Study for Functionally Graded Plate coupled with piezoelectric layer as a smart application. Static bending and deflection control is the aim of the study

1. DEFLECTION CONTROL OF HYBRID
PIEZOELECTRIC FUNCTIONALLY
GRADED PLATE
S. A. Patare & K. M. Bajoria
Department of Civil Engineering,
IIT Bombay

2. Introduction
● Shape control of FGM plate coupled with
Piezoelectric material
● Four variable plate theory for simple formulation
● Constant displacement feedback control algorithm
● Approximate analytical solution for simply supported
and clamped plate
● Static deflection control

3. Piezoelectric FGM Plate Model

4. Displacement Field [03]
u=uo−z
∂wb
∂ x
−
(4 z3
3ht
2
)
∂ws
∂x
v=vo−z
∂wb
∂ y
−
(4z3
3ht
2
)
∂ws
∂ y
w=wb+ws
{
σx
σ y
σxy
σxz
σ yz
}=
[
C11 C12 0 0 0
C12 C22 0 0 0
0 0 C66 0 0
0 0 0 C44 0
0 0 0 0 C55
]{
ϵx
ϵy
γxy
γxz
γ yz
}−
[
0 0 ¯e31
0 0 ¯e31
0 0 0
e15 0 0
0 e15 0
]{
Ex
Ey
Ez
}
{
Dx
Dy
Dz
}=
[
0 0 0 e15 0
0 0 0 0 e15
¯e31 ¯e31 0 0 0 ]{
ϵx
ϵy
γxy
γxz
γ yz
}+
[
Ω11 0 0
0 Ω11 0
0 0 Ω33
]{
Ex
Ey
Ez
}
Constitutive
Law[05]►
Electric
Displacement[05]►

5. Electric Potential Function
Φ=
{ Π T
[1−
(z−h/2−hp/2
hp/2 )
2
]+AT z+BT ;for⟨h
2
z⩽⩽
h
2
+hp ⟩
Π B
[1−
(− z−h/2−hp/2
hp/2 )
2
]+AB z+BB ;for⟨−
h
2
−hp z⩽⩽−
h
2 ⟩}
Φ(z=±
h
2 )=0, Dz
(z=±(h
2
+hp))=0Boundary Conditions for Open Circuit ►
BT =−(h
2)AT ;BB=(h
2)AB
Φ=
{ΠT
[1−
(z −h/2−hp /2
hp/2 )
2
]+ AT (z−
h
2) for⟨h
2
⩽z⩽
h
2
+hp ⟩
ΠB
[1−
(− z−h/2−hp/2
hp/2 )
2
]+ AB(z+
h
2) for ⟨−
h
2
−hp ⩽z ⩽−
h
2 ⟩

6. Hamilton’s Principle
0=∫
0
T
(δU+δV−δK)δt
δU=∫
A
∫
−
h
2
− hp
h
2
+hp
(σx δϵx +σy δϵy +σxy δγxy +σyz δγyz +σxz δγxz)dzdA
δV =−∫
A
q δ (wb+ws)dA
◄Strain Energy
Variation
Work Done Variation►

7. Governing Formulation
δu0 :
∂Nx
∂x
+
∂ Nxy
∂ y
=0 δv0:
∂ N y
∂ y
+
∂ Nxy
∂ x
=0
δ wb :
∂
2
Mx
b
∂ x
2
+
∂
2
M y
b
∂ y
2
+2
∂
2
M xy
b
∂ x ∂ y
+qw=0
δ ws:
∂
2
M x
s
∂ x
2
+
∂
2
M y
s
∂ y
2
+2
∂
2
M xy
s
∂ x ∂ y
+
∂Qxz
∂ x
+
∂Q yz
∂ y
+qw=0

8. ∫
h
2
h
2
+hp
(∂ Dx
∂x
+
∂Dy
∂ y
+
∂Dz
∂z )dz+ ∫
−
h
2
−hp
−
h
2
(∂Dx
∂ x
+
∂ Dy
∂ y
+
∂Dz
∂z )dz=0
λ1(∂2
ws
∂ x
2
+
∂2
ws
∂ y
2 )− λ2(∂2
wb
∂ x
2
+
∂2
wb
∂ y
2 )−λ3(∂2
ΠT
∂ x
2
+
∂2
ΠT
∂ y
2 )
− λ4(∂
2
ΠB
∂ x2
+
∂
2
ΠB
∂ y2 )− λ5 ΠT − λ6 ΠB }=0
Fifth Governing Equation►
Fifth Equation & Maxwell’s Law
◄Displacement Feedback
Control Law [02]
ΠT =Gd Π B+Gv ˙Π B

9. Approximate Analytical Solution [04]
u0=∑
m=1
∞
∑
n=1
∞
U eiwt ∂ XM (x)
∂x
YN ( y ) v0=∑
m=1
∞
∑
n=1
∞
V eiwt
XM (x)
∂YN ( y)
∂ y
wb=∑
m=1
∞
∑
n=1
∞
W b eiwt
XM (x)YN ( y) ws=∑
m=1
∞
∑
n=1
∞
W s eiwt
XM (x)YN ( y )
Π=∑
m=1
∞
∑
n=1
∞
Π eiwt
XM (x)YN ( y)

10. Numerical
● A square plate of 200mm size
● Thickness of FGM plate is 1 mm while Piezo layers
is 0.1 mm
● Uniform load of 100 N/m2
● FGM core of Ti-6Al-4V and Aluminium Oxide
● Power law is taken 2 (constant)

11. Shape control for simpy supported plate

12. Shape control for clamped Plate

13. Conclusion
● It is possible to consider simple four variable plate theory for
analysis of hybrid piezoelectric FGM plate
● Open circuit potential function for four variable plate theory
is proposed and validated
● Good results are obtained for static bending of plate
● Approximate analytical solution further simplies the analysis
for other boundary condition
● Smart hybrid piezoelectric FGM plate model is developed with
minimum computational efforts
● Change in power law, aspect ratio and FGM materials can be
considered for further study

14. References
1) P. A. Jadhav and K. M. Bajoria: Stability analysis of thick piezoelectric metal
based FGM plate using first order and higher order shear deformation theory,
Int. J. Mech. Mater. Des., 11, 2015, pp. 387-403
2) T. Nguyen-Thoi, K. Nguyen-Quang, H. Dang-Trung, V. Ho-Huu and H.
Luong-Van: Analysis and control of FGM plates integrated with piezoelectric
sensors and actuators using cell-based smoothed discrete shear gap method,
Compos. Struct., 165, 2016, pp. 115-129.
3) Seung-Eock Kim and Huu-Tai Thai: A simple higher-order shear deformation
theory for bending and free vibration analysis of functionally graded plates,
Compos. Struct., 96, 2013,pp. 165-173.
4) M. R. Barati, H. Shahverdi and A. M. Zenkour: Electro-mechanical vibration
of smart piezoelectric FG plates with porosities according to a refined four-
variable theory, Mech. Adv. Mater. Struct., 24 (12), 2017, pp. 987-998.
5) H. F. Tiersten: Linear piezoelectric plate vibrations, Plenum Press, New York,
1969.

15. Thank You