An electromechanical higher order model for piezoelectric ...tkant/papers/TKant-IITB-JP120.pdf · An electromechanical higher order model for piezoelectric functionally graded plates

An electromechanical higher order model for piezoelectricfunctionally graded plates

S. M. Shiyekar • Tarun Kant

Received: 19 March 2010 / Accepted: 31 March 2010 / Published online: 23 April 2010

� Springer Science+Business Media, B.V. 2010

Abstract Bidirectional flexure analysis of function-

ally graded (FG) plate integrated with piezoelectric fiber

reinforced composites (PFRC) is presented in this paper.

A higher order shear and normal deformation theory

(HOSNT12) is used to analyze such hybrid or smart

FG plate subjected to electromechanical loading. The

displacement function of the present model is approx-

imated as Taylor’s series in the thickness coordinate,

while the electro-static potential is approximated as

layer wise linear through the thickness of the PFRC

layer. The equations of equilibrium are obtained using

principle of minimum potential energy and solution is

by Navier’s technique. Elastic constants are varying

exponentially along thickness (z axis) for FG material

while Poisson’s ratio is kept constant. PFRC actuator

attached either at top or bottom of FG plate and analyzed

under mechanical and coupled mechanical and electri-

cal loading. Comparison of present HOSNT12 is made

with exact and finite element method (FEM).

Keywords Higher order theory � Piezoelectric

fiber reinforced composites � Functionally Graded

1 Introduction

Piezoelectric materials transform elastic field into the

electric field and converse behavior leads many

researchers to study their controlling capabilities

applicable to structures like plates and shells. Such

structures are called as smart, intelligent, adaptive as

well as hybrid structures. In conventional composites

failure occurs at interface due to abrupt change in

material properties. Elastic properties are varying

smoothly across the thickness of the FG material and

hence failure due to de lamination is avoided.

Piezoelectric materials show coupling phenome-

non between elastic and electric fields. Tiersten and

Mindlin (1962) initiated work on piezoelectric plates.

Further Tiersten (1969) contributed this work by

exploring the governing equations of linear piezo-

electric continuum by analyzing vibrations of a single

piezoelectric layer.

Monolithic piezoelectric materials exhibit very low

stress/strain coefficients and hence low controlling

capabilities. Smith and Auld (1991) presented micro-

mechanical analysis of vertically reinforced piezo-

electric composites with slight increase in the stress/

strain piezoelectric coefficients.

Mallik and Ray (2003) proposed the concept of

unidirectional piezoelectric fiber reinforced composite

(PFRC) materials and presented their effective elastic

and piezoelectric properties. Piezoelectric stress/strain

coefficients are improved considerably as compared

to monolithic piezoelectric materials. Vertically

S. M. Shiyekar (&)

Department of Civil Engineering, Sinhgad College

of Engineering, Vadgaon (Bk), Pune 411 041, India

e-mail: [email protected]

T. Kant

Department of Civil Engineering, Indian Institute

of Technology Bombay, Powai 400 076, India

e-mail: [email protected]

123

Int J Mech Mater Des (2010) 6:163–174

DOI 10.1007/s10999-010-9124-4

reinforced piezoelectric composites are not suitable for

the bending mode actuation (Mallik and Ray 2003).

Many investigators studied hybrid composite lam-

inates using various plate theories. Kapuria and Dumir

(2000) presented coupled first-order shear deforma-

tion theory for hybrid laminated plates subjected to

thermoelectrical loading. Elasticity solutions always

serve as benchmark for other approximate solutions.

Ray et al. (1993) developed three-dimensional (3D)

elasticity solutions for intelligent plate simply sup-

ported and perfectly bonded with distributed Polyvi-

nylidene Fluoride (PVDF) piezoelectric layers at top

and bottom and presented static displacement control

for different span to depth ratios. Heyliger (1994)

obtained exact solution for an unsymmetric cross ply

composite laminate attached with PZT-4 layers of

piezoelectric material at upper and lower surfaces. The

3D solution methodology used by Ray and Heyliger is

the same as the work of Pagano (1970) for laminated

composite plates. Later Heyliger (1997) provided the

3D exact solution for single and two layers of

piezoelectric materials. Vel and Batra (2000) used

Eshelby-Stroh formulation to obtained 3D elasticity

solution to analyze multilayered piezoelectric plate for

arbitrary boundary conditions. Batra and Vel (2001)

presented exact thermo elastic solutions for FG plates,

while Sankar (2001) presented 3D exact solutions for

functionally graded beams under mechanical pressure.

Ray and Sachade (2006a, b) reported 3D elasticity

solution and FEM for FG plate attached with PFRC

actuator. Reddy and Cheng (2001) also presented

elasticity solutions for smart FG plate.

Higher order shear deformation theories (HOST)

incorporate transverse shear and normal deformation

by expanding the primary displacement fields. Initially

Kant (1982) developed complete set of variationally

consistent governing equations of equilibrium and

presented first FE model based on HOST (Kant et al.

1982). Pandya and Kant (1987, 1988) and Kant and

Manjunatha (1988, 1994) extended the HOST for

unsymmetric laminates. Further Kant and Swaminathan

(2002) presented a refined higher order theory and

discussed analytical solution for sandwiches and lam-

inates. Reddy (1984) presented a simple third order

theory for laminates maintaining zero shear stress

conditions at boundaries of the thickness dimension.

In this paper, a higher order shear and normal

deformation theory (HOSNT12) is used to model the

elastic displacements of FG plate whereas; electric

potential in the PFRC actuator layer is modeled as

piece wise linear.

2 Formulation

Consider bidirectional flexure of functionally graded

(FG) plate. At x = 0, a and at y = 0, b the FG plate is

simply supported and attached with distributed PFRC

actuator of thickness tp at top or bottom as shown in

the Fig. 1. Span of the hybrid FG plate is a along x

axis and b along y axis in Cartesian coordinate

system. Thickness of FG plate is h along z axis and

located at -h/2 and ?h/2 from bottom and top of FG

plate, respectively. Electrostatic potential is applied

at top of PFRC actuator (h/2 ? tp) and FG plate is

electrically grounded at z = ?h/2.

Displacement components u(x,y,z), v(x,y,z) and

w(x,y,z) at any point in the plate are expanded in a

x

z

u0

w0

yh

Simple (Diaphragm) Supports

Functionally graded plate

Piezoelectric Fiber Reinforced Composite (PFRC) Layer

a

tp

Fig. 1 Geometry of

Functionally Graded (FG)

simply (diaphragm)

supported along all edges

plate attached with PFRC

actuator

164 S. M. Shiyekar, T. Kant

123

Taylor’s series to approximate the three-dimensional

(3D) elasticity problem as a two-dimensional (2D)

plate problem. The assumed displacement field of

present higher order shear and normal deformation

model is in the following form:

Model HOSNT12: (Kant and Swaminathan 2002)

uðx;y;zÞ¼ u0ðx;yÞþzhxðx;yÞþz2u�0ðx;yÞþz3h�xðx;yÞvðx;y;zÞ¼ v0ðx;yÞþzhyðx;yÞþ z2v�0ðx;yÞþ z3h�yðx;yÞwðx;y;zÞ¼ w0ðx;yÞþzhzðx;yÞþz2w�0ðx;yÞþz3h�z ðx;yÞ

ð1Þ

where the parameters u0, v0 are in-plane displacements

and w0 is transverse displacement at any point (x, y) on

the middle plane of the plate. hx & hy are the rotations of

the normal to the mid-plane about y and x axes,

respectively. Other parameters such as u�0;h�x ;v�0;h�y ;

w�0;h�z are the corresponding higher order terms in the

Taylor series expansion at mid-plane.

Strain displacement relationship as per classical

theory of elasticity is written in Eq. 2 as

ex

ey

ez

cxy

cyz

cxz

8>>>>>><

>>>>>>:

9>>>>>>=

>>>>>>;

¼ ouox

ovoy

owoz

ouoyþ ov

oxovozþ ow

oyouozþ ow

ox

n otð2Þ

Tiersten (1969) presented linear constitutive equa-

tion which couples the elastic and electric field for a

single piezoelectric layer as

frg ¼½C�feg � ½e�fEgL

fDg ¼½e�tfeg þ ½g�fEgLð3Þ

The electric field intensity vector E related to electro-

static potential n(x,y,z) in the Lth layer is given by

ELx ¼ �

onðx; y; zÞL

ox; EL

y ¼ �onðx; y; zÞL

oy;

ELz ¼ �

onðx; y; zÞL

oz

where r, Q, e, e, E, D and g are, stress vector, elastic

constant matrix, strain vector, piezoelectric constant

matrix, electric field intensity vector, electric dis-

placement vector and dielectric constant matrix,

respectively. Effective piezoelectric constant matrix

e and dielectric matrix g for PFRC layer aligned at

zero angle with respect to x-axis is given by Mallik

and Ray (2003) as

e½ � ¼

0 0 e31

0 0 e32

0 0 e33

0 0 0

0 e24 0

e15 0 0

2

6666664

3

7777775

; g½ � ¼g11 0 0

0 g22 0

0 0 g33

2

4

3

5

ð5ÞThe first set of Eq. 3 is divided in two components

of stresses. One is elastic stress component (es) and

other is piezoelectric stress component (pz).

First set of Eq. 3 is written as rf g ¼ rf ges� rf gpz

where

rf ges¼

C11 C12 C13 0 0 0

C12 C22 C23 0 0 0

C13 C23 C33 0 0 0

0 0 0 C44 0 0

0 0 0 0 C55 0

0 0 0 0 0 C66

2

6666664

3

7777775

L ex

ey

ez

cxy

cyz

cxz

8>>>>>><

>>>>>>:

9>>>>>>=

>>>>>>;

;

rf gpz¼

0 0 e31

0 0 e32

0 0 e33

0 0 0

0 e24 0

e15 0 0

2

6666664

3

7777775

L

� onðx;y;zÞox

� onðx;y;zÞoy

� onðx;y;zÞoz

8><

>:

9>=

>;

L

ð6Þ

and Cij are elastic constants of FG plate and defined

as per isotropic layer.

C11 ¼ C22 ¼ C33 ¼ð1� mÞEðzÞð1þ mÞð1� 2mÞ ;

C12 ¼ C13 ¼ C23 ¼mEðzÞ

ð1þ mÞð1� 2mÞ ;

C44 ¼ C55 ¼ C66 ¼EðzÞ

2ð1þ mÞ ¼ G:

ð7Þ

Young’s modulus of FG material is a function of z

coordinate and governed by exponential law (Sankar

2001), where as Poisson’s ratio is kept constant.

E zð Þ ¼ E0ekðzþh=2Þ ð8Þ

where E0 is Young’s modulus of FG material at

bottom and k is a non-homogeneous parameter of the

FG material across the thickness.

An electromechanical higher order model 165

123

Stress resultants are also defined as elastic and

piezoelectric stress resultants.

Elastic stress resultants [Q, S, M, N]es

Qesx ;Q

esy jQes�

x ;Qes�

y

h i

¼Xn

L¼1

Zþh=2

�h=2

sesxz; s

esyz

n o1jz2� �

dz; Sesx ; S

esy jSes�

x ; Ses�

y

h i

¼Xn

L¼1

Zþh=2

�h=2

sesxz; s

esyz

n ozjz3� �

dz

Mesx ;M

esy ;M

esxyjMes�

x ;Mes�

y ;Mes�

xy

h i

¼Xn

L¼1

Zþh=2

�h=2

resx ; r

esy ; s

esxy

n ozjz3� �

dz;Mesz

¼Xn

L¼1

Zþh=2

�h=2

rezdz;

Nesx ;N

esy ;N

esz ;N

esxyjNes�

x ;Nes�

y ;Nes�

z ;Nes�

xy

h i

¼Xn

L¼1

Zþh=2

�h=2

resx ; r

esy ; r

esz ; s

esxy

n o1jz2� �

dz: ð9Þ

Piezoelectric stress resultants [Q, S, M, N]pz

Qpzx ;Q

pzy jQpz�

x ;Qpz�

y

h i

¼Zh=2þtp

þh=2

spzxz ; s

pzyz

n o1jz2� �

dz; Spzx ; S

pzy jSpz�

x ; Spz�

y

h i

¼Zh=2þtp

þh=2

spzxz ; s

pzyz

n ozjz3� �

dz

Mpzx ;M

pzy ;M

pzxy jMpz�

x ;Mpz�

y ;Mpz�

xy

h i

¼Zh=2þtp

þh=2

rpzx ; r

pzy ; s

pzxy

n ozjz3� �

dz;Mpzz ¼

Zh=2þtp

þh=2

rpzz dz;

Npzx ;N

pzy ;N

pzz ;N

pzxy jNpz�

x ;Npz�

y ;Npz�

z ;Npz�

xy

h i

¼Zh=2þtp

þh=2

rpzx ; r

pzy ; r

pzz ; s

pzxy

n o1jz2� �

dz: ð10Þ

Total stress resultants (pz): Total stress resultants

[Q, S, M, N] are algebraic sum of elastic and piezo-

electric stress resultants.

½Q; S;M;N� ¼ ½Q; S;M;N�es þ ½Q; S;M;N�pz ð11ÞGoverning equations of equilibrium.

Using principal of minimum potential energy, the

equations of equilibrium are obtained as

du0 :oNx

oxþ oNxy

oy¼ 0 du�0 :

oN�xoxþ

oN�xy

oy� 2Sx ¼ 0

dv0 :oNy

oyþ oNxy

ox¼ 0 dv�0 :

oN�yoyþ

oN�xy

ox� 2Sy ¼ 0

dw0 :oQx

oxþ oQy

oyþ ðqþz Þ ¼ 0 dw�

0:oQ�xoxþ

oQ�yoy� 2M�z þ

h2

4ðqþz Þ ¼ 0

dhx :oMx

oxþ oMxy

oy� Q

x¼ 0 dh�x :

oM�xoxþ

oM�xy

oy� 3Q�x ¼ 0

dhy :oMy

oyþ oMxy

ox� Qy ¼ 0 dh�y :

oM�yoyþ

oM�xy

ox� 3Q�y ¼ 0

dhz :oSx

oxþ oSy

oy� Nz þ

h

2ðqþz Þ ¼ 0 dh�z :

oSx

oxþ oSy

oy� 3N�z þ

h3

8ðqþz Þ ¼ 0

ð12Þ


123

Total stress resultants are the addition of elastic

and piezoelectric stress resultants.

Following are the mechanical boundary conditions

used for simply supported plate

At edges x ¼ 0 and x ¼ a :

v0 ¼ 0; w0 ¼ 0; hy ¼ 0; hz ¼ 0;

Mx ¼ 0; Nx ¼ 0; v�0 ¼ 0; w�0 ¼ 0;

h�y ¼ 0; h�z ¼ 0; M�x ¼ 0; N�x ¼ 0:

At edges y ¼ 0 and y ¼ b :

u0 ¼ 0; w0 ¼ 0; hx ¼ 0; hz ¼ 0;

My ¼ 0; Ny ¼ 0; u�0 ¼ 0; w�0 ¼ 0;

h�x ¼ 0; h�z ¼ 0; M�y ¼ 0; N�y ¼ 0:

ð13Þ

Navier’s solution procedure is adopted to find the

solution of displacement variables, satisfying the

above boundary conditions and is expressed as follows:

where qþz is the mechanical loading term and n(x, z) is

the electrical loading term. Through thickness electric

potential nmnðzÞ is assumed as per the following.

The elastic FG layer is attached with distributed

actuator layer of PFRC. The thickness of the PFRC

layer is small as compared to the thickness of the

substrate, the electro-static potential in the actuator

layer is assumed to be linear through the thickness of

the PFRC layer.

nmnðzÞ ¼Vt

mn

tp

� �

z� Vtmnh

2tp

� �

ð15Þ

Equation 15 gives linear variation of thought thick-

ness electro-static potential in the PFRC actuating layer.

Vtmn represents amplitude of doubly sinusoidal electro-

static potential applied at top of the PFRC whereas; h

and tp are thicknesses of elastic FG plate and PFRC

layer, respectively. Assumed electrostatic potential

satisfies zero electric potential at interface. Similar

electrostatic potential can be assumed for PFRC layer

at bottom of FG plate.

The piezoelectric stress vectors are calculated

from second set of Eq. 6 by substituting actuating

electric function (Eq. 15) when either top or bottom

voltages (Vtmn, Vb

mn) are applied. Finally, piezoelectric

stress resultants are evaluated from Eq. 10. Similarly

elastic stress vectors and elastic stress resultants are

calculated from first set of Eqs. 6 and 9, respectively.

Displacement terms are obtained by solving linear

algebraic equations by substituting total stress resul-

tants from Eq. 11 in a set of equilibrium equations

(Eq. 12).

u0 ¼X1

m¼1

X1

n¼1

u0mncos

mpx

a

� �sin

npy

b

� �u�

0¼X1

m¼1

X1

n¼1

u�0mncos

mpx

a

� �sin

npy

b

� �

v0 ¼X1

m¼1

X1

n¼1

v0mnsin

mpx

a

� �cos

npy

b

� �v�

0¼X1

m¼1

X1

n¼1

v�0mnsin

mpx

a

� �cos

npy

b

� �

w0 ¼X1

m¼1

X1

n¼1

w0mnsin

mpx

a

� �sin

npy

b

� �w�0 ¼

X1

m¼1

X1

n¼1

w�0mnsin

mpx

a

� �sin

npy

b

� �

hx ¼X1

m¼1

X1

n¼1

hxmncos

mpx

a

� �sin

npy

b

� �h�x ¼

X1

m¼1

X1

n¼1

hxmn� cos

mpx

a

� �sin

npy

b

� �

hy ¼X1

m¼1

X1

n¼1

hymnsin

mpx

a

� �cos

npy

b

� �h�y ¼

X1

m¼1

X1

n¼1

h�ymnsin

mpx

a

� �cos

npy

b

� �

hz ¼X1

m¼1

X1

n¼1

hzmnsin

mpx

a

� �sinðnpy

bÞ h�z ¼

X1

m¼1

X1

n¼1

h�zmnsin

mpx

a

� �sin

npy

b

� �

qþz ¼X1

m¼1

X1

n¼1

qþzmnsin

mpx

a

� �sin

npy

b

� �nðx; zÞ ¼

X1

m¼1

X1

n¼1

nmnðzÞ sinmpx

a

� �sin

npy

b

� �

ð14Þ


123

3 Numerical results and discussions

Numerical evaluation of a FG plate attached with

piezoelectric layer either at top or at bottom is considered.

Property of piezoelectric fiber reinforced composite is

taken as follows (Ray and Sachade 2006b).

FG Material: E0 = 200 GPa and m = 0.3.

PFRC (PZT5H): C11 = 32.6 GPa, C11 = 4.3 GPa,

C22 = 7.2 GPa, C44 = 1.05 GPa, C55 = C44 = 1.29

GPa, e31 = -6.76 C/m2, e32 = e15 = e24 = e33 = 0.000,

e11 = e22 = 0.037 9 10-9 F/m2, e33 = 10.64 9 10-9 F/m2.

Results for displacements and stresses are obtained

for following cases.

Case I: Eh/E0 = 10, qþzmn= -40 N/m2, Vt

mn = 0,

100, -100, S = 10, 20, 100, PFRC at top

Case II: Eh/E0 = 0.1, qþzmn= -40 N/m2, Vt

mn = 0,

100, -100, S = 10, 20, 100, PFRC at top

Case III: Eh/E0 = 10, qþzmn= -40 N/m2, Vb

mn = 0,

100, -100, S = 10, 20, 100, PFRC at bottom

Case IV: Eh/E0 = 0.1, qþzmn= -40 N/m2, Vb

mn = 0,

100, -100, S = 10, 20, 100, PFRC at bottom.

Where Eh is Young’s modulus at top of FG plate.

Results are compared with 3D exact solution (Ray

and Sachade 2006a) and FE solutions (Ray and

Sachade 2006b). In-plane displacement, transverse

Table 1 FG plates (Eh/E0 = 0.1) with applied sinusoidal mechanical and electrical loadings | PFRC at top

Aspect ratio S = 10 S = 20 S = 100

Quantity Source V = 0 V = 100 V = -100 V = 0 V = 100 V = -100 V = 0 V = 100 V = -100

�w Presenta -8.6140 3164.12 -3181.35 -8.3577 790.3310 -807.046 -8.2750 23.7481 -40.2981

3Db -8.7397 3253.30 -3270.80 -8.5437 813.7105 -830.7979 -8.4806 23.9160 -40.5745

FEMc -8.6673 3212.90 -3230.30 -8.4154 797.9455 -814.7764 -8.3292 24.4974 -41.4587

�u Presenta -0.0905 -3.8486 3.6674 -0.0899 -1.14816 0.988288 -0.089722 -0.1336 -0.04584

0.1653 -112.9800 113.311 0.1689 -26.5921 26.9299 0.170091 -0.8807 1.22091

3Db -0.0914 -3.1845 3.0017 -0.0912 -0.9839 0.8015 -0.0912 -0.1285 -0.0539

0.1686 -114.0117 114.3489 0.1735 -27.0971 27.4441 0.175 -0.8991 1.2493

FEMc -0.0901 -3.0418 2.8616 -0.0902 -1.0641 0.8838 -0.0901 -0.1322 -0.0480

0.1714 -107.7880 108.1308 0.1716 -26.3653 26.7085 0.1715 -0.8840 1.2270

�rx Presenta 0.3908 -33.732 34.5138 0.3848 -7.74463 8.57438 0.38294 0.06279 0.703096

-0.0848 47.3070 -47.4767 -0.0846 11.26 -11.4293 -0.0845 0.3627 -0.53194

3Db 0.4042 -41.0915 41.8999 0.4031 -9.8557 10.6619 0.4027 -0.0061 0.8115

-0.0807 44.3075 -44.4689 -0.0796 10.6491 -10.8083 -0.0793 0.3451 -0.5036

FEMc 0.4066 -42.1225 42.9357 0.4068 -9.6579 10.4715 0.4066 0.0199 0.8013

-0.0793 43.0275 -43.1862 -0.0794 10.5905 -10.7493 -0.0793 0.3460 -0.5046

�ry Presenta 0.3763 -163.0770 163.83 0.3697 -41.0169 41.7565 0.36762 -1.29497 2.03021

-0.0866 30.8163 -30.9899 -0.0862 7.70831 -7.88081 -0.08611 0.22666 -0.398894

3Db 0.3900 -170.3199 171.0999 0.3883 -43.1562 43.9327 0.3877 -1.3654 2.1408

-0.0825 28.1032 -28.2681 -0.0812 7.1105 -7.2729 -0.0808 0.2088 -0.3704

FEMc 0.3916 -172.9020 173.6851 0.3913 -43.5671 44.3496 0.3909 -1.3748 2.1565

-0.0810 28.0609 -28.2229 -0.0810 7.0834 -7.2453 -0.0809 0.2072 -0.3689

�sxy Presenta -0.2116 55.3721 -55.7955 -0.20979 13.8615 -14.2811 -0.2091 0.356017 -0.77434

0.0408 -19.0577 19.1394 0.04162 -4.6504 4.7336 0.0418 -0.144775 0.228522

3Db -0.2138 56.9184 -57.3461 -0.2131 14.2724 -14.6986 -0.2128 0.3692 -0.7948

0.0416 -19.4500 19.5333 0.0427 -4.7790 4.8645 0.0431 -0.1491 0.2353

FEMc -0.2148 57.9332 -58.3627 -0.2148 14.3377 -14.7673 -0.2147 0.3667 -0.7962

0.0431 -19.1536 19.2399 0.0431 -4.7614 4.8477 0.0431 -0.1489 0.2352

a Present-HOSNT12 using linear variation of electrostatic potential through the thicknessb 3D-Exact (Ray and Sachade 2006a)c FEM-FOST based (Ray and Sachade 2006b)


123

Ta

ble

2F

Gp

late

s(E

h/E

0=

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ith

app

lied

sin

uso

idal

mec

han

ical

and

elec

tric

allo

adin

gs

|P

FR

Cat

top

Asp

ect

rati

oS

=1

0S

=2

0S

=1

00

Qu

anti

tyS

ou

rce

V=

0V

=1

00

V=

-1

00

V=

0V

=1

00

V=

-1

00

V=

0V

=1

00

V=

-1

00

� wP

rese

nta

-0

.95

75

18

7.8

14

0-

18

9.7

29

0-

0.9

25

10

74

5.5

33

6-

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57

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3D

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123

displacement, in-plane normal stresses and transverse

shear stress are evaluated and their comparison with

exact and FEM solutions are presented in Tables and

Figures.

Tables 1, 2, 3, and 4 show comparison of numer-

ical results of for the cases I, II, III and IV,

respectively.

Figure 2a and b demonstrates normalized variation of

transverse displacement ð�wÞ and in-plane displacement

(�u) through the thickness of thin FG plate (S = 100),

respectively. Transverse displacement is constant

through the thickness of the FG layer while, in-plane

displacement is linear. Normalized variations of in-

plane normal stresses (�rx, �ry) through the thickness of

thin FG plate are exhibited in Fig. 3a and b. Variation

of in-plane normal stress (�rx) at top of FG plate is more

than in-plane normal stress (�ry). This is due to

effectiveness of piezoelectric stress coefficient only

along x-axis. Present HOSNT12 results are closed to

exact solution for with or without applied voltages.

Figure 4a and b demonstrates normalized variation of

in-plane shear stress (�sxy) and transverse normal stress

(�rz) through the thickness of thin FG plate. Results of

transverse shear stress (�syz,�sxz) closely agree with exact

results for both the voltages as shown in Fig. 5a and b,

respectively. Traction free conditions for transverse

shear stress (�syz) can be observed at bottom as well as at

top of FG plate, but maximum shear traction is seen at

top of FG plate where actuator is placed.

4 Conclusions

In this paper a complete analytical solution for statics

of FG plate attached with distributed PFRC actuator

-0.50

-0.25

0.00

0.25

0.50

z/h

w (a/2,b/2,z)

Present HOSNT 12 100 V 3D Exact 0 V -100 V

S = 100

(a)

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08

-0.50

-0.25

0.00

0.25

0.50

z/h

u (0,b/2,z)


S = 100

(b)

Fig. 2 Variation of normalized (a) Transverse displacement �w(b) In-plane displacement �u through the thickness (z/h) of a

Functionally Graded (FG) plate (Eh/E0 = 10) simply supported

on all edges with or without applied voltage to the piezoelectric

layer

-0.50

-0.25

0.00

0.25

0.50

S = 100

z/h

σx (a/2,b/2,z)


(a)

(b)

-3 -2 -1 0 1 2

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4

-0.50

-0.25

0.00

0.25

0.50

z/h

σy (a/2,b/2,z)


S = 100

Fig. 3 Variation of normalized (a) In-plane normal stress �rx

(b) In-plane normal stress �ry through the thickness (z/h) of a

Functionally Graded (FG) plate ðEh=E0 ¼ 10Þ simply sup-

ported on all edges with or without applied voltage to the

piezoelectric layer


123

under electromechanical load is presented. A higher

order shear and normal deformation theory is used to

model the elastic responses of FG plate subjected to

voltages. Linear layer wise approximation of the

electrostatic potential is proposed in the present model.

Comparative numerical results and across the thick-

ness variations of displacements and stresses are

presented. Linear and constant variations of in-plane

and transverse displacements are observed. Consider-

able effect of actuation at the interface is observed in

case of in-plane normal stress and transverse shear

stress along x-axis as compared to their effects along y-

axis. Present model performs excellently to predict the

variations in these cases.

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123

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