Thermal postbuckling analysis of laminated cylindrical shells with piezoelectric actuators

Composite Structures 55 (2002) 13±22

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Thermal postbuckling analysis of laminated cylindrical shells with piezoelectric actuators Hui-Shen Shen School of Civil Engineering and Mechanics, Shanghai Jiao Tong University, 1954 Hua Shan Road, Shanghai 200030, People's Republic of China

Abstract Thermal postbuckling analysis is presented for a cross-ply laminated cylindrical shell with piezoelectric actuators subjected to the combined action of thermal and electric loads. The temperature ®eld considered is assumed to be a uniform distribution over the shell surface and through the shell thickness and the electric ®eld is assumed to be the transverse component EZ only. The material properties are assumed to be independent of the temperature and the electric ®eld. The governing equations are based on the classical shell theory with von Karman±Donnell-type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the e€ects of nonlinear prebuckling deformations, large de¯ections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of hybrid laminated cylindrical shells. A singular perturbation technique is employed to determine buckling temperatures and postbuckling load-de¯ection curves. The numerical illustrations concern thermal postbuckling behavior of perfect and imperfect, cross-ply laminated cylindrical thin shells with fully covered or embedded piezoelectric actuators under thermal and electric loads. The results show that the control voltage has a signi®cant e€ect on the buckling temperature as well as thermal postbuckling response of the shell. In contrast, it has a very small e€ect on the imperfection sensitivity of …0=90†2S laminated cylindrical shells with piezoelectric actuators. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Thermal postbuckling; Hybrid laminated cylindrical shell; Thermo±piezoelectric e€ect; Boundary layer theory of shell buckling; Singular perturbation technique

1. Introduction One of the recent advances in material and structural engineering is in the ®eld of smart structures which incorporates adaptive materials. By taking advantage of the direct and converse piezoelectric e€ects, piezoelectric composite structures can combine the traditional performance advantages of composite laminates along with the inherent capability of piezoelectric materials to adapt to their current environment. Therefore, hybrid laminated structures where a substrate made laminated material is coupled with surface-bonded or embedded piezoelectric actuator and/or sensor layers are becoming increasingly important. Numerous studies on the modeling and analysis of hybrid laminated cylindrical shells have been performed, see for example [1±8]. These studies were focused on the cases of linear bending analysis and/or vibration control. Due to boundary constraints, varying

E-mail address: [email protected] (H.-S. Shen).

temperature environments typically induce stresses, with ensuing buckling. However, relatively few studies have been reported for the thermal buckling of composite laminated cylindrical shells [9±13]. Recently, Oh et al. [14] gave a thermal postbuckling analysis of laminated plates with top and/or bottom surface-bonded actuators subjected to thermal and electric loads. In their analysis nonlinear ®nite element equations based on layerwise displacement theory were formulated, but their numerical results were only for thin plates and all plates were assumed to have perfect initial con®gurations. However, studies on thermal postbuckling of hybrid laminated cylindrical shells have not been seen in the literature. It has been shown in Shen [15] that in shell thermal buckling as well as in shell compressive buckling, there is a boundary layer phenomenon where prebuckling and buckling displacement vary rapidly. This phenomenon was previously reported by Bushnell and Smith [16]. Shen and Chen [17,18] suggested a boundary layer theory of shell buckling, which includes the e€ects of nonlinear prebuckling deformations, large de¯ections in

0263-8223/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 1 2 8 - 3

14

H.-S. Shen / Composite Structures 55 (2002) 13±22

the postbuckling range, and initial geometric imperfections of the shell. Based on this theory, the postbuckling analyses for perfect and imperfect, unsti€ened and sti€ened, laminated cylindrical shells under combined mechanical and thermal loads have been performed by Shen [19±21]. The present paper extends the previous works to the case of composite laminated cylindrical shells with piezoelectric actuators subjected to the combined action of thermal and electric loads. In the present study, the temperature ®eld considered is assumed to be a uniform distribution over the shell surface and through the shell thickness. The electric ®eld is assumed to be the transverse component EZ only. Note that temperature can a€ect the properties of ®berreinforced composites [22]. In addition, the properties of piezoelectric materials, including piezoelectric constants, vary with temperature [23]. The present analysis does not account for these e€ects, i.e., it is assumed that temperature variations do not a€ect material properties. Also, the material properties are assumed to be independent of the electric ®eld. The governing equations are based on the classical shell theory with von Karman± Donnell-type of kinematic nonlinearity and including thermo±piezoelectric e€ects. A singular perturbation technique is employed to determine the buckling temperatures and postbuckling load-de¯ection curves. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account but, for simplicity, the form of initial geometric imperfection is assumed to be the same as the initial buckling mode of the shell. The numerical illustrations show the full nonlinear thermal postbuckling response of hybrid laminated cylindrical shells subjected to the combined action of thermal and electric loads. 2. Theoretical development Consider a circular cylindrical shell with mean radius R, length L and thickness t, which consists of N plies. Some of the plies can be piezoelectric (see Fig. 1). The shell is referred to a coordinate system (X, Y, Z) in which X and Y are in the axial and circumferential directions of the shell and Z is in the direction of the inward normal to the middle surface, the corresponding displacement designated by U , V and W . The origin of the coordinate system is located at the end of the shell on the middle plane. The shell is assumed to be relatively thin and geometrically imperfect, and is subjected to the combined action of thermal and electric loads. Denoting  the initial geometric imperfection by W …X ; Y †, let W …X ; Y † be the additional de¯ection and F …X ; Y † be the stress function for the stress resultants de®ned by N x ˆ F ;yy , N y ˆ F ;xx and N xy ˆ F ;xy , where a comma denotes partial di€erentiation with respect to the corresponding coordinates.

Fig. 1. A hybrid laminated cylindrical shell with piezoelectric layers.

Based on classical shell theory (i.e., transverse shear deformation e€ects are neglected) with von K arm an± Donnell-type kinematic relations and including thermo± piezoelectric e€ects, the governing di€erential equations for a cross-ply laminated cylindrical shell with fully covered or embedded piezoelectric actuators can be derived in terms of a stress function F , and transverse displace ment W , along with initial geometric imperfection W . They are L~11 …W † ‡ L~12 …F †

P L~13 …N †

P L~14 …M †

 ~ ˆ L…W ‡ W ; F †;

L~21 …F † ˆ

L~22 …W †

1 F ;xx R …1†

1 P L~23 …N † ‡ W ;xx R

1~  L…W ‡ 2W ; W †; 2

where o4 o4 o4 L~11 … † ˆ D11 4 ‡ 2…D12 ‡ 2D66 † 2 2 ‡ D22 4 ; oX oX oY oY 4 o L~12 … † ˆ L~22 … † ˆ B21 4 ‡ …B11 ‡ B22 2B66 † oX o4 o4  2 2 ‡ B12 4 ; oX oY oY

…2†

H.-S. Shen / Composite Structures 55 (2002) 13±22

o2 o2 p p p p …B N † L~13 …N † ˆ …B11 N x ‡ B21 N y † ‡ 2 2 oX oX oY 66 xy o2 p p ‡ 2 …B12 N x ‡ B22 N y †; oY 2 o o2 o2 p p p p …M xy † ‡ 2 …M y †; L~14 …M † ˆ …M x † ‡ 2 2 oX oX oY oY o4 o4 o4 L~21 … † ˆ A22 4 ‡ …2A12 ‡ A66 † 2 2 ‡ A11 4 ; oX oX oY oY 2 2 o o p p p p    L~23 …N † ˆ …A N ‡ A22 N y † …A N † oX 2 12 x oX oY 66 xy o2 p p ‡ 2 …A11 N x ‡ A12 N y †; oY 2 2 o2 o2 o2 o2 ~ †ˆ o o L… ‡ 2 2 z: …3† 2 2 oX oY oX oY oX oY oY oX 2 In Eq. (3), [Aij ], [Bij ] and [Dij ] (i; j ˆ 1, 2, 6) are reduced sti€ness matrices, de®ned as in [21] as A ˆ A 1 , B ˆ A 1 B and D ˆ D BA 1 B, where A, B and D are de®ned in the standard way. It is noted that these shell equations show thermo±piezoelectric coupling as well as the interaction of stretching and bending. In the above equations, the equivalent thermo±piezoelectric loads are de®ned as  P  T  E N ˆ NT ‡ NE ; …4† P M M M T

T

E

E

where N , M and N , M are the forces and moments caused by elevated temperature and electric ®eld, respectively. The two end edges of the shell are assumed to be restrained against expansion longitudinally while temperature is increased steadily, so that the boundary conditions are X ˆ 0; L: W ˆ 0;

…5a†

U ˆ 0;

…5b† 2

2

2

oF oF oW Mx ˆ B11 2 D11 oX 2 oY oX 2 2 o W p D12 ‡ M ˆ 0; …simply supported† …5c† oY 2 W ;x ˆ 0; …clamped† …5d† B21

where M x is the bending moment. Also, we have the closed (or periodicity) condition Z 2pR oV dY ˆ 0 …6a† oY 0 or Z 2pR  0

  2 2 2 o2 F  o F  o W  o W B ‡ A ‡ B 12 21 22 oX 2 oY 2 oX 2 oY 2  2  W 1 oW oW oW ‡ R 2 oY oY oY  i p p   A12 N x ‡ A22 N y dY ˆ 0:

15

Because of Eqs. (6a),(6b), the in-plane boundary condition V ˆ 0 (at X ˆ 0, L) is not needed in Eqs. (5a)± (5d). The average end-shortening relationship is de®ned as Z 2pR Z L Dx 1 oU dX dY ˆ 2pRL 0 L 0 oX Z 2pR Z L  1 o2 F o2 F ˆ A11 2 ‡ A12 2 2pRL 0 oY oX 0  2   2 o2 W 1 oW  o W B11 ‡ B 12 oX 2 oY 2 2 oX   oW oW p p   …A11 N x ‡ A12 N y † dX dY : …7† oX oX The temperature ®eld is assumed to be a uniform distribution over the shell surface and through the shell thickness, i.e., T …X ; Y ; Z† ˆ DT . For the panel type piezoelectric material, only thickness direction electric ®eld EZ is dominant, and it is assumed that EZ ˆ

Vk ; tk

…8†

where Vk is the applied voltage across the kth ply and tk is the thickness of the ply. The forces and moments caused by elevated temperature or electric ®eld are de®ned by 2 T T 3 2 3 Nx Mx Ax N Z tk X T 7 6 T …1; Z†4 Ay 5 DT dZ …9a† 4 Ny My 5 ˆ tk 1 T T kˆ1 A xy k N M xy

xy

and 2 E Nx 6 E 4 Ny

E 3 2 3 Mx Bx N Z tk X Vk E 7 My 5 ˆ …1; Z†4 By 5 dZ; tk tk 1 E kˆ1 B xy M xy k

E N xy

in which 2 3 Ax 4 Ay 5 ˆ Axy

2

Q11 4Q 12 Q16

Q12 Q22 Q26

32 2 c Q16 Q26 54 s2 2cs Q66

…9b†

3   s2 a c2 5 11 ; a22 2cs …10a†

2

3

Bx 4 By 5 ˆ Bxy

2

Q11 4Q 12 Q16

Q12 Q22 Q26

32

c2 Q16 4 5 s2 Q26 2cs Q66

3

  s2 d c2 5 31 ; d32 2cs …10b†

A22

…6b†

where a11 and a22 are the thermal expansion coecients measured in the ®ber and transverse directions, respectively, d31 and d32 are piezoelectric strain constants of a single ply, and Qij are the transformed elastic constants, de®ned by

16

H.-S. Shen / Composite Structures 55 (2002) 13±22

3 2 4 c Q11 6 Q 7 6 c 2 s2 6 12 7 6 7 6 6 6 Q22 7 6 s4 7 6 6 6 Q 7 ˆ 6 c3 s 6 16 7 6 7 6 6 4 Q26 5 4 cs3 2

Q66

3

2c2 s2

s4

4c2 s2

c 4 ‡ s4 2c2 s2

c2 s2 c4

4c2 s2 4c2 s2

cs3

c3 s

cs3

c3 s

cs3

c3 s

c2 s2 2c2 s2 2 3 Q11 6Q 7 6 12 7 6 7; 4 Q22 5

c2 s2

2cs…c2 2cs…c2 …c2

Also let " T# " # N Z tk Ax Ax X dZ; ˆ tk 1 ATy Ay k kˆ1 " P# " # N Z tk Bx Bx V k X dZ: DV ˆ tk 1 BPy By k t k kˆ1

7 7 7 7 7 2 7 s †7 7 s2 † 5

s2 †2 …11a†

c14 F;xx

ˆ c14 b2 L…W ‡ W  ; F †;

where E11 Q11 ˆ ; …1 m12 m21 † m21 E11 ; Q12 ˆ …1 m12 m21 †

E22 Q22 ˆ ; …1 m12 m21 † Q66 ˆ G12 ;

L21 …F † …11b†

ˆ

1 c b2 L…W ‡ W  ; W †; 2 24

where

c ˆ cos h;

L11 … † ˆ

s ˆ sin h;

…11c†

where h is the lamination angle with respect to the shell X-axis.

Having developed the theory, we will try to solve Eqs. (1) and (2) with boundary conditions (5a)±(5d). Before proceeding, it is convenient ®rst to de®ne the following dimensionless quantities x ˆ pX =L; y ˆ Y =R; b ˆ L=pR; Z ˆ L2 =Rt;  1=4 e ˆ …p2 R=L2 † D11 D22 A11 A22 ; . 1=4  …W ; W  † ˆ e…W ; W † D11 D22 A11 A22 ; . 1=2 F ˆ e2 F D11 D22 ; c12 ˆ …D12 ‡ 2D66 †D11 ;   1=2 1 c22 ˆ …A12 ‡ A66 † A22 ; c14 ˆ D22 =D11 ; 2    1=2 c24 ˆ A11 =A22 ; c5 ˆ A12 =A22 ; …c30 ; c32 ; c34 ; c311 ; c322 † ˆ …B21 ; B11 ‡ B22

. 2B66 ; B12 ; B11 ; B22 † ‰D11 D22 A11 A22 Š1=4 ;

 1=4 …cT 1 ; cT 2 ; cP 1 ; cP 2 † ˆ …ATx =a0 ;ATy =a0 ; BPx ; BPy †R A11 A22 =D11 D22 ;  p …Mx ; Mxp † ˆ e2 …M x ; M x †…L2 =p2 † D11 ‰D11 D22 A11 A22 Š1=4 ;   Dx R ; kT ˆ a0 DT ; …12† dx ˆ L 2‰D11 D22 A11 A22 Š1=4

where a0 is an arbitrary reference value, and a22 ˆ a22 a0 :

…13†

L… † ˆ

…16†

o4 o4 o4 ‡ 2c12 b2 2 2 ‡ c214 b4 4 ; 4 ox ox oy oy

L12 … † ˆ L22 … † ˆ c30 L21 … † ˆ

3. Analytical method and asymptotic solutions

…15†

ec24 L22 …W † ‡ c24 W;xx

E11 , E22 , G12 , m12 and m21 have their usual meanings, and

a11 ˆ a11 a0 ;

…14b†

The nonlinear Eqs. (1) and (2) may then be written in dimensionless form as e2 L11 …W † ‡ ec14 L12 …F †

Q66

…14a†

o4 o4 o4 ‡ c32 b2 2 2 ‡ c34 b4 4 ; 4 ox ox oy oy

o4 o4 o4 ‡ 2c22 b2 2 2 ‡ c224 b4 4 ; 4 ox ox oy oy

o2 o2 ox2 oy 2

2

o2 o2 o2 o2 ‡ 2 2: oxoy oxoy oy ox

…17†

Because of the de®nition of egiven in Eq. (12), 1=4 for most of the composite materials D11 D22 A11 A22 ˆ …0:24 0:3†t, hence when Z ˆ …L2 =Rt† > 2:96, we have e < 1. Specially, pfor  isotropic cylindrical shells,1=2we have e ˆ p2 =Z B 12, where Z B ˆ …L2 =Rt†‰1 m2 Š is the Batdorf shell parameter, which should be greater than 2.85 in the case of classical linear buckling analysis (see Batdorf [24]). In practice, the shell structure will have Z P 0, so that we always have e  1. When e < 1, then Eqs. (15) and (16) are the equations of the boundary layer type, from which nonlinear prebuckling deformations, large de¯ections in the postbuckling range, and initial geometric imperfections of the shell can be considered simultaneously. The boundary conditions expressed by Eq. (5a)±(5d) become x ˆ 0, p; W ˆ 0;

…18a†

dx ˆ 0;

…18b†

Mx ˆ 0; W;x ˆ 0

…simply supported† …clamped†

and the closed condition becomes

…18c† …18d†

H.-S. Shen / Composite Structures 55 (2002) 13±22

Z 0

2p

"

o2 F ox2

‡ c24 W ‡ e…cT 2

X

17

   2 o2 F o2 W 2o W ec ‡ c c b c5 b 24 30 322 oy 2 ox2 oy 2  2 1 oW oW oW  c24 b2 c24 b2 2 oy oy oy #

The initial buckling mode is assumed to have the form

c5 cT 1 †kT ‡ e…cp2

w2 …x; y† ˆ A11 sin mx sin ny

2

c5 cp1 †DV dy ˆ 0:

…19†

c5 cP 2 †DV dx dy:

jˆ0

…2†

F~…x; n; y; e† ˆ

X

where l ˆ a11 =A11 is the imperfection parameter. Substituting Eqs. 21, 23a, 23b and 23c into Eqs. (15) and (16), collecting the terms of the same order of e, three sets of perturbation equations are obtained for the regular and boundary layer solutions, respectively. Then using Eqs. (24) and (25) to solve these perturbation equations of each order, and matching the regular solutions with the boundary layer solutions at the each end of the shell, so that the asymptotic solutions satisfying the clamped boundary conditions are constructed as    x a x x a p cos / p ‡ sin/ p exp e / e e     p x a p x p x …1† a p A00 cos/ p ‡ sin/ p exp e / e e  …2† …2† …2† ‡ e2 A11 sinmxsin ny ‡ A02 cos2ny …A02 cos2ny†     x a x x a p  cos/ p ‡ sin / p exp e / e e   p x a p x …2† …A02 cos2ny† cos / p ‡ sin/ p e / e   h i p x …3† …3†  exp a p ‡ e3 A11 sin mxsinny ‡ A02 cos2ny e h …4† …4† …4† ‡ e4 A00 ‡ A20 cos2mx ‡ A02 cos2ny i …4† …4† …26† ‡ A13 sin mxsin3ny ‡ A04 cos4ny ‡ O…e5 †;

 …1† W ˆ e A00

…21†

ej‡1 W~j‡1 …x; n; y†;

jˆ0

ej‡2 F~j‡2 …x; n; y†;

…25†

…2†

jˆ0

jˆ0

…24†

…2†

(This means for isotropic cylindrical shells the width of p  the boundary layers is of the order Rt.) In Eq. (21) the regular and boundary layer solutions are taken in the forms of perturbation expansions as X X w…x;y;e† ˆ ej wj …x; y†; f …x;y; e† ˆ ej fj …x;y†; …23a† W~ …x; n; y; e† ˆ

…23c†

jˆ0

where e is a small perturbation parameter (see beneath Eq. (17)) and w…x; y; e†, f …x; y; e† are called outer solutions or regular solutions of the shell, W~ …x; n; y; e†, F~…x; n; y; e† and W^ …x; 1; y; e†, F^…x; 1; y; e† are the boundary layer solutions near the x ˆ 0 and x ˆ p edges, respectively, and n and f are the boundary layer variables, de®ned as p p n ˆ x= e; 1 ˆ …p x†= e: …22†

X

ej‡2 F^j‡2 …x; 1; y†:

ˆ e2 lA11 sin mx sin ny;

By virtue of the fact that DV and DT are assumed to be uniform, the thermo±piezoelectric coupling in Eqs. (1) and (2) vanishes, but terms in DV and DT intervene in Eqs. (19) and (20). Applying Eqs. (15)±(17), (18a)±(18d), (19), (20), thermal postbuckling behavior of perfect and imperfect, laminated cylindrical shells with piezoelectric actuators subjected to the combined action of thermal and electric loads is determined by a singular perturbation technique. The essence of this procedure, in the present case, is to assume that

jˆ1

X

F^…x; 1; y; e† ˆ

W  …x; y; e† ˆ e2 a11 sin mx sin ny

…20†

W ˆ w…x; y; e† ‡ W~ …x; n; y; e† ‡ W^ …x; 1; y; e†; F ˆ f …x; y; e† ‡ F~…x; n; y; e† ‡ F^…x; 1; y; e†;

ej‡1 W^j‡1 …x; 1; y†;

and the initial geometric imperfection is assumed to have a similar form

The unit end-shortening relationship becomes  Z 2p Z p " 2 1 o2 F 1 2 2o F dx ˆ c5 2 e c24 b 4p2 c24 oy 2 ox 0 0  2   2 o2 W 1 oW 2o W ec24 c311 2 ‡ c34 b c ox oy 2 2 24 ox oW oW  ‡ e…c224 cT 1 c5 cT 2 †kT c24 ox ox # ‡ e…c224 cP 1

W^ …x; 1; y; e† ˆ

…23b†

Fˆ

…1†



A00

2



‡e

…1† y B00

2





2 …2† y B00 ‡e 2 2 2   x x …2† …1† …2† …2† ‡ B11 sin mx sin ny ‡ A00 b01 cos/ p ‡ b10 sin / p e e    x p x …1† …2†  exp a p ‡ A00 b01 cos/ p e e    p x p x …2† ‡ b10 sin / p exp a p e e  2 …3† y …3† …2† ‡ e3 B00 ‡ B02 cos 2ny ‡ …A02 cos2ny† 2 …0† y B00

2

18

H.-S. Shen / Composite Structures 55 (2002) 13±22

    x x x …3† …3†  b01 cos / p ‡ b10 sin / p exp a p e e e   p x p x …2† …3† …3† ‡ …A02 cos 2ny† b01 cos / p ‡ b10 sin / p e e    2 p x …4† y …4†  exp a p ‡ e4 B00 ‡ B11 sin mx sin ny 2 e  …4† …4† …4† ‡ B20 cos 2mx ‡ B02 cos 2ny ‡ B13 sin mx sin 3ny ‡ O…e5 †:

…27†

Note that, all of the coecients in Eqs. (26) and (27) are …2† related and can be written as functions of A11 , but for the sake of brevity the detailed expressions are not shown, whereas a and / are given in detail in Appendix A. Next, substituting Eqs. (26) and (27) into the boundary condition (18b) and into closed condition (19), the thermal postbuckling equilibrium path can be written as …0†

kT ˆ C11 ‰kT

…P †

kT

…2†

…2†

…4†

…2†

kT …A11 e†2 ‡ kT …A11 e†4 ‡


Š; …28†

…2† (A11 e)

in Eq. (28), is taken as the second perturbation parameter relating to the dimensionless maximum de¯ection. If the maximum de¯ection is assumed to be at the point …x; y† ˆ …p=2m; p=2n†, then …2†

A11 e ˆ Wm

H3 Wm2 ‡


;

…29a†

where Wm is the dimensionless form of maximum de¯ection of the shell that can be written as " # 1 t W Wm ˆ ‡ H4 : …29b† C3 ‰D11 D22 A11 A22 Š1=4 t All symbols used in Eqs. (28), (29a) and (29b) are also described in detail in Appendix A. Eqs. (28), (29a) and (29b) can be employed to obtain numerical results for full nonlinear thermal postbuck-

ling load-de¯ection curves of cross-ply laminated cylindrical shells with piezoelectric actuators subjected to the combined action of thermal and electric loads. In the next section, asymptotic solutions up to fourth-order, as given by Eqs. (26)±(28), are used. The initial buckling temperature of a perfect shell can readily be obtained  numerically, by setting W =t ˆ 0 (or l ˆ 0), while taking W =t ˆ 0 (note that Wm 6ˆ 0). In this case, the minimum thermal buckling load is determined by considering Eq. (28) for various values of the buckling mode (m, n), which determine the number of half waves in the X-direction and of full waves in the Y-direction. Note that because of Eq. (26), the nonlinear prebuckling deformation of the shell has been shown.

4. Numerical results and comments Numerical results are presented in this section for perfect and imperfect, cross-ply laminated cylindrical shells with symmetrically fully covered or embedded piezoelectric layers, where the outmost layer is the ®rst mentioned orientation. Graphite/epoxy composite material and PZT-5A were selected for the substrate orthotropic layers and piezoelectric layers, respectively. The material properties for Graphite/epoxy orthotropic 2 layers of the substrate were [14]: E11 ˆ 1:5  105 MN=m , 2 2 3 3 E22 ˆ 9:0  10 MN=m , G12 ˆ 7:1  10 MN=m , m12 ˆ 0:3, a11 ˆ 1:1  10 6 =°C, a22 ˆ 25:2  10 6 =°C, and for PZT-5A piezoelectric layers E11 ˆ E22 ˆ 6:3  104 MN=m2 , G12 ˆ 2:42  104 MN=m2 , m12 ˆ 0:3, a11 ˆ a22 ˆ 0:9  10 6 =°C and d31 ˆ d32 ˆ 2:54  10 10 m=V. However, the analysis is equally applicable to other types of composite materials as well. For these examples, the total thickness of the shell t ˆ 1:2 mm whereas the thickness of piezoelectric layers is 0.1 mm, and all other orthotropic layers are of equal thickness. A parametric study has been carried out and typical results are shown in Tables 1 and 2, and Figs. 2±6. It

Table 1 Comparisons of buckling temperatures DTcr (°C) for perfect piezolaminated cylindrical shells (R=t ˆ 300) under uniform temperature rise and three sets of electric loading conditions DTcr (°C) …P=…0=90†2 †S

a

a

…0=P=90=0=90†S

…P=…0=90†4 =P†T

…0=P=…90=0†3 =P=90†T

Z ˆ 100

VU ˆ VL ˆ 100 V VU ˆ VL ˆ 0 V VU ˆ VL ˆ ‡100 V

428.96 (2,14) 409.73 (2,14) 390.47 (2,14)

405.59 (2,14) 386.35 (2,14) 367.10 (2,14)

428.25 (3,14) 408.96 (3,14) 389.70 (3,14)

412.44 (3,14) 393.13 (3,14) 373.83 (3,14)

Z ˆ 200

VU ˆ VL ˆ 100 V VU ˆ VL ˆ 0 V VU ˆ VL ˆ ‡100 V

428.80 (3,14) 409.52 (3,14) 390.23 (3,14)

406.83 (3,14) 387.55 (3,14) 368.26 (3,14)

428.29 (4,14) 408.95 (4,14) 389.62 (4,14)

412.64 (4,14) 393.30 (4,14) 373.97 (4,14)

Z ˆ 500

VU ˆ VL ˆ 100 V VU ˆ VL ˆ 0 V VU ˆ VL ˆ ‡100 V

429.30 (5,14) 410.00 (5,14) 390.67 (5,14)

408.70 (5,15) 389.38 (5,15) 370.06 (5,15)

429.16 (6,14) 409.80 (6,14) 390.45 (6,14)

413.71 (6,14) 394.37 (6,14) 375.02 (6,14)

The number in brackets indicate the buckling mode (m; n).

H.-S. Shen / Composite Structures 55 (2002) 13±22

19

Table 2 Imperfection sensitivity k for imperfect piezolaminated cylindrical shells under uniform temperature rise and three sets of electric loading conditions (R=t ˆ 300 and Z ˆ 500) 

Lay-up

W =t

0.0

0.02

0.03

0.04

…P=…0=90†2 †S

VU ˆ VL ˆ 100 V VU ˆ VL ˆ 0 V VU ˆ VL ˆ ‡100 V

1.0 1.0 1.0

0.9399 0.9394 0.9388

0.916 0.914 0.913

0.896 0.894 0.891

…0=P=90=0=90†S

VU ˆ VL ˆ 100 V VU ˆ VL ˆ 0 V VU ˆ VL ˆ ‡100 V

1.0 1.0 1.0

0.937 0.936 0.935

0.912 0.910 0.908

0.892 0.889 0.886



should be appreciated that in all of these ®gures W =t denotes the dimensionless maximum initial geometric imperfection of the shell. Fig. 2 shows the variation of buckling temperature DTcr with shell geometric parameter Z for …0=90†2S symmetric cross-ply and …0=90†4T antisymmetric crossply laminated cylindrical shells with symmetrically fully covered or embedded piezoelectric layers, referred to as …P=…0=90†2 †S , …0=P=90=0=90†S , …P=…0=90†4 =P†T and …0=P=…90=0†3 =P=90†T , respectively. The control voltage with the same sign is also applied to both upper and

Fig. 3. E€ects of applied electric voltages on the thermal postbuckling of a …P=…0=90†2 †S cylindrical shell.

Fig. 2. Variation of buckling temperature with shell geometric parameter Z.

lower piezoelectric layer, referred to as VU and VL . Three electric loading cases are considered. Here, VU ˆ VL ˆ 0 V means the buckling under a grounding condition. It can be seen that for lower values of Z, the buckling temperature varies rapidly, whereas for higher values of Z, the buckling temperature approaches a constant value. It is noted that when Z ˆ 5 the …0=P=90=0=90†S shell has higher buckling temperature, but for other values of Z it has lower buckling temperature than the …P=…0=90†2 †S shell (see Fig. 2(a)). In contrast, the …0=P=…90=0†3 =P=90†T shell always has lower buckling temperature than the …P=…0=90†4 =P†T shell. It can also be seen that the applied voltages a€ect the buckling temperature signi®cantly. Numerical results for perfect …P=…0=90†2 †S , …0=P=90=0=90†S , …P=…0=90†4 =P†T and …0=P=…90=0†3 =P=90†T cylindrical shells with Z ˆ 100, 200 and 500 subjected to a uniform temperature rise and under three electric loading conditions are presented in Table 1 to show the di€erences. Fig. 3 shows the e€ect of applied voltages on the thermal postbuckling load-de¯ection curves for a …P=…0=90†2 †S cylindrical shell subjected to a uniform temperature rise. It can be seen that the minus control voltages VU ˆ VL ˆ 100 V make the shell panel contract so that the buckling temperature is increased and

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H.-S. Shen / Composite Structures 55 (2002) 13±22

Fig. 4. Comparisons of thermal postbuckling behavior of …P=…0=90†2 †S and …P=…0=90†4 =P†T cylindrical shells under thermal and electric loads.

Fig. 5. Comparisons of thermal postbuckling behavior of …P=…0=90†4 =P†T and …0=P=…90=0†3 =P=90†T cylindrical shells under thermal and electric loads.

the postbuckled de¯ection is decreased at the same temperature rise. In contrast, the plus control voltages VU ˆ VL ˆ ‡100 V decrease the buckling temperature and induce more large postbuckled de¯ections. It can also be seen that only a very weak ``snap-through'' phenomenon occurs in the postbuckling range. Fig. 4 compares the thermal postbuckling load-de¯ection curves for …P=…0=90†2 †S and …P=…0=90†4 =P†T cylindrical shells under thermal and electric loads. The results show that the thermal postbuckling load-de¯ection curves of …0=90†2S cylindrical shell is lower than that of …0=90†4T cylindrical shell with symmetrically fully covered piezoelectric layers, when W =t > 0:5. Fig. 5 compares the thermal postbuckling load-de¯ection curves for …P=…0=90†4 =P†T and …0=P=…90=0†3 =P=

Fig. 6. E€ects of shell geometric parameter on the thermal postbuckling of …0=P=90=0=90†S cylindrical shells.

90†T cylindrical shells under thermal and electric loads. The results show that the shell with embedded piezoelectric layers has lower buckling temperature and postbuckling equilibrium path. Fig. 6 shows the e€ect of shell geometric parameter Z ( ˆ 100 and 500) on the thermal postbuckling behavior of …0=P =90=0=90†S cylindrical shells under thermal and electric loads. It can be seen that the buckling temperatures compare very closely for these two shells. In contrast, the thermal postbuckling load-de¯ection curve of the shell with Z ˆ 100 is lower than that of the shell with Z ˆ 500, when W =t > 0:5. The e€ects of control voltages on the imperfection sensitivities of …0=90†2S laminated cylindrical shell with symmetrically fully covered or embedded piezoelectric layers are shown in Table 2, and only a very small e€ect can be seen. Note that …0=90†4T laminated cylindrical shell with piezoelectric layers is imperfection±insensitive. In Table 2, k is the maximum value of rx for the imperfect shell, made dimensionless by dividing by the critical value of rx for the perfect shell.

5. Concluding remarks Thermal postbuckling analysis has been presented for cross-ply laminated cylindrical shells with piezoelectric actuators subjected to thermal and electrical loads. The temperature ®eld considered is assumed to be a uniform distribution over the shell surface and through the shell thickness and the electric ®eld is assumed to be the transverse component EZ only. The material properties are assumed to be independent of the temperature and the electric ®eld. A boundary layer theory of shell buckling has been extended to the case of hybrid laminated cylindrical shells. A singular

H.-S. Shen / Composite Structures 55 (2002) 13±22

perturbation technique is employed to determine buckling temperatures and postbuckling load-de¯ection curves. The solutions presented give an insight into interaction between the thermal and electric ®elds. A parametric study for symmetric and antisymmetric cross-ply laminated shells with fully covered or embedded piezoelectric actuators under thermal and electric loads has been carried out. The results presented herein show that the minus control voltages increase the buckling temperature and decrease the postbuckled de¯ection at the same temperature rise, whereas the plus control voltages decrease the buckling temperature and induce more large postbuckled de¯ections. They also show that the control voltage has a very small e€ect on the imperfection sensitivities of …0=90†2S laminated cylindrical shells with piezoelectric actuators. Acknowledgements This work is supported in part by the National Natural Science Foundation of China under Grant 59975058. The author is grateful for this ®nancial support. Appendix A In Eqs. (27)±(29b),   4 1 c224 m …1 ‡ l† 1 H3 ˆ e C3 c14 c24 ‡ c234 16n2 b2 g2   1 c224 m2 ‡ 2 32 c14 c24 ‡ c34 c24 n2 b2   c g3  34 …1 ‡ 2l† 2c24 c24 g2   c5 1 g2 ‡ m2 …1 ‡ 2l ‡ e1=2 †e 2g3 e2 ‡ 32 e3 pa 8g8 m  c224 c25 cT 2 …2† ; ‡ k c24 gT T   c224 c25 cP 2 cT 2 …0† H4 ˆ DV ‡ kT ; c24 g8 gT 2 c m …2 ‡ l†g3 …0† kT ˆ 24 e 1 ‡ c24 …1 ‡ l†g2 …1 ‡ l†2 g2 " # 1 g1 g32 …1 l l2 † ‡ c24 e ‡ …1 ‡ l†m2 c14 g2 …1 ‡ l†2   l g3 g3 1‡ e …1 ‡ l†m2 …1 ‡ l†2 m4 " # g1 g2 …4 ‡ 4l ‡ l2 † 2 ‡ c24 3 e;  c14 g2 …1 ‡ l†2 gP …P † kT ˆ DV ; g8

…2†

kT ˆ

21



 c224 c24 m6 …2 ‡ l† 1 e 2 2g22 c14 c24 ‡ c34   4" c224 m c34 …1 ‡ l†2 ‡ …1 ‡ 2l† ‡ 1‡l c14 c24 ‡ c234 2g2 c24    g3 l…3 ‡ l† 1 c14 ‡ c24 4 c14 c24 ‡ c234 g2 …1 ‡ l†

c24 m2 n4 b4 g2 2 g2 …5 ‡ 11l ‡ 4l † ‡ 8m4 …1 ‡ l†…2 ‡ l†  g2 …1 ‡ l† 4m4   2 " c224 m g3 c34 …4 ‡ 12l ‡ 15l2 ‡ 4l3 † 2 c14 c24 ‡ c34 4g2 c24 …1 ‡ l†2 # g3 …4 ‡ l ‡ 2l2 ‡ l3 † e ‡ 2c24 g2 …1 ‡ l†2   c24 1 g2 m2 …1 ‡ 2l ‡ e1=2 †e 2g3 e2 ‡ 32 e3 ; pa 2g8 m  2 2 10 1 c24 c24 m …1 ‡ l† …4† kT ˆ 64 c14 c24 ‡ c234 g23  m2 …1 ‡ 2l†e ‡ e

g13 …6 ‡ 6l ‡ l2 † ‡ g2 …6 l2 †…1 ‡ l† 1 e g13 g2 …1 ‡ l†   2 8 1 c24 b c224 m …1 ‡ l†2 3=2 ‡ e 64 g8 32pa c14 c24 ‡ c234 n4 b4 g22  2 ) 4 2 3 g2 …1 ‡ 2l† ‡ 8m …1 ‡ l† 2 4 4 …A1† ‡ m n b …1 ‡ l† e g2 …1 ‡ l† 4m4 

in the above equations g1 ˆ m4 ‡ 2c12 m2 n2 b2 ‡ c214 n4 b4 ; g2 ˆ m4 ‡ 2c22 m2 n2 b2 ‡ c224 n4 b4 ; g3 ˆ c30 m4 ‡ c32 m2 n2 b2 ‡ c34 n4 b4 ; g13 ˆ m4 ‡ 18c22 m2 n2 b2 ‡ 81c224 n4 b4 ;  1=2 g3 c14 c24 e; b ˆ ; C3 ˆ 1 m2 1 ‡ c14 c24 c230  1=2  1=2 c14 c24 c30 b c b‡c ; aˆ ; /ˆ ; cˆ 2 2 1 ‡ c14 c24 c230 g8 4 a 2 1=2 ce ; C11 ˆ ; g8 ˆ c224 pb 5 gT 4a c …c c5 cT 1 †e1=2 ; gT ˆ …c224 cT 1 c5 cT 2 † ‡ p b 5 T2 4a c …c c5 cP 1 †e1=2 : …A2† gP ˆ …c224 cP 1 c5 cP 2 † ‡ p b 5 P2 References [1] Tzou HS, Gadre M. Theoretical analysis of a multi-layered thin shell coupled with piezoelectric shell actuators for distributed vibration controls. Journal of Sound and Vibration 1989;132(3):433±50.

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