Bending of exponentially graded plates integrated with piezoelectric fiber-reinforced composite actuators resting on elastic foundations

European Journal of Mechanics / A Solids 75 (2019) 461–471

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Bending of exponentially graded plates integrated with piezoelectric fiberreinforced composite actuators resting on elastic foundations

T

Ashraf M. Zenkoura,b,∗, Rabab A. Alghanmia,c a

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh, 33516, Egypt c Department of Mathematics, Rabigh College for Sciences and Arts, King Abdulaziz University, Rabigh, 21911, Saudi Arabia b

ARTICLE INFO

ABSTRACT

Keywords: PFRC actuator EG plate Elastic foundations Higher-order plate theory Navier's solution

This paper introduces analytical solution for exponentially graded (EG) plates attached with a single-layer of piezoelectric fiber reinforced composite (PFRC) actuator. The plate is undergoing sinusoidal electromechanical loads and resting on Pasternak-Winkler's elastic foundations. The displacement components are expressed via a higher-order shear deformation theory. The distributions of the electric potential are vary linearly along the thickness direction. The equations of equilibrium and boundary conditions are established by applying the principle of virtual displacements. Navier's method is used to obtain the problem solution. The influences of various parameters like voltage amount, side-to-thickness ratio, Young's modulus ratio and elastic foundations parameters are investigated. The current formulations are validated by comparison with published results.

1. Introduction Piezoelectric materials are widespread due to their capability of transforming an electric domain to mechanical domain and vice versa. These materials are used to design smart structures in various areas like industrial, aerospace, medical, automobile and space structures. Tiersten (1969) introduced the constitutive equations of piezoelectric plate. Crawley and de Luis (1987) have investigated a beam attached with piezoelectric actuators. Wang and Rogers (1991) used classical plate theory to study a plate with induced strain actuator. Crawley and Lazarus (1991) developed and experimented a model for the induced strain actuator of isotropic and anisotropic plates. Lee (1990) presented a consistent plate theory to investigate piezoelectric plates. Sze et al. (2004) investigated two forms for the distributions of electric potentials. One of them is that the electric field is piecewise linear in the transverse direction, and the other one is that the electric displacement is piecewise constant along the transverse direction. Feri et al. (2016) presented the static and free vibration of composite plate embedded with piezoelectric layers and discussed several boundary conditions. The traditional monolithic piezoelectric materials such as PVDF and PZT are used in smart structures as actuators and sensors but these materials have some shortcomings like low piezoelectric constants, shape control, and limited capability to conforming with curved surfaces. These limitations have been overcome using piezoelectric fiberreinforced composite (PFRC). In this type of composites, the ∗

piezoelectric fibers are embedded in non-piezoelectric materials. Mallik and Ray (2003) presented the unidirectional PFRC and introduced the effective coefficients. Kumar and Chakraborty (2009) developed a micromechanical model to evaluate the coupled thermo-electromechanical coefficients of the PFRC. Functionally graded materials (FGMs) are inhomogeneous materials where the material properties vary continuously in the desired direction, and so eliminate the undesired interface problem that usually happens in homogeneous composites. The concept of FGMs was first presented in 1984 in Japan, Yamanouchi et al. (1990). Zenkour (2006) analyzed the bending of FG plates via a generalized plate theory. Mantari et al. (2016) have discussed the bending of FG plates via Carrera's unified formulation (). Thom et al. (2017) developed an accurate computational approach using finite element method (FEM) and a new high-order shear deformation plate theory to analyze the static bending and buckling behaviors of bi-directional FG plates. Cong et al. (2018) applied Reddy's higher-order shear deformation plate theory to investigate the nonlinear buckling and post-buckling of FG plate with porosities resting on elastic foundations and subjected to thermal loads. Ray and Sachade (2006a) introduced a three-dimensional solution for the bending of an FG plate with PFRC layer. Ray and Sachade (2006b) developed FEM by using first-order shear deformation theory to study an FG plates with PFRC layer. Xiang and Shi (2009) used Airy stress function method to analyze functionally graded piezoelectric (FGPM) sandwich cantilever under the effect of electrical and thermal loading.

Corresponding author. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia. E-mail addresses: [email protected], [email protected] (A.M. Zenkour).

https://doi.org/10.1016/j.euromechsol.2019.03.003 Received 24 December 2018; Received in revised form 16 February 2019; Accepted 4 March 2019 Available online 11 March 2019 0997-7538/ © 2019 Elsevier Masson SAS. All rights reserved.

European Journal of Mechanics / A Solids 75 (2019) 461–471

A.M. Zenkour and R.A. Alghanmi

Based on the method of SaS located at Chebyshev polynomial node inside the plate, Kulikov and Plotnikova (2013) introduced an exact solution for piezoelectric FG laminated plates. Rouzegar and Abbasi (2017) investigated FG plates integrated with PFRC by using a refined finite element method. Panda and Ray (2008) introduced an FEM to analyze an FG plates with patches of PFRC with the presence/absence of thermal loads. Shiyekar and Kant (2010) used a higher- order shear and normal deformation theory to solve the problem of an FG plates integrated with PFRC actuator. Shen et al. (2016) studied the bending and thermal postbuckling of fiber-reinforced composites laminated beams with PFRC layers resting elastic foundation. The large amplitude vibration and postbuckling behavior of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plate integrated with piezoelectric layers were introduced by Keleshteri et al. (2017a, b, 2018) and Keleshteri et al. (2017). Using the theory of Love-Kirchhoff's, Fernandes and Pouget (2006) presented a model to investigate the conduct of an elastic plate with piezoelectric actuators. Depending on a layerwise theory, Shakeri and Mirzaeifar (2009) developed an FEM to analyze the static and dynamic of FG plates with piezoelectric layers. Duc et al. (2016) applied firstorder shear deformation theory to study the nonlinear dynamic response and vibration of piezoelectric eccentrically stiffened FG plates undergo thermal loads and resting on elastic foundation. Kim and Reddy (2013) applied a third-order plate theory to analyze FG plates. Her and Lin (2010) used an FEM to study a laminated plate that bonded by two piezoelectric actuators. Zenkour (2007) has introduced two-dimensional trigonometric and three-dimensional elasticity solutions to solve the bending problem exponentially graded plate (EGM) undergo a sinusoidal load. Mantari and Guedes Soares (2012) applied a higherorder shear deformation theory to analyze the bending of an EGM. The problem of plates resting on elastic foundations is widespread in structural engineering and have many applications in the rest fields of engineering. Zenkour (2009) used refined sinusoidal shear deformation theory for the analysis of a FGM plates resting on Pasternak elastic foundations and under the influence of thermomechanical loads. Zenkour et al. (2014) applied a refined shear and normal deformation plate theory to investigate an FG plates resting on elastic foundations and undergo hygrothermal and mechanicals loads. Akavci (2016) studied the bending, free vibration and buckling of FG sandwich plates resting on elastic foundations via a new hyperbolic shear and normal deformation plate theory. Liew et al. (2003) investigated the postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading. Duc et al. (2015) studied the nonlinear dynamic response and vibration of imperfect FGM double curved shallow shells integrated with piezoelectric actuators and resting on elastic foundations. Duc and Cong (2016) investigated the nonlinear dynamic response and vibration of piezoelectric functionally graded (PFGM) sandwich plates resting on elastic foundations and subjected to electrical, thermal, and damping loads. Duc et al. (2017) studied the thermal and mechanical stability of FG-CNTRC truncated conical shells resting on elastic foundations while Duc and Nguyen (2017) studied their dynamic response and vibration. Recently, Zenkour and Alghanmi (2018) applied a refined shear and normal deformation plate theory to investigate the displacement and stresses of FG plates. Zenkour and Alghanmi (2019) analyzed FG plates with piezoelectric layers via a four-unknown shear deformation plate theory. Keleshteri et al. (2019) adopted the generalized differential quadrature method in conjunction with the Newton-Raphson iterative method to study the nonlinear bending behavior of FG-CNTRC annular plates resting on an elastic foundation. In this study, a four-unknown shear deformation plate theory is considered for the analysis of an EG plates attached with a single-layer of PFRC actuator and resting on Pasternak's or Winkler's elastic foundation. The electric potential distributions are considered as a linear function along the thickness direction and the impact of voltage amount, side-to-thickness ratio, Young's modulus ratio and elastic

Fig. 1. Geometry of the EG plate resting on elastic foundation and attached with top PFRC actuator.

foundations parameters are investigated. Via comparisons with available results, the present formulation is validated. 2. Theoretical formulation The current study considers a simply supported EG plate of length a , width b and thickness h as shown in Fig. 1. The EG plate is attached with PFRC layer of thickness hp , acting as distributed actuator at top or bottom surface of the EG plate. The plate is lying on a two-parameter elastic foundation which known as Pasternak's foundation. The corner of the EG plate middle surface (z = 0 ) is located at orthogonal coordinate system (x , y , z ) This hybrid plate is subjected to electromechanical loadings. The Young's modulus E is a function of z and given in exponential form as (Sankar, 2001)

E (z ) = Eb e

(z+ h2 ),

=

1 E ln t , h Eb

(1)

where Eb and Et are the Young's modulus of the bottom and top surfaces of the EG plate, respectively, and is the parameter characterizing the material variation across the thickness of the EG plate. The Poisson's ratio is chosen to be constant across the thickness of the EG plate. The reaction-deflection relation of Pasternak's foundation fe can be expressed as

fe = Ku3

J

2u 3 x2

+

2u 3 y2

,

(2)

where K is Winkler's springs stiffness, J is Pasternak shear layer stiffness and u3 is the transverse displacement. Depending on a shear deformation theory, the displacement field is given as (Shimpi and Patel, 2006)

u1 (x , y, z ) = u (x , y )

z

u2 (x , y , z ) = v (x , y )

z

wb x wb y

f (z ) f (z )

ws , x ws , y

u3 (x , y, z ) = wb (x , y ) + ws (x , y ),

(3)

where u and v are the displacements of the middle plane in along x and y axes, and wb and ws are bending displacement and shear displacement. The shape function is taken as (Thai and Kim, 2012);

f (z ) =

z 4

+

( ) ( ), 5 3

z3 H2

where H = h + hp. Using the displacement

components in Eq. (3), the strain components are 462

European Journal of Mechanics / A Solids 75 (2019) 461–471

A.M. Zenkour and R.A. Alghanmi

x

=

u x

z yz

2w b x2

ws [1 y

= xy

=

2w s , x2

f (z )

y

f (z )], u y

v x

+

xz

2 z

v y

=

ws [1 x

=

2w b

z

f (z )

2w s , y2

c x

f (z )],

+ f (z )

x y

2w b y2

2w s

c y

.

x y

c c c11 c12 0 c c c12 c22 0 c 0 0 c44 0 0 0 0 0 0

y yz

=

xz xy

0 0 0 0 c c66

0 0 0 c c55 0

u x

c = c12 c yz

(4)

E (z ) , 2 1

c c12 =

p x

,

c11p c12p 0 c12p c22p 0 p 0 0 c44 0 0 0 0 0 0

y yz

=

xz xy

0 0 0 c55p 0

c c c c44 = c55 = c66 =

,

0 0 0 0 c66p

0 0 0 0 0 e24 e15 0 0 0

x y yz xz xy

y yz

11

+

xz

0 0

0 22

0 0

0

33

xy

E (z ) . 2(1 + )

p y

x

, Ey =

y

e31 e32 0 0 0

Ex Ey , Ez

Ex Ey , Ez

, Ez =

z

.

h 0 , 2 hp

h 2

z

u x

(z ) =

h 2

b

2w

b x2

c xz

2 z

2w b y2

z

2w b y2

c [1 = c55

2w b

x y

f (z )]

},

2w s

+ f (z )

x y

f (z )

2w s y2

,

f (z )

2w s y2

,

ws , x

(11)

f (z )

2w s x2

+ c11p

v y

z

2w b y2

f (z )

2w s y2

f (z )

2w s x2

+ c22p

v y

z

2w b y2

f (z )

2w s y2

,

hp

p yz

= c44p [1

f (z )]

ws , y

p xz

= c55p [1

f (z )]

ws , x

p xy

= c66p

u v + y x

z

2w 2w b s + f (z ) x y x y

2 z

.

The electric displacement relations can be expressed as: (8)

Dx = e15

ws [1 x

u x

2w b x2

Dz = e31 33

z

f (z )], 2w s x2

f (z )

ws [1 y

Dy = e24 + e32

v y

z

f (z )], 2w b y2

f (z )

2w s y2

( ). 0

(13)

hp

Equations of equilibrium are derived depending on the principle of virtual displacements as follows

(9)

v

(

ij

ij

Di Ei )dv

(q

fe )( wb + ws )d

= 0.

(14)

Substituting the strains and electric field components into Eq. (14) and integrating over the thickness direction and then rearranging variables yields u x

{Nxx u y

+ Nxy

Mxx

+ Nxy

2 w b x2

v x

Sxx

2 w s x2

2 w b

2Mxy

(q

x y

fe ) wb

(q

+ Nyy 2Sxy

v y 2 w s

x y

Myy

2 w b y2

Syy

ws y

+ Qxz

+ Q yz

fe ) ws}d

2 w s y2

ws x

(15)

= 0,

where the stress resultants can be written as

h + hp , 2

hp

(12)

(7)

{Nij , Mij , Sij} = h 0 z+ , 2 hp

ws , y v x

+

x2

z 0

+ e32

It is assumed that the surface of the PFRC layer being in contact with the EG plate is suitably grounded, i.e., having zero potential while the outer surfaces having nonzero potential. The electrodes on the PFRC layer is open-circuit. Because the thickness of the PFRC layer is very low, the variation of the electric potential function across the thickness may be considered as linear. Thus, the form of the electric potential function at the PFRC layer is taken as (Ray and Sachade, 2006b; Shen, 2009; Rouzegar and Abbasi, 2017, 2018; Samanta et al., 1996; Ray and Mallik, 2004; Shiyekar and Kant, 2011; Rouzegar and Abad, 2015)

(z ) = z

u y

v y

c + c22

z

,

hp

= c12p

(6)

in which cijp are the stiffness constants for the PFRC layer, Ex , E y and Ez are the vector of the electric field along x , y and z axis, eij are the piezoelectric constants and ij are the dielectric constants. The electric field is associated with electric potential as

Ex =

{

2w

z 0

+ e31

(5)

xy

2

0 0 0 e15 0 0 0 e24 0 0 e31 e32 0 0 0

=

f (z )

2w s x2

v y

c + c11

xz

x

Dx Dy Dz

2w s x2

f (z )]

c = c66

u x

= c11p

For the single piezoelectric layer, the stress field and the electric displacements are written as (Mallik and Ray, 2003) x

z

2w b x2

f (z )

and the stresses for the PFRC layer are

y yz

E (z ) 1

2w b x2

x

where the stiffness coefficients cijc for the EG plate are expressed as c c c11 = c22 =

z

c [1 = c44 c xy

The stress-strain relations for the EG plate are expressed as (Zenkour, 2006) x

u x

c = c11

h , 2

(10)

Qiz =

where 0 is the external electric voltage. The substitution of strains and electric field into the stress-strain relations and electric displacements yields the stresses for the EG plate

h 2 h 2

c iz [1

h 2 h 2

c ij {1,

z, f (z )}dz +

f (z )]dz +

h +h p 2 h 2

h +h p 2 h 2

p iz [1

p ij {1,

f (z )]dz ,

z, f (z )}dz ,

i, j = x , y .

(16)

After integrating by parts and setting the coefficients of u , v , wb and ws to zero, separately, the equations of equilibrium for the present formulation are obtained as follows 463

European Journal of Mechanics / A Solids 75 (2019) 461–471

A.M. Zenkour and R.A. Alghanmi Nx x

Nxy

2S

x x2

+2

2S

+2 xy

x y

2M xy

y y2

2M y y2

Q xz x

+

where { } and {F } denote the columns

= 0,

y

+

x y 2S

+

Ny

{ } = {U , V , Wb, Ws}, {F } = {F1, F2, F3, F4},

fe + q = 0, Q yz

+

fe + q = 0.

y

u x

Ny = A 4

u x

Mx = A2

u x

A10

My = A5

u x

A12

Sx = A3

u x

Sy = A 6

Mxy = A20

Sxy = A21 Qyz = A25

A3

2w

b x2

A5

2w

b x2

2w b x2

u v + y x

u v + y x ws , y

A11

A13

2w s x2 2w s x2 2w s x2

2 A20

2 A22

2 A23

Q xz = A26

v y

+ A7

2w s x2

A17

u v + y x

+ A4

2w s x2

A16

2w b x2

A13

2w s x2

A6

2w b x2

A11

u x

Nxy = A19

2w b x2

A2

2w

2w b y2

A5

v y

2w b y2

A8

+ A5

v y

A12

+ A8

v y

A14

+ A6 + A9 b

x y 2w b

x y

v y

A13

v y

+ A21

+ A23

A15 2w s

2w s y2

A6 A9

2w b y2

2w

b y2

2w b y2 2w b y2

a13 = a14 = [A3

+ Ny , + Mx ,

a33 = A10

A15

2w s y2

+ My ,

a34 = A11

x y

+ Sx ,

2w s y2

+ Sy ,

u = wb = ws =

F1 =

,

ws , x

=

ws y

= Nx = Mx = Sx =

= 0,

at x = 0, a,

=

ws x

= Ny = My = Sy =

= 0,

at y = 0, b .

2

+ µ2 )

4

+ (A26 + J ) J ) µ2

2

+ 2(A17 + 2A24 ) 2µ2

+ A18 µ 4 + K .

(24)

Ny Nx , F2 = , F3 = q0 + x y 2S x x2

+

2S

y

y2

2M

x x2

+

2M

y

y2

, F4

.

(25)

= 0.3.

The PFRC layer is composed of PZT5H (piezoelectric fiber) surrounded by epoxy (matrix). The value of the piezoelectric coefficient e31, that is in the fibers direction is so higher than the other coefficients (Mallik and Ray, 2003); due to that, the other piezoelectric coefficients e32 , e15 and e24 will not be considered in numerical results. The properties of the PFRC layer are taken as (Mallik and Ray, 2003);

c11p = 32.6 GPa, c12p = 4.3 GPa, c22p = 7.2 GPa, c44p = 1.05 GPa, c55p = c66p = 1.29 GPa, e31 =

6.76 C/m2 .

The amplitude of the sinusoidal mechanical load in all examples is q0 = 40 N/m2 . Different values of side-to-thickness ratios (S = a /h = 10 , 20 , 100 ) are considered with different values of applied voltage and elastic foundation parameters.

(20)

where = / a , µ = /b , q0 are the mechanical load intensity at the center of the plate and ¯0 is constant. The solution form for displacements u , v , wb and ws which satisfies the simply-supported boundary conditions are assumed to be as

U cos( x )sin(µy ) u v = V sin( x )cos(µy ) , (wb, ws ) (Wb, Ws )sin( x )sin(µy )

+ 2(A13 + 2A23 ) 2µ2 + A15 µ 4 + J (

Eb = 200 GPa,

Following Navier's method, the external force and applied voltage that satisfy the boundary conditions are supposed as 0

4

Several numerical results are presented for the EG plates attached at the top or bottom with a single-layer of PFRC actuator and resting on Pasternak's-Winkler's elastic foundation. The thickness of the EG plate is taken as 3mm while for the PFRC layer is considered as 250µm . The material properties of the EG plate are given as (Ray and Sachade, 2006a)

(18)

=

q0 ¯0 sin( x )sin(µy ),

+ µ2 )

4. Numerical results

2w 2w b s + A24 , x y x y

(19)

q

2

The elements {Fi} are obtained by

Consider an EG plates attached with a single-layer of PFRC actuator and resting on Pasternak-Winkler's elastic foundations. The hybrid plate is simply supported at its four edges. The following mechanical and electrical boundary conditions are considered

wb x

µ [A9 µ2 + (A6 + 2A21 ) 2],

+ (A25 +

3. Closed-form solution

wb y

µ [A8 µ2 + (A5 + 2A20 ) 2],

+ 2(A12 + 2A22 ) 2µ2 + A14 µ 4 + J (

a44 = A16

where the integration constants in Eq. (18) are defined in Appendix I.

v = wb = ws =

+ A7 µ2 ,

+ K,

2w s y2

= q0 +

2w s

4

2

+ K,

,

x y

[A2 2 + (A5 + 2A20 ) µ2 ], + (A6 + 2A21 ) µ2], a22 = A19

a24 =

A13

A18

2

+ A19 µ2 , a12 = (A 4 + A19 ) µ ,

a23 =

2w s y2

A17

2

a11 = A1

+ Nx ,

2w s y2

(23)

and the nonzero elements aij = aji of the coefficient matrix [A] are given by

(17)

The resultants defined in Eq. (17) can be written as

Nx = A1

(22)

[A]{ } = {F },

= 0,

y

+

x 2M x x2

Nxy

+

Example 1. For the purpose of verification, an EG plates attached at the top with a layer of PFRC actuator will considered without the elastic foundation. The results are compared with exact solution (Ray and Sachade, 2006a), FEM based on first-order shear deformation theory (Ray and Sachade, 2006b) and FEM based on higher-order shear deformation theory (Rouzegar and Abbasi, 2017). The nondimensional parameters used are

(21)

in which U , V , Wb and Ws are arbitrary parameters. Substitution of the expansions in Eq. (20) and Eq. (21) in Eq. (17), yields the system

u¯1 =

464

Eb b ±h 100E a b u1 0, , , u¯3 = 4 b u3 , , 0 , S 3h q0 2 2 S h q0 2 2

European Journal of Mechanics / A Solids 75 (2019) 461–471

A.M. Zenkour and R.A. Alghanmi

Table 1 The dimensionless displacements and stresses of EG square plates (Et / Eb = 10) integrated with top PFRC layer. (K 0 = 0 = J0 ) Theory

S = 10

S = 20

¯0 = 0

¯0 = 100

¯0 =

Present HSDTa FSDTb Exactc

−0.9577 −0.9503 −0.9485 −0.9553

186.4594 185.6516 183.9187 186.8222

Present

0.0093 −0.0196 0.0093 −0.0194 0.0093 −0.0195 0.0093 −0.0194

HSDTa FSDTb Exactc Present HSDT

a

FSDTb Exactc Present HSDTa FSDTb Exactc a

S = 100

¯0 = 0

¯0 = 100

¯0 =

−188.375 −187.552 −185.815 −188.733

−0.9254 −0.9184 −0.9233 −0.9252

45.4706 45.2894 45.2938 45.5393

−5.8813 −0.0897 −6.0439 0.0641 −5.9402 0.0459 −5.9373 −0.0307

5.8999 0.0506 6.0624 −0.1130 5.9588 −0.0849 5.9558 −0.0081

0.0093 −0.0195 0.0092 −0.0194 0.0093 −0.0195 0.0093 −0.0195

−0.4196 0.0877 −0.4144 0.0864 −0.4250 0.0893 −0.4201 0.0871

202.432 −5.7494 196.0864 −5.2596 208.1946 −6.1985 203.984 −5.8052

−203.271 5.9249 −169.913 5.4324 −209.044 6.3772 −204.824 5.9794

0.2262 −0.0472 0.2250 −0.0467 0.2290 −0.0481 0.2242 −0.0469

−70.3546 6.9605 −66.9552 7.3675 −71.7882 7.1027 −70.4771 6.8593

70.8069 −7.0549 67.4052 −7.4481 72.2462 −7.1989 70.9256 −6.9530

100

¯0 = 0

¯0 = 100

¯0 =

−47.321 −47.126 −47.140 −47.389

−0.9151 −0.9081 −0.9145 −0.9155

0.9349 0.9364 0.9328 0.9368

−2.7650 −2.7526 −2.7619 −2.7678

−1.4625 −0.0379 −1.4976 −0.0220 −1.4655 −0.0297 −1.4670 −0.0337

1.4810 −0.0011 1.5160 −0.0037 1.4840 −0.0039 1.4855 −0.0053

0.0093 −0.0195 0.0092 −0.0194 0.0093 −0.0195 0.0093 −0.0195

−0.0496 −0.0202 −0.0501 −0.0196 −0.0496 −0.0202 −0.0497 −0.0202

0.0681 −0.0187 0.0684 0.0191 0.0681 −0.0187 0.0682 −0.0188

−0.4167 0.0875 −0.4115 0.0861 −0.4250 0.0893 −0.4171 0.0873

50.2554 −1.3679 48.8083 −1.2609 51.4183 −1.4170 50.3911 −1.3742

−51.089 1.5429 −49.631 1.4331 −52.268 1.5957 −51.225 1.5489

−0.4158 0.0874 −0.4104 0.0860 −0.4247 0.0893 −0.4161 0.0874

1.6106 0.0292 1.5812 0.0297 1.6447 0.0299 1.6124 0.0291

−2.4421 0.1456 −2.4020 0.1423 −2.4941 0.1486 −2.4446 0.1457

0.2246 −0.0471 0.2234 −0.0465 0.2290 −0.0481 0.2243 −0.0470

−17.399 1.7028 −16.557 1.8100 −17.774 1.7394 −17.423 1.6982

17.8478 −1.7969 17.0036 −1.8905 18.2326 −1.8356 17.8717 −1.7922

0.2241 −0.0470 0.2229 −0.0465 0.2289 −0.0480 0.2243 −0.0470

−0.4805 0.0229 −0.4481 0.0237 −0.4907 0.0233 −0.4813 0.0230

0.9288 −0.1170 0.8940 −0.1141 0.9486 −0.1194 0.9298 −0.1171

100

100

HSDT (Rouzegar and Abbasi, 2017). FSDT (Ray and Sachade, 2006b). Exact (Ray and Sachade, 2006a).

b c

Table 2 The dimensionless displacements and stresses of EG square plates (Et / Eb = 10) integrated with top PFRC layer. K0

J0

(

a b , , b 2

(

b 2

w¯

0 10 0 10

u¯ 0, , 0 10 0 10

¯x

(

0 10 0 10

(

S = 20

¯0 = 0

¯0 = 100

¯0 =

0 0 10 10

−0.0877 −0.0806 −0.0321 −0.0311

17.0750 15.6983 6.2519 6.0574

0 0 10 10

0.0855 0.0786 0.0313 0.0303

0 0 10 10 0 0 10 10

)

S = 100

¯0 = 0

¯0 = 100

¯0 =

−17.25044 −15.8595 −6.3162 −6.1196

−0.0847 −0.0781 −0.0317 −0.0307

4.1639 3.8386 1.5579 1.5100

−53.8579 −52.5152 −43.3029 −43.1132

54.0290 52.6725 43.3655 43.1739

0.1698 0.1566 0.0636 0.0616

−0.0419 −0.0386 −0.0154 −0.0149

20.2432 19.5845 15.0655 14.9725

−20.3270 −19.6617 −15.0962 −15.002

0.2262 0.2079 0.0828 0.0802

−70.3546 −66.8043 −42.4451 −41.9435

70.8069 67.2202 42.6107 42.1039

100

¯0 = 0

¯0 = 100

¯0 =

−4.3334 −3.9949 −1.6213 −1.5714

−0.0838 −0.0773 −0.0316 −0.0306

0.0856 0.0789 0.0322 0.0313

−0.2532 −0.2336 −0.0954 −0.0925

−26.7852 −26.1331 −21.5609 −21.4649

27.1249 26.4463 21.6879 21.5881

0.8475 0.7820 0.3193 0.3096

−4.5423 −4.4754 −4.0027 −3.9927

6.2375 6.0395 4.6414 4.6119

−0.041.7 −0.0384 −0.0156 −0.0151

5.0255 4.8656 3.7442 3.7206

−5.1089 −4.9424 −3.7753 −3.7508

−0.0415 −0.0384 −0.0156 −0.0152

0.1610 0.1578 0.1346 0.1341

−0.2442 −0.2345 −0.1659 −0.1645

0.2246 0.2071 0.0840 0.0815

−17.3986 −16.5364 −10.4915 −10.3646

17.8478 16.9505 10.6595 10.5275

0.2241 0.2068 0.0844 0.0819

−0.4805 −0.4628 −0.3378 −0.3352

0.9288 0.8764 0.5067 0.4989

100

100

0

h 2

)

a b h , , 2 2 2

¯xy 0,0, 0 10 0 10

S = 10

h 2

)

)

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1 S 2 q0

¯x =

x

a b ±h , , , 2 2 2

¯xy =

1 S 2 q0

0,0,

xy

shear stress ¯xy of EG square plates attached to bottom PFRC actuator under distributed sinusoidally mechanical load in the presence/absence of positive or negative applied voltages for Et /Eb = 10 and Et /Eb = 0.1, respectively. The effect of bottom PFRC actuator on thick plate is more than thin plate and the deformations produced by the applied positive and negative voltages confirm the effectiveness of the actuator layer. It can be noted that the transverse displacement u¯3 and the in-plane shear stress ¯xy decreases with the presence of the elastic foundations for the EG plate with Et /Eb = 10 while the in-plane displacement u¯1 increases and the in-plane normal stress ¯x changed variably as shown in Table 4. Table 5 shows that the transverse displacement u¯3 and the in-plane shear stress ¯xy decreases with the presence of the elastic foundations while the in-plane normal stress ¯x decreases except when (S = 100 , ¯0 = 100 ), the stresses are increasing. The in-plane displacement u¯1 increases with the presence of the elastic foundations when both electrical and mechanical loads are applied and decreases when only mechanical loads are applied. It can be concluded from the tables that when the PFRC layer is subjected to negative voltage, the deformations are in the opposite direction of the deformations produced by the mechanical loading and when subjected to positive voltage it increases the deformations produced by the mechanical loading. The distribution of the non-dimensional displacements (u¯1 and u¯3 ) and the stresses ( ¯x and ¯xy ) through the thickness of EG plate with top PFRC actuator for Et /Eb = 10 and a/ h = 100 is shown in Fig. 2. It can be noted that the in-plane displacement u¯1 shows linear behavior through the thickness while the transverse displacement u¯3 is constant across thickness. The variations of the in-plane normal stress ¯x and the inplane shear stress ¯xy are nonlinear through the thickness of the EG plate. When the polarity of the voltage changing from negative to positive, the PFRC actuator makes reversal influence on the displacements and stresses of the EG plate. The variation of the dimensionless in-plane stress ¯xy through the thickness of the EG plate with top PFRC actuator and for K 0 = J0 = 5 are presented in Fig. 3a and Fig. 3b for Et /Eb = 10 and Et /Eb = 0.1, respectively. When the EG plate attached with PFRC actuator at the bottom of the plate, the variation of ¯xy are presented in Fig. 4a and b for Et /Eb = 10 and Et /Eb = 0.1, respectively. These figures reveal that the variation of the in-plane tangential stress ¯xy across the thickness of the EG plate is extremely influenced by the PFRC actuator. When the

±h . 2

The non-dimensional in-plane displacement u¯1, transverse displacement u¯3, in-plane normal stress ¯x and the in-plane shear stress ¯xy of EG square plates integrated with PFRC layer at the top surface with Et /Eb = 10 and under the influence of distributed sinusoidally mechanical and electrical loads is presented in Table 1. The results indicate good agreement with the exact solution and closer results to the exact solution than the results obtained using the FEMs. It is observed that the PFRC actuator effect thick plate more than thin plate. Example 2. The numerical results of the simply supported EG plates integrated at the top or bottom with a layer of PFRC actuator and resting on Pasternak-Winkler's elastic foundations are presented in this section. The non-dimensional parameters used are

u¯1 =

D0 b ±h 10D a b u1 0, , , u¯3 = 4 0 u3 , , 0 , S 2 q0 2 2 S q0 2 2

¯x =

1 10S 2 q0

x

1 S 2 q0

0,

=

xz

D0 K 0 , a4

K=

a b ±h , , , 2 2 2

J=

¯xy =

1 S 2 q0

xy

0,0,

±h , 2

¯xz

b ±h , , 2 2 D0 J0 , a2

D0 =

h3Eb 12(1

2)

.

The influence of top PFRC actuator and different elastic foundation parameters on the non-dimensional in-plane displacement u¯1, transverse displacement u¯3, in-plane normal stress ¯x and the in-plane shear stress ¯xy of EG square plates under distributed sinusoidally mechanical load in the presence/absence of positive or negative applied voltages are presented in Table 2 and Table 3 for Et /Eb = 10 and Et /Eb = 0.1, respectively. The tables show that the effect of top PFRC actuator on thick plate is more than thin plate. The actuation case with positive and negative voltages produces more deformations than the non-actuation case with zero voltage. The non-dimensional displacements and stresses decreases with the presence of the elastic foundations. Tables 4 and 5 show the non-dimensional in-plane displacement u¯1, transverse displacement u¯3, in-plane normal stress ¯x and the in-plane

Table 3 The dimensionless displacements and stresses of EG square plates (Et / Eb = 0.1) integrated with top PFRC layer. K0

J0

(

a b , , b 2

(

b 2

w¯

0 10 0 10

u¯ 0, , 0 10 0 10

¯x

(

0 10 0 10

(

S = 20

¯0 = 0

¯0 = 100

¯0 =

0 0 10 10

−0.0080 −0.0079 −0.0069 −0.0068

2.9676 2.9440 2.5623 2.5447

0 0 10 10

0.0157 0.0156 0.0136 0.0135

0 0 10 10 0 0 10 10

S = 100

¯0 = 0

¯0 = 100

¯0 =

−2.9836 −2.9599 −2.5761 −2.5584

−0.0077 −0.0076 −0.0067 −0.0066

0.7326 0.7270 0.6357 0.6315

−9.6599 −9.6135 −8.8624 −8.8278

9.6915 9.6448 8.8897 8.8545

0.0314 0.0312 0.0272 0.0271

−0.0078 −0.0077 −0.0067 −0.0066

4.1589 4.1359 3.7645 3.7475

−4.1745 −4.1513 −3.7781 −3.7609

0.0424 0.0421 0.0365 0.0363

−18.7795 −18.6547 −16.6356 −16.5425

18.8642 18.7388 16.7088 16.6151

100

¯0 = 0

¯0 = 100

¯0 =

−0.7481 −0.7423 −0.6491 −0.6448

−0.0076 −0.0075 −0.0066 −0.0065

0.0219 0.0217 0.0191 0.0189

−0.0372 −0.0369 −0.0323 −0.0321

−4.8053 −4.7824 −4.4107 −4.3935

4.8682 4.4845 4.4653 4.4477

0.1571 0.1559 0.1365 0.1356

−0.8102 −0.8068 −0.7511 −0.7485

1.1244 1.1186 1.0241 1.0197

−0.0077 −0.0077 −0.0067 −0.0066

1.0336 1.0279 0.9361 0.9318

−1.0492 −1.0434 −0.9496 −0.9452

−0.0077 −0.0077 −0.0067 −0.0067

0.0339 0.0337 0.0309 0.0308

−0.0494 −0.0491 −0.0444 −0.0442

0.0423 0.0419 0.0366 0.0364

−4.6617 −4.6309 −4.1313 −4.1082

4.7463 4.7149 4.2047 4.1810

0.0422 0.0419 0.0367 0.0365

−0.1459 −0.1449 −0.1300 −0.1293

0.2304 0.2288 0.2034 0.2022

100

100

0

h 2

)

a b h , , 2 2 2

¯xy 0,0, 0 10 0 10

)

S = 10

h 2

)

)

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Table 4 The dimensionless displacements and stresses of EG square plates (Et / Eb = 10) integrated with bottom PFRC layer. K0

J0

(

a b , , b 2

(

b 2

w¯

0 10 0 10

u¯ 0, , 0 10 0 10

¯x

(

0 10 0 10

)

0 0 10 10 h 2

)

0 0 10 10

)

0 0 10 10

(

S = 20

¯0 = 0

¯0 = 100

¯0 =

−0.0871 −0.0800 −0.0320 −0.0310

33.2318 30.5703 12.2239 11.8447

0.0858 0.0789 0.0315 0.0306

S = 100

¯0 = 0

¯0 = 100

¯0 =

−33.4059 −30.7305 −12.2880 −11.9067

−0.0841 −0.0776 −0.0316 −0.0306

8.2056 7.5689 3.0845 2.9899

−8.3738 −7.7241 −3.1477 −3.0512

−0.0832 −0.0768 −0.0315 −0.0305

0.2479 0.2289 0.0938 0.0909

−0.4142 −0.3824 −0.1568 −0.1520

2.9632 5.5848 23.6556 24.0292

−2.7917 −5.4270 −23.5925 −23.9681

0.1703 0.1571 0.0640 0.0620

1.6250 2.9143 11.9959 12.1874

−1.2849 −2.6000 −11.8679 −12.0632

0.8498 0.7846 0.3217 0.3119

1.1418 1.3363 2.7159 2.7452

0.5578 0.2329 −2.0726 −2.1215

−0.0419 −0.0386 −0.0154 −0.0149

4.6778 3.3953 −5.4452 −5.6279

−4.7618 −3.4725 5.4143 5.5981

−0.0417 −0.0384 −0.0157 −0.0152

1.1344 0.8190 −1.4024 −1.4492

−1.2177 −0.8959 1.3711 1.4189

−0.0416 −0.0384 −0.0157 −0.0153

0.2254 0.2074 0.0829 0.0803

−63.6978 −56.8063 −9.3027 −8.3205

64.1487 57.2210 9.4686 8.4812

0.2239 0.2065 0.0842 0.0816

−15.7359 −14.0413 −2.1048 −1.8531

16.1837 14.4544 2.2731 2.0163

100

¯0 = 0

100

¯0 = 100

¯0 =

100

0

a b h , , 2 2 2

¯xy 0,0, 0 10 0 10

S = 10

h 2

)

0 0 10 10

Young's modulus is maximum at the top surface of the EG plate (Et / Eb = 10) , the PFRC actuator make the stresses to be maximum at the top surface of the plate as shown in Figs. 3a and 4a. When the Young's modulus is maximum at the bottom surface of the EG plate (Et /Eb = 0.1), the maximum stresses occur at the bottom surface of the plate as shown in Figs. 3b and 4b. The variation of the non-dimensional displacements (u¯3 and u¯1) and the stresses ( ¯x and ¯xy ) through the thickness of EG plate with bottom PFRC actuator and with Et /Eb = 10, ¯0 = 100 and a/ h = 100 is presented in Fig. 5. The considered cases are

0.0054 −0.0041 −0.0716 −0.0730

0.2234 0.2062 0.0844 0.0819

−0.4147 −0.3636 −0.0009 0.0068

−0.0886 −0.0727 0.0401 0.0425 0.8615 0.7761 0.1700 0.1572

Case III: K 0 = 100, J0 = 0 . Case IV: K 0 = 100, J0 = 100. The non-dimensional transverse displacements u¯3 decreases with presence of the elastic foundation and the smallest deflections occurs with elastic foundations parameters in case IV as shown in Fig. 5a. The EG plate with non-elastic foundation gives the maximum in-plane displacement u¯1 at the bottom surface of the plate and becomes minimum at the top surface of the plate as shown in Fig. 5b. While the inclusion of Pasternak's foundation with the parameters values in case IV gives the minimum displacements at the bottom surface of the plate and becomes maximum at the top surface of the plate. It can be observed that the in-

Case I: K 0 = 0, J0 = 0. Case II: K 0 = 50, J0 = 10.

Table 5 The non-dimensional displacements and stresses of EG square plates (Et / Eb = 0.1) integrated with bottom PFRC layer. J0

K0

(

a b , , b 2

(

b 2

w¯

0 10 0 10

u¯ 0, , 0 10 0 10

¯x

(

0 10 0 10

(

S = 20

¯0 = 0

¯0 = 100

¯0 =

0 0 10 10

−0.0086 −0.0085 −0.0073 −0.0072

1.6235 1.6096 1.3885 1.3784

0 0 10 10

0.0178 0.0176 0.0152 0.0151

0 0 10 10

0 0 10 10

S = 100

¯0 = 0

¯0 = 100

¯0 =

−1.6406 −1.6267 −1.4032 −1.3929

−0.0083 −0.0082 −0.0071 −0.0070

0.3958 0.3925 0.3402 0.3378

0.1181 0.1467 0.6047 0.6256

−0.0826 −0.1115 −0.5743 −0.5954

0.0354 0.0351 0.0304 0.0302

−0.0087 −0.0086 −0.0074 −0.0073

0.5360 0.5221 0.2986 0.2884

−0.5533 −0.5393 −0.3134 −0.3031

0.0464 0.0460 0.0397 0.0394

−6.6172 −6.5425 −5.3449 −5.2902

6.7100 6.6345 5.4243 5.3689

100

¯0 = 0

¯0 = 100

¯0 =

−0.4123 −0.4089 −0.3544 −0.3519

−0.0082 −0.0081 −0.0070 −0.0069

0.0079 0.0078 0.0068 0.0067

−0.0243 −0.0241 −0.0209 −0.0208

0.0870 0.1009 0.3247 0.3349

−0.0162 −0.0307 −0.2638 −0.2746

0.1768 0.1754 0.1522 0.1512

0.1872 0.1886 0.2110 0.2121

0.1664 0.1621 0.0934 0.0902

−0.0086 −0.0085 −0.0074 −0.0073

0.1272 0.1238 0.0692 0.0667

−0.1445 −0.1409 −0.0840 −0.0814

−0.0086 −0.0085 −0.0074 −0.0073

−0.0032 −0.0033 −0.0043 −0.0044

−0.0141 −0.0139 −0.0104 −0.0103

0.0463 0.0459 0.0398 0.0395

−1.6178 −1.5996 −1.3069 −1.2934

1.7103 1.6914 1.3866 1.3726

0.0462 0.0459 0.0398 0.0395

−0.0203 −0.0199 −0.0141 −0.0138

0.1128 0.1117 0.0937 0.0929

100

100

0

h 2

)

a b h , , 2 2 2

¯xy 0,0, 0 10 0 10

)

S = 10

h 2

)

)

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European Journal of Mechanics / A Solids 75 (2019) 461–471

A.M. Zenkour and R.A. Alghanmi

Fig. 2. The distribution of the non-dimensional displacements and stresses through the thickness of EG plate with top PFRC layer (K 0 = J0 = 0, Et / Eb = 10) : (a) u¯3 , (b) u¯1, (c) ¯x , (d) ¯xy .

Fig. 3. The non-dimensional in-plane tangential stress through the thickness of EG plate with top PFRC layer (K 0 = J0 = 5 ): (a) Et / Eb = 10 , (b) Et / Eb = 0.1.

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European Journal of Mechanics / A Solids 75 (2019) 461–471

A.M. Zenkour and R.A. Alghanmi

Fig. 4. The non-dimensional in-plane tangential stress through the thickness of EG plate with bottom PFRC layer (K 0 = J0 = 5): (a) Et / Eb = 10 , (b) Et / Eb = 0.1.

Fig. 5. The variation of the dimensionless displacements and stresses through the thickness of EG plate with bottom PFRC layer ( ¯0 = 100 , Et / Eb = 10) : (a) u¯3 , (b) u¯1, (c) ¯x , (d) ¯xy .

plane displacement u¯1 is independent of the elastic foundation at z 0.18. The non-elastic foundation case gives the maximum tensile normal stress ¯x at the top surface of the EG plate while gives the maximum compressive stress at the bottom surface of the plate as shown in Fig. 5c. Fig. 5b shows that the non-elastic foundation case gives the maximum compressive shear stress ¯xy at the top surface of the EG plate and maximum tensile stress at the bottom surface of the plate.

Again, the last two figures show that these stresses are independent of the elastic foundation at z 0.18. Finally, The variation of the dimensionless stresses ¯x and ¯xz through the thickness of the EG plate (Et /Eb = 10 ) with top PFRC actuator and resting on Pasternak's foundation with the values of K 0 = 10 and J0 = 10 for different applied voltages are presented in Fig. 6. The non-dimensional normal stress ¯x gives the maximum stresses at the top surface of the EG plate due to the 469

European Journal of Mechanics / A Solids 75 (2019) 461–471

A.M. Zenkour and R.A. Alghanmi

Fig. 6. The distribution of the dimensionless stresses through the thickness of EG plate with top PFRC layer (K 0 = J0 = 10, Et /Eb = 10) : (a) ¯x , (b) ¯xz .

location of the PFRC actuator and the maximum value of Young's modulus as shown in Fig. 6a. The non-dimensional shear stress ¯xz gives the maximum stresses at z 0.26 for all the values of the applied voltage as shown in Fig. 6b.

for the current formulation is constructed by comparison with available results. From numerical examples, it is concluded that

• The numerical results indicate that the current formulation for the

5. Conclusions

• • • • •

Four-unknown shear deformation theory for the bending of an EG plates with a single-layer of PFRC actuator attached at the top or bottom surface of the plate is presented. The plate is resting on Pasternak's or Winkler's elastic foundation models and undergo sinusoidal electromechanical loadings. The electric potential distributions are vary linearly along the thickness direction. By considering the principle of virtual displacements, the equations of equilibrium are established. The problem is solved by following Navier's technique. The hybrid plate is simply-supported at its four edges. The obtained results due to the parameters ¯0 , S , K 0 , J0 and Et are all investigated. Validation

bending of an EG plate with non-elastic foundation is efficient compared with published results. The EG plate is highly influenced by the top or bottom PFRC layer that acts like an actuator. The PFRC actuator effects thick plate more than thin plate. The displacements and stresses are sensitive to the presence of the elastic foundations. Maximum stresses are located where the Young's modulus of the material is maximum. The negative applied voltage on the PFRC layer produces deformations in the opposite direction of the deformations caused by the downward mechanical loading while the positive voltage increases the mechanical deformations.

Appendix I

A1 A2 A3 A 4 A5 A6 = A7 A8 A9

h 2

h 2

A19 A20 A21 = A22 A23 A24

{A25 , A26 } =

h 2

c c66

h 2

c c {c44 , c55 }[1

h 2

N x Mx Sx N y M y Sy

h +h p 2 h 2

c11p c12p [1 z f (z )]dz, c22p

c c11 c c12 [ z 2 zf (z ) f 2 (z )]dz + c c22

h 2

A10 A11 A16 A12 A13 A17 = A14 A15 A18

h 2

c c11 c c12 [1 z f (z )]dz + c c22

h 2

1 z f (z ) dz + z 2 zf (z ) f 2 (z )

2

h +h p 2

f (z )] dz +

h +h p 2 h 2

h +h p 2

c66p

h 2

{c44p , c55p }[1

c11p c12p [ z 2 zf (z ) f 2 (z )]dz, c22p

1 z f (z ) dz , z 2 zf (z ) f 2 (z )

f (z )]2 dz,

h 2

h +h p 2

= h 2

0 e31 [1 z f (z )]dz. hp e32

foundation. Composites Part B 96, 136–152. Crawley, E.F., de Luis, J., 1987. Use of piezoelectric actuators as elements of intelligent structures. AIAA J. 25, 1373–1385.

References Akavci, S.S., 2016. Mechanical behavior of functionally graded sandwich plates on elastic

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European Journal of Mechanics / A Solids 75 (2019) 461–471

A.M. Zenkour and R.A. Alghanmi

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