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Composite Structures 84 (2008) 241–247 www.elsevier.com/locate/compstruct

Piezolaminated plates – Highly accurate solutions based on the extended Kantorovich method Lucy Edery-Azulay *, Haim Abramovich Faculty of Aerospace Engineering, Technion – Israel Institute of Technology, 32000 Haifa, Israel Available online 19 August 2007

Abstract Piezoelectric materials become more and more common in various industrial applications, which can be simulated as a piezoelectric plate. An exact mathematical solution solving ﬂexural behavior of a rectangular plate can be achieved, based on the Levy method, only when two parallel edges of the plate are simply supported. For any other combinations of boundary conditions, it becomes necessary to turn to one of the available approximate solutions, such as energy methods, and ﬁnite elements. In this paper, we present an original model that can be used to predict the ﬂexural behavior of piezoelectric plates on various boundary conditions. The analytical solution is based on the extended Kantorovich iterative procedure. The diﬀerential equations for the iterative procedure are derived using the Galerkin method. The solution was developed based on the classical plate’s theory (CLPT), and for this reason, it is limited only for the solution of extension type piezoelectric mechanism. This iterative procedure yields highly accurate solutions, which were compared against other available analytical results. 2007 Elsevier Ltd. All rights reserved. Keywords: Extended Kantorovich method; Galerkin method; Piezoelectric mechanism; Composite plate; Iterative procedure

1. Introduction One of the basic elements of adaptive structures is a thin composite plate with surface-induced or embedded patches actuators. Several plate models, based on various laminated plate theories like the classical plate theory (CLPT), and higher order theories like ﬁrst order shear deformation theory (FSDT) and higher order shear deformation theory (HSDT), have been developed to predict the ﬂexural response of laminated plates with layers of actuators. However, exact mathematical models, based on Navier or Levy method solutions, are limited only for a plate with continues piezoelectric layer and relies on at least two opposite simply-supported edges. Plates with various boundary conditions are usually solved using one of the available

*

Corresponding author. Tel.: +972 4 8293199; fax: +972 4 8292303. E-mail addresses: [email protected] (L. Edery-Azulay), [email protected] (H. Abramovich). 0263-8223/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2007.08.003

approximate solutions, such as energy methods, and ﬁnite elements [1–11]. The present study presents an original way of using the extended Kantorovich method to solve the ﬂexural behavior of a piezo–composite plate actuated by piezoelectric induced bending moments. Plates can be solved for any required boundary conditions. The Kantorovich method has two versions: the classical and the extended method [12–14]. The Kantorovich method, on its two versions, the classical and the extended method, is widely used to compute the buckling loads of compressed plates [15–19]. For example, Ungbhakorn and Singhatanadgid [15] employed the extended Kantorovich method to investigate the buckling problem of rectangular laminated composite plates with various edge supports. The solution is based on the principle of minimum total potential energy. Only few published studies solve the ﬂexural response of plates using the Kantorovich method, and yet they are all limited to the case of a plate with all around clamped edges.

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Dalei and Kerr [20] solved the case of a rectangular laminated plate based on CLPT theory. It was found that the convergence of the solution procedure based on the extended Kantorovich method, is very rapid and that the ﬁnal form of the generated solution is independent of the initial choice. Aghdam et al. [21] derived a solution for the bending of a rectangular Reissner type isotropic clamped plate. Aghdam and Falahatgar [22] extended this model to investigate the bending behavior of moderately thick rectangular laminated plates. Yuan et al. [23] used the extended Kantorovich method to solve the bending problem of rectangular Mindlin type plates using the multi-term trial functions. The present paper describes an original procedure of using the extended Kantorovich method to solve a piezo– composite plate actuated by piezoelectric induced bending moments. The procedure is presented in a general form and can also be used to solve the case of a plate under combined electric and mechanical loads. The reliability of the present analytical method was veriﬁed and approved when comparing to available exact solutions [1] and FE results based on the commercial ﬁnite elements code, ANSYS [28]. Typical results for piezo– composite plates with various boundary conditions are described. 2. The Kantorovich method 2.1. The classical Kantorovich method As already stated, the Kantorovich method has two versions: the classical and the extended method. The classical method was developed in 1958 by Kantorovich and Krylov [30] and refers to semi-analytical solutions which reduce the governing partial diﬀerential equations to a set of governing ordinary diﬀerential equations (ODE). These equations can be derived either by using the Galerkin method or the principle of minimum total potential energy. In this case, we assumed that the plate displacement has the following form wm ðx; yÞ ¼

m X

fn ðxÞ gn1 ðyÞ

ð1Þ

n¼1

where gn1(y) are a priori chosen functions. Substituting Eq. (1) into the corresponding plate equations of motion, and then applying the Galerkin method and utilizing the fundamental lemma of variational calculus, m ordinary diﬀerential equations would result for the determination of the m unknown fn(x) functions. These differential equations are solved in conjunction with the corresponding boundary conditions. Next, the generated fn functions are substituted back into Eq. (1). One should note that if the assumed initial solution in the ﬁrst step satisﬁes all the boundary conditions, the solution for the simple classical Kantorovich will yield a good result, depending on the form of the initial solution like in

the cases when either the Rayleigh–Ritz or Galerkin method are applied. 2.2. The extended Kantorovich method For the extended Kantorovich solution [31], the iterative procedure is repeated until the solution converges to a desired degree of accuracy. For this case, we assume that plate displacement has the following form m X fn ðxÞ gn ðyÞ ð2Þ wm ðx; yÞ ¼ n¼1

where fn(x) are the functions determined in the previous step (classical Kantorovich method), and the gn(y) are assumed unknown. The gn’s are determined from a set of m ordinary equations. This iterative procedure is continued by substituting the determined gn(y) expressions back into Eq. (2), determining the unknown fn(x) expressions, etc. Kerr [12] and Kerr and Alexander [13], found that the iterative process converges very rapidly, and that the ﬁnal form of the generated solution is independent of the initial solution choice, gn(y). Furthermore, although the extended Kantorovich method is based on a variational principle, the initial trial functions are neither required to satisfy the geometric nor the natural boundary conditions; the iterative procedure will force the solution to satisfy all boundary conditions eventually, Yuan and Jin [14]. 3. The mathematical model Let us consider a plate in a Cartesian coordinate system were x- and y-axes are parallel to the plate boundaries. For the simpliﬁcation of the model, let us assume a plate composed of laminated structural materials and piezoelectric layers (piezo-laminated plate), symmetric to its mid-plane, Bij = 0, and having no other coupling terms ( )16 = ( )26 = 0, and no surface shear stresses. Plane stress conditions are assumed for model, see Ref. [1]. The displacement ﬁeld is determined according to the classical plate theory, CLPT, where the transverse shear deformations are ignored.1,2 Based on these assumptions, and integrating the stress components across each lamina including the piezoelectric induced components, one can obtain the equivalent moments per unit width for the overall structure in terms of the out of plane displacements as

1

Although it is well known that the transverse shear deformations might eﬀect the lateral deﬂection of a laminated composite plate, it is appropriate for thin plates to use a simpler stress analysis based on a classical plate theory. 2 Using the CLPT, where the non-zero linear strains are only: ex, ey, cxy, the solution is limited to a plate actuated only by extension type piezoelectric mechanism.

L. Edery-Azulay, H. Abramovich / Composite Structures 84 (2008) 241–247

2 o2 o wðx; yÞ D wðx; yÞ þ F x ðx; yÞ 12 ox2 oy 2 2 2 o o M y ¼ D12 wðx; yÞ D wðx; yÞ þ F y ðx; yÞ 22 ox2 oy 2 2 o M xy :¼ 2 D66 wðx; yÞ ox oy M x ¼ D11

ð3Þ

where Dij is the bending stiﬀness of the composite plate and the piezoelectric induced moments Fii(x, y) stem from the piezoelectric constitutive equations. Governed by the plane stress assumption Fii(x, y) deﬁned as Z h=2 Qi3 e33 F ii ðx; yÞ ¼ ei3 E3 z dz i ¼ 1; 2 ð3aÞ Q33 h=2 E3 is the electric ﬁeld, Qij are the elasticity constants and eij are the piezoelectric constants. For more details see Ref. [1]. The well-known equation of motion for thin plates subjected to lateral pressure, q(x, y), is given by o2 M x o2 M xy o2 M y þ þ 2 ¼ qðx; yÞ ox2 ox oy oy 2

ð4Þ

Substituting Eq. (3) into Eq. (4) yields 4 4 o wðx; yÞ o wðx; yÞ D11 ð þ 4D Þ þ 2D 12 66 ox4 ox2 oy 2 4 o wðx; yÞ o2 F x ðx; yÞ o2 F y ðx; yÞ þ þ qðx; yÞ þ D22 ¼ oy 4 ox2 oy 2 ð5Þ Casting Eq. (5) into a simpler form yields D1

o4 wðx; yÞ o4 wðx; yÞ o4 wðx; yÞ þ D þ D ¼ P ðx; yÞ 3 2 ox4 ox2 oy 2 oy 4 ð6Þ

where D1 ¼ D11 ; P ðx; yÞ ¼

D3 ¼ ð2D12 þ 4D66 Þ;

Assuming that W(y) is a priori chosen known function, and substituting it in both Eq. (8) and the corresponding Galerkin equation (Eq. (7)) we get Z a Z b d4 W ðxÞ d2 W ðxÞ d2 W ðyÞ D1 W ðyÞ þ D 3 dx4 dx2 dy 2 0 0 4 d W ðyÞ þD2 W ðxÞ P ðx; yÞ W ðyÞ dy W ðxÞ dx ¼ 0 dy 4 ð9Þ To satisfy Eq. (9), the expression in the square brackets must be zero. This results in an ordinary diﬀerential equation for the determination of the unknown W(x) function,3 and can also be written in the following form H4

d4 W ðxÞ d2 W ðxÞ þ H2 þ H 0 W ðxÞ ¼ F ðxÞ 4 dx dx2

where Z

Z

d2 W ðyÞ W ðyÞ dy dy 2 0 0 Z b 4 Z b d W ðyÞ H 0 ¼ D2 W ðyÞ dy; F ðxÞ ¼ P ðx; yÞ W ðyÞ dy dy 4 0 0 ð10Þ H 4 ¼ D1

b

W 2 ðyÞ dy; H 2 ¼ D3

b

These integrals can be solved immediately using one of the available commercial mathematical programs.4 The solution of Eq. (10), which is the plate solution, is directly dependant on the loads pattern. For a simply-supported plate, the moments Mx and My along the edges should vanish. For a piezo-laminated plate, these moments would also include the induced piezoelectric bending moments, Fx(x, y) and Fy(x, y). Let consider a possible load case, where the piezoelectric induced bending moments and the mechanical loads are distributed in a double sinusoidal form according to the following expression.5 9 8 9 8 > = > = < qðx; yÞ > < q0 sinðk x xÞ sinðk y yÞ > F x ðx; yÞ ¼ f sinðk x xÞ sinðk y yÞ ; k x ¼ m p=a; k y ¼ n p=b > > > > ; : ; : F y ðx; yÞ f sinðk x xÞ sinðk y yÞ

D2 ¼ D22

o2 o2 F ðx; yÞ þ F y ðx; yÞ þ qðx; yÞ x ox2 oy 2

ð11Þ

3.1. The present iterative solution The solution to Eq. (6) is sought using the Galerkin method, namely Z a Z b o4 wðx; yÞ o4 wðx; yÞ o4 wðx; yÞ D1 þ D þ D P ðx; yÞ 3 2 ox4 ox2 oy 2 oy 4 0 0 wðx; yÞ dx dy ¼ 0 ð7Þ where a and b are the length and width of the plate, respectively. Applying the Kantorovich method, the solution w(x, y) is assumed to be separable as wðx; yÞ ¼ W ðxÞ W ðyÞ

243

ð8Þ

According to the classical Kantorovich method, the solution of one of the two directions is assumed to be known.

Ceramic PZT actuators are normally transversely isotropic where e31 = e32, therefore, it would yield equal induced bending moments in both x- and y-directions [24]. Eq. (10) transforms into the following simple form d4 W ðxÞ d2 W ðxÞ þ H þ H 0 W ðxÞ 2 dx4 dx2 ¼ ðq0 f ðk 2x þ k 2y ÞÞ sinðk x xÞ

H4

3

ð12Þ

In a similar manner, when W(x) is a priori known function one obtains an ordinary diﬀerential equation for the determination of the unknown W(y) function. 4 For this paper, the mathematical iterative procedure was programmed using the Maple code [26]. 5 For piezoelectric layers loaded with a DC electric voltage, the piezoelectric induced moments are constants all over the plate area.

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The solution for this equation having the homogenous and particular parts can be written in the brief form as W ðxÞ ¼

4 X

X i expðki xÞ þ

i¼1

f ðk 2x þ k 2y Þ q0 H 4 k 4x þ H 2 k 2x H 0

sinðk x xÞ ð13Þ

where the four roots of the corresponding characteristic equation are given by ﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ 2 2 H 4 H 2 þ H 2 4H 4 H 0 ki ¼

;

2H 4

i ¼ 1...4 ð14Þ

The four unknowns constants Xi, i=1 . . . 4 are determined by enforcing the boundary conditions in the xdirection. For a plate loaded by any other load patterns, the particular solution part of Eq. (15) will change. For example, in the case of plate under only uniform mechanical pressure the equation of the lateral displacement of plate will be W ðxÞ ¼

4 X i¼1

X i expðki xÞ þ

q0 H0

ð15Þ

For the extended Kantorovich solution, after this ﬁrst Kantorovich solution is obtained, we switch the plates’ direction to be solved, by using the obtained analytical solution as a speciﬁed function and freeing the other direction to be analytical solved. This iterative procedure can be repeated until the result converges to any desired degree of accuracy. 4. Results The mathematical iterative procedure was programmed using the Maple code [26]. To establish the accuracy of the present analytical solution, we ﬁrst compared the calculated solutions obtained using the present model with those known in the literature [1,25,27,29]. Two diﬀerent squared plates with 10 · 10 cm2 (a · b) and 0.8 mm thickness were investigated. One plate is assumed to be made of aluminum, E = 69 GPa, m = 0.32 and the second plate was assumed to be made of graphite–epoxy,6 with a symmetric layup [0/90]s (each layer has a thickness of 0.2 mm). The results of the iterative solution were compared with the well known Timoshenko solution [27] (only for the aluminum plate) and ANSYS ﬁnite element code results [28]. Timoshenko solutions has the following brief form (for a squared isotropic plate only): w = aq0a4/D, where the value of the parameter a depends on the plate’s boundary conditions, q0 is the lateral distributed loads, a is the plate’s length and width and D is its ﬂexural stiﬀness (for the pres6 Graphite–epoxy properties: E11 = 9.8 · 1010 [N/m2], E22 = 0.79 · 1010 [N/m2], G12 = 0.56 · 1010 [N/m2], m12 = 0.28.

ent aluminum plate D = 3.2774 [Nm2/m]7). For the ANSYS models, the aluminum plate was modeled using the SOLSH190 element, which is used for simulating shell structures with a wide range of thickness (from thin to moderately thick). The graphite–epoxy plate was modeled using the SOLID46 element, which is a layered element. The element size in x–y plane of both plates were 2.5 · 2.5 mm2. For the iterative calculations, an arbitrarily polynom was used for all the plates as an initial solution.8 The same initial solution was used for all the various boundary conditions. Table 1 summarizes the central deﬂection of a uniformly loaded plate. A good agreement was obtained while comparing the three diﬀerent solution results. Next, the piezoelectric actuation eﬀect was studied. Two piezoelectric layers of 0.1 mm thickness each were added as an upper and lower layer of the previous aluminum and graphite–epoxy plates.9 The properties of the piezoelectric material are: E11 = 8.84 · 1010 [N/m2], E22 = 7.6 · 1010 [N/m2], G12 = 2.1 · 1010 [N/m2], m12 = 0.37, e31 = 7.209 [C/m2], e33 = 15.118 [C/m2]. For the case of piezo–composite plates resting on at least two opposite simply-supported edges, the present results were compared with the results of the exact mathematical model, based on Navier or Levy method solutions [1]. The plate deﬂection was calculated for induced piezoelectric bending moments according to Eq. (11), (the value of parameter f equal 1). Figs. 1a–c present the central lateral displacements distribution for the section y = b/2, of the presented iterative method solution vs. Levy and Navier exact solutions for a plate with three various boundary conditions: (a) an all around simply-supported (SSSS); (b) simply–simply– clamped–clamped (SSCC) plate; and (c) simply–simply– clamped–simply (SSCS) plate. The two diﬀerent displacements lines refer to the two diﬀerent materials properties: graphite–epoxy or aluminum. According to the iterative process solution, for the three plates, SSSS, SSCC and SSCS, we considered that W(y) is a priori chosen known function in the form of sin(py/b). Results converged very rapidly, see Fig. 4. As can be seen, a very good agreement was received for the results calculated by the various solution methods, even though the exact solution (Levy and Navier) are based on the Reissner–Mindlin plate theory, while the iterative solution is based on the classical plate theory, Kirchhoﬀ–Love, see Table 2. 7

Flexural stiﬀness for graphite–epoxy plate with the symmetric layup [0/90]s, for a layer thickness of 0.2 mm: D11 = 3.724, D22 = 0.8228, D12 = 0.09497 and D66 = 0.2389 [Nm2/m]. 8 We used the polynom WY(y) = (y2 b2)2. This polynom was also used by Dalei and Kerr [20] for the solution of an all around clamped plate loaded with uniform mechanical loads. 9 For a piezo-laminated aluminum plate: ﬂexural stiﬀness D11 = 7.42, D22 = 6.85, D12 = 2.4 and D66 = 1.97 [Nm2/m]. Shear stiﬀness (for Levy solution) A44 = A55 = 0.2104 · 108. For a piezo-laminated graphite–epoxy plate: ﬂexural stiﬀness D11 = 7.79, D22 = 4.32, D12 = 1.39 and D66 = 1.105 [Nm2/m], A44 = A55 = 0.743 · 107.

L. Edery-Azulay, H. Abramovich / Composite Structures 84 (2008) 241–247

245

Table 1 The lateral deﬂection (w) of a squared uniformly loaded plate, q0 = 1 [N/m2] Aluminum plate

Graphite–epoxy plate

SSSS CCCC SSCC SSSC

w

0.00406 0.00126 0.00192 0.00279

0.12388 · 106 0.38444 · 107 0.58581 · 107 0.85126 · 107

Present iterative method, w

ANSYS w SOLSH190

Present iterative method, w

ANSYS w SOLID46

0.12386 · 106 0.38558 · 107 0.58370 · 107 0.84918 · 107

0.12387 · 106 0.38603 · 107 0.58454 · 107 0.83117 · 107

0.28433 · 106 0.65842 · 107 0.17757 · 106 0.22707 · 107

0.28464 · 106 0.66192 · 107 0.17782 · 106 0.22730 · 106

Graphite-epoxy

Graphite-epoxy

0.04

W [mm]

0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00

0.04

0.07

0.03

0.06 0.05

0.03

0.04

0.02

Aluminum

0.02

0.04

0.06

0.08

0.10

0.03

Aluminum

0.02

x[m]

Graphite-epoxy

W [mm]

Timoshenko a

W [mm]

B.C.

0.01

0.02

0.01

0.01

0.00 0.00

x[m] 0.02

Levy solution

0.04

0.06

0.08

Navier solution

0.10

Aluminum

0.00 0.00

x[m] 0.02

0.04

0.06

0.08

0.10

Present iterative method

Fig. 1. Lateral displacements along the x-direction for the line y = b/2. (a) A simply-supported plate SSSS, (b) a simply–simply–clamped–clamped plate, (c) a simply–simply–clamped–simply plate.

.

Graphite-epoxy

W [mm] 10-4

7

.

6 5 4

8 6

Aluminum

2

4

0

0.02

0.04

0.06

x[m]

0

0.08

0.1

Aluminum

2

1

x[m]

Graphite-epoxy

.

10

3

Aluminum

12

W [mm] 10-4

Graphite-epoxy

W [mm] 10-4

18 16 14 12 10 8 6 4 2 0

x[m]

0 0

0.02

0.04

0.06

Navier solution

Levy solution

0.08

0.1

0

0.02

0.04

0.06

0.08

0.1

Present iterative method

Fig. 2. Lateral displacements along the x-direction for the line y = b/2. (a) A simply-supported plate SSSS, (b) a simply–simply–clamped–clamped plate, (c) a simply–simply–clamped–simply plate.

Table 2 Boundaries conditions used in the present study

Simply-supported edge Clamped edge

Reissner–Mindlin

Kirchhoﬀ–Love

Wx, Mx, Uy Wx, Ux, oW/ox

Wx, Mx Wx, Ux

The solution sensitivity relative to the plate thickness was next investigated. The hosting plate’s thickness was increased to 4 mm.10 Fig. 2 demonstrated the lateral displacement distribution for aluminum and graphite–epoxy plate with various boundary conditions. A very minor dif10 For the laminated plate made of PZT and graphite–epoxy, we assumed that all graphite–epoxy layers are with equal thickness of 0.5 mm.

ference was recorded between the present iterative solution based on the CPL and the solution based on FSDT, even multiplying the plate thickness ﬁve times. In what follows, the actuation response of a piezo-laminated plate of CSCS, CCCC, CCCS, and CSCC boundaries conditions is considered. For these boundaries conditions no analytical solution is available. The aluminum and graphite–epoxy hosting plates, are assumed to be with the original thickness, 0.8 mm. Figs. 3a–d and Fig. 4 describe the transverse displacement pattern from a top view and the out of plane displacements along the x-direction for the line y = b/2, respectively, for a piezo– graphite–epoxy laminated plate. Fig. 5 demonstrates the results convergence for the piezo–aluminum plate, resting on various boundary condi-

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Fig. 3. Out of plane displacement patterns – a plate actuated by a continuous extension type piezoelectric layer. (a) Clamped–simply–clamped–simply, (b) all around clamped, (c) clamped–clamped–clamped–simply, (d) clamped–simply–clamped–clamped.

0.06

5. Conclusions

CSCS

Lateral displacement [mm]

CCCC 0.05 0.04

CCCS CSCC

0.03 0.02 0.01

x[m] 0.00 0.00

0.02

0.04

0.06

0.08

0.10

Fig. 4. The out of plane displacements – a piezo–graphite–epoxy plate with various boundary conditions.

Lateral displacement [mm]

0.08

0.06

0.04

0.02

SSSS

SSCC

SSSC

CCCS

CCSS CSCS

No. of iterations

0.00 1

2

3

Fig. 5. The convergence of the iterative process – actuation response of a piezo–aluminum plate with various boundary conditions.

tions. Each iteration contained the solution for the two direction of the plate’s plane, x and y. As a ﬁrst solution for the iterative process solution, we considered that W(y) has the form of sin(py/b). Although the same function was chosen for the various boundary conditions, only a few numbers of iterations were necessary for each plate and the results converged very rapidly. This is to conﬁrm that the initial trial functions are neither required to satisfy the geometric nor the natural boundary conditions because the iterative procedure forces the solution to satisfy all the boundary conditions of the plate.

An analytical model was developed and demonstrated to investigate the ﬂexural behavior of symmetrically laminated composite rectangular plates with various combinations of the boundary conditions. The analytical solution is based on the extend Kantorovich iterative procedure. The diﬀerential equations for the iterative procedure are derived using the Galerkin method. The solution was developed based on the classical plate’s theory (CLPT), and for this reason, it is limited only for the solution of extension piezoelectric mechanism type. The reliability of the present analytical method was veriﬁed and approved as compared to available exact solutions in the literature. The results show a high accuracy and a good agreement with exact analytical models that were calculated based on higher plate theories. The iterative process converges very rapidly. It was also found that the ﬁnal form of the generated solution is independent of the initial chosen solution. It is conﬁrmed that the initial trial functions are neither required to satisfy the geometric nor the force boundary conditions because the iterative procedure will eventually force the solution to satisfy all boundary conditions. References [1] Edery-Azulay L, Abramovich H. A reliable plain solution for rectangular plates with piezoceramic patches. J Intel Mat Syst Struct 2007;18. [2] Edery-Azulay L, Abramovich H. Actuation and sensing of shear type piezoelectric patches-closed form solutions. Compos Struct 2004;64:443–53. [3] Vel Senthil S, Batra RC. Exact solution for rectangular sandwich plates with embedded piezoelectric shear actuators. AIAA J 2001;39(7). [4] Vel Senthil S, Batra RC. Exact solution for the cylindrical bending of laminated plates with embedded piezoelectric shear actuators. Smart Mat Struct 2000;10:240–51. [5] Vel Senthil S, Batra RC. Analysis of piezoelectric bimorphs and plates with segmented actuators. Thin Wall Struct 2001;39:23–44. [6] Lin Chine-Chang, Hsu Chin-Yu, Huang Huang-Nan. Finite element analysis on deﬂection control of plate with piezoelectric actuators. Compos Struct 1996;35:423–33. [7] Robaldo A, Carrera E, Benjeddou A. A uniﬁed formulation for ﬁnite element analysis of piezoelectric adaptive plates. In: Proceedings of the seventh international conference on computational structures technology, 7–9 September, Lisbon, Portugal; 2004.

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