-
MODELLING OF SHEAR LAG EFFECT IN ADHESIVE
BOND LAYER FOR SMART STRUCTURE APPLICATIONS
BY Praveen Kumar
(2004CES2061)
Department Of Civil Engineering Submitted
In partial fulfilment of the requirement of degree of
MASTER OF TECHNOLOGY
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NewDelhi110016
MAY 2006
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CERTIFICATE This is to certify that the report titled “MODELLING
OF SHEAR LAG EFFECT IN ADHESIVE BOND LAYER FOR SMART STRUCTURE
APPLICATIONS” is a bona fide record of work done by PRAVEEN KUMAR
for the partial fulfillment of the requirement for the degree of
Masters in Structural Engineering, Civil
Engineering Department, Indian Institute of Technology, New
Delhi, India. He
has fulfilled the requirements for the submission of this
report, which to the best
of our knowledge has reached the requisite standard.
This project was carried out under our supervision and guidance
and has
not been submitted elsewhere for the award of any other
degree.
Dr. SURESH BHALLA Dr. T.K.DATTA Department of Civil Engg.
Department of Civil Engg. IIT DELHI IIT DELHI
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ACKNOWLEDGEMENT
I feel great pleasure and privilege to express my deep sense of
gratitude,
indebtedness and thankfulness towards my supervisors, Dr. SURESH
BHALLA and Dr. T.K.DATTA for their invaluable guidance, constant
supervision, and continuous encouragement and support throughout
the coursework. Their
recommendable suggestions and critical views have greatly helped
me in
successful completion of this work.
I must acknowledge the friendly attitude and valuable
suggestions made
by the faculty of Civil Engineering Department, IIT Delhi.
I also acknowledge with sincerity, the help rendered by my
colleagues at
various stages of this report.
My foremost thanks are due to my parents, my elder sisters and
my
younger brother for their encouragement, support, love and
affection and moral
boosting, which keep me going always.
I am also thankful to all those who helped directly or
indirectly in
completion of this work.
New Delhi PRAVEEN KUMAR MAY, 2006 2004CES2061
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ABSTRACT
The electromechanical impedance (EMI) technique for structural
health
monitoring (SHM) and non-destructive evaluation (NDE) employs
piezoelectric- ceramic
(PZT) patches, which are surface bonded to the monitored
structure using adhesives. The
adhesive forms a finitely thick, permanent interfacial layer
between the host structure and
the patch. Hence, the force transmission between the structure
and the patch occurs
through the bond layer, via shear mechanism, invariably causing
the shear lag. Bhalla and
Soh (2004) presented the step-by-step derivation to integrate
the shear lag effect into
impedance formulations, both 1D and 2D. But the solution
presented by them was a
rigorous solution and involved solving fourth order differential
equations. In this report, a
new simplified 1D impedance model to incorporate shear lag
effect has been developed
named as Kumar, Bhalla and Datta model or simply KBD model. The
conductance
signatures obtained using this model are compared with the 1D
impedance model of
Bhalla and Soh (2004).
It is found that the conductance signatures obtained using the
KBD model are in
close proximity with those given by the Bhalla and Soh model
(2004). Further, the effect
of the various parameters related to the bond layer viz. the
length of the PZT patch,
mechanical loss factor, shear modulus, thickness of bond layer
on the electromechanical
admittance response is studied by means of a detailed parametric
study. In addition, a
new method has been developed for predicting the shear stress in
the bonding layer for
different excitation frequencies based on the KBD model. A
comparison between the
shear stress obtained by the KBD model and the Bhalla and Soh
model (2004) revealed
reasonable agreement between the two models. Thus the new KBD
model, which is much
more simplified, can be used for carrying out preliminary design
in structural control
related problems.
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CONTENTS
Certificate ………… (i)
Acknowledgement ………..(ii)
Abstract ………..(iii)
List of Contents ……….(iv)
List of Figures ...……(vi)
List of Tables ………(vii)
List of Symbols ….(viii)
1. INTRODUCTION ………1
2. LITERATURE REVIEW
2.1 Structural Health Monitoring ……..3
2.2 Smart Systems/ Structures ……..4
2.3 Piezoelectric Materials ……5
2.4 Mechatronic Impedance Transducers ……7
2.6 Electro Mechanical Impedance (EMI) Technique ... …7
3 ANALYTICAL MODELLING OF SHEAR LAG EFFECT
3.1 Introduction …12
3.1.1 PZT patch as sensor …13
3.1.2 PZT patch as actuator …..16
3.2 Integration of shear lag effect into impedance models
…...17
3.3 Modified 1D Impedance Model by Xiu and Liu …..18
3.4 Inclusion of shear lag into 1D Impedance Model by Bhalla and
Soh …..19
4 KBD 1D IMPEDANCE MODEL
4.1 Introduction ……23
4.2 Determination of Real and Imaginary components of eqZ
……25
4.3 Verification of KBD Model
4.3.1 Generation of finite element model …26
4.3.2 Convergence Test …29
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4.3.3 Visual Basic Programs ….31
4.3.4 Matlab Program …31
4.3.5 Results …..32
5 PARAMETRIC STUDY
5.1 Introduction ……..35
5.2 Influence of bond layer shear modulus sG …….35
5.3 Influence of length of PZT patch …….37
5.4 Influence of mechanical loss factor …….38
5.5 Influence of bond layer thickness ……..39
6 SHEAR STRESS PREDICTION IN BOND LAYER
6.1 Introduction ………41
6.2 Shear stress by KBD model ……...41
6.3 Distribution of shear stress in bond layer using Bhalla and
Soh
1D impedance model (2004) ……..44
7 CONCLUSIONS AND RECOMMENDATIONS
7.1 Conclusions ……..49
7.2 Recommendations ……..50
REFERENCES
APPENDIX
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LIST OF FIGURES
Figure Description Page
2.1 A Piezoelectric Material Sheet with conventional 1, 2 ,3
directions 6
2.2 A PZT patch bonded to the Structure under electric
excitation 7
2.3 Interaction Model of PZT patch and the host structure 8
3.1 A PZT patch bonded to a beam using adhesive bond layer
12
3.2 Strain distribution across the length of the PZT patch
15
3.3 Variation of effective length with shear lag factor 15
3.4 Distribution of piezoelectric and beam strains 17
3.5 Modified Impedance model by Xiu and Liu 18
3.6 Deformation in bonding layer and PZT patch 21
4.1 Diagram showing the KBD model 23
4.2 A cantilever model in ANSYS 9 27
4.3 ANSYS model 28
4.4 Comparing Conductance Signatures for s pt t= 33
4.5 Comparing Conductance Signatures for / 3s pt t= 33
4.6 Comparing Conductance Signatures for 0.1s pt t= 34
4.7 Comparing Susceptance for s pt t= 34
5.1 Influence of shear modulus on Conductance Signatures 36
5.2 Influence of shear modulus on Susceptance 36
5.3 Influence of length of PZT patch on Conductance 37
5.4 Influence of length of PZT patch on Susceptance 37
5.5 Influence of mechanical loss factor on Conductance 38
5.6 Influence of mechanical loss factor on Susceptance 39
5.7 Influence of bond layer thickness on conductance 40
5.8 Influence of bond layer thickness on susceptance 40
6.1 Shear stress distribution along length of actuator using BSM
45
6.2 Comparing shear stress distribution for different
frequencies using
BSM 46
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LIST OF TABLES
Table Description Page
4.1 Physical Properties of Al-6061 – T6 27
4.2 Details of modes of vibrations of test structure 30
4.3 Physical Properties of PZT patch 32
6.1 Shear Stress Distribution for different frequencies
using BSM 47
6.2 Shear stress distribution for different frequencies
using
KBD Model 48
6.3 Comparing Shear stress distribution for different
frequencies
using KBD Model and BSM 48
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LIST OF SYMBOLS
SYMBOL DESCRIPTION
[ ]D Electric displacement vector
[ ]S Second order strain tensor Tε⎡ ⎤⎣ ⎦ Second order dielectric
permittivity tensor
[ ]E The applied electric field vector dd⎡ ⎤⎣ ⎦ ,
cd⎡ ⎤⎣ ⎦ The third order piezoelectric strain coefficient
tensors
ES⎡ ⎤⎣ ⎦ The fourth order elastic compliance tensor under
constant electric
field
d31 , d32 , d33 The normal strain in the 1, 2, and 3 directions
respectively.
d15 The shear strain in the 1-3 plane EY Young’s modulus of
elasticity of the PZT patch at constant electric
field
EY Complex Young’s modulus of elasticity of the PZT patch at
constant electric field
33Tε Complex electric permittivity
η Mechanical loss factor of the PZT material
δ Dielectric loss factor of the PZT material
κ Wave number
ω Angular frequency of excitation
sG Shear modulus of elasticity of the bonding layer
sG Complex shear modulus of elasticity of the bonding layer
ξ Strain lag ratio
Г Shear lag parameter
Z Impedance of the structure
aZ Impedance of the actuator
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η′ Mechanical loss factor of bonding layer
F Force transmitted to the structure
u Displacement in the structure
pu Displacement in the PZT patch
γ Shear strain in the bonding layer
τ Interfacial shear stress
st Thickness of the adhesive bond layer
pt Thickness of the PZT patch
eqZ Equivalent impedance apparent at the ends of the PZT
patch
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CHAPTER 1
INTRODUCTION
During the last decade, the electromechanical impedance (EMI)
technique has
emerged as a universal cost effective technique for structural
health monitoring (SHM)
and non destructive evaluation (NDE) of all types of engineering
structures and systems.
In this technique, a piezoceramic patch, surface bonded to the
monitored structure,
employs ultrasonic vibrations (typically in 30 – 400 kHz range)
to derive a characteristic
electrical signature of the structure (in the frequency domain),
containing vital
information concerning the phenomenological nature of the
structure. Electromechanical
admittance, which is the measured electrical parameter, can be
decomposed and analyzed
to extract the impedance parameters of the host structure
(Bhalla and Soh, 2004 b). In this
manner, the piezoceramic patch (commonly known as PZT patch),
acting as piezo
impedance transducer, enables structural identification, health
monitoring and NDE.
The PZT patches are made up of ‘piezoelectric’ materials, which
generate surface
charges in response to the mechanical stresses and conversely
undergo mechanical
deformations in response to electric fields. In the EMI
technique, the bonded PZT patch
is electrically excited by applying an alternating voltage
across its terminals using an
impedance analyzer. This produces deformations in the patch as
well as in the local area
of the host structure surrounding it. The response of this area
is transferred back to the
PZT wafer in the form of admittance (the electrical response),
comprising of the
conductance (real part) and the susceptance (imaginary part).
Hence, the same PZT patch
acts as an actuator as well as the sensor concurrently. Any
damage to the structure
manifests itself as a deviation in the admittance signature,
which serves as an indication
of the damage (Bhalla, 2004).
This report deals with the development and verification of a new
simplified 1D
impedance model incorporating the effect of finitely thick
adhesive bond layer between
the PZT patch and the host structure. The inherent cause of the
shear lag effect is the
flexibility associated with the adhesive bond layer due to which
same deformation is not
transferred to the PZT patch and the host structure. The effect
of this difference in
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deformation is that the absolute electromechanical admittance
signatures may not be
obtained unless the shear lag effect is incorporated into the
expression for the
electromechanical admittance. The new 1D impedance model
developed in this report is
named Kumar, Bhalla and Datta model or simply KBD model. This
KBD model is used
to predict the shear stresses in the adhesive bond layer.
However, the present study does
not cover the aspect of damage quantification.
In the present report Chapter 2 gives the review of available
literature on SHM.
Chapter 3 deals with the analytical modelling of shear lag
effect. A general theory related
to the shear lag effect and the 1D impedance models are covered.
In the chapter 4, the
new 1D impedance model, named as Kumar, Bhalla and Datta, is
developed. Chapter 4
covers the verification of the KBD model. Chapter 5 covers the
detailed parametric study
for the admittance signature using the KBD model. Chapter 6
provides the theory for the
prediction of the shear stress in the bond layer using the KBD
model. Chapter 7 provides
the major conclusions derived from the research conducted in
this work and the
recommendations. At the end of this chapter 7, list of
references used in the present work
are provided. At the end, the programs utilized in the analysis
are provided in the
appendix.
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CHAPTER 2
LITERATURE REVIEW
2.1 STRUCTURAL HEALTH MONITORING (SHM)
SHM is defined as the acquisition, validation and analysis of
technical data to
facilitate life cycle management decisions. SHM denotes a
reliable system with the
ability to detect and interpret adverse changes in a structure
due to damage or normal
operations (Bhalla, 2004). Such a system consists of sensors,
actuators, amplifiers and
signal conditioning circuits. While sensors are employed to
predict damage, the actuators
serve to excite the structure or decelerate/ arrest the
damage.
In the broad sense, the SHM/ NDE methodology can be classified
as global and
local. The global techniques rely on global structural response
for damage identification
whereas the local techniques employ localized structural
interrogation for this purpose.
2.1.1 Global SHM Techniques
The global SHM techniques can be further divided into two
categories, dynamic
and static. In global dynamic techniques, the test structure is
subjected to low frequency
excitations either harmonic or impulse and the resulting
vibration responses
(displacement, velocities or accelerations) are picked up at
specified locations along the
structure. The vibration pick up data is processed to extract
the first few mode shapes and
corresponding natural frequencies of the structure, which, when
compared with the
corresponding data for the healthy state, yields information
pertaining to the location and
the severity of the damages. In this connection, the impulse
excitation technique is much
more expedient than harmonic excitation (which is however much
more accurate) and
hence preferred for quick estimates (Bhalla, 2004).
Contrary to these vibration based global methods, many
researchers have
proposed methods based on global static structural response,
such as static displacement
response technique and the static strain measurement technique.
These techniques, like
the dynamic techniques essentially aim for structural system
identification, but employ
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static data (such as displacements and strains) instead of
vibration data. Although
conceptually sound, the application of the static response based
technique on real life
structure is not practically feasible. For example, the static
displacement technique
involves applying static forces at specified nodal points and
measuring the corresponding
displacements. Measurement of displacement on large structures
is a mammoth task. As a
first step, it warrants establishment of the frame of reference,
which, for contact
measurement, could demand the construction of a secondary
structure on an independent
foundation. Besides, the application of large load cause
measurable deflections (or
strains) warrants huge machinery and power input. As such, these
methods are too
tedious and expensive to enable a timely and cost effective
assessment of the health of
real life structures.
2.1.2 Local SHM Techniques
Another category of damage detection methods is formed by the so
called local
methods, which, as opposed to the global techniques, rely on
localized structural
interrogation for detecting damages. Some of the methods in this
category are the
ultrasonic techniques, acoustic emission, eddy currents, impact
echo testing, magnetic
field analysis, penetrant dye testing, and X-ray analysis.
2.2 SMART SYSTEMS/ STRUCTURES
The definition of smart structures was a topic of controversy
from the late
1970’s to the late 1980’s. In order to arrive at a consensus for
major terminology, a
special workshop was organized by the US Army Research Office in
1988, in which
sensors, actuators, control mechanism and timely response were
recognized as the four
qualifying features of any smart system or structure. In this
workshop, following
definition of smart systems/ structures was formally adopted
(Bhalla, 2004).
“A system or material which has built-in or intrinsic sensor(s),
actuator(s)
and control mechanism(s) whereby it is capable of sensing a
stimulus, responding to it
in a predetermined manner and extent, in a short and appropriate
time, and reverting
to its original state as soon as the stimulus is removed ”
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The sensor, actuator and controller combination can be realized
either at the
macroscopic (structure) level or microscopic (material) level.
Accordingly, we have
smart structures and materials respectively.
2.3 PIEZOELECTRIC MATERIALS
Piezoelectric materials are commonly used in smart structural
systems both as
sensors and actuators (Bhalla, 2004). A key characteristic of
these materials is the
utilization of the converse piezoelectric effect to actuate the
structure in addition to the
direct effect to sense structural deformations.
The constitutive relationships for piezoelectric materials,
under small field
conditions are (Bhalla, 2004)
T di ij j im mD E d Tε= + (2.1)
c E
k jk j km mS d E s T= + (2.2)
Equation (2.1) represents the direct effect (that is the stress
induced electric charge).
Equation (2.2) represents the converse effect (that is electric
field induced mechanical
strain).
When a sensor is exposed to stress field, it generates
proportional charge in
response, which can be measured. On the other hand, the actuator
is bonded to the
structure and an external field is applied to it, which results
in an induced strain field. In
more general terms above equations can be written in the tensor
form as (Bhalla, 2004).
T d
c E
D EdS Td s
ε⎛ ⎞⎡ ⎤ ⎡ ⎤= ⎜ ⎟⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦⎝ ⎠ (2.3)
where [ ]D (3x1) (C/m2) is the electric displacement vector, [
]S (3x3) the second order
strain tensor, [ ]E (3x1) (V/m) the applied electric field
vector and [ ]T (3x3) (N/m2) the
stress tensor. Accordingly, Tε⎡ ⎤⎣ ⎦ (F/m) is the second order
dielectric permittivity tensor
under constant stress, dd⎡ ⎤⎣ ⎦ (C/N) and cd⎡ ⎤⎣ ⎦ (m/V) the
third order piezoelectric strain
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coefficient tensors, and ES⎡ ⎤⎣ ⎦ (m2/N) the fourth order
elastic compliance tensor under
constant electric field.
The matrix cd⎡ ⎤⎣ ⎦ depends on the crystal structure. For PZT it
is given by
31
32
33
24
15
0 00 00 0
0 00 00 00
c
dd
d dd
d
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
(2.4)
where the coefficients d31 , d32 and d33 relate the normal
strain in the 1, 2, and 3 directions
respectively to an electric field along the poling direction 3
(see Fig. 2.1). For the PZT
crystals, the coefficients d15 relates the shear strain in the
1-3 plane to the field E1 and d24 relates the shear strain in the
2-3 plane to the electric field E2.
Fig. 2.1 A piezoelectric material sheet with conventional 1, 2
and 3 axes.
2
1
3
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2.4 MECHATRONIC IMPEDANCE TRANSDUCERS (MITs) FOR SHM
The term mechatronic impedance transducer (MIT) was coined by
Park (Bhalla,
2004). A mechatronic transducer is defined as a transducer which
can convert electrical
energy into mechanical energy and vice versa. The piezoceramic
(PZT) materials,
because of the direct (sensor) and converse (actuator)
capabilities are mechatronic
transducers.
2.5 ELECTRO MECHANICAL IMPEDANCE (EMI) TECHNIQUE
The EMI technique is very similar to the conventional global
dynamic response
techniques. The major difference is with respect to the
frequency range employed, which
is typically 30-400 kHz in EMI technique, against less than 100
kHz in the case of the
global dynamic methods.
In the EMI technique, a PZT patch is bonded to the surface of
the monitored
structure using a high strength epoxy adhesive, and electrically
excited via an impedance
analyzer. In this configuration, the PZT patch essentially
behaves as a thin bar
undergoing axial vibrations and interacting with the host
structure, as shown in Fig. 2.2.
(a)
Fig. 2.2 A PZT patch bonded to the structure under electric
excitation (Bhalla, 2004)
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Fig. 2.3 Interaction model of PZT patch and host structure
(Bhalla, 2004)
The PZT patch-host structure system can be modelled as a
mechanical impedance
(due to the host structure) connected to an axially vibrating
thin bar (the patch), as shown
in Fig.2.3. The patch in this figure expands and contracts
dynamically in the direction ‘1’
when an alternating electric field 3E (which is spatially
uniform i.e. 3 3/ /E x E y∂ ∂ = ∂ ∂ ) is
applied in the direction ‘3’.
The patch has half length ‘ l ’, width ‘ w ’ and thickness ‘ h
’. The host structure is
assumed to be a skeletal structure, that is, composed of one
dimensional members with
their sectional properties (area and moment of inertia) lumped
along their neutral axes.
Therefore, the vibrations of the PZT patch in the direction ‘2’
can be ignored. At the
same time, the PZT loading in direction ‘3’ is neglected by
assuming the frequencies
involved to be much less than the first resonant frequency for
thickness vibrations. The
vibrating patch is assumed infinitely small and to possess
negligible mass and stiffness as
compared to the host structure. The structure therefore can be
assumed to possess
uniform dynamic stiffness over the entire bonded area. The two
end points of the patch
can thus be assumed to encounter equal mechanical impedance, Z,
from the structure, as
shown in Fig.2.3. Under this condition, the PZT patch has zero
displacement at the mid
point ( 0x = ), irrespective of the location of the patch on the
host structure. Under these
assumptions, the constitutive relations (Eqs. 2.1 and 2.2) can
be simplified as (Bhalla,
2004)
3 33 3 31 1TD E d Tε= + (2.5)
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11 31 3ETS d EY
= + (2.6)
where S1 is the strain in direction ‘1’, D3 the electric
displacement over the PZT patch,
d31 the piezoelectric strain coefficient and T1 the axial stress
in the direction ‘1’.
(1 )E EY Y jη= + is the complex Young’s modulus of elasticity of
the PZT patch at
constant electric field and 33 33 (1 )T jε ε δ= − is the complex
electric permittivity (in
direction ‘3’) of the PZT material at constant stress, where 1j
= − . Here, η and δ
denote respectively the mechanical loss factor and the
dielectric loss factor of the PZT
material. The one-dimensional vibrations of the PZT patch are
governed by the following
differential equation (Bhalla, 2004), derived based on the
dynamic equilibrium of the
PZT patch.
2 2
2 2E u uY
x tρ∂ ∂=
∂ ∂ (2.7)
where ‘u’ is the displacement at any point on the patch in
direction ‘1’. Solution of the
governing differential equation by the method of separation of
variables yields
( sin cos )u A x B xκ κ= + (2.8) where κ is the wave number,
related to the angular frequency of excitation ω, the density
ρ and the complex Young’s modulus of elasticity of the patch
by
EYρκ ω= (2.9)
Application of the mechanical boundary condition that at x = 0 (
mid point of the PZT
patch ), u = 0 yields B = 0 .
Hence, strain in PZT patch
1( ) cosj tuS x Ae xx
ω κ κ∂= =∂
(2.10)
and velocity
( ) sinj tuu x Aj e xt
ωω κ∂= =∂
& (2.11)
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Further, by definition, the mechanical impedance Z of the
structure is related to the axial
force F in the PZT patch by ( ) 1( ) ( )x l x l x lF whT Zu= =
== = − & (2.12)
where the negative sign signifies the fact that a positive
displacement (or velocity) causes
compressive force in the PZT patch (Bhalla, 2004). Making use of
the Eq.(2.6) and
substituting the expressions for strain and velocity from Eqs.
(2.10) and (2.11)
respectively, we can derive
0 31cos( )( )
a
a
Z V dAh l Z Zκ κ
=+
(2.13)
where aZ is the short circuited mechanical impedance of the PZT
patch, given by
( ) tan( )
E
awhYZ
j lκω κ
= (2.14)
aZ is defined as the force required to produce unit velocity in
the PZT patch in the short
circuited condition ( i.e. ignoring the piezoelectric effect )
and ignoring the host structure.
The electric current, which is the time rate of change of
charge, can be obtained as
3 3A A
I D dxdy j D dxdyω= =∫∫ ∫∫&
(2.15)
Making use of the PZT constitutive relation (Eq. 2.5), and
integrating over the entire
surface of the PZT patch (-l to +l), Bhalla (2004) obtained an
expression for the
electromechanical admittance (the inverse of electro-mechanical
impedance) as
2 233 31 31tan2 ( )T E Ea
a
Zwl lY j d Y d Yh Z Z l
κω εκ
⎡ ⎤⎛ ⎞ ⎛ ⎞= − +⎢ ⎥⎜ ⎟ ⎜ ⎟+ ⎝ ⎠⎝ ⎠⎣ ⎦ (2.16)
In the EMI technique, this electro-mechanical coupling between
the mechanical
impedance Z of the host structure and the electro-mechanical
admittance Y is utilized for
Y damage detection. Z is a function of the structural
parameters-the stiffness, the
damping and the mass distribution. Any damage to the structure
will cause these
structural parameters to change, and hence alter the drive point
impedance Z. Assuming
that the PZT parameters remains unchanged, the electromechanical
admittance Y will
undergo a change and this serves as an indicator of the health
of the structure. Measuring
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11
Z directly may not be feasible, but Y can be easily measured
using any commercial
electrical impedance analyzer. Common damage types altering
local structural impedance
Z are cracks, debondings, corrosion and loose connections
(Esteban, 1996), to which the
PZT admittance signatures show high sensitivity.
The electromechanical admittance Y (unit Siemens or ohm-1)
consists of real and
imaginary parts, the conductance (G) and susceptance (B),
respectively. A plot of G over
a sufficiently wide frequency serves as a diagnosis signature of
the structure and is
called the conductance signature or simply signature. The sharp
peaks in the
conductance signature correspond to structural modes of
vibration. This is how the
conductance signature identifies the local structural system (in
the vicinity of the patch)
and hence constitutes a unique health signature of the structure
at the point of attachment.
Since the real part actively interacts with the structure, it is
traditionally preferred over
the imaginary part in the SHM applications. It is believed that
the imaginary part
(susceptance) has very weak interaction with the structure
(Bhalla, 2004). Therefore, all
investigators have so far considered it redundant and have
solely utilized the real part
(conductance) alone in the SHM applications.
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CHAPTER 3
ANALYTICAL MODELLING OF SHEAR LAG EFFECT
3.1 INTRODUCTION
Crawley and de Luis (1987) and Sirohi and Chopra (2000b)
respectively modelled
the actuation and sensing of a generic beam element using an
adhesively bonded PZT
patch. The typical configuration of the system modelling the
actuating and sensing of a
generic beam element using an adhesively bonded PZT patch is
shown in Fig. 3.1. The
patch has a length 2l , width pw and thickness pt while the
bonding layer has a thickness
st . The beam has depth bt and width bw . Let pT denote the
axial stress in the PZT patch
and τ the interfacial shear stress.
Fig. 3.1 A PZT patch bonded to a beam using adhesive bond layer
(Bhalla, 2004).
dx
pT pp
TT dx
x∂
+∂
PZT patch
Bond layer
l
BEAM
x
y
pt
st
dx Differential Element
l
-
13
3.1.1 PZT Patch as Sensor
Let the PZT patch be instrumented only to sense strain on the
beam surface and hence no
external field be applied across it. Considering the static
equilibrium of the differential
element of the PZT patch in the x-direction, as shown in Fig.
3.1, Bhalla (2004) derived
the following equation.
2
22 0xξ ξ∂ −Γ =
∂ (3.1)
where 1pb
SS
ξ⎛ ⎞
= −⎜ ⎟⎝ ⎠
(3.2)
and 23 s ps
p s p b b b p
G wGY t t Y w t t⎛ ⎞
Γ = +⎜ ⎟⎜ ⎟⎝ ⎠
(3.3)
In the above equations bY and pY respectively denote the Young’s
modulus of elasticity
of the beam and the PZT patch (at zero electric field for the
patch) respectively and bS
and pS are the corresponding strains. sG denotes the shear
modulus of elasticity of the
bonding layer.
The phenomenon of the difference in the PZT strain and the
host
structure’s strain is called as the shear lag effect. The
parameter Г (unit m-1) is called
the shear lag parameter. The ratio ξ , which is a measure of the
differential PZT strain
relative to surface strain on the host substrate, caused by the
shear lag is called as the
shear lag ratio. The general solution for Eq. (3.1) can be
written as
cosh sinhA x B xξ = Γ + Γ (3.4) Since the PZT patch is acting as
sensor, no external field is applied across it.
Hence, free PZT strain = d 31 E3 = 0. Thus, following boundary
conditions hold good:
(i) At x l= − , 0pS = 1ξ = −
(ii) At x l= + , 0pS = 1ξ = −
Applying these boundary conditions, we can obtain the constants
A and B as
-
14
1cosh
Al
−=
Γ and 0B = (3.5)
Hence, coshcosh
xl
ξ Γ= −Γ
(3.6)
Using Eq. (3.2), we can derive
cosh1cosh
p
b
S xS l
Γ⎛ ⎞= −⎜ ⎟Γ⎝ ⎠ (3.7)
Fig. 3.2 shows a plot of the strain ratio ( /p bS S ) across the
length of a PZT
patch ( 5l mm= ) for typical values of Γ = 10, 12, 30, 40, 50
and 60 (cm-1). From this
figure, it is observed that the strain ratio ( /p bS S ) is less
than unity near the ends of the
PZT patch. The length of this zone depends on Г, which in turn
depends on the stiffness
and thickness of the bond layer (Eq. 3.3). As sG increases and
st reduces, Γ increases,
and as can be observed from Fig. 3.3, the shear lag phenomenon
becomes less and less
significant and the shear is effectively transferred over very
small zones near the ends of
the PZT patch.
Thus, if the PZT patch is used as a sensor, it would develop
less voltage across
its terminals (than for perfectly bonded conditions) due to the
shear lag effect. In other
words, it will underestimate the strain in the substructure. In
order to quantify the effect
of shear lag, we can compute effective length of the sensor
(Bhalla, 2004)
0
1 tanh( )( / ) 1x leff
p bx
l lS S dxl l l
=
=
Γ= = −
Γ∫ (3.8)
which is nothing but the area under the curve ( Fig. 3.2 )
between 0x = and x l= .
-
15
Fig. 3.2 Strain distribution across the length of the PZT patch
for various values
of Г (Bhalla, 2004).
Fig. 3.3 Variation of effective length with shear lag factor
(Bhalla, 2004).
Fig. 3.3 shows a plot of the effective length (Eq. 3.8) for
various values of the
shear lag parameter Г. From the figure it can be observed that
typically, for 130cm−Γ > ,
-
16
( )/ 93%effl l > , suggesting that shear lag effect can be
ignored for relatively high
( )130cm−> values of Γ
3.1.2 PZT Patch as Actuator
If a PZT patch is employed as an actuator for a beam structure,
it can be shown
(Bhalla, 2004) that the strains pS and bS are given by
3 cosh(3 ) (3 )coshp
xSl
ψψ ψ∧ ∧ Γ
= ++ + Γ
(3.9)
3 3 cosh(3 ) (3 )coshb
xSlψ ψ
∧ ∧ Γ= −
+ + Γ (3.10)
where 31 3d E∧ = is the free piezoelectric strain , and ( /
)E
b b pY h Y hψ = is the product of
modulus and thickness ratios of the beam and the PZT patch. Fig.
3.4 shows the plots of
( / )pS ∧ and ( / )bS ∧ along the length of the PZT patch ( 5l
mm= ) for ψ =15 and
different values of Г. It is observed that as in the case of
sensor, as Г increases, the shear
is effectively transferred over the small zone near the ends of
the patch. Typically, for Г >
30 cm-1, the strain energy induced in the substructure by PZT
actuator is within 5% of the
perfectly bonded case.
-
17
Fig. 3.4 Distribution of piezoelectric and beam strains for
various values of Г: (a) strain
in PZT patch and (b) beam surface strain (Bhalla, 2004).
3.2 INTEGRATION OF SHEAR LAG EFFECT INTO IMPEDANCE MODELS
When acting as an actuator and/ or a sensor, there is a shear
lag phenomenon
associated with force transmission between the PZT patch and the
host structure through
the adhesive bond layer. This aspect was first investigated by
Xiu and Liu (2002) for the
EMI technique in which the same patch concurrently serves both
as a sensor as well as an
actuator. Later Bhalla and Soh (2004) developed a rigorous model
incorporating the
-
18
effect of bond layer on the EMI signatures. The next section
provides the brief
description of the two models.
3.3 MODIFIED 1D IMPEDANCE MODEL BY XIU AND LIU (2002) Xu and Liu
(2002) proposed a modified 1D impedance model in which the
bonding layer was modelled as a single degree of freedom (SDOF)
system connected in
between the PZT patch and the host structure, as shown in Fig.
3.5.
Fig. 3.5 Modified impedance model of Xu and Liu (2002) including
bond layer
(Bhalla, 2004).
The bonding layer was assumed to possess a dynamic stiffness bK
(or mechanical
impedance, /bK jω ) and the structure, a dynamic stiffness sK
(or mechanical impedance,
/sZ K jω= ). Hence the resultant mechanical impedance for this
series system can be
determined as
b bresb b s
Z Z KZ Z ZZ Z K K
ζ⎛ ⎞
= = =⎜ ⎟+ +⎝ ⎠ (3.11)
where
11 ( / )s bK K
ζ =+
(3.12)
The coupled electromechanical admittance, as measured across the
terminals of the PZT
patch and expressed earlier by Eq. 2.15, can thus be modified
as
-
19
2 233 31 31tan2 ( )T E Ea
a
Zwl lY j d Y d Yh Z Z l
κω εζ κ
⎡ ⎤⎛ ⎞ ⎛ ⎞= − +⎢ ⎥⎜ ⎟ ⎜ ⎟+ ⎝ ⎠⎝ ⎠⎣ ⎦ (3.13)
1ζ = implies a very stiff bond layer whereas 0ζ = implies a free
PZT patch. Xu and Liu
(2002) demonstrated numerically that for a SDOF system, as ζ
decreases (i.e. as the
bond quality degrades), the PZT system shows an increase in the
associated structural
resonant frequencies. It was stated that bK depends on the
bonding process and the
thickness of the bond layer. However, no closed form solution
was presented to
quantitatively determine bK and hence ζ (from Eq. 3.12). Also,
no experimental
verification was attempted.
3.4 INCLUSION OF SHEAR LAG INTO IMPEDANCE MODEL BY BHALLA
AND SOH (2004)
Bhalla and Soh (2004) included the shear lag effect, first into
1D model and then
extended it into 2D effective impedance-based model.
They derived the following fourth order differential
equation
0u pu qu′′′′ ′′′ ′′+ − = (3.14)
where u is the drive point displacement at the point in question
on the surface of the host
structure. p and q are given by
p ss
w Gp
Zt jω= − (3.15)
and
(1 )(1 )
s s sE E E
s p s p s p
G G j GqY t t Y t t j Y t t
ηη′+
= = ≈+
(3.16)
p and q are shear lag parameters, similar to the factor Γ in Eq.
3.3. The
parameter q is equivalent to the first component of Г and p to
the second component. As
seen from Eq. 3.16, q is directly proportional to the bond
layer’s shear modulus and
inversely proportional to the PZT’s Young’s modulus, the PZT
patch’s thickness and the
bond layer thickness. Examination of Eq. 3.15 similarly shows
that p is directly
proportional to the bond layer’s shear modulus and the PZT
patch’s width. It is inversely
-
20
proportional to the structural mechanical impedance and the bond
layer thickness. Being
a dynamic parameter, the frequency ω also comes into the
picture, influencing p
inversely. Further, it should be noted that p is a complex term
whereas the term q has
been approximated as a pure real term assuming η and η′ to be
very small in magnitude.
Substituting Z x yj= + , (1 )s sG G jη′= + and simplifying we
get p a bj= +
where
2 2( )
( )p s
s
w G y xa
t x yη
ω′−
=+
and 2 2( )
( )p s
s
w G x yb
t x yη
ω′+
=+
(3.17)
The solution of the governing differential equation (Eq. 3.14)
was derived by Bhalla and
Soh (2004) as
3 41 2x xu A A x Be Ceλ λ= + + + (3.18)
where
2
34
2p p q
λ− + +
= (3.19)
2
44
2p p q
λ− − +
= (3.20)
The constants A1, A2 , B and C were to be evaluated from the
boundary conditions as
4 21 31 4 2 3( )
k kBk kC k k k k−⎡ ⎤⎡ ⎤ ∧
= ⎢ ⎥⎢ ⎥ −−⎣ ⎦ ⎣ ⎦ (3.21)
1 ( )A B C= − + (3.22)
2 3 4( )A B Cλ λ= − + (3.23)
where
31 3 3 3(1 )lk n e λλ λ λ−= + − (3.24)
42 4 4 4(1 )lk n e λλ λ λ−= + − (3.25)
33 3 3 3(1 )lk n eλλ λ λ= + − (3.26)
44 4 4 4(1 )lk n eλλ λ λ= + − (3.27)
In general, the force transmitted to the host structure can be
expressed as
-
21
( )x lF Zj uω == − (3.28)
where ( )x lu = is the displacement at the surface of the host
structure at the end point of
the PZT patch. Conventional impedance models (e.g. Liang and
coworkers) assume
perfect bonding between the PZT patch and the host structure,
i.e. the displacement
compatibility ( ) ( )x l p x lu u= == , thereby approximating
Eq. 3.28 as ( )p x lF Zj uω == −
Fig. 3.6 Deformation in bonding layer and PZT patch.
However, due to the shear lag phenomenon associated with
finitely thick bond layer,
( ) ( )x l p x lu u= =≠ . According to Bhalla and Soh
(2004),
( )( ) ( ) ( )
11 ( / )( / )
x l
p x l s p s x l x l
uu Zt j w G u uω
=
= = =
=′−
(3.29)
( )( )
11 (1/ )( / )
x l
p x l o o
uu p u u
=
=
=′+
(3.30)
Where 0u is as shown in Fig. 3.6. The term ( ) ( )/x l x lu u=
=′ can be determined by using Eq.
(3.18). Making use of this relationship, the force transmitted
to the structure can be
written as
( )x lF Zu == − & (3.31)
-
22
( ) ( )0
0
11p x l eq p x l
ZF j u Z j uu
p u
ω ω= =−
= =⎛ ⎞′+⎜ ⎟
⎝ ⎠
(3.32)
where 0
0
11eq
ZZu
p u
=⎛ ⎞′+⎜ ⎟
⎝ ⎠
(3.33)
eqZ is the equivalent impedance apparent at the ends of the PZT
patch , taking
into consideration the shear lag phenomenon associated with the
bond layer. In the
absence of shear lag effect (i.e. perfect bonding), eqZ Z= .
From the above discussion, it can be observed that the model
presented by
Bhalla and Soh (2004) is quite a rigorous one. Extracting
conductance signatures and
susceptance by using their model is therefore going to be quite
cumbersome. This
necessitates the development of a simple model, which can
incorporate shear lag effect
into the 1D impedance model.
-
23
CHAPTER 4
KUMAR, BHALLA and DATTA (KBD) 1D IMPEDANCE MODEL
4.1 INTRODUCTION
Fig. 4.1 Diagram showing the KBD model
In the chapter 3 it was shown that the incorporating the shear
lag effect into the
1D impedance model of Bhalla and Soh (2004) method was quite a
rigorous procedure. It
involved solving fourth order differential equations and
obtaining the solution could be
very vigorous. In this section, a new simplified model named
Kumar, Bhalla and Datta
(KBD) model is developed.
Fig. 4.1 shows the proposed KBD model. The deformation in the
PZT patch is
denoted by pu . Due to the shear lag effect same deformation
would not be transferred to
the host structure. The deformation in the host structure is
denoted by u . The mechanical
impedance of the host structure is denoted by Z. The thickness
of the PZT patch is
denoted by pt while that of adhesive bond layer by st . It is
assumed that the force
transmission between the PZT patch and the host structure is
taking place via the simple
pure shear mechanism illustrated by Fig. 4.1.
Shear strain in the bonding layer is given by
ps
u ut
γ−⎛ ⎞
= ⎜ ⎟⎝ ⎠
(4.1)
PZT PATCHZ
pt pu
u BOND LAYER
st
STRUCTUREγ
-
24
p su u tγ= − (4.2)
Let the interfacial shear stress be denoted by ‘τ’ and ‘ sG ’ be
the complex shear modulus
of the bonding layer. Then,
sGτγ = (4.3)
Where (1 )s sG G jη′= + . Here ‘ sG ’ is the shear modulus of
bonding layer and ‘η′ ’ is
the mechanical loss factor associated with the bond layer. It is
strongly dependent on
temperature. It can vary from 5% to 30% at room temperature
depending on the type of
adhesive (Bhalla, 2004). Substituting Eq. (4.3) into Eq. (4.2)
we get
p ss
u u tGτ⎛ ⎞
= −⎜ ⎟⎝ ⎠
(4.4)
Let ‘ F ’ be the force transmitted to the host structure over
the area ‘ A ’. Then
we can write, /F Aτ = . Therefore,
p ss
Fu u tAG
⎛ ⎞= −⎜ ⎟⎝ ⎠
(4.5)
In terms of impedance ‘Z’, the force transmitted to the host
structure can be written as
( )x lF Zu jω== − (4.6) where ‘ω ’ is the excitation
frequency.
Substituting Eq. (4.6) into Eq. (4.5) and simplifying we get
sps
FtF Zj uAG
ω⎛ ⎞
= − −⎜ ⎟⎝ ⎠
(4.7)
Solving, we can derive
1
p
s
s
Zj uF
Z t jAG
ωω
−=⎛ ⎞−⎜ ⎟
⎝ ⎠
(4.8)
eq pF Z j uω= − (4.9)
-
25
where eqZ is the equivalent impedance apparent at the ends of
the PZT patch, taking into
consideration the shear lag phenomenon associated with the bond
layer. Thus,
1
eqs
s
ZZZ t jAGω
=⎛ ⎞−⎜ ⎟
⎝ ⎠
(4.10)
Considering that the force transmission is taking place over
unit width and considering
half the length of the PZT patch. For this condition A l= , and
Eq.(4.10) can be written as
1
eqs
s
ZZZ t jlGω
=⎛ ⎞−⎜ ⎟
⎝ ⎠
(4.11)
Once the value of eqZ is determined it can be used to extract
the conductance signatures
and susceptance by using Eq.2.16. To obtain the conductance and
susceptance signatures
eqZ should be used instead of Z in Eq.2.16
4.2 DETERMINATION OF REAL AND IMAGINARY COMPONENTS OF eqZ
The force transmitted to the host structure is given by
F Zj uω= (4.12)
FZj uω
= (4.13)
Since the present system is dynamic in nature, both force and
displacement are complex
numbers. Hence, they can be expressed as
r iF F jF= + (4.14)
r iu u ju= + (4.15)
( )r i
r i
F jFZj u juω
+=
+ (4.16a)
Rationalising the denominator and simplifying we get Z x jy= +
(4.16b)
where
-
26
2 2( )i r r i
r i
Fu F uxu uω−
=+
(4.17)
2 2( )r r i i
r i
F u Fuyu uω+
= −+
(4.18)
eqZ can be written as
( )( )1(1 )
eqs
s
x yjZ x yj j tAG j
ωη
+=
+−
′+
(4.19)
( )(1 )
1eq
s s
s s
x yj jZt ty x j
AG AG
ηω ωη
′+ +=⎛ ⎞ ⎛ ⎞
′+ + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
(4.20)
( )
( )(1 )1 ( )eq
x yj jZCy Cx j
ηη
′+ +=
′+ + − (4.21)
where /s sC t AGω= . Rationalising the denominator and
separating out the real and
imaginary components.
If eq eq eqZ X Y j= +
(4.22)
then
2 2( )(1 ) ( )( )
(1 ) ( )eqx y Cy x y CxX
Cy Cxη η η
η′ ′ ′− + + + −
=′+ + −
(4.23)
2 2( )(1 ) ( )( )
(1 ) ( )eqx y Cy x y CxY
Cy Cxη η η
η′ ′ ′+ + − − −
=′+ + −
(4.24)
4.3 VERIFICATION OF KBD MODEL
4.3.1 Generation of Finite Element Model
A Cantilever beam was generated in ANSYS 9. The beam was assumed
to be
made up of aluminium of grade Al 6061-T6 whose key mechanical
properties are listed
in Table 4.1.The beam was instrumented with a PZT patch between
points A & B as
-
27
shown in the Fig.4.2. Fig.4.3 shows the mesh generated using the
preprocessor of
ANSYS 9, with an element size of 2.0mm.
Fig. 4.2 A cantilever model in ANSYS 9
Table 4.1 Physical properties of Al 6061 – T6 (Bhalla, 2004)
Physical Parameter
Value
Density (kg /m3)
2715
Young’s Modulus, Y11E (N/m2 )
68.95 x 109
Poisson ratio
0.33
Mass damping factor , α
0
Stiffness damping factor, β
3 x 10-9
10 cm
1cm
1cm
1 KN -1 KN
4.2cm 0.6cm
A B
-
28
Fig.4.3 ANSYSModel
A B
-
29
An equal and opposite set of loads of 1 KN was applied at two
points, A and B (end
points of the PZT patch) 6mm apart on the top face of the model
as shown in Fig. 4.3.
Load of -1 KN is applied at node number 160 (at point A) and
load of 1 KN is applied at
node number 154 (at point B).
The material was assumed linear elastic and isotropic. Harmonic
analysis of the
model structure thus generated was carried out to determine the
real and imaginary parts
of the displacement at node 160. The frequency range considered
was 100 – 150 kHz.
By carrying out the above analysis we have the necessary data of
the force
transmitted to the host structure (1KN in the present model) and
the corresponding
displacement in the host structure at various frequencies of
excitation. This data was
processed further to extract the conductance and susceptance
signatures for different 1D
impedance models viz. without considering shear lag effect,
using Bhalla and Soh model,
and using the KBD model. Eq.4.16a was used to obtain the
structural mechanical
impedance at various frequencies in the range 100-150 kHz. Eq.
4.11 was used to obtain
the modified mechanical impedance. Finally, conductance and
susceptance signatures
were obtained using Eq. 2.16, by substituting eqZ in place of Z
.
4.3.2 Convergence Test
In dynamic harmonic problems, in order to obtain accurate
results, a sufficient
number of nodal points (3 to 5 per wavelength) should be present
in the finite element
mesh (Bhalla, 2004). In order to ensure this requirement, modal
analysis was additionally
performed. The frequency range of 0–150 kHz was found to contain
a total of 18 modes.
The modal frequencies are listed in Table 4.2, computed for 3
different element sizes,
5mm, 2mm and 1mm. It can be observed that good convergence of
the modal frequencies
is achieved at an element size of 2mm (which is the element size
used in the present
analysis). Thus, fairly accurate results are expected from the
analysis using FEM.
-
30
Table 4.2 Details of the modes of vibrations of the test
structure
MODE
MODAL FREQUENCIES (Hz)
5mm
2mm
1mm
1
860.59
858.49
857.78
2
5191.1
5136.8
5125.9
3
13410
13392
13388
4
13802
13494
13440
5
25400
24461
24308
6
39312
37244
36918
7
40255
40114
40089
8
55019
51236
50656
9
67164
66052
65123
10
72203
66637
66552
11
90665
81435
80056
12
94143
92783
92576
-
31
13
0.11026E+06
97207
95271
14
0.12114E+06
0.11323E+06
0.11062E+06
15
0.13080E+06
0.11830E+06
0.11788E+06
16
0.14800E+06
0.12934E+06
0.12597E+06
17
-
0.14282E+06
0.14106E+06
18
-
0.14523E+06
0.14206E+06
4.3.3 Visual Basic Programs
Two VB programs were used to generate conductance and
susceptance plots
from ANSYS output. The first program can determine the
conductance and susceptance
signatures for the 1D impedance model without incorporating
shear lag effect (Bhalla,
2004), i.e. Eq.2.16. The second program can determine the
conductance and susceptance
signatures for the KBD model developed in the present study.
These two programs are
listed in Appendix A and B respectively. The physical properties
of the PZT patch used
in the analysis are listed in Table 4.3.
4.3.4 MATLAB Program
A MATLAB program, listed in the Appendix C, can determine the
conductance
signatures and susceptance from the ANSYS output. The program is
based on the 1D
impedance model with shear lag effect incorporated into it, as
per Bhalla and Soh model
(2004).
-
32
Table 4.3 Physical Properties of PZT patch (Bhalla, 2004).
Physical Parameter
Value
Density (kg / m3)
7650
Thickness (m)
0.0002
Length (m)
0.006
31d
-1.66E-10
Young’s Modulus , 11
EY (N/m2)
6.3E+10
33Tε
1.5E-8
η
0.1
δ
0.012
4.3.5 Results
The conductance and susceptance signatures were extracted for
the three ID
impedance models viz model without incorporating shear lag
effect (denoted by wsle in
graphs), KBD model and Bhalla and Soh (2004) 1D impedance model
(denoted by BSM
in graphs). Fig. 4.4, 4.5 and 4.6 shows the conductance
signatures for different bond layer
thicknesses. The effect of changing the bond layer thickness on
the conductance
signatures given by the three models can be easily observed in
these figures. As the bond
layer thickness decreases, the conductance given by KBD model
and Bhalla and Soh
model (2004) are quite close.
-
33
Fig 4.7 shows the susceptance plots given by the three models.
The curves are
quite close to each other. This part has a weak interaction with
the structure and bond
layer does not seem to influence the susceptance signatures
much.
Fig. 4.4 Comparing conductance signatures obtained by three
models for bond layer
thickness s pt t= .
Fig. 4.5 Comparing conductance signatures obtained by three
models for bond layer
thickness / 3s pt t= . .
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
100000 110000 120000 130000 140000 150000 160000
FREQUENCY (Hz)
COND
UCTA
NCE
(S)
G(wsle)G(KBD)G(BSM)ts / tp = 1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
100000 110000 120000 130000 140000 150000 160000
FREQUENCY (Hz)
CON
DUC
TAN
CE (S
)
G(wsle)G(KBD)G(BSM)
ts / tp = 0.333
-
34
Fig. 4.6 Comparing the conductance signatures obtained by three
models for bond layer
thickness 0.1s pt t= .
Fig. 4.7 Comparing susceptance obtained by three models for bond
layer thickness
s pt t= .
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
100000 110000 120000 130000 140000 150000 160000
FREQUENCY (Hz)
COND
UCTA
NCE
(S)
G(wsle)G(KBD)G(BSM)
ts / tp = 0.1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
100000 110000 120000 130000 140000 150000 160000
FREQUENCY (Hz)
SUSC
EPTA
NCE
(S)
wsleKBDBSM
ts / tp = 1
-
35
CHAPTER 5
PARAMETRIC STUDY
5.1 INTRODUCTION
From Eq. 4.10 it can be observed that the electromechanical
admittance
signatures are influenced by the parameters related to the
adhesive bond layer viz.
modulus of shearing rigidity ‘ sG ’, half length of the PZT
patch ‘ l ’, mechanical loss
factor ‘η′ ’ and thickness of the bond layer ‘ st ’. The
influence of all these parameters on
the admittance signatures is studied using the KBD model and
presented in the following
sections.
5.2 INFLUENCE OF BOND LAYER SHEAR MODULUS Gs
Fig. 5.1 and Fig. 5.2 shows the influence of bond layer shear
modulus on the
conductance and susceptance plots. It is observed from the
Fig.5.1 that as the sG
decreases, the resonant peaks of conductance subside down and
shifts rightwards.
However, another important observation that can be made from the
Fig.5.1 is that at
0.5sG GPa= the conductance becomes negative at few frequencies.
Therefore, KBD
model cannot be used for smaller values of sG . In the Fig.5.1
and 5.2 legends G(PB)
stands for the conductance signatures for the perfect bonding
case, G(1), G(1.5), G(0.5)
stands for conductance signatures for shear modulus of 1GPa,
1.5GPa and 0.5GPa
respectively. Similar legend holds for the susceptance plot of
Fig.5.2.
-
36
Fig.5.1 Influence of shear modulus on conductance
signatures.
It can be observed from the Fig.5.2 that there is very marginal
difference in the
susceptance plots corresponding to 1.5sG = ,1 and 0.5GPa .
However the curve for
perfect bonding is quite distinct.
Fig.5.2 Influence of shear modulus on susceptance.
-0.05
0
0.05
0.1
0.15
0.2
100000 110000 120000 130000 140000 150000 160000
FREQUENCY (Hz)
COND
UCTA
NCE
(S)
G(PB)G(1)G(1.5)5(0.5)
00.10.20.30.40.50.60.70.80.9
1
100000 110000 120000 130000 140000 150000 160000
FREQUENCY (Hz)
SUSC
EPTA
NCE
(S)
B(PB)B(1)B(1.5)B(0.5)
-
37
5.3 INFLUENCE OF LENGTH OF PZT PATCH
Fig.5.3 and 5.4 shows the influence of length of PZT patch on
the conductance
and susceptance signatures respectively. The influence on
conductance and susceptance
signatures was studied for 2l mm= , 3mm and 5mm . It can be
observed from the Fig.5.3
that at higher resonant peaks, as the length of the actuator
increases, the peak shifts
upwards and rightwards. It can also be observed from Fig.5.4
that as the actuator length
increases, the susceptance shifts upwards.
Fig. 5.3 Influence of length of PZT patch on the
conductance.
Fig. 5.4 Influence of length of PZT patch on the
susceptance.
0
0.05
0.1
0.15
0.2
0.25
100000 110000 120000 130000 140000 150000 160000
FREQUENCY (Hz)
COND
UCTA
NCE
(S)
G(0.002)G(0.003)G(0.005)
00.10.20.30.40.50.60.70.80.9
1
100000 110000 120000 130000 140000 150000 160000
FREQUENCY (Hz)
SUSC
EPTA
NCE
(S)
B(0.002)B(0.005)B(0.003)
-
38
5.4 INFLUENCE OF MECHANICAL LOSS FACTOR
Mechanical loss factor is the measure of the damping of the
adhesive bond layer.
Fig.5.5 and 5.6 shows the influence of the mechanical loss
factor on the conductance and
susceptance signatures. The influence of mechanical loss factor
is studied for 0.1η′ = ,
0.15, 0.005. From Fig. 5.5 it can be observed that the
conductance is affected by the
mechanical loss factor η′ slightly, away from the resonant
peaks. At the resonant peaks
there is hardly any affect of η′ on the conductance. From Fig.
5.6 it can be seen that η′
has virtually no effect on the susceptance.
Fig. 5.5 Influence of mechanical loss factor on conductance.
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
100000 110000 120000 130000 140000 150000 160000
FREQUENCY (Hz)
COND
UCTA
NCE
(S)
G(0.1)G(0.15)G(0.005)
-
39
Fig. 5.6 Influence of mechanical loss factor on susceptance.
5.5 INFLUENCE OF BOND LAYER THICKNESS
Fig. 5.7 and 5.8 shows the influence of bond layer thickness on
the conductance
and susceptance signatures. The influence is studied for the
thickness ratio / 1s pt t = , 1/ 3
and 0.1 . As the thickness ratio /s pt t increases, the peaks in
the conductance signatures
shifts rightwards, i.e. the ‘apparent’ resonant frequency
increases. Also it can be observed
from Fig.5.8 that there is virtually no effect of change in the
thickness ratio on
susceptance.
00.050.1
0.150.2
0.250.3
0.350.4
0.450.5
100000 110000 120000 130000 140000 150000 160000
FREQUENCY (Hz)
SUSC
EPTA
NCE
(S)
B(0.1)B(0.15)B(0.005)
-
40
Fig.5.7 Influence of bond layer thickness on conductance
Fig.5.8 Influence of bond layer thickness on susceptance
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
100000 110000 120000 130000 140000 150000 160000
FREQUENCY (Hz)
COND
UCTA
NCE
(S)
tptp/ 3tp/ 10
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
100000 110000 120000 130000 140000 150000 160000
FREQUENCY (Hz)
SUSC
EPTA
NCE
(S)
tptp/ 3tp/ 10
-
41
CHAPTER 6
SHEAR STRESS PREDICTION IN BOND LAYER
6.1 INTRODUCTION
This chapter basically deals with the determination of shear
stress in the adhesive
bond layer. This is of critical importance in smart structures,
especially in “control”
related problems.
6.2 SHEAR STRESS BY KBD MODEL
Though Bhalla and Soh (2004) derived 1D impedance formulations,
analysis of shear
stress was left out. In this section the expressions for the
average shear stress using the
KBD model are derived.
The shear stress in the bond layer is given by
sGτ γ= (6.1)
Substituting Eq. (4.1) into Eq. (6.1) we get
( )p
ss
u uG
tτ
−= (6.2)
In Bhalla and Soh model (2004) explicit expressions were derived
for pu and u as
3 41 2x xu A A x Be Ceλ λ= + + + (6.3)
3 41 2 2 3 4( ) (1 ) (1 )x x
pu A nA A x B n e C n eλ λλ λ= + + + + + + (6.4)
where the constants 1A , 2A , B , C , 3λ and 4λ are given by
Eqs. (3.19) to (3.27), and
1np
= .
Now, the shear stress in the bond layer is also given by
FA
τ = (6.5)
where F is the total shear force transmitted and A is the area
over which the force
transmission is taking place. Substituting Eq. (4.8) in Eq.
(6.5) we get
-
42
1
p
s
s
Z juZ tA jAG
ωτ
ω−
=⎛ ⎞−⎜ ⎟
⎝ ⎠
(6.6)
Comparing Eq. (6.2) and Eq. (6.6) we get
( )
(1 )
p ps
ss
s
u u Zj uG Z tt A j
AG
ωω
− −=
− (6.7)
( )( )
s pp
s s
Zj t uu u
AG Z t jω
ω−
− =−
(6.8)
1( )
sp
s s
Z t ju uAG Z t j
ωω
⎡ ⎤+ =⎢ ⎥−⎣ ⎦
(6.9)
1
( )
ps
s s
uuZ t j
AG Z t jω
ω
=⎡ ⎤+⎢ ⎥−⎣ ⎦
(6.10)
Substituting r iu u ju= + and Z x yj= + , we get
( )( )1
(1 ) ( )
r ip
s
s s
u juux yj t j
AG j x yj t jω
η ω
+=⎡ ⎤++⎢ ⎥′+ − +⎣ ⎦
(6.11)
[ ]( ) (1 ) ( )(1 )
r i s sp
s
u ju AG j x yj t ju
AG jη ωη′+ + − +
=′+
(6.12)
[ ]2( ) ( ) ( )
(1 )(1 )
r i s s s sp
s
u ju AG t y AG x t ju j
AGω η ω
ηη
′+ + + −′= −
′+ (6.13)
[ ]2
( ) ) ( ) ( ) ( )(1 )
r i s s s s s s s sp
s
u ju AG t y AG x t j AG t y j AG x tu
AGω η ω η ω η η ω
η′ ′ ′ ′+ + + − − + + −
=′+
(6.14)
Separating out the real and imaginary components of pu .
If p pr piu u ju= +
(6.15)
then
-
43
[ ] [ ]2
( ) ( ) ( ) ( )(1 )
r s s s s i s s s spr
s
u AG t y AG x t u AG x t AG t yu
AGω η η ω η ω η ω
η′ ′ ′ ′+ + − − − − +
=′+
(6.16)
[ ] [ ]2 2(1 ) ( ) ( ) (1 )(1 ) (1 )ir
pruuu Cy Cx Cx Cyη η η η
η η′ ′ ′ ′= + + − − − − +
′ ′+ + (6.17)
where ss
tCAGω
= .
Similarly
[ ] [ ]2 2( ) (1 ) (1 ) ( )(1 ) (1 )ir
piuuu Cx Cy Cy Cxη η η η
η η′ ′ ′ ′= − − + + + + −
′ ′+ + (6.18)
Substituting Eq.(6.17) and Eq.(6.18) into Eq.6.6 and noting that
eq eq eqZ X Y j= + , we get
( ) ( )eq eq pr piX jY j u u j
Aω
τ− + +
= (6.19)
( ) ( )eq pi eq pr eq pi eq prX u Y u Y u X u jA Aω ωτ = + + −
(6.20)
If r ijτ τ τ= + (6.21)
Then
( )r eq pi eq prX u Y uAωτ = + (6.22)
( )i eq pi eq prY u X uAωτ = − (6.23)
The absolute value of shear stress in the bond layer is given
by
2 2r iτ τ τ= + (6.24)
Now, to determine the shear stress in the bond layer using the
KBD model for a
particular frequency of excitation we need to know the ru and iu
, which are obtained
from the ANSYS output. Using these, we can calculate the value
of pru and piu using Eq.
(6.17) and Eq. (6.18) respectively. Once these values are
determined we can put them in
the Eq. (6.22) and Eq. (6.23) to determine rτ and iτ and hence
finally getting τ using
Eq. (6.24).
The reason for getting explicit expressions in the Bhalla and
Soh model (2004) was that it
was developed using the elemental formulations of the bond
layer. However, in the case
-
44
of the KBD model, the overall deformation of the bond layer is
considered as
simplifications. Hence no explicit expressions are available for
u and pu . However, one
implicit expression involving u and pu is developed for the KBD
model as shown in the
preceding section.
6.3 DISTRIBUTION OF SHEAR STRESS IN BOND LAYER USING BHALLA
AND SOH 1D IMPEDANCE MODEL (2004)
The actual distribution of shear stress in the bond layer can be
very well
understood by using the expression developed by Bhalla and Soh
(2004) as follows
( ) ( )3 33 41 1x x
p
Zj B e C e
w
λ λω λ λτ
⎡ ⎤− − + −⎣ ⎦= (6.25)
From this expression, the average shear stress can be obtained
by calculating the area
under curve as shown in Fig. 6.1 and dividing it by the length
of the actuator. In the
present study, an attempt is made to correlate the average shear
stress in the bond layer
obtained using the Bhalla and Soh model (2004) and the KBD
model.
-
45
Fig.6.1 Shear stress distribution along length of actuator using
Bhalla and Soh
model (2004)
A MATLAB program was developed to obtain the values of shear
stress in the adhesive
bond layer. This program computes the values of shear stress at
thirty points along half
length of the actuator. To calculate the area under curve,
numerical integration technique
called Simpsons one third rule was used.
The effect of the different excitation frequencies on the shear
stress distribution was
studied for frequencies of 101 KHz, 110 KHz, and 150 KHz out of
which 150 KHz is the
resonant frequency. Table 6.1 shows the values of shear stress
at different points along
the length of the actuator for the frequencies of 101 KHz, 110
KHz and 150KHz. Fig.6.2
shows the plot of comparative shear stress distribution for
these three frequencies.
0 0.5 1 1.5 2 2.5 3
x 10-3
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
LENGTH (m)
SH
EA
R S
TRE
SS
( M
Pa)
-
46
Fig.6.2 Comparing shear stress distribution for different
frequencies using the
Bhalla and Soh model (2004)
It can be observed from Fig.6.2 that as the frequency approaches
the resonant frequency,
the curves becomes steeper and broader at the base. This
basically means that at resonant
frequencies of excitation the shear is transmitted mostly at the
ends. The most important
result derived from the above comparison is that the shear
stress distribution is
marginally affected by the frequency of excitation except near
resonance. This can
be seen clearly from the Fig.6.2 that the curves are very close
to each other. The
same result is obtained using the KBD model as can be seen in
Table 6.2. Hence, it
can be said that the shear stress distribution is practically
independent of excitation
frequency, except near resonance.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.001 0.002 0.003 0.004LENGTH OF ACTUATOR (m)
SHEA
R S
TRES
S (M
Pa)
F(101)F(110)F(150)
-
47
Table 6.1 Shear stress distribution for different frequencies
using Bhalla and Soh
model (2004)
SHEAR STRESS (MPa) LENGTH(m) F(101) F(110) F(150)
0 0 0 0 0.0001 0.00008 0.00008 0 0.0002 0.00016 0.00016 0.0001
0.0003 0.00024 0.00023 0.0001 0.0004 0.00031 0.0003 0.0001 0.0005
0.00039 0.00038 0.0002 0.0006 0.00046 0.00045 0.0002 0.0007 0.00053
0.00051 0.0002 0.0008 0.0006 0.00058 0.0003 0.0009 0.00066 0.00065
0.0003 0.001 0.00073 0.00071 0.0003
0.0011 0.00079 0.00077 0.0003 0.0012 0.00085 0.00083 0.0004
0.0013 0.00091 0.00089 0.0004 0.0014 0.00097 0.00095 0.0004 0.0015
0.00103 0.00101 0.0005 0.0016 0.00109 0.00106 0.0005 0.0017 0.00114
0.00111 0.0005 0.0018 0.00119 0.00117 0.0006 0.0019 0.00124 0.00122
0.0006 0.002 0.00129 0.00127 0.0006
0.0021 0.00134 0.00132 0.0006 0.0022 0.00139 0.00136 0.0007
0.0023 0.00144 0.00141 0.0007 0.0024 0.00151 0.00147 0.0007 0.0025
0.00166 0.00158 0.0007 0.0026 0.00223 0.00202 0.0008 0.0027 0.00483
0.00419 0.0008 0.0028 0.01679 0.01514 0.0008 0.0029 0.07214 0.07074
0.0097 0.003 0.32914 0.35422 1.4161
-
48
Table 6.2 Shear stress distribution for different frequencies
using KBD Model
FREQUENCY (kHz)
AVERAGE SHEAR STRESS (KBD)
(MPa) 101 0.33
110 0.33
150 0.33
Table 6.3 Comparing Shear stress distribution for different
frequencies using KBD
Model and BSM
FREQUENCY (kHz)
AREA UNDER CURVE (MPa-m)
AVERAGE SHEAR STRESS PEAK SHEAR STRESS(BSM)
(MPa) BSM (MPa)
KBD (MPa)
101 2.4732E-05 0.008244 0.33 0.3291
110 2.51173E-05 0.008372 0.33 0.3542
150 4.97167E-05 0.01657 0.33 1.4161
It can be observed from the Table 6.3 that the average shear
stress obtained by the Bhalla
and Soh model is very small compared with the average shear
stress obtained using the
KBD model. The reason for such a large difference is that most
of the shear is carried at
the ends of the PZT patch. Another important point to note is
that the peak shear stress
obtained by Bhalla and Soh model is only slightly higher than
the average shear stress
obtained using the KBD model. The difference in the average
shear stress values
predicted by the two models increases with the increase in the
frequency of excitation.
So, the value of shear stress obtained using the KBD model can
be correlated to the shear
stress value obtained for the different frequencies using the
Bhalla and Soh model.
-
49
CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
7.1 CONCLUSIONS
In the present research work a new simplified 1D impedance model
incorporating
the shear lag effect is developed and presented, named as Kumar,
Bhalla and Datta model
or simply KBD model. The conductance and susceptance signatures
obtained using the
KBD model are compared with those derived using the Bhalla and
Soh 1D impedance
model (2004). Further, a detailed parametric study on the
conductance and susceptance
signatures is done using the KBD model. In addition, a new
method is developed for
predicting the shear stress in the adhesive bond layer for
different excitation frequencies
based on the KBD model. The major research conclusions and
contributions can be
summarized as follows
(i) The KBD model developed in this report is found to predict
conductance and susceptance signatures in close proximity with
those given by the Bhalla and Soh
1D impedance model (2004). However this proximity is not
maintained at all
frequencies of excitation. Near the resonant peaks, there is
somewhat large
difference in the values of conductance predicted by these
models. But at higher
resonance peak frequencies, the difference in values of
conductance predicted by
KBD model and the Bhalla and Soh model (2004) is very small.
(ii) The susceptance signatures predicted by three models are
found to be in close proximity with each other for different
thicknesses of the bond layer. This part has
a weak dependence on the bond layer.
(iii) Parametric study conducted using KBD model suggests that
the apparent resonant frequency increases due to decrease in shear
modulus (i.e. degradation in bond
layer quality) and due to increase in bond layer thickness. It
is suggested that in
order to achieve best results