Investigation on buckling behaviors of elastoplastic functionally graded cylindrical shells subjected to torsional loads

Composite Structures 118 (2014) 234–240

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Investigation on buckling behaviors of elastoplastic functionally graded cylindrical shells subjected to torsional loads Huaiwei Huang ⇑, Biao Chen, Qiang Han School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, Guangdong 510640, PR China

a r t i c l e

i n f o

Article history: Available online 27 July 2014 Keywords: Functionally graded materials Cylindrical shells Buckling Elastoplastic Nonlinear analysis

a b s t r a c t In the present work, a semi-analytic solution is presented to analyze buckling behaviors of elastoplastic functionally graded circular cylindrical shells under torsional loads. The material properties vary smoothly through the shell thickness according to the power law distribution and a multi-linear hardening elastoplasticity of materials is included in the analysis. The Ritz energy method and both the flow and deformation constitutive theories help to develop the buckling government equation and the buckling critical condition. An iterative algorithm is resorted to derive the critical load and the buckling mode parameters. Numerical results reveal various effects of the constituent distribution of FGMs, dimensional parameters, and elastoplastic material properties. Meanwhile, the influences of material flow effect on buckling of elastoplastic FGM cylindrical shells are discussed. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introductions Functionally graded materials (FGMs) are new composites fabricated by mixing ceramic and metallic constituents. The continually varying characteristics of the mixing ratio enable continually grading in their material properties through the shell thickness. In the recent decades, elastic buckling behaviors of FGM plates and shells had been investigated intensively. Feldman and Aboudi [1] studied buckling behaviors of FGM plates under uniaxial in-plane load. Najafizadeh and Eslami [2] and Najafizadeh and Heydari [3] presented buckling analyses for buckling of circular plates under radial compression. Sofiyev and Schnack [4] investigated dynamic buckling of FGM cylindrical shells under linearly increasing torsional loads. Kadoli and Ganesan [5] considered buckling problems of clamped FGM cylindrical shells under thermal loads. Bagherizadeh et al. [6] investigated buckling issues of FGM cylindrical shells embedded in elastic medium and subjected to combined axial and radial compressive loads. Malekzadeh et al. [7,8] investigated respectively thermal and mechanical load induced buckling behaviors of FGM arbitrary straight-sided quadrilateral plates using differential quadrature method. Uymaz and Aydogdu [9] considered the effects of various boundary conditions on buckling behaviors of three dimensional shear buckling of FG plates. Wu et al. [10] presented linear buckling analysis of simply-supported, ⇑ Corresponding author. Tel.: +86 1366 030 2875. E-mail address: [email protected] (H. Huang). http://dx.doi.org/10.1016/j.compstruct.2014.07.025 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved.

multilayered FGM circular hollow cylinders under combined axial compression and external pressure. Yaghoobi and Fereidoon [11] investigated FGM plates resting on elastic foundation for the mechanical and thermal buckling responses. By using Finite element modeling method, Shariyat and Asemi [12] studied shear buckling behaviors of the orthotropic heterogeneous FGM plates resting on the Winkler-type elastic foundation. Meanwhile, some researches focused on the postbuckling issues including geometrical nonlinearity. For instance, Shen [13–16] investigated postbuckling behaviors of FGM cylindrical shells and plates with boundary layer theory considering various mechanical and thermal loads. Woo et al. [17] and Na and Kim [18] focused on postbuckling behaviors of FGM plates induced by thermal loads. Tung [19] presented an analytical approach to investigate the effects of tangential edge constraints on the postbuckling behaviors of FGM flat and cylindrical panels resting on elastic foundations and subjected to thermal, mechanical and thermomechanical loads. Buckling issue of structures including material nonlinearity is one of the most cumbersome and important components in structural stability theory. Especially in the field of elastoplastic buckling of homogeneous plates and shells, the researches have been very extensive [20–25]. Okada et al. [20] analyzed buckling behaviors of cylindrical vessels under shear forces in the elastic– plastic region. Ma et al. [21] experimentally investigated dynamic plastic buckling of circular cylindrical shells under impact torque. Mao and Lu [22,23] investigated elastoplastic buckling of isotopic cylindrical shells under axial compression and torsional loads with classical shell theory. Currently, literature reporting elastoplastic

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buckling performances of composite plates and shells is limited, especially for FGM ones. As far as the elastoplastic constitutive model was concerned, Tamura et al. [26] defined the rule of mixtures for metal alloy named TTO model, which was extended to ceramic/metal system by Bocciarelli [27] to describe the elastoplastic behaviors of FGMs. Meanwhile, an inverse analysis procedure based on indentation tests was proposed by Nakamura et al. [28] to identify the constitutive parameters of FGMs. With this model, some literatures were reported concerning thermal stress responses [29–33] and fracture performances of FGMs [34,35]. As FGMs are composed of metallic constituent materials, typically ductile ones, their plastic deformation may be of significant influence on material flow, and therefore, greatly affect buckling behaviors of FGM cylindrical shells. To investigate these effects, buckling analysis of elastoplastic FGM circular cylindrical shells subjected to torsional load is presented. 2. Basic description For FGM cylindrical shells with length L, mean radius measured from the middle plane R, and thickness h, the coordinate system is set as shown in Fig. 1. The origin is placed on the middle plane of the shell at one of the ends. The coordinate axes x, y, and z are respectively in the axial, circumferential, and the inward normal directions. The shells are assumed to be simply-supported at both ends, and subjected to a torsional moment M which would arouse an in-plane shear stress. Generally, ceramic constituents are usually brittle materials of relatively higher elastic modulus and strength than those of metallic constituents. As load increasing, the shear stress in the ceramic-rich side would rise faster, and results in plastic flow initiating in this area as shown in Fig. 1.

volume fraction and their subscripts c, m respectively correspond to the ceramic and metallic constituents. According to the TTO model, brittle ceramic constituents are assumed to be elastic throughout the deformation and material flow of FGMs is assumed to be primly induced by plastic deformation of ductile metallic constituents. Thus, multi-linear hardening elastoplastic material properties of FGMs [28] can be defined by introducing the ratio of stress to strain transfer parameter q in

E¼

V c ¼ ð0:5 þ z=hÞ ;

Vc þ Vm ¼ 1

ð1Þ

where k is the power law index, which is a positive real number. It is a critical parameter of constituent distribution. V denotes the



q þ Ec Vm þ Vc q þ Em





 q þ Em Ec Vc q þ Ec Em    q þ Ec q þ Ec Hm V m þ Ec V c Vm þ Vc H¼ q þ Hm q þ Hm

rY ¼ rYm V m þ

ð2Þ

where E(z) is elastic modulus, m the poison ratio, rY(z) yield limit, ~Ec , q ~ is the stress transfer parameter, H(z) the tangent modulus. q ¼ q ~ P 0. It should be noted that q ~ ¼ 0 represents the FGMs flow and q plastically once the metallic constituents reach their yield limit. The most popular elastoplastic constitutive relations of homogeneous materials are J2 flow theory and J2 deformation theory. In flow theory, the relation between stresses and strains is defined in the following increment form.

deij ¼

1 3m 3 S S dr drij  dij drm þ ^ ij ld ld 2G E 4J 2 H

ð3Þ

^ ¼ HE=ðE  HÞ; G ¼ E=½2ð1 þ mÞ. The mean stress rm = where H (rxx + ryy + rzz)/3. Sij is the tensor of stress deviator and the second invariant of stress deviator tensor J2 = SijSij/2. dij is unit matrix. In deformation theory, the constitutive relation of FGMs can be given as

eij ¼

k

q þ Ec Em V m þ Ec V c q þ Em

m ¼ V m mm þ V c mc

3. Material constitutive The constituent distribution of FGMs is usually given according to the power law rule [13]



  3 1 3 dij rm rij þ  2Es K 2Es

ð4Þ

in which, the secant modulus in complex stress state Es ¼ 3EE0s =½3E  ð1  2mÞE0s , E0s is the secant modulus in the uniaxial tension experiment and the elastic parameter K is defined as K = E/(1  2m). The corresponding incremental form of Eq. (4) reads

Fig. 1. Geometry of FGM cylindrical shell and the coordinate system.

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deij



 3 1 3 3 dij drm þ ¼ drij þ  Sij Sld drld 2Es K 2Es 4J 2 /

ð5Þ

in which / ¼ Et Es =ðEs  Et Þ. The tangent modulus in complex stress state Et = 3EH/[3E  (1  2m)H]. According to classic shell theory, the stresses out of the plane rxz, ryz, rzz can be neglected. Meanwhile, in the in-plane shear stress state, we also have rxx = ryy = 0, and the incremental form of flow theory and deformation theory can be simplified greatly as listed in Appendix A.

U in ¼ U inM þ U inB þ U inMB U inM ZZ h i 1 ¼ A11 e0xx2 þ A22 e0yy2 þ A33 e0xy2 þ 2A12 e0xx e0yy dxdy 2 ZZ hX i 1 U inB ¼ D11 K 2xx þ D22 K 2yy þ D33 K 2xy þ 2D12 K xx K yy dxdy 2 ZZX h 1 U inMB ¼ 2B11 K xx e0xx þ 2B22 K yy e0yy þ 2B33 K xy e0xy 2 X

i þ2B12 K yy e0xx þ K xx e0yy dxdy

4. Basic formulation

By using Eqs. (7), the work of external force reads

The strain component of cylindrical shells can be written as

e ¼ e þ zK L

ð6Þ T



0 xx

0 yy

 0 T ; xy

e e where e ¼ ½ exx eyy exy  ; e ¼ e K L ¼ ½ K xx K xy T and according to the von Kárman assumptions 0 L

K yy

1 w 1 2 R 2 K xx ¼ w;xx ; K yy ¼ w;yy ; K xy ¼ 2w;xy

e0xx ¼ u;x þ ðw;x Þ2 ; e0yy ¼ v ;y  þ ðw;y Þ2 ; e0xy ¼ u;y þ v ;x þ w;x w;y ð7Þ

e0L are the strain component on the shell middle plane, KL are the

curvature components. u, v, w are displacements along x, y, z, and the subscript comma denotes partial derivative. For a general form of material constitutive

r ¼ Ae

ð8Þ

T where r ¼ ½ rxx ryy rxy  and the matrix A = [aij], (i, j = 1, 2, 3). It

should be noted that aij are stress-dependent material parameters demonstrating anisotropy of elastoplastic FGMs and they can be defined by Eq. (3) or (5). For present torsional load case, we have

a13 ¼ a31 ¼ a23 ¼ a32 ¼ 0;

a21 ¼ a12

A13 ¼ A31 ¼ A23 ¼ A32 ¼ 0;

A21 ¼ A12

B13 ¼ B31 ¼ B23 ¼ B32 ¼ 0;

B21 ¼ B12

D13 ¼ D31 ¼ D23 ¼ D32 ¼ 0;

ð9Þ

rij dz; Mij ¼

Z

l

NL

" ¼

ML

rij zdz

~ B ~ A ~ ~ B D

#"

e0L

e0L

#

ML

"

¼

^ A ^ C

# ð11Þ

KL

# ^  NL  B ^ KL D

~ 1  B; ^ ¼ A ~ B

~ 1 ; ^¼B ~A C

~ 1  B ^ ¼D ~ B ~A ~ D

The strain energy of FGM cylindrical shells can be given as

U in ¼

1 2

ZZZ

X



e0xy  w;x w;y dxdy

ð15Þ

ð16Þ

To deducing the buckling government equation, all the displacements, strains, internal forces and moments should divided into the prebuckling and buckling parts.

u ¼ u0 þ u1 ; v ¼ v 0 þ v 1 ; w ¼ w0 þ w1 ; Nij ¼ Nij0 þ Nij1 ;

eij ¼ eij0 þ eij1

Mij ¼ M ij0 þ M ij1

ð17Þ

For buckling characteristics of structures, a load increment would induce a large displacement at the buckling state. Thus, the prebuckling displacements and strains should be much smaller than the buckling parts, while the buckling internal forces and moments much smaller than the prebuckling parts. Accordingly, Eqs. (7), (8), (10), (11), (12) and (14) can be rewritten in the similar incremental forms by replacing the displacements, strains with the buckling one, internal forces with the prebuckling one. For instance, Eq. (15) written as

U ex ¼ Nxy0

ZZ





e0xy1  w1;x w1;y dxdy

X

ð18Þ

w1;xx þ ðw1;xy Þ2 þ w1;xx w1;yy R

ð19Þ

Introducing the Airy’s stress function u1 ðx; yÞ which satisfied

Nxx1 ¼ u1;yy ;

Nyy1 ¼ u1;xx ;

Nxy1 ¼ u1;xy

ð20Þ

By eliminating the nonlinear terms, and substituting Eqs. (20) into the incremental form of Eq. (12), then, Eq. (19) turns into u

u

u

w1 1 H1 1 u1;yyyy þ H3 1 u1;xxyy þ H5 1 u1;xxxx þ Hw 1 w1;yyyy þ H3 w1;xxyy þ 1 Hw 5 w1;xxxx þ w1;xx =R ¼ 0

ð21Þ

where u  11 ; Hu1 ¼ A  12 þ A  21 þ A  33 ; Hu1 ¼ A  22 ; H1 1 ¼ A 5 3 w1 w1 w1      21 H1 ¼ B12 ; H3 ¼ B11  B22 þ 2B33 ; H5 ¼ B

ð12Þ

^ ¼ ½A  ij ; B ^ ¼ ½C  ij ; D ^ ¼ ½B  ij ; C ^ ¼ ½D  ij ; ði; j ¼ 1; 2; 3Þ are defined where A as

^¼A ~ 1 ; A



P ¼ U in  U ex

ð10Þ

T T in which NL ¼ ½ N xx Nyy Nxy  ; ML ¼ ½ Mxx Myy Mxy  . and the ~ B; ~ ¼ ½Aij ; B ~ D ~ are given as A ~ ¼ ½Bij ; D ~ ¼ ½Dij ; ði; j ¼ 1; 2; 3Þ. matrix A; By altering the position of e0L and NL the above equation turns into

"

ZZ

The potential energy of FGM cylindrical shell can be written as

l



ðu;y þ v ;x Þdxdy ¼ Nxy

e0xx1;yy þ e0yy1;xx  e0xy1;xy ¼ 

where l is the integral range in the normal direction. The above equations can be rewritten in the matrix form as



ZZ

From the incremental forms of Eq. (7), the compatible equation of incremental strains is given as

D21 ¼ D12

The internal force and moment components

Nij ¼

U ex ¼ Nxy

X

0 L

Z

ð14Þ

ðrT  eÞdzdxdy

ð13Þ

By substituting Eqs. (6) and (8) into the above equation, and integrate through the shell thickness, we have the strain energy divided into the following three part, i.e. stain energies aroused by membrane and bending deformation, as well as the coupling effect of them.

5. Prebuckling state In the present in-plane shear stress state, and under the thin shell assumption, the prebuckling in-plane strain exy0 is assumed to be uniform through the shell thickness, and plastic flow would induced initially in the ceramic-rich side of FGMs, as show in Fig. 1, in which s indicates the distant measuring from the middle plane of the shell. It should be noted that h=2 6 s 6 h=2. The elastic buckling is indicated by s = h/2 when buckling, while the plastic buckling by s = h/2 and the elastoplastic buckling by 0.5h < s < 0.5h. The equivalent stress and stain and their relation are

pffiffiffi

pffiffiffi

r ¼ 3rxy0 ; e ¼ exy0 = 3; r ¼ Es e

ð22Þ

H. Huang et al. / Composite Structures 118 (2014) 234–240

237

The elastoplastic interface can be derived from Mises yield criterion.

hardening elastoplastic material properties defined by Eqs. (1) and (2), and the elastoplastic material properties defined as

r  rY ¼ 0

Ec ¼ 375 GPa;

ð23Þ

To determine the position parameter s of material elastoplastic interface, the present in-plane shear stress state should be related to the uniaxial tensile experiment of material. In the uniaxial tensile experiment, and the stress is given as

rxx0 ¼



Eexx0

h=2 6 z < s

ð24Þ

rY þ Hðexx0  rY =EÞ s 6 z 6 h=2

By considering the first and the third equations of Eqs. (2) and the first equation of Eq. (24) in Eq. (23), and then replacing exx0 with e, the elastoplastic interface position s can be given as

( )  1 Ec þ q Em exy0 k 1 s¼h 1  pffiffiffi  Ec  Em 2 3rYm

ð25Þ

The prebuckling internal force of FGM cylindrical shells is derived by integrating rxy0 through the thickness according to the elastic or plastic material regions.

Nxy0 ¼

Z

h=2

rxy0 dz ¼

Z

h=2

s

h=2

Gdz þ

Z

h=2

! Es =3dz

exy0

ð26Þ

s

6. Solving the problem For FGM cylindrical shells, simply supported at both ends, the buckling deflection is chosen to be

w1 ¼

1 X 1 X

nmn sin am x cos bn ðy þ dxÞ

ð27Þ

m¼1 n¼1

where am = mp/L, bn = n/R. (m, n, d) represents the buckling wave parameters, and nmn is the unknown amplitude. It can be proven

that w1 jx¼0;L ¼ 0 but @ 2 w1 =@x2 x¼0;L –0. However, we have

R 2pR 2 @ w1 =@x2 x¼0;L dy ¼ 0. In other words, the simply support 0 boundary condition is satisfied in the integral sense. Substituting Eq. (27) into Eq. (21), one obtains

u1 ¼

1 X 1 X 2 X ð1Þi fimn sinðhi x þ bn yÞ

ð28Þ

m¼1 n¼1 i¼1

where fimn ¼ #i nmn ; hi ¼ ð1Þi1 am þ bn d; ði ¼ 1; 2Þ and.

#i ¼

1 b4n Hw þ h2 b2 Hw1 þ h4 Hw1  h2 =R 1 4 u1i n 23 2 u1i 5 4 u1i 2 bn H1 þ hi bn H3 þ hi H5

mc ¼ 0:14; Em ¼ 107 GPa; mm ¼ 0:34; rYm ¼ 450 MPa; Hm ¼ 14 GPa; q ¼ 4:5 GPa To simplify the result plot, non-dimensional load parameter k ¼ sxycr =sexycr are introduced in, with sxycr denoting the average critical in-plane shear stress and sexycr the average elastic critical in-plane shear stress. Sofiyev and Schnack [4] gave an analytic solution of torsional critical load for elastic FGM cylindrical shells as 3=4 Nxy0cr ¼ Scrs ¼ 3:435L0:5 B1 ðC 3 B1  C 2 B4 Þ5=8 1 R

ð31Þ

in which, B1, B4, C2, C3 are the material parameter of FGMs. In Fig. 1, comparisons made with those from Eq. (31) show an excellent agreement (see Fig. 2). To verify the present theories, comparisons are made with the experiments of homogeneous aluminium cylindrical shells presented by Ma et al. [21], with the inner radius 14 mm, the elastic modulus 64 GPa, the poison ratio 0.33 and the yield stress 37.155 MPa. In the following calculation, we chose the power law index to be a large value 1000, and Hm = 4 GPa to give approximate results. As shown in Table 1, the present deformation theory gives reasonable results while the flow theory is of enormous deviations. Herein, similar deviations between flow theory and deformation theory had previously been reported for axially compressed isotopic cylindrical shells [22] due to unknown paradox of the plastic constitutive theories. Accordingly, deformation theory would be used to investigate buckling behaviors of elastoplastic FGM cylindrical shells. To investigate the influences of dimensional parameters on buckling of FGM cylindrical shell, the power law index k is chosen to be 1, and R/h is set to be vary from 25 to 150, and L/R is chosen to be 2, 4, 10 for discussions. As shown in Table 2, the critical load sxycr of FGM cylindrical shells decreases dramatically with the increase of R/h and L/R. Yellow and blue backgrounds represent plastic and elastic buckling respectively. It shows elastic buckling usually occurs in long and thin shells with large value of R/h and L/R, while plastic buckling occurs in short and thick shells. Seeing that k varying from 0.456 to 1, the elastic results are much larger than the elastoplastic results in the plastic buckling shells. In the present torsional buckling case, m always equates to 1, while n increases with the increase of R/h, but decreases with L/R. Generally, the torsional mode parameter d is relatively larger in plastic buckling shells than that in elastic buckling shells. In other words, the buckling torsional angel is larger in shorter or thicker shells due to the plastic flow of FGMs.

By introducing Eqs. (27) and (28) into Eq. (20) and the incremental form of Eqs. (7) and (12), the incremental strain energy can be obtained by using Eqs. (14), (16) and (18). " # 1 X 1 X 2 2b4n ðK10 þ K12 a2i Þ þ 2b2n h2i ðK20 þ K22 a2i Þþ pRL X 2 DP ¼ n ð29Þ 8 m¼1 n¼1 i¼1 mn h4i ðK30 þ K31 ai þ K32 a2 Þ þ 2bn hi Nxy0 i By Ritz method, @ DP/@nmn should vanish, and the buckling critical condition is found as " 3    #, 2 ! 2 X X bn K10 þ K12 a2i þ bn h2i K20 þ K22 a2i þ Nxy0 ¼  hi ð30Þ   4 2 K þ K a þ K a Þ ð2b h 30 31 i 32 i i¼1 i¼1 n i 7. Numerical results The present numerical results are calculated through an iterative procedure by converging the results of Eqs. (26) and (30) under a given initial value of exy0 . The following numerical calculations are based on TiB/Ti FGMs [35], with the multi-linear

Fig. 2. Comparison of the present theoretical results with those of Sofiyev and Schnack.

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H. Huang et al. / Composite Structures 118 (2014) 234–240

Table 1 Comparison of the critical moment Mcr(N  m) with experiments. h (mm) 0.5 0.5 a

L (mm) 63 40

Flow theory a

86.87 (3) 103.36 (3)

Deformation theory

Experiments of Ma et al.

Deviation of J2 deform (%)

27.86 (4) 31.24 (3)

28.10 (3) 30.20 (3)

0.85 3.44

The number in bracket is n.

Table 2 Buckling critical loads of FGM cylindrical shells under different dimensional parameters.

The constituent distribution of FGMs is controlled by the power law index k. In this part, basic geometrical parameters L/R is chosen to be 2, and R/h set to be 50, 75, 100. Fig. 3 shows the effects of k varying from 0.1 to 50, with the coordinate axis plotted in the logarithmic form. As shown in Fig. 3(a), the critical load sxycr initially decrease with k dramatically, but tends to be steady when k P 5. The deviation of the elastoplastic critical loads from the elastic ones is reduced as R/h increasing. Meanwhile, the buckling mode (m, n) seems sensitive to the change of k. Fig. 3(b) shows the shifting elastoplastic interface with k. In the case of R=h ¼ 50; k 6 1, the shells occur plastic buckling,

(a)

which indicated by s = 0.5h, while in the other cases, 0.5h < s < 0.5h indicates elastoplastic buckling. According to TTO model, material flow of FGMs is primly induced by plastic deformation of metallic constituents. To investigate this effect, the yield limit rYm and tangent modulus Hm of metallic constituents are artificially assumed to be changeable. In the following discussions, rYm is chosen to be 350, 450, 550 MPa, and Hm varies from 7 to 42 GPa. The geometrical parameters are given as L/R = 2, R/h = 50, 75. As shown in Fig. 4(a), sxycr increases with increasing Hm, but decreases with rYm, and the buckling mode (m, n) seems insensitive

(b)

Fig. 3. Effects of the power law index on buckling of FGM cylindrical shells.

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H. Huang et al. / Composite Structures 118 (2014) 234–240

(a)

(b)

(c) Fig. 4. Effects of the elastoplastic material properties of metallic constituents on buckling of FGM cylindrical shells.

to the change of both of the parameters. It is shown from Fig. 4(b) that the value of s/h is much larger in a small value of rYm and steady with the change of Hm. Therefore, the plastic flow district would be dramatically enlarged when rYm decreases, but seem nearly invariable with Hm. Fig. 4(c) shows k increases with the increase of Hm and rYm. 8. Conclusions In the present work, elastoplastic buckling behaviors of FGM cylindrical shells subject torsional load are investigated by semianalytic method. Prime conclusions are drawn as following.  For torsion-loaded cylindrical shells, deformation theory gives reasonable results while flow theory is of enormous deviations due to unknown paradox of the plastic constitutive theory.  It shows elastic buckling usually occurs in long and thin FGM cylindrical shells with large value of the radius-to-thickness ratio and length-to-radius ratio, while plastic buckling occurs in short and thick shells.  The critical load of FGM cylindrical shells decreases dramatically with the increase of the power law index of FGMs, and the radius-to-thickness ratio or length-to-radius ratio of the shells. Meanwhile, the critical load increases with increasing the tangent modulus, but decreases with the yield limit of metallic constituents.  As the buckling mode as concerned, the axial half wave number always equates to 1 in the present torsional load case, while the circumferential wave number increases with the increase of radius-to-thickness ratio, but decreases with length-to-radius ratio of the shells. The torsional mode parameter d is

relatively larger in plastic buckling shells than that in elastic or elastoplastic buckling shells. In other words, the buckling torsional angel is larger in shorter or thicker shells due to the plastic flow of FGMs.  The plastic flow district would be dramatically enlarged when the yield limit of metallic constituents decreases, but seem nearly invariable with the tangent modulus of metallic constituents.

Acknowledgements The authors wish to acknowledge the supports from the Natural Science Foundation of China (11272123, 11132002), the Ph.D. Programs Foundation of Ministry of Education of China (20110172110031), the Natural Science Foundation of Guangdong Province (S2012040007894), and the Fundamental Research Funds for the Central Universities, SCUT (2013ZZ088). Appendix A Eq. (3) can be reduced as

1 ðdrxx  mdryy Þ E 1 ¼ ðdryy  mdrxx Þ E

dexx ¼ deyy

m

dezz ¼  ðdryy þ drxx Þ E   1 3 dexy ¼ deyx ¼ drxy þ ^ G H dexz ¼ dezx ¼ deyz ¼ dezy ¼ 0

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H. Huang et al. / Composite Structures 118 (2014) 234–240

and Eq. (5) reduced as



   1 1 1 1 dexx ¼ þ  drxx þ dryy 3K Es 3K 2Es     1 1 1 1 drxx þ dryy deyy ¼  þ 3K 2Es 3K Es   1 1 ðdryy þ drxx Þ dezz ¼  3K 2Es   3 3 drxy þ dexy ¼ deyx ¼ Es / dexz ¼ dezx ¼ deyz ¼ dezy ¼ 0 In Eq. (24), the material parameter is given as i 1 1h 2A12 B12 B22 þ A11 B222 þ A212 D22 þ A22 ðB212  A11 D22 Þ ðA212  A11 A22 Þ 2 " # 2 1 ðA12  A11 A22 ÞðD12 þ 2D33 Þ þ B12 ðA11 B22  A12 B12 Þþ K12 ¼ 2A22 ðA212  A11 A22 Þ K20 ¼ 2 1 2 B11 ðA22 B12  A12 B22 Þ  2B33 ðA12  A11 A22 ÞA33 h

i

1 1 2 2 1 2 ðA12  A11 A22 Þ K22 ¼ 2 2A12 þ A12  A11 A22 A33 A12  A11 A22 3 2 2A12 B11 B12 ðA222  A212 Þ þ 2A12 A22 B22 ðA11 B12  A12 B11 Þþ 7 6 K30 ¼ 4 A212 ðA11 B212 þ A22 B222 Þ þ ðA212  A11 A22 Þ2 D11 þ A22 B211 ð2A212  A11 A22 Þþ 5

K10 ¼

A222 B12 ðA22 B12  2A11 B12  2A12 B22 Þ

ðA212

1

 A11 A22 Þ K31 ¼ 4½A12 ðB11  B22 Þ  B12 ðA11  A22 ÞðA212  A11 A22 Þ

K32 ¼ 4A11 ðA212  A11 A22 Þ

1

1

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