2. What is Piezoelectric
Material?
• Piezoelectric Material is one that possesses
the property of converting mechanical energy
into electrical energy and vice versa.
3. Piezoelectric Materials
• Mechanical Stresses Electrical Potential
Field : Sensor (Direct Effect)
• Electric Field Mechanical Strain :
Actuator (Converse Effect)
Clark, Sounders, Gibbs, 1998
5. Piezoelectric Sensor
• When mechanical stresses are applied on the
surface, electric charges are generated
(sensor, direct effect).
• If those charges are collected on a conductor
that is connected to a circuit, current is
generated
6. Piezoelectric Actuator
• When electric potential (voltage) is applied to
the surface of the piezoelectric material,
mechanical strain is generated (actuator).
• If the piezoelectric material is bonded to a
surface of a structure, it forces the structure
to move with it.
14. Constitutive Relations
• The piezoelectric effect
appears in the stress
strain relations of the
piezoelectric material in
the form of an extra
electric term
• Similarly, the
mechanical effect
appears in the electric
relations
s11 1 d 31 E
D d 31 1 33 E
15. Constitutive Relations
•
•
•
•
‘S’ (capital s) is the strain
‘T’ is the stress (N/m2)
‘E’ is the electric field (Volt/m)
‘s’ (small s) is the compliance; 1/stiffness
(m2/N)
• ‘D’ is the electric displacement, charge per
unit area (Coulomb/m2)
17. Manipulating the
Equations
• The electric displacement is
the charge per unit area:
• The rate of change of the
charge is the current:
• The electric field is the
electric potential per unit
length:
Q
D
A
1
I
D Idt
A
As
V
E
t
18. Using those relations:
• Using the
relations:
• Introducing the
capacitance:
• Or the electrical
admittance:
d 31
1 s11 1 V
t
A 33 s
I Ad 31s 1
V
t
I Ad 31s 1 CsV
I Ad 31s 1 YV
19. For open circuit (I=0)
• We get:
• Using that into the
strain relation:
• Using the expression
for the electric
admittance:
Ad 31s
V
1
Y
2
31
Asd
1 s11 1
1
tY
2
d 31
1
1 s11 1
s
33 11
20. The electromechanical
coupling factor
• Introducing the factor ‘k’:
1 s11 1 k 1
2
31
• ‘k’ is called the electromechanical coupling factor
(coefficient)
• ‘k’ presents the ratio between the mechanical energy
and the electrical energy stored in the piezoelectric
material.
• For the k13, the best conditions will give a value of
0.4
21. Different Conditions
• With open circuit conditions, the stiffness of
the piezoelectric material appears to be higher
(less compliance)
1 s11 1 k 1 s 1
2
31
D
• While for short circuit conditions, the stiffness
appears to be lower (more compliance)
s11 s
E
22. Different Conditions
• Similar results could be obtained for the
electric properties; electric properties are
affected by the mechanical boundary
conditions.
23. Zero-strain conditions
(S=0)
• Using the
relations:
• Introducing the
capacitance:
• Or the electrical
admittance:
d 31
0 s11 1
V
t
2
As 33
d 31
1
V
I
t 33 s11
I Y 1 k V
2
31
26. Active Fiber Composites (AFC)
c eff 11 c E11
e
eff
31
v
C
2
v p e31
v p S 33
33
33e31
v C 33 v p S 33
33 S 33
eff 33 C
v 33 v p S 33
28. Axial Motion of Rods
• In this case, we will consider the case when
the PZT and the structure are deforming
axially only
29. Zero Voltage case
• If the structure is subject to axial force only,
we get:
a Ea a
s Es s
• And for the equilibrium:
F Aa a As s Aa Ea a As Es s
F Aa a As s Aa Ea As Es x
30. Zero Voltage case
• From that, we may write the force strain
relation to be:
F
F b
x
Aa Ea As Es 2ta Ea t s Es
31. Zero Force case
• In this case, the strain of the of the PZT will be
less than that induced by the electric field
only! E E E E d V
a
a s
a
p
a s
a
31
t
s Es s
• For equilibrium, F=0:
V
F Aa a As s Aa Ea s Aa Ea d31 As Es s 0
t
V
Aa Ea d 31
t
s
Aa Ea As Es
32. Homework #2
• Solve problems 1,2,&3 from textbook
• Due 27/11/2013 (11:59PM)
34. Review of Thin-Beam
Theory
• The Euler-Benoulli beam theory assumes that
the strain varies linearly through the thickness
of the beam and inversely proportional to the
radius of curvature.
d 2v
y 2
dx
d 2v
E Ey 2
dx
35. Equilibrium
• The externally applied moment has to be in
equilibrium with the internally generated
h/2
h/2
moment.
d 2v
M bydy
h / 2
Ey dx
2
h/2
• For homogeneous materials:
2
h/2
d v
d 2v
2
M E 2 y bdy EI 2
dx h / 2
dx
bydy
37. With piezoelectric
materials
• Introducing change in the material property:
h/2
M
ydy
b
h / 2
t s / 2
V
Ea a d 31 ydy Es s ydy
ta
h / 2
t s / 2
ts / 2
V
Ea a d 31 ydy
ta
ts / 2
h/2
38. With piezoelectric
materials
• Expanding the integral
2
M
d v
Ea 2
b
dx
ts / 2
V
/ 2y dy Ea d31 ta
h
ts / 2
ydy
2
2
ts / 2
h / 2
2
h/2
d v
d v
V
2
2
Es 2 y dy Ea 2 y dy Ea d 31
dx t s / 2
dx t s / 2
ta
h/2
ydy
ts / 2
39. With piezoelectric
materials
• Rearranging
ts / 2
h/2
ts / 2 2
M d v
2 Ea y dy Es y 2 dy Ea y 2 dy
b dx h / 2
t s / 2
ts / 2
2
V
Ea d 31
ta
t s / 2
V
/ 2ydy Ea d31 ta
h
h/2
ydy
ts / 2
40. With piezoelectric
materials
• Integrating
M
1 d 2v
3
3
3
Ea h 3 t s 2 E s t s Ea h 3 t s
b 24 dx 2
V 2
V 2
2
2
Ea d 31
t s h Ea d 31
h ts
8t a
8t a
Ea d 31V 2
M
1 d 2v
3
3
2
3
Ea h t s E s t s
h ts
2
b 12 dx
4ta
41. Remember:
• For homogeneous structures: Eh3 d 2v
M
2
12 dx
b
• Thus, in the absence of the voltage:
Ea h t s Es t s
EI Equivalent b
12
• OR:
M EI Equivalent
3
3
3
d 2 v Ea bd31V 2
2
h ts
2
dx
4t a
42. In the absence of load
2
Ea bd31V
d v
2
2
h ts
2
dx
4ta EI Equivalent
• Thus, the structure will feel a moment:
Es I s Eabd31V 2
d 2v
2
M s Es I s 2
h ts
dx
4ta EI Equivalent
43. Piezoelectric forces
• The above is equivalent of having a force
applied by the piezoelectric material that is
equal to:
Ms
Es I s Ea bd 31V 2
2
Fa
ts
4t s t a EI Equivalent
h
ts
44. Homework #3
• Solve problems 4,5,&6 from textbook
• Due 30/11/2013 (11:59PM)