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Sensing and actuating behaviours of piezoelectric layers with debonding in smart beams
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2001 Smart Mater. Struct. 10 713
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INSTITUTE OF PHYSICS PUBLISHING
SMART MATERIALS AND STRUCTURES
Smart Mater. Struct. 10 (2001) 713–723
PII: S0964-1726(01)24413-9
Sensing and actuating behaviours of
piezoelectric layers with debonding in
smart beams
Liyong Tong1,3 , Dongchang Sun1 and Satya N Atluri2
1
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney,
NSW 2006, Australia
2
Mechanical and Aerospace Engineering Department, 7704 Boelter Hall, University of
California, Los Angeles, Los Angeles, CA 90095-1600, USA
E-mail: ltong@aero.usyd.edu.au
Received 8 November 2000, in final form 23 March 2001
Published 18 July 2001
Online at stacks.iop.org/SMS/10/713
Abstract
This paper presents an analytical model for investigating the effect of
debonding between piezoelectric actuators/sensors and the host beam on the
sensing and actuating behaviour. In this model, both flexural and
longitudinal displacements of the host beam and the piezoelectric layers are
considered using classical beam theory. For the adhesive layer, only
through-thickness peel strain and transverse shear strain are taken into
account, and they are assumed to be constant across the adhesive thickness.
When a debonding occurs between the host beam and an actuator or sensor,
the stresses transferred via the adhesive layer are assumed to be zero. Using
this model, the effects of debonding on the sensing and actuating behaviour
are examined. Numerical results reveal that debonding can have remarkable
effects on the distributions of strains and internal forces as well as frequency
spectrum and charge output.
1. Introduction
Piezoelectric materials have been widely used as distributed
sensors and actuators in vibration and shape control of smart
structures due to their quick and effective transformation
between mechanical and electrical energy [1–14]. One
fundamental assumption in those studies is that piezoelectric
actuators/sensors are perfectly bonded onto a surface of a
host structure. While this assumption is valid for most
circumstances in active control of structural performance,
there exists a possibility of piezoelectric actuators/sensors
separating from host structures. This possibility increases
as new piezoelectric materials with high strain level and high
breakdown electric field become available to design engineers
of smart structures.
It is well known that piezoelectric ceramics are brittle and
susceptible to cracking, particularly when subjected to a cyclic
electric loading. For the case of conventional piezoelectric
ceramics used as actuators, debonding may be not an important
3
Corresponding author.
0964-1726/01/040713+11$30.00
© 2001 IOP Publishing Ltd
issue due to their low strain level (<0.2%) and low breakdown
electric field. However, for some new piezoelectric ceramics,
such as single crystals, debonding of piezoelectric ceramics
from a host structure can be an important problem for reliable
design and safe operation of smart structures due to their
high actuation authority. For example, piezoelectric single
crystals possess a high strain level up to 1.4% leading to high
strain energy density, and can sustain very high electric field
(of the order of 10 MV M−1 ) without electric breakdown.
The high strain induced in a single-crystal actuator creates
high differential straining in the actuator and high stresses in
adhesive along the edge of the actuator, and finally debonding
from the host structure. This is particularly important when the
actuator is subjected to a cyclic electric field; for example, in
structural vibration suppression, fatigue failure or debonding
may be induced.
When debonding exists in a smart structure, it is important
to understand its effects on the overall performance of the smart
structures. A literature search shows that there exist only
a few recent studies investigating debonding effects. Seeley
and Chattopadhyay [15] developed a finite-element model for
Printed in the UK
713
L Tong et al
the structures including actuator debonding by separating the
structures into debonded and non-debonded area based on the
refined high-order theory. In their model, the displacement
continuity in the interface between the debonded and nondebonded areas is imposed and implemented using a penalty
function approach. They also investigated the effects of the
debonding between the actuator and the host beam on the
closed-loop control by experiment [16], and found that the
debonding length is a critical factor to the closed-loop control.
Wang and Meguid [17] analytically investigated the static
coupled electromechanical behaviour of a thin piezoceramic
actuator embedded or bonded to an elastic medium under the
plane strain assumption. Based on a set of singular integral
equations in terms of an interfacial shear stress, they examined
the effect of interfacial debonding of an actuator on shear stress
by cutting off the part of the debonded actuator and applying
an equivalent stress on the remaining actuator parts.
It is evident that a full in-depth understanding of the
debonding effects has not yet been established. There are
a number of questions yet to be answered in relation to
debonding effects, for example identification of debonding,
debonding effects on functions of actuators/sensors, debonding
effects on controllability of structural vibration etc. In this
paper, we limit our study to the investigation of debonding
effects on actuating and sensing behaviour of actuators and
sensors in smart beams.
x
Host beam
xd
Ld
Figure 1. The beam and piezoelectric layer with debonding.
Ma+dMa
Ma
Ta
714
Qa+dQa
σ
τ
Qad
ρa Aa üa =
∂Ta
+ bτ
∂x
(1a)
ρa Aa ẅa =
∂Qa
+ bσ
∂x
(1b)
(1c)
(2a)
Qad
τ
σ
σ
τ
Mb+dMb
fl
Tb
We consider a slender composite beam, on which a
piezoelectric actuator layer is bonded with several debonded
regions, as shown in figure 1. It is assumed that the debonding
occurs throughout the width of the beam and the debonding
front lines are straight and perpendicular to the x-axis. Similar
to the case for adhesive bonded lap joints [18], both the
host beam and piezoelectric actuator layer are assumed to
undergo longitudinal and transverse deformations, which can
be modelled using classical beam theory; while the adhesive
layer is assumed to only carry constant transverse shear and
peel strains. For the adhesive segment with debonding, it is
assumed that there is no stress transferring between the host
beam and piezoelectric actuator layer. In addition, contact and
friction between the two debonded surfaces are not considered
for simplicity.
Figure 2 depicts the free-body diagram of the composite
beam including the host beam, the actuator layer and the
adhesive layer, and from this figure the following equations
of motion can be obtained:
∂Ma bha
+
τ − Qa = 0
∂x
2
∂Tb
− bτ + fl (x, t)
ρb Ab üb =
∂x
Ta+dTa
τ
σ
Qa
Mb
2.1. Assumptions and governing equations
Debonding
z
Qb
2. Actuator model with debonding
Piezoelectric patch
Adhesive layer
ft
Tb+dTb
Qb+dQb
Figure 2. Free-body diagram of the beam with actuator layer.
∂Qb
(2b)
− bσ + ft (x, t)
∂x
∂Mb bhb
+
τ − Qb = 0
(2c)
∂x
2
where the subscripts a and b represent the actuator layer and
the host beam, respectively, u is the longitudinal displacement
in the mid-plane, w is the transverse displacement, h denotes
the thickness, b is the width of the composite beam, τ and
σ are the shear and peel stress of the adhesive layer, T , Q
and M are the axial force, transverse shear force and bending
moment respectively, and fl (x, t) and ft (x, t) are the axial
and transverse loads per unit length. The equivalent mass
densities per unit length of the actuator and the host beam
are ρa Aa = ρa bha + ρad bhad /2 and ρb Ab = ρb bhb + ρad bhad /2
respectively, in which half the mass of the adhesive layer is
added.
The strain in the x-direction (the length direction) for a
point at a distance z from the neutral plane of the actuator
layer is
∂ua
∂ 2 wa
−z
.
(3)
εa =
∂x
∂x 2
The constitutive relation of the piezoelectric actuator layer is
given by
σa = Ya εa − e31a E
(4)
ρb Ab ẅb =
Sensing and actuating behaviours of piezoelectric layers with debonding in smart beams
where Y is the Young’s modulus that may be complex when the
damping is considered, E is the electric field density applied
on the actuator layer and e31a is the piezoelectric stress constant
of the actuator layer.
The axial stress resultant and the bending moment for the
actuator layer can be written as follows:
Ta = Āa
∂ua
∂ 2 wa
− B̄a
− be31a V
∂x
∂x 2
(5)
∂ua
∂ 2 wa
− D̄a
− be31a ra V
(6)
∂x
∂x 2
where ra is the z-coordinate value of the mid-plane of the
actuator layer from its neutral plane and V is the voltage applied
on the actuator alone in its thickness direction. In the above
equations, Āa and D̄a are the axial and bending stiffness, and
B̄a is the extension–bending coupling term given by
zau
zau
bYa dz;
B̄a =
bYa z dz;
Āa =
Ma = B̄a
zal
D̄a =
zal
zau
zal
(7)
bYa z dz
2
where zal and zau are the z-coordinates of the lower and upper
surfaces of the actuator layer measured from its own neutral
plane.
Similarly, for the host beam, we have
Tb = Āb
∂ub
∂ 2 wb
− B̄b
∂x
∂x 2
(8)
Mb = B̄b
∂ub
∂ 2 wb
− D̄b
∂x
∂x 2
(9)
where the property parameters Āb , B̄b and D̄b are defined
similarly to equation (7).
Using the constant shear and peel strain assumption [18],
the shear and peel stress in the adhesive layer are given by
τ = kb
σ =
Yad
γ
2(1 + νad )had
kb Yad (1 − νad )
(wb − wa)
(1 − 2νad )(1 + νad )had
(10)
(11)
where kb is a parameter characterizing the bonding condition

debonding

0
weak bonding
kb = >0 and <1
(12)


1
perfect bonding
had is the thickness of the adhesive layer, Yad and νad are the
Young’s modulus and Poisson’s ratio of the adhesive layer
respectively, and the shear strain takes the following form:
1 ∂wa ∂wb
γ =
+
2 ∂x
∂x
1
ub − ua
∂wa
∂wb
+
+ hb
+
ha
.
(13)
2had
∂x
∂x
had
In this paper, only the debonding and perfect bonding cases
are considered.
Eliminating all the axial, shear force, bending moments
from equations (1)–(11), the following equations of motion of
the composite beam can be obtained:
∂ 2 ua
∂ 3 wa
bkb Gad
−
ρa Aa üa − Āa 2 + B̄a
3
∂x
∂x
2
∂wa ∂wb
∂wa
∂wb
1
u b − ua
×
ha
+
+
+ hb
+2
∂x
∂x
had
∂x
∂x
had
∂V
= − be31a
(14a)
∂x
3
4
∂ ua
∂ wa
ha bkb Gad
ρa Aa ẅa − B̄a 3 + D̄a
−
4
∂x
∂x
4
2
∂ wa ∂ 2 wb
∂ 2 wa
∂ 2 wb
1
h
×
+
+
+
h
a
b
∂x 2
∂x 2
had
∂x 2
∂x 2
2 ∂ub
∂ua
kb Yad b
+
−
−
(wb − wa )
had ∂x
∂x
had
∂ 2V
= − be31a ra 2
(14b)
∂x
2
3
∂ ub
∂ wb bkb Gad
ρb Ab üb − Āb 2 + B̄b
+
3
∂x
∂x
2
∂wa ∂wb
1
∂wa
∂wb
ha
×
+
+
+ hb
∂x
∂x
had
∂x
∂x
ub − ua
+2
= fl (x, t)
(14c)
had
hb bkb Gad
∂ 3 ub
∂ 4 wb
ρb Ab ẅb − B̄b 3 + D̄b
−
∂x
∂x 4
4
2
2
∂ wa ∂ 2 wb
∂ 2 wa
∂ 2 wb
1
ha
+
×
+
+
+ hb
∂x 2
∂x 2
had
∂x 2
∂x 2
had
∂ub
∂ua
kb Yad b
×
(wb − wa ) = ft (x, t). (14d)
−
+
∂x
∂x
had
For the debonding area, where kb = 0, the equations of motion
become
∂ 2 ua
∂ 3 wa
a ∂V
+ B̄a
= −be31
2
3
∂x
∂x
∂x
(15a)
∂ 3 ua
∂ 4 wa
∂ 2V
+ D̄a
= −be31 ra 2
3
4
∂x
∂x
∂x
(15b)
∂ 2 ub
∂ 3 wb
+
B̄
= fl (x, t)
b
∂x 2
∂x 3
(15c)
∂ 3 ub
∂ 4 wb
+
D̄
= ft (x, t).
b
∂x 3
∂x 4
(15d)
ρa Aa üa − Āa
ρa Aa ẅa − B̄a
ρb Ab üb − Āb
ρb Ab ẅb − B̄b
2.2. Boundary conditions
For the actuator layer, there are six applicable boundary
conditions, namely the following.
Fixed end:
ua = 0;
wa = 0;
∂wa
= 0.
∂x
(16a)
Pinned end:
ua = 0;
B̄a
wa = 0;
∂ua
∂ 2 wa
− be31a ra V = 0.
− D̄a
∂x
∂x 2
(16b)
715
L Tong et al
Free end:
Āa
B̄a
∂ua
∂ wa
− B̄a
− be31a V = 0;
∂x
∂x 2
2
∂ua
∂ 2 wa
− be31a ra V = 0;
− D̄a
∂x
∂x 2
∂ 2 ua
∂ 3 wa
∂V kb bha
+
Gad γ = 0.
−
D̄
− be31a ra
a
2
3
∂x
∂x
∂x
2
(16c)
For the host beam, there are also six applicable boundary
conditions, i.e. the following.
Fixed end:
B̄a
ub = 0;
Pinned end:
wb = 0;
ub = 0;
∂wb
= 0.
∂x
(16d)
wb = 0;
B̄b
∂ub
∂ 2 wb
= 0.
− D̄b
∂x
∂x 2
Āb
∂ub
∂ 2 wb
= 0;
− B̄b
∂x
∂x 2
B̄b
∂ub
∂ 2 wb
− D̄b
= 0;
∂x
∂x 2
(16e)
Free end:
B̄b
(16f)
∂ 2 ub
∂ 3 wb kb bhb
−
D̄
+
Gad γ = 0.
b
∂x 2
∂x 3
2
2.3. Continuity conditions
To ensure continuity at the interface of the debonding and
bonding areas, all displacements and all stress resultants must
be identical, namely, the following continuity conditions are
imposed:
una = uda ,
Tan = Tad ,
unb
=
udb ,
Tbn = Tbd ,
wan = wad ,
Qna = Qda ,
wbn
=
wbd ,
Qnb = Qdb ,
∂wan
∂x
=
∂wad
∂x
,
Man = Mad
∂wbn
∂wbd
=
,
∂x
∂x
where zsu and zsl are the z-coordinates of the upper and lower
surfaces of the sensor layer.
Substituting equation (18) into (19) and neglecting the
reverse piezoelectric effect of the sensor layer, the output
charge of the piezoelectric sensor layer can be obtained as
∂us
∂ 2 ws
q(t) =
be31s
be31s rs
dx
dx −
∂x
∂x 2
Ln
Ln
∂us
∂ 2 ws
+
be31s
be31s rs
dx
(20)
dx −
∂x
∂x 2
Ld
Ld
where Ln and Ld represent the non-debonding and debonding
areas respectively, and rs is the z-coordinate value of the midplane of the sensor layer from its neutral plane. Equation (20)
shows that the output charge of the sensor layer is proportional
to the average strain in both debonding and non-debonding
areas. When the sensor layer is perfectly bonded onto the host
beam, the charge output is also proportional to the average
surface strain of the beam it covers. However, when debonding
occurs between the host beam and the sensor layer, as one part
of the whole output of the sensor, the charge contributed by
the debonding area of the sensor layer
∂us
∂ 2 ws
qdebond (t) =
be31s
be31s rs
dx (21)
dx −
∂x
∂x 2
Ld
Ld
may not be well related to the average strain of the part on the
host beam it covers.
The boundary conditions and the continuity conditions are
similar to the actuator case in the above section except that the
voltage should be zero.
(17)
Mbn = Mbd
where the superscripts n and d stand for the displacement
in non-debonded and debonded segments at the debonding
interface.
3. Sensor model with debonding
Following the same procedure as the actuator model, the
equations of motion of the composite beam with bonded
piezoelectric sensor layer with debonding can be derived,
which can be obtained by replacing subscript ‘a’ by ‘s’ in
equation (14), and equating the right-hand side to zero in
equations (14a) and (14b).
The constitutive equation of the piezoelectric sensor layer
is given by
(18)
D3 = e31s εs + ∈33 E
716
where D3 is the electric displacement in the thickness direction,
e31s is the piezoelectric stress constant of the sensor layer, ∈33
is the dielectric constant and E is the applied electric field
density. εs is the axial strain of the sensor layer, which can be
determined using equation (3) by replacing subscript a with s.
The electric charge accumulated on the electrodes of the
sensor layer can be obtained by
b l
q(t) =
(D3 |z=zsu + D3 |z=zsl ) dx
(19)
2 0
4. Solution scheme
The partial differential equations of motion of the smart beam
given in equation (14) may be directly used and solved by
several numerical methods. However, in order to simplify the
boundary conditions as well as the continuity conditions, we
choose the following procedure [18] to solve the problem.
Taking Fourier transformation with respect to time t in
equations (1)–(11), we have
dT̄a
+ bτ̄
dx
(22a)
dQ̄a
+ bσ̄
dx
(22b)
dM̄a bha
+
τ̄ − Q̄a = 0
dx
2
(22c)
−ρa Aa ω2 ūa =
−ρa Aa w̄a =
−ρb Ab ω2 ūb =
dT̄b
− bτ̄ + f¯l (x)
dx
(22d)
Sensing and actuating behaviours of piezoelectric layers with debonding in smart beams
Table 1. Physical properties and dimensions of the composite beam.
−ρb Ab ω2 w̄b =
Item
Host beam
(steel)
Piezo-patch
(PZT-5A)
Adhesive
layer
Mass density (kg m−3 )
Young’s modulus (GPa)
Loss factor
Poisson’s ratio
Piezo-constant d31 (m V−1 )
Thickness (mm)
Length (mm)
Width (mm)
7800
210
0.004
0.3
—
2
300
20
7600
63
0.011
0.3
370 × 10−12
0.4
300
20
1600
2.15
0.011
0.202
—
0.15
300
20
dQ̄b
− bσ̄ + f¯t (x)
dx
dM̄b bhb
+
τ̄ − Q̄b = 0
dx
2
(22f)
dūa
d2 w̄a
− be31a V̄
− B̄a
dx
dx 2
(22g)
dūa
d2 w̄a
− be31a ra V̄
− D̄a
dx
dx 2
(22h)
dūb
d2 w̄b
− B̄b
dx
dx 2
(22i)
T̄a = Āa
M̄a = B̄a
(22e)
T̄b = Āb
dūb
d2 w̄b
− D̄b
(22j)
dx
dx 2
where the over-bar represents the Fourier transformation with
respect to time, for example
∞
wa (x, t)e−iωt dt.
(23)
w̄a (x, ω) =
M̄b = B̄b
−∞
Equations (22) are simultaneous ordinary differential
equations with parameter ω. By introducing the variables
yi (i = 1, 2, . . . , 12) defined as
y1 = ūa
y2 = T̄a
dw̄a
y4 =
dx
y5 = Q̄a
y7 = ūb
y10 =
y8 = T̄b
dw̄b
dx
y11 = Q̄b
y3 = w̄a
y6 = M̄a
y9 = w̄b
(24)
y12 = M̄b
equations (22) can be written into the following state equation:
dY
= AY + B V̄
dx
(25)
where Y = (y1 , y2 , . . . , y12 )T is the state vector; the state
matrix A and vector B are given in the appendix.
After Fourier transformation of the boundary conditions in
equation (16), (25) and its corresponding boundary conditions
becomes a boundary value problem of ordinary differential
equations, which can be solved by the shooting method or the
relaxation method [19]. In this case, the continuity conditions
at the interface of the non-debonding and debonding regions
simply become
yin = yid ,
i = 1, 2, . . . , 12.
(26)
Since the governing equations are different in the bonded
and debonded areas, the multiple-shooting method [19] is
employed to solve this problem.
In this method, the
whole solution interval is first subdivided into a number of
subintervals, and the simple shooting method is used to obtain
solutions by adjusting the initial values for each subinterval.
The solutions for each subinterval can be combined to form
a set of linear algebraic equations by imposing the boundary
conditions at both ends of the overall interval and the continuity
conditions at the locations where two subintervals join. By
solving this set of algebraic equations, the solution for
equation (25) can be finally determined.
5. Results and discussion
In this section, dynamic changes of the smart beam with
debonding between the piezoelectric layer and the host beam
are numerically examined for a cantilever composite beam, and
particular attention will be given to the actuating and sensing
behaviour of the piezoelectric lamina with partial debonding.
The material properties and dimensions of the composite beam
are listed in table 1.
5.1. Comparison with the available results
To validate the model presented above, the frequencies of a
beam with debonded piezoelectric patches are compared with
the experimental result and the finite-element result presented
by Seeley and Chattopadhyay [15,16]. The host beam is made
of a [0◦ /90◦ ]3s composite material. The average thickness of
the beam is 1.94 mm, and the ply thickness is about 0.161 mm.
The beam is clamped at its left end and its effective length is
30 cm. Two 10.3 cm long and 0.0762 mm thick piezoelectric
patches are bonded on the upper and lower surfaces of the
composite beam and their left ends are 3.4 cm away from
the clamped end. The mass density of the beam and the
piezoelectric patch are 1507 and 5000 kg m−3 . The Young’s
modulus of the piezoelectric patches is 6.9 GPa. The equivalent
bending stiffness and extension stiffness for the composite
beam with unit width are calculated as D̄b = 45.5196 Nm2 ,
Āb = 1.19795×1010 N. In our calculation, the thickness of the
adhesive layer is taken as 0.1 mm and its physical properties
are given is table 1.
When the debonding length is taken as 0 (perfect bonding),
1.8, 3.6 and 5.4 cm, the first two modal frequencies of the beam
obtained by the present model are listed in table 2, in which
the experimental and finite-element analysis results by Seeley
and Chattopadhyay [15, 16] are also listed for the purpose of
717
L Tong et al
Table 2. Comparison of the natural frequencies (Hz) of the beam with debonded piezoelectric patches.
Present model
Experiment [15, 16]
Mode 1
Mode 2
Mode 1
Mode 2
Mode 1
Mode 2
0
1.8
3.6
5.4
25.48
25.19
24.96
24.80
115.95
115.85
118.71
116.58
25.1
24.5
24.6
24.5
120.6
118.7
119.4
120.3
25.4
24.6
24.2
23.8
116.3
115.3
113.2
116.3
4
60
2
Mode 1
Mode 2
Mode 3
40
Frequency change (%)
Amplitude change of sensor output (%)
80
20
0
-20
0
-2
-4
Mode 1
Mode 2
Mode 3
-6
-40
-8
-60
0
0.2
0.4
0.6
0.8
1
Normalized debonding location x d/L
Figure 3. Effect of debonding location of the sensor layer on
amplitude of charge output (debonding length Ld = 5 cm).
comparison. Table 2 shows that the first two modal frequencies
of the beam with debonded piezoelectric patch are in agreement
with the experiment and HOT results. The slight difference
may result from the following reasons: (1) a thin adhesive
layer (0.1 mm) with lower Young’s modulus is considered in
the calculation, which may affect the second modal frequency
more than the first modal frequency; (2) the equivalent bending
stiffness and extension stiffness of the composite host beam are
used in the calculation; (3) the small accelerometer (0.4 g) on
the free end of the host beam is not modelled in the current
calculation.
5.2. Example 1. Beam with fully covered piezoelectric
actuator and sensor layers
In this example, a cantilever composite beam with a sensor or
actuator covering its full length is considered. The beam itself
is clamped at one end and free at the other, while the boundary
conditions at both ends for the fully covered sensor or actuator
are free. The effect of debonding is considered by introducing
debondings of different length and locations. The debonding
effects on the sensing and actuating behaviour are discussed
as follows.
5.2.1. Sensing behaviour of actuator with debonding. To
study the debonding effects on the sensing behaviour, an initial
impulse ft (x, t) = 0.1δ(x − l)δ(t) N s is applied at the free
end of the clamped cantilever composite beam. Two cases are
considered. In case 1 a debonding of fixed length (50 mm)
718
High-order theory
(HOT) [15, 16]
Debonding
length
(cm)
0
0.2
0.4
0.6
0.8
1
Normalized debonding location x d/L
Figure 4. Effect of debonding location of the sensor layer on
frequencies of the first three modes (debonding length Ld = 5 cm).
is introduced and located at a different location from near the
clamped end to the free end. In case 2 the central location of
debonding is fixed, i.e. at the middle point of the beam, while
its length varies from 10 to 100 mm.
The frequency spectrum of the sensor output can be
calculated from the presented model. When a debonding
occurs between the piezoelectric sensor and the host beam,
both height and location of each peak in the frequency
spectrum, which reflect the amplitude and frequency of each
vibration mode of the beam, vary in different degrees. In
the following discussion, the amplitude and frequency change
of each peak in the frequency spectrum are used to examine
the effect of the debonding of the piezoelectric patches. For
case 1, figures 3 and 4 depict the amplitude changes of the
peaks in the spectrum of the charge generated by the sensor
layer and the natural frequency change, respectively, relative
to those without debonding. In both figures, xd denotes the
value of the x coordinates at the centre of the debonded
segment (see figure 1). As shown in figure 3, debonding can
significantly affect the amplitude of sensor output for the first
three modes when located near the clamped end. This effect
becomes relatively weak when the debonding area is far from
the clamped end. Figure 4 shows that a debonding near the
clamped end can reduce values of frequency for the first three
modes.
For case 2, figures 5 and 6 show the variation of the sensor
output and the natural frequencies relative to the non-debonded
cases are presented with different ratios of the debonding
length to the sensor length respectively. It is evident that
debonding barely affects the amplitude of sensor output and the
frequency for the first mode, and only a large debonding can
Sensing and actuating behaviours of piezoelectric layers with debonding in smart beams
(a)
Mode 1, debonding
-20
Mode 1
Mode 2
Mode 3
-40
1.5E-03
1.0E-03
5.0E-04
-60
0.0E+00
0
-80
5
10
15
20
25
30
x (cm)
0
0.1
0.2
0.3
0.4
Mode 2 and 3
Normalized debonding length L d/L s
4.E-04
(b)
Figure 5. Effect of the debonding length of the sensor layer on
charge output (xd = 15 cm).
Mode 2, perfect
Mode 2, debonding
Mode 3, perfect
Mode 3, debonding
3.E-04
|εs(x , ω)|
5
Frequency change (%)
Mode 1, perfect
2.0E-03
|εs(x , ω)|
Amplitude change of sensor output (%)
Mode 1
2.5E-03
0
2.E-04
0
1.E-04
0.E+00
-5
0
Mode 1
Mode 2
Mode 3
-15
0.2
0.3
0.4
15
x (cm)
20
25
30
Normalized debonding length L d/L
Figure 6. Effect of the debonding length of the sensor layer on
frequencies (xd = 15 cm).
have a noticeable effect in terms of reducing the amplitude of
sensor output and frequency. Since the sensor output is related
to the strain in the piezoelectric sensor layer (see equation (20)),
the strain distributions of the sensor layer for the first three
modes are shown in figure 7 for the case of a 5 cm long
debonding at the middle point of the beam. Clearly, the strain
distributions at the debonding area are remarkably different
from those of perfectly bonded sensors, which may be used to
potentially locate the debonding area.
5.2.2. Actuating behaviour of actuator with debonding.
In this case, an initial uniform voltage impulse V (x, t) =
100δ(t − 0) Vs is applied on the actuator layer to excite the
beam. For the case of different locations of a 3 cm long
debonding, figure 8 depicts the amplitude variation of the
transverse displacements of the free end of the beam. Once
again a debonding near the clamped end can significantly
change the transverse displacement at the free end. This
is further shown in figure 9, which plots the transverse
displacement at the free end versus the non-dimensional
60
Amplitude change of tip displacement (%)
0.1
10
Figure 7. Debonding effect on strain distribution in the mid-plane
of the sensor layer for the first three mode (xd = 15 cm, Ld = 5cm).
(a) mode 1; (b) mode 2 and 3.
-10
0
5
Mode 1
Mode 2
Mode 3
40
20
0
-20
-40
-60
0
0.2
0.4
0.6
0.8
Normalized debonding location x d/L
1
Figure 8. Effect of the debonding of the actuator layer on transverse
displacement at the free end (debonding length Ld = 3 cm).
debonding length for the case when the debonding occurs at
the clamped end. Similar to the sensor case, the actuating
behaviour of the actuator layer may be influenced when a
debonding locates at either end of the beam, especially at the
clamped end of the beam. In this case, a longer debonding can
lead to a more significant decrease of the actuator’s ability to
actuate the first and third modes of the beam.
719
L Tong et al
2.0E-03
1.8E-03
60
1.6E-03
40
Axial strain | εa|
Amplitude change of tip displacement (%)
80
Mode 1
mode 2
Mode 3
20
0
8.0E-04
6.0E-04
2.0E-04
-40
0.07
0.08
0.09
0.1
0.11
Sensor span (m)
0
0.1
0.2
0.3
Figure 11. Effect of debonding located in the middle of the beam on
axial strain distribution in sensor patch.
Figure 9. Effect of the debonding length of the actuator layer on
transverse displacement at the free end (debonding at left end).
2.5E+03
3.2E-05
2.8E-05
(a)
1.5E+03
|q (ω1)|
Mode 1, perfect
Mode 1, debonding
Mode 2, perfect
Mode 2, debonding
2.0E+03
Axial force (N)
Mode 1, perfect
Mode 1, debonding
Mode 2, perfect
Mode 2, debonding
1.0E-03
0.0E+00
0.06
Normalized debonding length L d/L s
2.4E-05
2.0E-05
1.0E+03
had/hs=0.25
had/hs=0.5
5.0E+02
1.6E-05
0.1
0.2
had/hs=0.375
had/hs=0.75
0.3
0.4
0.5
Normalized debonding length L d/L s
0.0E+00
0
0.1
0.2
Beam span (m)
0.3
Figure 12. Effect of debonding on sensor output at the first mode
with different bonding thickness (debonding located at the right end
of the sensor patch).
3.5E-02
(b)
Mode 1, perfect
Mode 1, debonding
Mode 2, perfect
Mode 2, debonding
3.0E-02
Bending moment (Nm)
1.2E-03
4.0E-04
-20
-60
1.4E-03
2.5E-02
5.3. Example 2. Beam bonded with piezoelectric actuators
and sensors covering a portion
In this example, we consider the same cantilever beam as in
example 1 except that the piezoelectric actuator and sensor
only cover a small portion of its length; that is, both sensor and
actuator are assumed to be 5 cm long, i.e. Ls = La = 5 cm.
The excitations for the sensor and actuator cases are also the
same as those used in example 1.
2.0E-02
1.5E-02
1.0E-02
5.0E-03
0.0E+00
0
0.05
0.1
0.15
0.2
0.25
0.3
Beam span (m)
Figure 10. Debonding effect on internal force distribution in
actuator layer.
Figure 10 shows the axial force and the bending moment
distributions induced by the applied actuating voltage impulse
for the case when there is a 3 cm debonding located at the
middle of the beam. It is clearly shown that for the three modes
the distributions of the axial force and the bending moment
along the actuator deviate remarkably from those of a perfectly
bonded actuator at the debonding region.
720
5.3.1. Sensing. The sensor patch is bonded on the surface of
the host beam and its left and right ends are 6 and 11 cm away
from the clamped end. When a 1 cm long debonding occurs
at the mid-point of the sensor patch, its effect on the strain
distribution for the first and the second modes of the beam can
be obtained and is shown in figure 11. It is noted that the axial
strain becomes a constant in most of the debonding region for
both modes 1 and 2. This is similar to the results observed in
example 1.
Figure 12 depicts the amplitude variation of the sensor
output for the first mode with different debonding length and
bondline thickness when a debonding takes place at the right
end of the sensor patch (11 cm from the clamped end). In
figure 12, Ld and Ls denote the debonding length and the
Sensing and actuating behaviours of piezoelectric layers with debonding in smart beams
1st Mode
3.0E-02
3.5E-05
2.5E-02
(a)
perfect bonding
2.0E-02
|w tip(ω1)| (m)
3.0E-05
left end debonding
2.5E-05
|q (ω)|
mid debonding
1.5E-02
2.0E-05
right end debonding
1.5E-05
5.0E-03
1.0E-05
0.0E+00
5.0E-06
0.0E+00
18
19
Frequency (Hz)
20
perfect
Frequency (Hz)
|q (ω)|
1.5E-06
1.0E-06
18.85
18.8
18.75
18.7
18.65
5.0E-07
0.0E+00
114
0.5
ha/hb=0.1
ha/hb=0.2
ha/hb=0.3
ha/hb=0.4
18.9
left end
debonding
mid
debonding
right end
debonding
2.0E-06
0.1
0.2
0.3
0.4
Normalized debonding length L d/L a
18.95
3.0E-06
(b)
0
(a)
2nd Mode
2.5E-06
ha/hb=0.1
ha/hb=0.2
ha/hb=0.3
ha/hb=0.4
1.0E-02
18.6
115
116
Frequency (Hz)
117
Figure 13. Debonding effect on frequency spectra of the transverse
displacement at the free end: (a) first mode; (b) second mode.
length of the sensor, respectively. Figure 12 clearly shows
that the sensor output for the first mode linearly decreases
with the non-dimensional debonding length. The sensor output
drops almost 50% when the non-dimensional debonding length
increases from 10 to 50%. It can also be seen that the bondline
thickness slightly affects the sensor output; however, it does
not affect the overall trend.
The effect of debonding location on the frequency
structure of the charge output of the sensor patch is investigated
by considering the following three debonding cases:
(a) 1 cm debonding measured from the left end of the sensor
patch,
(b) 3 cm debonding located in the middle of the sensor patch
and
(c) 1 cm debonding measured from the right end of the sensor
patch.
Figure 13 depicts the frequency spectra near the first and
second modes for the three debonding cases. Figure 13(a)
shows that the debonding at both ends of the sensor patch
reduces the peak values of the sensor output for mode 1 and
debonding in the middle (case (b)) does not have any effect.
The reduction in sensor output for case (a) is slightly more than
that for case (c). This may be because the debonding of case (a)
is located closer to the clamped end than that of case (c) for
the first mode. Figure 13(b) shows that for mode 2 debonding
of case (c) yields a significant reduction in the sensor output
0
0.1
0.2
0.3
0.4
Normalized debonding length Ld/La
0.5
(b)
Figure 14. Effect of debonding length on the first mode with
different thicknesses of actuator patch (debonding located at the
right end of the actuator patch): (a) effect on amplitude of transverse
displacement; (b) effect on the first frequency of the beam.
while debonding of cases (a) and (b) have very little effect. It
is believed that the significant reduction in sensor output for
mode 2 in case (c) is primarily due to the location of debonding.
It is thus clear that debonding at both ends may cause a
significant reduction in the amplitude of the sensor’s output,
and consequently affect the sensing ability of the sensor patch.
Figure 13 also demonstrates that the debonding at both ends
can also slightly change the frequencies of the beam, which
may affect the control robustness in a closed-loop control.
5.3.2. Actuating. For the case of an actuator patch, we
consider a patch located 10 cm away from the clamped end
and with a debonding at its right end. The debonding length
varies from zero to half the length of the actuator, and the
thickness of actuator patch is also changed. Figure 14 depicts
the effect of debonding length and the thickness of actuator
patch on the displacement at the free end and the natural
frequency for the first vibration mode. In this figure, ha
and hb denote the thickness of the actuator and host beam
respectively, while Ld and La represent the length of the
debonding region and actuator. Figure 14(a) clearly indicates
that a debonding at the right end can reduce the tip deflection
of the beam, which implies loss of actuating ability if tip
deflection is used as a measure of the actuating authority. It
721
L Tong et al
non-debonded region similar to that of perfect debonding, but
with a smaller magnitude. This is probably because the actual
left end of the actuator with debonding is slightly further from
the clamped end. Figure 15(b) shows that a middle debonding
creates another two small peaks at the ends of the debonding
region in addition to the original two peaks, but the original
two peak values are hardly affected. It is also observed that the
left end debonding can remarkably reduce values of the two
peaks.
800
(a)
Axial force |T a(ω1)| (N)
700
600
500
400
perfect bonding
300
200
mid debonding
100
left end
debonding
0
0.02
0.03
0.04
0.05
6. Conclusions
0.06
0.07
Actuator patch span (m)
70
(b)
Shear force |Q a(ω1)| (N)
60
perfect bonding
50
mid debonding
40
left end
debonding
30
20
10
0
0.02
0.03
0.04
0.05
0.06
0.07
Actuator span (m)
Figure 15. Debonding effect on stress resultants in the actuator
patch (first mode).
is also found that reduction in the thickness of actuator patch
leads to an increased tip deflection, and the reduction in the tip
deflection due to debonding for the case of the thicker actuator
is slightly smaller than that for the case of the thinner one.
The results indicate that a thinner actuator patch may be more
profoundly affected by the debonding in terms of actuated tip
deflection. It can also be seen that a thicker actuator patch will
not excite a larger vibration at the same actuating voltage level
probably due to the stiffness increase of the composite beam.
Figure 14(b) shows that the effect of debonding length on the
frequency of mode 1 is similar to that on the tip deflection;
however, the magnitude of the reduction is not as remarkable
as that in the tip deflection. The effect of actuator thickness on
the natural frequency is opposite to that on the tip deflection;
namely, the frequency for the case with the thicker actuator is
more sensitive to debonding than with the thinner one.
Consider another example in which the left end of the
actuator patch is located 2 cm away from the clamped end
with two debonding cases: (a) 1 cm debonding at the left
end of the actuator patch and (b) 3 cm debonding located at
the middle of the actuator patch. Figure 15 depicts the axial
force and shear force distributions along the actuator span.
Figure 15(a) shows that the axial force becomes a constant
in the debonding region when it occurs in the middle of the
actuator patch. It also shows that the axial force for the case
of debonding at the left end has a distribution pattern in the
722
This paper presents analytical models for smart beams with
debonded piezoelectric actuators and sensors. Numerical
results illustrate the following points:
(a) The distributions of strains and stress resultants, such
as axial force, shear force and bending moment, in
the piezoelectric layers/patches in the debonding region
are remarkably different from those without debonding.
However, the effect of debonding on displacement
distributions in the debonding region is very weak.
(b) Debonding at the end of the sensor/actuator may have a
more noticeable effect in terms of reducing the sensor
output charge, frequency and tip deflection of the beam
than that in the middle area, which in turn affect the sensing
ability and actuating authority, particularly for the first
vibration mode.
(c) Debonding at the end of an actuator leads to reduction
of the actuated tip deflection, and this effect for a thin
actuator may be more severe than that for a thick actuator.
Appendix
The state matrix A and vector B in equation (25) have the
following form:
0
bG
h ad
Aa
0
0
0
0
0
0
Ba
0
0
0
0
a
bGh a
2h ad
0
bG
h ad
0
0
=
0
bGh b
2h ad
bYad
h ad
0
0
0
0
0
0
0
0
0
0
Yad
h ad
0
0
0
0
bGh a
2h ad
1
0
0
a
bGh 2a
4h ad
0
bGh a
2h ad
0
0
0
bGh a h b
4h ad
0
Ba
0
0
0
0
0
Da
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
bG
h ad
0
0
0
0
0
0
0
0
0
0
bYad
h ad
0
0
bGh a
0
2h ad
0
Ab
bG
b 0
h ad
0
0
0
Bb
0
0
bGh b
0
2h ad
0
bGh b
2h ad
0
0
0
0
0
0
0
b
0
B = (A∗a be31a − Ba∗ be31a ra , 0, 0, Ba∗ be31a
−Da∗ be31a ra , 0, 0, 0, 0, 0, 0, 0, 0)T
where
αa = ρa Aa ω2 ,
αb = ρb Ab ω2 ,
G̃ = kb Gad ,
Ỹad = kb Yad
D̄
B̄a
a
A∗a =
,
Ba∗ =
,
Āa D̄a − B̄a2
Āa D̄a − B̄a2
Da∗ =
Āa
Āa D̄a − B̄a2
0 0
0 0
0 0
0 0
bGh a h b
0
4h ad
0
0
bGh b
0
2h
1
0
0
0
0
bYad
h ad
0 0
0
0
Bb
0
0
Db
0 0
bGh 2b
1 0
4h ad
Sensing and actuating behaviours of piezoelectric layers with debonding in smart beams
A∗b =
D̄b
,
Āb D̄b − B̄b2
Db∗ =
Āb
.
Āb D̄b − B̄b2
Bb∗ =
B̄b
,
Āb D̄b − B̄b2
The matrix A has different entries in the debonding and
perfect bonding areas since kb has different values as shown in
equation (13).
Acknowledgment
The authors are grateful for the support of the Australia
Research Council through a large grant (no A 10009074).
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