Advanced models for smart multilayered plates based on Reissner Mixed Variational Theorem

Accepted Manuscript Advanced models for smart multilayered plates based on Reissner Mixed Variational Theorem I. Benedetti, A. Milazzo PII:

S1359-8368(16)32934-1

DOI:

10.1016/j.compositesb.2017.03.007

Reference:

JCOMB 4941

To appear in:

Composites Part B

Received Date: 1 December 2016 Revised Date:

27 February 2017

Accepted Date: 7 March 2017

Please cite this article as: Benedetti I, Milazzo A, Advanced models for smart multilayered plates based on Reissner Mixed Variational Theorem, Composites Part B (2017), doi: 10.1016/ j.compositesb.2017.03.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

I. Benedetti, A. Milazzo∗

RI PT

Advanced models for smart multilayered plates based on Reissner Mixed Variational Theorem

SC

Dipartimento di Ingegneria Civile, Ambientale, Aerospaziale, dei Materiali - DICAM Universit` a degli Studi di Palermo, Viale delle Scienze, 90128, Palermo, Italy

Abstract

M AN U

In the present work, families of equivalent singe layer and layer-wise models for the static and free vibrations analysis of magneto-electro-elastic multilayered plates are developed. The models are defined in the framework of a unified formulation, which offers a systematic approach for generating refined plate theories through suitable expansions of the through-the-thickness components of the relevant fields, considering the expansion order as a free parameter. The key features of the developed formulation are: a) the condensation of the elec-

D

tric and magnetic description into the mechanical representation, based on the

TE

quasi-static electric-magnetic approximation, which allows to reduce the computation of the analysis for both layer-wise and equivalent single layer models; b) the employment of the Reissner Mixed Variational Theorem, in which the

EP

displacements and transverse stress components are used as primary variables, thus allowing the explicit fulfilment of the transverse stress interface equilibrium. The proposed methodology is assessed by generating different layer-wise

AC C

and equivalent single layer models for the analysis of thick magneto-electroelastic layer and comparing their solution against available three-dimensional analytic results.

Keywords: smart plates, magneto-electro-elastic materials, advanced plate theories ∗

Corresponding author Email addresses: [email protected] (I. Benedetti), [email protected] (A. Milazzo)

Preprint submitted to Journal Name

March 7, 2017

ACCEPTED MANUSCRIPT

2

1. Introduction

RI PT

1

Multi-physics materials, such as piezo-electric (PZ) or magneto-electro-elastic (MEE) materials, are today largely employed in several scientific and techno-

4

logical applications, as the coupling between different fields, e.g. mechanical,

5

electric, magnetic, can be advantageously used for different purposes, such as

6

structural health monitoring (SHM), vibration control, structural morphing, en-

7

ergy harvesting [1]. In aerospace and mechanical engineering applications, the

8

employment of piezo-electric or MEE materials has led to the concept of smart

9

structure, in which the structural components are endowed with multifunctional

10

sensing or acting capabilities. In this framework, the use of piezoelectric ma-

11

terials is today prevalent, while MEE materials are emerging as an interesting

12

alternative with extended functionalities, due to the possibility of exploiting the

13

additional coupling with the magnetic field [2]. Among the different possible

14

applications, much research has been focused on the development of compos-

15

ite laminates with embedded multi-functional layers, often referred to as smart

16

laminates [3–5].

17

One of the factors supporting the successful design of smart structural systems is

18

the availability of reliable modelling tools, essential for supporting experimental

19

testing and reducing its cost. In the case of piezoelectric or MEE materials, the

20

models must accurately represent the complex interactions among the different

21

fields. Several analytical, numerical and computational techniques have been

22

developed for the study of systems involving attached or embedded piezoelec-

23

tric and MEE elements [6–19]). The present work focuses of the modelling of

24

smart composite plates with general lay-ups of MEE layers.

25

Apart from classical models worthy for thin, single-layer configurations, e.g.

26

[20], several two-dimensional (2D) multilayered smart plate advanced theories

27

have been developed, with the aim of enabling suitable accuracy at affordable

28

computational cost. both layer-wise (LW) [21, 22] and equivalent-single layer

29

(ESL) [23, 24] approaches have been proposed in the literature. LW techniques

AC C

EP

TE

D

M AN U

SC

3

2

ACCEPTED MANUSCRIPT

offer high accuracy, but their computational cost increases with the number of

31

layers. On the other hand, the computational cost of ESL models is independent

32

from the number of layers, thus resulting in more affordable computations; how-

33

ever, their accuracy is generally reduced with respect to that obtained through

34

LW approaches, especially for thick laminates, whose reliable analysis generally

35

requires higher order theories.

36

Both LW and ESL theories for smart plates have been generally formulated

37

using the electric and magnetic primary quantities as independent state vari-

38

ables of the problem, see Ref. [25] and references therein; recently, models for

39

smart MEE laminates have been developed employing the concept of effective

40

plate, resulting from the condensation of the electric and magnetic state into

41

the mechanical variables [26–28]. Such theories are usually based on generalised

42

variational statements extending the application of the variational theorems of

43

classical elasticity [29, 30]. Most of the proposed models are based on the ex-

44

tension of the principle of virtual displacements (PVD), where the generalised

45

displacement components are seen as primary variables and the in-plane stress

46

components are calculated using Hooke’s law. While these theories might pro-

47

duce accurate predictions for the displacement and in-plane stress components,

48

they might fail to yield the same accuracy for the transverse shear and normal

49

stresses. To overcome this drawback, the Reissner mixed variational theorem

50

(RMVT) can be invoked [31, 32], where the displacement and transverse stress

51

components are employed as primary variables and the in-plane stress compo-

52

nents are derived from these.

53

Starting from the PVD and RMVT for the formulation and implementation of

54

both LW and ESL refined higher order theories for multilayered plates, Carrera

55

proposed a powerful approach known as Carrera Unified Formulation (CUF),

56

whose underlying ideas, principles and implementation issues for mechanical

57

problems is reviewed in Refs. [33] and [34]. The CUF offers a systematic

58

methodology for generating refined plate models, considering the order of the

59

theory as a free parameter of the formulation. The CUF has been used for

60

smart laminates with both piezoelectric, see Ref. [25] and references therein,

AC C

EP

TE

D

M AN U

SC

RI PT

30

3

ACCEPTED MANUSCRIPT

and MEE layers [35–37] .

62

By using the unified formulation and the generalised PVD for smart materials,

63

Milazzo and co-workers presented families of ESL and LW models for MEE mul-

64

tilayered plates involving only mechanical kinematical variables, as the electric

65

and magnetic state is preliminarily condensed into the mechanical description

66

[38, 39]. The formulation has been extended to geometric non-linearities [40]

67

and finite elements have been formulated based on the proposed models [41].

68

Starting from the sketched context, in the present work, families of ESL and

69

LW models for MEE multilayered plates, with the condensation of the electric

70

and magnetic fields into the mechanical description, are developed. The nov-

71

elty of the formulation consists in the use of the generalised RMVT variational

72

statement in the framework of the CUF, which enhances the model reliability

73

through the fulfilment of transverse stress interface continuity.

M AN U

SC

RI PT

61

74

2. Basic equations

D

75

Let us consider a multilayered plate comprising N homogeneous orthotropic

77

MEE layers, all with poling direction along the plate thickness. The plate is

78

referred to a three-dimensional coordinate system, with the x1 and x2 axes lying

79

on the plate mid-plane Ω, bounded by the curve ∂Ω, and the x3 axis directed

80

along the plate thickness (see Fig. ??).

EP

TE

76

In the following, we will refer to vector and tensor components directed

82

along x1 and x2 as in-plane components and to x3 components as out-of-plane

AC C

81

83

components. The plate may be generally subjected to mechanical loads and to

84

electric and magnetic actions applied on the top and bottom surfaces.

85

2.1. Gradient equations

86

87

The mechanical gradient equations, namely the strain-displacement rela-

tions, are ε = Dp u,

γ = Dn u + Dx3 u

4

(1)

ACCEPTED MANUSCRIPT

88

89

n o n o where uT = u1 u2 u3 is the displacement vector and εT = ε11 ε22 ε12 n o and γ T = ε31 ε32 ε33 are the vectors collecting the in-plane and out-ofplane components of the strain field, respectively.

91

The differential operators appearing in Eq. (1) are defined as   0 0 ∂1     Dn = 0 0 ∂2  ,   0 0 0

Dx3

  ∂3 0 0     =  0 ∂3 0  = Di ∂3   0 0 ∂3 (2)

SC

  ∂1 0 0     Dp =  0 ∂2 0 ,   ∂2 ∂1 0

RI PT

90

where ∂i = ∂(·)/∂xi and Di is the 3 × 3 identity matrix.

M AN U

As the elastic waves propagate several order of magnitude slower than the electromagnetic ones, the approximation of quasi-static electro-magnetic behaviour is assumed: introducing the electric potential Φ and the magnetic scalar potential Ψ , the electric and magnetic gradient equations are written as

94

95

H p = −∇p Ψ,

H n = −∇n Ψ,

(4)

D

(3)

n o n o T E1 E2 and H p = H1 H2 are the vector collecting the inn o plane components of the electric field E and the magnetic field H, E Tn = E3 n o and H Tn = H3 are the vectors of the corresponding out-of-plane components where E Tp =

and

TE

93

E n = −∇n Φ,

EP

92

E p = −∇p Φ,

∂2

iT

,

AC C

h ∇p = ∂1

5

h i ∇n = ∂3 .

(5)

ACCEPTED MANUSCRIPT

98

The constitutive law for an orthotropic MEE composite with electric and magnetic poling directions parallel to the x3 −axis is compactly written as      σ              τ        D  



C pn

0

C nn −epn

T

−enp −enn

T

−q np

0 −q pn

T

T

−q nn T

epn

pp

0

dpp

0

enn

0

nn

0

dnn

q pn

dpp

0

µpp

0

q nn

0

dnn

0

µnn

     ε              γ           Ep      En             H p        H   n

(6)

M AN U

C  pp   C np    0 p =     enp  Dn                0  B p         B  q np n

T

RI PT

97

2.2. Constitutive equations

SC

96

99

where C ij are matrix blocks containing the elastic stiffness coefficients, the

100

ij and µij blocks collect the dielectric constants and magnetic permeabilities

103

104

105

106

D

102

respectively, whereas eij , q ij and dij collect the piezo-electric, piezo-magnetic n o and magneto-electric coupling coefficients. Consistently: σ T = σ11 σ22 σ21 n o and τ T = σ31 σ32 σ33 are vectors collecting the in-plane and out-of-plane n o n o components of stress respectively; D Tp = D1 D2 and D Tn = D3 collect the in-plane and out-of-plane components of the electric displacements; B Tp = n o n o T B1 B2 and B n = B3 collect the in-plane and out-of-plane components of

TE

101

the magnetic induction vector. It is worth noting that, for multilayered plates,

108

the constitutive matrices depend on the x3 coordinate. In the present work,

109

they are considered constant within each laminate layer. For the application

110

of variational theorems to the MEE problem, mixed forms of the constitutive

111

equation are often invoked [42, 43]; in the present formulation, employing the

AC C

EP

107

112

extension of the Reissner Mixed Variational Theorem (RMVT) [31, 32], the

6

ACCEPTED MANUSCRIPT

following mixed form is used              



C σε

   C γε    0 p =      C Dn ε Dn              Bp    0           Bn  C Bn ε

C στ

0

C γτ

C γEp

C σEn

0

0

C Dp Hp

0

C Dn En

0

C Dn Hn

0

C Bp Hp

0

C Bn En

0

C Bn Hn

C Bp τ C Bp Ep C Bn τ

0

C σHn

C γn En C γn Hp C γn Hn

C Dp τ C Dp Ep C Dn τ

0

     ε              τ           Ep   ,    En           H p          H   n

RI PT

   σ       γ     D

(7)

SC

113

where the involved matrices have the form given in Appendix A.

115

2.3. Governing equations

M AN U

114

The governing equations for a MEE body with volume V bounded by the

117

frontier ∂V are obtained by generalising the Reissner Mixed Variational Prin-

118

ciple (RMTV). Assuming that the gradient and constitutive equations are sat-

119

isfied, it states that the solution of the MEE problem is given in terms of the

120

displacement field u, the out-of-plane stress field τ , the electric potential Φ and

121

the magnetic potential Ψ, i.e. the primary variables, which make stationary the

122

functional

TE

D

116

Z


εT σ + τ T γ − E Tp D p − E Tn D n − H Tp B p − H Tn B n dV + V Z Z ˆ ) dV − + τ T (γ − γ ρuT u ¨ dV + V V Z Z

¯ d∂V − uT f¯ − Φ¯ q dV − uT t¯ − ΦQ

EP

Π=

∂V

AC C

V

(8)

123

where t¯ are the applied surface tractions, f¯ are body forces, ρ is the mass

124

¯ and q¯ are the surface and body free electric charge densities and density, Q

125

the overdot denotes the derivative with respect to the time t. In Eq. (8) the

126

superimposed hat denotes quantities evaluated via the constitutive equations

127

whereas the other quantities depending on the primary variables are evaluated

128

via the gradient equations. Taking the first variation of Eq. (8) and integrating

129

by parts, through standard calculus of variations, one obtains:

7

ACCEPTED MANUSCRIPT

131

i) The equilibrium equations and the Gauss’ laws for electro-statics and magnetostatics, holding in the volume V δu :

RI PT

130

DTp σ + DTn τ + DTx3 τ + f¯ − ρ¨ u=0

(9a)

∇Tp D p + ∇n D n + q¯ = 0

(9b)

132

δΦ : 133

∇Tp B p + ∇n B n = 0

(9c)

SC

δΨ :

(the conjugated varied fields are indicated for the sake of completeness).

135

ii) The compatibility equation for the out-of-plane strains holding in the volume

136

V

M AN U

134

ˆ=0 γ−γ

δτ :

iii) The natural boundary conditions holding on the boundary ∂V eTσ + D eTτ + D e T τ = t¯ D p n x3

(11a)

D

137

(10)

δΦ :

e T Dp + ∇ e n Dn = Q ¯ ∇ p

(11b)

δΨ :

e nBn = 0 e T Bp + ∇ ∇ p

(11c)

δu : 138

TE

139

e (·) and ∇ e (·) are obtained from the differenIn Eqs.(11) the boundary operators D

141

tial operators D(·) and ∇(·) by substituting the partial derivatives ∂i = ∂(·)/∂xi

142

with the corresponding boundary outward direction cosines ni .

AC C

EP

140

143

3. Formulation

144

To develop the advanced refined theories for multilayered smart plates pro-

145

posed in the present paper, extensive use of the indicial notation is adopted,

146

according to the following specifications. A superscript hki is used to denote

147

quantities associated with the k-th layer of the laminate. Unless otherwise spec-

148

ified, it is assumed that α, β ∈ {0, 1, . . . , Mu } where Mu + 1 is the number of

149

terms of the displacement expansion, λ, µ ∈ {0, 1, . . . , Mτ } where Mτ + 1 is the 8

ACCEPTED MANUSCRIPT

151

number of terms of the transverse stress expansion, k, i ∈ {1, 2, . . . , N } where X XX N is the number of layers in the laminate. Finally, the notation = is

152

used to indicate multiple summation with respect to the indexes r and s raging

153

over the associated standard set.

154

3.1. Mechanical variables modeling

150

r

s

RI PT

r,s

Within each layer k of the laminate, the displacement field and the out-of-

156

plane stresses are modelled as an expansion along the plate thickness of so-called

157

thickness functions Fα (x3 ) and F˜α (x3 ), respectively [33, 34]. One writes Mu X

M AN U

uhki (x1 , x2 , x3 ) =

SC

155

hki uhki α (x1 , x2 )Fα (x3 ) =

τ hki (x1 , x2 , x3 ) =

Mτ X

Fαhki uhki α

(12a)

hki hki F˜λ τ λ

(12b)

α

α=0 158

X

hki

hki

τ λ (x1 , x2 )F˜λ (x3 ) =

λ=0

X λ

159

Using the gradient relationships, Eqs.(1), the corresponding mechanical strain

160

field is written as

X

Fαhki Dp uhki α

D

εhki =

(13a)

161

TE

α

γ hki =

X α

! uhki α

(13b)

3.2. Electro-magnetic variables modeling

EP

162

hki

∂Fα Di Fαhki Dn + ∂x3

The two-dimensional nature of plates allows simplifying assumptions on the

164

electromagnetic behaviour of smart plates [38]. For both thin and thick plates,

AC C

163

165

the in-plane derivatives of the in-plane components of the electric displacement

166

and magnetic induction can be assumed negligible with respect to the out-

167

of-plane components. Thus in particular it holds ∂Dj /∂xj = ∂Bj /∂xj = 0

168

for j = 1, 2 and using Eqs.(7), (1), (3) and (4) into the Gauss’ equations for

169

electrostatics and magnetostatics, namely Eqs. (9b) and (9c), one infers the

9

ACCEPTED MANUSCRIPT

170

following system of equations holding for each k-th layer

RI PT

 hki hki ∂ 2 Ψhki ∂ 2 Φhki  hki hki ∂ε hki ∂τ hki    C Dn En ∂x2 + C Dn Hn ∂x2 = C Dn ε ∂x + C Dn τ ∂x 3 3 3 3 2 hki 2 hki hki hki  ∂ Φ ∂ Ψ ∂ε ∂τ   + C Bn Hn = C Bn ε + C Bn τ  C Bn En ∂x23 ∂x23 ∂x3 ∂x3

(14)

Taking the mechanical variable approximation into account, namely Eqs. (12b)

172

and (13a), one obtains

SC

171

X ∂Fαhki hki X ∂ F˜ hki hki hki ∂ 2 Φhki hki λ = A D u + A Φτ τ λ p α Φε ∂x23 ∂x ∂x 3 3 α

M AN U

λ

173

(15a)

hki

hki

X ∂ F˜ X ∂Fα ∂ 2 Ψhki hki hki hki λ = A Ψε Dp uhki A Ψτ τ λ α + 2 ∂x3 ∂x ∂x 3 3 α

(15b)

λ

174

where for r ∈ {ε, τ }

Integrating twice Eqs.(15) gives Φhki =

X

176

hki

hki Ghki α A Φε Dp uα +

EP

α

TE

175

(16)

D

−1      hki hki hki hki C Dn r A Φr C Dn En C Dn Hn     = hki hki hki hki C Bn En C Bn Hn C Bn r A Ψr

Ψhki =

X

hki

hki Ghki α A Ψε Dp uα +

α

Z

X

˜ hkiA hki τ hki + x3 ahki + bhki G Φτ λ Φ Φ λ

(17a)

˜ hkiA hki τ hki + x3 ahki + bhki G Ψτ λ Ψ Ψ λ

(17b)

λ

X λ

˜ hki (z) = Fαhki (z)dz and G λ

Z

hki F˜λ (z)dz. The integration

where Ghki α (z) =

178

constants aΦ , bΦ , aΨ , and bΨ are determined by enforcing the electric and

179

magnetic continuity conditions at the N −1 layers interfaces and external electric

180

and magnetic boundary conditions at the top and bottom faces of the laminate.

181

The electric and magnetic interface conditions are given by the continuity of

182

the electric and magnetic potentials, normal electric displacement and mag-

183

netic induction; the electric and magnetic boundary conditions at the upper

AC C 177

hki

hki

hki

hki

10

ACCEPTED MANUSCRIPT

and bottom laminate surfaces consist of prescribed potentials or normal electric

185

displacement and magnetic induction, whose values are collected in the vector

186

ξ. According with the form of enforced electric and magnetic interface and

187

boundary conditions, a system of equations of the form Ax=

X

B iα Dp uhii α +

i,α

X

RI PT

184

hii C iλ τ λ + W ξ

i,λ

(18)

is obtained, where x is a vector collecting the layers integration constants and

189

A , B iα , C iλ and W are matrices involving the layers material properties and

190

thickness functions only. Details on the procedure used to derive Eq.(18) are

191

given in Ref.[39]; they are not reported in the present paper for the sake of

192

conciseness. Therefore, the integration constants can be compactly written in

193

the general form hki

zΛ =

M AN U

SC

188

X hkii X hkii hii hki z Λε α Dp uhii z Λτ λ τ λ + z Λξ ξ α + i,α

(19)

i,λ

where the symbol z stands for a or b and the symbol Λ stands for Φ or Ψ , so

195

that Eq.(19) holds for aΦ , bΦ , aΨ , and bΨ . It is worth remarking that in

196

Eq.(19) the (1 × 3) row arrays z Λε α and z Λτ λ collect the coefficients expressing

197

the effects on the k-th layer’s potential Λ of the i-th layer strains terms Dp uhii α

198

and transverse stresses terms τ hii α . Finally, the (1 × 4) row array z Λξ collects

199

the coefficients expressing the influence on the potential Λ of the electric and

200

magnetic boundary conditions ξ. In conclusion, the electric and magnetic state

201

is described by the following expression of the potentials

D

194

hki

hki

hki

hkii

AC C

EP

TE

hkii

hki

Φhki =

X

hkii A˜ Φε α Dp uhii α +

i,α

X

hkii hii hki A˜ Φτ λ τ λ + A˜ Φξ ξ

(20a)

hkii hii hki A˜ Ψτ λ τ λ + A˜ Ψξ ξ

(20b)

i,λ

202

Ψ hki =

X

hkii A˜ Ψε α Dp uhii α +

i,α

X i,λ

11

ACCEPTED MANUSCRIPT

203

where, for Λ = Φ, Ψ , the following relations hold hkii hki hkii hkii A˜ Λε α = GαA Λε δki + x3 aΛε α + bΛε α

RI PT

(21)

204

hkii ˜ λA hki δki + x3 ahkii + bhkii A˜ Λτ λ = G Λτ Λτ λ Λτ λ

(22)

205

hki hki hki A˜ Λξ = x3 aΛξ + bΛξ

(23)

By using Eqs.(3) and (4), the expressions for the in- and out-of-plane compo-

207

nents of the electric and magnetic fields are obtained as

hii o hii Xn hkii ∂Dp uα hkii ∂Dp uα + I 2A˜ Φε α − I 1A˜ Φε α ∂x1 ∂x2 i,α

M AN U

Ep = −

SC

206

hii hii o Xn hkii ∂τ hkii ∂τ hki ∂ξ hki ∂ξ I 1A˜ Φτ λ λ + I 2A˜ Φτ λ λ − I 1A˜ Φξ − I 2A˜ Φξ ∂x1 ∂x2 ∂x1 ∂x2 i,λ

208

E hki n =−

hkii X ∂A˜ Φ α ε

i,α

∂x3

209

τ

i,λ

∂x3

hii

τλ −

(24a)

hki ∂A˜ Φξ

∂x3

ξ

(24b)

hii o hii Xn hkii ∂Dp uα hkii ∂Dp uα + I 2A˜ Ψε α − I 1A˜ Ψε α ∂x1 ∂x2 i,α

D

Hp = −

Dp uhii α −

X ∂A˜ hkii Φ λ

i,λ

X ∂A˜ hkii Ψ α

EP

210

TE

hii hii o Xn hkii ∂τ hkii ∂τ hki ∂ξ hki ∂ξ I 1A˜ Ψτ λ λ + I 2A˜ Ψτ λ λ − I 1A˜ Ψξ − I 2A˜ Ψξ ∂x1 ∂x2 ∂x1 ∂x2

H hki n =−

ε

i,α

∂x3

(24c)

Dp uhii α −

X ∂A˜ hkii Ψ λ τ

i,λ

∂x3

hii

τλ −

hki ∂ A˜ Ψξ

∂x3

ξ

(24d)

h

AC C

211

i h i where I T1 = 1 0 and I T2 = 0 1 .

212

3.3. Layer governing equations

213

Considering that the Gauss’ laws for electrostatics and magnetostatics are

214

fulfilled in their strong form by the expressions of Φ and Ψ determined in the

215

preceding section via Eqs.(14), the first variation of the functional Π reduces to

216

its mechanical part only. For the k-th layer, the stationarity condition is then

12

ACCEPTED MANUSCRIPT

Z

Z

Ωhki

Z

hk

δτ hki

T



 ˆ hki dx3 dΩ+ γ hki − γ

hk

T

ρhki δuhki u ¨hki dx3 dΩ−

hk−1

Z

Ωhki

Z

i T T δεhki σ hki + δγ hki τ hki dx3 dΩ+

hk−1

Z

Ωhki

Z

h

hk−1

Z

Ωhki

Z

hk

hk

T hki δuhki f¯ dx3 dΩ −

Ωhki

Z

∂Ωhki

hk−1



Z

T hki δuhki t¯x3



+ hk



T hki δuhki t¯x3



hk

T

hki δuhki t¯n dx3 d∂Ω−

SC

δΠhki =

RI PT

written as

hk−1



dΩ = 0

hk−1

M AN U

217

(25)

218

hki where the integration domain Ωhki ≡ Ω is the reference plane of the lamina, ¯tn

219

are the tractions applied on the layer lateral surfaces and ¯tx3 denotes tractions

220

acting on planes parallel to Ω. The notation f |z is used to indicate that the

221

h1i function f is evaluated at x3 = z. It is worth noting that t¯x3 |h0 =−H/2 and

222

hN i t¯x3 |hN =H/2 coincide with the external tractions applied over the plate bottom

223

and top surfaces, respectively.

224

The discrete form of Eq.(25) is obtained according to the following steps. First,

225

the constitutive relationships from Eqs.(7) are substituted into Eq.(25). In turn,

226

the gradient equations, namely Eqs.(13) and (24), are introduced and eventu-

227

ally the primary variables are expressed through their approximations, namely

228

Eqs.(12).

229

Integrating by parts the resulting expression, one obtains the RMVT stationar-

AC C

EP

TE

D

hki

13

ACCEPTED MANUSCRIPT

ity statement for the k-th layer that reads  Z

X

− Ωhki

T δuhki α

 X



α

u hkii hii δu Kαβ uβ

+

X

Z X

∂Ω

231

δuhki α

δuhki α

X

+

X

T

i,µ

T

hkki hki u ¨β dΩ δu Mαβ u

−

Z X Ω

β T hki δuhki Tα dΩ α

α

hki T ξ hki δτ Wλ ξdΩ+

δτ λ

λ

X

∂Ωhki

α

= DTp

τ hkii δε Qαµ

ε hkii δε Qαβ Dp

+ DTn

τ hkii δγ Qαµ

− δuτ Qhkii αµ

τ hkii δu Kαµ

= DTp

u hkii δτ Kλβ

=δτε Qλβ Dp + δτγ Qλβ Dn + δτu Qλβ +

233

∂ε ∂x1

hkii

hki δuα

T ξ f hki δu Wα ξ d∂Ω+

α

(26)

∂τ

hkii

hkii

hkki u δu Mαβ u e hkii δu Kαβ

∂τ ∂ hkii ∂ ∂x1 + δτ Qλµ ∂x1 ∂x2

hki

= J αβ

hkii

(27a) (27b)

hkii

∂ε ∂ hkii ∂ ∂x2 Dp + δτ Dp Qλβ ∂x1 ∂x2

TE

Qλβ

∂x1 = δττ Qλµ + δτ Qλµ

EP

hkii τ δτ Kλµ

hkii

D

hkii

δτ

236

T ξ hki δu Wα ξ dΩ+

where the so-called fundamental nuclei [33] are defined as

232

235

X Ωhki

δuhki α

h i hki T ¯ hki ¯ hki dΩ− δuα Fhki α + Pα + Pα

=0

α

u hkii δu Kαβ

234

Z +

i,µ

i,β

X

hkii hii  τ dΩ δτ Kλµ τ µ

  Z X u hkii hii X τ hkii hii e e  d∂Ω +  δu Kαµ τ µ δu Kαβ uβ +

α

α

u hkii hii δτ Kλµ uµ

i,β

X ∂Ωhki

Z

X 

λ

Z

Ω



M AN U

Ωhki

hki T

δτ λ

Ωhki

SC

X

X

dΩ −

i,µ

i,β

 Z

Z

τ hkii hii  δu Kαµ τ µ

RI PT

230

(27c)

(27d) (27e)

e T ε Q Dp =D p δε αβ

(27f)

e T τ Qhkii + D e T τ Qhkii =D p δε αµ n δγ αµ

(27g)

AC C

237

238

τ e hkii δu Kαµ

In Eq.(26), the electric and magnetic effective loading vectors are given by ξ hki δu Wα

= DTp δεξ Qhki α

(28a)

ξ f hki δu Wα

e T ξ Qhki =D p δε α

(28b)

239

14

ACCEPTED MANUSCRIPT

hki ξ δτ Wλ

241

∂ξ

hki

hki

∂x1 Qλ = δτξ Qλ + δτ

∂ξ ∂ hki ∂ ∂x2 + δτ Qλ ∂x1 ∂x2

and the mechanical loading vectors read as ¯ hki = F hki¯thki P x3 α α

242

(29a)

hk−1

¯ hki = F hki¯thki P x3 α α

(29b)

243

hk

Z

hκi

Fαhki f¯

hk−1 244

hk

dx3

M AN U

Z

SC

hk

Fhki α = Tαhki =

(28c)

RI PT

240

hki

Fαhki t¯ dx3

(29c)

(29d)

hk−1

245

The expressions of the smart layer’s effective characteristic matrices sr Qhpqi ως , ef-

246

fective inertia characteristics J hki ως and electric-magnetic effective loading char-

247

f acteristics sr Qhpi ω are given in Appendix B. In general, in the notation g B αβ ,

248

f and g indicate specific fields, α and β indicate specific terms of the expansions

249

of the above fields and m and n indicate specific layers within the laminate; in

250

particular, the expansion term α of the field g within the layer m and the expan-

251

sion term β of the field f within the layer n are being considered. Therefore, the

252

fundamental nucleus sr Khpqi ως accounts for the contribution of the expansion term

253

ω of the field r within the layer p to the work associated with the variation of the

254

term ς of the field s within the layer q. On the other hand, the electric-magnetic

255

loading kernel ξr Whpi ω accounts for the contribution of the electric-magnetic load-

256

ing term ξ within the layer p to the work associated with the variation of the

AC C

EP

TE

D

hmni

257

expansion term ω of the field r within the layer p itself.

258

Through standard calculus of variations on the stationarity statement given by

259

Eq.(26), the layer governing equations holding in Ωhki are eventually obtained

15

ACCEPTED MANUSCRIPT

260

as

RI PT

X X u hkii hii τ hkii hii   Kαβ uβ +  δu δu Kαµ τ µ +    i,µ i,β    X  hki hki u hki ¯ hki ¯ hki ¨β + δuξ Whki δu Mαβ u α ξ − Fα − Pα − Pα = 0 (30)  β     X X  hkii hii ξ  u hkii hii τ hki   δτ Kλβ uβ + δτ Kλµ τ µ + δτ Wα ξ = 0 

SC

i,µ

i,β

with α = 1, 2, ...., Mu and λ = 1, 2, ...., Mτ ; the associated natural boundary

262

conditions, holding on ∂Ωhki , are given by X

u b hkii hii δu Kαβ uβ

264

265

X

τ b hkii hii δu Kαµ τ µ

c hki ξ − T hki = 0. + δuξ W α α

(31)

i,µ

i,β

263

+

M AN U

261

4. Multilayered plate

The governing equations for multilayered plates are obtained by enforcing the stationarity condition of the functional Π for all the laminate layers

D

n X

δΠhki = 0

(32)

TE

k=1

with the additional requirements on the interface displacements continuity and

267

tractions equilibrium

EP

266

AC C

 hki hk+1i  u − u =0  hk hk  hki hk+1i  t¯ + t¯ =0 x3 x3 hk

k = 1, 2, . . . , (N − 1) (33) κ = 1, 2, . . . , (N − 1)

hk

268

Enforcing the stationarity and interface conditions in Eqs.(32-33) leads to the

269

plate resolving system ¨ K uu U + K uτ T = F u − M uu U K τ uU + K τ τ T = F τ

16

(34)

ACCEPTED MANUSCRIPT

270

where U and T are vectors collecting the the unknown coefficients of the dis-

271

placement and transverse stress expansions uhκi α and τ λ and the operator ma-

272

trices K rs (r, s = u, τ ), M uu and the loading vectors F r (r = u, τ ) are obtained

273

via assembly procedures of the fundamental nuclei, which are described in the

274

next sections.

275

4.1. Layer-wise theories

RI PT

hκi

hκi

SC

Assuming the primary variable expansion coefficients uhκi α and τ λ

as dis-

tinct for each individual layer leads to layer-wise plate models. In this case, a suitable choice of the thickness functions Fα is a linear combination of Legendre

hκi

F0

=

M AN U

polynomials Lα (ζk ) specified as

L0 (ζk ) − L1 (ζk ) 2

Fαhκi = Lα+1 (ζk ) − Lα−1 (ζk ) hκi

FM =

α = 1, ...(M − 1)

L0 (ζk ) + L1 (ζk ) 2

(35) (36) (37)

where ζk = (2x3 − hκ − hκ−1 ) / (hκ − hκ−1 ) (see Fig. ??). The same choice is

277

introduced for the transverse stress thickness functions F˜λ . Due to the prop-

278

erties of the Legendre polynomials, the displacements continuity and tractions

279

equilibrium at the layer interfaces reduce to the following relationships

TE

D

276

hk+1i

hki

hk+1i

EP

hki

uMu = u0

k = 1, . . . , (N − 1).

(38)

AC C

τ Mτ = τ 0

280

Taking Eqs.(38) into account, the unknown primary variables can be collected

281

into the global column vectors U , whose rows with indexes ranging from 3[(k −

282

1)Mu +α+1]−2 to 3[(k−1)Mu +α+1] contain the coefficients uhki α , and T , whose

283

rows with indexes ranging from 3[(k − 1)Mτ + α + 1] − 2 to 3[(k − 1)Mτ + α + 1]

284

contain the coefficients τ hki α .

285

According with this global indexing of the unknown coefficients, the resolving

286

system matrices K rs are built by summing the fundamental nuclei δrs Khkii ως at the

17

ACCEPTED MANUSCRIPT

rows with indexes ranging from 3[(k−1)Mr +ω+1]−2 to 3[(k−1)Mr +ω+1] and

288

the columns with indexes ranging from 3[(i−1)Ms +ς+1]−2 to 3[(i−1)Ms +ς+1].

289

Similarly, the matrix M uu is built by summing the fundamental nuclei δuu Mhkii ως

290

at the rows with indexes ranging from 3[(k − 1)Mu + ω + 1] − 2 to 3[(k − 1)Mu +

291

ω +1] and the columns with indexes ranging from 3[(i−1)Mu +ς +1]−2 to 3[(i−

292

1)Mu + ς + 1]. The right-hand-side vector F u is built by: a) summing the nuclei

293

ξ hki δu Wω

294

¯ h1i to 3[(k − 1)Mu + ω + 1]; b) summing at the rows from 1 to 3 the nucleus P 0

295

h1i accounting for the external mechanical load ¯tx3 |−h/2 applied at the laminate

296

bottom surface; c) summing at the rows from 3(N Mu + 1) − 2 to 3(N Mu + 1)

297

¯ hN i accounting for the external mechanical load ¯thN i |h/2 applied the nucleus P x3 Mu

298

at the laminate top surface Finally, the right-hand-side vector F τ is built by

299

summing the nuclei

300

3[(k−1)Mr +ω+1]. It is worth noting that ¯tx3 |−h/2 = τ 0 and ¯tx3 |h/2 = τ Mτ ;

301

this represents a constraint for the transverse stress expansion coefficients, which

302

can be forced in the resolving system through penalty techniques.

RI PT

287

ξ hki δτ Wω

M AN U

SC

and the domain load terms F hki ω at the rows from 3[(k −1)Mu +ω +1]−2

at the rows from from 3[(k − 1)Mr + ω + 1] − 2 to h1i

hN i

4.2. Equivalent-single-layer theories

TE

304

hN i

D

303

h1i

For the equivalent-single-layer plate models, the primary variable expansion

306

coefficients uhκi = uα are assumed common to the all of the layers whereas α

307

the expansion coefficients τ λ are again assumed as distinct for each individual

308

layer. In this case, a suitable choice of the thickness functions Fαhki is provided

309

by the adoption of Taylor polynomials

AC C

EP

305

hκi

Fαhκi = xα 3,

α = 0, ..., Mu .

(39)

310

hki The thickness functions F˜λ are defined by Eq.(35). Due to the thickness

311

functions selections, the displacements continuity and tractions equilibrium at

18

ACCEPTED MANUSCRIPT

the layer interfaces reduce to the following relationships hk+1i uhki α = uα

with

hk+1i

hki

α = 0, . . . , Mu

(40)

RI PT

312

k = 1, . . . , (N − 1)

τ Mτ = τ 0

The displacement expansion coefficients are then collected within the global col-

314

umn vector U , whose rows with indexes ranging from 3[α + 1] − 2 to 3[α + 1]

315

contain the coefficients uα and the transverse stress expansion coefficients are

316

collected within the vector T , whose rows with indexes ranging from 3[(k −

317

1)Mτ + α + 1] − 2 to 3[(k − 1)Mτ + α + 1] contain τ hki α .

318

According with this global indexing of the unknown coefficients the resolving

319

system matrices are built as follows. The matrices K uu and M uu are built by

320

u hkki summing the fundamental nuclei δuu Khkii ως and δu Mως , respectively, at the rows

321

with indexes ranging from 3[ω + 1] − 2 to 3[ω + 1] and the columns with indexes

322

ranging from 3[ς + 1] − 2 to 3[ς + 1].

323

The matrix K uτ is built by summing the fundamental nuclei

324

rows with indexes ranging from 3[(ω + 1] − 2 to 3[ω + 1] and the columns with

325

indexes ranging from 3[(i − 1)Mτ + ς + 1] − 2 to 3[(i − 1)Mτ + ς + 1].

326

The matrix K τ u is built by summing the fundamental nuclei

327

rows with indexes ranging from 3[(k − 1)Mτ + ω + 1] − 2 to 3[(k − 1)Mτ + ω + 1]

328

and the columns with indexes ranging from 3[ς + 1] − 2 to 3[ς + 1].

329

The matrix K τ τ is built by summing the fundamental nuclei δττ Khkii ως at the rows

330

with indexes ranging from 3[(k −1)Mτ +ω +1]−2 to 3[(k −1)Mτ +ω +1] and the

331

columns with indexes ranging from 3[(i−1)Mτ +ς +1]−2 to 3[(i−1)Mτ +ς +1].

AC C 332

333

334

τ hkii δu Kως

at the

u hkii δτ Kως

at the

EP

TE

D

M AN U

SC

313

The right-hand-side vector F u is built by summing the nuclei

ξ hki δu Wω ,

main load terms F hki and the external surface load characteristics ω hN i ¯ P ω at the rows with indexes ranging from 3[ω + 1] − 2 to 3[ω + 1].

the do-

¯ h1i P ω

and

335

Finally, the right-hand-side vector F τ is built by summing the nuclei δτξ Whki ω at

336

the rows with indexes ranging from 3[(k−1)Mr +ω+1]−2 to 3[(k−1)Mr +ω+1].

337

Also in this case, the transverse stress expansion coefficients constraints, ex-

338

pressed by ¯tx3 |−h/2 = τ 0

h1i

h1i

hN i

hN i

and ¯tx3 |h/2 = τ Mτ , have to be be enforced.

19

ACCEPTED MANUSCRIPT

339

5. Solution and results

RI PT

340

Let us consider a simply-supported rectangular plate with MEE layers with

342

principal material axes aligned with the plate reference system. In this case,

343

a closed form solution can be found assuming the following expressions for the

344

primary variable expansion terms              

 XX mπ nπ hki  σ31 λ mn cos x1 sin x2   L Ly  x  m n         XX nπ mπ hki x1 cos x2 σ32 λ mn sin = L Ly x  m n         XX  mπ nπ hki   σ33 λ mn sin x1 sin x2  Lx Ly m n

             

D

EP

TE

hki

τλ

M AN U

 XX mπ nπ  u1 hki x1 sin x2  α mn cos  Lx Ly   m n        XX  nπ mπ x1 cos x2 u2 hki = α mn sin L Ly x  m n         XX  mπ nπ   u3 hki x1 sin x2  α mn sin L Ly x m n

uhki α

345

SC

341

exp

√
−1$t ,

(41a)

            

exp

√


−1$t .

(41b)

            

Substituting Eqs.(41) into the plate governing equations and applying the Bubnov-

347

Galerkin method, an algebraic resolving system is obtained for the unknown

AC C

346

hki

348

coefficients uj hki α mn and σ3j λ mn associated with each of the harmonic terms of

349

the primary variables approximation series. In the present work, representative

350

results obtained using the mentioned closed-form solution are presented and

351

discussed. Different LW and ESL theories have been implemented considering

352

variable kinematics with through-the-thickness power expansion orders up to

353

the fourth.

354

Results are presented for smart MEE multilayered plates with plies of piezo-

20

ACCEPTED MANUSCRIPT

electric barium titanate BaT iO3 and piezomagnetic cobalt ferrite CoF e2 O4 ,

356

shortly denoted as B and F respectively, whose material properties can be

357

found in Ref.[26]. Attention is devoted to thick plates, as these allow to high-

358

light the possible lack of accuracy and the effectiveness of 2D plate theories.

359

Thus, simply-supported square plates with side length Lx = 1m, total thickness

360

H = 0.3m and equal thickness layers B/F/B and F/B/F stacking sequences

361

have been investigated as representative problem. However it is worth mention-

362

ing that the scheme has been used to analyse different lay-ups, obtaining similar

363

results.

364

5.1. Static analysis under mechanical load

SC

M AN U

365

RI PT

355

The plates undergo a bi-sinusoidal pressure p3 = p¯3 sin (πx1 /Lx ) sin (πx2 /Lx ) applied over the top surface with peak p¯3 = 1N/m2 . The electric and magnetic

367

potentials are set to zero on both the top and bottom plate surfaces, which cor-

368

respond to the closed circuit condition. For this problem the exact 3D solution

369

can be evaluated by the approach given in Ref.[6]. First, a set of numerical

370

experiments have been performed by selecting transverse stress expansions of

371

order greater than, equal to and lower than that adopted for the displacements

372

expansions. Fig.?? and Fig.?? show representative results for these analyses

373

highlighting how the choice of the same expansion order for both displacements

374

and transverse stresses provides reliable results for all the investigated theories.

375

Indeed, it is observed that, for LW models, selecting for the transverse stresses

376

expansions an order lower than that selected for the displacements provides

377

good results for higher order theories only. On the other hand, for ESL mod-

AC C

EP

TE

D

366

378

els, selecting for the transverse stresses expansions an order greater than that

379

selected for the displacements provides good results for higher order theories

380

only.

381

To appraise the approach accuracy, Fig.?? and Fig.?? show the through-the-

382

thickness distribution of representative mechanical and electric and magnetic

383

quantities at the point of coordinates x1 = x2 = 0.25Lx for different plate mod-

384

els and their comparison with the 3D exact solution. These results have been 21

ACCEPTED MANUSCRIPT

obtained using the same expansion order for both displacements and transverse

386

stresses. As expected, better results are obtained employing higher order layer-

387

wise models, which provide accurate distributions of mechanical quantities and

388

electric potential, whereas a slight loss of accuracy with respect to the analytic

389

solution is shown for the magnetic potentials. Higher order ESL models can give

390

satisfactory results and also in this case the magnetic potential results slightly

391

overestimated with respect to the analytic solution.

392

5.2. Static analysis under electromagnetic load

SC

RI PT

385

A plate with H/Lx = 0.1 and the same lay-up as that considered in the

394

previous example, subjected to a bi-sinusoidal electric voltage with peak ∆Φ =

395

100V , has been analysed; the voltage is symmetrically applied between the

396

upper and bottom surfaces and zero magnetic potential is assumed over both

397

the top and bottom plate surfaces. In this case the plate can be considered

398

as an actuator under the action of the external electric excitation. Results for

399

representative quantities of the plate response are shown in Fig.?? and Fig.??

400

for layer-wise and equivalent-single-layer theories, respectively. The analytic

401

3D solution, computed using the approach given in Ref.[6], is also shown. It is

402

observed that the obtained results are generally in good agreement with those

403

of the 3D solution with a slight overestimation of the magnetic quantities. As

404

expected, also in this case, layer-wise theories generally deliver more reliable

405

results with respect to those produced by equivalent-single-layer models and

406

they are adequately accurate if higher order expansions are employed.

AC C

EP

TE

D

M AN U

393

407

5.3. Free vibrations

408

Tables ?? and ?? list the first five natural frequency $ of the B/F/B and

409

F/B/F simply-supported square plates related to spatial modes with m =

411

n = 1 in Eq. (41). The plate circular frequencies are normalised as ω ¯ = p $Lx ρmax /Cmax , where Cmax and ρmax are the maximum values of the elas-

412

tic coefficient and density among the materials of the plate layers. Results are

413

presented for both LW and ESL models, which are labeled by the acronyms

410

22

ACCEPTED MANUSCRIPT

LW n and ESLn being n the order of the employed thickness expansions, se-

415

lected with the same order for both displacements and transverse stresses. Two

416

different aspect ratios, namely H/Lx = 0.1 and H/Lx = 0.3, are investigated

417

and closed circuit conditions, i.e. zero electric and magnetic potentials on both

418

top and bottom plate surfaces, are enforced. Results are compared with those

419

produced using the 3D approach given in Ref. [7]. The agreement between the

420

two different solutions confirms the accuracy of the proposed approach. Both

421

LW and ESL models give reliable natural frequencies and little or no difference

422

is found between the predictions of the two techniques, which can be motivated

423

by the fact that natural frequencies characterise the global behaviour of the

424

plate that is not affected by slight differences in the local fields.

425

6. Conclusions

M AN U

SC

RI PT

414

In the present work, advanced multilayered models for smart plates have

427

been developed in the framework of an axiomatic unified formulation where the

428

order of the through-the-thickness expansion is assumed as a model parameter

429

at layer level. The proposed models are based on the condensation of the electric

430

and magnetic states into the plate mechanics, performed via the strong form of

431

the electrostatics and magnetostatics governing equations. Consequently, the

432

plate governing equations are inferred by using the generalised Reissner Mixed

433

Variational Theorem. The resulting layer model corresponds to an effective

434

plate, with mechanical behaviour only, described in terms of suitable stiffness,

435

inertia and loading characteristics that take the magneto-electro-elastic coupling

AC C

EP

TE

D

426

436

into account. The layers equations are then coupled through the interface con-

437

ditions that provide displacement and transverse stress continuity, which can

438

be explicitly enforced thanks to the use of RMVT. The method naturally trans-

439

lates into through-the-thickness algorithmic assembly procedures that allow to

440

build models based on both equivalent single layer and layer-wise kinematics.

441

The approach has been validated against solutions available in the literature for

442

static problems and free vibrations. The obtained results confirm good accuracy

23

ACCEPTED MANUSCRIPT

and the potential of the approach for the construction of general smart laminate

444

models.

445

7. References

446

447

RI PT

443

[1] I. Chopra, Review of state of art of smart structures and integrated systems, AIAA Journal 40 (11) (2002) 2145–2187.

[2] C.-W. Nan, M. Bichurin, S. Dong, D. Viehland, G. Srinivasan, Multifer-

449

roic magnetoelectric composites: Historical perspective, status, and future

450

directions, Journal of Applied Physics 103 (3).

M AN U

SC

448

451

[3] S. V. Gopinathan, V. V. Varadan, V. K. Varadan, A review and critique of

452

theories for piezoelectric laminates, Smart Materials and Structures 9 (1)

453

(2000) 24.

[4] J. Zhai, Z. Xing, S. Dong, J. Li, D. Viehland, Magnetoelectric laminate

455

composites: An overview, Journal of the American Ceramic Society 91 (2)

456

(2008) 351–358.

458

[5] S. Kapuria, P. Kumari, N. J.K., Efficient modeling of smart piezoelectric

TE

457

D

454

composite laminates: a review, Acta Mechanica 214 (2010) 31–48. [6] E. Pan, Exact solution for simply supported and multilayered magneto-

460

electro-elastic plates, Journal of Applied Mechanics, Transactions ASME

461

68 (4) (2001) 608–618.

EP

459

[7] E. Pan, P. R. Heyliger, Free vibrations of simply supported and multilay-

463

ered magneto-electro-elastic plates, Journal of Sound and Vibration 252 (3)

464

(2002) 429 – 442.

AC C 462

465

[8] J. Wang, L. Chen, S. Fang, State vector approach to analysis of multi-

466

layered magneto-electro-elastic plates, International Journal of Solids and

467

Structures 40 (7) (2003) 1669–1680.

24

ACCEPTED MANUSCRIPT

[9] R. Bhangale, N. Ganesan, Static analysis of simply supported functionally

469

graded and layered magneto-electro-elastic plates, International Journal of

470

Solids and Structures 43 (10) (2006) 3230–3253.

RI PT

468

471

[10] A. Milazzo, I. Benedetti, C. Orlando, Boundary element method for mag-

472

neto electro elastic laminates, CMES - Computer Modeling in Engineering

473

and Sciences 15 (1) (2006) 17–30.

[11] C.-P. Wu, Y.-C. Lu, A modified pagano method for the 3d dynamic re-

475

sponses of functionally graded magneto-electro-elastic plates, Composite

476

Structures 90 (3) (2009) 363–372.

M AN U

SC

474

477

[12] I. Benedetti, A. Milazzo, M. Aliabadi, Structures with surface-bonded pzt

478

piezoelectric patches: a bem investigation into the strain-transfer mecha-

479

nism for shm applications, SDHM: Structural Durability & Health Moni-

480

toring 5 (3) (2009) 251–274.

[13] I. Benedetti, M. Aliabadi, A. Milazzo, A fast {BEM} for the analysis of

482

damaged structures with bonded piezoelectric sensors, Computer Methods

483

in Applied Mechanics and Engineering 199 (912) (2010) 490 – 501.

D

481

[14] G. Dav´ı, A. Milazzo, A regular variational boundary model for free vi-

485

brations of magneto-electro-elastic structures, Engineering Analysis with

486

Boundary Elements 35 (3) (2011) 303–312.

EP

TE

484

487

[15] M.-F. Liu, An exact deformation analysis for the magneto-electro-elastic

488

fiber-reinforced thin plate, Applied Mathematical Modelling 35 (5) (2011) 2443–2461.

AC C 489

490

[16] Z. Yifeng, C. Lei, W. Yu, Z. Xiaopin, Z. Liangliang, Asymptotical construc-

491

tion of a reissner-like model for multilayer functionally graded magneto-

492

electro-elastic plates, Composite Structures 96 (2013) 786–798.

493

[17] D. Sladek, V. Sladek, S. Krahulec, E. Pan, The mlpg analyses of large de-

494

flections of magnetoelectroelastic plates, Engineering Analysis with Bound-

495

ary Elements 37 (2013) 673–682. 25

ACCEPTED MANUSCRIPT

[18] J. Liu, P. Zhang, G. Lin, W. Wang, S. Lu, Solutions for the magneto-

497

electro-elastic plate using the scaled boundary finite element method, En-

498

gineering Analysis with Boundary Elements 68 (2016) 103–114.

RI PT

496

499

[19] F. Zou, I. Benedetti, M. H. Aliabadi, A boundary element model for struc-

500

tural health monitoring using piezoelectric transducers, Smart Materials

501

and Structures 23 (1) 015022.

[20] A. Apuzzo, R. Barretta, R. Luciano, Some analytical solutions of func-

503

tionally graded kirchhoff plates, Composites Part B: Engineering 68 (2015)

504

266–269.

M AN U

SC

502

505

[21] R. Garcia Lage, C. M. Mota Soares, C. A. Mota Soares, J. N. Reddy,

506

Layerwise partial mixed finite element analysis of magneto-electro-elastic

507

plates, Computers & Structures 82 (17-19) (2004) 1293–1301.

508

509

[22] S. Phoenix, S. Satsangi, B. Singh, Layer-wise modelling of magneto-electroelastic plates, Journal of Sound and Vibration 324 (3-5) (2009) 798–815. [23] J. Sim˜ oes Moita, C. Mota Soares, C. Mota Soares, Analyses of magneto-

511

electro-elastic plates using a higher order finite element model, Composite

512

Structures 91 (4) (2009) 421 – 426.

TE

D

510

[24] M. Rao, R. Schmidt, K.-U. Schrder, Geometrically nonlinear static fe-

514

simulation of multilayered magneto-electro-elastic composite structures,

515

Composite Structures 127 (2015) 120–131. [25] E. Carrera, S. Brischetto, P. Nali, Plates and Shells for Smart Structures:

AC C

516

EP

513

517

518

519

Classical and Advanced Theories for Modeling and Analysis, Wiley, 2011.

[26] A. Milazzo, An equivalent single-layer model for magnetoelectroelastic multilayered plate dynamics, Composite Structures 94 (6) (2012) 2078–2086.

520

[27] A. Alaimo, A. Milazzo, C. Orlando, A four-node mitc finite element for

521

magneto-electro-elastic multilayered plates, Computers and Structures 129

522

(2013) 120–133. 26

ACCEPTED MANUSCRIPT

[28] A. Alaimo, I. Benedetti, A. Milazzo, A finite element formulation for large

524

deflection of multilayered magneto-electro-elastic plates, Composite Struc-

525

tures 107 (2014) 643–653.

RI PT

523

526

[29] G. Altay, M. Dkmeci, On the fundamental equations of electromagnetoelas-

527

tic media in variational form with an application to shell/laminae equations,

528

International Journal of Solids and Structures 47 (3-4) (2010) 466–492.

[30] Z.-B. Kuang, Physical variational principle and thin plate theory in electro-

530

magneto-elastic analysis, International Journal of Solids and Structures

531

48 (2) (2011) 317–325.

M AN U

SC

529

532

[31] E. Reissner, On a certain mixed variational theorem and a proposed appli-

533

cation, International Journal for Numerical Methods in Engineering 20 (7)

534

(1984) 1366–1368.

[32] E. Reissner, On a mixed variational theorem and on shear deformable plate

536

theory, International Journal for Numerical Methods in Engineering 23 (2)

537

(1986) 193–198.

D

535

[33] E. Carrera, Theories and finite elements for multilayered plates and shells:

539

A unified compact formulation with numerical assessment and benchmark-

540

ing, Archives of Computational Methods in Engineering 10 (3) (2003) 215–

541

296.

EP

TE

538

[34] E. Carrera, L. Demasi, Classical and advanced multilayered plate elements

543

based upon pvd and rmvt. part 1: Derivation of finite element matrices,

AC C

542

544

International Journal for Numerical Methods in Engineering 55 (2) (2002)

545

191–231.

546

[35] E. Carrera, M. Di Gifico, P. Nali, S. Brischetto, Refined multilayered plate

547

elements for coupled magneto-electro-elastic analysis, Multidiscipline Mod-

548

eling in Materials and Structures 5 (2) (2009) 119–138.

27

ACCEPTED MANUSCRIPT

[36] E. Carrera, S. Brischetto, C. Fagiano, P. Nali, Mixed multilayered plate

550

elements for coupled magneto-electro-elastic analysis, Multidiscipline Mod-

551

eling in Materials and Structures 5 (3) (2009) 251 – 256.

RI PT

549

[37] E. Carrera, P. Nali, Multilayered plate elements for the analysis of multifield

553

problems, Finite Elements in Analysis and Design 46 (9) (2010) 732 – 742.

554

[38] A. Milazzo, Refined equivalent single layer formulations and finite elements

555

for smart laminates free vibrations, Composites Part B: Engineering 61

556

(2014) 238–253.

558

[39] A. Milazzo, Layer-wise and equivalent single layer models for smart multi-

M AN U

557

SC

552

layered plates, Composites Part B: Engineering 67 (2014) 62–75.

559

[40] A. Milazzo, Variable kinematics models and finite elements for nonlinear

560

analysis of multilayered smart plates, Composite Structures 122 (2015)

561

537–545.

[41] A. Milazzo, Unified formulation for a family of advanced finite elements for

563

smart multilayered plates, Mechanics of Advanced Materials and Structures

564

23 (9) (2016) 971–980.

TE

D

562

[42] M. D’Ottavio, B. Krplin, An extension of reissner mixed variational theo-

566

rem to piezoelectric laminates, Mechanics of Advanced Materials and Struc-

567

tures 13 (2) (2006) 139–150.

EP

565

[43] E. Carrera, S. Brischetto, P. Nali, Variational statements and computa-

569

tional models for multifield problems and multilayered structures, Mechan-

AC C

568

570

571

572

573

ics of Advanced Materials and Structures 15 (3-4) (2008) 182–198.

Appendix A. Matrices of the mixed form MEE constitutive law The matrices involved in the the mixed form of the MEE constitutive law,

Eq.(7), are defined as C σε = C pp − C Tnp C −1 nn C np 28

(A.1)

ACCEPTED MANUSCRIPT

C στ = C Tnp C −1 nn

(A.2)

T T C σEn = C Tnp C −1 nn enn − enp

(A.3)

RI PT

574

575

T T C σHn = C Tnp C −1 nn q nn − q np

(A.4)

576

C γε = −C −1 nn C np

(A.5)

577

C γτ = C −1 nn T C γEp = C −1 nn epn 579

M AN U

T C γEn = C −1 nn enn

(A.6)

SC

578

580

(A.7) (A.8)

T C εn Hp = C −1 nn q pn

(A.9)

T C γHn = C −1 nn q nn

(A.10)

C Dp τ = epn C −1 nn

(A.11)

T C Dp Ep = epn C −1 nn epn + pp

(A.12)

T C Dp Hp = epn C −1 nn q pn + η pp

(A.13)

581

582

583

D

584

585

587

AC C

588




(A.14)

C Dn τ = enn C −1 nn

(A.15)

T C Dn En = enn C −1 nn enn + nn

(A.16)

T C Dn Hn = enn C −1 nn q nn + η nn

(A.17)

C Bp τ = q pn C −1 nn

(A.18)

T C Bp Ep = q pn C −1 nn epn + η pp

(A.19)

T C Bp Hp = q pn C −1 nn q pn + µpp

(A.20)

EP

586

TE

C Dn ε = − enn C −1 nn C np − enp

589

590

591

592

C Bn ε = − q nn C −1 nn C np − q np




(A.21)

593

C Bn τ = q nn C −1 nn 29

(A.22)

ACCEPTED MANUSCRIPT

T C Bn En = q nn C −1 nn enn + η nn

(A.23)

T C Bn Hn = q nn C −1 nn q nn + µnn

(A.24)

596

597

Appendix B. Effective MEE layer characteristics

The MEE layer effective characteristic matrices involved in Eq.(27) and (28) are defined as hk

Z

ε hkii δε Qαβ

= hk−1

h   hki hki hki hki hki Fαhki Fβ δik C hki − C A − C A σε σEn Φε σHn Ψε − hki hkii Fαhki C σEn aΦε β

hk

Z =

hk−1

−

i

599

Z

hk

=− hk−1

hki hkii Fαhki C σHn aΨτ µ

−

(B.1)

dx3

 h  hki hki hki hki Fαhki F˜µhki δik C hki στ − C σEn A Φτ − C σHn A Ψτ − hki hkii Fαhki C σEn aΦτ µ

ξ hki δε Qα

hki hkii Fαhki C σHn aΨε β

M AN U

598

τ hkii δε Qαµ

SC

595

RI PT

594

(B.2)

i dx3

h i hki hki hki hki Fαhki C σEn aΦξ + C σHn aΨξ dx3

(B.3)

600

hk

Z

D

τ hkii δγ Qαµ

= Di

h

i Fαhki F˜µhki δki dx3

(B.4)

601

TE

hk−1

τ hkii δu Qαµ

= Di

Z

h ∂F hki

hk

α

∂x3

hk−1

602

Z

hk

h

  hki hki hki hki hki hki F˜λ Fβ −C hki γε + C γEn A Φε + C γHn A Ψε δki +

EP

ε hkii δτ Qλβ

=

hk−1

hki hki hkii F˜λ C γEn aΦε β

AC C

i F˜µhki δki dx3

hki hki hkii F˜λ C γHn aΨε β

+

i

(B.5)

(B.6)

dx3

603

γ hkii δτ Qλβ

= Di

Z

hk

hk−1

hki hki F˜λ Fβ δki

(B.7)

604

u hkii δτ Qλβ

= Di

hk

Z

hk−1

hki

hki F˜λ

∂Fβ

∂x3

δki

(B.8)

605

τ hkii δτ Qλµ

Z

hk

= hk−1

h

  hki hki hki hki hki F˜λ Fµhki −C hki γτ + C γEn A Φτ + C γHn A Ψτ δki + hki hki hkii F˜λ C γEn aΦτ µ

+

hki hki hkii F˜λ C γHn aΨτ µ

30

i

dx3

(B.9)

ACCEPTED MANUSCRIPT

∂ε ∂x1

δτ

hkii Qλβ

Z

hk

h   hki hki hki hki hki hki F˜λ Gβ C γEp A Φε + C γHp A Ψε δki +

= I1 hk−1

606 ∂ε ∂x2

δτ

Z

hkii

Qλβ = I 2

hk

h   hki hki hki hki hki hki F˜λ Gβ C γEp A Φε + C γHp A Ψε δki +

hk−1

607

hk

h   hki hki hki F˜λ Ghki C A + C A γEp Φτ γHp Ψτ δki + α

M AN U

δτ

Z

hkii

Qλµ = I 1

hk−1

  hki hkii hkii F˜λ x3 C γEp aΦτ µ + C γHp aΨτ µ +  i hki hkii hkii F˜λ C γEp bΦτ µ + C γHp bΨτ µ dx3 608 ∂τ ∂x2

δτ

hkii Qλµ

Z

(B.11)

SC

  hki hki hkii hki hkii F˜λ x3 C γEp aΦε β + C γHp aΨε β +  i hki hki hkii hki hkii F˜λ C γEp bΦε β + C γHp bΨε β dx3 ∂τ ∂x1

(B.10)

RI PT

  hki hki hkii hki hkii F˜λ x3 C γEp aΦε β + C γHp aΨε β +  i hki hki hkii hki hkii F˜λ C γEp bΦε β + C γHp bΨε β dx3

hk

(B.12)

h   hki hki hki hki hki F˜λ Ghki C γEp A Φτ + C γHp A Ψτ δki + µ

= I2 hk−1

(B.13)

TE

D

  hki hki hkii hki hkii F˜λ x3 C γEp aΦτ µ + C γHp aΨτ µ +  i hki hki hkii hki hkii F˜λ C γEp bΦτ µ + C γHp bΨτ µ dx3

609

ξ hki δτ Qλ 610

hki Qλ

hk

=

hk−1

Z

hk

EP

∂ξ ∂x1

Z

AC C

δτ

= I1

h

hk−1

  hki hki hki hki hki F˜λ C γEn aΦξ + C γHn aΨξ dx3

  hki hki hki hki hki F˜λ x3 C γEp aΦξ + C γHp aΨξ +

(B.14)

(B.15)

 i hki hki hki hki hki F˜λ C γEp bΦξ + C γHp bΨξ dx3

611

∂ξ ∂x2

δτ

hki Qλ

Z

hk

= I2

h

hk−1

  hki hki hki hki hki F˜λ x3 C γEp aΦξ + C γHp aΨξ +

(B.16)

 i hki hki hki hki hki F˜λ C γEp bΦξ + C γHp bΨξ dx3

612

hki J αβ

Z

hk

= hk−1

hki

ρhki Fαhki Fβ dx3

31

(B.17)

ACCEPTED MANUSCRIPT

where δki denote the Kronecker delta. It is remarked that the generic effective

614

characteristic matrix sr Qhpqi ως accounts for the cross-work between the expansion

615

term ς of the field s within the layer q and the expansion term ω of the field

616

r within the layer p; on the other hand, the electric-magnetic effective load-

617

ing sr Qhpi ω accounts for the contribution of the electric-magnetic loading field s

618

within the layer p to the work associated with the variation of the term ω of the

619

field r within the layer p itself.

SC

RI PT

613

AC C

EP

TE

D

M AN U

620

32

TE

D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

Figure 1: Plate geometrical scheme.

33

ACCEPTED MANUSCRIPT

-0.5 -6

-4

-2

0

-0.5 -0.1

2

u3 (Lx /4, Ly /4, x3 ) [m]

× 10

0.5

= = = =

1 2 3 4

x3 /H

Mτ Mτ Mτ Mτ

0

5.6

5.7

5.8

5.9

6

u3 (Lx /4, Ly /4, x3 ) [m]

0.1

× 10

0

-0.5 -0.1

6.1

3D Mu Mu Mu Mu

0

0.2

0.3

0.4

= = = =

Mτ Mτ Mτ Mτ

0.1

= = = =

1 2 3 4

0.2

0.3

0.4

τxz (Lx /4, Ly /4, x3 ) [N/m2 ]

-12

TE

0.5

0

M AN U

= = = =

D

x3 /H

3D Mu Mu Mu Mu

3D Mu = Mτ + 1 = 2 Mu = Mτ + 1 = 3 Mu = Mτ + 1 = 4

τxz (Lx /4, Ly /4, x3 ) [N/m2 ]

-10

0.5

-0.5 5.5

0

SC

x3 /H

x3 /H

3D Mu = Mτ + 1 = 2 Mu = Mτ + 1 = 3 Mu = Mτ + 1 = 4

0

RI PT

0.5

0.5

0.5

x3 /H

EP 0

AC C

x3 /H

3D Mu = Mτ − 1 = 1 Mu = Mτ − 1 = 2 Mu = Mτ − 1 = 3

-0.5 5.2

5.4

5.6

5.8

6

u3 (Lx /4, Ly /4, x3 ) [m]

6.2

0

-0.5 -0.1

6.4

3D Mu = Mτ − 1 = 1 Mu = Mτ − 1 = 2 Mu = Mτ − 1 = 3

0

0.1

0.2

0.3

0.4

τxz (Lx /4, Ly /4, x3 ) [N/m2 ]

× 10 -12

Figure 2: B/F/B simply-supported square plate with H/Lx = 0.3 loaded by bi-sinusoidal pressure: through-the thickness distributions at x1 = x2 = 0.25Lx for layer-wise models with different variable approximation schemes, namely Mu > Mτ , Mu = Mτ and Mu < Mτ .

34

ACCEPTED MANUSCRIPT

0.5

0.5

-0.5 5.2

5.4

5.6

5.8

6

6.2

u3 (Lx /4, Ly /4, x3 ) [m]

Mτ Mτ Mτ Mτ

= = = =

1 2 3 4

x3 /H

= = = =

5.6

5.8

6

6.2

3D Mu Mu Mu Mu

0

6.4

× 10

-0.2

0.3

0.4

= = = =

Mτ Mτ Mτ Mτ

= = = =

1 2 3 4

0

0.2

0.4

τxz (Lx /4, Ly /4, x3 ) [N/m2 ]

-12

TE

u3 (Lx /4, Ly /4, x3 ) [m]

0.5

0.2

-0.5

D

0

5.4

0.1

τxz (Lx /4, Ly /4, x3 ) [N/m2 ]

M AN U

3D Mu Mu Mu Mu x3 /H

0

-12

0.5

0.5

-0.5 5.2

-0.5 -0.1

6.4

× 10

RI PT

0

3D Mu = Mτ + 1 = 2 Mu = Mτ + 1 = 3 Mu = Mτ + 1 = 4

0

SC

x3 /H

x3 /H

3D Mu = Mτ + 1 = 2 Mu = Mτ + 1 = 3 Mu = Mτ + 1 = 4

0.5

x3 /H

EP 0

AC C

x3 /H

3D Mu = Mτ − 1 = 1 Mu = Mτ − 1 = 2 Mu = Mτ − 1 = 3

0

-0.5

-0.5

4

3D Mu = Mτ − 1 = 1 Mu = Mτ − 1 = 2 Mu = Mτ − 1 = 3

4.5

5

5.5

u3 (Lx /4, Ly /4, x3 ) [m]

0

6

× 10 -12

0.1 0.2 0.3 τxz (Lx /4, Ly /4, x3 ) [N/m2 ]

0.4

Figure 3: B/F/B simply-supported square plate with H/Lx = 0.3 loaded by bi-sinusoidal pressure: through-the thickness distributions at x1 = x2 = 0.25Lx for equivalent-single-layer models with different variable approximation schemes, namely Mu > Mτ , Mu = Mτ and Mu < M τ .

35

ACCEPTED MANUSCRIPT

0.5

0.5

0

= = = =

Mτ Mτ Mτ Mτ

= = = =

1 2 3 4

-0.5 -2

-1

0

1

-0.5 5.5

2

u1 (Lx /4, Ly /4, x3 ) [m]

× 10

Mτ Mτ Mτ Mτ

= = = =

1 2 3 4

0.1

0.2

0.3

0.4

0

6

6.1

× 10 -12

3D Mu Mu Mu Mu

= = = =

Mτ Mτ Mτ Mτ

= = = =

1 2 3 4

= = = =

Mτ Mτ Mτ Mτ

= = = =

AC C

0.6

0.8

1

1 2 3 4

0

0

3D Mu Mu Mu Mu

= = = =

Mτ Mτ Mτ Mτ

= = = =

1 2 3 4

-0.5

-0.5

0.2

0.4

0.5

x3 /H

3D Mu Mu Mu Mu

0.2

σzz (Lx /4, Ly /4, x3 ) [N/m2 ]

TE

EP

0.5

x3 /H

5.9

-0.5 0

τxz (Lx /4, Ly /4, x3 ) [N/m2 ]

0

5.8

1 2 3 4

0

D

-0.5 -0.1

= = = =

x3 /H

0

5.7

M AN U

0.5

3D Mu Mu Mu Mu

5.6

= = = =

u3 (Lx /4, Ly /4, x3 ) [m]

-12

0.5

x3 /H

0

Mτ Mτ Mτ Mτ

SC

3D Mu Mu Mu Mu

= = = =

RI PT

x3 /H

x3 /H

3D Mu Mu Mu Mu

0.4

0.6

0.8

Φ(Lx /4, Ly /4, x3 ) [V ]

1

-2

1.2

× 10

-1

0

1

Ψ(Lx /4, Ly /4, x3 ) [C/s]

-3

2

3

× 10 -7

Figure 4: B/F/B simply-supported square plate with H/Lx = 0.3 loaded by bi-sinusoidal pressure: through-the thickness distributions at x1 = x2 = 0.25Lx for different layer-wise models.

36

ACCEPTED MANUSCRIPT

0.5

= = = =

3D Mu Mu Mu Mu

1 2 3 4

0

-0.5 -2

-1

0

1

2

u1 (Lx /4, Ly /4, x3 ) [m]

× 10

Mτ Mτ Mτ Mτ

= = = =

1 2 3 4

-0.5

5.8

6

6.2

6.4

× 10 -12

= = = =

Mτ Mτ Mτ Mτ

= = = =

1 2 3 4

0

-0.5 0

0.2

0.4

-1

D

-0.2

-0.5

3D Mu Mu Mu Mu

= = = =

Mτ Mτ Mτ Mτ

= = = =

3D Mu Mu Mu Mu

1 2 3 4

AC C

0

0.2

1

= Mτ = Mτ = Mτ = Mτ

=1 =2 =3 =4

0

-0.5

-0.5

0

0.5

0.5

x3 /H

EP

0.5

0

σzz (Lx /4, Ly /4, x3 ) [N/m2 ]

TE

τxz (Lx /4, Ly /4, x3 ) [N/m2 ]

x3 /H

5.6

3D Mu Mu Mu Mu

x3 /H

= = = =

1 2 3 4

u3 (Lx /4, Ly /4, x3 ) [m]

0.5

0

= = = =

M AN U

x3 /H

5.4

-12

0.5

3D Mu Mu Mu Mu

Mτ Mτ Mτ Mτ

0

-0.5 5.2

3

= = = =

SC

Mτ Mτ Mτ Mτ

x3 /H

x3 /H

= = = =

RI PT

0.5

3D Mu Mu Mu Mu

0.4

0.6

0.8

Φ(Lx /4, Ly /4, x3 ) [V ]

1

-4

1.2

× 10

-2

0

Ψ(Lx /4, Ly /4, x3 ) [C/s]

-3

2

× 10 -7

Figure 5: B/F/B simply-supported square plate with H/Lx = 0.3 loaded by bi-sinusoidal pressure: through-the thickness distributions at x1 = x2 = 0.25Lx for different equivalentsingle-layer models.

37

0.5

0.4

0.4

0.3

0.3

0.2

0.2

-0.1

= Mτ = Mτ = Mτ = Mτ

0.1

=1 =2 =3 =4

x3 /H

x3 /H

0

-0.2

3D Mu Mu Mu Mu

-0.2

-0.3

-0.3

-0.4

-0.4

-0.5 -15

-10

-5

-0.5 -12

0

u1 (Lx /4, Ly /4, x3 ) [m]

-10

-8

-6

= Mτ = Mτ = Mτ = Mτ

=1 =2 =3 =4

-4

-2

0

2

u3 (Lx /4, Ly /4, x3 ) [m]

-10

×10

4 ×10-11

0.5

0.5

0.4

0.4

0.3

0.2 0.1

TE

0 -0.1 -0.2 -0.3

EP

-0.4

-20

-10

0

3D Mu Mu Mu Mu

= Mτ = Mτ = Mτ = Mτ

0.2 0.1

x3 /H

D

0.3

x3 /H

0 -0.1

M AN U

3D Mu Mu Mu Mu

0.1

-0.5 -30

SC

0.5

RI PT

ACCEPTED MANUSCRIPT

3D Mu Mu Mu Mu

= Mτ = Mτ = Mτ = Mτ

=1 =2 =3 =4

0 -0.1

=1 =2 =3 =4

-0.2 -0.3 -0.4 -0.5

10

20

-1

30

-0.5

0

0.5

Ψ(Lx /4, Ly /4, x3 ) [C/s]

AC C

Φ(Lx /4, Ly /4, x3 ) [V ]

1 ×10-5

Figure 6: B/F/B simply-supported square plate with H/Lx = 0.1 loaded by bi-sinusoidal electric potential: through-the thickness distributions at x1 = x2 = 0.25Lx for different layerwise models.

38

0.5

0.4

0.4

0.3

0.3

0.2

0.2

-0.1

= Mτ = Mτ = Mτ = Mτ

0.1

=1 =2 =3 =4

x3 /H

x3 /H

0

-0.2

3D Mu Mu Mu Mu

-0.2

-0.3

-0.3

-0.4

-0.4

-0.5 -15

-10

-5

-0.5 -12

0

u1 (Lx /4, Ly /4, x3 ) [m]

-10

-8

-6

= Mτ = Mτ = Mτ = Mτ -4

=1 =2 =3 =4 -2

0

2

4

u3 (Lx /4, Ly /4, x3 ) [m]

-10

×10

×10-11

0.5

0.5

0.4

0.4

0.3

0.2 0.1

TE

0 -0.1 -0.2 -0.3

EP

-0.4

-20

-10

0

3D Mu Mu Mu Mu

= Mτ = Mτ = Mτ = Mτ

0.2 0.1

x3 /H

D

0.3

x3 /H

0 -0.1

M AN U

3D Mu Mu Mu Mu

0.1

-0.5 -30

SC

0.5

RI PT

ACCEPTED MANUSCRIPT

3D Mu Mu Mu Mu

= Mτ = Mτ = Mτ = Mτ

=1 =2 =3 =4

0 -0.1

=1 =2 =3 =4

-0.2 -0.3 -0.4 -0.5

10

20

-1

30

0

Ψ(Lx /4, Ly /4, x3 ) [C/s]

AC C

Φ(Lx /4, Ly /4, x3 ) [V ]

1 ×10-4

Figure 7: B/F/B simply-supported square plate with H/Lx = 0.1 loaded by bi-sinusoidal electric potential: through-the thickness distributions at x1 = x2 = 0.25Lx for different equivalent-single-layer models.

39

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

H/Lx = 0.1 2

3

4

5

1

2

3

4

5

ESL1 ESL2 ESL3 ESL4

0.406 0.393 0.393 0.391

1.885 1.885 1.885 1.885

3.431 3.391 3.276 3.397

12.874 13.716 13.042 13.043

13.256 14.030 13.400 13.358

0.972 0.973 0.956 0.961

1.885 1.884 1.884 1.884

3.396 3.335 3.232 3.349

4.628 4.890 4.681 4.682

5.534 5.634 5.538 5.424

LW 1 LW 2 LW 3 LW 4

0.400 0.399 0.399 0.399

1.885 1.885 1.885 1.885

3.432 3.431 3.430 3.430

12.862 13.052 13.042 13.042

13.205 13.378 13.373 13.374

0.976 0.975 0.973 0.973

1.884 1.884 1.884 1.884

3.401 3.399 3.389 3.389

4.628 4.684 4.681 4.681

5.439 5.447 5.455 5.459

3D

0.395 1.858 3.294 12.528 13.248

D

1

TE

Mode n.

H/Lx = 0.3

0.961 1.856 3.228 4.497 5.215

AC C

EP

Table 1: Normalized natural circular frequencies ω ¯ for the B/F/B simply-supported square plate.

40

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

H/Lx = 0.1 2

3

4

5

1

2

3

4

5

ESL1 ESL2 ESL3 ESL4

0.474 0.437 0.436 0.435

1.932 1.932 1.932 1.932

3.468 3.529 3.488 3.488

12.316 13.292 12.499 12.500

12.846 13.682 12.880 12.878

1.065 1.042 1.030 1.027

1.932 1.931 1.931 1.931

3.421 3.451 3.431 3.429

4.506 4.804 4.561 4.561

5.721 5.708 5.431 5.419

LW 1 LW 2 LW 3 LW 4

0.439 0.435 0.435 0.435

1.932 1.932 1.932 1.932

3.467 3.467 3.467 3.467

12.361 12.500 12.492 12.492

12.733 12.878 12.870 12.870

1.040 1.027 1.027 1.027

1.931 1.931 1.931 1.931

3.407 3.406 3.407 3.407

4.517 4.561 4.558 4.558

5.363 5.418 5.416 5.416

3D

0.444 1.960 3.478 12.997 13.961

D

1

TE

Mode n.

H/Lx = 0.3

1.067 1.960 3.388 4.742 5.895

AC C

EP

Table 2: Normalized natural circular frequencies ω ¯ for the F/B/F simply-supported square plate.

41