Maximum Allowed Stress Theoretical Analysis of Multilayer Structures
52 The resultant stress on the clamped area of a beam for each layer of a unimorph is
proportional to the input moment divided by the inertia across the length of the beam as,
3-23
To find the exact value of stress of each layer of the material, the moment inertia of the beam,
I
b
has to be defined. The transformed cross-section of a unimorph is redrawn in Figure 3-5 with parallel-axis passing through the centroid of the beam.
Figure 3-5: Parallel-axis for a transformed cross-section of a unimorph.
From the parallel-axis theorem for the moment of inertia [93]
3-24
where
d
is the distance from the centroid of the layer to the neutral axis of the structure and
A
is the cross-section area of the layer. The integration of the second term at the right hand side of equation 3-24 is zero, therefore, the total moment of inertia for a
unimorph as shown in Figure 3-5 is,
3-25
dl I
d l
M l
b
l unimor ph
b l
1
½
h
e1
½
h
e2
h
2
h
3
h
N
Neutral axis
½h
p
h
w
h h
h h
h h
h h
dA d
dA h
d dA
h dA
d h
I
2 1
2 1
2 2
1 2
1 2
1 2
1 2
2 1
2 1
2
2
2 2
3 2
2 1
2 1
3 2
3 1
2 3
2 1
2 1
12 1
2 1
12 1
e e
e e
e ep
p N
p p
unimor ph
h h
h h
h he
h h
w n
h h
wh wh
I
53 where
n
ep
is the elastic modular ratio as defined in equation 3-20. By substituting
h
2
= h
p
+ h
e1
–
h
N
and
h
3
= h
N
+ h
e2
in equation 3-25, we get
3-26 If the neutral axis is passing through the PZT centroid,
p N
h h
2 1
and the upper electrode and lower electrode are made of same material and with same thickness,
e e
e
h h
h
2 1
, equation 3-26 can be simplified as
3-27
Comparing equation 3-27 to the bending modulus per unit width in equation 3-13, we get
3-28
The input moment as a function of length from the clamped area of a beam,
Mx
is
3-29
By substituting equation 3-4, 3-18 and 3-28 into 3-23, we get
3-30
p e
e p
e ep
p unimor ph
h h
h h
h n
h w
I
2 2
3 3
2 1
3 2
12 1
p unimor ph
e Dw
I
l l
z y
M x
M
b eff
2 2
2 2
1 2
2 1
1 1
236 .
118 .
r r
D y
l w
M h
h h
d l
e
unimor ph b
b pm
p p
e e
e b
p l
1 1
1 2
1 2
2 2
1 2
3 2
3 1
2 3
1
e p
p e
p e
e e
N e
e N
e e
ep
h h
h h
h h
h h
h h
h h
h h
wn
N p
N p
p unimor ph
h h
h h
h w
I
2 2
3
3 1
54 We can see from equation 3-30 that the resultant stress is proportional to the distance
from the neutral axis, and therefore the thicknesses of the upper and lower electrodes are very critical in determining the resultant stress of the beam. The equation also shows
that the width of the cantilever does not affect the resultant stress, though there is a practical limit to the size of the width. A beam with a small ratio of width to its
thickness when subjected to shear force will be twisted and become unstable and is therefore not suitable to be operated as a resonant device.
Maximum stresses are produced on the upper and lower electrodes compared to the PZT layer when the beam is vibrating. Thus the elastic modulus of the electrodes has to be
high in order to support the brittle ceramic layer at the centre of the structure.