MODELLING OF SHEAR LAG EFFECT IN ADHESIVE BOND...
Transcript of MODELLING OF SHEAR LAG EFFECT IN ADHESIVE BOND...
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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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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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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
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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