Exact solutions of functionally graded piezoelectric material sandwich cylinders by a modified Pagano method

Applied Mathematical Modelling 36 (2012) 1910–1930

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Exact solutions of functionally graded piezoelectric material sandwich cylinders by a modified Pagano method Chih-Ping Wu ⇑, Tsu-Chieh Tsai Department of Civil Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 26 October 2010 Received in revised form 13 July 2011 Accepted 27 July 2011 Available online 2 August 2011 Keywords: The Pagano method Exact solutions Coupled electro-elastic effects Functionally graded materials Piezoelectric materials Cylinders

a b s t r a c t The three-dimensional (3D) coupled analysis of simply-supported, functionally graded piezoelectric material (FGPM) circular hollow sandwich cylinders under electro-mechanical loads is presented. The material properties of each FGPM layer are regarded as heterogeneous through the thickness coordinate, and obey an exponent-law dependent on this. The Pagano method is modified to be feasible for the study of FGPM sandwich cylinders. The modifications are as follows: a displacement-based formulation is replaced by a mixed formulation; a set of the complex-valued solutions of the system equations is transferred to the corresponding set of real-valued solutions; a successive approximation method is adopted to approximately transform each FGPM layer into a multilayered piezoelectric one with an equal and small thickness for each layer in comparison with the mid-surface radius, and with the homogeneous material properties determined in an average thickness sense; and a transfer matrix method is developed, so that the general solutions of the system equations can be obtained layer-by-layer, which is significantly less time-consuming than the usual approach. A parametric study is undertaken of the influence of the aspect ratio, open- and closed-circuit surface conditions, and material-property gradient index on the assorted field variables induced in the FGPM sandwich cylinders. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction In recent decades, a class of multilayered hybrid piezoelectric and elastic structures with beam-, plate- and shell-forms has been developed, known as intelligent (or smart) structures, for use in various engineering applications, such as sensors for monitoring and actuators for controlling structural responses. Because this class of intelligent structures is commonly utilized in a changing thermal environment, some problems may occur due to sudden changes in the material properties at the interfaces between adjacent layers, such as huge thermal residual stresses and de-lamination. Consequently, an emerging class of functionally graded piezoelectric material (FGPM) structures, the material properties of which are heterogeneous and gradually and continuously vary through the thickness coordinate, has been developed to overcome these issues. This feature, however, also increases the complexity and difficulty of analyzing such functionally graded (FG) structures, and the literature with regard to the accurate analysis of these is small in comparison with that examining multilayered homogeneous ones. This literature survey will focus on the three-dimensional (3D) analysis of multilayered homogeneous and FG piezoelectric plates and shells, because this may not only serve as a standard for assessing various approximate two-dimensional (2D) theories of plates and shells, but also provide a reference for making the appropriate kinetic and kinematical assumptions

⇑ Corresponding author. Fax: + 886 6 2370804. E-mail address: [email protected] (C.-P. Wu). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.07.077

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

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prior to developing more advanced 2D theories. In general, there are four major approaches, namely the Pagano, state space, power series expansion, and asymptotic approaches, drawn from the literature with regard to the various 3D analyses for single-layer homogeneous, multilayered composite and FG material plates and shells. The Pagano approach was initially proposed for the 3D static, free vibration and buckling analyses of multilayered composite plates by Pagano [1,2] and Srinivas and Rao [3], and then used to study the static behaviors of laminated piezoelectric cylinders, plates and strips by Heyliger [4– 6] and Heyliger and Brooks [7], as well as the free vibration and cylindrical bending vibration of multilayered ones by Heyliger and Saravanos [8] and Heyliger and Brooks [9]. The state space approach is also known as the transfer matrix [10] and initial functions [11] methods, which are used to determine the exact solutions of multilayered composite and isotropic thick plates, respectively. These have been extended to the 3D analysis of multilayered composite plates by Ye [12], of multilayered hollow cylinders by Ye and Soldatos [13–15], and of doubly curved multilayered composite and FG piezoelectric shells by Fan and Zhang [16] and Wu and Liu [17]. Using the power series expansion approach, Ren [18,19] studied the 3D static analysis of laminated cylindrical shells under cylindrical bending loads, Kapuria et al. [20,21] investigated the 3D piezothermo-elastic analysis of multilayered piezoelectric cylindrical shells under axisymmetric and asymmetric thermoelectric loads, and Vel and Batra [22] and Vel et al. [23] presented the 3D solutions of free and force vibrations of FG elastic and piezoelectric composite plates. Based on electro-magneto-thermo-elasticity, Dai et al. [24] presented exact solutions of FGPM hollow cylinders placed in a uniform magnetic field and subjected to electric, thermal and mechanical loads. Wu et al. [25–32] developed a series of asymptotic theories for the static, free vibration, nonlinear geometric, thermal buckling and dynamic instability analyses of multilayered composite and FG piezoelectric plates and shells. Finally, a comprehensive literature review with regard to these 3D approaches for the analyses of multilayered composite and FG piezoelectric plates and shells was undertaken by Wu et al. [33], in which the derivation of the formulation of each method and a comparison of their results were presented. Some research into the 3D approximate analysis of multilayered composite and FGM plates and shells has also been carried out. Ramirez et al. [34,35] developed a discrete layer approach, combined with the Ritz method, for the 3D static and free vibration analyses of multilayered and FG elastic, piezoelectric and magneto-electro-elastic structures. Based on the principle of virtual displacement (PVD), Cheung and Jiang [36] and Akhras and Li [37,38] developed the PVD-based finite layer method (FLM) for the quasi-3D static, vibration and buckling analyses of piezoelectric composite plates. Based on the Reissner mixed variational theorem (RMVT), instead of the PVD, Wu and Li [39] and Wu et al. [40–45] developed the RMVT-based finite layer, meshless collocation, and element-free Galerkin methods for the quasi-3D static and free vibration analyses of multilayered FGM plates and multilayered FG piezoelectric and magneto-electro-elastic materials plates and shells. Alibeigloo and Kani [46] developed a 3D state space-based differential quadrature method for the 3D free vibration analysis of laminated cylindrical shells, which were bonded with piezoelectric layers on their outer surfaces. Sheng and Wang [47] presented thermo-elastic vibration and buckling analyses of FGPM cylindrical shells using a first-order shear deformation theory. After a close literature survey, we found that to date there are few 3D coupled electro-elastic analyses of FG plates and shells in comparison with those of multilayered homogeneous ones. Among the 3D approaches mentioned above, the Pagano method is the most widely applied for multilayered plates and shells, although it is not feasible for FG smart ones without further modifications. In order to achieve this, Wu et al. [48] and Wu and Lu [49] developed a modified Pagano method for the static and free vibration analyses, respectively, of simply supported, FG magneto-electro-elastic plates. Their modifications to the original Pagano method are as follows: (a) A mixed formulation, rather than the displacement-based one, is used, so that both the lateral boundary conditions on the outer surfaces and the continuity conditions at the interface between adjacent layers can be directly applied. (b) The sets of complex-valued solutions of system equations are transferred to the corresponding sets of real-valued solutions by means of Euler’s formula for the purpose of computational efficiency. (c) A successive approximation (SA) method, which was first proposed by Soldatos and Hadjigeorgiou [50] for the analysis of homogeneous isotropic cylindrical shells using the state space approach, is adopted, where the FG plate or shell is artificially divided into a certain number of individual layers with an equal and small thickness, compared with the in-plane dimensions of the plate or the mid-surface radius of the shell, for each layer. By the refinement manipulation, one may reasonably approximate the variable material coefficients of each layer to the constant material coefficients in an average thickness sense, so that the system of thickness-varying differential equations for each individual layer can be reduced to a system of thickness-invariant differential equations. (d) A transfer matrix method is developed, so that the general solutions of system equations can be obtained layer-by-layer. These modifications mean that the Pagano method can be used for the 3D analysis of FG plates and shells, and their earlier implementations show that the computation thus becomes less time-consuming and independent of the total number of layers, constituting the plates and shells. Due to the benefits of the modified Pagano method, as noted above, in this paper it is extensively applied to the 3D coupled electro-elastic analysis of simply-supported, FGPM circular hollow sandwich cylinders under electro-mechanical loads, in which the material properties of each FGPM layer are assumed to obey an exponent-law distribution through the thickness coordinate, and four different loading conditions acting on the lateral surfaces of the sandwich cylinders are considered, namely the closed- or open-circuit surface conditions with either electric or mechanical loads. Because the FGPM sandwich cylinder is transformed into a multilayered homogeneous piezoelectric one in this formulation using the SA method, the static behavior of multilayered hybrid elastic and piezoelectric cylinders can thus be included as a special case and be studied using this formulation. A parametric study is thus undertaken of the influence of the aspect ratio, open- and closed-circuit

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surface conditions, and material-property gradient index on the through-thickness distributions of assorted field variables induced in the sandwich cylinders. 2. Basic equations of 3D piezoelectricity A simply-supported, FGPM circular hollow sandwich cylinder with heterogeneous material properties through the thickness coordinate is considered, as shown in Fig. 1. A set of the cylindrical coordinates (x, h, r) is located on the middle surface of the cylinder. The total thickness, length, radius to the middle surface, and the thickness coordinate of the cylinder are 2h, L, R and f, respectively, and r = R + f. The linear constitutive equations, valid for the nature of the symmetry class of the piezoelectric material considered, are given by

8 rx > > > > > rh > > > <

9 > > > > > > > > =

2

c11

c12

c13

0

0

6c 6 12 6

c22

c23

0

0

6 c13 c23 c33 0 rr 0 ¼6 6 > > 0 0 0 c44 0 > > shr > > 6 > > 6 > > > > 4 0 0 0 0 c55 s > > xr > > : ; 0 0 0 0 0 sxh

8 9 2 0 > < Dx > = 6 Dh ¼ 4 0 > : > ; e31 Dr

0

0

0

e15

0

0

e24

0

e32

e33

0

0

38 2 ex 9 0 > > > > > > 6 > > > > e 0 7 h > 7> 6 0 > > > 6 7> 0 7< er = 6 0 7 6 > 6 0 7 > 7> 6 0 > chr > > > 7> 6 > > > 4 e15 c 0 5> > > xr > > : ; cxh c66 0 0

8 ex 9 > > > > > > eh > > > 3> 2 > > > > g11 0 > = r 7 6 05 þ4 0 > > chr > > > 0 0 > > >c > > > > > > xr > > : ;

e31

0

e32 7 78 9 7> Ex > e33 7< = 7 Eh ; 0 7 : > ; 7> 7 Er 0 5

0 e24 0 0

0

g22 0

3

0

ð1Þ

0

38 9 > < Ex > = 7 0 5 Eh ; > > g33 : Er ; 0

ð2Þ

cxh

where (rx rh rr shr sxr sxh) and (ex eh er chr cxr cxh) are the stress and strain components, respectively, Di and Ei(i = x, h, r) denote the electric displacement and electric field components, respectively; cij(i, j = 1–6), elj(l = 1  3, j = 1–6) and glk(l, k = 1–3) are the elastic, piezoelectric and dielectric permeability coefficients, respectively. These material properties are considered as heterogeneous through the thickness coordinate (i.e., cij(f), elj(f), glk(f)).

Fig. 1. The configuration and coordinates of an FGPM sandwich cylinder.

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

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The strain–displacement relationships are:

8 > > > > > > > > <

3 2 ex 9 0 0 @x > > > 6 0 > eh > ð1=rÞ@ h ð1=rÞ 7 > 78 9 6 > > 7> ux > 6 = 6 0 er 0 @ r 7< = 7 uh ; ¼6 6 0 > chr > ð1=rÞ þ @ r ð1=rÞ@ h 7 : > > ; > 7> 6 > > > > 7 ur 6 > > > > 5 4 c @ 0 @ > > r x > : xr > ; cxh ð1=rÞ@ h @x 0

ð3Þ

where @ i ¼ @=@ i ði ¼ x; h; rÞ, and ux, uh and ur are the displacement components. The stress equilibrium equations without body forces are given by

rx ;x þ ðsxh ;h =rÞ þ sxr ;r þ ðsxr =rÞ ¼ 0; sxh ;x þ ðrh ;h =rÞ þ shr ;r þ ð2shr =rÞ ¼ 0; sxr ;x þ ðshr ;h =rÞ þ rr ;r þ ðrr =rÞ  ðrh =rÞ ¼ 0:

ð4Þ ð5Þ ð6Þ

The equations of electrostatics for the piezoelectric material without the electric charge density are:

Dx ;x þ ðDh ;h =rÞ þ Dr ;r þ ðDr =rÞ ¼ 0:

ð7Þ

The relations between the electric field and electric potential are:

Ek ¼ U;k =ck

k ¼ ðx; h; rÞ;

ð8Þ

where U denotes the electric potential, and cx = cr = 1 and ch = r. Four different loading conditions applied on the lateral surfaces of the hollow cylinders are considered, and these are specified as follows: Case 1. In the case of closed-circuit and prescribed mechanical loads:

sxr ¼ shr ¼ U ¼ 0 on f ¼ h;

ð9aÞ

rr ¼ qr on f ¼ h:

ð9bÞ

and

Case 2. In the case of closed-circuit and prescribed electric potential:

sxr ¼ shr ¼ rr ¼ 0 on f ¼ h; 

U¼U

on f ¼ h:

ð10aÞ ð10bÞ

Case 3. In the case of open-circuit and prescribed mechanical loads:

sxr ¼ shr ¼ Dr ¼ 0 on f ¼ h; rr ¼ qr on f ¼ h:

ð11aÞ ð11bÞ

Case 4. In the case of open-circuit and prescribed normal electric displacement:

sxr ¼ shr ¼ rr ¼ 0 on f ¼ h; Dr ¼

Dr

on f ¼ h:

ð12aÞ ð12bÞ

The edge boundary conditions of the cylinder are considered as fully simple supports, suitably grounded, and are given as:

rx ¼ uh ¼ ur ¼ U ¼ 0; at x ¼ 0 and x ¼ L:

ð13Þ

There are twenty-two basic equations for 3D piezoelectricity, as listed in Eqs. (1)–(8), and these are essentially a system of simultaneously partial differential equations with variable coefficients. In the later work of this paper, a modified Pagano method will be developed for this coupled analysis of simply supported, FGPM sandwich cylinders under electro-mechanical loads. 3. Nondimensionalization In order to scale all the field variables within a close order of magnitude and prevent unexpected numerical instability in the computation process, we define a set of dimensionless coordinates and variables, as follows:

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qffiffiffiffiffiffiffiffi pffiffiffiffiffiffi x1 ¼ x= Rh; x2 ¼ h= h=R; x3 ¼ f=h; pffiffiffiffiffiffi pffiffiffiffiffiffi u1 ¼ ux = Rh; u2 ¼ uh = Rh; u3 ¼ ur =R;

r1 ¼ rx =Q ; r2 ¼ rh =Q ; s12 ¼ sxh =Q ;

 qffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffi Q h=R ; s23 ¼ shr Q h=R ; qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi D1 ¼ Dx h=R=e; D2 ¼ Dh h=R=e; D3 ¼ Dr =e;

s13 ¼ sxr

r3 ¼ rr R=ðQhÞ;

ð14a-pÞ

/ ¼ Ue=Qh; where Q and e stand for the reference elastic and piezoelectric coefficients, respectively. In this formulation, the elastic displacements ðux ; uh ; ur Þ, the transverse shear and normal stresses ðsxr ; shr ; rr Þ and the normal electric displacement and electric potential components ðDr ; UÞ are selected as the primary field variables. The other field variables are the secondary ones, and can be expressed in terms of the primary ones. Introducing the set of dimensionless coordinates and variables given in Eq. (14a-p) and using the method of direct elimination, we obtain one set of state space equations in terms of the primary field variables for the coupled analysis of FGPM sandwich cylinders, and they are given as follows:

3 2 u1 0 6 u2 7 6 0 7 6 6 7 6 6 6 D3 7 6 0 7 6 6 6 6 @ 6 r3 7 7 6 k41 7¼6 6 @x3 6 s13 7 6 k51 7 6 6 6s 7 6k 6 23 7 6 61 7 6 6 4 / 5 4 k53 u3 k54 2

0

0

k17

0

k26

k27

k17

k27

k37

k18 k55

k28 0

0 0

k64

0

k66

0

k74

0

0

0

3 u1 7 6 k28 76 u2 7 7 76 7 0 76 D3 7 76 7 7 6 k48 76 r3 7 7 76 7; 7 6 k58 7 76 s13 7 7 6 k68 76 s23 7 7 76 7 k78 54 / 5

k84

0

0

0

k88

0

0

k22

0

0

0

k33

0

k42 k52

k43 k53

k44 k54

k62

k63

k63

k73

k64

k74

k15

k18

32

ð15Þ

u3

where

k15 ¼ h=ð~c55 RÞ; k22 k33 k41 k48 k53 k61 k64

k17 ¼ ð~e15 hÞ@ 1 =ð~c55 RÞ;

k18 ¼ @ 1 ;

¼ h=ðc2 RÞ; k26 ¼ h=ð~c44 RÞ; k27 ¼ ð~e24 hÞ@ 2 =ð~c44 c2 RÞ; k28 ¼ ð1=c2 Þ@ 2 ;       ~ 11 h=R @ 11 þ ~e224 =~c44 þ g ~ 22 h= c22 R @ 22 ; ¼ h=ðRc2 Þ; k37 ¼ ~e215 =~c55 þ g



 ~ 21 =c @ 1 ; k42 ¼ Q ~ 22 =c2 @ 2 ; k43 ¼ a22 e=ðQ c Þ; k44 ¼ ða12  1Þh=ðc RÞ; ¼ Q 2 2 2 2 h

 i h

 i 2 2 ~ ~ ~ ~ ~ ¼ Q 22 =c2 ; k51 ¼  Q 11 @ 11 þ Q 66 =c2 @ 22 ; k52 ¼  Q 12 þ Q 66 =c2 @ 12 ;

 ~ 12 =c @ 1 ; ¼ ða21 e=Q Þ@ 1 ; k54 ¼ ða11 h=RÞ@ 1 ; k55 ¼ h=ðRc2 Þ; k58 ¼  Q 2 h

 i h

 i ~ 21 þ Q ~ 66 =c @ 12 ; k62 ¼  Q ~ 66 @ 11 þ Q ~ 22 =c2 @ 22 ; k63 ¼ ½a22 e=ðQ c Þ@ 2 ; ¼ Q 2 2 2

 2 ~ 22 =c @ 2 ; ¼ ½a12 h=ðRc2 Þ@ 2 ; k66 ¼ 2h=ðRc2 Þ; k68 ¼  Q 2

k73 ¼ e2 b2 =Q; 2

k74 ¼ ehb1 =R; 2

k84 ¼ a1 Qh =R ;

k78 ¼ a22 e=ðQ c2 Þ;

k88 ¼ a12 h=ðc2 RÞ;

c2 ¼ ch =R;

the relevant coefficients in the previous terms of kij are given in Appendix A. The in-surface stress and electric displacement components are dependent field variables, which can be calculated using the primary variables, which are determined as follows:

rp ¼ B1 u þ B2 u3 þ B3 r3 þ B4 D3 ; d ¼ B5 rs þ B6 /; where

 3 2 ~ 11 @ 1 ~ 12 =c @ 2 8 9 Q Q 2 > 6

< r1 > =

 7 7 6 D s u ~ 21 @ 1 ~ 22 =c @ 2 7; rp ¼ r2 ; u ¼ 1 ; rs 13 ; d ¼ 1 ; B1 ¼ 6 Q Q 2 7 6 > > u s D 2 23 2 : 5 4

 s12 ; ~ 66 =c @ 2 ~ 66 @ 1 Q Q 2

ð16Þ ð17Þ

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

1915

2 3 3 2 3 ~ 12 =c a21 e=Q a11 h=R Q 2 6 7 6~ 7 6 7 ; B3 ¼ 4 a12 h=R 5; B4 ¼ 4 a22 e=Q 5; B2 ¼ 4 Q 22 =c2 5 2

0 "  #   ~ 11 ðh=RÞ@ 1  ~e215 =~c55 þ g ð~e15 hÞ=ð~c55 RÞ 0   B5 ¼ ; B6 ¼ : ~ 22 h@ 2 =ðRc2 Þ  ~e224 =~c44 þ g 0 ð~e24 hÞ=ð~c44 RÞ 0

0



The dimensionless forms of the boundary conditions of the problem are specified as follows:

Case 1:

s13 ¼ s23 ¼ / ¼ 0 on x3 ¼ 1; r3 ¼ q3 on x3 ¼ 1;

ð18aÞ ð18bÞ

  where q 3 ¼ qr R=ðQhÞ.

Case 2:

s13 ¼ s23 ¼ r3 ¼ 0 on x3 ¼ 1;

ð19aÞ

 /¼/

ð19bÞ

on x3 ¼ 1;

  ¼ U e=ðQhÞ. where /

Case 3:

Case 4:

s13 ¼ s23 ¼ D3 ¼ 0 on x3 ¼ 1; r3 ¼ q3 on x3 ¼ 1:

ð20aÞ ð20bÞ

s13 ¼ s23 ¼ r3 ¼ 0 on x3 ¼ 1;

ð21aÞ

 D3 ¼ D 3

ð21bÞ

on x3 ¼ 1;

 ¼ D   =e. where D 3 r At the edges, the following quantities are satisfied:

pffiffiffiffiffiffi

r1 ¼ u2 ¼ u3 ¼ / ¼ 0; at x1 ¼ 0 and x1 ¼ L= Rh:

ð22Þ

4. The modified Pagano method 4.1. The double Fourier series expansion method The double Fourier series expansion method is applied to reduce the system of partial differential Eqs. (15)–(17) to a system of ordinary differential equations, such that the mechanical or electric loads, acting on the lateral surfaces of the hollow cylinder, are expressed by the double Fourier series, as follows:



1 X 1  X     ^ ^ ; D^ ^ sinðmx=L r ; U ; Dr ¼ rm^ n^ ; U ^ ^ hÞ: q q Þcosðn mn r mn

ð23Þ

^ ^ ¼0 m¼1 n

The dimensionless form of Eq. (23) is:



1 X 1  X      ; D ¼  ^ ^ ; D^ ^ sin mx 3 ; / 3m^ n^ ; / ~ x2 ; ~ 1 cos n q q 3 mn 3mn

ð24Þ

^ ^ ¼0 m¼1 n

pffiffiffiffiffiffi pffiffiffiffiffiffiffiffi   =e; m   ¼ U e=ðQhÞ; D ¼ D  ^ n^ R=ðQhÞ; / ~¼n ^ h=R, in which m ^ and n ^ are zero or positive ~ ¼m ^ p Rh=L; n where q ^n ^ ¼ qr m ^n ^ ^n ^ ^n ^ ^n ^ 3m m m 3m rm integers. By satisfying the edge boundary conditions, we express the primary variables in the following form:

ðu1 ; s13 Þ ¼ ðu2 ; s23 Þ ¼

1 X 1 X ^ ^ ¼0 m¼1 n 1 X 1 X

ðu1m^ n^ ðx3 Þ;

~ 1 cos n ~ x2 ; s13m^ n^ ðx3 ÞÞ cos mx

ð25Þ

ðu2m^ n^ ðx3 Þ;

~ 1 sin n ~ x2 ; s23m^ n^ ðx3 ÞÞ sin mx

ð26Þ

^ ^ ¼0 m¼1 n 1 X

ðu3 ; r3 ; /; D3 Þ ¼

1 X

ðu3m^ n^ ðx3 Þ;

~ 1 cos n ~ x2 ; r3m^ n^ ðx3 Þ; /m^ n^ ðx3 Þ; D3m^ n^ ðx3 ÞÞ sin mx

ð27Þ

^ ^ ¼0 m¼1 n

For brevity, the symbols of summation are omitted in the following derivation. Using the set of dimensionless coordinates and field variables, given in Eq. (14), and substituting Eqs. (25)–(27) in Eq. (15), we have the resulting equations, as follows:

dFðx3 Þ=dx3 ¼ KFðx3 Þ;

ð28Þ

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where F ¼ fu1m^ n^ u2m^ n^ D3m^ n^ r3m^ n^

2

0

6 6 0 6 6 0 6 6  6 k41 K¼6 6 k  6 51 6  6 k61 6 6  4 k53 ~ k54

0 22 k

s13m^ n^ s23m^ n^ /m^ n^ u3m^ n^ gT : 0

0 26 k

17 k 27 k

17 k 18 k

27 k 28 k

37 k

54 k 64 k

55 k 0

0 66 k

74 k 84 k

0

0

0

0

0

0

0

0

0 33 k 43 k

0

0 42 k

0 44 k

52 k 62 k

53 k ~ k63

63 k 64 k

73 k 74 k

15 k

0 0 0

3 18 k 28 7 7 k 7 0 7 7 48 7 7 k 7;  k58 7 7 68 7 7 k 7 78 7 k 5 88 k

ij are given in Appendix B. and k 4.2. Theories of the homogeneous linear systems Eq. (28), which is a system of eight simultaneously homogeneous ordinary differential equations in terms of eight primary variables, represents the state space equations for the static behaviors of a simply-supported, FG piezoelectric hollow cylinder under electro-mechanical loads, applied on the lateral surfaces, and the general solution of this is:

F ¼ XL;

ð29Þ

where L is an 8  1 matrix of arbitrary constants; O is a fundamental matrix of Eq. (29), and is formed by eight linearly independent solutions in the form of X ¼ ½X1 ; X2 ; . . . ; X8 ; Xi ¼ Ki eki x3 ði ¼ 1; 2; . . . ; 8Þ; ki and Ki are the eigenvalues and their corresponding eigenvectors of the coefficient matrix K in Eq. (28), respectively.  has a complex eigenvalue k1 (i.e., k1 ¼ Reðk1 Þ þ iImðk1 Þ), then its complex conjugate k2 (i.e., If the coefficient matrix K  due to the fact that all of the coefficients of K  are real. In addition, k2 ¼ Reðk1 Þ  iImðk1 Þ) is also an eigenvalue of K K1;2 ¼ ReðK1 Þ  iImðK1 Þ are the corresponding eigenvectors of the complex conjugate pair k1;2 . Although there is nothing wrong with these two complex-valued solutions in Eq. (29), we replace them with two linearly independent solutions involving only real-valued quantities in order to achieve more efficient computational performance. Using Euler’s formula these complex-valued solutions are replaced with the real-valued solutions, and are given by

X1 ¼ eRe ðk1 Þx3 ½ReðK1 Þ cosðImðk1 Þx3 Þ  ImðK1 Þ sinðImðk1 Þx3 Þ

ð30Þ

X2 ¼ eRe ðk1 Þx3 ½ReðK1 Þ sinðImðk1 Þx3 Þ þ ImðK1 Þ cosðImðk1 Þx3 Þ:

ð31Þ

On the basis of the previous set of linearly independent real-valued solutions, a transfer matrix method can be developed for the analysis of multilayered piezoelectric hollow cylinders, and it can be extended to the analysis of FGPM sandwich cylinders using an SA method [50], where each FGPM layer of the sandwich cylinder is artificially divided into a finite number (NL) of individual layers with an equal and small thickness for each layer, compared with the mid-surface radius, as well as with constant material properties, determined in an average thickness sense. The exact solutions of the assorted field variables induced in the FGPM sandwich cylinder can thus be gradually approached by increasing the number of individual layers. It is noted that this solution process can be performed layer-by-layer, and the computational performance is independent of the total number of individual layers. Consequently, the implementation of the present approach is much less time-consuming than usual. 4.3. The successive approximation method This paper undertakes the 3D static analysis of FGPM sandwich cylinders, which are one of the widely-utilized multilayP ered FGPM cylinders, in which the thickness of each layer is hi (i = 1, 2 and 3, counted from the bottom layer), 3i¼1 hi ¼ 2h, ðkÞ and the material properties of each layer, which are g ij ðfÞ (k = 1, 2 and 3, are assumed to be symmetric with respect to the mid-surface of the sandwich cylinder, and obey an exponent-law distribution through the thickness coordinate, as follows: ð1Þ

ðcÞ







g ij ðfÞ ¼ g ij ej½ðff1 Þ=ðf0 f1 Þ ð2Þ g ij ðfÞ ð3Þ g ij ðfÞ

¼

ðcÞ g ij

¼

   ðcÞ g ij ej½ðff2 Þ=ðf3 f2 Þ

f0 6 f 6 f1 ;

f1 6 f 6 f2 ;

ð32aÞ ð32bÞ

f2 6 f 6 f3 ;

ð32cÞ

where the superscript c in the parentheses denotes the core (or middle) layer of the cylinder, f0 ¼ f3 ¼ h; f1 ¼ h2 =2; f2 ¼ h2 =2, and j denotes the material-property gradient index, which represents the degree of the material gradient along the thickness coordinate. It is noted that j ¼ 0 corresponds to the homogeneous material, j < 0 to the graded soft material, and j > 0 to the graded stiff material. Because the material properties of each individual layer, constituting the cylinder, vary along the thickness coordinate, resulting in a variant coefficient matrix in the system equation (i.e., Eq. (28)), the conventional Pagano method can not be directly applied to this coupled analysis of an FGPM cylinder. An

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

1917

SA method is thus adopted to make the present approach feasible. In this method, each individual FGPM layer is artificially divided into an NL-layered cylinder with a small thickness, compared with the mid-surface radius, with homogeneous material properties for each layer, and the FGPM sandwich cylinder is thus transformed into an N-layered homogeneous piezoelectric cylinder, in which N ¼ 3N L . For a typical kth-layer (k ¼ 1; 2; . . . ; N L ) in the upper FGPM layer of the cylinder, the ðkÞ material properties g ij are regarded as constants, and determined in an average thickness sense, as follows: ðkÞ gij ¼

1 Dfk

Z

ðcÞ

fk

g ij ðfÞdf ¼

fk1



g ij h3 eðjf2 =h3 Þ  ðjf =h Þ e k 3  eðjfk1 =h3 Þ ; jðDfk Þ

ð33Þ

where fk1 and fk are the thickness coordinates, measured from the middle surface of the cylinder to the bottom and top surfaces of the kth-layer in the upper FGPM layer, respectively, and Dfk denotes the thickness of the kth-layer, which is Dfk ¼ fk  fk1 . By means of Eq. (33), the modified Pagano method can be extensively applied to the coupled analysis of FGPM sandwich cylinders. Increasing the number of artificial layers (NL), we can approximate the exact solutions of this coupled analysis of the FGPM sandwich cylinders to any desired accuracy. 4.4. The transfer matrix method As we noted above, the modified Pagano method can be applied to the coupled analysis of FGPM sandwich cylinders under electro-mechanical loads using Eq. (33). The through-thickness distributions of material properties are modified as the layerwise Heaviside functions, and the upper half of these are given by

g ij ðfÞ ¼

NL X

ðkÞ gij ½Hðf  fk1 Þ  Hðf  fk Þ f2 6 f 6 f3 ;

ð34aÞ

k¼1 ðcÞ

g ij ðfÞ ¼ g ij

0 6 f 6 f2 ;

ð34bÞ

ðkÞ

where g ij refers to the coefficients of cij ; eij and gij of the kth-layer in general, HðfÞ is the Heaviside function, and the material properties of the lower half of the cylinder are symmetric with those of the upper half with respect to the mid-surface of the cylinder, which were given above and thus are not repeated. A transfer matrix method for the coupled analysis of the N -layered piezoelectric cylinders is then developed as follows, in which N ¼ 3N L . According to Eq. (29), we obtain the general solution for the coupled electro-elastic equations of the mthlayer (m ¼ 1; 2; . . . ; N) in the form of:

FðmÞ ðx3 Þ ¼ XðmÞ ðx3 ÞLðmÞ ;

ð35Þ

when x3 ¼ x3ðm1Þ , in which x3ðm1Þ ¼ fm1 =h, according to Eq. (35) we obtain:

 1 LðmÞ ¼ XðmÞ ðx3ðm1Þ Þ Fm1 ;

ð36Þ

where x3ðm1Þ is the dimensionless thickness coordinate measured from the middle surface to the bottom surface of mthlayer, Fðm1Þ denotes the vector of primary field variables at the interface between the (m-1)th- and mth-layers, and Fðm1Þ ¼ FðmÞ ðx3 ¼ x3ðm1Þ Þ. When x3 ¼ x3ðmÞ , in which x3ðmÞ ¼ fm =h, using Eqs. (35) and (36), we obtain:

FðmÞ ¼ RðmÞ Fðm1Þ ; ðmÞ

ð37Þ 1

ðmÞ

where RðmÞ ¼ X ðx3ðmÞ Þ½X ðx3ðm1Þ Þ . By analogy, the vectors of primary variables in the elastic and electric fields between the top and bottom surfaces of the plate (i.e., FðNÞ and Fð0Þ ) are linked by

FðNÞ ¼ RðNÞ FðN1Þ ¼

N Y

! RðmÞ Fð0Þ :

ð38Þ

m¼1

where PNm¼1 RðmÞ ¼ RðNÞ RðN1Þ    Rð2Þ Rð1Þ . Eq. (38) represents a set of eight simultaneous algebraic equations. Imposing the prescribed loading conditions on the lateral surfaces, we may determine the other unknown primary variables in the elastic and electric fields on the lateral surfaces. The values of these primary variables through the thickness coordinate of the hollow cylinder can then be obtained by

h i1 Fð1Þ ðx3 Þ ¼ Xð1Þ ðx3 Þ Xð1Þ ð1Þ Fð0Þ ;

ð39aÞ

h i1 FðmÞ ðx3 Þ ¼ XðmÞ ðx3 Þ XðmÞ ðx3ðm1Þ Þ Fðm1Þ ;

ð39bÞ

and

where Fðm1Þ ¼ Rðm1Þ Fðm2Þ and m ¼ 2; 3; . . . ; N.

1918

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

Once the primary variables, varied through the thickness of the cylinder, are determined, the corresponding set of dependent variables in the elastic and electric fields can then be obtained using Eqs. (16), (17). 5. Illustrative examples 5.1. Multilayered hybrid piezoelectric and elastic hollow cylinders As mentioned above, the present formulation for the analysis of FGPM sandwich cylinders can also be used for that of multilayered hybrid elastic and piezoelectric cylinders, and thus a coupled analysis of these with closed-circuit surface con2  þ ditions and subject to the nonaxisymmetric mechanical load (i.e., q r ¼ q0 sinðpx=LÞ cos 4h; qr ¼ 0, and q0 ¼ 1N=m ) and electric potential (i.e., Uþ ¼ /0 sinðpx=LÞ cos 4h; U ¼ 0, and /0 ¼ 1V) are considered in Tables 2 and 3, respectively, for comparison purposes. The cylinder is considered to be composed of a [00 =900 =00 ] laminated composite cylinder bonded with piezoelectric layers of PZT-4 on the outer and inner surfaces (see Fig. 1). The thickness ratio for each layer constituting the cylinder is as follows: PZT-4 layer: 00 -layer: 900 -layer: 00 -layer: PZT-4 layer = 0.1h: 0.6h: 0.6h: 0.6h: 0.1h, the geometric parameters are taken as R/2h = 5 and L/R = 4, and the material properties of the composite material (graphite/epoxy) and piezoelectric material (PZT-4) are given in Table 1. A set of normalized variables is defined as follows: In the cases of applied mechanical loads,

 ; v ; w  Þ ¼ ðux c =q0 ð2hÞ; uh c =q0 ð2hÞ; ur c =q0 ð2hÞÞ; ðu  x; r  h; s xh ; r  z; s xz Þ ¼ ðrx ; rh ; sxh ; rr ; sxr Þ=q0 ; ðr      ¼ Ue =q ð2hÞ; Dx ; Dz ¼ ðDx c =q e ; Dr c =q e Þ; / 0

0

ð40a-dÞ

0

where c ¼ 1N=m2 ; e ¼ 1C=m2 , and ðux ; sxr Þ are at the positions of ðx ¼ 0; h ¼ 0Þ; ðuh ; shr Þ ^ h ¼ p=2Þ; ður ; rx ; rh ; U; Dr Þ at those of ðx ¼ L=2; h ¼ 0Þ, and sxh at ðx ¼ 0; n ^ h ¼ p=2Þ. ðx ¼ L=2; n In the cases of applied electrical potential:

; w  Þ ¼ ðux c =/0 e ; ur c =/0 e Þ; ðu  x ; rh ; s xh ; r  z; s xz Þ ¼ ðrx ; rh ; sxh ; rr ; sxr Þð2h=/0 e Þ; ðr   

 ¼ U=/ : Dx ; Dz ¼ 2hDx c =/0 ðe Þ2 ; 2hDr c =/0 ðe Þ2 ; / 0

at

those

of

ð41a-dÞ

The values of elastic and electric field variables induced at some crucial positions of the multilayered piezoelectric cylin^ ¼ 1 and n ^ ¼ 4, are preder with closed-circuit surface conditions and under the loading conditions of Cases 1 and 2 with m sented in Tables 2 and 3, respectively, such as at the interfaces between adjacent layers, outer and inner surfaces, and the mid-surface of the cylinder. The solutions obtained using the modified Pagano method are compared with the those obtained using the 3D asymptotic theory in Wu et al. [30] and the meshless collocation method on the basis of the differential reproducing kernel interpolation in Wu et al. [41], and it is seen in Tables 2 and 3 that they are in excellent agreement with one another. Figs. 2 and 3 show the through-thickness distributions of assorted field variables of the five-layered hybrid piezoelectric and elastic cylinders with the open-circuit surface conditions and under the mechanical loads (Case 3) and electric normal 2 þ ¼ þ  displacement (Case 4), respectively, which are q in Fig. 2 and D r ¼ q0 sinðpx=LÞ cos 4h; qr ¼ 0, and q0 ¼ 1N=m r

Table 1 Elastic, piezoelectric and dielectric properties of composite and piezoelectric materials. Moduli

Graphite/epoxy

PZT-4

c11 (GPa) c22 c33 c12 c13 c23 c44 c55 c66 e24 ðC=m2 Þ e15 e31 e32 e33 g11 ðF=mÞ

183.433 11.662 11.662 4.363 4.363 3.918 2.870 7.170 7.170 0.000 0.000 0.000 0.000 0.000 1.53e08 1.53e08 1.53e08

138.499 138.499 114.745 77.371 73.643 73.643 25.6 25.6 30.6 12.72 12.72 5.2 5.2 15.08 1.306e08 1.306e08 1.151e08

g22 g33

Table 2 The elastic and electric field variables at crucial positions of multilayered hybrid piezoelectric and elastic cylinders with closed-circuit surface conditions and under mechanical loads (Case 1). Theories

v ðfÞ

 wðfÞ

r h ðfÞ

sxh ðfÞ

shz ðfÞ

r z ðfÞ

 /ðfÞ

Dz ðfÞ

h

Present Meshless collocation 3D asymptotic

1.3761e11 1.3749e11 1.3750 e11

1.2580e09 1.2580e09 1.2579e09

24.8090 24.8101 24.8090

2.3349 2.3349 2.3349

0.0000 0.0000 0.0000

1.0000 1.0000 1.0000

0.0000 0.0000 0.0000

1.7218e11 1.7231e11 1.7236 e11

0.9h±

Present Meshless collocation 3D asymptotic

5.8720e11 5.8709e11 5.8709 e11

1.2631e09 1.2631e09 1.2631e09

21.3882 (2.2587) 21.3892 (NA) 21.3880 (2.2587)

1.9217 (0.4503) 1.9218 (NA) 1.9218 (0.4503)

0.8685 0.8685 0.8685

7.8032e01 7.8032e01 7.8033e01

1.1618e02 1.1618e02 1.1617e02

2.2560e11 2.2570e11 2.2575e11

0.3h±

Present Meshless collocation 3D asymptotic

2.1769e10 2.1768e10 2.1768e10

1.2615e09 1.2615e09 1.2615e09

0.9607 (13.9863) 0.9607 (NA) 0.9606 (13.9880)

0.0357 (0.0357) 0.0357 (NA) 0.0357 (0.0357)

1.3682 1.3683 1.3682

4.5428e01 4.5429e01 4.5430e01

1.1471e02 1.1471e02 1.1471e02

8.3966e12 8.3877e12 8.3808e12

0

Present Meshless collocation 3D asymptotic

3.1802e10 3.1801e10 3.1800e10

1.2611e09 1.2611e09 1.2611e09

0.3712 0.3704 0.3704

0.1269 0.1269 0.1268

2.2654 2.2655 2.2654

2.5116e02 2.5111e02 2.5134e02

1.1638e02 1.1638e02 1.1638e02

2.6013e11 2.6005e11 2.5996e11

0.3h±

Present Meshless collocation 3D asymptotic

4.1826e10 4.1826e10 4.1824e10

1.2594e09 1.2594e09 1.2594e09

15.4891 (0.8670) 15.4878 (NA) 15.4880 (0.8673)

0.2930 (0.2930) 0.2930 (NA) 0.2930 (0.2930)

1.4483 1.4484 1.4483

3.7396e03 3.7277e03 3.7437e03

1.1988e02 1.1988e02 1.1987e02

4.5634e11 4.5625e11 4.5616e11

0.9h±

Present Meshless collocation 3D asymptotic

5.6558e10 5.6557e10 5.6555e10

1.2480e09 1.2480e09 1.2480e09

2.4013 (24.1381) 2.4013 (NA) 2.4018 (24.1360)

0.7356 (3.1394) 0.7356 (NA) 0.7356 (3.1392)

1.1875 1.1876 1.1875

2.6396e01 2.6396e01 2.6393e01

1.3329e02 1.3328e02 1.3327e02

9.3355e11 9.3346e11 9.3333e11

h

Present Meshless collocation 3D asymptotic

6.1305e10 6.1304e10 6.1302e10

1.2419e09 1.2419e09 1.2419e09

29.0414 29.0410 29.0390

3.6323 3.6323 3.6322

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

8.5731e11 8.5713e11 8.5701e11

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

f

1919

1920

Table 3 The elastic and electric field variables at crucial positions of multilayered hybrid piezoelectric and elastic cylinders with closed-circuit surface conditions and under electric potential (Case 2). Theories

v ðfÞ

 wðfÞ

r h ðfÞ

sxh f

shz ðfÞ

r z ðfÞ

 /ðfÞ

 z ðfÞ D

h

Present Meshless collocation 3D asymptotic

8.7323e11 8.7324e11 8.7323 e11

1.7218e11 1.7228e11 1.7236e11

11.2042 11.2037 11.2040

0.9998 0.9998 0.9998

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

1.0000 1.0000 1.0000

1.6838e08 1.6837e08 1.6837 e08

0.9h±

Present Meshless collocation 3D asymptotic

6.7781e11 6.7782e11 6.7781 e11

2.6907e11 2.6917e11 2.6925e11

12.0475 (0.6141) 12.0471 (NA) 12.0470 (0.6143)

0.8204 (0.1922) 0.8204 (NA) 0.8204 (0.1922)

0.4212 0.4212 0.4212

1.1455e01 1.1455e01 1.1455e01

0.9418 0.9418 0.9418

1.6460e08 1.6460e08 1.6460e08

0.3h±

Present Meshless collocation 3D asymptotic

1.9085e11 1.9084e11 1.9082e11

2.7156e11 2.7166e11 2.7173e11

0.2793 (0.7321) 0.2794 (NA) 0.2792 (3.7324)

0.1251 (0.1251) 0.1251 (NA) 0.1251 (0.1251)

0.3568 0.3568 0.3568

1.8709e01 1.8709e01 1.8709e01

0.6322 0.6322 0.6322

1.5219e08 1.5219e08 1.5219e08

0

Present Meshless collocation 3D asymptotic

1.1997e11 1.1995e11 1.1992e11

2.6263e11 2.6273e11 2.6283e11

2.7521 2.7521 2.7521

0.1137 0.1137 0.1137

0.0086 0.0086 0.0086

1.1843e01 1.1843e01 1.1842e01

0.4850 0.4850 0.4850

1.4836e08 1.4835e08 1.4835e08

0.3h±

Present Meshless collocation 3D asymptotic

1.2150e11 1.2146e11 1.2142e11

2.6300e11 2.6309e11 2.6319e11

2.8216 (0.1731) 2.8213 (NA) 2.8213 (0.1732)

0.1153 (0.1153) 0.1153 (NA) 0.1153 (0.1153)

0.3508 0.3508 0.3508

1.6679e02 1.6681e02 1.6682e02

0.3407 0.3407 0.3407

1.4637e08 1.4637e08 1.4637e08

0.9h±

Present Meshless collocation 3D asymptotic

4.6158e11 4.6152e11 4.6147e11

3.0745e11 3.0755e11 3.0749e11

0.4594 (11.2693) 0.4594 (NA) 0.4597 (11.2700)

0.1639 (0.6993) 0.1638 (NA) 0.1638 (0.6992)

0.4883 0.4883 0.4884

1.1300e01 1.1300e01 1.1301e01

0.0527 0.0527 0.0527

1.4906e08 1.4906e08 1.4906e08

h

Present Meshless collocation 3D asymptotic

4.4012e11 4.4006e11 4.4001e11

3.9314e11 3.9324e11 3.9317e11

11.3251 11.3255 11.3260

0.6861 0.6861 0.6860

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

1.5057e08 1.5057e08 1.5057e08

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

f

1921

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

a

-1.2E-008 -8E-009 -4E-009 1

0

4E-009 1

0.5

ζ/h

-2E-007 1

0.5

0

0

R/2h

-0.5

0

-1 4E-009

100

200

u(ζ)

-200 1

-100

0

1

0

2E-007

4E-007 1

0.5

ζ/h

-0.5

5 10 20

-1 -1.2E-008 -8E-009 -4E-009

c

b

0.5

0

0

-0.5

d

-0.5

R/2h

5 10 20

-1 -2E-007

0

-80

-40

w(ζ)

2E-007

-1 4E-007

0

40

1

1 R/2h

0.5

0.5

ζ/h 0

0

R/2h

-0.5

-1 -200

0

σx(ζ)

0

100

0.9

0

R/2h

-1

-0.14

τxz(ζ) -0.07

0.5

0.5

0

-0.5

-1 -0.21

0

-0.5

R/2h

5 10 20

-0.14

-0.07 φ(ζ)

0

-1 0.07

τxθ(ζ)

-0.6

0

40

0.6

1.2 1

ζ/h

0.5

0

0

-0.5

-0.5

R/2h

5 10 20

-1

-1 -1.8

0.07 1

-1.2

0

1

1.8

0

-40

0.5

-1

-0.21 1

ζ/h

f

-0.5

5 10 20

0.9

-1

-1.8

ζ/h 0

g

-1

0.5

0

-0.5

1.8

0.5

-0.9

0

-80

1

-0.5

0

200

1

0.5

-0.5

-1 -100

-0.9

e

ζ/h

-0.5

5 10 20

5 10 20

0.5

h

-1.2

-0.6

σz(ζ)

-2E-011 -1E-011 1

0

0

0.6

1E-011

0.5

2E-011 1

0.5

ζ/h 0

0

R/2h

-0.5

-0.5

5 10 20

-1 -2E-011 -1E-011

0

1.2

0

0

0

Dz(ζ)

1E-011

-1 2E-011

Fig. 2. The through-thickness distributions of assorted field variables in the ½PZT  4=0 =90 =0 =PZT  4 cylinders with open-circuit surface conditions and under the applied mechanical loads (Case 3).

1922

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930 -0.004 1

a

0

0.004

0.008

0.012

0.016 1

0.5

-0.2

b

0.5

ζ/h 0

0

-0.5

0

0.1 1

0.5

ζ/h

-0.5

R/2h

-0.1

1

0.5

0

0

-0.5

-1 -0.004

-4E+008 1

c

0

0.004

0.008

u(ζ)

0

4E+008

5 10 20

-1 0.016

0.012

8E+008 1.2E+009 1

0.5

ζ/h

0

-1 -4E+008

-3E+006 1

e

0

R/2h 5 10 20

0

4E+008 σx(ζ)

0

3E+006

0

-0.5

-1 -3E+006

-2E+009 1

g

5 10 20

0

τxz(ζ)

-1E+009

0

1E+009 1

0.5

0

0

-0.5

-1 -2E+009

R/2h 5 10 20

-1E+009

φ(ζ)

0

-0.5

-1 1E+009

0

0.1

4E+007

8E+007 1

0.5

0

0

-0.5

-0.5

R/2h 5 10 20

-1 -8E+007 -4E+007

-8E+006 1

f

0

τxθ(ζ)

-4E+006

4E+007

0

-1 8E+007

4E+006 1

0.5

ζ/h

0.5

0

0

-0.5

-1 6E+006

3E+006

0.5

ζ/h

ζ/h

-0.5

R/2h

w(ζ)

0

0.5

0.5

0

-0.1

-8E+007 -4E+007 1

d

-0.5

6E+006 1

-1 -0.2

-1 8E+008 1.2E+009

0.5

ζ/h

-1

0.5

-0.5

-0.5

R/2h

5 10 20

5 10 20

-1 -8E+006

h

-0.5

R/2h

-1

-4E+006

σz(ζ)

-0.5

0

0.5

-1 4E+006

0

1

1.5

1

1

0.5

0.5

ζ/h 0

0

R/2h

-0.5

-0.5

5 10 20

-1

-1 -1

0

-0.5

0

0

0

0.5

Dz(ζ)

1

1.5

Fig. 3. The through-thickness distributions of assorted field variables in the ½PZT  4=0 =90 =0 =PZT  4cylinders with open-circuit surface conditions and under the applied electric normal displacement (Case 4).

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

1923

  ¼ 0, and D0 ¼ 1C=m2 in Fig. 3. The dimensionless variables used in the cases of applied mechanical D0 sinðpx=LÞ cos 4h; D r loads (Case 3) are identical to those used in Example 5.1 given in Eq. (40), and those used in the case of applied electric normal displacement (Case 4) are defined as:

; w  Þ ¼ ðux e =D0 ð2hÞ; ur e =D0 ð2hÞÞ; ðu x ; r  h; s xh ; r  z; s xz Þ ¼ ðrx ; rh ; sxh ; rr ; sxr Þe =D0 c ; ðs    z ¼ ðDx =D0 ; Dr =D0 Þ;  x; D D

ð42a-dÞ

 ¼ Uðe Þ2 =D0 c ð2hÞ: / The influence of the aspect ratio, which is S = R/(2 h), on these distributions of assorted field variables is studied, in which S = 5, 10 and 20, and L/R = 4. It is seen in Figs. 2 and 3 that the through-thickness distributions of in- and out-of-surface displacements, in-surface stresses, transverse stresses, and electric potential and normal displacement appear to be linear, layerwise linear and layerwise nonlinear, and layerwise linear, respectively. The transverse shear and normal stresses increase when the cylinder becomes thinner in the case of applied mechanical load (Case 3), while they increase when the cylinder becomes thicker in the case of applied electric normal displacement (Case 4). It is noted that the peak values of transverse shear stresses occur at the interfaces between adjacent layers in the loading conditions of Cases 3 and 4, and those of transverse normal stresses occur in the vicinity of the middle surface of the 90°-layer in the loading conditions of Case 3, and at the interfaces between adjacent layers in the loading conditions of Case 4. In addition, it is also shown that the present analysis leads to continuous values for the primary variables at the interfaces between adjacent layers, and the boundary conditions on the lateral surfaces are exactly satisfied. 5.2. FGPM sandwich cylinders In these examples we undertake the 3D static analysis of an FGPM circular hollow sandwich cylinder, subjected to the loading conditions of Cases 1_4 on the lateral surfaces. The dimensionless variables used in the cases of applied mechanical loads (Cases 1 and 3) are identical to those used in Example 5.1 and given in Eq. (40), and those used in the cases of applied electric potential and normal displacement (Cases 2 and 4) are given in Eqs. (41) and (42), respectively. The material properties of each FGPM layer are assumed to obey the exponent-law, exponentially varying along the thickness coordinate, and these are given as Eq. (32), in which the material-property gradient index j is taken as j ¼ 3:0; 1:5; 0:0; 1:5; 3:0 and the material properties of the core layer (the layer 2) of the cylinder are considered to be the same as those of PZT-4, which are given in Table 1. Table 4 shows the convergence of the present results of the electric and elastic field variables, which are induced at the critical positions of the sandwich cylinder with closed-circuit surface conditions and under the mechanical þ  loads, q and q0 ¼ 1N=m2 , in which the thickness ratio for each layer is r ¼ q0 sinðpx=LÞ cos h; qr ¼ 0, h1 : h2 : h3 ¼ 2h=3 : 2h=3 : 2h=3, the aspect ratio R/2h = 5, and L/R = 4. It is seen in Table 4 that although the convergence rate of the FGPM sandwich cylinders ðj–0Þ is slower than that of homogeneous piezoelectric ones ðj ¼ 0Þ, and slows down when the sandwich cylinders become stiffer or softer, the deviation between the time needed for the homogeneous piezoelectric and FGPM sandwich cylinders is minor, due to the use of the transfer matrix method. It is also seen that the convergent solutions are yielded at N L ¼ 20 for the ranges of j from j ¼ 3 to j ¼ 3, and this is thus used in the subsequent examples. The through-thickness distributions of assorted variables in the elastic and electric fields induced in the sandwich cylinders under four different loading conditions (Cases 1–4) are presented in Figs. 4–7, respectively, in which ^ ¼n ^ ¼ 1. It is shown in Figs. 4 and 6 that the through-thickness dish1 : h2 : h3 ¼ 2h=3 : 2h=3 : 2h=3, R/2h = 5, L=R ¼ 4 and m tributions of various elastic variables induced in the sandwich cylinder with closed-circuit surface conditions are almost identical to those induced in the one with open-circuit surface conditions in the case of applied mechanical loads, whereas these distributions of electric variables are significantly different from each other. Figs. 4(a)–7(a) and Figs. 4(b)–7(b) show that the through-thickness distributions of the elastic in-surface displacement appear to be linear for both the homogeneous (i.e., j ¼ 0) and FGPM sandwich cylinders (j–0); and the through-thickness distributions of the elastic out-of-surface displacement appear to be linear for the homogeneous cylinders, while slightly nonlinear for FGPM sandwich ones. Figs. 4(c)– 7(c) and Fig. 4–7d show that the in-surface stresses appear to be linear variations through the thickness coordinate for homogeneous cylinders, whereas they become layerwise nonlinear variations for FGPM sandwich cylinders, and dramatically change through the thickness coordinate when the absolute value of j becomes larger. Fig. 4(e and f) and Fig. 6(e and f) show that the through-thickness distributions of the transverse shear and normal stresses appear to be parabolic functions in homogeneous cylinders, while these become piecewise higher-degree polynomials for FGPM sandwich cylinders. In addition, the magnitude of the transverse shear stress increases when the cylinder becomes stiffer, and that of the transverse normal stress increases in the bottom half (1 6 x3 6 0) and decreases in the top half (0 6 z 6 1) of the cylinder, in comparison with the results for the homogeneous cylinders, when j becomes a positive value. Fig. 5(e and f) and Fig. 7(e and f) show that the through-thickness distributions of transverse shear and normal stresses appear to be parabolic and higher-degree polynomials for the homogeneous and FGPM sandwich cylinders when the cylinders are subjected to the electric loads. These distributions change dramatically when the material-property gradient index j becomes a positive value in Cases 2 and 4. Figs. 4–7 show that the distributions of elastic and electric variables through the thickness of FGPM sandwich

1924

Table 4 The elastic and electric field variables at crucial positions of FGPM sandwich cylinders with closed-circuit surface conditions and under mechanical loads. Theories

v ðhÞ

 wðhÞ

r h ðhÞ

sxh ðhÞ

shz ðh=3Þ

r z ðh=3Þ

 /ðh=3Þ

Dz ðh=3Þ

3.0

Present NL = 4 NL = 8 NL = 12 NL = 20 NL = 40

3.8056e09 3.8038e09 3.8035e09 3.8033e09 3.8032e09

4.0258e09 4.0226e09 4.0220e09 4.0217e09 4.0216e09

0.1831 0.1598 0.1556 0.1534 0.1525

0.7156 0.7157 0.7157 0.7157 0.7157

0.0229 0.0234 0.0234 0.0235 0.0235

0.8369 0.8364 0.8363 0.8363 0.8363

0.0185 0.0190 0.0191 0.0191 0.0191

5.1934e10 4.9923e10 4.9557e10 4.9370e10 4.9291e10

1.5

Present NL = 4 NL = 8 NL = 12 NL = 20 NL = 40

3.0983e09 3.0976e09 3.0975e09 3.0974e09 3.0974e09

3.3090e09 3.3078e09 3.3076e09 3.3075e09 3.3074e09

0.5013 0.5104 0.5121 0.5130 0.5133

2.5701 2.5702 2.5703 2.5703 2.5703

0.0225 0.0226 0.0226 0.0226 0.0226

0.7746 0.7745 0.7745 0.7745 0.7745

0.0040 0.0041 0.0041 0.0041 0.0041

1.1321e09 1.1220e09 1.1201e09 1.1192e09 1.1188e09

0.0

Present NL = 4 NL = 8 NL = 12 NL = 20 NL = 40

2.1132e09 2.1132e09 2.1131e09 2.1131e09 2.1131e09

2.2702e09 2.2701e09 2.2701e09 2.2701e09 2.2701e09

4.3836 4.3836 4.3836 4.3836 4.3836

7.7751 7.7747 7.7747 7.7746 7.7746

0.0445 0.0445 0.0445 0.0445 0.0445

0.6704 0.6704 0.6704 0.6704 0.6704

0.0065 0.0065 0.0065 0.0065 0.0065

1.5199e09 1.5199e09 1.5199e09 1.5199e09 1.5199e09

1.5

Present NL = 4 NL = 8 NL = 12 NL = 20 NL = 40

1.1013e09 1.1007e09 1.1006e09 1.1006e09 1.1006e09

1.1759e09 1.1752e09 1.1751e09 1.1750e09 1.1750e09

12.3941 12.3817 12.3794 12.3782 12.3777

18.3164 18.3098 18.3086 18.3080 18.3077

0.0622 0.0621 0.0621 0.0621 0.0621

0.5706 0.5707 0.5707 0.5707 0.5707

0.0080 0.0079 0.0079 0.0079 0.0079

1.1066e09 1.1001e09 1.0988e09 1.0982e09 1.0980e09

3.0

Present NL = 4 NL = 8 NL = 12 NL = 20 NL = 40

4.4164e10 4.4127e10 4.4120e10 4.4116e10 4.4115e10

4.6685e10 4.6640e10 4.6631e10 4.6627e10 4.6625e10

21.5434 21.5055 21.4984 21.4948 21.4932

33.2706 33.2430 33.2378 33.2352 33.2340

0.0524 0.0522 0.0522 0.0522 0.0522

0.5235 0.5236 0.5236 0.5236 0.5236

0.0044 0.0043 0.0043 0.0043 0.0043

4.8447e10 4.7618e10 4.7464e10 4.7385e10 4.7351e10

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

j

1925

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930 -2E-009 -1.5E-009 -1E-009 -5E-010 1

a

0 1

0.5

ζ/h

0.5

0

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

u(ζ)

-60

-30

0

30

1

0.5

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-30

-0.15 1

e

0

σx(ζ)

0

30

0.15

1

0.5

ζ/h

-0.5

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-1 -0.15

g

ζ/h

-0.06 1

-0.03

τxz(ζ)

0

0.03

0.5

-1 -0.06

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.03

-0.5

0

φ(ζ)

0.03

-1 0.06

w(ζ) -20

-10

0

-1 8E-009 10 1

0.5

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-1 -40

-30

-20

-10

0

10

-1

-0.5

0

0.5

1

1.5

τxθ(ζ)

1

1

0.5

ζ/h

0.5

0

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-0.5

-1

-1 -1

0.06 1

4E-009

0

0.3

0.5

ζ/h 0 -0.5

f

-0.5

0.15

0

-1

-1 0

-30

-0.5

0.5

0

-0.5

0.5

60

0.3

κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

1

-1 -60

0

-40

d

-0.5

-1

8E-009 1

0

-1 -8E-009 -4E-009

0.5

0

4E-009

0.5

-0.5

-1

1

ζ/h

ζ/h

0 60

0

0.5

-0.5

-1 -2E-009 -1.5E-009 -1E-009 -5E-010

c

-8E-009 -4E-009 1

b

h

-3E-009 1

-0.5

0

0.5

σz(ζ)

-2E-009

-1E-009

1

1.5

0 1

0.5

0.5

ζ/h 0 -0.5

-1 -3E-009

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-1 -2E-009

Dz(ζ)

-1E-009

0

Fig. 4. The through-thickness distributions of assorted field variables in an FGPM sandwich cylinder with the closed-circuit surface conditions and under the applied mechanical loads (Case 1).

1926

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

a -1.2E-009

-8E-010

-4E-010

0

1

1

0.5

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-1 -1.2E-009 -120 1

-60

u(ζ) 0

60

120

-10

ζ/h 0

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

e

-1.5

-1

60

σx(ζ)

-0.5

0

0.5

1

1.5 1

0.5

ζ/h 0

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-0.5

-1

-1 -1.5

-1

-1

-0.5

-0.5

0

τxz(ζ)

0

0.5

0.5

1

1

1

0.5

0.5

ζ/h 0

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-0.5

-1

-1 -1

-0.5

0

φ(ζ)

0.5

1

1.5

w(ζ)

-5

0

5

-1 2E-009 10 1

0.5

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-0.5

-1

f

-1 -10

-5

-2

-1

0

5

10

0

1

2

τxθ(ζ)

1

1

0.5

0.5

ζ/h 0

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-0.5

-1

-1 -2

1.5

1E-009

ζ/h 0

1.5

1

0

1

120

1

-0.5

0.5

-1 0

0.5

g

d

-0.5

-60

κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

0.5

-1 -120

0

-1 -1E-009

0.5

-0.5

0.5

0

1

2E-009 1

ζ/h 0

-1 -4E-010

1E-009

0.5

-0.5

-8E-010

0

1

0.5

ζ/h 0

c

b -1E-009

h -6E-008

-1

0

σz(ζ)

-4E-008

1

2

-2E-008

0

1

1

0.5

0.5

ζ/h 0 -0.5

-1 -6E-008

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-1 -4E-008

Dz(ζ)

-2E-008

0

Fig. 5. The through-thickness distributions of assorted field variables in an FGPM sandwich cylinder with the closed-circuit surface conditions and under the applied electric potential (Case 2).

1927

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

a -2E-009

-1.5E-009 -1E-009 -5E-010

1

0.5

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

u(ζ)

-60

-30

0

30

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-0.15 1

-30

0

σx(ζ)

0

30

0.15

g

κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

0.5

ζ/h 0

-1 -0.12 -0.08 -0.04

φ(ζ)

0.04

0.08

-1

0.3

-1

-0.5

0

0.5

1

1.5

f

0.12 1

0.5

-1 0.12

τxθ(ζ)

1

1

0.5

0.5

ζ/h 0

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-0.5

-1

-1 -1

-0.5

0

-0.5

10

0

-0.5

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

0

0.3

0.08

0.5

-1

-1

0.04

1

-0.5

-0.5

0

10

-10

0

-0.12 -0.08 -0.04 1

0

-20

κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

τxz(ζ)

-10

ζ/h 0

0.5

0.15

-20

-1 8E-009

-30

1

0

-30

w(ζ)

4E-009

-40

ζ/h 0

-1 -0.15

-40

0

60

0.5

-0.5

-0.5

0.5

-1 -60

κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

1

-0.5

-1

e

d

0.5

ζ/h 0

0

-1 -8E-009 -4E-009

1

0.5

8E-009 1

0.5

-0.5

-1

60

4E-009

ζ/h 0

0

1

0

0.5

-0.5

-1 -2E-009 -1.5E-009 -1E-009 -5E-010

-4E-009

1

0.5

ζ/h 0

c

b -8E-009

0

1

h

-6E-011 1

-0.5

0

0.5

1

σz(ζ)

-3E-011

0

0.5

-1 -6E-011

3E-011 1

0.5

ζ/h 0 -0.5

1.5

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-3E-011

Dz(ζ)

0

-1 3E-011

Fig. 6. The through-thickness distributions of assorted field variables in an FGPM sandwich cylinder with the open-circuit surface conditions and under the applied mechanical loads (Case 3).

1928

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

a

-0.12 -0.08 -0.04 1

0

0.04

0.08

0.12 1

0.5

0.5

ζ/h 0 -0.5

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-6E+009 1

0

0.04

u(ζ)

-3E+009

0.08

0

0.5

-0.5

-1 -6E+009

e

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

σx(ζ)

-1.5E+008 -1E+008 -5E+007 1

0

5E+007 1

0.5

-0.5

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

τxz(ζ)

g

-3E+009 1

-2E+009

0

-1E+009

0 1

0.5

ζ/h 0

-1 -3E+009

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-1 -2E+009

φ(ζ)

-1E+009

0

-2E+009 -1E+009 1

0

1E+009

-1 0.04 2E+009 1

0.5

0.5

ζ/h 0

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-0.5

-1 -2E+009 -1E+009

f

0

1E+009

0

5E+007

τxθ(ζ)

-1E+008 -5E+007 1

-1 2E+009

1E+008 1

0.5

0.5

ζ/h 0

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-1 5E+007

0.5

-0.5

d

-0.5

-1 -1.5E+008 -1E+008 -5E+007

0

w(ζ)

0.5

ζ/h 0

-0.5

-0.2 -0.16 -0.12 -0.08 -0.04

-1 3E+009

0

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-1

-0.5

-3E+009

0.04 1

0.5

-0.5

0.5

ζ/h 0

0

ζ/h 0

-1 0.12 3E+009 1

-0.2 -0.16 -0.12 -0.08 -0.04 1

0.5

-0.5

-1 -0.12 -0.08 -0.04

c

b

-0.5

-1 -1E+008 -5E+007

h

-1

-0.5

0

σz(ζ)

0

0.5

5E+007

1

-1 1E+008

1.5

1

1

0.5

0.5

ζ/h 0

0 κ=-3.0 κ=-1.5 κ=0.0 κ=1.5 κ=3.0

-0.5

-0.5

-1

-1 -1

-0.5

0

0.5

Dz(ζ)

1

1.5

Fig. 7. The through-thickness distributions of assorted field variables in an FGPM sandwich cylinder with the open-circuit surface conditions and under the applied electric normal displacement (Case 4).

C.-P. Wu, T.-C. Tsai / Applied Mathematical Modelling 36 (2012) 1910–1930

1929

cylinders reveal different patterns from these distributions for homogeneous cylinders, and the influence of the materialproperty gradient index j on assorted field variables is significant. In addition, it is seen from Figs. 4–7(c, f, g and h) that the prescribed boundary conditions on the lateral surfaces of the cylinder in Cases 1_4, respectively, are exactly satisfied. 6. Concluding remarks In this paper, we have developed a modified Pagano method for the 3D static analysis of simply-supported, FGPM circular hollow sandwich cylinders under electro-mechanical loads, and the analysis of multilayered hybrid piezoelectric and elastic cylinders can be included in this formulation as a special case. The accuracy and convergence rate of this formulation are evaluated in comparison with the available 3D solutions, with which the present solutions are shown to converge rapidly and be in excellent agreement with. In addition, the influences of the aspect ratio, closed- and open-circuit surface conditions and material-property gradient index on the assorted variables in the elastic and electric fields, as induced in the FGPM sandwich cylinders under electro-mechanical loads, are studied. It is shown that the peak values of transverse shear and normal stresses occur at the interfaces between adjacent layers for the multilayered cylinders subjected to electric loads, and the influence of the material-property gradient index on the through-thickness distributions of the FGPM sandwich cylinders is significant. Acknowledgment This work was supported by the National Science Council of Taiwan, the Republic of China through Grant NSC 97-2221E006-128-MY3. Appendix A The relevant coefficients of kij in Eq. (15) are given by

~cij ¼ cij =Q;

~eij ¼ eij =e;

g~ ij ¼ gij Q=e2 ;

~ ij ¼ Q =Q ; Q ij ¼ cij  ci3 a1j  e3i a2j ði; j ¼ 1; 2; 6Þ; Q ij     a1j ¼ cj3 g33 þ e3j e33 =D; a2j ¼ e33 cj3  e3j c33 = Dðj ¼ 1; 2Þ; a1 ¼ g33 =D;

a2 ¼ b1 ¼ e33 =D;

and D ¼ c33 g33 þ e233 ;

ðA:1Þ

b2 ¼ c33 =D:

Appendix B ij in Eq. (28) are given by The coefficients k

 ¼k ; k 15 15 22 ¼ k22 ; k 33 ¼ k33 ; k 41 k 51 k 54 k 61 k 64 k

 ¼ m  ¼ m; ~ ðh~e15 =R~c55 Þ; k ~ k 17 18   28 ¼ n ~ ~ ~ ð1=c2 Þ; ~ k26 ¼ k26 ; k27 ¼ nðhe24 Þ=ðRc2 c44 Þ; k         2 2 2 2 37 ¼ m ~ ~e15 =~c55 þ g ~ 11 h=R  n ~ 22 h= c22 R ; ~ ~e24 =~c44 þ g k



 44 ¼ k44 ; k 48 ¼ k48 ; 42 ¼ n 43 ¼ k43 ; k ~ 21 =c ; k ~ 22 =c2 ; k ~ Q ~ Q ¼ m 2 2



 53 ¼ m 52 ¼ m ~ 12 þ Q ~ 66 =c ; k ~ 11 þ n ~ 66 =c2 ; k ~ 2Q ~2 Q ~n ~ Q ~ ða21 e=Q Þ; ¼m 2 2

 55 ¼ k55 ; k 58 ¼ m ~ 12 =c ; ~ ða11 h=RÞ; k ~ Q ¼ m 2 h

 i

 2 2  63 ¼ n ~ ~ 22 =c2 ; k ~ ~ ~n ~ Q 21 þ Q 66 =c2 ; k62 ¼ m ~ Q 66 þ n ~ Q ~½a22 e=ðQ c2 Þ; ¼ m 2

 66 ¼ k66 ; k 68 ¼ n ~ 22 =c2 ; ~ ½a12 h=ðRc2 Þ; k ~ Q ¼n 2

73 ¼ k73 ; k

74 ¼ k74 ; k

78 ¼ k78 ; k

84 ¼ k84 ; k

ðB:1Þ

88 ¼ k88 : k

References [1] N.J. Pagano, Exact solutions for composite laminates in cylindrical bending, J. Compos. Mater. 3 (1969) 398–411. [2] N.J. Pagano, Exact solutions for rectangular bidirectional composites and sandwich plates, J. Compos. Mater. 4 (1970) 20–34. [3] S. Srinivas, A.K. Rao, Bending, Vibration and buckling of simply supported thick orthotropic rectangular plates and laminates, Int. J. Solids Struct. 6 (1970) 1463–1481. [4] P. Heyliger, Static behavior of laminated elastic/piezoelectric plates, AIAA J. 32 (1994) 2481–2484. [5] P. Heyliger, A note on the static behavior of simply-supported laminated piezoelectric cylinders, Int. J. Solids Struct. 34 (1997) 3781–3794. [6] P. Heyliger, Exact solutions for simply supported piezoelectric plates, J. Appl. Mech. 64 (1997) 299–306. [7] P. Heyliger, S. Brooks, Exact solutions for laminated piezoelectric plates in cylindrical bending, J. Appl. Mech. 63 (1996) 903–910.

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