Nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial compression and radial pressure

Nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial compression and radial pressure

International Journal of Non-Linear Mechanics 44 (2009) 209 -- 218 Contents lists available at ScienceDirect International Journal of Non-Linear Mec...

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International Journal of Non-Linear Mechanics 44 (2009) 209 -- 218

Contents lists available at ScienceDirect

International Journal of Non-Linear Mechanics journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / n l m

Nonlinear buckling and postbuckling of heated functionally graded cylindrical shells under combined axial compression and radial pressure Huaiwei Huang, Qiang Han ∗ Department of Engineering Mechanics, School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510640, PR China

A R T I C L E

I N F O

Article history: Received 11 June 2008 Received in revised form 24 October 2008 Accepted 12 November 2008

Keywords: FGMs Cylindrical shells Non-linear buckling Postbuckling Axial compression Lateral pressure Thermal environment

A B S T R A C T

The nonlinear large deflection theory of cylindrical shells is extended to discuss nonlinear buckling and postbuckling behaviors of functionally graded (FG) cylindrical shells which are synchronously subjected to axial compression and lateral loads. In this analysis, the non-linear strain–displacement relations of large deformation and the Ritz energy method are used. The material properties of the shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. Meanwhile, by taking the temperature-dependent material properties into account, various effects of external thermal environment are also investigated. The non-linear critical condition is found by defining the possible lowest point of external force. Numerical results show various effects of the inhomogeneous parameter, dimensional parameters and external thermal environments on non-linear buckling behaviors of combine-loaded FG cylindrical shells. In addition, the postbuckling equilibrium paths are also plotted for axially loaded pre-pressured FG cylindrical shells and there is an interesting mode jump exhibited. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Functionally graded materials (FGMs) were born in 1984 [1]. This new type of composite material is usually a mixture of ceramic and metallic constituent materials. By continuously changing in the volume fraction of the constituent materials, the material properties of FGMs vary smoothly through the thickness. This effectively avoids thermal stress concentration seen in the traditional laminated composite materials. The prime advantage of FGMs is that the ceramic component provides high temperature resistance due to its low thermoconductivity while the ductile metal component prevents fracture. FGMs were initially applied as a kind of thermal barrier materials for aerospace structures and fusion reactors where extremely high temperature or temperature gradient exists. Due to the increasing demands for high heat-resisting, lightweight structures, the studies on functionally graded (FG) structures, such as FG plates and FG cylindrical shells, have attracted much attention. As a heat-resisting material, FGMs were initially studied in the thermal response of FG structures (see [2–4]). Noda and Praveen pointed out that it is important to consider the temperaturedependent material properties in thermo-elastic analyses.

∗ Corresponding author. Tel./fax: +86 20 87114460. E-mail address: [email protected] (Q. Han). 0020-7462/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2008.11.016

In the literature available, many discussed vibration characters. Loy et al. [5] and Pradhan et al. [6] presented the importance of the boundary condition effect on free vibration characters of FG cylindrical shells. Ng et al. [7] processed a forced vibration analysis for simply supported FG cylindrical shells under axial periodic load. Chen et al. [8] and Patel et al. [9] studied free vibration problems of FG cylindrical shells and FG elliptic cylindrical shells. Rajesh et al. [10] investigated buckling behaviors and free vibration characters of FG conical shells. Hiroyuki [11] discussed the free vibration and stability problems of FG plates by the high order shear deformation theory. There were many presented for buckling behaviors. By using the first order shear deformation theory, Shahsiah and Eslami [12] and Wu et al. [13] considered the effects of various temperature distributions on thermal buckling of simply supported FG cylindrical shells, but the temperature-dependent material properties were not included. Kadoli and Ganesan [14] investigated thermal buckling behaviors and vibration characters of FG cylindrical shells. In the paper, the one-dimensional steady thermal conduction and the temperature-dependent material properties were simultaneously taken into account. Li and Batrab [15] studied buckling behaviors of an axial compressed three-layer circular cylindrical shell with the middle layer made of FGMs. Their results showed that the critical load was markedly influenced by the average Young's modulus of FGMs and the radius-to-thickness ratio of the shell. Employing Donnell shell theory and a three-dimensional finite element code, Najafizadeh et al. [16] studied linear buckling behaviors of axially

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H. Huang, Q. Han / International Journal of Non-Linear Mechanics 44 (2009) 209 -- 218

compressed stiffened FG cylindrical shells. Based on the non-linear prebuckling consistent theory and considering the temperaturedependent material properties, Huang and Han [17] analyzed linear buckling behaviors of imperfect FG cylindrical shells under axial loads. The results showed axially compressed FG cylindrical shells are of imperfect sensitivity as observed in their isotropic counterparts. The literatures on dynamic buckling of FG structures are still limited. Sofiyev et al. [18–21] covered a range of linear dynamic buckling studies of FG cylindrical shells and conical shells under time-dependent torsional, lateral and axial loads. But it is regretful that thermal effects were not included in these papers. Besides, by using the high order shear deformation theory and the finite element method, Shariyat [22] considered the coupling thermo-elastic effects on buckling and postbuckling of pre-stressed imperfect FG cylindrical shells under a thermal impact. Thin-walled cylindrical shells in engineering applications are usually loaded by not simply one of the three buckling relevant membrane forces, i.e. axial compression, circumferential compression and shear, but a combination of them. In this paper, the non-linear large deflection theory of cylindrical shells is extended to discuss nonlinear buckling and postbuckling behaviors of FG cylindrical shells which are synchronously subjected to axial and lateral loads. By taking the temperature-dependent material properties into account, various effects of external thermal environment are also investigated. 2. Formulation of the problem Consider a FG cylindrical shell with mean radius R, thickness h, and length L. The middle surface of the cylindrical shell is referred to the coordinates x, y, z as shown in Fig. 1. 2.1. Material properties of FGMs The material properties of FGMs' constituent materials Pc and Pm (where the subscripts “c” and “m” denote ceram and metal, respectively) are usually expressed as the following function of temperature T: Pc (T)

Pm (T) = c0 (c−1 T −1 + 1 + c1 T + c2 T 2 + c3 T 3 ),

or

(1)

where c0 , c−1 , c1 , c2 , c3 are temperature coefficients which are unique to FGMs' constituents. The material properties of FGMs P are related not only to the material properties of the constituents, but also to their volume fractions Vc and Vm . We assume that Vc follows a simple power law distribution according to the power law exponent k, which represents the inhomogeneity of FGMs: P = Pc Vc + Pm Vm ,

Vc + Vm = 1,

Vc = (0.5 + z/h)k .

(2)

Using Eq. (2), the material properties of FGMs are written as P(z) = (Pc − Pm )(0.5 + z/h)k + Pm .

o

R

(3)

y

z

x

Then P = Pc when z = h/2 and P = Pm when z = −h/2. Accordingly, the Young's modulus E(z), Poisson's ratio (z), thermal expansion coefficient ¯ (z) and thermoconductivity (z) of FGMs can be written in a similar form of Eq. (3): E(z) = (Ec − Em )(0.5 + z/h)k + Em ,

(z) = (c − m )(0.5 + z/h)k + m , ¯ (z) = (¯ c − ¯ m )(0.5 + z/h)k + ¯ m , (z) = (c − m )(0.5 + z/h)k + m .

(4)

2.2. Temperature fields in FGMs FGMs usually serve in thermal environment, thus, effects of thermal environment should be included. The following three thermal cases are to be discussed in this paper. Case 1: Temperature rise T uniformly, and the temperature field through the shell thickness is expressed as T(z) = T0 + T where T0 is the initial temperature of environment. Case 2: Temperature rises Tc on the ceramic surface and keeps unchanged on the metallic surface. The temperature field is assumed to be linear distribution through the shell thickness, i.e. T(z) = T0 + Tc (0.5 + z/h). Case 3: Temperature rises Tc on the ceramic surface and keeps unchanged on the metallic surface. The temperature field is given by solving the following one-dimensional steady thermal conduction equation: 



j jT(z) (z) = 0, jz jz

T(−h/2) = T0 ,

T(h/2) = T0 + Tc .

(5)

For simplification, a series expansion method according to [13] is applied to Eqs. (5) and the temperature field is approximately expressed as +∞ T(z) = T0 + Tc

i=0







c − m i z ik+1 1 0.5 + m h ik + 1 .   +∞ c − m i 1 − i=0 m ik + 1 −

(6)

During the process of thermal conduction, material properties are coupled with thermal environment. In other words, FGMs' material properties and the temperature field interact to each other. Thus, the temperature field in FGMs should be obtained by following a similar iterative method reported in [14], and the iteration procedures are as follows. Step 1: According to Eqs. (1) and (4), under a given initial temperature T0 , (z) is obtain. Step 2: Using (z) calculated in Step 1, substituting it into Eq. (6) (with i = 0, 1, . . . , 10) yields a temperature field T1 (z). Step 3: By Step 1, recalculate a new (z) under T1 (z), and then following Step 2 derives a new temperature field T2 (z). Step 4: Repeating the above procedures for r times derives Tr (z) until the expression Max(|Tr (z) − T(r−1) (z)|) < 0.001 holds for convergence. 2.3. Basic equations According to the non-linear strain–displacement relations of cylindrical shells, the mid-surface strain components are

h L Fig. 1. Geometry and the coordinate system of FG cylindrical shells.

1 2

0x = U,x + W,x2 ,

0y = V,y −

0xy = U,y + V,x + W,x W,y ,

1 2 W + W,y , R 2 (7)

H. Huang, Q. Han / International Journal of Non-Linear Mechanics 44 (2009) 209 -- 218

in which U(x, y), V(x, y), and W(x, y) are the displacements along x-, y-, and z-axes, respectively. Subscripts following a comma stand for partial differentiations. Assume there is a temperature rise Tri (z) through the shell thickness. Including thermal effects, the strain components can be written as follows:

x = 0x − zW ,xx − ¯ Tri ,

y = 0y − zW ,yy − ¯ Tri ,

xy = 0xy − 2zW ,xy ,

(8)

The stress–strain relations are given as

x = K(z)[x + (z)y ], y = K(z)[y + (z)x ], xy = 12 K(z)[1 − (z)]xy ,

(9)

where K(z) = E(z)/(1 − (z)2 ). For thin cylindrical shells (h/R>1), the approximate forms of internal force and moment resultants are {(Nx , Ny , Nxy ), (Mx , My , Mxy )} =

 h/2 −h/2

{x , y , xy }(1, z) dz.

(10)

Substituting of Eqs. (13) into Eqs. (11) obtains

0x = J0 [A10 ,yy − A20 ,xx + J1 W,xx + J2 W,yy − J3 1 ], 0y = J0 [A10 ,xx − A20 ,yy + J2 W,xx + J1 W,yy − J3 1 ], 0xy = (2A31 W,xy − ,xy )/A30 ,



where C1 = J2 /A10 , C2 = 1/(A10 J0 ). For thin cylindrical shells, the strain energy is approximately given as Uin =

A10

⎢A ⎢ 20 ⎢ ⎢ 0 ⎢ ⎢ ⎢ A11 ⎢ ⎢ ⎣ A21

A20

0

A11

A21

A10

0

A21

A11

0

A30

0

0

A21

0

A12

A22

A11

0

A22

A12

A31 0 0 ⎫ ⎧ ⎫ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ⎬ 0 xy + , × ⎪ ⎪ ⎪ ⎪ ⎪ −W,xx ⎪ ⎪ ⎪ 2⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −W,yy ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ⎩ ⎭ 0 −2W,xy 0 ⎧ 0 x ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ y ⎪ ⎪ ⎪ ⎨ 0

0

0

 h/2 −h/2

K(z)zj dz,

A2j =

 h/2 −h/2

−h/2

K1 = (11)

(17)

K2 =

A10 (A211 + A221 ) − A210 A12 + A20 (A20 A12 − 2A11 A21 ) 2(A220 − A210 ) A20 (A211 + A221 ) − A220 A22 + A10 (A10 A22 − 2A11 A21 ) A210 − A220

,

,

K4 = A10 /[2(A210 − A220 )], K5 = A20 /(A220 − A210 ),

K6 = 1/(A10 − A20 ),

K01 = 1/(A10 + A20 ), K02 = (A11 + A21 )/(A10 + A20 ),  h/2

3 = [1 + (z)]K(z)¯ (z)2 Tri (z)2 z dz. Assume that the shell is synchronously subjected to axial and lateral loads. With the aid of Eqs. (7) and (14), the work done by the external forces is Uex = q =q

 2 R  L 0

0

 2 R  L 0

0

W dx dy − 0x h W dx dy − 0x h

 2 R  L 0

0

0

0

 2 R  L

(U,x + J0 J3 1 ) dx dy [J0 (A10 ,yy

2 ] dx dy, − A20 ,xx + J1 W,xx + J2 W,yy ) − 12 W,x

(12)

Introduce Airy's stress function (x, y) satisfying Nxy = − ,xy .

2 + W2 ) + K W W [K1 (W,xx 2 ,xx ,yy ,yy

−h/2

[1 + (z)]K(z)¯ (z)Tri (z)z dz.

Ny = ,xx ,

0

K3 = A12 − A22 + (A11 − A21 )2 /(A20 − A10 ),

From Eqs. (7), the compatible equation is obtained as

Nx = ,yy ,

0

where

(z)K(z)zj dz,

1 R

(16)

+ (K02 1 − 2 )(W,xx + W,yy )] dx dy,

A32

2 −W W . 0x,yy + 0y,xx − 0xy,xy = − W,xx + W,xy ,xx ,yy

(x x + y y + xy xy ) dx dy dz,

+ K5 ,xx ,yy + K6 2,xy + 3 − K01 21

−h/2

 h/2



2 + K ( 2 + 2 ) + K3 W,xy 4 ,xx ,yy

 1 h/2 [1 − (z)]K(z)zj dz, A3j = 2 −h/2  h/2

1 = − [1 + (z)]K(z)¯ (z)Tri (z) dz,

2 = −



 2 R  L

0 ⎥ ⎥ ⎥ A31 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎦

where Aij (i = 1, 2, 3, j = 0, 1, 2) are stiffness coefficients and j(0, 1, 2) represent the coefficients of membrane, flexural, and coupled stiffness, respectively. For isotropic cylindrical shells, we have Ai1 =0. 1 and 2 are additional internal force and moment caused by Tri (z). They are defined as follows: A1j =

1 2

where is the volume field of the shell. By using Eqs. (8), (9) and (14), Eq. (16) becomes Uin =



(14)

where J0 = 1/(A210 − A220 ), J1 = A10 A11 − A20 A21 , J2 = A10 A21 − A20 A11 , J3 = A10 − A20 . Considering Eqs. (14) in Eq. (12), we have the compatible equation rewritten as   1 2 +W W (15) W,xx − W,xy ∇ 4 + C1 ∇ 4 W + C2 ,xx ,yy = 0, R

Considering Eqs. (8) and (9) in Eq. (10) yields ⎧ ⎫ Nx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ny ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Nxy ⎪ ⎬ = ⎪ ⎪ Mx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ My ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ Mxy

211

(13)

(18)

where 0x is the average axial stress on the shell's end sections, positive when the shell is axially compressed. q is uniform radial pressure. Then, the total potential energy of the system is UTPE = Uin − Uex .

(19)

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H. Huang, Q. Han / International Journal of Non-Linear Mechanics 44 (2009) 209 -- 218

With the aid of Eqs. (7) and (14), the following circumferential closed condition should be satisfied for a cylindrical shell:  2 R  L 0

0

V,y dx dy =

 2 R  L  J0 (A10 ,xx − A20 ,yy + J2 W,xx 0

0

+J1 W,yy − J3 1 ) +

Uin = 12 RL{K1 f12 (2 + 2 )2 + 4(2K1 f22 4 − K01 21 + 3 )

 1 2 W dx dy = 0. (20) − W,y R 2

¯ x can be Similarly, the average end-shortening ratio of the shell given as  2 R  L

1 U,x dx dy 2 RL 0 0  2 R  L  1 J0 (A10 ,yy − A20 ,xx + J1 W,xx = − 2 RL 0 0  1 2 dx dy. +J2 W,yy − J3 1 ) − W,x 2

¯ x = −

Using Eq. (22) and (24), and noting the relations: K5 + K6 = 2K4 , K2 + K3 = 2K1 , we have Eqs. (17) and (18) written as

+ K4 [32b21 4 + 32b22 4 + b23 (2 + 2 )2 + b24 (92 + 2 )2 ] + 4h2 [K5 0x 0y + K4 (20x + 20y )]}, Uex = RL{q(2f0 + f2 ) +

(25)

1  h[(f12 + 2f22 )2 4 0x

+ 8J0 h(A10 0x − A20 0y )]}.

(26)

Substituting Eqs. (25) and (26) into Eq. (19), we obtain the total potential energy, to which the Ritz energy method is then applied as (21)

3. Solution of the problem

jUTPE jUTPE jUTPE = = = 0. jf0 jf1 jf2

(27)

By noting Eqs. (22) and (24), Eq. (20) becomes According to [23], the deflection of axially and laterally combineloaded shells can be expressed as follows: W(x, y) = f0 + f1 sin x sin y + f2 sin2 x,

(22)

where  = m /L,  = n/R, and m, n are the axial half-wave numbers along x-axis and the wave numbers along y-axis. f0 , f1 , and f2 are unknown amplitudes. f0 denotes the uniform deflection in the prebuckling state. f1 sin x sin y represents the linear buckling shape, while f2 sin2 x represents the non-linear diamond buckling shape of large deflection. This buckling mode had been justified by a modal analysis for postbuckling of isotropic cylindrical shells reported by Goncalves and Prado [24]. Substitution of Eq. (22) into Eq. (14) yields ∇ 4 = b01 cos 2x + b02 cos 2y + b03 sin x sin y + b04 sin 3x sin y, where

b04 = C2 f1 f2 2 2 . Then, the general solution of is given as

y2 x2 − 0y h , 2 2

(24)

where 0y is the average circumferential stress, positive when the shell is circumferentially compressed, and

a4 = − a6 =

C2 2 2 2 2

(2 +  )

,

C2  2 2 . 2 2 2

(9 +  )

a2 =

a5 =

b3 = a4 f1 f2 + a5 f1 , 322

,

C2  2 R(2 + 2 )2

a3 =

+ [4A10 J0 (A20 J0 + K5 )0x h − 4A210 J02 qR − K4 (f12 2 − 8A20 J0 0x h + 8J0 J3 1 )]} = 0.

0y = qR/h.

(29)

(30)

C2  2 322

− C1 ,

1 2  0x h 2 ,

(31)

q = H06 f2 + H07 f12 + H08 f12 f2 − 2 f2 0x h,

(32)

f12 = −

= b1 cos 2x + b2 cos 2y + b3 sin x sin y

C2 2

jUTPE L {4K4 (2f0 + f2 ) = jf0 2A210 J02 R

circumferential stress should be related only to the radial pressure when the shell can freely move in radial direction. Then, considering Eq. (30) in the last two partial derivatives of Eq. (27) and noting that f1  0, we have

b03 = [C2 2 /R − C2 f2 2 2 − C1 (2 + 2 )2 ]f1 ,

  1 C a1 = 4C1 − 2 , 8 R2

Substituting of the above equation into Eq. (19), then, from Eq. (27), we have

jUTPE / jf0 =0 hold simultaneously. This indicates that the prebuckling

b02 = C2 f12 2 2 /2,

b2 = a3 f12 ,

(28)

So as the above equation is satisfied, the close condition and

b01 = (16C1 f2 2 + C2 f12 2 − 4C2 f2 /R)2 /2,

b1 = a1 f2 + a2 f12 ,

⎞ f12 2 2f + f 0 2 ⎝ 0y = + A20 0x h − J3 1 ⎠ . − 2J0 R 8J0 A10 h

Considering Eqs. (28) and (29) obtains (23)

+ b4 sin 3x sin y − 0x h



1

,

b4 = a6 f1 f2 ,

H01 + H04 f22 + H05 f2 − H03

where H01 = (K1 + K4 a25 )(2 + 2 )2 ,

H03 = 64K4 (a22 4 + a23 4 ),

H04 = K4 [a24 (2 + 2 )2 + a26 (92 + 2 )2 ], H05 = 2K4 [32a1 a2 4 + a4 a5 (2 + 2 )2 ], H06 = 84 (K1 + 4K4 a21 ), H07 = K4 [32a1 a2 4 + a4 a5 (2 + 2 )2 ], H08 = K4 [a24 (2 + 2 )2 + a26 (92 + 2 )2 ].

H. Huang, Q. Han / International Journal of Non-Linear Mechanics 44 (2009) 209 -- 218

From Eqs. (31) and (32), we obtain

0x h =

2 [H07 + (H08 − 2H03 )f2 ]2

By introducing Eq. (30) in, Eq. (37) degenerates into the linear critical load of isotropic shells which is quite consistent with the classical linear result (see [25]) as   4 h2 (2 + 2 )2 0ycl = E + . R2 2 (2 + 2 )2 122 (1 − 2 )

[H01 H07 + H03 q + (H01 H08

+ H05 H07 − H03 H06 )f2 + (H04 H07 + H05 H08 )f22 + H04 H08 f23 ],

(33)

which can be used to derive the non-linear critical condition for axially and laterally combine-loaded FG cylindrical shell. By omitting the non-linear buckling shape in Eqs. (22) and (33), i.e. f2 = 0. Eq. (33) turns into

0x h =

2H01

2

+

H03 q

H07 2

.

⎡  2 2 2  1 ⎣ A10 − A20 0x h = + (A10 A12 − A211 ) A10 R2  2 + 2

 2 + 2 × 

2 ⎤ ⎦ + 2(A20 A11 − A10 A21 ) . R

(35)

By minimizing 0x with regard to ((2 + 2 )/ )2 , the linear critical axial compression of FG cylindrical shells is obtained as

0xcl =

 2 ( (A210 − A220 )(A10 A12 − A211 ) A10 Rh + (A20 A11 − A10 A21 )).

(36)

By setting k = 0, Eq. (36) degenerates into theclassic critical load of axially loaded isotropic shells, i.e. cl = Eh/[R 3(1 − 2 )]. If 0x = 0, from Eq. (34), the linear critical radial pressure of FG cylindrical shells is thus obtained as q=

2 2 C2 R3 (4 + 4 )(2 + 2 )2

With the aid of Eq. (30), substituting Eqs. (22) and (24) into Eq. (21) yields the expression of the end-shortening ratio of the shell

¯ x = =

[C22 K4 4 − 2C1 C2 K4 R2 (2 + 2 )2

+ (K1 + C12 K4 )R2 (4 + 4 )][C2 (94 + 22 2 + 4 ) − 12C1 R2 (2 + 2 )2 ].

(37)

2 32

2 32

(4f12 + 3f22 ) + J0 (A10 0x h − A20 0y h + J3 1 ) 

3f22 −

4 H03

  1 H01 + H04 f22 + H05 f2 − 2 0x h 2

+ J0 (A10 0x h − A20 0y h + J3 1 ).

(34)

Eq. (34) can be used to obtain the interaction relation between axial and lateral linear critical loads, by minimizing 0x (or q) under a given q (or 0x ) value under various combination of (m, n). If q = 0, then we have



213

(38)

4. Numerical results In the following discussions, the constituents of FGMs are chose to be ceramic zirconium oxide (referred to as Zirconia) and metallic titanium alloy (referred to as Ti6Al4V ). The temperature coefficients c0 , c−1 , c1 , c2 , c3 for these constituents in Eq. (1) are listed in Table 1 from [3]. Seeing that the Poisson's ratios  of the two constituent materials are insensitive to variation of temperature, we assume  to be a constant 0.3. 4.1. Non-linear buckling results and discussions As the aforementioned, the non-linear critical condition of large deflection can be derived from Eq. (33). We define the non-linear critical condition as the possible lowest point of external force. Thus, the specific solution procedures are exhibited as follows: by using Eq. (33) with a given q, a series of 0x versus f2 curves can be drawn under various combinations of the mode (m, n) (see Fig. 2(a)). Alternately, we can also derive a series of q versus f2 curves, under a given 0x and various combinations of (m, n) (see Fig. 2(b)). From the lowest of these curves, an envelope curve is obtained. The lowest point of the envelope curve is regarded as the non-linear critical condition with the non-linear critical axial stress cr or radial pressure qcr and the corresponding non-linear buckling mode (m, n). Fig. 3(a) plots the relation curves of cr versus k for pre-pressured FG cylindrical shells, while Fig. 3(b) plots the relation curves of qcr versus k for pre-compressed FG cylindrical shells. The parameters used in the calculation and the non-linear buckling modes are given in the figure as well. k-axes are in the logarithmic form. As shown, the critical load (either cr or qcr ) decreases with the increase of k. The prime reason for the fall of the critical loads is that: higher value of k corresponds to a metal-richer shell which usually has less stiffness than a ceram-richer one. It is clear that cr decreases when

Table 1 Temperature coefficients for the material properties of FGMs' constituents. Material properties Zirconia Ec (Pa)

c ¯ c (1/K) c (W/m K) Ti6Al4V Em (Pa)

m ¯ m (1/K) m (W/m K)

c0

c−1

c1

c2

c3

244.26596 × 109 0.2882 12.7657 × 10−6 1.7

0 0 0 0

−1.3707 × 10−3 1.13345 × 10−4 −0.00149 0.0001276

1.21393 × 10−6 0 0.1 × 10−5 0.66485 × 10−5

−3.681378 × 10−10 0 −0.6775 × 10−11 0

122.55676 × 109 0.28838235 7.57876 × 10−6 1.20947

0 0 0 0

−4.58635 × 10−4 1.12136 × 10−4 0.00065 0.0139375

0 0 0.31467 × 10−6 0

0 0 0 0

214

H. Huang, Q. Han / International Journal of Non-Linear Mechanics 44 (2009) 209 -- 218

0x (MPa) q=50kPa,T=300,k=1, L/R=1,R/h=200, m=1,n=9-4

400 The linear critical condition

n=9 n=8

300

n=4 n=5

n=7 n=6

200 The envelope curves 100

0

The nonlinear critical condition with the nonlinear critical load cr and the nonlinear buckling mode(m,n)=(1,6)

The envelope curves 2

4

6

8

10

f2/h q(kPa) 250

The linear critical condition

n=10

n=9 n=8

n=6

n=7

n=5

n=11

200

150 The envelope curves 100

50

0

The envelope curves s0x=50MPa, T=300K,k=1, L/R=1,R/h=200, m=1,n=11-5

1

The nonlinear critical condition with the nonlinear critical load qcr and the nonlinear buckling mode(m,n)=(1,6) 2

3 f2h

4

5

6

Fig. 2. Diagrammatic sketch of solving the non-linear critical load and the buckling mode: (a) the non-linear critical axial stress; (b) the non-linear critical lateral pressure.

p increases (see Fig. 3(a)) and similarly pcr decreases when 0x increases (see Fig. 3(b)). Besides, the non-linear buckling mode (m, n) seems insensitive to variation of k. Fig. 3(c) shows the interaction curves of qcr and cr . The inclosed area by the qcr –cr curve and the two coordinate axes is referred to as stability ensuring region, in which the structure would be stable sufficiently and out of which the structure would be likely to buckle. It is interesting to find that the qcr –cr curve exhibits a character of broken line consisting of straight sections according to change of the non-linear buckling mode (m, n) from (1, 6) to (1, 7). With the increase of k, the area of the stability region reduces. Fig. 4(a)–(c) show dimensional effects. Fig. 4(a) plots the relation curves of cr versus R/h for FG cylindrical shells under given radial pressures (q=0, 25, 50 kPa), while Fig. 4(b) plots the relation curves of qcr versus R/h for FG cylindrical shells under given axial compressions

(0x = 0, 25, 50 MPa). The length-to-radius ratios L/R is chosen to be 0.5, 1,1.5, and 2. From Fig. 4(a) and (b), It is obvious that either cr or qcr decreases markedly with the increase of R/h. Form Fig. 4(a), 0x decreases with the increase of L/R, especially in the cases of q = 25, 50 kPa and when L/R increases from 0.5 to 1. Also, it is interesting to find that, L/R − cr relation curves are of little difference in some cases which are q = 0, L/R = 1, 1.5, 2, 100  R/h  500 and q = 25 kPa, L/R = 1, 1.5, 2, 100  R/h  250. Fig. 4(b) shows qcr always decreases with the increase of L/R. Besides, from Fig. 4(c), the area of the stability region reduces with the increase of R/h and L/R. Both R/h and L/R considerably affect the non-linear buckling mode. The three aforementioned thermal environments are considered in the following discussions. Fig. 5(a) shows the relation of cr versus 2T or Tc under given radial pressures (q = 0, 25, 50 kPa), while Fig. 5(b) shows the relation of qcr versus 2T or Tc under given axial

H. Huang, Q. Han / International Journal of Non-Linear Mechanics 44 (2009) 209 -- 218

250

100 T=300K, L/R=1, R/h=200, (m,n)=(1,6)

80

T=300k, L/R =1, R/h=200 (m,n) = (1,6)

200

0x=0 0x=20MPa 0x=40MPa 0x=60MPa

150

60 qcr(kPa)

cr(MPa)

215

40

q=0

20

q=20KPa q=40KPa q=60KPa

100

50

q=80KPa 0 0.01

0 0.01

0.1

1 k

10

100

0.1

1 k

10

100

100 (1,6)* T=300K, L/R =1, R/h=200

(1,6) 80

(1,6)

k=0.1 k=0.5 k=1 k=2 k=5

(1,6) cr(MPa)

60

(1,6)

(1,6)

(1,6)

(1,6) 40

k=10 (1,7)

(1,6) (1,7)

(1,6) 20

(1,7)

(1,7) (1,7) *the numbers in the parentheses denote the nonlinear buckling modes (m,n)

0 0

50

100

(1,7) 150

200

qcr(kPa) Fig. 3. Effects of the power law exponent on non-linear buckling behaviors: (a) relation curves of cr versus k; (b) relation curves of qcr versus k; (c) interaction curves of cr and qcr under various k.

compressions (0x =0, 25, 50 MPa). As shown, temperature rise leads to fall of the non-linear critical load (cr or qcr ). This is due to less stiffness of the shell aroused by higher temperature rise. With the uniform temperature rise in Case 1 viewed as an average temperature rise of the thermal gradient cases (Cases 2 and 3), i.e. T = Tc /2, the non-linear critical loads in the three thermal cases can be arranged in a sequence of Case 1 < Case 2 < Case 3. As temperature rises, the discrepancies of the non-linear critical loads in the three thermal cases enlarge, but the error between Cases 2 and 3 is still relatively small (within 1.5%). The interaction curves of qcr –cr is plotted in Fig. 5(c) for Case 1 and Fig. 5(d) for Cases 2 and 3. With the increase of T or Tc , the area of the stability region reduces. 4.2. Postbuckling results and discussions From Eqs. (33) and (38), we obtain a series postbuckling response curves of load versus end-shortening ratios under various combinations of the mode (m, n) (see Fig. 6 for the postbuckling equilibrium path of a pre-pressured shell). The lowest of these curves depict the postbuckling equilibrium path. For the prebuckling state, there should be no buckling deflection. Hence, the prebuckling equilibrium path is defined by substituting Eq. (30) into Eq. (38) without

¯ x = J (A  h − A qR + J ). As shown in Fig. 6, f1 and f2 , i.e. 0 10 0x 20 3 1 the state of the system initially follows the prebuckling equilibrium path. Until the upper value of the critical load or the linear critical load is reached, it follows the postbuckling equilibrium path in which a continuous mode jump (i.e. m = 1, n = 9, 8, . . . , 4) occurs, and the carrying capacity of the structure falls dramatically until the lower value of the critical load or the non-linear critical load, and then goes up gently. The continuous mode jump can also be observed in both the experiments of Yamaki [26] and the analytic results of Simitses [27]. According to the reported theory of Simitses, in which shell's imperfection sensitivity was considered, the upper value of the critical load in imperfect shells should be lower than that in perfect shells and when the imperfection amplitude is large enough the upper value of the critical load falls rapidly to the lower value of the critical load due to imperfection sensitivity of the structure. In other words, the lower value of the critical load is the boundary point, below which buckling will definitely not occur and beyond which buckling will be possible. Generally, geometric imperfection of cylindrical shells is usually small and immeasurable, so it is out of our consideration. However, as imperfection sensitivity does exist, the lower value of the critical load or the non-linear critical load seems

H. Huang, Q. Han / International Journal of Non-Linear Mechanics 44 (2009) 209 -- 218

200

150 cr(MPa)

1000

L/R=0.5,q=0 L/R=0.5,q=25kPa L/R=0.5,q=50kPa L/R=1.0,q=0 L/R=1.0,q=25kPa L/R=1.0,q=50kPa L/R=1.5,q=0 L/R=1.5,q=25kPa L/R=1.5,q=50kPa L/R=2.0,q=0 L/R=2.0,q=25kPa L/R=2.0,q=50kPa

T=300K, k=1

100

L/R=0.5,0x=0 L/R=0.5,0x=25MPa L/R=0.5,0x=50MPa L/R=1.0,0x=0 L/R=1.0,0x=25MPa L/R=1.0,0x=50MPa L/R=1.5,0x=0 L/R=1.5,0x=25MPa L/R=1.5,0x=50MPa L/R=2.0,0x=0 L/R=2.0,0x=25MPa L/R=2.0,0x=50MPa

T=300K, k=1 800

qcr(kPa)

216

600

400

50 200

0

0 100

150

200

250

300 R/h

350

400

450

100

500

150

200

250

300 R/h

350

400

450

500

120 T=300K, k=1, m=1

n=9 100

cr(MPa)

80 6

9

9

60

L/R=0.5,R/h=200 L/R=0.5,R/h=300 L/R=0.5,R/h=400 L/R=1,R/h=200 L/R=1,R/h=300 L/R=1,R/h=400

10

10

6 40

10

7

11

10 20

11

7

0 0

50

100

150

200

250

300

350

qcr(kPa) Fig. 4. Effects of dimensional parameters on non-linear buckling behaviors: (a) relation curves of cr versus R/h; (b) relation curves of qcr versus R/h; (c) interaction curves of cr and qcr under various R/h and L/R.

more significant in engineering applications than the upper value of the critical load or the linear critical load. For this reason, instead of the linear critical load, we merely give the non-linear critical load in the foregoing discussions. From Fig. 7, the existence of radial pressure arouses a slight translation of the prebuckling equilibrium path and leads to fall of the linear critical load and the postbuckling equilibrium path of axially loaded FG cylindrical shells. With the increase of k, the linear critical load and the postbuckling equilibrium path descend. In Fig. 8(a) and (b), effects of dimensional parameters on postbuckling of axially loaded FG cylindrical shells under an invariable lateral pressure are considered. As shown, in the both cases of q = 0 and 25 kPa the linear critical loads and the postbuckling equilibrium paths markedly descend with the increase of R/h, but they seems nearly invariable with the variation of L/R. Fig. 9 shows the effects of thermal environments. The aforementioned three thermal cases are considered. It is clear that, temperature rise leads to decrease of the linear critical load and descend of the postbuckling equilibrium path. With the uniform temperature rise in Case 1 viewed as an average temperature rise of the thermal gradient cases (Cases 2 and 3), i.e. T = Tc /2, the postbuckling equilibrium paths corresponded to these thermal cases can be compared and some discrepancies are observed.

Seeing that nearly uniform postbuckling mode jumps occurred (i.e. m = 1, n = 9, 8, . . . , 4) in Figs. 7, 8(a), and 9, we conclude that the postbuckling mode is little affected by the power law exponent, the radius-to-thickness ratio and temperature rise. But it is affected by the length-to-radius ratio because of the non-uniform jump seen in Fig. 8(b). Shells with larger value of L/R tends to jump at a larger value of m. For instance, m = 3 for L/R = 3, m = 2 for L/R = 2 and m = 1 for L/R = 1. 5. Conclusions This paper deals with the non-linear buckling and postbuckling problems of axially and laterally combine-loaded FG cylindrical shells by using the non-linear large deflection theory of cylindrical shells. Effects of the inhomogeneous parameter of FGMs, dimensional parameters of structure and various thermal environments are investigated. Numerical results reveal two prime conclusions as follows. (i) The interaction curve qcr –cr , which defines the stability region of the structure, exhibits a character of broken line consisting of straight sections according to variation of the nonlinear buckling mode. Increasing in the power law exponent,

H. Huang, Q. Han / International Journal of Non-Linear Mechanics 44 (2009) 209 -- 218

200

100 90

Case1,q=0

Case1,q=25kPa

Case1,q=50kPa

Case2,q=0 Case3,q=0

Case2,q=25kPa Case3,q=25kPa

Case2,q=50kPa Case3,q=50kPa

Case1,ox=25MPa Case2,ox=25MPa

Case1,ox=0 Case2,ox=0 Case3,ox=0

qcr(kPa)

70 60 50

Case2,ox=50MPa Case3,ox=50MPa

Case3,ox=25MPa

100

50 T0=300K, k=1, R/h=200,L/R=1, (m,n)=(1,6∼7)

T0=300K, k=1, R/h=200,L/R=1, (m,n)=(1,6)

40 30

0 0

100

200

300

400 500 600 ΔTc or 2ΔT(K)

700

800

900

0

1000

100

300

200

400

500

600

700

800

900

1000

ΔTc or 2ΔT(K)

100

100 ΔT =0

Case1, T0=300K, k=1, R/h=200,L/R=1

80

T0=300K, k=1, R/h=200,L/R=1

ΔT =100K

80

ΔT =200K ΔT =300K

60 cr(MPa)

60 cr(MPa)

Case1,ox=50MPa

150

80

cr(MPa)

217

40

20

ΔTc =0 Case3, ΔTc =100K Case3, ΔTc =200K Case3, ΔTc =300K Case2, ΔTc =100K Case2, ΔTc =200K Case2, ΔTc =300K

40

20

0

0 0

20

40

60

80

100

120

140

160

0

qcr(kPa)

20

40

60

80 100 qcr(kPa)

120

140

160

Fig. 5. Effects of external thermal environment on non-linear buckling of combine-loaded FG cylindrical shells: (a) relation curves of cr versus Tri ; (b) relation curves of qcr versus Tri ; (c) interaction curves of cr and qcr under various T for Case 1; (d) interaction curves of cr and qcr under various Tc for Cases 2 and 3.

5

5

The upper value of the critical load q=25kPa, T=300K,k=0.5, L/R=1,R/h=200, m=1,n=9-4

3

0x (100MPa)

0x (100MPa)

4

4

n=9 4

2 5

8

1

0

2

4

Δx (x10-3)

6

k=0.5 k=1

q=50kPa q=0

k=5

3 2 1

6 The lower value of the critical load

7

T=300K, R/h=200,L/R=1, m=1,n=9-4

8

10

Fig. 6. Diagrammatic sketch of the load-shortening ratio response curve.

radius-to-thickness ratio, length-to-radius ratio, and temperature rise, leads to reducing in the area of the stability region. Among these influencing factors, the radius-to-thickness ratio is the most critical one. The length-to-radius ratio seems of little significance to the non-linear critical load (cr or qcr ), when its value larger than 1.

0

2

4

Δx (x10-3)

6

8

10

Fig. 7. Postbuckling responses of non-pressured and pre-pressured FG cylindrical shells under various power law exponents.

(ii) The postbuckling results show there is a mode jump after buckling of axially and laterally combine-loaded FG cylindrical shells. The postbuckling equilibrium path descends with the increase of radial pressure, the power law exponent, radius-to-thickness ratio or temperature rise, but the postbuckling mode jump seems

218

H. Huang, Q. Han / International Journal of Non-Linear Mechanics 44 (2009) 209 -- 218

5

5

0x (100MPa)

3

R/h=200

k=1,R/h=200,T=300K, n=9-3

4 q=0 q=25kPa

0x (100MPa)

k=1,T=300K,L/R=1, m=1,n=9-4

4

2 R/h=300

q=0 q=25kPa

3 2

L/R=1, m=1

L/R=3,m=3

1

1

L/R=2,m=2

R/h=400

0

4

2

6

8

10

0

2

4

6

8

10

Δx (x10-3)

Δx (x10-3)

Fig. 8. Postbuckling responses of non-pressured and pre-pressured FG cylindrical shells under different dimensional parameters: (a) effects of radius-to-thickness ratio; (b) effects of length-to-radius ratio.

5

0x (100MPa)

4 3

k=1,T0=300K, R/h=200, L/R=1, q=50kPa, ΔTc =400K m=1,n=9-4 Tri=0 ΔT=200K ΔTc =800K

Case1 Case2 Case3

2 ΔT=400K 1 0

-4

-2

0

2

4

6

8

10

Δx (x10-3) Fig. 9. Postbuckling responses of non-pressured and pre-pressured FG cylindrical shells under various thermal cases.

independent of these influencing factors. However, the postbuckling mode jump is affected by length-to-radius ratio. Acknowledgments The authors wish to acknowledge the supports from the National Natural Science Foundation of China (10672059) and the Natural Science Foundation of Guangdong Province (8151064101000002). References [1] M. Koizumi, The concept of FGM, Ceramic Transactions, Functionally Gradient Materials 34 (1993) 3–10. [2] N. Noda, Thermal stresses in functionally graded materials, Journal of Thermal Stresses 22 (1999) 477–512. [3] G.N. Praveen, C.D. Chin, J.N. Reddy, Thermoelastic analysis of functionally graded ceramic-metal cylinder, Journal of Engineering Mechanics 10 (1999) 1259–1267. [4] K.M. Liew, S. Kitipornchai, et al., Analysis of thermal stress behaviour of functionally graded hollow circular cylinders, International Journal of Solids and Structures 40 (2003) 2355–2380. [5] C.T. Loy, K.Y. Lam, J.N. Reddy, Vibration of functionally graded cylindrical shells, International Journal of Mechanical Sciences 41 (1999) 309–324. [6] S.C. Pradhan, C.T. Loy, K.Y. Lam, J.N. Reddy, Vibration characteristics of functionally graded cylindrical shells under various boundary conditions, Applied Acoustics 61 (2000) 111–129.

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