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

������� ���� ������������

����������������������������

NewDelhi110016

MAY 2006

i

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

ii

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

iii

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.

iv

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

v

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

vi

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

vii

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

viii

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

ix

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

1

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

2

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.

3

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

4

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 ”

5

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

6

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

7

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)

8

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)

9

11 31 3E

TS 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)

10

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 31

cos( )( )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 31

tan2 ( )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

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.

12

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 0

xξ ξ∂−Γ =

∂ (3.1)

where 1p

b

SS

ξ⎛ ⎞

= −⎜ ⎟⎝ ⎠

(3.2)

and 2 3 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 ( / )Eb 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 bres

b 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 31

tan2 ( )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 s

s

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 2

x 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 2

1 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

p

s

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

sp

s

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 ) ( )eq

x y Cy x y CxXCy Cx

η η ηη

′ ′ ′− + + + −=

′+ + − (4.23)

2 2

( )(1 ) ( )( )(1 ) ( )eq

x y Cy x y CxYCy Cx

η η ηη

′ ′ ′+ + − − −=

′+ + − (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, Y11

E (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 2

x 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 s

ps

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 )

irpr

uuu Cy Cx Cx Cyη η η ηη η

′ ′ ′ ′= + + − − − − +′ ′+ +

(6.17)

where s

s

tCAGω

= .

Similarly

[ ] [ ]2 2( ) (1 ) (1 ) ( )(1 ) (1 )

irpi

uuu Cx Cy Cy Cxη η η ηη 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 3

3 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, the PZT patch should be bonded to the structure

using an adhesive of high shear modulus and smallest practicable thickness.

(iv) The most important result derived from the study of the shear stress distribution

in the adhesive bond layer is that shear stress distribution is marginally affected

50

by the frequency of excitation except near resonance peaks. This result has been

verified both by the KBD model and the Bhalla and Soh 1D impedance model.

Hence, it can be said that the shear stress distribution is practically independent of

excitation frequency, except near resonance.

(v) Another important result obtained from the analysis of shear stress distribution in

the adhesive bond layer is that the peak shear stress obtained using the Bhalla and

Soh model (2004) is close to the average shear stress value obtained using the

KBD model. This result shows that the KBD model can be made use of for

carrying out the preliminary design in structural control related problems.

(vi) The advantages of the KBD model developed in this report are quite apparent.

This model is a quite simplified model of the complex shear lag phenomenon

associated with the force transmission between the PZT patch and the host

structure bonded to each other by adhesive bond layer. The main advantage of this

model is that it does not involve solving the complex differential equations.

However, simple it may be, it is not a substitute for Bhalla and Soh model (2004)

particularly when accuracy is of utmost importance.

7.2 RECOMMENDATIONS

The present research work can be extended as follows

(i) The KBD model can be used for carrying out preliminary design in structural

control related problems.

(ii) It is noticed that there is somewhat large difference in the conductance

obtained using the KBD model and the Bhalla and Soh model (2004) at the

resonant frequencies. Therefore it is suggested that peak correction factors

related to the excitation frequency may be developed for the KBD model to

minimize that difference.

(iii) Further, research can be conducted to arrive at a expression relating the

average shear stress in the bond layer obtained using KBD model and the

shear stress obtained using Bhalla and Soh model (2004) at any excitation

frequency including the resonant frequency.

51

REFERENCES

1. Bhalla, S. and Soh, C.K. (2004), “Impedance Based Modeling for Adhesively

Bonded Piezo- Transducers”, Journal of Intelligent Material Systems and

Structures, Vol. 15, No. 12, pp. 955-972.

2. Bhalla, S. and Soh, C.K. (2004a), “Structural Health monitoring by Piezo-

Impedance Transducers: Applications”, Journal of Aerospace Engineering,

ASCE, Vol. 17, No.4, pp. 166-175.

3. Bhalla, S. and Soh, C.K. (2004b), “Structural Health monitoring by Piezo-

Impedance Transducers: Modeling”, Journal of Aerospace Engineering, ASCE,

Vol. 17, No.4, pp. 154-165.

4. Crawley, E.F.,de Luis, J. (1987), “Use of Piezoelectric Actuators as Elements of

Intelligent Structures”, AIAA Journal, Vol.25, No. 10, pp.1373-1385.

5. Bhalla, S. (2004), “A Mechanical Impedance Approach for Structural

Identification, Health Monitoring and Non Destructive Evaluation using Piezo

Impedance Transducers”, Ph.D Thesis, Nanyang Technological University.

6. Sirohi, J. and Chopra, I. (2000b), “Fundamental Understanding of Piezoelectric

Strain Sensors”, Journal of Intelligent Material Systems and Structures, Vol. 11,

No.4, pp.246-257.

52

APPENDIX A

'Determination of signatures from ANSYS outputs for 1D impedance model by Liang et

al. (1994),'

'A factor of two is used, since the model considers only one-half

'the patch

Public Const LA As Double = 0.003

Public Const WA As Double = 1#

Public Const HA As Double = 0.0002

Public Const RHO As Double = 7650#

Public Const D31 As Double = -0.000000000166

Public Const Y11E As Double = 63000000000#

Public Const E33T As Double = 0.000000015

Public Const ETA As Double = 0.001

Public Const DELTA As Double = 0.012

Dim f As Double

Dim k_real As Double

Dim k_imag As Double

Dim x, y As Double

Dim xa, ya As Double

Dim r, t As Double

Dim Yr, Yi As Double

Dim Fr, Fi, Ur, Ui As Double

Sub main()

Dim index As Integer

Dim kl_real, kl_imag As Double

For index = 8 To 57

f = Cells(index, 1)

Fr = Cells(index, 6)

53

Fi = Cells(index, 7)

Ur = Cells(index, 4)

Ui = Cells(index, 5)

Call calc_str_impd

Call calc_k(f)

kl_real = k_real * LA

kl_imag = k_imag * LA

Call tanz_by_z(kl_real, kl_imag)

Call Z_actuator

Call calc_Y

Cells(index, 9) = x

Cells(index, 10) = y

Cells(index, 11) = xa

Cells(index, 12) = ya

Cells(index, 13) = 2 * Yr

Cells(index, 14) = 2 * Yi

Next index

End Sub

Sub tanz_by_z(rl, im As Double)

Dim a, b, C, d, u, v, q As Double

a = (Exp(-im) + Exp(im)) * Sin(rl)

b = (Exp(-im) - Exp(im)) * Cos(rl)

C = (Exp(-im) + Exp(im)) * Cos(rl)

d = (Exp(-im) - Exp(im)) * Sin(rl)

u = C * rl - d * im

v = d * rl + C * im

q = u * u + v * v

54

r = (a * u - b * v) / q

t = (-1#) * (a * v + b * u) / q

End Sub

Sub calc_k(freq)

Dim w, cons As Double

w = 2# * 3.14 * freq

cons = Sqr(RHO / (Y11E * (1 + ETA * ETA)))

k_real = cons * w

k_imag = cons * w * (-0.5 * ETA)

End Sub

Sub calc_Y()

Dim p, q, Big_p, Big_q, Big_R, Big_T, Big_pq As Double

Dim temp_r, temp_i As Double

p = x + xa

q = y + ya

Big_p = xa * p + ya * q

Big_q = ya * p - xa * q

Big_R = r - ETA * t

Big_T = ETA * r + t

Big_pq = p * p + q * q

temp_r = (Big_p * Big_T + Big_q * Big_R) / Big_pq

temp_i = (Big_p * Big_R - Big_q * Big_T) / Big_pq

t_r = ETA - temp_r

t_i = temp_i - 1

multi = (WA * LA * 2# * 3.14 * f) / HA

Yr = multi * (DELTA * E33T + t_r * D31 * D31 * Y11E)

Yi = multi * (E33T + t_i * D31 * D31 * Y11E)

55

End Sub

Sub Z_actuator()

Dim multi As Double

multi = (WA * HA * Y11E) / (2 * 3.14 * LA * f)

Big_rt = r * r + t * t

xa = multi * (ETA * r - t) / Big_rt

ya = multi * (-1#) * (r + ETA * t) / Big_rt

End Sub

Function calc_str_impd()

Dim div As Double

div = 2# * 3.14 * f

Big_U = Ur * Ur + Ui * Ui

x = (Fi * Ur - Fr * Ui) / (div * Big_U)

y = (-1#) * (Fr * Ur + Fi * Ui) / (div * Big_U)

End Function

56

APPENDIX B

'Determination of signatures from ANSYS outputs considering shear lag effect by KBD

model.'

'A factor of two is used, since the model considers only one-half

'the patch

Public Const LA As Double = 0.003

Public Const WA As Double = 1#

Public Const HA As Double = 0.0002

Public Const RHO As Double = 7650#

Public Const D31 As Double = -0.000000000166

Public Const Y11E As Double = 63000000000#

Public Const E33T As Double = 0.000000015

Public Const ETA As Double = 0.001

Public Const DELTA As Double = 0.012

Public Const ETAP As Double = 0.1

Public Const ts As Double = 0.0002

Dim f As Double

Dim k_real As Double

Dim k_imag As Double

Dim Xeq, Yeq As Double

Dim x, y As Double

Dim xa, ya As Double

Dim r, t As Double

Dim Yr, Yi As Double

Dim Fr, Fi, Ur, Ui As Double

Sub main()

Dim index As Integer

Dim kl_real, kl_imag As Double

For index = 8 To 57

57

f = Cells(index, 1)

Fr = Cells(index, 6)

Fi = Cells(index, 7)

Ur = Cells(index, 4)

Ui = Cells(index, 5)

Call calc_str_impd

Call calc_k(f)

kl_real = k_real * LA

kl_imag = k_imag * LA

Call tanz_by_z(kl_real, kl_imag)

Call Z_actuator

Call calc_Y

Cells(index, 9) = x

Cells(index, 10) = y

Cells(index, 11) = xa

Cells(index, 12) = ya

Cells(index, 13) = 2 * Yr

Cells(index, 14) = 2 * Yi

Cells(index, 15) = Xeq

Cells(index, 16) = Yeq

Next index

End Sub

Sub tanz_by_z(rl, im As Double)

Dim a, b, C, d, u, v, q As Double

a = (Exp(-im) + Exp(im)) * Sin(rl)

b = (Exp(-im) - Exp(im)) * Cos(rl)

C = (Exp(-im) + Exp(im)) * Cos(rl)

d = (Exp(-im) - Exp(im)) * Sin(rl)

58

u = C * rl - d * im

v = d * rl + C * im

q = u * u + v * v

r = (a * u - b * v) / q

t = (-1#) * (a * v + b * u) / q

End Sub

Sub calc_k(freq)

Dim w, cons As Double

w = 2# * 3.14 * freq

cons = Sqr(RHO / (Y11E * (1 + ETA * ETA)))

k_real = cons * w

k_imag = cons * w * (-0.5 * ETA)

End Sub

Sub calc_Y()

Dim p, q, Big_p, Big_q, Big_R, Big_T, Big_pq As Double

Dim temp_r, temp_i As Double

p = Xeq + xa

q = Yeq + ya

Big_p = xa * p + ya * q

Big_q = ya * p - xa * q

Big_R = r - ETA * t

Big_T = ETA * r + t

Big_pq = p * p + q * q

temp_r = (Big_p * Big_T + Big_q * Big_R) / Big_pq

temp_i = (Big_p * Big_R - Big_q * Big_T) / Big_pq

t_r = ETA - temp_r

59

t_i = temp_i - 1

multi = (WA * LA * 2# * 3.14 * f) / HA

Yr = multi * (DELTA * E33T + t_r * D31 * D31 * Y11E)

Yi = multi * (E33T + t_i * D31 * D31 * Y11E)

End Sub

Sub Z_actuator()

Dim multi As Double

multi = (WA * HA * Y11E) / (2 * 3.14 * LA * f)

Big_rt = r * r + t * t

xa = multi * (ETA * r - t) / Big_rt

ya = multi * (-1#) * (r + ETA * t) / Big_rt

End Sub

Function calc_str_impd()

Dim div As Double

Dim C As Double

div = 2# * 3.14 * f

Big_U = Ur * Ur + Ui * Ui

x = (Fi * Ur - Fr * Ui) / (div * Big_U)

y = (-1#) * (Fr * Ur + Fi * Ui) / (div * Big_U)

C = 366.667 * div * ts * 0.000000001

Xeq = (x * (1 + ETAP * C * x + C * y) - C * y * (x + ETAP * y)) / ((1 + ETAP * C * x +

C * y) ^ 2 + C * C * (x + ETAP * y) ^ 2)

Yeq = (x * C * (x + ETAP * y) + y * (1 + ETAP * C * x + C * y)) / ((1 + ETAP * C * x +

C * y) ^ 2 + C * C * (x + ETAP * y) ^ 2)

End Function

60

APPENDIX C

%sig_shearp2.m

%MATLAB program to obtain PZT electro-mechanical signatures from ANSYS outputs,

%taking into account shear lag effect associated with adhesive layer.

%This program does correction for peak displacement (rather single peak displacement)

%It is based on the 1D impedance model given by Bhalla and Sow (2004) based on

effective impedance.

%PZT parameters are based on measurement. Since the model considers only

%one-half of the PZT, a factor of 2 used in final answer

data=dlmread('output.txt','\t'); %Data-matrix, stores the ANSYS output

%Parameters of PZT (averaged, as in Chapter 6)

LA=0.003; HA= 0.0002; RHO=7650; D31= -0.000000000166;

Y11E= 63000000000; E33T=1.5e-8; ETA= 0.001; DELTA= 0.012;WA=1;

GE = 1.0e9; HE = 0.0000667; BETA = 0.1;V=1.4;

f = data(:,1); %Frequency in Hz

Fr = data(:,2); %Real component of

effective force

Fi = data(:,3); %Imaginary component of

effective force

Ur = data(:,4); %Real component of

effective displacement

Ui = data(:,5); %Imaginary component of

effective displacement

N=size(f); %No of data points

for I = 1:N,

61

%Calculation of structural impedance

omega(I) = 2* pi * f(I); %Angular frequency

in rad/s

Big_U(I)= Ur(I)*Ur(I) + Ui(I)*Ui(I);

x(I) = (Fi(I) * Ur(I) - Fr(I) * Ui(I)) / (omega(I) * Big_U(I));

y(I) = (-1.0) *(Fr(I) * Ur(I) + Fi(I) * Ui(I)) / (omega(I) * Big_U(I));

%Consideration of shear lag effect

PS = D31*V/HA; %Free PZT strain

qe = GE/(Y11E*HA*HE); % qeff

ae = WA*GE*(y(I)-BETA*x(I))/(omega(I)*HE*(x(I)^2+y(I)^2));

be = WA*GE*(x(I)+BETA*y(I))/(omega(I)*HE*(x(I)^2+y(I)^2));

pe=complex(ae,be); %peff

root3=(-pe/2)+sqrt(pe*pe/4+qe);

root4=(-pe/2)-sqrt(pe*pe/4+qe);

ne=1/pe;

E3=exp(root3*LA);

E4=exp(root4*LA);

E3m=exp(-1*root3*LA);

E4m=exp(-1*root4*LA);

%Determination of Constants from boundary conditions

k1 = (1+ne*root3)*root3*E3m-root3;

k2 = (1+ne*root4)*root4*E4m-root4;

k3 = (1+ne*root3)*root3*E3-root3;

k4 = (1+ne*root4)*root4*E4-root4;

Be=PS*(k4-k2)/(k1*k4-k2*k3);

Ce=PS*(k1-k3)/(k1*k4-k2*k3);

A1=-Be-Ce;

A2=-Be*root3-Ce*root4;

62

%End displacement on surface of host structure

ue=A1+A2*LA+Be*E3+Ce*E4;

%End strain surface of host structure

se=A2+Be*root3*E3+Ce*root4*E4;

Z=complex(x(I),y(I));

Zeq=Z/(1+ne*se/ue);

x(I)=real(Zeq);

y(I)=imag(Zeq);

%Calculation of wave number

cons = (RHO / (Y11E * (1 + ETA * ETA)))^0.5;

k_real(I) = cons * omega(I);

k_imag(I) = cons * omega(I) * (-0.5 * ETA);

%Calculation of tan(kl)/kl

rl(I) = k_real(I) * LA ;

im(I) = k_imag(I) * LA ;

a(I) = (exp(-im(I)) + exp(im(I))) * sin(rl(I));

b(I) = (exp(-im(I)) - exp(im(I))) * cos(rl(I));

c(I) = (exp(-im(I)) + exp(im(I))) * cos(rl(I));

d(I) = (exp(-im(I)) - exp(im(I))) * sin(rl(I));

u(I) = c(I) * rl(I) - d(I) * im(I);

v(I) = d(I) * rl(I) + c(I) * im(I);

h(I) = u(I)^2 + v(I)^2;

r(I) = (a(I) * u(I) - b(I) * v(I)) / h(I);

t(I) = (-1.0) * (a(I) * v(I) + b(I) * u(I)) / h(I);

%Calculation of actuator impedance

multia(I) = (WA*HA*Y11E) / (2*pi*LA*f(I));

63

Big_rt(I) = r(I) * r(I) + t(I) * t(I);

xa(I) = multia(I) * (ETA * r(I) - t(I)) / Big_rt(I);

ya(I) = multia(I) * (-1.0) * (r(I) + ETA * t(I)) / Big_rt(I);

%Calculation of conductance and susceptance

p(I) = x(I) + xa(I);

q(I) = y(I) + ya(I);

Big_p(I) = xa(I) * p(I) + ya(I) * q(I);

Big_q(I) = ya(I) * p(I) - xa(I) * q(I);

Big_R(I) = r(I) - ETA * t(I);

Big_T(I) = ETA * r(I) + t(I);

Big_pq(I) = p(I) * p(I) + q(I) * q(I);

temp_r(I) = (Big_p(I) * Big_T(I) + Big_q(I) * Big_R(I)) / Big_pq(I);

temp_i(I) = (Big_p(I) * Big_R(I) - Big_q(I) * Big_T(I)) / Big_pq(I);

t_r(I) = ETA - temp_r(I);

t_i(I) = temp_i(I) - 1;

multi(I) = (WA * LA * omega(I)) / HA;

G(I) = 2 * multi(I) * (DELTA * E33T + t_r(I)*D31*D31*Y11E);

B(I) = 2 * multi(I) * (E33T + t_i(I)*D31*D31*Y11E);

end

plot(f,G);

figure;

plot(f,B);

64

APPENDIX D

%sig_shearp2.m

%MATLAB program to obtain Shear Stress in the adhesive bond layer using the Bhalla

and Soh model(2004) and the KBD model taking into account shear lag effect associated

with adhesive layer.

%This program does correction for peak displacement (rather single peak displacement)

%It is based on the new 1D model based on effective impedance.

%PZT parameters are based on measurement. Since the model considers only

data=dlmread('output.txt','\t'); %Data-matrix, stores the ANSYS output

%Parameters of PZT

LA=0.003; HA= 0.0002; RHO=7650; D31= -0.000000000166;

Y11E= 63000000000; E33T=1.5e-8; ETA= 0.001; DELTA= 0.012;WA=1;ETAP=0.1;

GE = 1.0e9; HE = 0.00002; BETA = 0.1;V=1.4;

f = 101000

Fr = -1000

Fi = 0

Ur = -3.02E-07

Ui = 1.46E-09

X = data(:,6); %Length along actuator

N=size(f); %No of data points

for I = 1:N,

%Calculation of structural impedance

omega(I) = 2* pi * f(I); %Angular frequency

in rad/s

Big_U(I)= Ur(I)*Ur(I) + Ui(I)*Ui(I);

x(I) = (Fi(I) * Ur(I) - Fr(I) * Ui(I)) / (omega(I) * Big_U(I));

65

y(I) = (-1.0) *(Fr(I) * Ur(I) + Fi(I) * Ui(I)) / (omega(I) * Big_U(I));

%Consideration of shear lag effect

PS = D31*V/HA; %Free PZT strain

qe = GE/(Y11E*HA*HE); % qeff

ae = WA*GE*(y(I)-BETA*x(I))/(omega(I)*HE*(x(I)^2+y(I)^2));

be = WA*GE*(x(I)+BETA*y(I))/(omega(I)*HE*(x(I)^2+y(I)^2));

pe=complex(ae,be); %peff

root3=(-pe/2)+sqrt(pe*pe/4+qe);

root4=(-pe/2)-sqrt(pe*pe/4+qe);

ne=1/pe;

E3=exp(root3*LA);

E4=exp(root4*LA);

E3m=exp(-1*root3*LA);

E4m=exp(-1*root4*LA);

E3x=exp(root3*X);

E4x=exp(root4*X);

%Determination of Constants from boundary conditions

k1 = (1+ne*root3)*root3*E3m-root3;

k2 = (1+ne*root4)*root4*E4m-root4;

k3 = (1+ne*root3)*root3*E3-root3;

k4 = (1+ne*root4)*root4*E4-root4;

Be=PS*(k4-k2)/(k1*k4-k2*k3);

Ce=PS*(k1-k3)/(k1*k4-k2*k3);

A1=-Be-Ce;

A2=-Be*root3-Ce*root4;

%End displacement on surface of host structure

ue=A1+A2*LA+Be*E3+Ce*E4;

66

%End strain surface of host structure

se=A2+Be*root3*E3+Ce*root4*E4;

Z=complex(x(I),y(I));

Zeq=Z/(1+ne*se/ue);

x(I)=real(Zeq);

y(I)=imag(Zeq);

%Calculation of wave number

cons = (RHO / (Y11E * (1 + ETA * ETA)))^0.5;

k_real(I) = cons * omega(I);

k_imag(I) = cons * omega(I) * (-0.5 * ETA);

%Calculation of tan(kl)/kl

rl(I) = k_real(I) * LA ;

im(I) = k_imag(I) * LA ;

a(I) = (exp(-im(I)) + exp(im(I))) * sin(rl(I));

b(I) = (exp(-im(I)) - exp(im(I))) * cos(rl(I));

c(I) = (exp(-im(I)) + exp(im(I))) * cos(rl(I));

d(I) = (exp(-im(I)) - exp(im(I))) * sin(rl(I));

u(I) = c(I) * rl(I) - d(I) * im(I);

v(I) = d(I) * rl(I) + c(I) * im(I);

h(I) = u(I)^2 + v(I)^2;

r(I) = (a(I) * u(I) - b(I) * v(I)) / h(I);

t(I) = (-1.0) * (a(I) * v(I) + b(I) * u(I)) / h(I);

%Calculation of actuator impedance

multia(I) = (WA*HA*Y11E) / (2*pi*LA*f(I));

Big_rt(I) = r(I) * r(I) + t(I) * t(I);

xa(I) = multia(I) * (ETA * r(I) - t(I)) / Big_rt(I);

ya(I) = multia(I) * (-1.0) * (r(I) + ETA * t(I)) / Big_rt(I);

67

%Calculation of conductance and susceptance

p(I) = x(I) + xa(I);

q(I) = y(I) + ya(I);

Big_p(I) = xa(I) * p(I) + ya(I) * q(I);

Big_q(I) = ya(I) * p(I) - xa(I) * q(I);

Big_R(I) = r(I) - ETA * t(I);

Big_T(I) = ETA * r(I) + t(I);

Big_pq(I) = p(I) * p(I) + q(I) * q(I);

temp_r(I) = (Big_p(I) * Big_T(I) + Big_q(I) * Big_R(I)) / Big_pq(I);

temp_i(I) = (Big_p(I) * Big_R(I) - Big_q(I) * Big_T(I)) / Big_pq(I);

t_r(I) = ETA - temp_r(I);

t_i(I) = temp_i(I) - 1;

multi(I) = (WA * LA * omega(I)) / HA;

G(I) = multi(I) * (DELTA * E33T + t_r(I)*D31*D31*Y11E);

B(I) = multi(I) * (E33T + t_i(I)*D31*D31*Y11E);

SSR=-Z*complex(0,1)*(Be*root3*(E3x-1)+Ce*root4*(E4x-1))/WA

SSRR=real(SSR)

SSRI=imag(SSR)

SSRA=(SSRR.^2+SSRI.^2).^0.5

A=LA

U=complex(Ur,Ui)

Up=U/(1+Z*omega*HE*complex(0,1)/(LA*GE*(1+ETAP*complex(0,1))-

Z*omega*HE*complex(0,1)))

STRESS=-Z*complex(0,1)*omega*Up/(A-

Z*omega*HE*complex(0,1)/GE*(1+ETAP*complex(0,1)))

STR1=real(STRESS)

STR2=imag(STRESS)

STR_ABS=(STR1.^2+STR2.^2).^0.5

h=ones(50,1)

qqq=A2*h+Be*root3*E3x+Ce*root4*E4x

TTT=-Z*complex(0,1)*omega*qqq/WA

68

TTT_real=real(TTT)

TTT_imag=imag(TTT)

TTT_ABS=(TTT_real.^2+TTT_imag.^2).^0.5

end

TTT

TTT_ABS

STRESS

STR_ABS

69

APPENDIX E

Output.txt

1.01E+05 -1.00E+03 0.00E+00 -3.02E-07 1.46E-09 0.00E+00

1.02E+05 -1.00E+03 0.00E+00 -3.11E-07 1.52E-09 1.00E-04

1.03E+05 -1.00E+03 0.00E+00 -3.21E-07 1.69E-09 2.00E-04

1.04E+05 -1.00E+03 0.00E+00 -3.34E-07 2.19E-09 3.00E-04

1.05E+05 -1.00E+03 0.00E+00 -3.55E-07 4.01E-09 4.00E-04

1.06E+05 -1.00E+03 0.00E+00 -4.34E-07 2.52E-08 5.00E-04

1.07E+05 -1.00E+03 0.00E+00 -2.04E-07 2.75E-08 6.00E-04

1.08E+05 -1.00E+03 0.00E+00 -2.88E-07 4.14E-09 7.00E-04

1.09E+05 -1.00E+03 0.00E+00 -3.09E-07 2.28E-09 8.00E-04

1.10E+05 -1.00E+03 0.00E+00 -3.21E-07 1.77E-09 9.00E-04

1.11E+05 -1.00E+03 0.00E+00 -3.31E-07 1.09E-09 1.00E-03

1.12E+05 -1.00E+03 0.00E+00 -3.39E-07 1.64E-09 1.10E-03

1.13E+05 -1.00E+03 0.00E+00 -3.47E-07 1.72E-09 1.20E-03

1.14E+05 -1.00E+03 0.00E+00 -3.55E-07 1.86E-09 1.30E-03

1.15E+05 -1.00E+03 0.00E+00 -3.65E-07 2.09E-09 1.40E-03

1.16E+05 -1.00E+03 0.00E+00 -3.76E-07 2.48E-09 1.50E-03

1.17E+05 -1.00E+03 0.00E+00 -3.91E-07 3.16E-09 1.60E-03

1.18E+05 -1.00E+03 0.00E+00 -4.13E-07 4.52E-09 1.70E-03

1.19E+05 -1.00E+03 0.00E+00 -4.51E-07 7.84E-09 1.80E-03

1.20E+05 -1.00E+03 0.00E+00 -5.34E-07 2.01E-08 1.90E-03

1.21E+05 -1.00E+03 0.00E+00 -9.28E-07 1.85E-07 2.00E-03

1.22E+05 -1.00E+03 0.00E+00 1.40E-07 1.22E-07 2.10E-03

1.23E+05 -1.00E+03 0.00E+00 -1.66E-07 1.77E-08 2.20E-03

1.24E+05 -1.00E+03 0.00E+00 -2.41E-07 7.55E-09 2.30E-03

1.25E+05 -1.00E+03 0.00E+00 -2.77E-07 4.72E-09 2.40E-03

1.26E+05 -1.00E+03 0.00E+00 -3.00E-07 3.64E-09 2.50E-03

70

1.27E+05 -1.00E+03 0.00E+00 -3.18E-07 3.20E-09 2.60E-03

1.28E+05 -1.00E+03 0.00E+00 -3.33E-07 3.11E-09 2.70E-03

1.29E+05 -1.00E+03 0.00E+00 -3.48E-07 3.28E-09 2.80E-03

1.30E+05 -1.00E+03 0.00E+00 -3.65E-07 3.80E-09 2.90E-03

1.31E+05 -1.00E+03 0.00E+00 -3.86E-07 4.92E-09 3.00E-03

1.32E+05 -1.00E+03 0.00E+00 -4.17E-07 7.59E-09 0.00E+00

1.33E+05 -1.00E+03 0.00E+00 -4.76E-07 1.66E-08 0.00E+00

1.34E+05 -1.00E+03 0.00E+00 -6.90E-07 1.09E-07 0.00E+00

1.35E+05 -1.00E+03 0.00E+00 -1.53E-07 1.21E-07 0.00E+00

1.36E+05 -1.00E+03 0.00E+00 -6.49E-07 3.93E-07 0.00E+00

1.37E+05 -1.00E+03 0.00E+00 -9.35E-08 4.48E-08 0.00E+00

1.38E+05 -1.00E+03 0.00E+00 -2.12E-07 1.20E-08 0.00E+00

1.39E+05 -1.00E+03 0.00E+00 -2.57E-07 6.52E-09 0.00E+00

1.40E+05 -1.00E+03 0.00E+00 -2.82E-07 4.66E-09 0.00E+00

1.41E+05 -1.00E+03 0.00E+00 -3.01E-07 3.87E-09 0.00E+00

1.42E+05 -1.00E+03 0.00E+00 -3.16E-07 3.55E-09 0.00E+00

1.43E+05 -1.00E+03 0.00E+00 -3.30E-07 3.51E-09 0.00E+00

1.44E+05 -1.00E+03 0.00E+00 -3.44E-07 3.72E-09 0.00E+00

1.45E+05 -1.00E+03 0.00E+00 -3.59E-07 4.23E-09 0.00E+00

1.46E+05 -1.00E+03 0.00E+00 -3.77E-07 5.28E-09 0.00E+00

1.47E+05 -1.00E+03 0.00E+00 -4.02E-07 7.50E-09 0.00E+00

1.48E+05 -1.00E+03 0.00E+00 -4.44E-07 1.33E-08 0.00E+00

1.49E+05 -1.00E+03 0.00E+00 -5.39E-07 3.67E-08 0.00E+00

1.50E+05 -1.00E+03 0.00E+00 -9.57E-07 4.61E-07 0.00E+00