Nonlinear buckling of torsion-loaded functionally graded cylindrical shells in thermal environment

European Journal of Mechanics A/Solids 29 (2010) 42–48

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Nonlinear buckling of torsion-loaded functionally graded cylindrical shells in thermal environment Huaiwei Huang, Qiang Han* Department of Engineering Mechanics, College of Traffic and Communications, South China University of Technology, Guangzhou 510640, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 May 2008 Accepted 15 June 2009 Available online 23 June 2009

Based on the nonlinear large deflection theory of cylindrical shells, this paper deals with the nonlinear buckling problem of functionally graded cylindrical shells under torsion load by using the energy method and the nonlinear strain–displacement relations of large deformation. The material properties of the functionally graded shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. Meanwhile, on the base of taking the temperature-dependent material properties into account, various effects of external thermal environment on the critical state of the shell are also investigated. Numerical results show various effects of the inhomogeneous parameter, the dimensional parameters and external thermal environment on nonlinear buckling of functionally graded cylindrical shells under torsion. The present theoretical results are verified by those in literature. Ó 2009 Elsevier Masson SAS. All rights reserved.

Keywords: FGMs Cylindrical shells Nonlinear buckling Torsion Thermal environment

1. Introduction Functionally Graded Materials (FGMs) were firstly invented in 1984 by a group of Japanese scientists (see Koizumi (1993)). This new type of composite material is usually mixed by the specific ceramic and metallic constituent materials. By continuously changing in the volume fraction of the constituent materials, the material properties vary smoothly through the thickness. This effectively avoids the 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 induced by thermal stresses. 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. With increase in demand, FGMs have been widely used in general structures. Hence, many functionally graded (FG) structures have been extensively studied, such as FG plates, FG cylindrical shells etc. Because FGMs were generally used in thermal environment, investigating their thermal responses is very essential. Noda (1999) presented an extensive review covering a wide range of topics from thermo-elasticity to thermo-inelasticity. Praveen et al. (1999) investigated the thermo-elastic responses of FG cylinders under

* Corresponding author. Fax: þ86 20 87114460. E-mail address: [email protected] (Q. Han). 0997-7538/$ – see front matter Ó 2009 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2009.06.002

transient thermal conduction. They concluded that more factors predominating thermal stresses in a FG cylinder than those in its isotropic counterpart. Both Noda and Praveen clarified the essentiality of considering the temperature-dependent material properties in thermo-elastic analyses. In the field of static buckling of FG cylindrical shells, Shen (2002, 2003, 2004), Shen and Noda (2005) worked systematically on postbuckling behaviors of FG cylindrical shells in thermal environments. Using the boundary layer theory and considering the temperature-dependent material properties, he and his coworkers investigated thermal postbuckling behaviors and the postbuckling behaviors of uniformly heated FG cylindrical shells under axial and lateral loads. But, his theory is very cumbersome in mathematical calculation for practical applicants. By using the first order shear theory, Shahsiah and Eslami (2003) considered the effects of various temperature distribution, such as uniform temperature rise, linear and nonlinear temperature distribution through the shell thickness, on buckling of simply supported FG cylindrical shells. Wu et al. (2005) presented a linear thermal buckling analysis for FG cylindrical shells under various temperature fields. In his paper, the temperature-dependent material properties were not included. Kadoli and Ganesan (2006) investigated thermal buckling behaviors as well as vibration characters of FG cylindrical shells. In the paper, the one-dimensional thermal conduction and the temperature-dependent material properties were simultaneously taken into account. Li and Batra (2006) studied buckling behaviors of an axial compressed three-layer circular cylindrical shell with the middle layer made of FGMs.

H. Huang, Q. Han / European Journal of Mechanics A/Solids 29 (2010) 42–48

Their results showed that the critical load was markedly influenced by the average Young’s modulus of FGMs and the radius-tothickness ratio of the shell. Employing Donnell shell theory and a three-dimensional finite element code, Najafizadeh et al. (2008) studied linear buckling behaviors of axially compressed stiffened FG cylindrical shells. At present, the study on nonlinear buckling of FG cylindrical shells subjected to torsion is yet not seen. In this paper, the nonlinear buckling problem of torsion-loaded FG cylindrical shells is investigated by using the energy method and the nonlinear strain–displacement relations of large deformation. It is assumed that the material properties of the functionally graded shells vary smoothly through the shell thickness according to a power law distribution of the volume fraction of the constituent materials. Meanwhile, on the base of taking the temperature-dependent material properties into account, various effects of external thermal environment on the critical condition of the shell are also considered. 2. Formulation of the problem Consider a cylindrical shell with mean radius R, thickness h, and length L made of FGMs (Fig. 1). The origin o is located at the left end of the shell and the middle length of thickness. The coordinate axes x, y, and z, are in the longitudinal, circumferential and inward radial directions respectively. 2.1. Material properties of FGMs Because FGMs are typically made from a mixture of metal and ceram, their material properties are related to both the material properties and the continual distribution of the constituent materials. Meanwhile, the material properties of both metal and ceram are related to environmental temperature. Thus, FGMs’ material properties vary smoothly through their thickness and exhibit temperature dependency. The material properties of the 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 ðTÞ or Pm ðTÞ ¼ c0 c1 T 1 þ 1 þ c1 T þ c2 T 2 þ c3 T 3

(1)

where c0, c1, c1, c2, c3 are temperature coefficients which are unique to the constituent materials. The material properties of FGMs P are related not only to the material properties of the constituent materials, but also to their volume fraction Vc and Vm.

P ¼ Pc Vc þ Pm Vm ;

Vc þ Vm ¼ 1

43

We assume that the volume fraction of ceram follows the simple power law distribution as

Vc ¼ ð0:5 þ z=hÞk

(3)

where k is power law exponent satisfying 0  k < þN. Using Eqs. (2) and (3), the material properties of FGMs are written as

PðzÞ ¼ ðPc  Pm Þð0:5 þ z=hÞk þPm

(4)

Then, we have P ¼ Pc when z ¼ h/2, and P ¼ Pm when z ¼ h/2. FGMs’ properties vary smoothly from Pc to Pm through the thickness according to the power law exponent k. Accordingly, the effective Young’s modulus E(z), Poisson’s ratio n(z), thermal expansion coefficient aðzÞ and thermoconductivity k(z)of FGMs can be written in the similar form of Eq. (4).

EðzÞ nðzÞ aðzÞ kðzÞ

¼ ¼ ¼ ¼

ðEc  Em Þ ð0:5 þ z=hÞk þEm ðnc  nm Þ ð0:5 þ z=hÞk þnm ðac  am Þ ð0:5 þ z=hÞk þam ðkc  km Þ ð0:5 þ z=hÞk þkm

(5)

2.2. Basic equations The present analysis is based on the thin shell assumption and the Kirchhoff–Love hypothesis (see Yamaki (1984)) upon which the normal stress acting in the direction normal to the middle surface may be neglected in comparison with the stresses acting in the direction parallel to the middle surface. For cylindrical shells, the nonlinear strain–displacement relations on the middle surface are

30x ¼ U;x þ 12W;x2 W

1

30y ¼ V;y  þ W;y2 R 2 g0xy ¼ U;y þ V;x þ W;x W;y

(6)

in which 30x , 30y , and g0xy are the strain components on the reference surface. 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. The strain components can be written as following.

3x ¼ 30x  zW;xx 3y ¼ 30y  zW;yy gxy ¼ g0xy  2zW;xy

(7)

Assume that the shell is subjected to a temperature rise Tri(z). Including thermal effects, the stress–strain relations are given by

(2)





sx ¼ KðzÞ 3x þ nðzÞ3y  ½1 þ nðzÞaTri   sy ¼ KðzÞ 3y þ nðzÞ3x  ½1 þ nðzÞaTri sxy ¼ 12KðzÞ½1  nðzÞgxy

(8)

where K(z) ¼ E(z)/[1  n(z)2]. Under the thin shell assumption, we have h/R  1 and the approximate form of internal force and moment resultants are given as h



   Nx ; Ny ; Nxy ; Mx ; My ; Mxy ¼

Z2



h2

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



sx ; sy ; sxy ð1; zÞdz

Considering Eqs. (7) and (8) in Eq. (9) yields

(9)

44

H. Huang, Q. Han / European Journal of Mechanics A/Solids 29 (2010) 42–48

8 Nx > > > Ny > > >
9 > > > > > > > > =

2

A10 6A 6 20 6 6 0 xy ¼6 6A > Mx > > > 6 11 > > > > 6 > > > 4 A21 > : My > > ; Mxy 0

A20 0 A10 0 0 A30 A21 0 A11 0 0 A31

9 8 9 38 0 > 3 f1 > A21 0 > > > > x > > > > > > > > 7 > > > 0 > > > > 3 f1 > A11 0 7> > > > y > > > > > > > 7> = < < 0 7 0 A31 7 gxy 0 = þ > f2 > A22 0 7 > > > > W;xx > 7> > > > > > > > 7> > > > > > > > A12 0 5> f W > > > > 2 ;yy > > > > ; : > > ; : 0 A32 0 2W;xy

A11 A21 0 A12 A22 0

(10) where Aij (i ¼ 1,2,3, j ¼ 0,1,2) are stiffness coefficients and j (0,1,2) represent the membrane, bending-tension coupling, and bending stiffness coefficients respectively. For isotropic cylindrical shells, the coupled stiffness coefficients Ai1 should vanish. f1 and f2 are the additional internal force and moment caused by temperature rise Tri(z). They are defined as the following

A1j ¼

Z

h 2

KðzÞzj dz;

h2

A3j ¼

1 2

Z

A2j ¼

h 2

nðzÞKðzÞzj dz;

h2

Z

Z

K1 ¼ K2 ¼ K3 ¼ K4 ¼

Assume that the shell is subjected to a torsional load. With the aid of Eqs. (6) and (13), the work done by the external force during buckling is

½1 þ nðzÞKðzÞaðzÞTri ½z dz; Uex ¼ s0xy h

h 2

¼ s0xy h

1 R

2 30x;yy þ 30y;xx  g0xy;xy ¼  W;xx þ W;xy  W;xx W;yy

(11)

Ny ¼ 4;xx ;

Nxy ¼ 4;xy

i

h

i

30y ¼ J0 A10 4;xx  A20 4;yy þ J2 W;xx þ J1 W;yy  J3 f1 .  A30 ¼ 2A31 W;xy  4;xy



1 2 þ W;xx W;yy W;xx  W;xy R



¼ 0

(14)

2



sx 3x þ sy 3y þ sxy gxy dx dy dz

(15)

U

where U is the volume field. Using Eqs. (7) and (8), Eq. (15) turns into the following form:

(17)

(18)

V;y dx dy ¼ 0

(19)

0

Combining Eqs. (6) and Eqs. (13), the closed condition becomes

(13)

where C1 ¼ J2/A10, C2 ¼ 1/(A10J0). For thin cylindrical shells, the strain energy is approximately given as

ZZZ  1

0

  1  2A31 W;xy  4;xy W;x W;y dx dy A30

For a cylindrical shell, the closed (or periodicity) condition in the circumferential direction should be satisfied.

0

where J0 ¼ 1/(A210  A220), J1 ¼ A10A11  A20A21, J2 ¼ A10A21  A20A11, J3 ¼ A10  A20. Considering Eqs. (13) in Eq. (11), we have the compatible equation rewritten as

V4 4 þ C 1 V4 W þ C 2

0

where s0xy is the average shear stress on the end sections of the shell. Then, the total potential energy of the system is

2pR Z L Z

Substituting of Eqs. (12) into Eqs. (10) obtains

h

 U;y þV;x dx dy

UTPE ¼ Uin  Uex (12)

30x ¼ J0 A10 4;yy  A20 4;xx þ J1 W;xx þ J2 W;yy  J3 f1



2pR Z L

Z 0

Introducing Airy’s stress function 4(x,y) which satisfies

Nx ¼ 4;yy ;

2pR Z L Z 0

½1 þ nðzÞKðzÞaðzÞTri ½zz dz

From Eqs. (6), the compatible equation is obtained as

Uin ¼

2ðA220 A210 Þ A20 ðA211 þA221 ÞA220 A22 þA10 ðA10 A22 2A11 A21 Þ A210 A220 2 A12  h A22 þ ðA11 iA21 Þ =ðA20  A10 Þ; 2 2 A10 = 2 A10  A20

  K5 ¼ A20 = A220  A210 ; K6 ¼ 1=ðA10  A20 Þ K01 ¼ 1=ðA10 þ A20 Þ; K02 ¼ ðA11 þ A21 Þ=ðA10 þ A20 Þ h Z2 f3 ¼ ½1 þ nðzÞKðzÞaðzÞ2 Tri ½z2 z dz

h2

g0xy

A10 ðA211 þA221 ÞA210 A12 þA20 ðA20 A12 2A11 A21 Þ

h2

h 2

Z

0

where

½1  nðzÞKðzÞzj dz

2h

f2 ¼ 

0

  2 2 2 K1 W;xx þ K2 W;xx W;yy þ K3 W;xy þ W;yy

  þ K4 42;xx þ 42;yy þ K5 4;xx 4;yy þ K6 42;xy þ f3  K01 f21  i þ ðK02 f1  f2 Þ W;xx þ W;yy dx dy ð16Þ

h 2

h2

f1 ¼ 

Uin ¼

2pR Z L Z h

2pR Z L

Z   J0 A10 4;xx  A20 4;yy þ J2 W;xx þ J1 W;yy  J3 f1 0

0

 W 1 2 þ  W;y dx dy ¼ 0 R 2

ð20Þ

3. Solution of the problem Assume the buckling deflection is in the following form (see Wu (1996)).

px nðy  gxÞ px Wðx; yÞ ¼ x0 h þ xh sin sin  d sin2 L R L

(21)

where n is the circumferential wave number. g is the tangent of the included angle between the wave shape and x-axis. x0, x, and d are unknown amplitude parameters. x0h denotes the uniform

H. Huang, Q. Han / European Journal of Mechanics A/Solids 29 (2010) 42–48

deflection. xh sinpx=L sinnðy  gxÞ=R represents the linear buckling shape, while dxh sin2 px=L reflects the nonlinear buckling shape of large deformation. From Eq. (21), Eq. (14) becomes

2px 2pðy  gxÞ V4 4 ¼ b01 cos þ b02 cos L Lq p½y þ ðq  gÞx p½  y þ ðq þ gÞx þ b03 cos þ b04 cos Lq Lq p½y þ ð3q  gÞx p½  y þ ð3q þ gÞx þ b05 cos þ b05 cos Lq Lq (22)

4 ¼ s0xy hxy



q ¼

nL

b02 ¼

b01 ¼

;

C2



p2 h C2 L2  4C1 p2 R xd L4 R

þ

C2 p4 h2 x

4 ¼ 4* þ 4

2L4 q

2

    2 K1 p4 h2 x q1 þ2d2 þ4K6 L4 h2 s20xy þ4L4 f3 K01 f21  i þ2K4 p2 16b21 þ16q2 b22 þ q3 b23 þ q4 b24 þ q5 b25 þ q6 b26 ð26Þ h i pRh2 s0xy A30 p2 hx2 g þ 4L2 q2 s0xy ¼ (27) 2 2A30 Lq

Uin ¼

;

pR h

2L3

where

p4 h2 x2



2

  2 2 2 b03 ¼ p h4 x4 C1 p2 R 1 þ g2  2gq þ q 2RL q

2 2 2 C2 L2 q ðg þ qÞ 12C2 p2 Rhq xd   2 2 2 b04 ¼ p h4 x4  C1 p2 R 1 þ g2  2gq þ q 2RL q

2 2 2 þC2 L2 q ðg þ qÞ þ12C2 p2 Rhq xd

2

h

i2

h

i2

    2 2 q4 ; q2 ¼ 1þ g2 =q4 ; þ2g2 1þ3q

q3 ¼ 1þðg  qÞ2 =q4 i2

h

q4 ¼ 1þðg þ qÞ2 =q4 ; q5 ¼ 1þðg 3qÞ2 =q4 ; i2

h

q6 ¼ 1þðg þ3qÞ2 =q4 From Eqs. (26), (27), and (25), the total potential energy UTPE can be expressed as a function of x, d, g, n, and s0xy.

4 2 2

C2 p h x d 2 4L4 q

  UTPE ¼ UTPE x; d; g; n; s0xy

Then, the particular solution of the stress function 4* is obtained as

p½yþðq  gÞx 2px 2pðy gxÞ þb3 cos þb2 cos L Lq Lq p½yþðq þ gÞx p½yþð3q  gÞx þb4 cos þb5 cos Lq Lq p½yþð3q þ gÞx þb5 cos Lq

where

By Ritz energy method, we have

(23)

(29)

From Eq. (28), the expression vUTPE =vx0 ¼ 0 has been automatically satisfied, and the expressions vUTPE =vx ¼ 0 and vUTPE =vd ¼ 0 yield respectively

p5 Rxh L3

2

(28)

vUTPE vUTPE vUTPE ¼ ¼ ¼ 0 vd vx 0 vx

4* ¼ b1 cos

2



q1 ¼ g4 þ 1þ q2

q 2L4

b05 ¼ b06 ¼

(25)

Using Eqs. (21) and (25), and noting the relationsK5 þ K6 ¼ 2K4, K2 þ K3 ¼ 2K1, we have the strain energy expression Eq. (16) and the external work Eq. (17) rewritten as following.

Uex

2

(24)

Thus, the general solution of the stress function is given as

where

pR

45

  i 2 2 H01 þ H02 x þ 6H03 xd þ 2d2 H04 þ 2H05 x  H06 s0xy ¼ 0 (30)

2

b1 ¼ a1 xd þ a2 x ; b2 ¼ a3 x ; b3 ¼ a4 x d þ a5 x 2 2 2 b4 ¼ a6 x d þ a7 x; b!5 ¼ a8 x d; b6 ¼ a9 x d 2 h C2 L2 C2 h2 C2 h2 q a1 ¼ ; a ¼ ; a ¼  4C  2 1 2 3 2 16 p2 R 32q 32 1 þ g2 2

C2 h2 q

a4 ¼  h i ; 2 2 4 1 þ ðg  qÞ i o n h 2 2 2 h C1 p2 R 1 þ ðg  qÞ  C2 L2 q ðg  qÞ a5 ¼ i h 2 2 2p2 R 1 þ ðg  qÞ

2 2p5 Rx h

i  2 H03 x þ d H04 þ H05 x ¼ 0

L3

(31)

where

    H01 ¼ K1 q1 h2 þ 2K4 a25 q3 þ a27 q4 ; H02 ¼ 64K4 a22 þ a23 q2 H03 ¼ K4 ð16a1 a2 þ a4 a5 q3 þ a6 a7 q4 Þ; H04 ¼ K1 h2 þ 16K4 a21   2 3 H05 ¼ K4 a24 q3 þ a26 q4 þ a28 q5 þ a29 q6 ; H06 ¼ gL2 h2 p q

Noting that x s 0, Eq. (31) becomes

2

C2 h2 q

a6 ¼

h i ; 2 2 4 1 þ ðg þ qÞ n h i o 2 2 2 h C1 p2 R 1 þ ðg þ qÞ  C2 L2 q ðg þ qÞ a7 ¼ i h 2 2 2p2 R 1 þ ðg þ qÞ 2

a8 ¼

h

2

C2 h2 q

2

4 1 þ ðg  3qÞ

i2 ;

C2 h2 q a9 ¼  h i 2 2 4 1 þ ðg þ 3qÞ

The homogeneous solution of the stress function is

H03 x d ¼  2 H04 þ H05 x

(32)

With the aid of the above equation, Eq. (30) becomes

s0xy

" 2 x2 6H03 1 2 H01 þ H02 x  ¼ 2 H06 H04 þ H05 x   2 þ 2 H04 þ 2H05 x

!2 #

H03 x 2

H04 þ H05 x

ð33Þ

46

H. Huang, Q. Han / European Journal of Mechanics A/Solids 29 (2010) 42–48

Table 1 Temperature coefficients for the material properties of ceram and metal. Materials properties Zirconia Ec (Pa)

nc ac ð1=KÞ kc (W/mK) Ti6Al4V Em (Pa)

nm am ð1=KÞ km (W/mK)

c0

c1

c1

c2

c3

244.26596  109 0.2882 12.7657  106 1.7

0 0 0 0

1.3707  103 1.13345  104 0.00149 0.0001276

1.21393  106 0 0.1  105 0.66485  105

3.681378  1010 0 0.6775  1011 0

122.55676  109 0.28838235 7.57876  106 1.20947

0 0 0 0

4.58635  104 1.12136  104 0.00065 0.0139375

0 0 0.31467  106 0

0 0 0 0

Eq. (33) can be used to derive the nonlinear critical condition for FG cylindrical shells. In this equation, the linear buckling load s*0xy of torsion-loaded FG cylindrical shells can be obtained by omitting the second or higher order items of x.

s*0xy

h  i p2 q2 K1 q1 h2 þ 2K4 a25 q3 þ a27 q4 H01 ¼ ¼ H06 gL2 h3

(34)

Substituting Eqs. (21) and (25), the closed condition Eq. (20) yields the relation between x0 and x. 2

p2 Rhx2 J0 J3 Rf1 þ x0 ¼   þ 2 2 h 8L2 q 4 H04 þ H05 x H03 x

(35)

4. Numerical results and discussion In this paper, FGMs are made from a mixture of ceramic material Zirconia and metallic material Ti6Al4V. The temperature coefficients c0, c1, c1, c2, c3 for these constituent materials in Eq. (1) are listed in Table 1 (from Praveen et al. (1999)). Seeing that Poisson’s ratios of the two constituent materials are closed to each other, and insensitive to variation of temperature, we set Poisson’s ratio of FGMs n as a constant 0.3. As the aforementioned, the nonlinear critical condition of large deflection can be derived from Eq. (33). We define the critical condition as the possible lowest point of external force. Thus, the specific solving procedures are presented as following: by using Eq.

Fig. 2. Diagrammatic sketch of solving the nonlinear critical load and the buckling mode.

(33), a series of s0xy versus x curves can be drawn under various combination of the mode (n,g) (see Fig. 2 in which the basic calculating parameters are listed). From the lowest of these curves, an envelope curve is obtained. The lowest point of the envelope curve is regarded as the nonlinear critical condition with the average critical shear stress scr and the corresponding buckling mode (n,g). 4.1. Validation of the present results The present linear and nonlinear results are compared with the experimental and linear theoretical results of isotropic cylindrical shells from Wang et al. (1992). The present linear and nonlinear results are respectively obtained by using Eqs. (33) and (34). After introducing the critical end-torque Mcr ¼ 2pR2hscr and the shell pgeometric parameter (or the Batdorf shell parameter) ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ¼ 1  n2 L2 =ðRhÞ, the relation curves of Mcr versus Z are plotted in Fig. 3. As shown, the nonlinear results of the present theory are in good agreement with those of experiment with the average error less than 1.5%. The present linear results are slightly lower than those in the literature. Meanwhile, the present linear prediction is averagely 17.4% higher than the experimental results. 4.2. Effects of the power law exponent Fig. 4 shows the effects of the power law exponent k on the buckling loads of FG cylindrical shells. In the figure, k-axis is plotted in the logarithmic form. The shells with L/R ¼ 1 and R/h ¼ 100, 200, 300 are exposed to a room temperature 300K without temperature rise. It is apparent that the critical load scr decreases with k

Fig. 3. Comparison of the present results with the experimental and theoretical results of isotropic cylindrical shells reported by Wang et al. (1992).

H. Huang, Q. Han / European Journal of Mechanics A/Solids 29 (2010) 42–48

increasing. The prime reason for falling of scr is that: higher value of k corresponds to a metal-richer shell which usually has less stiffness than a ceram-richer one. Besides, It is remarkable that the buckling mode (n,g) seems insensitive to variation of k.

4.3. Effects of dimensional parameters Consider a FG cylindrical shell with k ¼ 1 under room temperature 300K. Table 2 lists the critical loads and the corresponding buckling modes under various dimensional parameters. It is apparent that the dimensional parameters greatly affected both the critical load and the buckling mode. With increase of R/h or L/R, scr initially has a rapid decrease, and then gradually levers off. In addition, the circumferential wave number n increases with R/h increasing or L/R decreasing.

4.4. Effects of thermal environments Because FGMs are born to serve in thermal environment, several thermal cases are considered and listed as following. Case 1: Temperature rise DT uniformly, and the temperature field T(z) ¼ Ti þ DT where Ti is the initial temperature of environment. Case 2: Temperature rises DTc on the ceramic surface and keeps unchanged on the metallic surface. The temperature field is assumed to be linear distribution through shell thickness. i.e. T(z) ¼ Ti þ DTc(0.5 þ z/h). Case 3: Temperature rises DTc on the ceramic surface and keeps unchanged on the metallic surface. The temperature field is given by the following static one-dimensional thermal conduction equation.

Table 2 Effects of dimensional parameters on the buckling load scr (MPa) and the buckling mode. R/h 100 150 200 300 400 500 *

L/R ¼ 0.5 494.92 279.03 187.46 108.46 73.85 54.97

L/R ¼ 1

L/R ¼ 1.5



v vTðzÞ kðzÞ ¼ 0 vz vz

(36)

With thermal boundary condition: T(h/2) ¼ Ti, T(h/2) ¼ Ti þ DTc, temperature field can be numerically given by solving Eq. (36). For simplification, a series expansion method according to Wu et al. (2005) is employed as following to approximately depict the temperature field under thermal conduction.

PþN  TðzÞ ¼ Ti þ DTc

i¼0

 kckmkm  P þN i¼0

i  ikþ1 1 0:5 þ hz ikþ1 i 1  kckmkm ikþ1

(37)

During the process of thermal conduction, material properties are coupled with thermal environment. In other words, the thermoconductivity k(z) and temperature field affect each other in the process of thermal conduction. Thus, the temperature field in FGMs should be obtained by following a similar iterative method as given by Na and Kim (2006). The specific procedures of this iteration are as following. Step 1: According to Eqs. (1) and (5), calculating the thermoconductivity of FGMs under a given initial temperature Ti.

220

Case3,k=0.5,(8,0.41) Case3,k=1,(8,0.41) Case3,k=5,(7,0.35) Case2,k=0.5,(8,0.41) Case2,k=1,(8,0.41) Case2,k=5,(7,0.35)

R/h=100, L/R=2 200

180

160

L/R ¼ 2

230.66 (8,0.41) 197.74 (8,0.41) (11,0.45)* 295.40 (9,0.43) (13,0.45) 173.75 (11,0.43) 137.95 (10,0.41) 117.35 (9,0.38) (14,0.45) 119.27 (12,0.42) 95.43 (11,0.40) 81.70 (10,0.37) (17,0.45) 70.84 (14,0.41) 56.52 (12,0.35) 48.61 (11,0.33) (18,0.43) 48.90 (15,0.39) 39.25 (13,0.33) 33.82 (12,0.31) (20,0.43) 36.78 (16,0.36) 29.61 (14,0.32) 25.58 (13,0.30)

The numbers in the parentheses denote the buckling mode (n,g).

Fig. 5. Effects of uniform temperature rise on buckling of torsion-loaded FG cylindrical shells.

τ cr(MPa)

Fig. 4. Effects of the power law exponent on buckling of FG cylindrical shells. *The numbers in the parentheses denote the buckling mode (n,g).

47

140

Δ Tc(K)

0

100

200

300

400

500

600

Fig. 6. Effects of various temperature gradients on buckling of torsion-loaded FG cylindrical shells.

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H. Huang, Q. Han / European Journal of Mechanics A/Solids 29 (2010) 42–48

Step 2: Using the thermoconductivity calculated in Step 1, substituting it into Eqs. (36) and (37) yields a temperature field T1(z). Step 3: By the same procedure as Step 1, calculating the thermoconductivity of FGMs under the temperature field T1(z), and then following Step 2 derives a new temperature field T2(z). Repeating the above procedures for r times derives the temperature field Tr(z). When Max(jTr(z)  T(r1)(z)j) < 0.001 is satisfied, the iteration of the temperature field converges. In general, this iteration converges rapidly. To simplify the following calculation, Eq. (37) with i ¼ 0,1,/,5 is used. In the following discussion on thermal effects, a set of basic parameters Ti ¼ 300K, R/h ¼ 100, L/R ¼ 2, k ¼ 0.5, 1, 5 are used. As shown in Fig. 5, the critical load scr decreases with increase of the uniform temperature rise DT. When DT ¼ 300K and 600K, the averagely falls of scr reach 19.5% and 30.8%, respectively. Fig. 6 plots the relation curves of the critical load scr versus temperature rise on the ceramic surface DTc. Both Case 2 (assuming linear temperature distribution) and Case 3 (considering thermal conduction) are considered. As shown in the figure, the critical load scr decreases with DTc increasing. When DTc ¼ 300K and 600K, the averagely decrease of scr respectively reaches 11.4% and 17.9% for Case 2, and 10.6% and 17% for Case 3. The critical load of Case 2 is slightly higher than that of Case 3. The discrimination between the two cases gradually enlarges with DTc increasing, but even in the case of DTc ¼ 600K, the error between them is still smaller than 1.2%. Therefore, in a certain range of temperature rise (i.e. DTc  600K), we can use the assumed linear temperature distribution instead of exactly considering the thermal conduction in buckling analysis of torsion-loaded FG cylindrical shells for simplification. From the three thermal cases discussed above, we also note that the buckling mode (n,g) seems not to be affected by temperature change. 5. Conclusion This paper deals with the nonlinear buckling problem of FG cylindrical shells under torsion by the nonlinear large deflection theory. Effects of the inhomogeneous parameter of FGMs, dimensional parameters of structure and various thermal environments on the critical condition of the shell are investigated. Numerical results reach several conclusions as following. i. The critical load of torsion-loaded FG cylindrical shells decreases with increase of the inhomogeneous parameter or the power law exponent. The buckling mode seems insensitive to variation of the power law exponent. ii. Dimensional parameters have great effects on both the critical load and the buckling mode of torsion-loaded FG cylindrical shells. The critical load decreases greatly with the

radius-to-thickness ratio or the length-to-radius ratio increasing. The circumferential wave number increases with increase of the radius-to-thickness ratio. But it decreases with increase of the length-to-radius ratio. iii. When temperature rises, the critical load of torsion-loaded FG cylindrical shell falls. In thermal gradient environment, we can use the assumed linear temperature distribution instead of exactly considering the thermal conduction in buckling analysis of torsion-loaded FG cylindrical shells for simplification. The buckling mode seems not to be affected by temperature change. Acknowledgements 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 Kadoli, R., Ganesan, N., 2006. Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition. Journal of Sound and Vibration 289, 450–480. Koizumi, M., 1993. The concept of FGM. Ceramic Transactions, Functionally Gradient Materials 34, 3–10. Li, S.L., Batra, R.C., 2006. Buckling of axially compressed thin cylindrical shells with functionally graded middle layer. Thin-Walled Structures 44, 1039–1047. Na, K.S., Kim, J.H., 2006. Thermal postbuckling investigations of functionally graded plates using 3-D finite element method. Finite Elements in Analysis and Design 42, 749–756. Najafizadeh, M.M., et al., 2008. Mechanical stability of functionally graded stiffened cylindrical shells. Applied Mathematical Modelling 33, 1151–1157. Noda, N., 1999. Thermal stresses in functionally graded materials. Journal of Thermal Stresses 22, 477–512. Praveen, G.N., Chin, C.D., Reddy, J.N., 1999. Thermoelastic analysis of functionally graded ceramic-metal cylinder. Journal of Engineering Mechanics 10, 1259–1267. Shahsiah, R., Eslami, M.R., 2003. Thermal buckling of functionally graded cylindrical shell. Journal of Thermal Stresses 26, 277–294. Shen, H.S., 2002. Postbuckling analysis of axially-loaded functionally graded cylindrical shells in thermal environments. Composites Science and Technology 62, 977–987. Shen, H.S., 2003. Postbuckling analysis of pressure-loaded functionally graded cylindrical shells in thermal environments. Engineering Structures 25, 487–497. Shen, H.S., 2004. Thermal postbuckling behavior of functionally graded cylindrical shells with temperature-dependent properties. International Journal of Solids and Structures 41, 1961–1974. Shen, H.S., Noda, N., 2005. Postbuckling of FGM cylindrical shells under combined axial and radial mechanical loads in thermal environments. International Journal of Solids and Structures 42, 4641–4662. Yamaki, N., 1984. Elastic Stability of Circular Cylindrical Shells. North-Holland Press. Wang, D.Y., Ma, H.W., Yang, G.T., 1992. Studies on the torsional buckling of elastic cylindrical shells. Applied Mathematics and Mechanics 13, 193–197 (in Chinese). Wu, L.H., Jiang, Z.Q., Liu, J., 2005. Thermoelastic stability of functionally graded cylindrical shells. Composite Structures 70, 60–68. Wu, L.Y., 1996. Stability Theory of Plates and Shells. Huazhong University of Science & Technology Press (in Chinese, pp. 185–186).