Computational approach of piezoelectric shells by the GDQ method

Composite Structures 92 (2010) 811–816

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Computational approach of piezoelectric shells by the GDQ method C.C. Hong * Department of Mechanical Engineering, Hsiuping Institute of Technology, Taichung, 412 Taiwan, ROC

a r t i c l e

i n f o

Article history: Available online 19 August 2009 Keywords: GDQ PVDF Shear deformation Piezoelectric shells

a b s t r a c t A piezoelectric laminated cylindrical shell with shear rotations effect under the electromechanical loads and four sides simply supported boundary condition was studied by using the two-dimensional generalized differential quadrature (GDQ) computational method. The typical hybrid composite shells with 3layered cross-ply [90°/0°/90°] graphite–epoxy laminate and bounded PVDF layers are considered under the sinusoidal pressure loads and electric potentials on the shell. The governing partial differential equation with first-order shear deformation theory in terms of mid-surface displacements and shear rotations can be expressed in series equations by the GDQ formulation. Thus we obtain the GDQ numerical solutions of non-dimensional displacement and stresses at center position of laminated piezoelectric shells. Displacement is generally affected by the thickness of laminated piezoelectric shells under the action of mechanical load. Stresses are generally affected by the thickness and the length of laminated piezoelectric shells under the actions of mechanical load and electric potential. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Piezoelectric materials had been utilized to many applications such as: transducer, actuator, sensor, smart structure, nanotube, resonant ultrasonic device and medical imaging field. Skatulla et al. used a nonlinear generalized continuum approach to study the electro-mechanically coupled behaviour of electro-active polymers (EAP) [1]. Brodal et al. made the discontinuous Galerkin finite element studies for the piezoelectric transducer [2]. Frankel and Chisholm made the sensitivity analyses for the PVDF thin films [3]. Salehi-Khojin and Jalili presented the numerical results for a PVDF reinforced with carbon nanotube [4]. Wang et al. presented a numerical result for the response of piezoelectric films [5]. Jiang and Li used a new finite element model to study the thermal deformation of PVDF bimorph and piezolaminated graphite–epoxy beam [6]. Dong and Wang presented an analytical method to study the wave propagation in piezoelectric cylindrical laminated shells [7]. Hong used the GDQ method to study the thermal vibration of Laminated Magnetostrictive Plates [8]. Tornabene and Viola analyzed the dynamical manner for spherical structure by using the GDQ method [9]. Choi et al. made the numerical simulation for the thin walled beams with MFC actuators and PVDF sensors [10]. Hong et al. also made the numerical analyses of piezoelectric strip under the symmetric pressure, voltage on the upper and lower edges and the traction-free boundary condition with the GDQ method [11]. Hong et al. studied the thermally induced vibra-

* Tel.: +886 9 19037599; fax: +886 4 24961187. E-mail address: [email protected] 0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2009.08.026

tion of a thermal sleeve with the GDQ method [12]. Hong and Jane made the thermal bending and thermal vibration analyses of shear deformable laminated plates by using the GDQ method, respectively [13,14]. Kapuria et al. had some solutions about the three-dimensional piezoelectric cylindrical shell, such as Naviertype approach [15] and exact analytical Fourier series solution [16]. Alibeigloo and Madoliat used the differential quadrature method to make the static analysis of cross-ply laminated plates with integrated surface piezoelectric layers [17]. Liao and Yu used the variational asymptotic method to construct an electromechanical Reissner–Mindlin model for laminated piezoelectric plates [18]. It is interesting to compute the static solution for the piezoelectric laminated circular cylindrical shell. The purpose of this paper is to use the GDQ method to compute and study the distribution of stresses and deflection of laminated piezoelectric shells, under the electromechanical static load. The governing partial differential equation in terms of displacements and shear rotations are derived into the series form of discretized equations by using the GDQ method. 2. Governing equations The governing partial differential equilibrium equation in terms of mid-surface displacements u0, v0, w0 and shear rotations u1, u2 under the first-order shear deformation theory and the Flugge-type thin-shell approximation for a piezoelectric laminated circular cylindrical shell can be re-written in the following matrix forms by Kapuria et al. [15]:

½Lij U ¼ P

ð1Þ

812

C.C. Hong / Composite Structures 92 (2010) 811–816

where U = [u0v0w0u1u2]T, P = [P1P2P3P4P5]T is the load vector, Lij are the elements of matrix of differential operators in symmetric forms Lij = Lji, listed as follows:

P2 ¼ ph  ðQ eh þ Neh;h Þ=R;

P 1 ¼ px  Nex;x ;

 N M N X aðA12 þ A66 Þ=R X h ðA66 þ B66 Þ X ð1Þ ð1Þ ð2Þ Ai;‘ Bj;m U ‘;m þ Ai;‘ V ‘;j aw a2 ‘¼1 m¼1 ‘¼1

þ

P 3 ¼ pz þ Q ex;x þ ðQ eh;h  N eh Þ=R; P 4 ¼ mx þ Q ex  Mex;x ;

P5 ¼ mh þ Q eh  M eh;h =R

L11 ¼ f1 ðÞ;xx þ f2 ðÞ;hh ;

L12 ¼ f3 ðÞ;xh ;

L14 ¼ f5 ðÞ;xx þ f6 ðÞ;hh ;

L15 ¼ f7 ðÞ;xh

L22 ¼ f8 ðÞ;xx þ f9 ðÞ;hh þ f10 ;

L23 ¼ f11 ðÞ;h ;

L33 ¼ f16 ðÞ;xx þ f17 ðÞ;hh þ f18 ; L45 ¼ f23 ðÞx;h ;

f2 ¼ ðA66  B66 þ D66 Þ=R2 ; f 5 ¼ B11 þ D11 =R; f8 ¼ A66 þ B66 ;

N M N X ðB12 þ B66Þ X B66 þ D66 =R X ð1Þ ð1Þ ð2Þ Ai;‘ Bj;m /1‘;m þ Ai;‘ /2‘;j 2 a aw ‘¼1 m¼1 ‘¼1

f 4 ¼ A12 =R;

f 7 ¼ f12 ¼ B12 þ B66

f 9 ¼ f18 ¼ ðA22  B22 þ D22 Þ=R2 ;

Bij ¼ Bij =R;

þ

f 19 ¼ B12  B55  A55 ;

Dij ¼ Dij =R

f 26 ¼ D22 ; þ

2

in which Aij, Bij and Dij are the in-plane, coupling and bending stiffness of the laminate, respectively. R is the mid-surface radius. mx, mx, mh, ph and pz are the equivalent distributed load on the mid-surface, respectively. Mx, Mh, Mxh and Mhx are the moment resultants. Nx, Nh, Nxh and Nhx are the in-plane force resultants. Qx and Qh are the transverse shear forces. x, h and z are the axial, circumferential and radial coordinates, respectively. Superscript ()e is the contribution to force and moment resultants due to applied electric field, []T is the matrix transpose. Subscript (),x and (),xx, etc. are the order one and order two partial differentiations with respective to x. 3. Discrete equations

ð1Þ

Bj;m V i;m þ

w2

N M ðA11 þ B11 Þ X aðA66  B66 þ D66 Þ=R2 X ð2Þ ð2Þ Ai;‘ U ‘;j þ Bj;m U i;m 2 a w ‘¼1 m¼1

¼ Pzi;j

 N N h A12 =R X B11 þ D11 =R X ð1Þ ð2Þ Ai;‘ W ‘;j þ Ai;‘ /1‘;j 2 a a ‘¼1 ‘¼1

¼ P xo;j

w

ð2Þ

Bj;m /1i;m þ

m¼1

N 1X ð1Þ  A Ne a ‘¼1 i;‘ x‘;j



ð2Þ

Bj;m W i;m þ h ðA22  B22

m¼1

N M 1X 1 1X ð1Þ ð1Þ þ A Qe þ B Q e  Nehi;j a ‘¼1 i;‘ x‘;j R w m¼1 j;m hi;m

!

N M ðB11 þ D11 =RÞ X aðB66  D66 Þ=R X ð2Þ ð2Þ Ai;‘ U ‘;j þ Bj;m U i;m 2 a w ‘¼1 m¼1

þ

  N M X h ðB12 þ B66 Þ X h ðB12  B55  A55 Þ ð1Þ ð1Þ Ai;‘ Bj;m V ‘;m þ aw a ‘¼1 m¼1 N X

ð1Þ

Ai;‘ W ‘;j þ

N M D11 X D66 X ð2Þ ð2Þ Ai;‘ /1‘;j þ 2 Bj;m /1i;m 2 a ‘¼1 w m¼1

þ ðA55  B55 Þ/1 ¼ mxi;j þ Q exi;j 

i;j

þ

N M X ðA55  B55 Þ X ð1Þ ð1Þ Ai;‘ Bj;m /2‘;m aw ‘¼1 m¼1

N 1X ð1Þ A Me a ‘¼1 i;‘ x‘;j

ð2dÞ

 N M N X aðB12 þ B66 Þ X h ðB66 þ D66 =RÞ X ð1Þ ð1Þ ð2Þ Ai;‘ Bj;m U ‘;m þ Ai;‘ V ‘;j 2 aw a ‘¼1 m¼1 ‘¼1  M h ðB22  D22 Þ=R X 2

w

m¼1



ð2Þ

Bj;m V i;m þ

h ðB44  D44  A44 Þ V i;j R

M ðB22 þ B44  A44  D22  D44 Þ=R X ð1Þ þ Bj;m W i;m w m¼1

N M X h ðA12 þ A66 Þ=R X ð1Þ ð1Þ þ Ai;‘ Bj;m V ‘;m aw ‘¼1 m¼1

2

ð2bÞ

ð2cÞ

þ



M ðB66  D66 Þ=R X

/2i;j

M ðB22 þ B44  A44  D22  D44 Þ=R X ð1Þ Bj;m /2i;m w m¼1

‘¼1

þ

R2

 N h ðA55  B55 Þ X ð2Þ Ai;‘ W ‘;j 2 a ‘¼1

 M h ðB44  D44  A44 Þ=R2 X



We use the GDQ method to approximate the first-order and the second-order derivative of function. The common procedure of two-dimensional discretized formulation of this paper is presented by Hong et al. [8,11–14]. The governing partial differential equation (1) is discretized into the series form and the following non0 0 0 dimensional parameters: X ¼ ax ; Y ¼ wh ; U ¼ ua , V ¼ vh ; W ¼ wh are introduced, where a is length of shell, w is span angle of panel, h is total thickness of shell. Thus we can obtain the discrete governing equations of piezoelectric shell as follows:

þ

!

N ðB12  B55  A55 Þ X ð1Þ þ D22 Þ=R2 W i;j þ Ai;‘ /1‘;j a ‘¼1

f 21 ¼ D11

f 25 ¼ D66 ;

m¼1

m¼1

f 14 ¼ ðB22  D22 Þ=R

f 24 ¼ ðD12 þ D66 Þ=R;

M X



f11 ¼ ðA44  B44 þ D44 þ A22  B22 þ D22 Þ=R2 ;

f 20 ¼ ðB22 þ B44  A44  D22  D44 Þ=R;

B44  D44  A44

V i;j

 N aA12 =R X h ðA44  B44 þ D44 þ A22  B22 þ D22 Þ=R2 ð1Þ Ai;‘ U ‘;j þ a w ‘¼1

R f10 ¼ Rf15 ¼ R2 f17 ¼ f27 ¼ B44  D44  A44

f22 ¼ D66 ;

ð2Þ

Bj;m /2i;m þ

M 1 1X ð1Þ B Ne Q ehi;j þ R w m¼1 j;m hi;m

and f 1 ¼ A11 þ B11

f 3 ¼ ðA12 þ A66 Þ=R;

f 6 ¼ ðB66  D66 Þ=R;

f16 ¼ f23 ¼ A55  B55 ;

w2

L35 ¼ f20 ðÞ;h ;

2

f 13 ¼ B66 þ D66 =R;

M ðB22  D22 Þ=R X

¼ Phi;j 

L55 ¼ f25 ðÞ;xx þ f26 ðÞ;hh þ f27

R2

m¼1

þ þ

L44 ¼ f21 ðÞ;xx þ f22 ðÞ;hh þ f23

h ðB44  D44  A44 Þ

 M h ðA44  B44 þ D44 þ A22  B22 þ D22 Þ=R2 X ð1Þ Bj;m W i;m w m¼1

L24 ¼ f12 ðÞx;h ;

L34 ¼ f19 ðÞ;x ;

w2



ð2Þ

Bj;m V i;m þ

þ

L13 ¼ f4 ðÞ;x ;

L25 ¼ f13 ðÞ;xx þ f14 ðÞ;hh þ f15

 M h ðA22  B22 þ D22 Þ=R2 X

þ

N M X B12 þ B66 X ð1Þ ð1Þ Ai;‘ Bj;m /2‘;m aw ‘¼1 m¼1



N M N X ðA55  B55 Þ X D66 X D22 ð1Þ ð1Þ ð2Þ Ai;‘ Bj;m /1‘;m þ 2 Ai;‘ /2‘;j þ 2 aw a w ‘¼1 m¼1 ‘¼1 M X

ð2Þ

Bj;m /2i;m þ ðB44  D44  A44 Þ/2i;j

m¼1

ð2aÞ

¼ mhi;j þ Q ehi;j 

M 1 X ð1Þ B Me wR m¼1 j;m hi;m

ð2eÞ

813

C.C. Hong / Composite Structures 92 (2010) 811–816

And the terms of the contribution to force and moment resultants due to applied electric field E = [Ex, Eh, Ez]T are given for the Nk layers. Ex = u,x, Eh = u,h/(R + z), Ez = u,z, where u1 and u2 are the electric potentials at the inner and outer surfaces of the shell. The potentials u are assumed to vary linearly across the layers of shell, u = u1 + zu2 in the two-dimensional analysis. 4. Numerical results and discussions

The displacement ratio W ¼ Wj W 

r h ¼ rh jhr h R

h ¼0:01 R

 x ¼ r j r x ; and stress ratios r x h ¼0:01 R

are defined and used for the following Figs. 2, 4, 6

¼0:01

and 8. All the non-dimensional deflection and stress parameters  are compared to the corresponding values at hR ¼ 0:01. Fig. 1 shows that the variation of the non-dimensional h  C1  C1 parameters W C1 ; r x and rh with the R at center position of hybrid shell Type 1 under Load Case 1 with Ra ¼ 4; W C1 jh ¼0:01 ¼ R

In order to obtain some numerical GDQ results in personal computer, we consider the hybrid composite shell made of cross-ply graphite–epoxy laminate and PVDF (polyvinyledene fluoride) layer bonded to its surface. The typical types of hybrid composite shell are listed as follows: Type 1: 3-layered cross-ply graphite–epoxy laminate [90°/0°/ 90°] and an outer PVDF layer. Type 2: 3-layered cross-ply graphite–epoxy laminate [90°/0°/ 90°] and PVDF layers bonded to its inner and outer surfaces. The material properties of graphite–epoxy are given as follows [15]:

EL ¼ 172:5 GPa; ET ¼ 6:9 GPa; GLT ¼ 3:45 GPa; GTT ¼ 1:38 GPa; cLT ¼ cTT ¼ 0:25;

di ¼ 0

    h sin pax cos Case 1: Pz ¼ P 0 1 þ 2R  h;  /1 ¼ /2 ¼ 0. Case 2: Pz = 0, u1 = 0, /2 ¼ /0 sin pax cos h.

   i1 xi ¼ 0:5 1  cos p a; i ¼ 1; 2; . . . ; N; N1    j1 ¼ 0:5 1  cos p w; j ¼ 1; 2; . . . ; M M1

hi ð3Þ

C2

C2 x C2 j  x h ¼0:01 R

r C2 x ¼ r rx =P0 S

;

r

,

; W C2 ¼ W C2Wj 

r C2 h ¼

rh =P0 rC1 , h ¼ S

C2 rC2 h  r h jhR ¼0:01 ,

10W W C2 ¼ Sd ; T /0

YT = 6.9 GPa, P0 = 1.0 MPa, S ¼

C1

C1

R

R

rx r C1 x ¼ rC1 j  x h

,

h ¼0:01 R

rh  C1 ; r , h ¼ rC1 j  h h ¼0:01 ¼0:01

T =P 0 W C1 ¼ 0:1WY ; S2

where





rh h rx h C2 rC2 x ¼ Y T dT /0 ; rh ¼ Y T dT /0 ,

R ; h

2

dT ¼ 30  10

R

Stresses are generally affected by the length of laminated piezoelectric shells under the action of mechanical load. Fig. 3 shows that the variation of the non-dimensional parameh  C2  C2 ters W C2 ; r x and rh with the R at center position of hybrid shell a R

¼ 4; W C2 jh ¼0:01 ¼ 0:433519; R

C2   rC2 x jhR ¼0:01 ¼ 0:00117221, rh jhR ¼0:01 ¼ 0:0308589. We find that  C2  C2  C2 W is almost constant with hR , but r x and rh are all increasing 

with hR increasing. Stresses are generally affected by the thickness of laminated piezoelectric shells under the action of electric potential. Fig. 4 shows that the variation of the non-dimensional parame x and r  h with the Ra at center position of hybrid shell Type ters W; r  1 under Load Case 2 and hR ¼ 0:01; Wjh ¼0:01 ¼ 0:390167;

rx jhR ¼0:01 ¼ 0:727941 MPa, rh jhR ¼0:01 ¼ 19:1634 MPa. We find that  x and r  h are all decreasing with W is almost constant with Ra, but r a R

increasing. Stresses are generally affected by the length of laminated piezoelectric shells under the action of electric potential. Fig. 5 shows that the variation of the non-dimensional parame  C1  C1 and r with the hR at center of x–h plane and ters W C1 ; r x h z ¼ 0:1 of hybrid shell Type 2 under Load Case 1 and Ra ¼ 4; h W C1 jh ¼0:01 ¼ 0:00212052;

C1   rC1 x jhR ¼0:01 ¼ 0:000471482, rh jhR ¼0:01 ¼   C1 0:0112159. We find that W C1 increases as the hR increasing, r x inR

The convergence study of deflection W at center position for Type 1 hybrid shell under Case 1 load with P0 = 1.0 MPa is made firstly, all plies of hybrid shell have equal thickness, span angle of the panel w = 120°, length-to-mid-surface radius ratio Ra ¼ 1; 2; 3; 4 and thick ness-to-mid-surface radius ratio hR ¼ 0:01; 0:05; 0:1; 0:17; 0:25. The non-dimensional deflections W at center position Type 1 hybrid shell in GDQ method for the grid points N  M = 5  5, 9  9, 13  13, 15  15, 19  19, 23  23, 27  27 and 31  31 are presented in Table 1. We find that the 31  31 grid point have a good convergence of result and can be used further for the hybrid shell including shear deflection analyses under the electromechanical loads and four sides simply supported. The non-dimensional parameters of displacements W C1 ; W C2  C1  C1  C2  C2 and stresses r x ; rh , rx ; rh in Case 1 and Case 2, respectively, are defined as follows and used for Figs. 1, 3, 5 and 7. C1

R

rh jhR ¼0:01 ¼ 1:24117 MPa. We find that W is almost  x and r  h are all decreasing with Ra increasing. constant with Ra, but r 0:044555 MPa,

R

And the following coordinates of grid points are used in the GDQ computation:

h ¼0:01 R

The displacement W C1 center position value is almost similar with the FSDT edge position value by Kapuria et al. [15]. Deflection and stresses are generally affected by the thickness of laminated piezoelectric shells under mechanical load. Fig. 2 shows that the variation of the non-dimensional parame x and r  h with the Ra at center position of hybrid shell Type ters W; r  1 under Load Case 1 with hR ¼ 0:01; Wjh ¼0:01 ¼ 0:02525, rx jh ¼0:01 ¼

Type 1 under Load Case 2 and

And the material properties of PVDF are given as follows: Ei ¼ 2:0 GPa; cij ¼ 13 ; d1 ¼ 3  1012 CN1 , d2 ¼ 23  1012 CN1 ; d3 ¼ 30  1012 CN1 ; d4 ¼ 0; d5 ¼ 0. We consider the typical hybrid shell including shear deformation under four sides simply supported for the following electromechanical Load Cases.

W C1 ¼ W C1Wj 

C1   rC1 x jhR ¼0:01 ¼ 0:00044555, rh jhR ¼0:01 ¼ 0:0124117. We   C1  C1 and r are all increasing with hR increasing. find that W C1 ; r x h

0:00174225;

1

rC1 x ¼ with

CN ; /0 ¼ 1:0dT .





creases firstly, up to hR ¼ 0:05 then decreases as the hR increasing, h  C1 but r h decreases firstly, down to R ¼ 0:05 then increases as the  h increasing. Stresses are generally affected by the length of lamiR nated piezoelectric shells under the action of mechanical load. Fig. 6 shows that the variation of the non-dimensional parame x and r  h with the Ra at center of x–h plane and hz ¼ 0:1 of ters W; r  hybrid shell Type 2 under Load Case 1 and hR ¼ 0:01; Wjh ¼0:01 ¼ 0:0307321, rx jh ¼0:01 ¼ 0:0471482 MPa; rh jh ¼0:01 ¼ R

R

R

x 1:12159 MPa. We find that W is almost constant with Ra, but r  h are all decreasing with Ra increasing. Stresses are generally and r affected by the length of laminated piezoelectric shells under the action of mechanical load. Fig. 7 shows that the variation of the non-dimensional parame  C2  C2 and r with the hR at center of x–h plane and ters W C2 ; r x h z ¼ 0:1 of hybrid shell Type 2 under Load Case 2 and Ra ¼ 4; h W C2 jh ¼0:01 ¼ 0:843983; R

 rC2 x jhR ¼0:01 ¼ 0:00202604,

 rC2 h jhR ¼0:01 ¼ 

0:0447108. We find that W C2 is almost constant as the hR increash h  C2 ing, r x increases firstly, up to R ¼ 0:05 then decreases as the R h  C2 increasing, but r decreases firstly, down to ¼ 0:05 then inh R  creases as the hR increasing. Stresses are generally affected by the

814

C.C. Hong / Composite Structures 92 (2010) 811–816

Table 1     h sin pax cos h; P0 ¼ 1:0 MPa, /1 ¼ /2 ¼ 0. Convergence of center deflection W for Type 1 hybrid shell: [90°/0°/90°] and an outer PVDF, Case 1 load: Pz ¼ P0 1 þ 2R a/R

GDQ method N  M grid points

Center deflection, W 

h R



h R

¼ 0:01



h R

¼ 0:05



h R

¼ 0:1

¼ 0:17



h R

¼ 0:25

1

55 99 13  13 15  15 19  19 23  23 27  27 31  31

0.03573 0.03468 0.03213 0.02555 0.02693 0.02737 0.02763 0.02753

0.00146 0.00142 0.00132 0.00106 0.00111 0.00113 0.00114 0.00114

0.000381 0.000369 0.000343 0.000278 0.000292 0.000297 0.000298 0.000299

0.000139 0.000135 0.000126 0.000103 0.000108 0.000110 0.000110 0.000110

0.0000692 0.0000673 0.0000630 0.0000523 0.0000544 0.0000551 0.0000553 0.0000553

2

55 99 13  13 15  15 19  9 23  23 27  27 31  31

0.03248 0.03161 0.02953 0.02356 0.02495 0.02549 0.02568 0.02581

0.00132 0.00128 0.00120 0.00096 0.00102 0.00104 0.00105 0.00105

0.000339 0.000325 0.000308 0.000264 0.000261 0.000267 0.000269 0.000269

0.000121 0.000117 0.000110 0.000088 0.000093 0.000095 0.000096 0.000096

0.0000581 0.0000564 0.0000528 0.0000423 0.0000448 0.0000457 0.0000460 0.0000461

3

55 99 13  13 15  15 19  19 23  23 27  27 31  31

0.03187 0.03104 0.02904 0.02314 0.02459 0.02511 0.02529 0.02539

0.00130 0.00126 0.00118 0.00094 0.00100 0.00102 0.00103 0.00103

0.000332 0.000323 0.000301 0.000241 0.000256 0.000262 0.000264 0.000264

0.000118 0.000115 0.000107 0.000085 0.000091 0.000093 0.000093 0.000094

0.0000564 0.0000547 0.0000508 0.0000408 0.0000434 0.0000443 0.0000447 0.0000448

4

55 99 13  13 15  15 19  19 23  23 27  27 31  31

0.03166 0.03085 0.02887 0.02302 0.02446 0.02499 0.02519 0.02525

0.00129 0.00125 0.00117 0.00094 0.00099 0.00102 0.00102 0.00102

0.000330 0.000320 0.000298 0.000239 0.000254 0.000260 0.000262 0.000262

0.000117 0.000114 0.000106 0.000085 0.000090 0.000092 0.000093 0.000093

0.0000558 0.0000543 0.0000509 0.0000402 0.0000430 0.0000436 0.0000442 0.0000444

σ xC1

W C1 1.50 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00

σ θC1

1.8

2.5

1.6 2.0

1.4 GDQ

1.2

FSDT *

h 0.01 0.05 0.1 0.17 0.25 R Fig. 1.

h R

1.5

1.0 0.8

h* 0.01 0.05 0.1 0.17 0.25 R

1.0

 C1  C1 vs. W C1 ; r x ; rh at center position of hybrid shell Type 1 under Load Case 1,

σx

σθ

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

W

0.0

1

Fig. 2.

2

a R

0.01 0.05 0.1 0.17 0.25

3

4

a R

0.0

1

2

3

4

a R

1

2

3

x; r  h at center position of hybrid shell Type 1 under Load Case 1, vs. W; r

length of laminated piezoelectric shells under the action of electric potential.

a R

4

h R

h* R

¼ 4.

a R

¼ 0:01.

Fig. 8 shows that the variation of the non-dimensional parame x and r  h with the Ra at center of x–h plane and of hybrid shell ters W; r

815

C.C. Hong / Composite Structures 92 (2010) 811–816

σ xC 2 W C2

1.8

1.5

1.6

σ θC 2 2.5 2.0

1.4

1.0

1.2 0.5

1.5

Fig. 3.

h R

h* R

0.25

0.8

0.17

h* R

0.05

0.01 0.05 0.1 0.17 0.25

0.01

0.0

0.1

1.0

2.0

1.0

1.5

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0

1.0

0.5

0.5

0.0 2

a R

3

a R

4

0.0

1

2

3

4

a R

1

2

3

x; r  h at center position of hybrid shell Type 1 under Load Case 2, vs. W; r

1.2

1.2

1.1

1.1

¼ 4.

4

h R

a R

¼ 0:01.

σ θC1

σ xC1

W C1

a R

h* R

σθ

σx

W

Fig. 4.

0.01 0.05 0.1 0.17 0.25

 C2  C2 vs. W C2 ; r x ; rh at center position of hybrid shell Type 1 under Load Case 2,

1.5

1

1.0

1.1 1.0

1.0 1.0

0.9

0.7

0.9

0.9

0.8 0.01 0.05 0.1 0.17 0.25 h

*

0.8

0.01 0.05 0.1 0.17 0.25 h

Fig. 5.

h R

 C1  C1 vs. W C1 ; r x ; rh at center of x–h plane and

1.5 1.0 0.5 0.0

Fig. 6.

z h

1

a R

2

3

a R

4

*

R a R

¼ 4.

σθ

2.5 2.0 1.5 1.0 0.5 0.0 1

x; r  h at center of x–h plane and vs. W; r



0.01 0.05 0.1 0.17 0.25 h

¼ 0:1 of hybrid shell Type 2 under Load Case 1 and

σx

W

0.8

R

R 

*

2

z h

3

4

a R

6.0 5.0 4.0 3.0 2.0 1.0 1

2

3

¼ 0:1 of hybrid shell Type 2 under Load Case 1 and

Type 2 under Load Case 2 and hR ¼ 0:01; Wjh ¼0:01 ¼ R 0:759585; rx jh ¼0:01 ¼ 1:25817 MPa, rh jh ¼0:01 ¼ 27:7654 MPa. We R R a  x and r  h are all decreasfind that W is almost constant with R, but r ing with Ra increasing. Stresses are generally affected by the length of laminated piezoelectric shells under the action of electric potential.

a R

4 h R

¼ 0:01.

The GDQ method provides a good efficiency of computation to study the laminated piezoelectric shells under electromechanical loads. Some of the static computed GDQ solutions might be used as the basic state and applied to the field of functionally graded piezoelectric actuators or sensors.

816

C.C. Hong / Composite Structures 92 (2010) 811–816

W C2

σ θC 2

σ xC 2

1.1

1.1

1.1

1.0

1.0

1.0

0.9

0.9

0.9

0.8

0.01 0.05 0.1 0.17 0.25 h

*

0.8

0.01 0.05 0.1 0.17 0.25

R Fig. 7.

h R

 C2  C2 on W C2 ; r x ; rh at center of x–h plane and

z h

2.0

1.0

1.5

0.0

Fig. 8.

a R

0.5 1

2

3

4

a R

0.0

*

R a R

¼ 4.

σθ

1.0 0.5

0.01 0.05 0.1 0.17 0.25 h

¼ 0:1 of hybrid shell Type 2 under Load Case 2 and

σx

W 1.5

0.8

h* R

1

x; r  h at center of x–h plane and vs. W; r

2

z h

3

4

a R

6.0 5.0 4.0 3.0 2.0 1.0 1

2

3

4

a R

¼ 0:1 of hybrid shell Type 2 under Load Case 2 and

5. Conclusions The two-dimensional GDQ method gives us a computational solution of the laminated PVDF hybrid shell under the electromechanical loads and four sides simply supported. We obtain the variations of the non-dimensional parameters of displacement and stresses with respect to thickness-to-mid-surface radius ratio and length-to-mid-surface radius ratio at center position of hybrid shell. Deflection is generally affected by the thickness of laminated piezoelectric shells under the action of mechanical load. Stresses are generally affected by the thickness and the length of laminated piezoelectric shells under the actions of mechanical load and electric potential. Acknowledgement The completion of this study was made possible by a Grant NSC-92-2212-E-164-007 from National Science Council, Taiwan, ROC. References [1] Skatulla S, Arockiarajan A, Sansour C. A nonlinear generalized continuum approach for electro-elasticity including scale effects. J Mech Phys Solids 2009;57:137–60. [2] Brodal E, Hesthaven JS, Melandso F. Numerical modeling of double-layered piezoelectric transducer systems using a high-order discontinuous Galerkin method. Comput Struct 2008;86:1747–56. [3] Frankel JI, Chisholm C. Integral equation and sensitivity analyses of creep behavior for PVDF thin films. Mech Mater 2008;40:594–601.

h R

¼ 0:01.

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