classical theory of shell, are [38, 39]

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

R

N R

x

y y

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

(9)

It follows from Eq (8) that

ε0x=D I N0 10( x−I N20 y+D w1 ,xx+D w2 ,yy),

ε0y =D I N0 10( y−I N20 x+D w1 ,yy+D w2 ,xx), (10)

γ0

xy = I (N xy+ I w,xy),

where

D

0

102 202 1 10 11 20 21

1

=

− , = − ,

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

Inserting Eq (10) into the expression of M ij (8) and then M ij into the (9) leads to

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

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

∇ + ∇ + , + , + , + N

R

N

x x

y y

+ + − 1 + ∇2 2 =0,, where

P D D1= 0 2, P2 =D I D I D0 11 1( + 21 2)−I12,

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

For an imperfect FGM shell, Eqs (11) are modified to the form

P1 4f P w f yy w xx w xx f xy w xy w xy

∇ + ∇ + , ( , + ,* )− ( + )+

R

f

x

xx y

, ( , + ,* )+ , + , + − 1 + ∇2 2 =0, (13)

where w x y*( , ) is a known function representing the initial small imperfection of the shell The geometrical compatibility equation for the imperfect doubly curved shell is written as

εx yy0, +εy xx0, −γxy xy0, =w,2xy−w w,xx ,yy + 2w w w w w w w

R

w R

x

xx y

From constitutive relations (10), in conjunction with Eq (12), one has

ε0x=D I f0 10( ,yy−I f20 ,xx+D w1 ,xx+D w2 ,yy),

ε0y =D I f0 10( ,xx−I f20 ,yy+D w1 ,yy+D w2 ,xx), (15)

γ0

xy = I (−f,xy+ I w,xy).

Inserting Eq (15) into Eq (14) gives the compatibility equation of the imperfect doubly curved FGM shell as

∇ + ∇ −  − + −







4

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

R

w R

x

xx y

, ,* − , ,* − , − ,  ,



 = 0 (16)

1

= , =

Equations (13) and (16), which are nonlinear with respect to the variables w and f , are used to investigate the

sta-bility of thin doubly curved shallow FGM shells resting on elastic foundations and subjected to mechanical, thermal, and thermomechanical loads

In the present study, the edges of curved shallow shells are assumed to be simply supported Depending on in-plane restraints at the edges, three cases of boundary conditions, named Cases 1, 2 and 3, will be considered [17, 18, 25-27]

Case 1 All four edges of the shell are simply supported and freely movable (FM) The associated boundary

condi-tions are

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

w N= xy =M y=0 , N y=N y0 aty=0,b

Case 2 All four edges of the shell are simply supported and immovable (IM) In this case, boundary conditions are

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

w v M= = y =0 , N y=N y0 aty=0,b (18)

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

immovable For this case, the boundary conditions are defined as

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

w v M= = y=0 , N y=N y0 at y=0,b

where N x0 and N y0are in-plane compressive loads at the movable edges (i.e., Case 1 and the first of Case 3) or fictitious compressive loads at the immovable edges (i.e., Case 2 and the second of Case 3)

The approximate solutions for w and f satisfying boundary conditions (16)-(18) are assumed to be [19, 20, 38]

w W= sinλm xsinδn 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 cos λ cos δ sinλ sinδ , where λm=m aπ , δn =n bπ , W is the deflection amplitude, and µ− is the imperfection parameter The coefficients A i,

i = 1-3, are determined by inserting Eqs (19) into Eq (16):

m

2 2

= δ +

λ ( µ ) , A P m W W h

n

2 2

= λ +

δ ( µ ), A P

n x

m y

2 2 2

3

= +





  −

λ δ

δ λ

W Introducing Eqs (19) into Eq (13) and applying the Galerkin procedure to the resulting equation yield

m n

n x

m

π

λ δ

2

3 1 4

4 ( + ) +





  + −( ) ( + )









−

+





  − ( + )−







 +

P

n x

m

4

2 2 2

λ δ

3 2 4 2 2

3

λ δ

λ δ

m n

n x

m y

P

+





 −



















 +

+  +



 −















 ( + )−

P

m n

m x

n

1 4 6

4

λ δ

64 (λ +δ ) ( +µ ) ( +2µ )

−ab N( x m+N y n) (W+ h)

4 0λ2 0δ2 µ + +





 + =

λ δm n x x λ δ

y

N R

N R

where m and n are odd numbers This is the basic equation governing the nonlinear response of imperfect doubly curved

shal-low FGM shells in mechanical, thermal, and thermomechanical loadings In what folshal-lows, some common mechanical loading conditions will be considered in this paper

4 Nonlinear stability analysis

Consider a simply supported doubly curved shallow FGM shell, with all its edges movable, resting on an elastic foundation Two cases of mechanical loads will be analyzed

4.1 doubly curved shallow fgm shell under a uniform external pressure

Consider a doubly curved FGM shell subjected only to a uniform external pressure in the absence of thermal and

compressive edge loads In this case, N x0 =N y0=0, and Eq (20) is reduced to

q b W b W W= 11 + 21 ( +µ)+b W W31 ( +2µ)+b W W41 ( +µ) (W+2µ), (21) where

b mn B P K

B

mn B P n R m B R

a h

11

2 4

2 1 4

2 2

2 2 2

+

22 2

4 2

2 2

16

( ) +mn B P K B (m B +n )

a

π

mn P P P

a

3 1 4

6

2 1 3

aa2+n2 2

B m B n

m n B

a

2 1

2 2 4 3

3 2 2 2 2

2 2 4 2 2

3

2

+

B h

3 4

3 ,

B

m n B P P B

h

a h

3 1

2

3

2 2 4 2

1 4 4

=−π ( + )+ π

B h m B a n

4

256

= π ( + )

with

B h =b h/ , B a =b a/ , W W h= / , R ax =a R/ x, R by =b R/ y,

K k a P

4 2

= , K k a

P

2 2

= , P P

h

1= 1 , P P

h

2= 23 , P P

h

3= 32 , P P

h

4 = 4 Equation (21) is a closed-form relation for the pressure–deflection curves of doubly curved shallow FGM shells under

a uniform external pressure

For a perfect shallow shell (µ = 0 , Eq (21) leads to)

q b W= 11 + b b W+ +b W

2

1 3

1 2

4

1 3 ( )

4.2 cylindrical shallow fgm shells under axial compressive loads

Consider a cylindrical shallow FGM shell (R x→ ∞)supported by an elastic foundation and subjected to an axial

compressive load F x uniformly distributed at the curved edges x=0, in the absence of external pressure In this case, q = 0, a

N y0 =0, and N x0 = −F h x , and Eq (20) leads to

W W

x =

+ + +

+

+ + ( + )

2

2 µ

µ

where

B

P P P m B n

m B B

by h

a

a h

2

2 1 3 2 2 2 2

2 2 2

= −( + ) −π ( − )( + ) + P m B n K

m B

P m B R

m B n

B P K

a h

a by a

a h

2 2

2

2 1

2 2 +

+

B m B n

nP mB

22

2 4

3 2

32 3

32 3

= −

+

m B B

P P n mB

by

32 3 2 24 1 42

2 3

16 3

m B B

a

a h

42

2

2 2 2 16

=π ( + )

For a perfect cylindrical shell (µ = 0 subjected only to an axial compressive load F) x, Eq (22) leads to

F x=b12+ b +b W b W+

22 32 42 2 ( ) ,

from which the upper buckling compressive load can be obtained at W → 0 as

F x=b12=−(P P P n R+ )( +m B R )− ( − )( + )

m B B

m B

a h

a a

2

2

2

π

22B h4

+ ( + ) + +

+

P m B n K

m B

P n R m B R

B a

h

a

a

2 2

2 2 2 2 2 2

2

π

PP K

m2 2π 2 1B h2

If n =const, then P P1= 3 =0, and Eq (22) becomes

W W W

x =

+ + +

+

+

2 µ

µ

µ , where

m B B

m B

P n R a

a h

a h

ax

13

2

2 2 2 2 2

2 2 2

2 2

4 2

P B K

a by a

a h

2 2 2 2 2 2

2 2 1

2 2 2

)

π ( + ) + π

mB m B n

32 3

+

,

m n B B

a h

33

3 2 2

2 3

= − ( + )

m B B

a

a h

43

2

2 2 2 16

=π ( + )

4.3 Numerical results and discussion

In this section, results for a shallow ceramic-metal shell with Em = 70 GPa and Ec = 380 GPa [19, 20, 38] are presented

Figure 2 shows the effect of the volume fraction index N on the postbuckling of spherical shallow FGM shells under

a uniform external pressure q As seen, the mechanical load q is higher at lower values of N

Figure 3 illustrates the effects of elastic foundation on the nonlinear response of spherical shallow FGM shells a under

uniform external pressure The effect of the modulus of Pasternak foundation K2on the critical uniform external pressure is

greater than that of the modulus of Winkler foundation K1 This conclusion was also made in [31-38]

As a part of the effects of imperfection, the postbuckling load–deflection curves of spherical FGM shells are shown

in Fig 4 Figures 5-8 illustrate the effects of geometrical parameters on the postbuckling behavior of spherical shallow FGM shells when ν ν= ( )z All results in Figs 2-6 are obtained from Eq (21) for spherical shallow FGM shells The results in

Figs 7-8 are obtained from Eq (21) for doubly curved shallow FGM shells

The postbuckling load–deflection curves of cylindrical FGM shells are shown in Fig 9 For cylindrical shallow shells

in the case K1=K2 =0 (without an elastic foundation) and n = const the results obtained are seen in Fig 10 and coincide with those presented in [19]

0

1.4 1.2 1.0 0.8 0.6 0.4 0.2

q, GPa

1 2

N = 0

W h/

0

1.4 1.2 1.0 0.8 0.6 0.4 0.2

q, GPa

1 2

W h/

3 4

Fig 2 Pressure–deflection curves q–W/h of spherical shallow FGM shells at m = n = 1, b/a = 1, b/h = 20,

a R x = b R y = 0.5, m = 0.1, K1 = 100, K2 = 30, and ν ν= ( )z

Fig 3.The same at m (K1, K2) = (50, 20) (1), (100, 10) (2), (100, 0) (3), and (0, 0) (4)

0

1.4 1.2 1.0 0.8 0.6 0.4 0.2

q, GPa

= 0.5

W h/

0.2

 0 0.2 0.5

Fig 4 Postbuckling curves of spherical shallow FGM shells The values of parameters as in Fig 2

Figures 11 and 12 show the effect of the volume fraction index N and the Poisson ratio on the postbuckling behavior

of cylindrical shallow FGM shells with movable edges under axial compressive loads F x In the case K1=K2 =0 (without

an elastic foundation) and n = const, these results coincide with those presented in [19]

Figure 13 illustrates the effects of elastic foundations on the postbuckling behavior of cylindrical shallow FGM shells

with movable edges subjected to an axial compression F x Obviously, both the buckling loads and the postbuckling equilibrium paths of shallow FGM shells become considerably higher due to the support of elastic foundations, especially of the Pasternak one However, the severity of snap-through instability is almost unchanged at different values of foundation parameters

Figures 14-16 depict the effects of geometrical parameters on the postbuckling behavior of cylindrical shallow FGM shells with ν ν= ( )z The effects of Poisson ratio on the pressure–deflection curves of cylindrical shallow FGM shells are

illustrated in Fig 17

0

1.4 1.2 1.0 0.8 0.6 0.4 0.2

q, GPa

b a/ = 0.75

W h/

1.5 1.0

0

0.7 0.6 0.5 0.4 0.3 0.2 0.1

q, GPa

b h/ = 20

W h/

40

30

Fig 5 Pressure–deflection curves q–W/h of spherical shallow FGM shells at m = n = 1, N = 1,

b/h = 20, K1 = 100, K2 = 30, a R x = b/R y = 0.5, m = 0.1, and ν ν= ( )z with b/a = 0.75 (1), 1 (2), and

1.5 (3)

Fig 6 The same at b/a = 1.

0

2.5 2.0 1.5 1.0 0.5

q, GPa

W h/

a R/ x= 1.5

0.75 1.0

0

2.5 2.0 1.5 1.0 0.5

q, GPa

W h/

b R/ y= 1.5

0.75 1.0

Fig 7 Pressure–deflection curves q–W/h of doubly curved shallow FGM shells at various values of

a R x The values of parameters as in Fig 2

Fig 8 The same at various values of b R y