An electromechanical higher order model for piezoelectric functionally graded plates S. M. Shiyekar • Tarun Kant Received: 19 March 2010 / Accepted: 31 March 2010 / Published online: 23 April 2010 Ó 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
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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/
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)
168 S. M. Shiyekar, T. Kant
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An electromechanical higher order model 169
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20
06
b)
170 S. M. Shiyekar, T. Kant
123
Ta
ble
4F
Gp
late
s(E
h/E
0=
10
)w
ith
app
lied
sin
uso
idal
mec
han
ical
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elec
tric
allo
adin
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|P
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bo
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1
aP
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nt-
HO
SN
T1
2u
sin
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aria
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ectr
ost
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ug
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ick
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3D
-Ex
act
(Ray
and
Sac
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e2
00
6a)
cF
EM
-FO
ST
bas
ed(R
ayan
dS
ach
ade
20
06
b)
An electromechanical higher order model 171
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)
Present HOSNT 12 100 V 3D Exact 0 V -100 V
S = 100
(b)
Fig. 2 Variation of normalized (a) Transverse displacement �w(b) In-plane displacement �u through the thickness (z/h) of a