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Accident involving American Airlines Airbus A300-600 ………….2. (LAMSS, 2003). Fig. 1.3. Image of Columbia about a minute before it broke apart……..……3.
A MECHANICAL IMPEDANCE APPROACH FOR STRUCTURAL IDENTIFICATION, HEALTH MONITORING AND NON-DESTRUCTIVE EVALUATION USING PIEZO-IMPEDANCE TRANSDUCERS

SURESH BHALLA

SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING NANYANG TECHNOLOGICAL UNIVERSITY

2004

A Mechanical Impedance Approach for Structural Identification, Health Monitoring and Non-Destructive Evaluation Using Piezo-Impedance Transducers

Suresh Bhalla

School of Civil and Environmental Engineering

A thesis submitted to the Nanyang Technological University in fulfillment of the requirements for the degree of Doctor of Philosophy

2004

ACKNOWLEDGEMENTS First and foremost, I would like to extend my sincere thanks and gratitude towards my supervisor, Professor Soh Chee-Kiong, for his continuous guidance, encouragement and strong support during the course of my Ph.D. research. I am forever grateful for his kindness and contributions, not only towards my research, but towards my professional growth as well. I am also grateful to other members of Professor Soh’s research team, namely Prof Yang Yaowen, Xu Jianfeng, Akshay Naidu, Ong Chin Wee, Jin Zhanli and Wang Chao, for giving numerous suggestions during the weekly research meetings. Often, during presentations, the team members would pose questions that would immensely help in improving my work. My special thanks go towards Prof Lu Yong, who not only provided extremely useful suggestions as the examiner of my first year report, but also personally oversaw the execution of many critical experiments. Thanks are also due to Mr Lim Say Ian and Mr Goo Kian Tiong (Jimmy), who provided assistance in performing many of the experiments as a part of their final year projects. I express my special thanks to Mrs Koh, Mrs Ho, Ms May Sim, Mr Subhas, Mr Tan and other technicians, who provided their technical support generously during the lab work. Without their support and practical tips as well as the good work environment in the Structures Lab, it would not have been possible to finish the experimental work so smoothly. I also express my gratitude towards my colleagues and other supporting staff at the School of Civil & Environmental Engineering, who directly or indirectly contributed towards my research. I am very thankful to my parents for their encouragement and sacrifices, and I wish to mention a very special acknowledgement to Rupali, my wife, for her continued support and co-operation. She maintained amazing home in-spite of her own research programme.

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TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS...……………………………………………………….i TABLE OF CONTENTS…………………………………………………………...ii SUMMARY…...…………………………………………………………………….x LIST OF TABLES…………………..…………………………………………….xiii LIST OF FIGURES…………………………………………………………...…...xiv LIST OF SYMBOLS……………………………………………………………....xxi LIST OF ACRONYMS…………………………………………….……………...xxv CHAPTER 1: INTRODUCTION…………………………………………………1 1.1 Structural Damages and Failures…………..…………………………….1 1.2 An Overview of Recent Structural Failures……………………………..2 1.3 Structural Health Monitoring ………………………………………..….6 1.4 Requirements for any SHM System…………………………………….7 1.5 SHM by Electro-mechanical Impedance (EMI) Technique…………….9 1.6 Research Objectives …………………………………...………………11 1.7 Research Originality and Contributions..……………………………...11 1.8 Thesis Organisation...………………………………………………….12 CHAPTER 2: ELECTRO-MECHANICAL IMPEDANCE (EMI)…………...14 TECHNIQUE FOR SHM AND NDE 2.1 State of the Art in SHM/ NDE………………………………………...14 2.1.1

Global SHM Techniques………………………………………14

2.1.2

Local SHM Techniques……………………………………..…18

2.1.3

Advent of Smart Materials, Structures and Systems for.………21 SHM and NDE

2.2 Smart Systems/ Structures………………….………………………….22 2.2.1

Definition of Smart Systems/ Structures………………………22

2.2.2

Smart Materials…...……………………………………………24 ii

2.2.3

Active and Passive Smart Materials………………..………..….25

2.2.4

Applications of Piezoelectric Materials ……………..………....26

2.2.5

Smart Materials: Future Applications…………………………...27

2.3 Piezoelectricity and Piezoelectric Materials………………………...…..27 2.3.1

Constitutive Relations…………………………………………...28

2.3.2

Second Order Effects………………………….....……………...32

2.3.3

Pyroelectricity and Ferroelectricity.………………………….…33

2.3.4

Commercial Piezoelectric Materials………………………...…..33

2.4 Piezoelectric Materials as Mechatronic Impedance Transducers…….…36 (MITs) for SHM 2.4.1

Physical Principles…………………..…………………………..37

2.4.2

Method of Application…………………………..……………...42

2.4.3

Major Technological Developments During Last Nine Years.…42

2.4.4

Details of PZT Patches …………………………..…………….44

2.4.5

Selection of Frequency Range………………………….…..…...45

2.4.6

Sensing Zone of Piezo-Impedance Transducers………………...46

2.4.7

Modes of Wave Propagation………………………..…………...47

2.4.8

Effects of Temperature…………………………………...……..48

2.4.9

Effects of Noise and Other Miscellaneous Factors…....………...49

2.4.10 Thermal Stresses in Piezo-Impedance Transducers………….….50 2.4.11 Multiple Sensor Requirements…..……………………………...50 2.4.12 Signal Processing Techniques and Damage Quantification…….51 2.5 Advantages of EMI Technique………………………………………….54 2.6 Limitations of EMI Technique………………………………………….56 2.7 Needs for Further Research in EMI Technique…………………..….….57 2.7.1

Theoretical and Data Processing Considerations..………..……..57

2.7.2

Hardware/ Technology Considerations….………………..…….58

2.8 Concluding Remarks………………………………………..………......59

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CHAPTER 3: PZT-STRUCTURE ELECTRO-MECHANICAL……………..60 INTERACTION 3.1 Introduction………………………………………….……………..…60 3.2 Mechanical Impedance of Structures….……...………………………60 3.3 Mechanical Impedance of PZT Patches.……...………………………62 3.4 Electro-Mechanical Interaction in Single Degree of Freedom………..65 (SDOF) Systems 3.5 Structure-PZT Interaction in Complex Systems...……………………80 3.6 Implications of Structure-PZT Interaction……………………………84 3.7 Decomposition of Coupled Electro-Mechanical Admittance…………84 3.8 Concluding Remarks………………………………………...…….…..88 CHAPTER 4: DAMAGE ASSESSMENT OF SKELETAL STRUCTURES…89 VIA EXTRACTED MECHANICAL IMPEDANCE 4.1 Introduction………………………………………………………...…89 4.2 Analogy Between Electrical and Mechanical Systems…...……....….89 4.3 Measurement of Mechanical Impedance………………………….…..91 4.4 Decomposition of Admittance Signatures……………….. …………..92 4.5 Extraction of Structural Mechanical Impedance of Skeletal…..……...94 Structures 4.5.1

Computational Procedure………………………………………94

4.5.2

Determination of (tan κl/ κl)……………………………..…….96

4.5.3

Physical Interpretation of Drive Point Impedance……………..97

4.6 Definition of Damage Metric Based on Extracted Structural ….….....98 Impedance 4.7 Proof of Concept Application: Diagnosis of Vibration Induced..…….98 Damages 4.7.1

Flexural Damage Prediction by PZT Patch #2……..…..….…100

4.7.2

Shear Damage Prediction by PZT Patch #1……..……..…….103

4.7.3

Damage Sensitivity of the Proposed Methodology…………..104

4.8 Discussions………………………………………..………………...106 iv

4.9 Concluding Remarks………………………………………...……...106 CHAPTER 5: GENERALIZED ELECTRO-MECHANICAL ……………..107 IMPEDANCE FORMULATIONS: THEORETICAL DEVELOPMENT AND SHM APPLICATIONS 5.1 Introduction…………………...……………………………….…...107 5.2 Existing PZT-Structure Interaction Models…….……………….…107 5.3 Limitations of Existing Modelling Approaches...……………….…110 5.4 Definition of Effective Mechanical Impedance……………....….…111 5.5 Electro-Mechanical Formulations Based on Effective Impedance....113 5.6 Experimental Verification……..…….……….……………………..117 5.6.1

Details of Experimental Set-up…………………………..117

5.6.2

Determination of Structural EDP Impedance by FEM…..118

5.6.3

Modelling of Structural Damping………………………...121

5.6.4

Wavelength Analysis and Convergence Test………….…122

5.6.5

Comparison Between Theoretical and Experimental….….122 Signatures

5.7 Refining the Model of PZT Sensor-Actuator Patch………………...126 5.8 Decomposition of Coupled Electro-Mechanical Admittance…….…134 5.9 Extraction of Structural Mechanical Impedance………………….…136 5.10 System Parameter Identification from Extracted Impedance Spectra.138 5.11 Damage Diagnosis in Aerospace and Mechanical Systems………...143 5.12 Extension to Damage Diagnosis in Civil- Structural Systems……...151 5.13 Concluding Remarks………………………………………………...153 CHAPTER 6: CALIBRATION OF PIEZO-IMPEDANCE ………………….155 TRANSDUCERS FOR STRENGTH PREDICTION AND DAMAGE ASSESSMENT OF CONCRETE 6.1 Introduction………………………………………………………..…...155 6.2 Conventional NDE Methods in Concrete……………………………...155 6.2.1

Surface Hardness Methods………………………………….156 v

6.2.2

Rebound Method……………………………………………156

6.2.3

Penetration Techniques……………………………………...157

6.2.4

Pullout Test………………………………………………….157

6.2.5

Resonant Frequency Method………………………………..157

6.2.6

Ultrasonic Pulse Velocity Method…………………………..158

6.3 Concrete Strength Evaluation Using EMI Technique………………….160 6.4 Extraction of Damage Sensitive Concrete Parameters from……………164 Admittance Signatures 6.5 Monitoring Concrete Curing Using Extracted Impedance…….………..169 Parameters 6.6 Establishment of Impedance-Based Damage Model for Concrete……...173 6.6.1

Definition of Damage Variable…………………..……..……173

6.6.2

Theory of Statistics and Probability…………..……………..174

6.6.3

Theory of Fuzzy Sets……………….………………………..176

6.6.4

Statistical Analysis of Damage Variable for Concrete……….178

6.6.5

Fuzzy Probabilistic Damage Calibration of Piezo-…………..178 Impedance Transducers

6.7 Discussions………….…………………………………………………..183 6.8 Concluding Remarks…………………………………………………....185 CHAPTER 7: INCLUSION OF INTERFACIAL SHEAR LAG EFFECT…..186 IN IMPEDANCE MODELS 7.1 Introduction……………………………………………………….….186 7.2 Shear Lag Effect……………………………………………………...186 7.2.1

PZT Patch as Sensor……………………………………....188

7.2.2

PZT Patch as Actuator….…………………………………192

7.3 Integration of Shear Lag Effect into Impedance Models…………....194 7.4 Inclusion of Shear Lag Effect in 1D Impedance Model..…………....196 7.5 Extension to 2D-Effective Impedance Based Model….…………….201 7.6 Experimental Verification…………………………………………...203

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7.7 Parametric Study on Adhesive Layer Induced Admittance………...207 Signatures 7.7.1

Influence of Bond Layer Shear Modulus (Gs)………..…..207

7.7.2

Influence of Bond Layer Thickness (ts)…………………..209

7.7.3

Influence of Damping of Adhesive Layer ( η ′ )…………...210

7.7.4

Overall Influence of Parameter peff ……………………...211

7.7.5

Overall Influence of Parameter qeff……………………….212

7.7.6

Influence of Sensor Length (l)……………………………213

7.7.7

Quantification of Overall Influence of Bond Layer………214

7.8 Summary and Concluding Remarks…………………………….…..214 CHAPTER 8: PRACTICAL ISSUES RELATED TO EMI TECHNIQUE….215 8.1 Introduction……………………………………………………….….215 8.2 Evaluation of Long term Repeatability of Signatures…….………… 215 8.3 Protection of PZT Transducers Against Environment………………. 216 8.4 Multiplexing of Signals from PZT Arrays………………………...…220 8.5 Concluding Remarks…………………………………………………222 CHAPTER 9: CONCLUSIONS AND RECOMMENDATIONS………….…223 9.1 Introduction……………………………………………………….…223 9.2 Research Conclusions and Contributions………..……………….….223 9.3 Recommendations for Future Work………………………………....228 AUTHOR’S PUBLICATIONS………………………………………..….…….230 REFERENCES…………………………………………………………..…..…..234

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APPENDICES Appendix A

Visual Basic program to derive conductance and

252

susceptance plots from ANSYS output. This program is based on 1D impedance model of Liang et al. (1994), Eq. (2.24) Appendix B

Visual Basic program to derive real and imaginary

254

components of structural impedance from admittance signatures. This program is based on 1D impedance model of Liang et al. (1994), Eq. 2.24 Appendix C

MATLAB

program

to

derive

electro-mechanical

256

admittance signatures from ANSYS output. The program is based on the new 2D model based on effective impedance, covered in Chapter 5 (Eq. 5.30). Appendix D

MATLAB

program

to

derive

electro-mechanical

258

admittance signatures from ANSYS output, using updated PZT model (twin-peak). The program is based on the new 2D model based on effective impedance, covered in Chapter 5 (Eq. 5.56). Appendix E

MATLAB program to derive structural mechanical impedance from experimental admittance signatures, using updated PZT model (twin-peak). The program is based on the new 2D model based on effective impedance, covered in Chapter 5 (Eq. 5.56).

viii

260

Appendix F

MATLAB program to compute fuzzy failure probability.

262

Appendix G

MATLAB

263

program

to

derive

electro-mechanical

admittance signatures from ANSYS output, taking shear lag in the adhesive layer into account. The program is based on the new 2D model based on effective impedance, covered in Chapter 5 (Eq. 5.30).

ix

SUMMARY The last few decades have witnessed construction of vast infrastructural facilities in Singapore and other parts of the world. Now, the ageing of these structures is creating maintenance problems and increasingly prompting the development of automated structural health monitoring (SHM) and non-destructive evaluation (NDE) systems, which can provide cost-effective alternative to traditional visual inspection. Similar necessity is increasingly felt for civil and military aircraft, spaceships, heavy machinery, trains, and so on, where long endurance combined with intensive usage causes gradual but unnoticed deterioration, often leading to unexpected disasters, such the as the Columbia Shuttle breakdown. The recent advent of ‘smart’ or ‘intelligent’ materials and structures concept and technologies has ushered a new avenue for the development SHM/ NDE systems. Smart piezoelectric-ceramic (PZT) materials, for example, have emerged as high frequency mechatronic impedance transducers (MITs) for SHM and NDE. As MIT, the PZT patches are not only robust, cost-effective, and show high damage sensitivity, but are also ideal for already constructed infrastructures and currently operational machinery because they only require non-intrusive external installation. The piezoimpedance transducers, acting as collocated actuators and sensors, employ ultrasonic vibrations (typically in 30-400 kHz range) to read the characteristic ‘signature’ of the structure, which contains vital information governing the phenomenological nature of the structure, and can be analysed to predict the onset of structural damages. High operational frequency ensures a sensitivity high enough to capture any damage at the incipient stage itself, much before it acquires detectable macroscopic dimensions. This new SHM/ NDE technique is popularly called the electro-mechanical impedance (EMI) technique in the literature. In spite of enormous potential due to its low-cost and high sensitivity, the EMI technique is still in the infancy stage as far as damage severity assessment or access to

x

the inherent damage mechanism is concerned. Changes in the diagnostic signature and the nature, severity and type of damage are not well correlated. Till date, all the existing approaches are non-parametric and statistical in nature and are able to utilize only the real part of signature. The information concerning damage carried by the imaginary part is therefore lost. Besides, no attempt has been made to extract the mechanical impedance of the interrogated structure from the electro-mechanical signatures, partly due to the non-existence of suitable impedance models. This research has focused on utilizing the underlying PZT-structure electromechanical interaction for an impedance based structural identification and SHM/ NDE using the EMI technique. A new concept of active signatures has been introduced to extract the damage-sensitive information from the raw signatures and a new PZTstructure interaction model has been developed based on the concept of ‘effective impedance’. The proposed formulations can be conveniently employed to extract the hidden damage sensitive structural parameters for any ‘unknown’ structure by means of surface-bonded PZT patches. A new experimental technique has been developed to ‘update’ the model of the PZT patch, so as to enable it extract the host structure’s impedance information much more accurately. A unified impedance approach has been developed to ‘identify’ the host structure from the extracted mechanical impedance spectra and carry out quantitative and parametric damage prediction. This has made possible greater information about the nature of damage in terms of stiffness, damping and mass changes, which was so far lacking. As proof-of-concept, the new diagnostic approach has been applied on representative aerospace and civil structural components. Further, in order to rigorously calibrate the piezo-impedance transducers for damage assessment, comprehensive tests were carried out on concrete specimens. An empirical fuzzy probabilistic damage model has been proposed for predicting damage level in concrete using piezo-impedance transducers. In addition, a new experimental technique has been developed to predict in situ concrete strength non-destructively using the EMI technique, thereby imparting it further edge over the contemporary NDE techniques. Finally, the intermediate bond layer between the PZT patch and the xi

structure has been integrated into the impedance models, thereby enabling a rigorous analysis of the shear lag effect associated with the bond layer. It is hoped that this research will make significant contributions in the field of SHM and NDE and will enable the maintenance engineers to make much more timely and accurate prediction of damages in any structural component.

xii

LIST OF TABLES

Page Table 2.1

Sensitivities of common local NDE techniques

21

(Boller, 2003). Table 3.1

Key parameters of PZT patch .

66

Table 3.2

Key material properties of structure.

82

Table 4.1

Key properties of PZT patches (PI Ceramic, 2003).

100

Table 4.2

Typical base motions and time-histories to which test

101

frame was subjected. Table 5.1

Physical properties of Al 6061-T6.

117

Table 5.2

Details of modes of vibration of test structure.

123

Table 5.3

Mechanical impedance of combinations of spring, mass

139

and damper. Table 6.1

Averaged parameters of test sample of PZT patches.

165

Table 6.2

Common probability distributions.

175

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LIST OF FIGURES Page Fig. 1.1

Accident involving Aloha Airlines (LAMSS, 2003)….………….…2

Fig. 1.2

Accident involving American Airlines Airbus A300-600 ………….2 (LAMSS, 2003).

Fig. 1.3

Image of Columbia about a minute before it broke apart……..……3 (AWST, 2003).

Fig. 1.4

Shuttle left wing cutaway diagrams (NASA, 2003)………….……..4

Fig. 1.5

Damage identified on RCC panel 8 in Discovery after a mission…..5 in 2000 (CAIB, 2003).

Fig. 1.6

The Mianus river bridge collapse (USDT, 2003)…………………...6

Fig. 1.7

Illustrating the components and operation of typical SHM system…7 (Boller, 2002).

Fig. 2.1

Classification of smart structures (Rogers, 1990)…..…………..….24

Fig. 2.2

Common smart materials and associated stimulus-response………25

Fig. 2.3

Centro-symmetric crystals: the act of stretching does not cause…..28 any dipole moment (µ = dipole moment).

Fig. 2.4

Noncentro-symmetric crystals: the act of stretching causes dipole..28 moment in the crystal (µ = dipole moment)

Fig. 2.5

A piezoelectric material sheet with conventional 1, 2 and 3 axes...30

Fig. 2.6

Strain vs electric field for PZT (piezoelectric) and………….……..32 PMN (electrostrictive).

Fig. 2.7

Polarization vs electric field for ferroelectric crystals………...…...33

Fig. 2.8

Modelling PZT-structure interaction…………………………...….37

Fig. 2.9

Conductance and susceptance plots of a PZT patch bonded to…....41 bottom flange of a steel beam.

Fig. 2.10

A typical commercially available PZT patch…………….………..45

xiv

Fig. 2.11

Modes of wave propagation associated with PZT patch…………..48 (Giurgiutiu and Rogers, 1997).

Fig. 3.1

Representation of harmonic force and velocity by rotating phasors.61

Fig. 3.2

Determination of mechanical impedance of a PZT patch...…..……62

Fig. 3.3

Variation of actuator impedance with frequency………....…..……65

Fig. 3.4

A PZT patch coupled to a spring-mass-damper system...………….66

Fig. 3.5

Signatures for SDOF-Case I, m = 2.0 kg, k = 1.974x107N/m, c = 125.7Ns/m……………………….…………………………..…68

Fig. 3.6

Signatures for SDOF-CaseII, m = 200 kg, k = 1.974x109N/m, c = 12566.4Ns/m…………………….……………………………..71

Fig. 3.7

Signatures for SDOF-CaseIII, m = 0.2 kg, k = 1.974x106N/m, c = 12.57Ns/m……………………….………………………..……73

Fig. 3.8

Signatures for SDOF-CaseIV, m = 2500 kg, k = 2.46x1010N/m, c = 3927Ns/m..……………………..…………………………...….74

Fig. 3.9

Signatures for caseV..………..…….…………………………...….76

Fig. 3.10

Signatures for SDOF-caseVI, m = 0.0002 kg, k = 197.4N/m, c = 0.01257Ns/m…………………….………………………….….78

Fig. 3.11

Appearance of large number of ‘false’ peaks.……………………...79

Fig. 3.12

A MDOF system considered for PZT-structure interaction………..81

Fig. 3.13

Graphical representation of Mode 48 (f = 162.46 kHz)…..………..82

Fig. 3.14

Signatures for MDOF system considered in Fig. 3.12…..……….…83

Fig. 3.15

Active-conductance and active-susceptance (modified…………….87 signatures after filtering out the PZT contribution).

Fig. 3.16

Active-susceptance plot for Case-II………………………………..87

Fig. 4.1

(a) A SDOF system under dynamic excitation……………….....….90 (b) Phasor representation of spring force (Fs), damping force (Fd) and inertial force (Fi)

Fig. 4.2

(a) Details of test frame………………………………..……………99 (b) Test frame just before applying loads

Fig. 4.3

Raw-signatures of PZT patch #2 at various damage states (1,2..6).102

Fig. 4.4

Damage prediction by patch #2…………………………….….….102 xv

Fig. 4.5

Raw-signatures of PZT patch #1 at various damage states (1,2..8).104

Fig. 4.6

Damage prediction by patch #1…………………………….….….104

Fig. 4.7

(a) Natural frequency of vibration of floor #2 beam at various…...105 damage states. (b) Evaluation of damage based on natural frequency, rawconductance and extracted mechanical impedance.

Fig. 5.1

Modelling of PZT-structure interaction by static approach………108

Fig. 5.2

Modelling PZT-structure 2D physical coupling by …………….109 impedance approach (Zhou et al., 1996).

Fig. 5.3

A PZT patch bonded to an ‘unknown’ host structure…………….112

Fig. 5.4

A square PZT patch under 2D interaction with host structure…...113

Fig. 5.5

Experimental set-up to verify effective impedance based new….117 electro-mechanical formulations.

Fig. 5.6

Finite element model of one-quarter of test structure……………119

Fig. 5.7

Examination of mode 24 to check adequacy of mesh size ………124 of 1mm.

Fig. 5.8

Comparison between experimental and theoretical signatures…..125

Fig. 5.9

Plots of quasi-static admittance functions of free PZT patches….127 to obtain electric permittivity and dielectric loss factor.

Fig. 5.10

Experimental and analytical plots of free PZT signatures…….…129

Fig. 5.11

Plots of free-PZT admittance signatures using an updated………131 PZT model.

Fig. 5.12

Comparison between experimental and theoretical signatures…..133 based on updated PZT model.

Fig. 5.13

(a) PZT effective impedance, based on idealised and updated…. 134 Models. (b) Error in extracted structural impedance in the absence of updated PZT model. (c) Relative magnitudes of structure and PZT impedances.

Fig. 5.14

Comparison between |Zeff|-1 obtained experimentally and………..137 numerically. xvi

Fig. 5.15

Impedance plots of basic structural elements- spring, damper……138 and mass.

Fig. 5.16

Mechanical impedance of aluminium block in 25-40 kHz………..140 frequency range.

Fig. 5.17

Mechanical impedance of aluminium block in 180-200 kHz…….142 frequency range. The equivalent system plots are obtained for system 11 (Table 5.3).

Fig. 5.18

Refinement of equivalent system by introduction of………….….142 additional spring K* and additional damper C*.

Fig. 5.19

Mechanical impedance of aluminium block in 180-200 kHz…….143 frequency range for refined equivalent system ( shown in Fig. 5.18).

Fig. 5.20

Levels of damage induced on test specimen (aluminium block)…144

Fig. 5.21

Effect of damage on extracted mechanical impedance…………...145 in 25-40 kHz range.

Fig. 5.22

Effect of damage on equivalent system parameters………………145 in 25-40kHz range

Fig. 5.23

Effect of damage on extracted mechanical impedance…………...147 in 180-200kHz range.

Fig. 5.24

Plot of mechanical impedance of aluminium block in 180-200….148 for various damage states.

Fig. 5.25

Effect of damage on equivalent system parameters………………149 in 180-200kHz range

Fig. 5.26

Plot of residual specimen area versus equivalent spring constant..150

Fig. 5.27

Damage diagnosis of a prototype RC bridge using proposed…….151 methodology.

Fig. 5.28

Mechanical impedance of RC bridge in 120-140 kHz frequency..152 range. The equivalent system plots are obtained for a parallel spring damper combination.

Fig. 5.29

Effect of damage on equivalent system parameters of RC bridge..153

Fig. 6.1

(a) Determining natural frequency of specimen using sonometer..158 xvii

(b) Correlation between dynamic modulus and concrete strength. (Source: Malhotra, 1976) Fig. 6.2

(a) Determining velocity of sound in concrete using PUNDIT…...159 (b) Correlation between ultrasonic pulse velocity and strength. (Source: Malhotra, 1976)

Fig. 6.3

Admittance spectra for free and fully clamped PZT patches…….160

Fig. 6.4

(a) Optical fibre pieces laid on concrete surface before applying..161 adhesive. (b) Bonded PZT patch.

Fig. 6.5

Effect of concrete strength on first resonant frequency of PZT….162 patch.

Fig. 6.6

Correlation between concrete strength and first resonant………..163 frequency.

Fig. 6.7

Concrete cube to be ‘identified’ by piezo-impedance……………164 transducer.

Fig. 6.8

Equivalent system ‘identified’ by PZT patch………………….…165

Fig. 6.9

Impedance plots for concrete cube C43.…………..…….…….….166

Fig. 6.10

Experimental set-up for inducing damage on concrete cubes……167

Fig. 6.11

Load histories of four concrete cubes…..…………..………….…167

Fig. 6.12

Correlation between loss in secant modulus and loss in ..…….…168 equivalent spring stiffness with damage progression.

Fig. 6.13

Changes in equivalent damping and equivalent stiffness for….…169 concrete cube C43.

Fig. 6.14

Monitoring concrete curing using EMI technique………….……170

Fig. 6.15

Short-term effect of concrete curing on conductance signatures...171

Fig. 6.16

Long-term effect of concrete curing on conductance signatures...171

Fig. 6.17

Effect of concrete curing on equivalent spring stiffness……..….172

Fig. 6.18

Different types of membership functions for fuzzy sets…..…….177

Fig. 6.19

Effect of damage on equivalent spring stiffness………….……..179

Fig. 6.20

Theoretical and empirical probability density functions….…….180 near failure. xviii

Fig. 6.21

Fuzzy failure probabilities of concrete cubes at incipient………182 damage level and at failure stage

Fig. 6.22

Fuzzy failure probabilities of concrete cubes at various..………182 load levels.

Fig. 6.23

Typical stress-strain plot for PZT (Cheng and Reece, 2001)..….183

Fig. 6.24

Cubes after the test……………………………………..……….184

Fig. 7.1

A PZT patch bonded to a beam using adhesive bond layer….….186

Fig. 7.2

Deformation in bonding layer and PZT patch…..…………..….187

Fig. 7.3

Strain distribution across the length of PZT patch……………..190 for various values of Γ.

Fig. 7.4

Variation of effective length with shear lag factor…………..…191

Fig. 7.5

Distribution of piezoelectric and beam strains for various……..193 values of Γ.

Fig. 7.6

Modified impedance model of Xu and Liu (2002) including ….194 bond layer.

Fig. 7.7

Stresses acting on an infinitesimal PZT element…………...….201

Fig. 7.8

Theoretical normalized conductance…………...…………..….204

Fig. 7.9

Experimental normalized conductance for ts/tp = 0.417……….205 and ts/tp = 0.838.

Fig. 7.10

Theoretical normalized susceptance……………..………….….205

Fig. 7.11

Experimental normalized susceptance for ts/tp =0.417………....205 and ts/tp = 0.838.

Fig. 7.12

Analytical and experimental plots for ts/tp equal to 1.5……….. 206

Fig. 7.13

Influence of shear modulus of elasticity of bond layer..……….208

Fig. 7.14

Influence of bond layer thickness……………….…….………..209

Fig. 7.15

Influence of damping of bond layer…………………………….210

Fig. 7.16

Influence of Parameter peff ……….………………..…....……..211

Fig. 7.17

Influence of Parameter qeff…………………..………………………...212

Fig. 7.18

Influence of sensor length……….………………………….......213

Fig. 8.1

Test specimen for evaluating repeatability of ………………….217

xix

admittance signatures. Fig. 8.2

A set of conductance signatures of PZT patch #1spanning over…217 two months.

Fig. 8.3

A set of susceptance signatures of PZT patch #1spanning over….217 two months.

Fig. 8.4

Effect of humidity on signature………………………………..…219

Fig. 8.5

Effect of damage on signatures………………………………..…219

Fig. 8.6

Test specimen for evaluating signature multiplexing………....….220

Fig. 8.7

Experimental set-up consisting of impedance analyzer,….. ....….221 controller PC and multiplexer.

Fig. 8.8

Effect of damage on collective signature of 20 PZT patches….…222

xx

LIST OF SYMBOLS

A

Area

B

Raw susceptance

BA,

Active susceptance

BP

Passive susceptance

C, C1, C2

Correction factor(s) to update model of PZT

c

Damping constant

[C]

Damping matrix

D1, D2, D3

Electric displacement across surfaces normal to 1, 2, 3 axes respectively

[D]

Electric displacement vector

d31 (dik)

Piezoelectric strain coefficient of PZT patch corresponding to axes 3(i) and 1(k)

Di

Damage variable at ith frequency point

Dc

Critical value of damage variable

DU, DL

Upper and lower limits of damage variable in the fuzzy interval

E3 (Ei)

Electric field along axis 3 (i) of PZT patch

[E]

Electric field vector

f

Frequency f

Boundary traction (per unit length)

F

(Effective) Force

fm

Membership function of a fuzzy set



Empirical cumulative distribution function

G

Raw conductance

GA

Active conductance

GP

Passive conductance

xxi

Gs

Shear modulus of elasticity of bond layer

h

Thickness of PZT patch

I

Complex Electric current

j

−1

k

Spring constant

[K]

Stiffness matrix

l

Half-length of PZT patch

m

Mass

M

Bending moment

Mmn

Electrostriction coefficient

[M]

Mass matrix

po

Perimeter of PZT patch in undeformed condition

p(D)

Probability density function of damage variable D

S1 (Si)

Mechanical strain along axis 1 (i)

E skm

An element of the elastic compliance matrix at constant electric field

T1 (Ti)

Mechanical stress along axis 1 (i) of PZT patch

T

Complex tangent function

tp

Thickness of PZT patch

ts

Thickness of bond layer

u

Displacement

V

Complex electric voltage

wp

Width of PZT patch

x

Real part of the mechanical impedance of structure

xa

Real part of the mechanical impedance of PZT patch

y

Imaginary part of the mechanical impedance of structure

ya

Imaginary part of the mechanical impedance of PZT patch

Y

Complex electro-mechanical admittance

YE

Complex Young’s modulus of elasticity at constant electric field

YA

Active component of complex admittance

xxii

YP

Passive components of complex admittance

Z

Complex mechanical impedance of structure (Z = x + yj)

Za

Complex Mechanical impedance of PZT patch (Za = xa + yaj)

T ε 33

Complex permitivity of PZT patch along axis 3 at constant stress

ω

Angular frequency (rad/s)

δ

Dielectric loss factor

ρ

Material density

η

Mechanical loss factor of PZT patch

η′

Mechanical loss factor of adhesive

τ

Shear stress

α

Mass damping factor

β

Stiffness damping factor

φ

Phase lag

Γ

Shear lag parameter

ξ

Strain lag ratio

ξd

Damping ratio

ν

Poisson’s ratio

Λ

Free piezoelectric strain (= E3d31)

ψ

Product of beam to PZT modulus and thickness ratios Structural mechanical impedance correction factor

κ

Wave number

γ

Shear strain

µ

Mean

σ

Standard deviation

φ

Phase lag {=tan-1(y/x)}

Gio , Gi1

Pre-damage and post-damage raw conductance respectively for ith frequency point

G o , G1

Mean value of pre-damage and post-damage raw conductance

xxiii

Subscripts A

Active

eff

Effective

o

Amplitude of a quantity

eq

Equivalent; Equilibrium

f, free

Free

i

Imaginary

P

Passive

p

Relevant to PZT patch

qs

Quasi-static

r

Real

res

Resultant

s

Under static conditions

1,2,3 or x,y,z Coordinate axes Superscripts T

Quantity at constant stress

E

Quantity at constant electric field

xxiv

LIST OF ACRONYMS

ACS

Active Conductance Signature

ASS

Active Susceptance Signature

ASTM

American Society for Testing and Materials

ATM

Adaptive Template Matching

AWST

Aviation Week and Space Technology

CC

Correlation Coefficient

CAIB

Columbia Accident Investigation Board

EC

Eddy Currents

EDP

Effective Drive Point

ELODS

Equivalent Level of Degradation System

EMI

Electro-Mechanical Impedance

ER

Electro-Rheological (Fluid)

FFP

Fuzzy Failure Probability

FEM

Finite Element Method

IDT

Inter Digital Transducers

LAMSS

Laboratory for Active Materials and Smart Structures

LCR

Inductance (L) Capacitance (C) and Resistor (R) (Circuit)

MAPD

Mean Absolute Percent Deviation

MDOF

Multiple Degree of Freedom (System)

MEMS

Micro-Electro Mechanical Systems

MIT

Mechatronic Impedance Transducer

NASA

National Astronautics and Space Administration

NDE

Non-Destructive Evaluation

xxv

NDT

Non-Destructive Testing

PC

Personal Computer

PCS

Passive Conductance Signature

PSS

Passive Susceptance Signature

PUNDIT

Portable Ultrasonic Non-Destructive Digital Indicating Tester

PVDF

Polyvinvylidene Fluoride

PZT

Lead (Pb) Zirconate Titanate

RC

Reinforced Concrete

RCC

Reinforced Carbon Carbon

RCS

Raw Conductance Signature

RD

Relative Deviation

RMS

Root Mean Square

RMSD

Root Mean Square Deviation

SAC

Signature Assurance Criteria

RSS

Raw Susceptance Signature

SDOF

Single Degree of Freedom (System)

SHM

Structural Health Monitoring

SMA

Shape Memory Alloy

USDT

United States Department of Transport

UTM

Universal Testing Machine

WCC

Waveform Chain Code (Technique)

xxvi

Chapter 1: Introduction

Chapter 1 INTRODUCTION

1.1 STRUCTURAL DAMAGES AND FAILURES Structures are assemblies of load carrying members capable of safely transferring the superimposed loads to the foundations. They are constructed (e.g. buildings, bridges, dams, transmission towers, etc.) or manufactured (e.g. machines, trains, ships, aircraft, etc.) to serve specific functions during their design lives. Each structure forms an integral component of civil, mechanical or aerospace systems. In order to serve their designated functions, the structures must satisfy both strength and serviceability criteria throughout their stipulated design lives. However, with the passage of time, some amount of deterioration and damages are bound to occur, due to a variety of factors; such as environmental degradation, fatigue, excessive loads, natural calamities or simply due to long endurance combined with intensive usage. Even the best designed structures, constructed from advanced high strength materials, are not 100% immune from damage. According to Yao (1985), ‘damage’ is defined as a deficiency or deterioration in the strength of a structure, caused by external loads, environmental conditions, or human errors. Physically, a damage may be visible as a crack, delamination, debonding, reduction in thickness/ cross-section, or exfoliation. The term ‘damage’ carries much different meaning from the term ‘failure’. In most general terms, ‘failure’ refers to any action leading to an inability on the part of a structure or machine to function in the intended manner (Ugural and Fenster, 1995). Fracture, permanent deformation, buckling and even excessive linear elastic deformation may be regarded as modes of failure. Failure results when a particular type of damage exceeds its threshold value, thereby impairing the safety and/ or the functioning of the structure seriously.

1

Chapter 1: Introduction

1.2 AN OVERVIEW OF RECENT STRUCTURAL FAILURES On April 28, 1988, Boeing 737 of Aloha Airlines met with a severe mid-flight accident in which entire fuselage panels were ripped apart from the main body, as shown in Fig. 1.1. Fortunately, the passengers remained held against air pressure by their safety belts. The underlying cause of this accident was later found to be the appearance of multi-site cracks in the skin joints, which led to the unzipping of large portions of the fuselage (LAMSS, 2003). However, these cracks could not be detected during the routine pre-flight inspections.

Fig. 1.1 Accident involving Aloha Airlines (LAMSS, 2003). Similarly, on November 12, 2001, the mid air crashing of the American Airlines Airbus A 300-600 (Flight 587) was one of the deadliest accidents in the American aviation history. From preliminary investigations, it was found that the tail (vertical stabilizer) broke off during take off, right from the root of the connection to the main body, as shown in Fig. 1.2(a). The investigators found the

Fig. 1.2 Accident involving American Airlines Airbus A300-600 (LAMSS, 2003). (a) Breakaway tail component. (b) Close-up view of breakaway composite joint. 2

Chapter 1: Introduction

existence of an undetected damage in the tail, caused by previous mid air events involving severe loading, which had resulted in the weakening of the composite joint. Surprisingly, the conventional NDE techniques, including visual inspections, had failed to detect the presence of the previous damages. This incipient damage was further aggravated by the aerodynamic loads and the tail finally broke apart, as shown in Fig. 1.2(b). Another recent aerospace disaster, which attracted worldwide attention, was the crashing of the NASA space shuttle Columbia, on February 1, 2003, during its re-entry into earth’s atmosphere. Fig. 1.3 shows the US Air Force image of Columbia taken about a minute before it broke apart (AWST, 2003). This image shows that the left inboard wing was jagged near the location where it begins to intersect the fuselage. This location houses reinforced carbon-carbon (RCC) composites, which constitute critical structural and thermal protection components of any shuttle. The right wing, on the other hand, can be seen to be smooth along its entire length. The ragged edge on the left leading wing indicated that either a structural breach occurred there, or that a small portion of the leading edge fell off, allowing the 2000oF re-entry heat to erode the additional structure there. Comprehensive investigation into the disaster was carried out by Columbia Accident Investigation Board (CAIB, 2003) and the findings were made public on August 26, 2003. The CAIB report confirmed that the physical cause of the loss of Columbia was a breach in its thermal protection system on the leading edge of the left wing. This breach was initiated by a piece of insulating foam, separated from the left bipod ramp, that struck the left wing in the vicinity of the lower half of RCC Right wing

Left wing Wing distortion Flow distortion Fig. 1.3 Image of Columbia about a minute before it broke apart. (AWST, 2003). 3

Chapter 1: Introduction

panel 8, 81.9 seconds after the launch (Chapter 3, page 49 of CAIB report). As shown in Fig. 1.4, each wing’s leading edge consists of 22 RCC panels. RCC is a hard structural material characterized by high strength over extreme temperatures ranging from –250oF to 3000oF. During re-entry, this breach allowed the superheated air to penetrate into and melt the aluminium structure (melting point: 1200oF) of the left wing, thereby weakening it, until the aerodynamic forces caused failure of the wing and total break-up of the orbiter. Ironically, although the event of foam striking the left wing had caught the attention of the ground team, the space shuttle was not equipped with any NDE system on-board to assess the level of damage caused. Conclusions of ground team based on computational analysis that the impact was not so severe proved wrong. Following additional findings of CAIB are worth taking note of: (i)

The RCC is vulnerable to damage due to oxidation if oxygen penetrates the microscopic fissures of the silicon-carbide protective coating. The loss of mass due to oxidation reduces the load capacity of the structure. Currently, the mass loss cannot be directly measured (Finding F 3.3-4, page 58 CAIB report). This weakening can eventually lead to significant deterioration, for example, as shown in Fig. 1.5 for panel 8 of space shuttle Discovery, after a mission in January 2000.

(1-10)

(16-17)

RCC Panels (1-10 and 16-17)

(b)

(a) Fig. 1.4 Shuttle left wing cutaway diagrams (NASA, 2003).

(a) Complete view of spaceship Columbia. (b) Left-wing showing RCC panels.

4

Chapter 1: Introduction

Damage

Fig. 1.5 Damage identified on RCC panel 8 in Discovery after a mission in 2000 (CAIB, 2003).

(ii)

During manufacturing, the integrity of production composites used in the RCC system is checked by physical tap, ultrasonic, radiographic, eddy currents, visual tests and also by limited number of destructive tests. However, no rigorous test plan is followed after assembly in the shuttle. Post flight inspection is primarily visual and tactile (poking with finger). The board noted that the current inspection techniques are not adequate to assess structural integrity of RCC, the supporting structure and the attached hardware. (Findings F 3.2-2 and F 3.2-3, page 58 of CAIB report).

(iii)

There are no qualified NDE techniques to determine the characteristics of the foam in the as-installed condition before flight (finding F 3.2-2, page 55 of CAIB report). In view of the above findings, the CAIB recommended NASA to develop

and implement a comprehensive inspection plan to determine the structural integrity of all RCC system components, taking advantage of the “advanced NDE technology” (recommendation R 3.3.1, page 59). Military aircraft also suffer similar mid flight accidents due to damages. In the past 10 years, the Indian Air force has lost more than 100 MiG fighter aircraft with over 80 pilots dead. This amounts to billions of dollars worth of equipment and human resources. During the past 3 years alone, 52 such fighter planes have been lost (based on Defence Minister’s statement in parliament on 25 July 2003). No

5

Chapter 1: Introduction

Fig. 1.6 The Mianus River Bridge collapse (USDT, 2003).

sophisticated SHM system is presently in place to monitor the planes during flight and prevent loss of the aircraft and the pilot. Besides the above aerospace failures, numerous instances of civil-structural failures have occurred. Many buildings and bridges constructed during the economical boom of the eighties are now showing the problems of ageing, for which the maintenance engineers are not logistically prepared. The Mianus river bridge collapse (see Fig. 1.6), in Greenwich, during June 1983, resulted from a hangar pin connection failure due to excessive corrosion accumulation (USDT, 2003). This failure emphasized that special inspection techniques are necessary for civil-structures also, since visual inspection is likely to miss out many critical incipient damages. There are a total of 127,154 railway bridges in India, taking freight traffic of over 550 million tonnes and passenger load of more than 500 billion passenger kilometers every year. Out of these bridges, 56,169 (44.17%) are more than 80 years old and hence prone to disaster any time (Hindustan Times, 20 July 2003). Hence, a rigorous inspection and test plan is necessary to ensure passenger safety and prevent unexpected losses. 1.3 STRUCTURAL HEALTH MONITORING The brief overview of recent catastrophic accidents in the preceding section has clearly shown the destructive power of any structural damage when it starts to grow from the incipient level. Hence, even a minor damage of incipient nature should not

6

Chapter 1: Introduction

be ignored since it carries the potential to grow and cause failure, either leading to wide scale loss of life and property or halting some revenue earning activity or both. It is this possibility which calls for a rigorous inspection of the structures on a regular basis or in other words, 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 (Kessler et al., 2002). The idea of SHM is pictorially illustrated in Fig. 1.7 (Boller, 2002). Such a system typically 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. Impator

Structure

Actuator

Sensor

Amplifier

Filter

Signal Analyzer

Signal Generator

Fig. 1.7 Illustrating the components and operation of typical SHM system (Boller, 2002).

1.4 REQUIREMENTS FOR ANY SHM SYSTEM In the aviation sector, the aircraft are designed for specific number of flight hours based on a specified usage under predefined load spectrum. However, often the airline or the air force continues to fly the aircraft much beyond their initial design life. Presently, the average age of the US air force fleet stands at 22 years. It is expected to increase to 25 years in 2007 and 30 years in 2020 (Boller, 2002). Since the US Air Force cannot boost its purchases by 170 aircraft per year, this problem is expected to be more severe in the long run. The same holds true for the

7

Chapter 1: Introduction

civil aircraft as well. In general, aircraft demand large amount of inspection at well defined intervals, ranging from daily checks to over 120 months, especially when they are highly loaded or when they reach older days. Till date, visual inspection supplemented with magnifying glass, tap test and some primitive non-destructive tests (dye-penetrant, magnetic particle etc.) has been the most prevalent method of pre-emptive structural inspections for the aircraft. Usually, trained personnel conduct these inspections, and the procedure is not only very tedious and time consuming, but also characterised by high implementation costs. It is estimated that about 27% of an aircraft’s life cycle cost is spent on inspections and repair, excluding the opportunity cost associated with the time it remains grounded (Kessler et al., 2002). This is not due to a large effort in detecting damage via non-destructive testing (NDT) equipment, but owing to the fact that many critical components such as the main lading gear fitting need to be dismantled before inspections and reassembled afterwards. Rather, this process (dismantling and reassembling) eats up to 45% of the entire inspection time (Boller, 2002). Hence, an unobtrusive automated inspection mechanism to detect the onset of damages in such inaccessible components can significantly enhance flight safety besides reducing the operating costs. Similarly, in civil-structures, often the critical parts are not be readily accessible and demand removal of the existing finishes (such as false ceilings), which makes the inspection process extremely laborious as well as costly. Most of the existing non-destructive evaluation (NDE) techniques (such as ultrasonic, penetrant dye testing, acoustic emission etc.) demand physically moving a probe, which proves impractical for the large-sized civil structures. These considerations call for a means of SHM that should avoid the dismantling and reassembling process or removal of the finishes and should also avoid physically moving heavy equipment. Such a system can achieve a significant reduction in the inspection time, effort and cost. The need to develop this kind of SHM system has recently attracted a large number of academic and industrial researchers from various disciplines. The ultimate goal of all SHM related research is to enable systems and structures monitor their own integrity while in operation and throughout their design lives.

8

Chapter 1: Introduction

Such system should preferably be real time and online. By real-time, it is implied that the level of responsiveness of such a system should be immediate or quick enough to enable appropriate remedial action or evacuation. By on-line, it is implied that the alerting system must use user friendly on-screen imaging and audible alarms. The application areas for SHM techniques are aerospace systems, mechanical and chemical pressure vessels, nuclear power plants, dams, bridges and buildings. In general, adoption of automated SHM is highly justified in the case of components for which the loads are less predictable and maintenance is restricted and costly. It may be unwarranted for low-cost components or if the loading and component’s behaviour are well understood and do not show significant variation. Although SHM has been shown feasible by numerous researchers, it has still not developed to the stage of being generally recognized as an element of the overall engineering system. The main reasons for this, according to Boller (2002) are: (i)

Benefits resulting from such system have not been carefully quantified.

(ii)

This is still not statutory requirement.

(iii)

Validation and certification needs to be done on a broader basis.

(iv)

Rapid emergence of new technologies and obsolescence of the old ones, leading to confusion in general. In general, SHM can enable taking greater advantage of structural material

potential, thereby saving natural as well as financial resources. 1.5 SHM BY ELECTRO-MECHANICAL IMPEDANCE (EMI) TECHNIQUE The recent developments in the area of smart materials and systems have ushered new openings for SHM and NDE. Smart materials, such as the piezoceramics, the shape memory alloys and the fibre-optic materials can facilitate the development of non-obtrusive miniaturized systems with higher resolution, faster response and far greater reliability than the conventional NDE techniques. Especially, the so-called ‘active’ smart materials possess immense capabilities of damage diagnosis because of their inherent stimulus-response and energy

9

Chapter 1: Introduction

transduction capabilities. These materials can be easily embedded or bonded unobtrusively on locations inaccessible for physical inspection. Hence, they meet the requirements outlined in the previous section for any viable SHM system. Among the so many smart materials available today, the piezoelectric-ceramic (PZT) materials have emerged as high frequency mechatronic impedance transducers (MITs) for SHM during the last nine years (Sun et al., 1995; Ayres et al., 1998; Soh et al., 2000; Park, 2000; Bhalla, 2001). In this application, a PZT patch is bonded to the structure to be monitored and its electro-mechanical conductance signature across a high frequency band serves as a diagnostic-signature of the structure. The technique is popularly called as the electro-mechanical impedance (EMI) technique. The EMI technique has been shown to be extremely sensitive to incipient damages, is practically immune to mechanical noise and demands a low implementation cost (Park et al., 2000a). The PZT patches can be easily bonded to inaccessible locations of structures and aircraft and can be interrogated as and when required, without necessitating the structures to be placed out of service or any dismantling/ re-assembling of the critical components. All these features definitely give an edge to the EMI technique over other existing passive sensor systems. However, the EMI technique is presently in the developmental stage as far as understanding the underlying damage mechanism or quantitative damage prediction are concerned. The changes in the electro-mechanical signatures are not well correlated with the changes in the underlying structural parameters. Till date, all the methods utilize raw signatures alone and make use of statistical indicators to quantify damage, which is rather a crude way of analysis. Hence, no structural parameter based damage quantification and damage severity prediction approach is presently available. This research was carried out with the objective of upgrading the EMI technique from its present state-of-the-art and expanding its NDE capabilities. The following sections highlight the objectives and contributions to the EMI technique by this research.

10

Chapter 1: Introduction

1.6 RESEARCH OBJECTIVES The primary objective of this research was to investigate and suitably model the key electro-mechanical interaction between the PZT transducer, the intermediate bonding layer and the host structure in PZT-based smart systems. This was pursued to enable an impedance-based structural identification and extraction of damage sensitive structural parameters for any ‘unknown’ system from the interrogation of the bonded PZT patch alone, without warranting any information a priori. These parameters are expected to govern the phenomenological nature and behavior of the structure. Hence, this process is expected to enable a more rigorous and quantitative evaluation of structural damages, besides providing a greater insight into the underlying damage mechanism. Further, this research aimed at rigorously calibrating the impedance parameters with damage and extending the technique for more meaningful applications such as in situ material strength assessment. 1.7 RESEARCH ORIGINALITY AND CONTRIBUTIONS This research programme aimed to expand the present capabilities of the EMI technique for experimental structural identification as well as NDE/ SHM. This research has attempted to balance theoretical developments with practical applications in order to maximize the potential benefits of the EMI technique. The original contributions of this research can be summarized as follows. (i)

A new concept of active-signature has been introduced to facilitate the extraction of damage sensitive signature component using signature decomposition.

(ii)

A new PZT-structure interaction model has been developed based on the concept of ‘effective impedance’. The new impedance formulations can be conveniently employed to extract the 2D mechanical impedance of any ‘unknown’ structure from the admittance signatures of a surface-bonded PZT

patch.

The

hidden

structural

parameters

governing

the

phenomenological nature of the structure can thus be identified by this process.

11

Chapter 1: Introduction

(iii)

A new experimental technique has been developed to ‘update’ the model of the PZT patch to enable it extract the impedance information of the host structure much more accurately. The new impedance formulations are employed in conjunction with the ‘updated’ PZT model to ‘identify’ the host structure and to carry out a parametric damage assessment, thereby revealing more information about the associated damage mechanism. Many proof-of-concept applications of the proposed methodology, ranging from precision machine and aerospace components to civil-structures, are presented.

(iv)

An empirical fuzzy probabilistic damage model has been proposed to calibrate the identified damage-sensitive structural parameters with damage progression for concrete. Besides, a new experimental technique has been developed to predict in situ concrete strength non-destructively.

(v)

Inclusion and rigorous analysis of the adhesive bond layer (between the PZT and the host structure) into impedance formulations and its implications on the accuracy of structural identification have been rigorously dealt with.

(vi)

Practical issues in the widespread application of the EMI technique, such as signature repeatability, sensor protection and sensor multiplexing have been duly addressed. The findings of the present research work have been published in many

international refereed journals and conferences, as detailed on page 230. 1.8 THESIS ORGANISATION This thesis consists of a total of nine chapters including this introductory chapter. Chapter 2 presents a detailed review of state-of-the art in SHM, introduction to the concept of smart systems and materials, description of the EMI technique and the current challenges facing the effective implementation of the technique on real-life structures. Chapter 3 deals with the important issues of structure-transducer electro-mechanical interaction, which is key to effective implementation of the technique for structural identification as well as NDE/ SHM. It also provides a rigorous mathematical analysis of the coupling between the PZT

12

Chapter 1: Introduction

patch and the host structure and motivations for signature decomposition. Significant deductions are made from this interaction and utilized in the subsequent chapters. Chapter 4 presents a mathematical analysis to extract the real and imaginary parts of the structural impedance of skeletal structures from the measured admittance signatures. Based on these parameters, a new methodology is developed for parametric quantification of the damage. Proof-of-concept application of the methodology on a model RC frame is presented. Chapter 5 presents the theoretical derivation, experimental verification and NDE applications of new generalized impedance formulations based on the concept of ‘effective impedance’. Chapter 6 presents the results from comprehensive tests conducted on concrete cubes to calibrate the extracted structural parameters with damage severity. Chapter 7 deals with modelling the behaviour of interfacial bond layer and its implications on the admittance signatures. Chapter 8 deals with key practical issues governing the application of the EMI technique. Finally, conclusions and recommendations are presented in Chapter 9, which is followed by a list of author’s publications, a comprehensive list of references, and appendices.

13

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

Chapter 2 ELECTRO-MECHANICAL IMPEDANCE TECHNIQUE FOR SHM AND NDE

2.1 STATE-OF-THE ART IN SHM/ NDE The prime motivations behind the ongoing research on SHM and NDE were elaborately covered in Chapter 1. This chapter primarily deals with a critical review of the various available SHM/ NDE techniques with regard to the EMI technique. For any critical structure under service, it is very important to monitor (a) load spectrum; and/ or (b) occurrence of damages. Whereas monitoring the load spectrum and the corresponding deflections/ strains helps in validating key design considerations, monitoring the occurrence of damages is key to ensure safety by preventing catastrophic failures. This thesis is concerned with part (b) only, by means of the EMI technique. In a broad sense, the SHM/ NDE methodologies 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 (displacements, 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 the corresponding natural frequencies of the structure, which, when compared with the corresponding data for the healthy state, yield information pertaining to the locations and the severity of the damages. In this

14

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

connection, the impulse excitation technique is much more expedient than harmonic excitation (which is however much more accurate) and hence preferred for quick estimates (Giurgiutiu and Zagrai, 2002). Application of this principle for damage detection can be found as early as in the 1970’s (e.g. Adams et al., 1978). Subsequently, this concept was employed for structural system identification, which is to establish a mathematical model of the structure from the experimental input-output data (e.g. Yao, 1985; Oreta and Tanabe, 1994; Loh and Tou, 1995). It may be mentioned that many of these techniques consist of ‘updating’ a numerical model of the structure from the test measurements. In the 1990’s, with the development of improved sensors, testing hardware and data acquisition and processing techniques, many researchers developed ‘quick’ SHM algorithms (mainly for bridge type structures), such as the change in curvature mode shapes method (Pandey et al., 1991), the change in stiffness method (Zimmerman and Kaouk, 1994), the change in flexibility method (Pandey and Biswas, 1994) and the damage index method (Stubbs and Kim, 1994). A comparative evaluation of these algorithms on an actual bridge structure, by Farrar and Jauregui (1998), showed the damage index method to be the most sensitive among these methods. Many related publications can be found, reporting the use of improved algorithms, modern wireless technology and high speed data processing (Singhal and Kiremidjian, 1996; Skjaerbaek et al., 1998; Pines and Lovell, 1998; Aktan et al., 1998, 2000; Lynch et al., 2003a). However, in spite of rapid progress in the hardware and the software technologies, the basic principle remains the same, which is to identify changes in the modal and the structural parameters (or their derivatives) resulting from damages. The main limitations of the global dynamic techniques can be summarized as follows (i)

These techniques typically rely on the first few mode shapes and the corresponding natural frequencies of structures, which, being global in nature, are not sensitive enough to be altered by localized incipient damages. For example, Pandey and Biswas (1994) reported that a 50% reduction in the Young’s modulus of elasticity, over the central 3% length of a 2.44m long

15

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

beam (used by the investigators as an example), only resulted in about 3% reduction in the observed first natural frequency. Changes of such small order of magnitude may not be considered as reliable damage indicators in real-life structures, in light of experimental errors of about the same order of magnitude. The global parameters (on which these techniques heavily rely) do not alter significantly due to local damages. In physical terms, the reason for this is attributed to the fact that the long wavelength stress waves associated with the low-frequency modes may cross a local damage (such as a crack), without sensing it. It is for this reason that Farrar and Jauregui (1998) found that the global dynamic techniques failed to identify damage locations for less severe damage scenarios in their experiments. It could be possible that a damage, just large enough to be detected by global dynamic techniques, may already be critical for the structure in question. (ii)

These techniques demand expensive hardware and sensors, such as inertial shakers, self-conditioning accelerometers and laser velocity meters. Typically, the cost of a single accelerometer is of the order of US$ 1000 (Lynch et al., 2003b). For a large structure, the overall cost of such sensor systems could easily run into millions of dollars. For example, the Tsing Ma suspension bridge in Hong Kong was instrumented with only 350 sensors in 1997 with a total cost of over US$ 8 million.

(iii)

A major limitation of these techniques is the interference caused by the ambient mechanical noise, besides the electrical and the electromagnetic noise associated with the measurement systems. Due to low frequency, the techniques are highly susceptible to ambient noise, which also happens to be in the low frequency range, typically less than 100Hz.

(iv)

For small miniature structural components (such as precision machinery or computer parts), the sensors involved in these techniques prove not only bulky, but also likely to interfere with structural dynamics due to their own mass and stiffness. Laser vibrometers are suitable for small structures, but are highly expensive and need to scan the entire structure for measuring mode shapes, which proves very tedious (Giurgiutiu and Zagrai, 2002).

16

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

(v)

The pre-requisite of a high fidelity ‘model’ of the test structure restricts the application

of

the

methods

to

relatively

simple

geometries

and

configurations only. Because evaluation of stiffness and damping at the supports (which are often rusted during service), is extremely difficult, reliable identification of a ‘model’ is quite difficult in practice. (vi)

Often, the performance of these techniques deteriorates in multiple damage scenarios (Wang et al., 1998). Contrary to these vibration-based global methods, many researchers have

proposed methods based on global static structural response, such as the static displacement response technique (Banan et al., 1994) and the static strain measurement technique (Sanayei and Saletnik, 1996). These techniques, like the dynamic techniques, essentially aim for structural system identification, but employ static data (such as displacements or strains) instead of vibration data. Although conceptually sound, the application of the static-response-based techniques on real life-sized structures is not practically feasible. For example, the static displacement technique (Banan et al., 1994) involves applying static forces at specific nodal points and measuring the corresponding displacements. Measurement of displacements on large structures is a mammoth task. As a first step, it warrants the establishment of a frame of reference, which, for contact measurement, could demand the construction of a secondary structure on an independent foundation (Sanayei and Saletnik, 1996). Besides, the application of large loads to 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. Many researchers have integrated the global static or dynamic methods with neural networks (e.g. Szewczyk and Hajela, 1994; Elkordy et al., 1994; Rhim and Lee, 1995; Jones et al., 1997; Nakamura et al., 1998; Barbosa et al., 2000; Hung and Cao, 2002). Neural networks offer several advantages, such as ability to generalise solutions (Flood and Kartam, 1994a, 1994b), not demanding a priori information concerning phenomenological nature of the structure (Masri et al., 1996), and can produce solutions within a very short time irrespective of the problem complexity.

17

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

Thus neural networks can reduce huge processing times involved in static and dynamic techniques. However, they are characterised by few limitations, such as lack of precision and limited ability to rationalise solutions. Above all, they lack rigorous theory to assist their design and training in a well-defined manner. In summary, the global techniques (static/ dynamic) provide only little information about local damages unless very large numbers of sensors are employed. They also require intensive computations to process the measurement data. Not much information about the specifics of location/ type of damage can be inferred without the use of high fidelity numerical models and intensive data processing. 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. The ultrasonic methods are based on elastic wave propagation and reflection within the material for non-destructive strength characterization and for identifying field inhomogeneities caused by damages. In these methods, a probe (a piezoelectric crystal) is employed to transmit high frequency waves into the material. These waves reflect back on encountering any crack, whose location is estimated from the time difference between the applied and the reflected waves. These techniques exhibit higher damage sensitivity as compared to the global techniques, due to the utilization of high frequency stress waves. Shah et al. (2000) reported a new ultrasonic wave based method for crack detection in concrete from one surface only. Popovics et al. (2000) similarly developed a new ultrasonic wave based method for layer thickness estimation and defect detection in concrete. In spite of high sensitivity, the ultrasonic methods share few limitations, such as: (i)

They typically employ large transducers and render the structure unavailable for service throughout the length of the test.

18

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

(ii)

The measurement data is collected in time domain that requires complex processing.

(iii)

Since ultrasonic waves cannot be induced at right angles to the surface, they cannot detect transverse surface cracks (Giurgiutiu and Rogers, 1997).

(iv)

These techniques do not lend themselves to autonomous use since experienced technicians are required to interpret the data. In acoustic emission method, another local method, elastic waves generated

by plastic deformations (such as at the tip of a newly developed crack), moving dislocations and disbonds are utilized for analysis and detection of structural defects. It requires stress or chemical activity to generate elastic waves and can be applied on the loaded structures also (Boller, 2002), thereby facilitating continuous surveillance. However, the main problem to damage identification by acoustic emission is posed by the existence of multiple travel paths from the source to the sensors. Also, contamination by electrical interference and mechanical ambient noise degrades the quality of the emission signals (Park et al., 2000a; Kawiecki, 2001). The eddy currents perform a steady state harmonic interrogation of structures for detecting surface cracks. A coil is employed to induce eddy currents in the component. The interrogated component, in-turn induces a current in the main coil and this induction current undergoes variations on the development of damage, which serves an indication of damage. The key advantage of the method is that it does not warrant any expensive hardware and is simple to apply. However, a major drawback of the technique is that its application is restricted to conductive materials only, since it relies on electric and magnetic fields. A more sophisticated version of the method is magneto-optic imaging, which combines eddy currents with magnetic field and optical technology to capture an image of the defects (Ramuhalli et al., 2002). In impact echo testing, a stress pulse is introduced into the interrogated component using an impact source. As the wave propagates through the structure, it is reflected by cracks and disbonds. The reflected waves are measured and analysed to yield the location of cracks or disbonds. Though the technique is very good for

19

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

detecting large voids and delaminations, it is insensitive to small sized cracks (Park et al., 2000a). In the magnetic field method, a liquid containing iron powder is applied on the component to be interrogated, subjected to magnetic field, and then observed under ultra-violet light. Cracks are detected by appearance of magnetic field lines along their positions. The main limitation of the method is that it is applicable on magnetic materials only. Also, the component must be dismounted and inspected inside a special cabin. Hence, the technique not very suitable for in situ application. In the penetrant dye test, a coloured liquid is brushed on to the surface of the component under inspection, allowed to penetrate into the cracks, and then washed off the surface. A quick drying suspension of chalk is thereafter applied, which acts as a developer and causes coloured lines to appear along the cracks. The main limitation of this method is that it can only be applied on accessible locations of structures since it warrants active human intervention. In X-ray method, the test structure is exposed to X-rays, which are then recaught on film, where the cracks are delineated as black lines. Although the method can detect moderate sized cracks, very small surface cracks (incipient damages) are difficult to be captured. A more recent version of the X-ray technique is computer tomography, whereby a cross-sectional image of solid objects can be obtained. Although originally used for medical diagnosis, the technique is recently finding its use for structural NDE also (e.g. Kuzelev et al., 1994). By this method, defects exhibiting different density and/ or contrast to the surroundings can be identified. Table 2.1 summarises the typical damage sensitivities of the local NDE methods described above. A common limitation of the local methods is that usually, probes, fixtures and other equipment need to be physically moved around the teststructure for recording data. Often, this not only prevents autonomous application of the technique, but may also demand the removal of finishes or covers such as false ceilings. Moving the probe everywhere being impractical, these techniques are often applied at very selected probable damage locations (often based on preliminary visual inspection or past experience), which is almost tantamount to knowing the damage location a priori. Generally, they cannot be applied while the component is under service, such as in the case of an aircraft during flight. Computer tomography

20

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

and X-ray techniques, due to their high equipment cost, are limited to very high performance components only (Boller, 2002). Table 2.1 Sensitivities of common local NDE techniques (Boller, 2002). Method Ultrasonic

Minimum detectable crack length 2mm

High probability detectable crack length (>95%) 5-6mm

Eddy currents (low-frequency) Eddy currents (high-frequency) X – Ray

2mm

4.5-8mm

2mm (surface) 0.5mm (bore holes) 4mm

2.5mm (surface) 1.0mm (bore holes) 10mm

Magnetic particle Dye penetrant

2mm 2mm

4mm surface 10mm surface

2.1.3

Remarks Dependent upon structure geometry and material Suitable for thickness 12mm

Advent of Smart Materials, Structures and Systems for SHM and NDE The SHM/ NDE methods described so far are the conventional monitoring

techniques. They typically rely on the measurement of stresses, strains, displacements, accelerations or other related physical responses to identify damages. The conventional sensors, which these techniques employ, are passive and bulky, and can only extract secondary information such as load and strain history, which may not lead to any direct information about damages (Giurgiutiu et al., 2000). However, the past few years have witnessed the emergence of ‘smart’ materials, systems and structures, which have shown new possibilities for SHM and NDE. Due to their inherent ‘smartness’, the smart materials work on fundamentally different principles and exhibit greater sensitivities to any changes in the environment. The next section briefly describes the principles and the recent developments in SHM/ NDE based on smart structures and materials.

21

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

2.2 SMART SYSTEMS/ STRUCTURES 2.2.1

Definition of 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 organised by the US Army Research Office in 1988, in which ‘sensors’, ‘actuators’, ‘control mechanism’ and ‘timely response’ were recognised as the four qualifying features of any smart system or structure (Rogers, 1988). In this workshop, following definition of smart systems/ structures was formally adopted (Ahmad, 1988). “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/ appropriate time, and reverting to its original state as soon as the stimulus is removed” According to Vardan and Vardan (2000), smart system refers to a device which can sense changes in its environment and can make an optimal response by changing its material properties, geometry, mechanical or electromagnetic response. Both the sensor and the actuator functions with their appropriate feedback must be properly integrated. It should also be noted that if the response is too slow or too fast, the system could lose its application or could be dangerous (Takagi, 1990). Previously, the words ‘intelligent’, ‘adaptive’ and ‘organic’ were also used to characterize smart systems and materials. For example, Crawley and de Luis (1987) defined ‘intelligent structures’ as the structures possessing highly distributed actuators, sensors and processing networks. Similarly, Professor H. H. Robertshaw preferred the term ‘organic’ (Rogers, 1988) which suggests similarity to biological processes. The human arm, for example, is like a variable stiffness actuator with a control law (intelligence). However, many participants at the US Army Research Office Workshop (e.g. Rogers et al., 1988) sought to differentiate the terms ‘intelligent’, ‘adaptive’ and ‘organic’ from the term ‘smart’ by highlighting their subtle differences with the term ‘smart’. The term ‘intelligence’, for example, is

22

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

associated with abstract thought and learning, and till date has not been implemented in any form of adaptive and sensing material or structure. However, still many researchers use the terms ‘smart’ and ‘intelligent’ almost interchangeably (e.g. In the U.S.-Japan Workshop: Takagi, 1990; Rogers, 1990), though ‘adaptive’ and ‘organic’ have become less popular. The idea of ‘smart’ or ‘intelligent’ structures has been adopted from nature, where all the living organisms possess stimulus-response capabilities (Rogers, 1990). The aim of the ongoing research in the field of smart systems/ structures is to enable such a structure or system mimic living organisms, which possess a system of distributed sensory neurons running all over the body, enabling the brain to monitor the condition of the various body parts. However, the smart systems are much inferior to the living beings since their level of intelligence is much primitive. In conjunction with smart or intelligent structures, Rogers (1990) defined following additional terms, which are meant to classify the smart structures further, based on the level of sophistication. The relationship between these structure types is clearly explained in Fig. 2.1. (a) Sensory Structures: These structures possess sensors that enable the determination or monitoring of system states/ characteristics. (b) Adaptive Structures: These structures possess actuators that enable the alteration of system states or characteristics in a controlled manner. (c) Controlled Structures: These result from the intersection of the sensory and the adaptive structures. These possess both sensors and actuators integrated in feedback architecture for the purpose of controlling the system states or characteristics. (d) Active Structures: These structures possess both sensors and actuators that are highly integrated into the structure and exhibit structural functionality in addition to control functionality. (e) Intelligent Structures: These structures are basically active structures possessing highly integrated control logic and electronics that provides the cognitive element of a distributed or hierarchic control architecture.

23

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

A

C E

B

D A: Sensory structures; B: Adaptive structures; C: Controlled structures; D: Active structures; E: Intelligent structures.

Fig. 2.1 Classification of smart structures (Rogers, 1990).

It may be noted that the sensor-actuator-controller combination can be realised either at the macroscopic (structure) level or microscopic (material) level. Accordingly, we have smart structures and materials respectively. The concept of smart materials is introduced in the following section. 2.2.2

Smart Materials Smart materials are new generation materials surpassing the conventional

structural and functional materials. These materials possess adaptive capabilities to external stimuli, such as loads or environment, with inherent intelligence. In the US Army Research Office Workshop, Rogers et al. (1988) defined smart materials as materials, which possess the ability to change their physical properties in a specific manner in response to specific stimulus input. The stimuli could be pressure, temperature, electric and magnetic fields, chemicals or nuclear radiation. The associated changeable physical properties could be shape, stiffness, viscosity or damping. This kind of ‘smartness’ is generally programmed by material composition, special processing, introduction of defects or by modifying the microstructure, so as to adapt to the various levels of stimuli in a controlled fashion. Like smart structures, the terms ‘smart’ and ‘intelligent’ are used interchangeably for smart materials. Takagi (1990) defined intelligent materials as the materials which respond to environmental changes at the most optimum conditions and manifest

24

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

their own functions according to the environment. The feedback functions within the material are combined with properties and functions of the materials. Optical fibres, piezo-electric polymers and ceramics, electro-rheological (ER) fluids, magneto-strictive materials and shape memory alloys (SMAs) are some of the smart materials. Fig. 2.2 shows the associated ‘stimulus’ and ‘response’ of common smart materials. Because of their special ability to respond to stimuli, they are finding numerous applications in the field of sensors and actuators. A very detailed description of smart materials is covered by Gandhi and Thompson (1992). 2.2.3

Active and Passive Smart Materials

Smart materials can be either active or passive. Fairweather (1998) defined active smart materials as those materials which possess the capacity to modify their geometric or material properties under the application of electric, thermal or magnetic fields, thereby acquiring an inherent capacity to transduce energy. Piezoelectric materials, SMAs, ER fluids and magneto-strictive materials are active smart materials. Being active, they can be used as force transducers and actuators. For example, the SMA has large recovery force, of the order of 700 MPa (105 psi) (Kumar, 1991), which can be utilized for actuation. Similarly piezoelectric materials, which convert electric energy into mechanical force, are also ‘active’. (1) Stress (2) Electric field

Piezoelectric Material

Heat

Shape Memory Alloy

Electric field

Electro-rheological Fluid

Temperature, pressure, mechanical strain

Magnetic field

Optical Fibre

Magneto-strictive material

(1) Electric Charge (2) Mechanical strain

Original Memorized Shape

Change in viscosity (Internal damping) Change in OptoElectronic signals

Mechanical Strain

Fig. 2.2 Common smart materials and associated stimulus-response.

25

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

The smart materials, which are not active, are called passive smart materials. Although smart, these lack the inherent capability to transduce energy. Fibre optic material is a good example of a passive smart material. Such materials can act as sensors but not as actuators or transducers. 2.2.4

Applications of Piezoelectric Materials

Since this thesis is primarily concerned with piezoelectric materials, some typical applications of these materials are briefly described here. Traditionally, piezoelectric materials have been well-known for their use in accelerometers, strain sensors (Sirohi and Chopra, 2000b), emitters and receptors of stress waves (Giurgiutiu et al., 2000; Boller, 2002), distributed vibration sensors (Choi and Chang, 1996; Kawiecki, 1998), actuators (Sirohi and Chopra, 2000a) and pressure transducers (Zhu, 2003). However, since the last decade, the piezoelectric materials, their derivative devices and structures have been increasingly employed in turbomachinery actuators, vibration dampers and active vibration control of stationary/ moving structures (e.g. helicopter blades, Chopra, 2000). They have been shown to be very promising in active structural control of lab-sized structures and machines (e.g. Manning et al., 2000; Song et al., 2002). Structural control of large structures has also been attempted (e.g. Kamada et al., 1997). Other new applications include underwater acoustic absorption, robotics, precision positioning and smart skins for submarines (Kumar, 1991). Skin-like tactile sensors utilizing piezoelectric effect for sensing temperatures and pressures have been reported (Rogers, 1990). Very recently, the piezoelectric materials have been employed to produce micro and nano scale systems and wireless inter digital transducers (IDT) using advanced embedded system technologies, which are set to find numerous applications in microelectronics, bio-medical and SHM (Varadan, 2002; Lynch et al., 2003b). Recent research is also exploring the development of versatile piezo-fibres, which can be integrated with composite structures for actuation and SHM (Boller, 2002). The most striking application of the piezoelectric materials in SHM has been in the form of EMI technique. This is the main focus of the present thesis and details will be covered in the subsequent sections.

26

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

2.2.5

Smart Materials: Future Applications

Seasoned researchers often share visionary ideas about the future of smart materials in conferences and seminars. According to Prof. Rogers (Rogers, 1990), following advancements could be possible in the field of smart materials and structures. •

Materials which can restrain the propagation of cracks by automatically producing compressive stresses around them (Damage arrest).



Materials, which can discriminate whether the loading is static or shock and can generate a large force against shock stresses (Shock absorbers).



Materials possessing self-repairing capabilities, which can heal damages in due course of time (Self-healing materials).



Materials which are usable up to ultra-high temperatures (such as those encountered by space shuttles when they re-enter the earth’s atmosphere from outer space), by suitably changing composition through transformation (thermal mitigation). Takagi (1990) similarly projected the development of more functional and

higher grade materials with recognition, discrimination, adjustability, selfdiagnostics and self-learning capabilities. 2.3 PIEZOELECTRICITY AND PIEZOELECTRIC MATERIALS The word ‘piezo’ is derived from a Greek word meaning pressure. The phenomenon of piezoelectricity was discovered in 1880 by Pierre and Paul-Jacques Curie. It occurs in non-centro symmetric crystals, such as quartz (SiO2), Lithium Niobate (LiNbO3), PZT [Pb(Zr1-xTix)O3)] and PLZT [(Pb1-xLax)(Zr1-yTiy)O3)], in which electric dipoles (and hence surface charges) are generated when the crystals are loaded with mechanical deformations. The same crystals also exhibit the converse effect; that is, they undergo mechanical deformations when subjected to electric fields. In centro-symmetric crystals, the act of deformation does not induce any dipole moment, as shown in Fig. 2.3. However, in non-centro symmetric crystals, this

27

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

µ=0

µ=0

Fig. 2.3 Centro-symmetric crystals: the act of stretching does not cause any dipole moment (µ = Dipole moment).

µ=0

µ≠0

Fig. 2.4 Noncentro-symmetric crystals: the act of stretching causes dipole moment in the crystal (µ = Dipole moment). leads to a net dipole moment, as illustrated in Fig. 2.4. Similarly, the act of applying an electric field induces mechanical strains in the non-centro symmetric crystals. 2.3.1

Constitutive Relations The constitutive relations for piezoelectric materials, under small field

condition are (IEEE standard, 1987) Di = ε ijT E j + dimd Tm

(2.1)

E S k = d cjk E j + skm Tm

(2.2)

Eq. (2.1) represents the so called direct effect (that is stress induced electrical charge) whereas Eq. (2.2) represents the converse effect (that is electric field induced mechanical strain). Sensor applications are based on the direct effect, and actuator applications are based on the converse effect. When the sensor is exposed to a 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

28

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

field is applied to it, which results in an induced strain field. In more general terms, Eqs. (2.1) and (2.2) can be rewritten in the tensor form as (Sirohi and Chopra, 2000b)  D  ε T S  =  c   d

d d E    s E  T 

(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 external electric field vector and [T] (3x3) (N/m2) the stress tensor. Accordingly, [ ε T ] (F/m) is the second order dielelectric permittivity tensor under constant stress, [dd] (C/N) and [dc] (m/V) the third order piezoelectric strain coefficient tensors, and [ s E ] (m2/N) the fourth order elastic compliance tensor under constant electric field. Taking advantage of the symmetry of the stress and the strain tensors, these can be reduced from a second order (3x3) tensor form to equivalent vector forms, (6x1)

in

size.

Thus,

[ S ] = [ S11 , S 22 , S 33 , S 23 , S 31 , S12 ]T

and

similarly,

[T ] = [T11 , T22 , T33 , T23 , T31 , T12 ]T . Accordingly, the piezoelectric strain coefficients can be reduced to second order tensors (from third order tensors), as [dd] (3x6) and [dc] (6x3). The superscripts ‘d’ and ‘c’ indicate the direct and the converse effects respectively. Similarly, the fourth order elastic compliance tensor [ s E ] can be reduced to (6x6) second order tensor. The superscripts ‘T’ and ‘E’ indicate that the parameter has been measured at constant stress (free mechanical boundary) and constant electric field (short-circuited) respectively. A bar above any parameter signifies that it is complex in nature (i.e. measured under dynamic conditions). The piezoelectric strain coefficient d cjk defines mechanical strain per unit electric field under constant (zero) mechanical stress and d imd defines electric displacement per unit stress under constant (zero) electric field. In practice, the two coefficients are numerically equal. In d cjk or d imd , the first subscript denotes the direction of the electric field and second the direction of the associated mechanical strain. For example, the term d31 signifies that the electric field is applied in the direction ‘3’ and the strain is measured in direction ‘1’.

29

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

If static electric field is applied under the boundary condition that the crystal is free to deform, no mechanical stresses will develop. Similarly, if the stress is applied under the condition that the electrodes are short-circuited, no electric field (or surface charges) will develop. For a sheet of piezoelectric material, as shown in Fig. 2.5, the poling direction is usually along the thickness and is denoted as 3-axis. The 1-axis and 2-axis are in the plane of the sheet. The matrix [dc] depends on crystal structure. For example, it is different for PZT and quartz, as given by (Zhu, 2003)

0 0  0 dc =  0 d15   0

0 0 0 d 24 0 0

d 31  d32  d33   (PZT) 0 0  0 

,

0  d11 − d 0  11  0 0  0  d14  0 − d14  − 2d11  0

0 0 0  (quartz) 0 0  0

(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. For PZT crystals, the coefficient 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. It is not possible to produce shear in the 1-2 plane purely by the application of an electric field, since all terms in the last row of the matrix [dc] are zero (see Eq. 2.4). Similarly, shear stress in the 1-2 plane does not generate any electric response. In all poled piezoelectric materials, d31 is negative and d33 is positive. For a good sensor, the algebraic sum of d31 and d33 should be the maximum and at the same time, ε33 and the mechanical loss factor should be minimum (Kumar, 1991). 3

2

1 Fig. 2.5 A piezoelectric material sheet with conventional 1, 2 and 3 axes.

30

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

The compliance matrix has the form sE  11E  s21 s E s E =  31 E  s41  E  s51 E  s61

s12E

s13E

s14E

s15E

E s22

E s23

E s24

E s25

s32E

s33E

s34E

s35E

E s42

E s43

E s44

E s45

s52E

s53E

s54E

s55E

E s62

E s63

E s64

E s65

s16E   E s26  E  s36  E  s46 E  s56  E  s66 

(2.5)

From energy considerations, the compliance matrix is symmetric, which leaves only 21 independent coefficients. Further, for isotropic materials, there are only two independent coefficients, as expressed below (remaining terms are zero) E s11E = s22 = s33E =

1

E E = s23 = s31E = s32E = s12E = s13E = s21

E E s44 = s55E = s66 =

(2.6)

YE

1 GE

−ν YE

(2.7) (2.8)

where Y E is the complex Young’s modulus of elasticity (at constant electric field), G E the complex shear modulus (at constant electric field) and ν the Poisson’s ratio. It may be noted that the static moduli, YE and GE, are related by GE =

YE 2(1 + ν )

(2.9)

The electric permittivity matrix can be written as ε T ε T ε T  13  11 12  T T T  [ε T ] = ε 21 ε 22 ε 23  T  T T ε 31 ε 32 ε 33 

(2.10)

From energy arguments, the permittivity matrix can also be shown to be symmetric, which reduces the number of independent coefficients to 6. Further, taking advantage of crystal configurations, more simplifications can be achieved. For example, it takes following simple forms for monoclinic, cubic and orthorhombic crystals (Zhu, 2003)

31

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

ε T 0 ε T  31  11  T 0 ε 0   22 ε T 0 ε T  33   13  monoclinic

[ε T ]

=

2.3.2

Second Order Effects

,

ε T 0 0  11 T   0 ε 22 0  T  0 0 ε 33   orthorhombic (e.g. PZT)

ε T 0 0   11 T   0 ε11 0   0 0 εT  11    cubic

,

(2.11)

It should be noted that Eqs. (2.1) and (2.2) are valid under low electric fields only. At high electric fields, the second order terms in electric fields make significant contributions. This effect is called the electrostrictive effect. As a result of this effect, Eq. (2.1) need to be modified as Di = ε ijT E j + dimd Tm + M mn Em En

(2.12)

where Mmn is called the electrostriction coefficient. The electro-strictive effect is independent of the direction of the electric field (Sirohi and Chopra, 2000a). A very common electrostrictive crystal is PMN [Pb(Mg1/3Nb2/3)O3]. The main advantage of the electrostrictive materials is that they exhibit negligible hysteresis (which is significant in piezoelectric crystals), making them the first choice for high voltage applications or where precision positioning of components is warranted (Zhu, 2003). Besides, due to non-linear dependence, they can generate larger motions, as shown in Fig. 2.6. It is for this reason that PMN is used in actuators in the hubble space telescope. S PMN PZT

E

Fig. 2.6 Strain vs electric field for PZT (piezoelectric) and PMN (electrostrictive).

32

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

2.3.3

Pyroelectricity and Ferroelectricity These phenomenon are very similar to piezoelectricity, and the three are inter-

coupled in many crystals. Pyroelectricity is the development of surface charge upon heating. Ferroelectricity is the spontaneous presence of an electric polarization in the absence of an applied field, as shown in Fig. 2.7. Ferroelectic materials are used in random access memory chips. All ferroelectric crystals are simultaneously pyroelectric and piezoelectric as well, however the converse is not necessarily true. P

Spontaneous polarization E

Fig. 2.7 Polarization vs electric field for ferroelctric crystals.

2.3.4

Commercial Piezoelectric Materials Previously, piezoelectric crystals, which used to be brittle and of large

weight, were used in practice. However, now the commercial piezoelectric materials are available as ceramics or polymers, which can be cut into a variety of convenient shapes and sizes and can be easily bonded. (a) Piezoceramics Lead zirconate titanate oxide or PZT, which has a chemical composition [Pb(Zr1xTix)O3)],

is the most widely used type piezoceramic. It is a solid solution of lead

zirconate and lead titanate, often doped with other materials to obtain specific properties. It is manufactured by heating a mixture of lead, zirconium and titanium oxide powders to around 800-1000oC first to obtain a perovskite PZT powder, which is mixed with a binder and sintered into the desired shape. The resulting unit cell is

33

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

elongated in one direction and exhibits a permanent dipole moment along this axis. However, since the ceramic consists of many such randomly oriented domains, it has no net polarization. Application of high electric field aligns the polar axes of the unit cells along the applied electric field, thereby reorienting most of the domains. This process is called poling and it imparts a permanent net polarization to the crystal. This also creates a permanent mechanical distortion, since the polar axis of the unit cell is longer than other two axes. Due to this process, the material becomes piezoelectrically transversely isotropic in the plane normal to the poling direction i.e. d31 = d32 ≠ d33; d15 = d24, but remains mechanically isotropic (Sirohi and Chopra, 2000b). PZT is a very versatile smart material. It is chemically inert and exhibits high sensitivity of about 3µV/Pa, that warrants nothing more sophisticated than a charge amplifier to buffer the extremely high source impedance of this largely capacitive transducer. It demonstrates competitive characteristics such as light weight, low-cost, small size and good dynamic performance. Besides, it exhibits large range of linearity (up to electric field of 2kV/cm, Sirohi and Chopra, 2000a), fast response, long term stability and high energy conversion efficiency. The PZT patches can be manufactured in any shape, size and thickness (finite rectangular shapes to complicated MEMS shapes) at relatively low-cost as compared to other smart materials and can be easily used over a wide range of pressures without serious non-linearity. The PZT material is characterized by a high elastic modulus (comparable to that of aluminum). However, PZT is somewhat fragile due to brittleness and low tensile strength. Tensile strength measured under dynamic loading is much lower (about one-third) than that measured under static conditions. This is because under dynamic loads, cracks propagate much faster, resulting in much lower yield stress. Typically, G1195 (Piezo Systems Inc., 2003) has a compressive strength of 520 MPa and a tensile strength of 76 MPa (static) and 21 MPa (dynamic) (Zhou et al., 1995). The PZT materials have negative d31, which implies that a positive electric field (in the direction of polarization) results in compressive strain on the PZT sheet. If heated above a critical temperature, called the Curie temperature, the crystals lose their piezoelectric effect. The Curie temperature typically varies from 150oC to 350oC for most commercial PZT crystals. When exposed to high electric fields (>12 kV/cm), opposite to the poling direction, the PZT

34

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

loses most of its piezoelectric capability. This is called deploing and is accompanied by a permanent change in the dimensions of the sample. Due to high stiffness, the PZT sheets are good actuators. They also exhibit high strain coefficients, due to which they can act as good sensors also. These features make the PZT materials very suitable for use as collocated actuators and sensors. They are used in deformable mirrors, mechanical micropositioners, impact devices and ultrasonic motors (Kumar, 1991), sonic and ultrasonic sensors, filters and resonators, signal processing devices, igniters and voltage transformers (Zhu, 2003), to name only a few. For achieving large displacements, multi layered PZT systems can be manufactured, such as stack, moonie and bimorph actuators. However, due to their brittleness, the PZT sheets cannot withstand bending and also exhibit poor conformability to curved surfaces. This is the main limitation with PZT materials. In addition, the PZT materials show considerable fluctuation of their electric properties with temperature. Also, soldering wires to the electroded piezoceramics requires special skill and often results in broken elements, unreliable connections or localized thermal depoling of the elements. As a solution to these problems, active piezoceramic composite actuators (Smart Materials Corporation), active fibre composites (Massachusetts Institute of Technology) and macro fibre composites, MFCs (NASA, Langley Centre) have been developed recently (Park et al., 2003a). The MFCs have been commercially available since 2003. These new types of PZTs are low-cost, damage tolerant, can conform to curved surface and are embeddable. In addition, Active Control eXperts, Inc. (ACX), now owned by Mide Technology Corporation, has developed a packaging technology in which one or more PZT elements are laminated between sheets of polymer flexible printed circuitry. This provides the much robustness, reliability and ease of use. The packaged sensors are commercially called QuickPack® actuators (Mide Technology Corporation, 2004). These are now widely used as vibration dampers in sporting goods, buzzer alerts, drivers for flat speakers and more recently in automotive and aerospace components (Pretorius et al., 2004). However, these are presently many times expensive than raw PZT patches.

35

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

(b) Piezopolymers The most common commercial piezopolymer is the Polyvinvylidene Fluoride (PVDF). It is made up of long chains of the repeating monomer (-CH2-CF2-) each of which has an inherent dipole moment. PVDF film is manufactured by solidification from the molten phase, which is then stretched in a particular direction and poled. The stretching process aligns the chains in one direction. Combined with poling, this imparts a permanent dipole moment to the film. Because of stretching, the material is rendered piezoelectrically orthotropic, that is d31 ≠ d32, where ‘1’ is the stretching direction. However, it still remains mechanically isotropic. The PVDF material is characterized by low stiffness (Young’s modulus is 1/12th that of aluminum). Hence, the PVDF sensors are not likely to modify the stiffness of the host structure due to their own stiffness. Also, PVDF films can be shaped as desired according to the intended application. Being polymer, it can be formed into very thin sheets and adhered to curved surfaces also due to its flexibility. These characteristics make PVDF films more attractive for sensor applications, in spite of their low piezoelectric coefficients (approximately 1/10th of PZT). It has been shown by Sirohi and Chopra (2000b) that shear lag effect is negligible in PVDF sensors. Piezo-rubber, which consists of fine particles of PZT material embedded in synthetic rubber (Rogers, 1990), has appeared as an alternative for PVDF. The piezo-rubber shows much higher electrical output due to larger thickness, which is not possible in PVDF. The piezo-rubber is used in piezoelectric coaxial cable as a vehicle sensor. It has much longer life and is immune to rain water. 2.4 PIEZOELECTRIC MATERIALS AS MECHATRONIC IMPEDANCE TRANSDUCERS (MITs) FOR SHM The term mechatronic impedance transducer (MIT) was coined by Park (2000). 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. When used as MIT, their electromechanical impedance characteristics

36

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

are utilized for diagnosing the condition of the structures and the same patch plays the dual roles, as an actuator as well as a sensor. The technique utilizing the PZT based MIT for SHM/ NDE has evolved during the last nine years and is called as the electro-mechanical impedance (EMI) technique in the literature. The following sections describe the various aspects of this technique in detail. 2.4.1

Physical Principles The EMI technique is very similar to the conventional global dynamic

response techniques described previously. The major difference is with respect to the frequency range employed, which is typically 30-400kHz in EMI technique, against less than 100Hz 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. 2.8 (a). The PZT patch-host structure system can be modelled as a mechanical impedance (due the host structure) connected to an axially vibrating thin bar (the patch), as shown in Fig. 2.8 (b). The patch in this figure expands and 3 (z) 2 (y) 1 (x)

PZT Patch l

l

Alternating electric field source

3

Host structure

2

Z

Point of mechanical fixity

E3

1

PZT patch

Z

w h l

(a)

l

Structural Impedance

(b)

Fig. 2.8 Modelling PZT-structure interaction. (a) A PZT patch bonded to structure under electric excitation. (b) Interaction model of PZT patch and host structure.

37

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

contracts dynamically in direction ‘1’ when an alternating electric field E3 (which is spatially uniform i.e. ∂E3/∂x = ∂E3/∂y = 0) 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 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 infinitesimally small and to possess negligible mass and stiffness as compared to the host structure. The structure can therefore 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. 2.8 (b). Under this condition, the PZT patch has zero displacement at the mid-point (x= 0), 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 (Ikeda, 1990) T D3 = ε 33 E3 + d 31T1

S1 =

T1 YE

+ d 31E3

(2.13) (2.14)

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 direction ‘1’.

Y E = Y E (1 + ηj ) is the complex Young’s modulus of elasticity of the PZT patch at T T = ε 33 (1 − δj ) is the complex electric permittivity (in constant electric field and ε 33

direction ‘3’) of the PZT material at constant stress, where j = − 1 . 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 (Liang et al., 1994), derived based on dynamic equilibrium of the PZT patch.

YE

∂ 2u ∂ 2u ρ = ∂x 2 ∂t 2

38

(2.15)

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

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

u = ( A sin κx + B cos κx)e jωt

(2.16)

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

κ =ω

ρ YE

(2.17)

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

S1 ( x) =

∂u = Ae jωtκ cos κx ∂x

(2.18)

and velocity

u& ( x) =

∂u = Ajωe jωt sin κx ∂t

(2.19)

Further, by definition, the mechanical impedance Z of the structure is related to the axial force F in the PZT patch by F( x = l ) = whT1( x = l ) = − Zu&( x = l )

(2.20)

where the negative sign signifies the fact that a positive displacement (or velocity) causes compressive force in the PZT patch (Liang et al., 1993, 1994). Making use of Eq. (2.14) and substituting the expressions for strain and velocity from Eqs. (2.18) and (2.19) respectively, we can derive A=

Z aVo d 31 hκ cos(κl )( Z + Z a )

(2.21)

where Za is the short-circuited mechanical impedance of the PZT patch, given by Za =

κwhY E ( jω ) tan(κl )

(2.22)

Za is defined as the force required to produce unit velocity in the PZT patch in 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

39

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

I = ∫∫ D& 3 dxdy = jω ∫∫ D3 dxdy A

(2.23)

A

Making use of the PZT constitutive relation (Eq. 2.13), and integrating over the entire surface of the PZT patch (-l to +l), we can obtain an expression for the electromechanical admittance (the inverse of electro-mechanical impedance) as Y = 2ωj

 Z a  2 E  tan κl  wl  T E 2 d 31Y   (ε 33 − d 31Y ) +  h   κl   Z + Za 

(2.24)

This equation is same as that derived by Liang et al. (1994), except that an additional factor of 2 comes into picture. This is due to the fact that Liang et al. (1993, 1994) considered only one-half of the patch in their derivation. 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 in 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 mechanical impedance Z. Assuming that the PZT parameters remain unchanged, the electromechanical admittance Y will undergo a change and this serves as an indicator of the state of health of the structure. Measuring 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. Contrary to low-frequency vibration techniques, damping plays much more significant role in the EMI technique due to the involvement of ultrasonic frequencies. Most conventional damage detection algorithms (in low-frequency dynamic techniques), on the other hand are based on damage related changes in structural stiffness and inertia, but rarely in damping (Kawiecki, 2001). It is worthwhile to mention here that traditionally, in order to achieve selfsensing, a complicated circuit was warranted (Dosch et al., 1992). This was so because in the traditional approach, an actuating signal was first applied and the sensing signal was then picked up and separated from the actuating signal. But due to the high voltage, and also due to the strong dependence of the capacitance on

40

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

temperature, the signal was mixed with the input voltage as well as noise and was therefore not very accurate. The EMI technique, on the other hand, offers a much hassle free, simplified, and more accurate self-sensing approach. At low frequencies ( 500oC), such as steam pipes and boilers in power plants. Besides, he also developed practical statistical cross-correlation based methodology for temperature compensation. This paved way for application of the technique to real situations, where the effects of damage and temperature are mixed. (5) Soh et al. (2000) established the damage detection and localization ability of piezo-impedance transducers on real-life RC structures by successfully monitoring a 5m span RC bridge during its destructive load testing. Besides, criteria were outlined for transducer positioning, damage localization and transducer validation. (6) Park et al. (2000b) were the first to integrate the EMI technique with wave propagation modelling for thin beams (1D structures) under ‘free-free’ boundary conditions, by utilizing axial modes. The conventional statistical indices of the EMI technique were used for locating the damages in the frequency range 70-90 kHz. The damage severity was determined by spectral finite element based wave propagation approach, in the frequency range 10-40 kHz. However, this combination necessitated the use of some additional hardware and sensors, such as accelerometers, which are not accurate at ultrasonic frequencies. Also, the application of the wave propagation approach demands additional computational effort, which could restrict the application to simple structures only. Besides, the integration of the EMI technique with wave propagation approach was not seamless in true sense. (7) After the year 2000, numerous papers appeared in the literature demonstrating successful extension of the technique on sophisticated structural components

43

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

such as restrengthened concrete members (Saffi and Sayyah, 2001) and jet engine components under high temperature condtions (Winston et al., 2001). (8) Inman et al. (2001) proposed a novel technique to utilize a single PZT patch for health monitoring as well as for vibration control (9) Abe et al. (2002) developed a new stress monitoring technique for thin structural elements (such as strings, bars and plates) by applying wave propagation theory to the EMI measurement data in the moderate frequency range (1-10kHz). This has paved way for the application of the technique for load monitoring, besides damage detection. The major advantage is that owing to localized wave propagation, the technique is insensitive to boundary conditions and can make accurate stress identification. However, the suitable frequency band for this application is very narrow, and generally difficult to identify. Also, the method is prone to high errors, especially in 2D components, due to imprecise modelling of the interfacial bonding layer. (10) Giurgiutiu et al. (2002) combined the EMI technique with wave propagation approach for crack detection in aircraft components. While the EMI technique was employed for near field damage detection, the guided ultrasonic wave propagation technique (pulse echo) was used for far field damage detection. (11) Peairs et al. (2003) developed a novel low-cost and portable version of impedance analyzer, the major hardware used in the EMI technique, paving way for significant cost-reduction. Integration of the EMI technique with wireless technology and development of stand-alone sensor cum processor cum transmission units based on MEMS and inter digital transducers (IDT) is also underway (Park et al., 2003b) which would enable large-scale instrumentation and monitoring of civil-structures. 2.4.4

Details of PZT Patches In the EMI technique, the same PZT patch serves the actuating as well as the

sensing functions. Fig. 2.10 shows a typical commercially available PZT patch suitable for this particular application (PI Ceramic, 2003). The characteristic feature of the patch is that the electrode from the bottom edge is wrapped around the thickness, so that both the electrodes are available on one side of the PZT patch,

44

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

while the other side is bonded to the host structure. PZT patches of sizes ranging from 5mm to 15mm and thickness from 0.1mm to 0.3mm are best suited for most structural materials such as steel and RC. Such thin patches usually have thickness resonance frequency of the order of few MHz. Hence, the frequency response signature is relatively flat in 30-400 kHz frequency range.

10mm

10mm

Top electrode film

Bottom electrode film wrapped to top surface

Fig. 2.10 A typical commercially available PZT patch.

2.4.5

Selection of Frequency Range The operating frequency range must be maintained in hundreds of kHz so that

the wavelength of the resulting stress waves is smaller than the typical size of the defects to be detected (Giurgiutiu and Rogers, 1997). Typically, for such high frequencies, wavelengths as small as few mm are generated. Contrary to the large wavelength stress waves in the case of low frequency techniques, these are substantially attenuated by the occurrence of any incipient damages (such as cracks) in the local vicinity of the PZT patch. Sun et al. (1995) recommended that a frequency band containing major vibrational modes of the structure (i.e. large number of peaks in the signature), such as the one shown in Fig. 2.9, serves as a suitable frequency range. Large number of peaks signifies greater dynamic interaction between the structure and the PZT patch. Park et al. (2003b) recommended a frequency range from 30 kHz to 400 kHz for PZT patches 5 to 15mm in size. According to Park and coworkers, a higher frequency range (>200 kHz) is favourable in localizing the sensing range, while a lower frequency range (< 70 kHz) covers a large sensing area. Further, frequency

45

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

ranges higher than 500Hz are found unfavourable, because the sensing region of the PZT patch becomes too small and the PZT signature shows adverse sensitivity to its own bonding condition rather than any damage to the monitored structure. It should also be noted that the piezo-impedance transducers do not behave well at frequencies less than 5kHz. Below 1kHz, the EMI technique is not at all recommended (Giurgiutiu and Zagrai, 2002). 2.4.6

Sensing Zone of Piezo-Impedance Transducers As MIT, the PZT patches have a localized sensing zone of influence. This is

because a PZT patch vibrating at high frequencies excites ultrasonic modes of vibration the structure, which are essentially local in nature. Besides, damping is much more significant at high ultrasonic frequencies, leading to localization of the waves generated by the vibrating PZT patch. Esteban (1996) carried out extensive numerical modelling based on wave propagation theory, as well as conducted comprehensive parameteric studies to identify the sensing zone of the piezoimpedance transducers. However, at such high frequencies, exact quantification of energy dissipation proved very difficult and hence the sensing zone could not be exactly identified. However, it was found that this zone depends on the material of the host structure, its geometry, the frequency of excitation and the presence of structural discontinuities. It was concluded that structural discontinuities acting as the sources of multiple reflections cause maximum attenuation to the propagating waves. However, based on experimental data from a large number of case studies, Park et al. (2000a) claimed that the sensing radius of a typical PZT patch might vary anywhere from 0.4m on composite reinforced structures to about 2m on simple metal beams. Tseng and Naidu (2001) reported the sensing range to be greater than 1m in their experiments on thin aluminum beams. Therefore, for effective damage localization, in general, the structures must be instrumented with an array of PZT patches. Due to a localized sensing region, the technique shares a rare ability to detect damages without being affected by far field boundary conditions, external loading

46

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

or normal operating conditions (Esteban, 1996). However, this advantage comes at the cost of a limited sensing area. 2.4.7

Modes of Wave Propagation In an unbounded 3-D elastic solid, two basic wave types exist: dilatational

and rotational. Dilational waves, (Kolsky, 1963) are described by the equation

ρ

∂ 2∆ = (λ + 2G )∇ 2 ∆ ∂ 2t 2

(2.26)

where ∆ = ε xx + ε yy + ε zz (sum of principal strains) is the dilation of the medium, λ the Lame’s constant, G the shear modulus, and ρ the mass density. In seismic studies, the dilatational waves are called P-waves, or ‘Principal’ or ‘Pressure’ waves. The rotational waves, on the other hand, are described by ∂ 2ϖ ρ 2 2 = (2µ )∇ 2ϖ ∂t

(2.27)

where ϖ is the rotation vector. Rotational waves correspond to incompressible distortion of solids, like shear, and are often referred to as S-waves or ‘Secondary’ or ‘Shear’ waves in seismic studies. When the solid medium is not infinite, two additional aspects need to be considered (i) wave reflection and refraction on account of boundary, and (ii) existence of additional wave types closely related to the boundary effects. When a pure P-wave (or S-wave) travelling at an oblique angle hits a boundary, both pressure and shear waves are generated in the reflection process. A free boundary, on the other hand, gives rise to two new wave types - Rayleigh waves and Lamb waves. Rayleigh wave amplitude decreases rapidly with depth, and becomes almost zero at a depth of approximately 1.6λ. At surface, it is the Rayleigh waves which represents maximum proportion of wave energy. Lamb waves are only confined to a superficial layer existing on the top of a homogeneous solid. The wave propagation dynamics (reflection, refraction and transmission) determines the drive-point mechanical impedance of the structure and its modification with degradation of the material on account of damages. In the EMI technique, typically surface waves (mainly Rayleigh waves) are generated due to

47

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

PZT vibrations, as shown in Fig. 2.11, and these travel radially outwards from the patch. They play crucial role in determining the drive point impedance and in detecting any defects which tend to obstruct their path. 2D surface of structure PZT Patch PZT Patch Structure under examination

(a)

(b)

Fig.2.11 Modes of wave propagation associated with PZT patches (Giurgiutiu and Rogers, 1997) (a) PZT transducer patch affixed to the host structure. (b) Surface waves generated by the vibrating PZT patch.

2.4.8

Effects of Temperature The conductance signatures of piezo-impedance transducers have been found

to be temperature sensitive (Sun et al, 1995; Park et al., 1999). In real situations, the effects of damage and temperature are bound to be mixed. This necessitates a method to decouple the two. Fortunately, over a small frequency band, the overall effect of temperature has been observed to be a superposition of uniform horizontal and vertical translations of the signature (Sun et al., 1995). This is absolutely different from the signature deviation resulting from any damages, which causes an abrupt and local variation. It was observed by Pardo De Vera and Guemes (1997) that the horizontal shift is not uniform and depends on frequency. However, if the frequency band is rather narrow, it can be assumed to be uniform. Park et al. (1999) proposed statistical cross-correlation based methodologies for temperature compensation. Bhalla (2001) studied temperature effects using finite element simulation. It was found that the major effects of temperature on the signatures are the horizontal shift, due to change in the host material’s Young’s

48

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

modulus, and the vertical shift, due to variations in ε33 and d31 of the PZT patch. All the shifts were found to vary linearly with temperature over narrow frequency bands. Out of these, the most critical was the vertical shift due to change in ε33. A simple temperature compensation methodology was proposed which required the acquisition of the baseline signatures at two different temperatures. 2.4.9

Effects of Noise and Other Miscellaneous Factors Most low frequency vibration based SHM/ NDE methods on real-life

structures are likely to encounter the presence of noise. The noise could be (a) mechanical noise, caused by sources such as vehicle movement or wind; (b) electrical noise, generated by variations in the power supply; or (c) electromagnetic noise, caused by communication waves, which affect the signal acquisition and transmission through cables and other susceptible circuitry (Samman and Biswas, 1994a). The greatest advantage of the high frequency EMI technique is that the signal (in few hundred kHz frequency range) is not likely to be affected by mechanical noise, since this type of noise is dominant in the low frequency ranges only (typically less than 100Hz). Electrical noise too is not crucial in the EMI technique since the power required by each PZT patch is in the low milliwatt range, which does not call for the deployment of high power generating sets. Rather, it makes possible the development of battery operated sensors (Park, 2000). The only possible noise could be the electromagnetic noise, which can be minimized by using coaxial cables. Another source of error could be the parasitic electrical admittance of the connection wires. It can be accounted for by performing zero-correction in the impedance analyzer, prior to taking measurements. However, it could be problematic for large arrays where each PZT patch may have a different wire length. It is recommended that the same set of connection wire be used for recording both the baseline signature as well as the signature at any future point of time, so that the residual conductance (if not properly accounted for in the zero correction) is the same in both cases. The change in signature, if any, will be due to structural damage alone. It should also be noted that extensive experimental study

49

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

by Raju (1998) found that the method can still work well in-spite of variable test wire lengths. Park et al. (2000a) demonstrated that the technique is insensitive to distant boundary condition changes and mass loading. The technique is also insensitive to arbitrary ambient inputs to the structure. This is very important, especially for the in-flight monitoring of aircraft or bridges, while under service. However, it should be noted that care must be exercised in applying the EMI technique on structures which are instrumented with ultrasonic transducers for purposes of NDE. The high frequency excitations from these transducers could generate a high frequency noise for the EMI technique. Hence, it should be made sure that these are turned off before applying the EMI technique. 2.4.10 Thermal Stresses in Piezo-Impedance Transducers During vibrations, thermal stresses are produced owing to the presence of electrical and mechanical damping. In many applications, the thermal stresses could be significant. Zhou et al. (1995) carried a detailed analysis of the problem and found the internal thermal stresses to increase with the thickness of the PZT patch and the rate of internal heat generation. However, they also found that for very thin PZT patch, such as up to 0.2-0.3 mm, the thermal stress may be ignored in the overall stress analysis, since the thickness is small enough to let the generated heat dissipate quickly. It could be significant in the case of stacked actuators or high voltage actuation applications, which is not the case of for the EMI technique. 2.4.11 Multiple Sensor Requirements Since the EMI technique is essentially acousto-ultrasonic in nature, the number of sensors necessary depends upon the geometry and material of the monitored component. The number of sensors is small in thin beams and plates where the acoustic waves can easily travel long distances through the material medium. However, in complex structures with holes, notches, discontinuities and thickness variations, a large number of sensors may be required due to greater losses on account of energy dissipation. Also, the same would be true for materials such as composites or concrete, which are characterized by high material damping.

50

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

In such scenarios, it is important that such a multi-sensor architecture to have a built-in redundancy such that one or more sensors may be allowed to fail without making the entire system ineffective (Boller, 2002). Also, it is important to consider issues like sensor validation, data pre-processing, feature extraction and pattern recognition. Suitable locations for bonding the patches can be easily determined from the geometry and loading conditions to which the structure is likely to be subjected during the course of its service by preliminary structural analysis (Soh et al, 2003). It is recommended to locate the patches at the points of maximum bending moments and shear, which can be ascertained by the theory of structures. It may be mentioned here that given an array of PZT patches, it can either be excited in self-impedance fashion (The EMI technique) or transfer impedance fashion (Esteban, 1996). In the transfer function method, one PZT patch acts as actuator and emits acoustic signal into the structure. The signals are picked by another patch acting as sensor. The main advantage of the transfer impedance method (or the gain-phase) method is that it provides greater sensing range and hence reduces the number of sensors required. Besides, this can also enable the determination of mechanical properties of the monitored component. Impedance analyzer can be easily utilized for the transfer impedance approach also. However, the ‘gain’ levels encountered in the transfer impedance approach are much smaller since the waves have to travel longer distance, besides encountering higher noise (Park et al., 2003b). Increasing the excitation level could help overcome this problem and this could help the two methods to supplement each other, since the same sensor array can be utilized for both the techniques. 2.4.12 Signal Processing Techniques and Damage Quantification The prominent effects of structural damages on the conductance signature are the appearance of new peaks in the signature and lateral and vertical shifting of the peaks (Sun et al., 1995), which are the main damage indicators. Samman and Biswas (1994a, 1994b) reported many pattern recognition techniques to quantify the variations occurring in the structural signatures (similar to conductance signatures) due to damages; such as the waveform chain code (WCC) technique, the signature

51

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

assurance criteria (SAC), the equivalent level of degradation system (ELODS) and the adaptive template matching (ATM). Similar statistical techniques have been employed by the investigators researching on the EMI technique; such as the root mean square deviation or RMSD (Giurgiutiu and Rogers, 1998), relative deviation or RD (Ayres et al., 1998; Sun et al., 1995), the difference of transfer function between damaged and undamaged conditions (Pardo de Vera and Guemes, 1997) and the mean absolute percent deviation or MAPD (Tseng and Naidu, 2001). The RMSD index is defined as (Giurgiutiu and Rogers, 1998; Giurgiutiu et al., 1999) N

RMSD (%) =

1 0 2 ∑ (Gi − Gi )

i =1

N

0 2 ∑ (Gi )

x 100

(2.28)

i =1

where Gi1 is the post-damage conductance at the ith measurement point and Gi0 is the corresponding pre-damage value. Similarly, RD is based on the sum of mean square algorithm, normalized with respect to an arbitrarily chosen maximum amount of damage, and is defined for the ith patch (in an array) as (Sun et al., 1995) N

RDi =

1 0 2 ∑ (Gik −Gik )

k =1 N

0 2 1 ∑ (G1k −G1k )

(2.29)

k =1

where the numerator represents the mean square deviation at the ith location and the denominator represents the deviation for the chosen reference maximum damage location ‘1’. The MAPD index is defined as (Tseng and Naidu, 2001)

MAPD =

100 N Gi1 − Gi0 ∑ N i =1 Gi0

(2.30)

The covariance (Cov) and correlation coefficient (CC) are respectively defined as (Tseng and Naidu, 2001)

Cov (G o , G1 ) =

CC =

1 N (Gi0 − G 0 )(Gi1 − G1 ) ∑ N i =1

Cov (G 0 , G1 )

σ 0σ 1

52

(2.31)

(2.32)

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

where σ0 and σ1 are the standard deviations of the baseline signature and the signature after damage respectively. G 0 and G1 respectively are the mean values of the baseline signature and the signature after damage. Following observations by different investigators regarding statistical indices are worth being taken note of. (1) The author performed a comparative study of the RMSD, the SAC, the WCC and the ATM techniques, as a part of M. Eng. Research (Bhalla, 2001) and found the RMSD algorithm as the most robust and most representative of damage progression among these indices. (2) Tseng and Naidu (2001) demonstrated the use of MAPD, covariance (Cov) and correlation coefficient (CC) to quantify damages in thin aluminium beams. They found Cov and CC to be very good indicators when quantifying increase in damage size at one particular location. When the peaks of one signature match with the peaks of the other signature, the covariance value obtained is positive. When valleys of one signature match with peaks of the other, and vice versa, covariance is negative. When values in both signatures are unrelated, covariance is nearly zero. Thus, the damages can be characterized by the fact that when the deviation between the signatures is large, the covariance is closer to zero or is negative. (3) Giurgiutiu et al. (2002) reported comprehensive investigations of CC as damage index in their experiments on thin circular aluminium plates. It was experimentally found by these researchers that (1-CC)3 decreased linearly as the distance between the sensor and the damage (a simulated crack) increased. Although the statistical methods are easy to implement and share the advantage of being non-parametric (Soh et al., 2000), their main drawback is that they do not provide any clear picture of the associated damage mechanism or any change in mechanical parameters of the structure under question. For example, in many situations, incipient damage and the high order damage can be found to lead to an RMSD index of the same order of magnitude. As such, the particular “threshold value” demanding an alarm could vary from structure to structure (Soh et al., 2000). In such situations, one needs to rely on the slope of the RMSD curve rather than its

53

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

absolute magnitude. However, this may also prove unreliable. It is probably for this reason that Giurgiutiu et al. (2002) have remarked “…Further work is needed to systematically investigate the most appropriate damage metric that can be used for processing the frequency spectra successfully…”. 2.5 ADVANTAGES OF EMI TECHNIQUE The major advantages of the EMI techniques over the prevalent global and local SHM techniques are summarized below (i)

The EMI technique shows far greater damage sensitivity than the conventional global methods. Typically, the sensitivity is of the order of the local ultrasonic techniques (Park et el., 2003b). Yet the technique is very straightforward to implement on large structures as compared to the local methods, whose application is quite cumbersome. It does not warrant very expensive hardware like the ultrasonic techniques and also does not warrant any probe to be physically moved from one location to other. The data acquisition is much more simplified as compared to the traditional accelerometer-shaker combination in the global vibration techniques since the data is directly obtained in the frequency domain. Thus, the EMI technique provides a very nice interface between global vibration based techniques and local ultrasonic techniques.

(ii)

The PZT patches are bonded non-intrusively on the structure, possess negligible weight and demand low power consumption. Small and nonintrusive sensors can monitor inaccessible locations of structures and components. Hence, this could save the expensive time and effort involved in dismantling machines and structural components for inspection purposes. Easy installation (no sub-surface installation) makes the piezo-impedance transducers equally suitable for existing as well as to-be-built structures.

(iii)

The use of the same transducer for actuating as well as sensing saves the number of transducers and the associated wiring.

(iv)

The limited sensing area of the PZT patches helps in isolating changes due to far field variations such as boundary conditions and normal operational vibrations. Also, multiple damages in different areas can be picked easily.

54

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

(v)

The technique is practically immune to mechanical, electrical and electromagnetic noise. This makes the technique very suitable for implementation during operating conditions, such as in aircraft during flight.

(vi)

The PZT patches can be produced at very low costs, typically US$1 (Peairs et al., 2003) to US$10 (Giurgiutiu and Zgrai, 2002), in contrast to conventional force balance accelerometers, which may be as expensive as US$1000 (Lynch et al., 2003b) and at the same time bulky and narrowbanded.

(vii)

The technique is very favourable for autonomous and online implementation since the requirements for data processing are minimal. The data is directly recorded in the frequency domain thereby saving expensive domain transform efforts.

(viii) The method can be implemented at any time in the life of a structure. For example, the PZT patches can be installed on structures after an earthquake to monitor the growing cracks or loosening connections. Many other methods warrant installation of the sensors at the time of construction and hence not suitable for existing structures. However, it should be noted that the PZT patches would be able to detect any structural damages appearing in the post-installation period only. Hence, they cannot detect “existing” damages in the structures. (ix)

Being non-model based, the technique can be easily applied to complex structures.

(x)

The PZT patches are orders of magnitude below the stiffness and mass of the monitored structures. Hence the dynamics of the host structure are not modified and accurate structural identification is possible.

(xi)

PZT sensors are non-resonant devices with wide bad capabilities and exhibit large range of linearity, fast response, light weight, high conversion efficiency and long-term stability.

(xii)

Commercial availability of portable and low-cost impedance analyzers will further enhance the applicability of the technique.

55

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

It is therefore needless to say that the EMI technique has evolved as a universal NDE method, applicable to almost all engineering materials and structures. If the damage location could be predicted in advance (i.e. ‘where to expect damage’), the EMI technique would be most powerful technique in such applications (Park et al., 2003b). 2.6 LIMITATIONS OF EMI TECHNIQUE In spite of many advantages over other techniques, the EMI technique shares several limitations as outlined below (i)

A PZT patch is sensitive to structural damages over a relatively small sensing zone, ranging from 0.4m to 2m only, depending upon the material and geometrical configuration. Though sufficient for monitoring miniature components and mechanical/ aerospace systems, the small sensing zone warrants the deployment of several thousands of PZT patches for real-time monitoring of large civil-structures, such as bridges or high rise buildings. The large number of PZT patches would warrant significant cost and effort for laying out the wiring system, data collection and data processing. Hence, critical locations must be judiciously decided based on the theory of structures.

(ii)

Since all civil and mechanical structures are statically indeterminate, cracking of a few joints might not necessarily affect the overall safety and stability of the monitored structure. Thus, a drawback of the EMI technique as compared to the global SHM techniques is its inability to assess the overall structural stability. Rather, in this respect, global SHM techniques and the EMI techniques could easily complement each other.

(iii)

PZT materials and the related technologies are only supplementary steps in addition to good designs of structures and machines. Many academicians argue that more research should be focused on improving material strength and design rather than on sensors. But even the best-designed structures could have problems, therefore it is justified to explore the application smart materials to sense or detect damages in advance (Reddy, 2001).

56

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

2.7 NEEDS FOR FURTHER RESEARCH IN EMI TECHNIQUE 2.7.1

Theoretical and Data Processing Considerations

(i)

In spite of key advantages over other NDE technologies, the difficulties in developing a theoretical model at high frequencies renders the EMI technique unable to correlate the changes in signature with specific changes in structural properties (Lopes et al., 1999). Hence, no comparison can be found in the literature between theoretical and experimental electrical admittance spectra, especially in 100-200 kHz frequency range. Giurgiutiu et al. (2000) acknowledged that the main barrier to the widespread industrial application of the EMI technique is the meager understanding of the multidomain interaction between the PZT patch and the host structure. The wave propagation dynamics associated with vibrating PZT patches has also not been thoroughly investigated so far.

(ii)

Till date, all the existing damage quantification approaches are nonparametric and statistical in nature and are able to utilize the real part of signature only. The information about damage possessed by the imaginary part is therefore lost. Giurgiutiu and Zagrai (2002) employed the imaginary part to check the sensor bonding conditions, but not for any damage related information.

(iii)

No attempt has been made to extract the mechanical impedance of the interrogated structure from the electro-mechanical signatures, partly due to the non-existence of suitable impedance models.

(iv)

No ready calibration is currently available so as to realistically predict damage level based on the measured signatures.

(v)

The influence of shear lag caused by finite thickness adhesive layer used for bonding the PZT patches to the surface of host structures has not been thoroughly investigated so far.

(vi)

For practical application of the technique, it is very important to address the issues of sensor calibration, validation and self-diagnostics.

(vii)

Uncertainties in signature deviation due to damage have also been noted by various investigators. These need to be taken into consideration more scientifically, using the tools of probability and statistics.

57

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

(viii) At the present moment, the sensing region of the patch can only be estimated very crudely. 2.7.2 Hardware/ Technology Considerations (i)

Presently, the commercial impedance analyzers used in the EMI technique are very expensive (> US$16000) and bulky (typically, HP 4192A impedance analyzer measures 425x235x615mm in size and weighs 19kg). Besides, the requirements of wiring could seriously limit the practical application of the technique on real-life structures. Although Peairs et al. (2003) developed a novel low-cost and portable impedance analyzer, the data acquisition, processing and signal transmission are still elementary. This calls for the integration of the EMI technique with wireless technologies and development of stand-alone sensor cum processor cum transmission units based on MEMS and IDT. Efforts for developing stamp sized chips capable of replacing impedance analyzers are also underway Park et al. (2003a).

(ii)

Park et al. (2003b) suggested the integration of local computing units with sensor systems so as to save energy consumption in data transmission to any central processing unit. Utilizing ambient vibrations for deriving necessary operational power can also be of great practical advantage since this would eliminate the requirement of replacing batteries periodically in wireless applications.

(iii)

Many practical aspects such as protection of PZT patches against harsh environmental conditions for long serviceability and the reliability of the adhesive bonding under extreme conditions need to be investigated.

(iv)

Signal multiplexing can significantly reduce sensor interrogation times, especially for critical large-sized structures. Suitable algorithms and technological solutions for multiplexing and de-multiplexing the signatures need to be developed.

58

Chapter 2: Electro-Mechanical Impedance Technique for SHM and NDE

2.8 CONCLUDING REMARKS This chapter has presented a detailed review of the state of the art in SHM, with a special emphasis on the EMI technique. The chapter also introduced the concept of smart materials and structures. The physical principles underlying the EMI technique and the details of the previous work undertaken by prominent research groups of the world (Liang, Rogers and coworkers; Inman, Park and co-workers; Giurgiutiu and coworkers) have been presented. The needs for further research to improve this technique were also highlighted. This research has primarily focused on understanding the structure-PZT interaction mechanism to develop analytical tools for realistically calibrating the piezo-impedance transducers for damage prediction. The next chapter will deal with the structure-PZT interaction mechanism inherent in the piezo-impedance transducers.

59

Chapter 3: PZT-Structure Electro-Mechanical Interaction

Chapter 3 PZT-STRUCTURE ELECTRO-MECHANICAL INTERACTION

3.1

INTRODUCTION The electro-mechanical interaction between the piezo-impedance transducer

and the host structure is key to damage detection in the EMI technique. On the application of an alternating voltage across a bonded PZT patch, deformations are produced in the patch as well as in the local area of the host structure surrounding it. The response of this area to the imposed mechanical vibrations is transferred back to the PZT wafer in the form of electrical response, as conductance and susceptance signatures. As a result of this interaction, the structural characteristics are reflected in the signatures. An understanding of the PZT-structure electro-mechanical interaction is therefore very vital for an effective implementation of the EMI technique for NDE. Important aspects of PZT-structure interaction are addressed in this chapter. 3.2

MECHANICAL IMPEDANCE OF STRUCTURES A harmonic force, acting upon a structure, can be represented by a rotating

phasor on a complex plane (to differentiate it from a vector), as shown in Fig. 3.1. Let Fo be the magnitude of the phasor and let it be rotating anti-clockwise at an angular frequency ω (same as the angular frequency of the harmonic force). At any instant of time ‘t’, the angle between the phasor and the real axis is ‘ωt’. The instantaneous force (acting upon the structure) is equal to the projection of the phasor on the real axis i.e. Focosωt. The projection on the ‘y’ axis can be deemed as the ‘imaginary’ component. Hence, the phasor can be expressed, using complex notation as

60

Chapter 3: PZT-Structure Electro-Mechanical Interaction

Y (Imaginary Axis) Force phasor Fo ωt X (Real Axis)

φ uo

Velocity phasor

Fig. 3.1 Representation of harmonic force and velocity by rotating phasors.

F (t ) = Fo cos ωt + jFo sin ωt = Fo e jωt

(3.1)

The resulting velocity response, u& , at the point of application of the force, is also harmonic in nature. However, it lags behind the applied force by a phase angle φ, due to the ‘mechanical impedance’ of the structure. Hence, velocity can also be represented as a phasor, as shown in Fig. 3.1, and expressed as u& = u&o cos(ωt − φ ) + ju&o sin(ωt − φ ) = u&o e j (ωt −φ )

(3.2)

The mechanical impedance of a structure, at any point, is defined as the ratio of the driving harmonic force to the resulting harmonic velocity, at that point, in the direction of the applied force. Mathematically, the mechanical impedance, Z, can be expressed as Z=

F F e jωt F = oj (ωt −φ ) = o e jφ u& u&o e u& o

(3.3)

Based on this definition, the mechanical impedance of a pure mass ‘m’ can be derived as ‘mωj’ (Hixon, 1988). Similarly, the mechanical impedance of an ideal spring possessing a spring constant ‘k’ can be derived as ‘–jk/ω’, and that of a damper can be obtained as ‘c’ (the damping constant). For a parallel combination of ‘n’ mechanical systems, the equivalent mechanical impedance is given by (Hixon, 1988)

61

Chapter 3: PZT-Structure Electro-Mechanical Interaction n

Z eq = ∑ Z i

(3.4)

i =1

Similarly, for a series combination, n 1 1 =∑ Z eq i =1 Z i

(3.5)

The main advantage of the impedance approach is that the differential equations of Newtonian mechanics are reduced to simple algebraic equations and a black-box concept is introduced. Critical forces and velocities only at one or two points of interest alone need to be considered, thereby eliminating the need of a complex analysis of the system. 3.3 MECHANICAL IMPEDANCE OF PZT PATCHES 3

PZT patch

2 1

h

w

Stress = T1 Resultant force = F

l

Fig. 3.2 Determination of mechanical impedance of a PZT patch. As a general practice, the mechanical impedance of the PZT patches is determined in short circuited condition, as shown in Fig. 3.2, so as to eliminate the piezoelectric effect and to invoke pure mechanical response alone. If F is the force applied on the PZT patch, then from Eq. (3.3), the short-circuited mechanical impedance of the patch, Za, can be determined as F( x = l ) whT1( x = l ) whY E S1( x = l ) Za = = = u&( x = l ) u&( x = l ) u&( x = l )

(3.6)

where T1 is the axial stress in the patch, S1 the corresponding strain, Y E the complex Young’s modulus of elasticity of the patch, u& the velocity response and l, w and h the patch dimensions as shown in Fig. 3.2. It should be noted that we are considering one symmetrical half of the PZT patch (l = half length) in accordance with the developments in section 2.4.1). As derived in Chapter 2, the displacement response of a vibrating PZT patch is given by

62

Chapter 3: PZT-Structure Electro-Mechanical Interaction

u = ( A sin κx)e jωt

(3.7)

Calculating S1(x=l) and u& (x=l) with the aid of Eq. (3.7) (by differentiation with respect to ‘x’ and ‘t’ respectively), and substituting in Eq. (3.6), we can derive the mechanical impedance of the PZT patch as E  wh  κl  Y Za =     l  tan κl  ( jω )

(3.8)

Za, which is a function of frequency, is a complex quantity, and can therefore be expressed as Z a = xa + ya j

(3.9)

 tan κl  E E On substituting   by (r + tj) and Y by Y (1 + ηj ) in Eq. (3.8), and l κ   simplifying, we can obtain following expressions for xa and ya xa =

whY E (ηr − t )

ωl (r 2 + t 2 )

and ya = −

whY E (r + ηt )

ωl (r 2 + t 2 )

(3.10)

If the operating frequency is very low as compared to the first resonant frequency, 1  tan kl  (typically, ω |Z|), with the exception that the real part of the structural impedance is frequency dependent and the imaginary part is constant (which is possible in real world over small frequency intervals). The relative order of magnitude of the impedance of the host structure and the PZT patch are similar to case III, i.e. the patch is much stiffer than the host structure. In this case, the magnitude of ‘y’ is very small as compared to ‘ya’ (Fig. 3.9d). It can be clearly observed that the G-plot exhibits similar variation as the damping constant ‘c’ (or ‘x’). In this case, two types of damages were induced: (i) increasing ‘x’ increased by 20%, (b) Reducing ‘y’ reduced by 50%. It is found that the G-plot is only sensitive to variation in ‘x’ (which is increased only by 20%) rather than ‘y’ (which is reduced by 50%). At the same time, the B-plot exhibits an extremely feeble sensitivity to damages, and it can be deemed useless from NDE point of view

75

Chapter 3: PZT-Structure Electro-Mechanical Interaction

0.000004

0.0004

20% increase in ‘c’ 0.00035

20% increase in ‘c’

0.0003 B (S)

0.000003

0.00025

Pristine State

0.0000025

Pristine State

0.0002

650

600

550

400

650

600

550

500

450

400

500

0.00015

0.000002

450

G (S)

0.0000035

Frequency (Hz)

Frequency (Hz)

(a)

(b) 500

20

Structure, x

y, y a (Ns/m)

x, x a (Ns/m)

12

8

PZT patch, xa

650

600

550

500

450

-500

400

0

16

Structure, y

-1000 -1500

PZT patch, ya

-2000 -2500

4

-3000 -3500

650

600

550

500

450

400

0

Frequency (Hz)

Frequency (Hz)

(d)

(c)

PZT patch

1000

50% reduction in the imaginary part 20% increase in ‘c’

100

650

600

450

400

550

Pristine State

10

500

|Z|, |Za| (Ns/m)

10000

Frequency (Hz)

(e)

Fig. 3.9 Signatures for Case V. (a) Conductance vs Frequency.

(b) Susceptance vs Frequency.

(c) Real impedance vs Frequency (pristine). (d) Imaginary impedance vs Frequency (pristine). (e) Absolute Impedance vs Frequency (pristine).

76

Chapter 3: PZT-Structure Electro-Mechanical Interaction

Case Study VI: All the case studies described so far were characterized by low operating frequencies as compared to the first resonant frequency of the PZT patch. Consider case VI, where the structural parameters are ‘k’= 197.4 N/m, ‘m’= 0.0002 kg, and ‘c’= 0.01257 Ns/m

(damping ratio ξd = 0.03), thus implying a system resonant

frequency of 158 Hz. The frequency range chosen for this case is from 5 kHz to 40 kHz, which includes the first resonant frequency of the patch (14.123 kHz) and the second resonant frequency is also quite close (42.369kHz). The plots for Case VI are shown in Fig. 3.10. It is observed that the G-plot and the B-plot exhibit two very sharp peaks, although the structure does not have any resonant frequency in this particular range (structural resonant frequency = 158 Hz!). The peaks occur at the points where the condition ‘y = -ya’ (see Fig. 3.10d) is satisfied, however not at structural resonant frequency. Thus the structural modes are identified falsely. Both the real part and the imaginary part of admittance are very feebly affected by any changes in the structural parameters. From Figs. 3.10 (a) and (b), it is observed that only the change in ‘m’ is detectable whereas for all other simulated damages, the plots virtually coincide with the plot for the pristine state. This particular case study is characterized by two features: (i)

The order of magnitude of the structural impedance and the PZT mechanical impedance are similar over certain frequency ranges, such as around 15 000 Hz and 40 000 Hz (see Fig. 3.10e).

(ii)

The frequency range under consideration includes the resonant frequencies of the PZT patch. These two situations occurring concurrently must be avoided under all

circumstances. This case shows that for highly stiff PZT patches, the peaks of the signatures could actually be near the natural frequencies of the patch, rather than the structure. In order to illustrate the extent to which false peaks can dominate the electromechanical admittance spectrum, consider a hypothetical case of a PZT patch, 30x30m wide and 0.25mm thick, actuating a SDOF system with parameters: ‘m’= 200 kg, ‘k’=1.97x109 N/m, and ‘c’=12566.4 Ns/m (ξd = 0.01). This system also has a natural frequency of 500 Hz. The plots for this case are shown in Fig. 3.11. It is

77

Chapter 3: PZT-Structure Electro-Mechanical Interaction

20% increase in ‘m’ 1

0.3

0.1

0.2

Pristine State

0.1

B (S)

40000

35000

30000

25000

-0.1

0.0001

20000

0 15000

0.001

10000

Pristine State

5000

0.01

G (S)

20% increase in ‘m’

-0.2 40000

35000

30000

25000

20000

15000

10000

5000

0.00001 -0.3

Frequency (Hz)

Frequency (Hz)

(b)

(a) 3.00E+04 100000

PZT patch, ya

PZT patch, xa

Structure, y

0.1

40000

-3.00E+04

40000

35000

30000

25000

20000

15000

10000

5000

35000

y =-ya

y =-ya

0.001

30000

25000

20000

15000

-1.00E+04

10000

Structure, x

1.00E+04

5000

10

y, y a (Ns/m)

x, x a (Ns/m)

1000

Frequency (Hz)

Frequency (Hz)

(d)

(c) PZT patch

|Z|, |Za| (Ns/m)

10000

100

Pristine State 1

20% increase in ‘m’ 40000

35000

30000

25000

20000

15000

10000

5000

0.01

Frequency (Hz)

(e)

Fig. 3.10 Signatures for SDOF-Case VI, m= 0.0002 kg, k= 197.4 N/m, c= 0.01257 Ns/m. (a) Conductance vs Frequency.

(b) Susceptance vs Frequency.

(c) Real impedance vs Frequency (pristine). (d) Imaginary impedance vs Frequency (pristine). (e) Absolute impedance vs Frequency (pristne).

78

7

800

6

700

5

600

4

500

B (S)

3

400 300

2

200

1

100

Frequency (Hz)

2000

Frequency (Hz)

(b)

(a)

10000000

6000000

Structure, x 4000000

100000

2000000

1000

PZT patch, xa 2000

1600

1200

800

Structure, y

-6000000

Frequency (Hz)

Frequency (Hz)

(d)

(c) 100000000

Structure

|Z|, |Za| (Ns/m)

10000000 1000000 100000 10000 1000

PZT patch 2000

1600

1200

800

400

0

100

Frequency (Hz)

(e)

Fig. 3.11 Appearance of large number of ‘false’ peaks. (a) Conductance vs Frequency.

(b) Susceptance vs Frequency.

(c) Real impedance vs Frequency.

(d) Imaginary impedance vs Frequency.

(e) Absolute impedance vs Frequency.

79

2000

1600

1200

800

-2000000

-4000000

100 400

0 400

10000

PZT patch, ya

0

y, y a (Ns/m)

1000000

0

x, x a (Ns/m)

1600

1200

0

800

0

1200

1100

1000

900

800

700

600

500

400

300

200

0

400

G (S)

Chapter 3: PZT-Structure Electro-Mechanical Interaction

Chapter 3: PZT-Structure Electro-Mechanical Interaction

observed that instead of one peak (at 500 Hz, the natural frequency of the structure), there are a total of 9 peaks in the G-plot (Fig. 3.11a). These peaks appear at all those frequencies which are characterised by the condition ‘y = -ya’, typically near the natural frequencies of the PZT patch. 3.5

STRUCTURE-PZT INTERACTION IN COMPLEX SYSTEMS In the previous section, various cases were investigated for structure-PZT patch

interaction for a simple SDOF system. This section considers a multiple degree of freedom (MDOF) system for understanding the structure-PZT interaction. Consider a 2D-MDOF structure, driven by a PZT patch, as shown in Fig. 3.12(a). The PZT patch is assumed to be 10 mm long and 0.2 mm thick, and extending along the width of the host structure, thereby ensuring plain strain conditions. The patch is assumed to possess the properties listed in Table 3.1. The mechanical impedance for this system was determined using Eq. (3.3), by computing the drive point harmonic velocity corresponding to a finite harmonic actuating force, using dynamic harmonic finite element method (FEM). Taking advantage of symmetry, the finite element model of one half of the structure, shown in Fig. 3.12(b), is sufficient for analysis. Fine meshing was carried out in the region surrounding the PZT patch in order to realistically simulate the transfer of the PZT forces (Liang et al., 1993). Material properties of unalloyed and low-alloyed steels at 25oC (source: Richter, 1983) were considered for the host structure (see Table 3.2). The real and the imaginary parts of the electrical admittance were then determined by using Eq. (2.24) in the frequency range 115-165 kHz, at an interval of 200Hz. A Visual Basic program listed in appendix A was used to perform computations. In order to ensure adequacy of finite element meshing, modal analysis was additionally performed. According to Makkonen et al. (2001), while carrying out dynamic harmonic analysis by FEM, the element size should be sufficiently small (typically 3 to 5 nodal points per half-wavelength) to ensure accuracy of solution. All the modes of vibration in the frequency range of interest were analysed, from which the wavelengths of the excited modes were found to be quite large as compared to the element size considered. Fig. 3.13 typically shows mode 48 (highest excited mode), characterised by a natural frequency of 162.46 kHz. From

80

Chapter 3: PZT-Structure Electro-Mechanical Interaction

the figure, the wavelength of the excitation can be seen to be quite large as compared to the element size. Hence, the criteria of Makkonen et al. (2001) is clearly satisfied.

PZT patch

10 mm = = A

Structure

50mm

B

200 mm = =

(a)

50mm

B (Point of attachment of PZT patch.)

100mm

(b) Fig. 3.12 A MDOF system considered for PZT-structure interaction. (a) 2D host structure. (b) Finite element model of the right half of structure. 81

Chapter 3: PZT-Structure Electro-Mechanical Interaction

Table 3.2 Key material properties of structure. Physical Parameter

Value

Density (kg/m3)

7850

Young’s Modulus (N/m2)

2.1267 x 1011

Shear Modulus (N/m2)

8.2815 x 1010

Poisson’s Ratio

0.2840

Wavelength of excited mode

Fig. 3.13 Graphical representation of Mode 48 (f = 162.46 kHz).

Fig. 3.14 shows the interaction plots for this structure. As can be seen from Fig. 3.14(e), the mechanical impedance of the structure varies with frequency, attaining minimum values at the points of resonance and maximum at the points of antiresonance. From Fig. 3.14(f), it is seen that the structural impedance is much higher as compared to the PZT patch. Both the real as well as the imaginary parts of the structural mechanical impedance are of very high order of magnitude as compared to their PZT counterparts (Fig.3.14c and 3.14d). As such, like in Case II, the B-plot is a straight line and the G-plot captures the variation in the real part of the structural impedance. The magnitude of ‘x’ (the real component of structural impedance) shows many peaks (e.g. points x1 and x2 in Fig.3.14c). The G-plot exhibits peaks at almost same frequencies (e.g. points G1 and G2 in Fig.3.14a, corresponding to x1 and x2). Thus the G-plot reflects the dynamic characteristics of the structure. Although the frequency range considered includes the

82

resonant

Chapter 3: PZT-Structure Electro-Mechanical Interaction

0.012

1

0.011

0.8

G2 G1

0.6

B (S)

G (S)

0.01

0.009

0.4 0.008 0.2

165000

20000

(a) 2 –D structure used in the study. Structure, y (b) Finite element model of the right half of-40000 the structure. y, y a (Ns/m)

60000

PZT patch, xa

165000

160000

y2

y1

-60000 -80000

-100000

20000

-120000

165000

160000

155000

150000

145000

140000

135000

130000

115000

165000

160000

155000

150000

145000

140000

135000

130000

125000

120000

115000

125000

-140000

0

120000

Frequency (Hz)

Frequency (Hz)

(c)

(d) 1000000

160000

100000

140000

10000

|Z|, |Za| (Ns/m)

180000

120000 100000 80000

PZT patch

Structure

1000 100 10

165000

160000

155000

150000

145000

135000

130000

125000

115000

120000

1

165000

160000

155000

150000

145000

140000

135000

130000

125000

120000

115000

60000 140000

x, x a (Ns/m)

155000

PZT patch, ya

x2 0 x 100000Fig. 3.12 Mode 1shape 48 (f = 162.459 kHz). Structure, x -20000

|Z| (Ns/m)

150000

(b)

120000

40000

145000

Frequency (Hz)

(a)

80000

140000

135000

130000

125000

Frequency (Hz)

120000

0 115000

160000

155000

150000

145000

140000

135000

130000

125000

120000

115000

0.007

Frequency (Hz)

Frequency (Hz)

(e)

(f)

Fig. 3.14 Signatures for MDOF system considered in Fig. 3.12. (a) Conductance vs Frequency.

(b) Susceptance vs Frequency.

(c) Real impedance vs Frequency.

(d) Imaginary impedance Vs Frequency.

(e) Structural absolute impedance vs Frequency. (f) Relative absolute impedance of structure and PZT patch. 83

Chapter 3: PZT-Structure Electro-Mechanical Interaction

frequency of the PZT patch (143.6 kHz, see Fig. 3.14f), no false peak is visible in the G and B plots. This is because the PZT mechanical impedance is sufficiently low as compared to its structural counterpart. This is the desirable criteria for any real-life structural system. 3.6

IMPLICATIONS OF STRUCTURE-PZT INTERACTION It is apparent from the case studies discussed in Sections 3.4 and 3.5 that the

nature of the interaction between the PZT patch and the host structure, and the resulting electrical admittance spectra, are both governed entirely by the relative magnitudes of x, xa, and y, ya. As a guideline, the PZT patch should typically possess negligible mass and stiffness as compared to the structure (to ensure |Za| > |Za|. Otherwise, resonance peaks could be shifted in the conductance plot and in worst case, false peaks might also occur. It is shown that the raw-conductance and the raw-susceptance signatures contain passive contribution from the PZT patch, which is insensitive to damage. Especially, the imaginary part is drowned by the passive PZT contribution. Filtering out the PZT contribution using signature decomposition could significantly improve the quality of signature, particularly the susceptance signature, which has so far not been utilized by researchers. Subsequent chapters will show how both the conductance as well as the susceptance could be employed to derive more meaningful information about the structural parameters and for improved SHM/ NDE.

88

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

Chapter 4 DAMAGE ASSESSMENT OF SKELETAL STRUCTURES VIA EXTRACTED MECHANICAL IMPEDANCE

4.1 INTRODUCTION This chapter presents a new method of damage diagnosis, based on changes in structural mechanical impedance at high frequencies. The mechanical impedance is extracted from the electro-mechanical admittance signatures of piezo-impedance transducers, by means of signature decomposition, which was introduced in the preceding chapter. The main feature of this approach is that both the real and the imaginary components of the admittance signature are utilized in damage quantification. A complex damage metric is proposed to quantify damage parametrically, based on the extracted structural parameters. As proof of concept, the chapter reports a damage diagnosis study conducted on a model RC frame subjected to base vibrations on a shaking table. The proposed methodology was found to perform better than the existing damage quantification approaches i.e. the low frequency vibration methods as well as the traditional raw-signature based damage quantification using EMI technique. 4.2 ANALOGY BETWEEN ELECTRICAL AND MECHANICAL SYSTEMS The concept of mechanical impedance, introduced in the previous chapter, is analogous to the concept of electrical impedance in electrical circuits (Halliday et al., 2001). The impedance approach allows a simplified analysis of complex mechanical systems by reducing the differential equations of Newtonian mechanics into simple algebraic equations, as in the electrical circuits.

89

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance k m c

Fi

F(t) = Focosωt

FResultant

(Fi - Fs) x(t) = xocos(ωt - φ)

φ Fd

Fs

(a)

x

(b)

Fig. 4.1 (a) A single degree of freedom (SDOF) system under dynamic excitation. (b) Phasor representation of spring force (Fs), damping force (Fd) and inertial force (Fi).

Consider a SDOF spring-mass-damper system, subjected to a dynamic excitation force Fo at an angular frequency ω, as shown in Fig. 4.1(a). Let the instantaneous velocity response (which is same for each component of the system due to parallel connection) be

x& = x& o cos(ωt − φ )

(4.1)

where x&o is the velocity amplitude and φ the phase lag of the velocity with respect to the applied force. Displacement and acceleration can be determined from Eq. (4.1) by integration and differentiation respectively. Hence, the force associated with each structural element i.e. the spring (the elastic force), the damper (the damping force) and the mass (the inertial force) can be determined. Thus, Damping force,

Fd = cx& = cx& o cos(ωt − φ )

(4.2)

Inertial force,

π  Fi = m&x& = mx& oω cos ωt − φ +  2 

(4.3)

Spring force,

π  k x&   Fs = k x =  s o  cos ωt − φ −  2  ω  

(4.4)

From Eqs. (4.2) to (4.4), it is observed that this system is mathematically analogous to a series LCR circuit in the classical electricity. The term

x&

is

analogous to the current (which is same for each element of the LCR circuit) and the mechanical force is analogous to the electro-motive force (voltage). The damper is analogous to the resistor, since Fd is in phase with x& (Eq. 4.2). The mass is analogous to the inductor, since Fi leads x& by 90o (Eq. 4.3). Similarly, the spring is

90

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

analogous to the capacitor, since Fs lags behind x& by 90o (Eq. 4.4). These terms can be analogously represented by a phasor diagram, as shown in Fig. 4.1(b). Hence the amplitude of the resultant force (analogous to voltage across the entire LCR circuit) is given by Fo = Fdo2 + ( Fio − Fso ) 2

(4.5)

where the subscript ‘o’ denotes amplitude of the concerned force. Substituting expressions for the amplitudes from Eqs. (4.2) to (4.4) and solving, we can obtain the amplitude of the mechanical impedance of the structure as  mω 2 − k c +  ω 

F | Z |= o = x& o

2

  

2

(4.6)

The quantity Z is analogous to the electrical impedance (ratio of voltage to current) of an LCR circuit. Using complex number notation, analogous to that used in classical electricity, it may be expressed in Cartesian and polar coordinates as  mω 2 − k   j = Z e jφ Z = x + yj = c +   ω 

(4.7)

The phase lag ‘φ’ of the velocity ‘ x& ’with respect to the resultant driving force ‘F’ is given by (Fig. 1b) tan φ =

Fi − Fs mω 2 − k = Fd cω

(4.8)

Here, ‘x’ is the dissipative or real part and ‘y’ is the reactive or imaginary part of the mechanical impedance. It should be noted that damping can be included in the stiffness itself, by adopting complex stiffness, as given by k = k (1 + ηj )

(4.9)

where the term η, commonly known as mechanical loss factor, can be expressed as

η=

cω k

(4.10)

4.3 MEASUREMENT OF MECHANICAL IMPEDANCE The concept of mechanical impedance, introduced above for SDOF systems, can be easily extended to any complicated real-life MDOF system. Although Eqs. (4.7) and (4.8) have been derived here for the SDOF system, complex structural

91

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

systems too essentially possess a mechanical impedance consisting of the real (dissipative) and imaginary (reactive) components. These two terms can be considered to represent a purely resistive element (such as damper) connected in parallel to a purely reactive element (such as spring or mass or a combination of the two). The two terms can be considered to be the “equivalent SDOF” representation of the actual system. However, analytical determination of mechanical impedance for complex MDOF systems is very tedious. It can be measured experimentally by applying a sinusoidal force at a point on the structure and measuring the resulting velocity at that point in the direction of the force. Conventionally, this is done by using impedance head, which consists of a force transducer and an accelerometer (Hixon, 1988). The force transducer is an electro-magnetic shaker, which produces a sinusoidal force proportional to the input sinusoidal voltage. The accelerometer measures acceleration at the point of interest, again in the form of a proportional sinusoidal voltage signal. Being harmonic, velocity can be deduced from acceleration by integration. The magnitude of the mechanical impedance is thus determined from the ratio of the measured force and the velocity amplitudes (Eq. 4.6), and the phase difference between the two is given by the phase difference between the corresponding measured voltage signals.

However, the conventional

impedance heads possess very small operational bandwidth, which prohibits their application for high frequencies. The same holds equally true for conventional accelerometers. Even the high-tech miniaturized accelerometers share the disadvantages of high cost and small operational bandwidth (Giurgiutiu and Zagrai, 2002; Lynch et al., 2003b). The next sections demonstrate how this difficulty can be overcome with the aid of the EMI technique. 4.4 DECOMPOSITION OF ADMITTANCE SIGNATURES In the previous chapter, the coupled electro-mechanical admittance signature, acquired by using EMI technique, was decomposed into active and passive components. At this point, the author would like to introduce the following definitions.

92

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

(a) Raw Complex Admittance The complex electro-mechanical admittance of a piezo-impedance transducer bonded to a structure, obtained from direct measurement through the EMI technique, is called the raw complex admittance. It is denoted by Y and can be expressed as Y = G + Bj

(4.11)

The real part, G, is called the raw conductance and the imaginary part, B, is called the raw susceptance. A frequency plot of the raw conductance is called the raw conductance signature (RCS) and that of the raw susceptance is called the raw susceptance signature (RSS). (b) Passive Complex Admittance The contribution of the PZT patch in the complex electro-mechanical admittance, or in other words the passive component, is called the passive complex admittance or the PZT admittance. It is denoted by YP and can be expressed as YP = GP + BP j

(4.12)

The real part, GP, is called the passive conductance and the imaginary part, BP, is called the passive susceptance. Expressions for GP and BP are given by (From Eq. 3.22)

{

GP = 2ω

wl T δε 33 + d 312 Y Eη h

BP = 2ω

wl T ε 33 − d312 Y E h

{

}

}

(4.13) (4.14)

(c) Active Complex Admittance Active complex admittance is that part of the raw complex admittance, which arises from the electro-mechanical interaction between the PZT patch and the host structure. It is denoted by YA and can be expressed as YA = GA + BA j

(4.15)

The real part, GA, is called the active conductance and the imaginary part, BA, is called the active susceptance. The active conductance and susceptance can be

93

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

obtained from Eqs. (3.26) and (3.27) respectively. A frequency plot of active conductance is called the active conductance signature (ACS) and that of active susceptance is called the active susceptance signature (ASS). Traditionally, the researchers working in the field of EMI technique have utilized the deviation in the RCS alone for damage assessment, using statistical indices such as RMSD, RD, MAPD etc., which have been described in detail in Chapter 2, with their shortcomings highlighted. As seen in the analysis presented in Chapter 3, the raw-conductance is mixed with the non-interactive passiveconductance of the PZT patch, which masks its damage detection ability. Most of the published work related to the EMI technique has been focused on relatively light structures. In majority of the reported investigations, the damage was typically simulated non-destructively such as by loosening bolts or similar components (Sun et al., 1995; Ayres et al., 1998; Park et al., 2001). Only few destructive tests on the structures instrumented with PZT patches have been reported (Soh et al., 2000; Park et al., 2000a).

In many structures, simply the

‘detection’ of damage might be more than sufficient, which can be done conveniently by means of the conventional statistical indices. However, in civilstructures, we often need to find out whether the damage is ‘incipient’ or ‘severe’. We might even tolerate an incipient damage without endangering the lives or properties. This fact has motivated us to extract the drive point structural impedance from measured raw signatures for damage quantification. 4.5 EXTRACTION OF STRUCTURAL MECHANICAL IMPEDANCE OF SKELETAL STRUCTURES 4.5.1 Computational Procedure Electro-mechanical admittance relations were derived in Chapter 1 for piezoimpedance transducers bonded to ‘skeletal’ structures. Using the principle of signature decomposition introduced in Chapter 3, for a skeletal structure, the active complex admittance, YA , can be expressed as a function of structural impedance, PZT impedance and frequency as (using Eq. 3.20)

94

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

YA = GA + BA j = 2ωj

 tan κl wl Z a d 312 Y E  h (Z + Z a )  κl

  

(4.16)

Substituting Za = xa + yaj (actuator impedance), Z = x + yj (structure impedance), tan κl = r + tj , Y E = Y E (1 + ηj ) , and after eliminating the imaginary term from the κl denominator, we obtain 0.5YA = ωj

wl ( xa + ya j )[( x + xa ) − ( y + ya ) j ] 2 E d 31Y [(r − ηt ) + (t + ηr ) j ] h [( x + xa ) 2 + ( y + ya ) 2 ]

(4.17)

Denoting (x + xa) by xT and (y + ya) by yT, this equation can be rewritten in a simplified form as 0.5YA = Kωj

( P + Qj )( R + Tj ) ( xT2 + yT2 )

(4.18)

where the terms K, P, Q, R and T are defined as wl 2 E d31Y h

K=

(4.19)

P = xa xT + ya yT

Q = ya xT − xa yT

(4.20)

R = r − ηt

T = t + ηr

(4.21)

YA can now be decomposed into the real and imaginary parts, GA and BA respectively as

0.5G A =

− K (QR + PT )ω ( xT2 + yT2 )

(4.22)

0.5BA =

K ( PR − QT )ω ( xT2 + yT2 )

(4.23)

Dividing Eq. (4.22) by Eq. (4.23) and solving, we can obtain the ratio c = Q / P as c=

Q (GA / BA ) R + T = P (GA / BA )T − R

(4.24)

Further, by using Eq. (4.20), we can obtain the ratio Ct = yT/xT as Ct =

yT y − cxa = a xT cya + xa

(4.25)

Hence, we can solve Eqs. (4.20) and (4.22) using the constants Ct (determined from PZT parameters) and c (determined using GA and BA) to obtain following expression for xT and yT

95

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

xT =

− Kω (cR + T )( yaCt + xa ) , 0.5GA (1 + Ct2 )

and yT = Ct xT

(4.26)

It should be noted that xa and ya can be determined from the PZT parameters, as given by Eq. (3.10). Hence, xT and yT , can be determined from ‘x’ and ‘y’ as x = xT − xa

and

y = yT − ya

(4.27)

Following this computational procedure, ‘x’ and ‘y’ can be extracted from the measured conductance and susceptance signatures alone. Only the PZT parameters are assumed known. No a-priori information about the structure is warranted. It is important to predict the PZT mechanical impedances xa and ya accurately. In this chapter, this is done using Eq. (3.10), on the basis of the data provided by the manufacturer. Methods for more accurate predictions will be covered in the subsequent chapters. Further, in order to ensure smooth computations, |x| > |xa| and |y| > |ya|. Otherwise, false peaks could appear in the impedance spectra. This is consistent with the findings reported in Chapter 2. 4.5.2 Determination of (tan κl/κl) In these computations, the quantity tan κl / κl = r + tj must be determined precisely using the theory of complex algebra (Kreyszig, 1993). This term was approximated by Liang et al. (1994) as unity under the assumption that the operational frequency is much lower than the first resonant frequency of the PZT patch (or in other words “quasi-static sensor approximation”). However, this is not the case in SHM applications where the frequency range is typically in few hundred kHz. Denoting κl by z, we can write tan κl sin z = κl z cos z

(4.28)

Noting from the theory of complex numbers (Kreyszig, 1993) that e jz + e − jz cos z = 2

e jz − e − jz sin z = , 2j

and

and substituting z = rl + (im)j, we can derive, after algebraic manipulations,

96

(4.29)

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

tan κl  au − bv   av + bu  = 2 − 2 j 2  2  κl u +v  u +v  where

(4.30)

a = [e − jm + e jm ] sin(rl )

(4.31)

b = [e − jm − e jm ] cos(rl )

(4.32)

c = [e − jm + e jm ] cos(rl )

(4.33)

d = [e − jm − e jm ] sin( rl )

(4.34)

u = c(rl ) − d (im)

(4.35)

v = d (rl ) + c(im)

(4.36)

4.5.3 Physical Interpretation of Drive Point Impedance As mentioned before, all the previous reported works so far utilized only the real part of the raw complex admittance to quantify structural damages. However, in the newly developed methodology, both the real and imaginary parts are utilized. They are first filtered, using signature decomposition, to remove the PZT contribution and to yield active signatures. The active signatures

are further

processed to extract the real and the imaginary parts of the drive point mechanical impedance, which are direct functions of the structural parameters. The drive point impedance is typically a function of frequency. The absolute mechanical impedance, | Z |=

x 2 + y 2 , attains minimum values at the points of structural

resonance and maximum values at the points of anti-resonance. The real part, ‘x’, is the equivalent SDOF damping of the structure and the imaginary part, ‘y’, is the equivalent SDOF stiffness- mass factor (see Eq. 4.7) of the host structure, at the ‘drive point’ of the PZT patch. In other words they are the structural parameters ‘apparent’ to the PZT patch at its ends. Being direct structural parameters, these should be more sensitive to structural damages than stresses or strains, which are secondary effects. In this manner, a piezo-impedance transducer identifies the ‘equivalent’ parameters of the ‘black-box’ host structure in the form of SDOF damping and stiffness-mass factors.

97

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

4.6 DEFINITION OF DAMAGE METRIC BASED ON EXTRACTED STRUCTURAL IMPEDANCE As pointed out in the preceding section, a damage index based on drive point structural impedance is expected to be more realistic than those based on RCS, as in the conventional approaches. This has motivated the author to define a complex damage metric as D = Dx + Dy j

(4.37)

where Dx denotes the damage metric of the real part of the structural impedance (equivalent SDOF damping) and Dy the corresponding value for the imaginary part (equivalent SDOF stiffness-mass factor) . Dx is defined as the average of Dxi, which is the value of the metric at the ith frequency point, defined as follows. If ( xi / xio ) < 1 , then, Dxi = 1 − ( xi / xio ) else,

Dxi = 1 − ( xio / xi ) . Here, xio is the baseline value at the ith frequency point and xi is the value at the current state. The other component, Dy, is similarly defined using ‘y’ in place of ‘x’. This definition of the damage metric quantifies the damage on a uniform 0-1 (fractional) or 0-100 (percentage) scale. Hence, Dx measures the changes in equivalent SDOF damping associated with the drive point of the PZT patch and Dy similarly measures the variation in the equivalent mass-stiffness factor. Being based on the extracted impedance rather than the raw-signatures, this method quantifies the damage parametrically. 4.7 PROOF OF CONCEPT APPLICATION: DIAGNOSIS OF VIBRATION INDUCED DAMAGES The proposed mechanical impedance based methodology was tested for damage detection on a model RC frame subjected to base vibrations. The test structure was a two-storeyed portal frame, made of reinforced concrete, as shown in Fig. 4.2. This model represented a prototype frame with storey height of 2.9m and span length of 3.3m, at a scale of 1:10. The shaker was an electromagnetic shaking table, rated to a maximum acceleration of 120g and a maximum frequency of 3000Hz.

98

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

130 12 33.5

Patch #2

20 BEAM

Patch #1 25 30 COLUMN

(a)

(b)

Fig. 4.2 (a) Details of test frame (All dimensions are in mm). (b) Test frame just before applying loads.

The test frame was instrumented with two PZT patches, shown in Fig. 4.2 as patch #1 and patch #2, which were bonded to the structure using RS 850-940 epoxy adhesive (RS Components, 2003). Patch #1 was instrumented on the first floor beam, very close to the beam-column joint, a location very critical from the point of view of shear cracks. Patch #2, on the other hand, was instrumented at the bottom face of the second floor beam, near the mid point, a location very critical from the point of view of flexural cracks. Both the patches were 10mm square and 0.2mm thick and conformed to grade PIC 151 (PI Ceramic, 2003). The electrical and mechanical parameters of the PZT patches are as listed in Table 4.1. The test frame was a typical skeletal structure and hence signature decomposition and mechanical impedance extraction outlined in the preceding sections can be conveniently applied. The test loads were applied in the form of vertical base motions of varying frequencies and amplitudes. The buildings are normally subjected to such base motions during earthquakes and underground explosions (Lu et al., 2001). The test was performed in eight phases according to the range of the imposed base motion

99

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

frequencies and the velocity and acceleration amplitudes. The induced base motions are graphically shown in Table 4.2. After each excitation, the patches were scanned to acquire the raw-signatures in the frequency range of 100-150 kHz, at an interval of 100Hz, using HP 4192A impedance analyzer (Hewlett Packard, 1996). The signatures were decomposed to obtain active components first, which were then processed to extract the drive point mechanical impedance. Damage metric was determined by the procedure outlined in the previous sections. A program written in Visual Basic and listed in Appendix B was used for computations. The test structure was also instrumented with conventional sensors such as accelerometers, LVDTs and strain gauges. This part of the instrumentation was carried out by another research group (Lu et al., 2000), which was interested in monitoring the condition of the frame by low frequency vibration techniques. 4.7.1 Flexural Damage Prediction by PZT Patch #2 The raw-signatures of PZT patch #2 are shown in Fig. 4.3. Fig. 4.4 shows the components Dx and Dy of the complex damage metric at various states. It also shows the RMSD index (conventional index in the EMI technique) for comparison. From State 1 to State 3, only minor deviations could be noticed in the raw-signatures. This observation was consistent with previous prediction (Lu et al., 2000) that flexural cracks will start from State 4 onwards. At State 4, a prominent shift was observed in the conductance signature (Fig. 4.3a). The inherent cause of the shift can be correlated with damage indices shown in Fig. 4.4(a). This shift in the signature is accompanied by a prominent rise in the value of Dx. This signifies a change in the Table 4.1 Key properties of PZT patches (PI Ceramic, 2003). Physical Parameter Density (kg/m3)

Value 7800

T Electric Permittivity, ε 33 (farad/m)

2.124 x 10-8

Piezoelectric Strain Coefficient, d31 (m/V)

-2.10 x 10-10

Young’s Modulus, Y E (N/m2)

6.667 x 1010

Dielectric loss factor, δ

0.015

100

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

Table 4.2 Base motions and time-histories to which test frame was subjected. PHASE

LOAD DESCRIPTION

TYPICAL BASE MOTION TIME HISTORIES

BASELINE 10.0

Phase1

Freq.(850~200)Hz Acceleration12.48g / Velocity0.027m/s

Base Acceleration

0.0 -10.0 0.00

0.05

0.10

0.15

0.20

Time (s)

STATE 1 2.0

Phase 2

(150-15)Hz 3.016g / 0.057m/s

Base Acceleration

0.0 -2.0 0.00

0.20

0.40

0.60

0.80

Time (s)

STATE 2 50

Phase 3

700Hz 20.36g / 0.131m/s

Base Acceleration

0 -50 0.00

0.10

0.20

0.30

Time (s)

STATE 3 50

Phase 4

700Hz 25.62g / 0.203m/s

Base Acceleration

0 -50 0.00

0.10

0.20

0.30

Time (s)

STATE 4 50

Phase 5

200Hz 23.67g / 0.443m/s

Base Acceleration

0 -50 0.00

0.20

0.40

0.60

Time (s)

STATE 5 50

Phase 6

200Hz 13.46g / 0.376m/s

Base Acceleration

0 -50 0.00

0.20

0.40

0.60

Time (s)

STATE 6 50

Phase 7

200Hz 25.12g / 0.744m/s

Base Acceleration

0 -50 0.00

0.20

0.40

0.60

Time (s)

STATE 7 50 Base Acceleration

Phase 8

200Hz 25.12g / 0.744m/s

0 -50 0.00

0.20

0.40

Time (s)

STATE 8

101

0.60

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

0.008 5

4

0.00105

6

Susceptance (S)

Conductance (S)

0.00125 3

0.00085 0.00065

Baseline

0.00045 0.00025 100

1

110

120

130

140

2

4, 5

0.007 0.006

1,2,3

0.005 0.004

6

0.003 0.002 100

150

110

120

130

140

150

Frequency (kHz)

Frequency (kHz)

(a)

(b)

Fig. 4.3 Raw-signatures of PZT patch #2 at various damage states (1, 2, 3, .., 6). (a) Raw-conductance. (b) Raw-susceptance.

Patch found damaged

Visible cracks 140

80

120 Dx

RMSD (%)

Dx, Dy

60

Dy

40 20

100 80 60 40 20 0

0 1

2

3

4

5

0

6

Damage States

1

2

3

4

5

6

7

8

Damage States

(a)

(b)

Fig. 4.4 Damage prediction by patch #2. (a) Real and imaginary components of complex damage metric. (b) RMSD (%) in raw-conductance.

equivalent SDOF damping associated with the drive point impedance of the PZT patch. An increase in damping is an expected phenomenon associated with crack development. At State 5, further upward shift of the conductance signature (Fig. 4.3a) as well as increase of Dx (Fig. 4.4a) can be observed. No major change in Dy is observed from State 1 to 5. This is because damping is much more sensitive to

102

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

damage at high frequencies as compared to the stiffness or inertia related effects (Esteban, 1996). The area around the patch was continuously monitored and observable flexural cracks could only be detected at State 6. The patch however provided the necessary warning much earlier, at State 4 itself. State 6 was accompanied by a reduction of Dx and a rise in Dy. A reduction in the ‘apparent damping’ could be due to the development of disbonding between the patch and the host structure and possible damage to the patch itself. This is also reflected in the equivalent stiffnessmass factor since the associated index Dy shows an abrupt rise in magnitude, signalling a reduction in the equivalent spring stiffness. This is further supported by the fact that the patch was found to be damaged at State 7 (a crack was detected running through the patch). However, the patch provided the necessary warning much earlier, at State 4. As can be observed from Fig. 4.4(a), the conventional processing approach, the RMSD, failed to respond to damage to the patch itself at State 6 and continued to show a rising trend. 4.7.2 Shear Damage Prediction by PZT Patch #1 Fig. 4.5 shows the raw-signatures of PZT patch #1 and Fig. 4.6 shows the components of the complex damage metric and the RMSD index at various states. From the Baseline State to State 6, the raw-conductance signature of patch #1 did not undergo any substantial change. The indices Dx and Dy also did not display any prominent rise (Fig. 4.6a). Again, higher sensitivity of the associated equivalent damping to incipient damage was confirmed by the relatively large magnitude of Dx as compared to Dy (State 1 to State 6). At State 7, an observable shift was observed in the conductance signature (Fig. 4.5a). This can be seen to be accompanied by rise of Dx (Fig. 4.6a). At State 8, a sudden and more prominent vertical shift of the signature was observed. From Fig. 4.6(a), it is observed that at this stage, both Dx and Dy attained relatively large values, suggesting development of an abrupt damage. Close examination of the region surrounding the patch in fact showed the development of a hairline shear crack near the beam-column joint. The patch however provided the information of the imminent damage at State 7 itself, in the form of an abrupt and significant variation in signature.

103

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

Baseline, 1,2,3,4,5,6

0.007 8

0.0008

7

6

0.0006 0.0004

Baseline, 1,2,3,4,5

0.0002 100

110

120

130

140

Susceptance (S)

Conductance (S)

0.001

7

0.006 0.005 0.004

8

0.003 100

150

110

120

130

140

150

Frequency (kHz)

Frequency (kHz)

(b)

(a)

Fig. 4.5 Raw-signatures of PZT patch #1 at various damage states (1, 2, 3, .., 8). (a) Raw-conductance. (b) Raw-susceptance.

80

100 Dx

80

RMSD (%)

Dx, Dy

60

Dy 40 20 0

60 40 20 0

1

2

3

4

5

6

7

8

0

1

Damage States

2

3

4

5

6

7

8

Damage States

(b)

(a) Fig. 4.6 Damage prediction by PZT patch #1.

(a) Real and imaginary components of complex damage metric. (b) RMSD (%) in raw-conductance.

4.7.3 Damage Sensitivity of the Proposed Methodology It is worthwhile to compare the sensitivity of the proposed damage diagnosis methodology with the low frequency vibration based methods as well as conventional approach based on raw-conductance signatures utilizing statistical quantifiers. Shown in Fig. 4.7(a) are the reductions in the natural frequency associated with the local

104

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

vibrations of the second floor beam (on which PZT patch #2 was instrumented). These were obtained using conventional accelerometers, which have a relatively small frequency bandwidth, generally the upper limit is of the order of 100-200Hz. From overall structural point of view, these frequencies correspond to a ‘higher’ mode. These are compared with the RMSD of the raw-conductance (traditional approach in EMI technique) as well as with the RMSD of the extracted real part of structural impedance, ‘x’ in Fig. 4.7(b). The higher sensitivity of ‘x’ to damage as compared to the low frequency vibration techniques as well as the conventional damage quantification approach (based on G) in EMI technique is clearly evident from Fig. 4.7(b). Thus the new methodology enables us to derive greater information about the nature of damages occurring in the vicinity of the PZT patches, viz. the equivalent SDOF stiffness, the damping and the mass associated with the drive point of the PZT patch. It predicts the damage on a uniform 0-1 (fractional) or 0-100 (percent) scale. It is therefore more pragmatic than the previously reported non-parametric statistical approaches. It is recommended that (Dx+Dy) ≤ 20% indicates incipient nature of damage and (Dx+Dy) ≥ 50% indicates severe nature of damage. Tests will be reported in the Chapter 6 for calibrating damage with specific changes in ‘x’ and ‘y’. 250 200

150

RMSD (%)

Frequency (Hz)

200

100

150 100

50

0

%Reduction in natural frequency RMSD Based on G RMSD based on x

50 0

0

1

2

3

4

5

6

7

8

1

Damage States

2

3

4

5

Damage States

(b)

(a)

Fig.4.7 (a) Natural frequency of vibration of floor #2 beam at various damage states. (b) Evaluation of damage based on natural frequency, raw-conductance and extracted mechanical impedance.

105

Chapter 4: Damage Assessment of Skeletal Structures via Extracted Mechanical Impedance

4.8 DISCUSSIONS To author’s best knowledge, this is the first attempt to extract structural parameters from the measured electrical admittance signatures of piezo-impedance transducers. In the proposed derivation, only 1D vibrations have been considered. The electro-mechanical coupling in the other direction has been neglected. This is justifiable in the present case due to the skeletal nature of the test structure. In other structures, significant coupling could be present in the other direction. Analysis for structures in which two-dimensional interaction is dominant will be presented in next chapter. Nonetheless, the present method can still be applied to structures where 2D coupling is significant. In this case, the extracted parameters will represent the ‘equivalent 1D parameters’. The 1D analysis, as presented in this chapter, offers a simple and convenient approach to make meaningful interpretations about damage. 4.9 CONCLUDING REMARKS In this chapter, a new method of analyzing the electro-mechanical admittance signatures obtained from the PZT patches bonded to structures has been presented. The proposed method extracts the ‘apparent’ drive point structural impedance associated with the bonded PZT patch. A complex damage metric has been proposed to quantify structural damages, based on changes in the drive point mechanical impedance of the host structure. The real part of the damage metric measures changes in the equivalent SDOF damping caused by damages, and the imaginary part similarly represents the changes in the equivalent SDOF stiffnessmass factor associated with the drive point of the PZT patch. The proposed method was tested on a model frame structure that was subjected to base vibrations on a shaking table. The instrumented PZT patches were found to provide a meaningful insight into the changes taking place in the structural parameters as a result of damages. The patches were successful in identifying flexural and shear cracks, two prominent types of incipient damages in RC frames. The proposed method was found to have a higher sensitivity to damages as compared to the existing approaches.

106

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

Chapter 5 GENERALIZED ELECTRO-MECHANICAL IMPEDANCE FORMULATIONS: THEORETICAL DEVELOPMENT AND SHM APPLICATIONS

5.1 INTRODUCTION It was demonstrated in Chapter 4 that structural mechanical impedance is far more reliable for SHM as compared to raw admittance signatures. However, the methodology based on signature decomposition covered in Chapter 4 is in principle valid for skeletal structures only. This chapter introduces new generalized PZTstructure electro-mechanical formulations valid for the more general class of structures where significant 2D coupling exists between the PZT patch and the host structure. The proposed formulations can be easily employed to extract the mechanical impedance of any ‘unknown’ structural system from the admittance signatures of a surface bonded PZT patch. The chapter also outlines a new methodology to quantify structural damages using the extracted impedance spectra, suitable for diagnosing damages in structures ranging from miniature precision machine components to large civil-structures. 5.2 EXISTING PZT-STRUCTURE INTERACTION MODELS Two well-known approaches for modelling the behaviour of PZT-based electro-mechanical smart systems are the static approach and the impedance approach. The static approach, proposed by Crawley and de Luis (1987), assumes frequency independent actuator force, determined from static equilibrium and strain compatibility between the PZT patch and the host structure. In this approach, the patch is assumed to be a thin bar (length ‘l’, width ‘w’ and thickness ‘h’), under

107

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

static equilibrium with the structure, which is represented by its static stiffness Ks, 3(z)

Static electric field

2(y)

E3

1(x)

PZT patch

w

Structure KS x

h l

Fig. 5.1 Modelling of PZT-structure interaction by static approach.

as shown in Fig. 5.1. In this configuration, owing to static condition, the imaginary component of the complex terms in the PZT constitutive relations (Eqs. 2.13 and 2.14) can be dropped. Hence, from Eq. (2.14), the axial force in the PZT patch can be expressed as FP = whT1 = wh (S1 − d31 E3 )Y E

(5.1)

Similarly, the axial force in the structure can be determined as FS = − K S x = − K S lS1

(5.2)

The negative sign signifies that a positive displacement ‘x’ causes compressive force in the spring (the host structure). Force equilibrium in the system implies that FP and FS should be equal, which leads to the equilibrium strain, Seq, given by S eq =

d 31E3   1 + K S l   Y E wh   

(5.3)

Hence, from Eq. (5.2), the magnitude of the force in the PZT (or the structure) can be worked out as Feq = K S lS eq . In order to determine the response of the system under an alternating electric field, the static approach recommends that a dynamic force with amplitude Feq = K S lS eq be applied to the host structure, irrespective of the frequency of

actuation. Since the static approach employs only static PZT properties, the effects of damping and inertia, which significantly affect PZT output characteristics, are completely ignored. Because of these reasons, the static approach often leads to

108

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

significant errors, especially near the resonant frequency of the structure or the patch. (Liang et al., 1993; Fairweather 1998). In order to alleviate this inaccuracy, impedance approach was proposed by Liang et al. (1993, 1994), based on dynamic equilibrium rather than static equilibrium and by rigorously including dynamic PZT properties and structural stiffness. In the impedance approach, the host structure is represented by mechanical impedance Z, rather than a pure spring, as shown in Fig. 2.8(b). The force-displacement relationship for the structure (Eq. 5.2) is replaced by impedance based force-velocity relationship (Eq. 2.20). Further, instead of actuator’s static stiffness, the impedance approach considers actuator impedance Za, similar in principle to structural impedance. Impedance model based electromechanical formulations have already been derived for 1D structures in Chapter 2. Zhou et al. (1995, 1996) extended 1D impedance approach to model the interactions of a generic PZT element coupled to a 2D host structure. The analytical model of Zhou et al. is schematically shown in Fig. 5.2. In this approach, the structural impedance is represented by direct impedances Zxx and Zyy, and the cross impedances Zxy and Zyx, which are related to the planar forces F1 and F2 (in directions 1 and 2 respectively) and the corresponding planar velocities u&1 and u&2 by  Z xx  F1   F  = − Z  2  yx

Z xy   u&1  Z yy  u&2 

(5.4)

Applying D’Alembert’s principle along the two principal axes and after imposing boundary conditions, Zhou et al. (1995) derived following expression for the electro-mechanical admittance across PZT terminals

Zyy

E3

Zyx

z, 3

Zxy y, 2 x, 1

w Zxx

l

Fig. 5.2 Modelling PZT-structure 2D physical coupling by impedance approach (Zhou et al., 1995). 109

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

wl  T 2d312 Y E d312 Y E  sin κl Y = G + Bj = jω + ε 33 −  h  (1 − ν ) (1 − ν )  l

sin κw  −1 1  N   w  1

(5.5)

where κ, the 2D wave number, is given by

κ =ω

ρ (1 − ν 2 ) YE

(5.6)

and N is a 2x2 matrix, given by   l Z yx Z yy    w Z xy Z  κ cos(κw) + xx  −ν κ cos(κl )1 − ν   l Z axx Z axx  Z ayy    w Z ayy   N = (5.7)  Z xy Z yx Z yy    l w Z xx  κ cos(κl ) −ν +   κ cos(κw)1 − ν  l Z Z w Z Z   axx axx ayy ayy    

where Zaxx and Zayy are the two components of the mechanical impedance of the PZT patch in the two principal directions, derived in the same manner as in the 1D impedance approach. 5.3 LIMITATIONS OF EXISTING MODELLING APPROACHES The inability of the static approach in accurately modelling system behaviour has already been pointed out in the previous section. Although Liang et al. (1993, 1994) proposed more accurate formulations using impedance approach, they however ignored the two-dimensional effects associated with PZT vibrations. Their formulations are strictly valid for skeletal structures only, such as the test frame described in Chapter 4. In other structures, where 2D coupling is significant, Liang’s model might introduce serious errors. Zhou et al. (1995, 1996) addressed this problem by extending Liang’s approach to planar vibrations, assuming a fourparameter

impedance model for the host structure (Eq. 5.4). Although the

analytical derivations (Eqs. 5.4-5.7) of Zhou et al. (1995, 1996) are accurate in themselves, the experimental difficulties prohibit their direct application for extraction of host structure’s mechanical impedance. For example, using the EMI technique, we can only obtain two quantities- G and B (Eq. 5.5). If we need to acquire complete information about the structure, we need to solve Eq. (5.5) for 4 complex unknowns- Zxx, Zyy, Zxy, Zyx (or 8 real unknowns). Thus, the system of equations is highly indeterminate (8 unknowns with 2 equations only).

110

As such,

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

the method could not be employed for experimental determination of the drive point mechanical impedance. To alleviate the shortcomings inherent in the existing models, a new concept of ‘effective impedance’ is introduced in the next section, followed a step-by-step derivation of electro-mechanical admittance across the PZT terminals. The new formulations aim to bridge the gap between 1D model of Liang et al. (1993, 1994) and the 2D model of Zhou et al. (1995, 1996). 5.4 DEFINITION OF EFFECTIVE MECHANICAL IMPEDANCE Conventionally, the mechanical impedance at a point on the structure is defined as the ratio of the driving harmonic force (acting on the structure at the point in question) to the resulting harmonic velocity at that point. The existing models are based on this definition, the point considered being the PZT end point. The corresponding impedance is called the ‘drive point mechanical impedance’. However, the true fact is that the mechanical interaction between the patch and the host structure is not restricted at the PZT end points alone, rather it extends all over the finite sized PZT patch. This section introduces a new definition of mechanical impedance based on ‘effective velocity’ rather than ‘drive point velocity’. In the derivations that follow, we assume that the force transmission between the PZT patch and the structure occurs along entire boundary of the patch, and that plane stress conditions exist within the patch. Besides, the patch is assumed square shaped and infinitesimally small as compared to the host structure, so as to possess negligible mass and stiffness. Opposite edges of the patch therefore encounter equal dynamic stiffness from the structure, irrespective of the location of the patch on the host structure. Hence, the nodal lines invariably coincide with the two axes of symmetry in the PZT patch. At the same time, we ignore the effects of the PZT vibrations in the thickness direction, assuming the frequency range of interest to be much lower than the dominant modes of thickness vibration. Consider a finite sized square PZT patch, surface bonded to an unknown host structure, as shown in Fig. 5.3, subjected to a spatially uniform electric field

(∂E / ∂x = ∂E / ∂y = 0) ,

undergoing harmonic variations with time. The patch has

111

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications Boundary S

PZT patch f (Interaction force at boundary)

E3

l l

l

l ‘Unknown’ host structure

Fig. 5.3 A PZT patch bonded to an ‘unknown’ host structure.

half-length equal to ‘l’. Its interaction with the structure is represented in the form of boundary traction ‘f’ per unit length, varying harmonically with time. This planar force causes planar deformations in the PZT patch, leading to variations in its overall area. The ‘effecive mechanical impedance’ of the patch is hereby defined as r f ∫S .nˆ ds F Z a , eff = = (5.8) u& eff u& eff where nˆ is a unit vector normal to the boundary and ‘F’ represents the overall planar force (or effective force) causing area deformation in the PZT patch. ueff = δA/po is defined as ‘effective displacement’, where δA is the change in the surface area of the patch and po its perimeter in the undeformed condition. More precisely, po is equal to the summation of the lengths of ‘active boundaries’, i.e. the boundaries

undergoing

mechanical

interaction

with

the

host

structure.

Differentiation with respect to time of the effective displacement yields the effective velocity, u&eff . It should be noted that in order to ensure overall force equilibrium,



r f ds = 0

(5. 9)

S

The effective drive point (EDP) impedance of the host structure can also be defined on similar lines. However, for determining structural impedance, force needs to be

112

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

applied on the surface of the host structure along the boundary of the proposed location of the PZT patch. 5.5 ELECTRO-MECHANICAL FORMULATIONS BASED ON EFFECTIVE IMPEDANCE

y l T2

Area ‘A’

u2o

T1 Nodal line

l x

u1o

Nodal line

Fig. 5.4 A square PZT patch under 2D interaction with host structure.

Consider a square PZT patch, as shown in Fig. 5.4, under in-plane excitation, caused by a spatially uniform and harmonic electric field, with an angular frequency ω. Since the nodal lines coincide with the axes of symmetry, it suffices to consider the interaction of one quarter of the patch with the corresponding one-quarter of host structure, since only the ratio of the two mechanical impedances that will govern the electrical admittance across the terminals of the PZT patch. Let the patch be mechanically and piezoelectrically isotropic in the x-y plane. Hence, Y11E = Y22E = Y E

and d31 = d32 . Therefore, the PZT constitutive

relations (Eqs. 2.1 and 2.2) can be reduced to T D3 = ε 33 E3 + d31 (T1 + T2 )

S1 =

T1 − νT2 YE

+ d 31E3

113

(5.10) (5.11)

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

S2 =

T2 − νT1 YE

+ d 31E3

(5.12)

where ν is the Poisson’s ratio of the PZT patch. By algebraic manipulation, we can obtain

( S1 + S2 − 2d31E3 )Y E T1 + T2 = 1 −ν

(5.13)

If the PZT patch is in short-circuited condition (i.e. zero electric field), Eq. (5.13) can be reduced to

(T1 + T2 ) short − circuited

( S1 + S 2 )Y E = 1 −ν

(5.14)

As derived by Zhou et al. (1996), the displacements of the PZT patch in the two principal directions are given by

u1 = ( A1 sin κx)e jωt

u2 = ( A2 sin κy )e jωt

and

(5.15)

where the wave number κ is given by Eq. (5.6) and A1 and A2 are constants to be determined from boundary conditions.

The corresponding velocities can be

obtained by differentiating these equations with respect to time. Hence, u&1 =

∂u1 = ( A1 jω sin κx )e jωt ∂t

u&2 =

and

∂u2 = ( A2 jω sin κy )e jωt ∂t

(5.16)

Similarly, corresponding strains can be obtained by differentiation with respect to the two coordinate axes. Hence, S1 =

∂u1 = ( A1κ cos κx)e jωt and ∂x

S2 =

∂u2 = ( A2κ cos κy )e jωt ∂y

(5.17)

From Fig. 5.4, the effective displacement of the PZT patch, considering displacements at the active boundaries of one-quarter of the patch (the boundaries along the nodal axes are ‘inactive’ boundaries) can be deduced as ueff =

δA u1ol + u2 ol + u1ou2o u1o + u2o = ≈ po 2l 2

(5.18)

where u1o and u2o are edge displacements, as shown in Fig. 5.4. Differentiating with respect to time, we obtain the effective velocity as u&eff =

u&1o + u&2o u&1( x =l ) + u&2( y =l ) = 2 2

114

(5.19)

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

From Eqs. (5.8) and (5.19), we can obtain the short-circuited effective mechanical impedance of the quarter PZT patch as

Z a , eff =

(T1( x = l )lh + T2( y = l )lh) short − circuited  u&1( x = l ) + u&2( y = l )    2  

(5.20)

Making use of Eq. (5.14), we obtain or

Z a , eff

( S1( x = l )lh + S 2( y = l )lh)Y E = + u&2( x =l )   u&  (1 − ν ) 1( x =l ) 2  

(5.21)

Substituting the values of the velocities and strains (Eqs. 5.16 and 5.17 respectively) at the two active edges of the PZT patch, and upon solving, we obtain Z a , eff =

2κlhY E jω (tan kl )(1 − ν )

(5.22)

The overall planar force (or the effective force), F, is related to the EDP impedance of the host structure by F = ∫ f .nˆds = − Z s , eff u&eff

(5.23)

S

As in the 1D case, negative sign signifies that a positive effective displacement causes compressive force on the patch (due to reaction from the host structure). Since we are considering a square patch, Eq. (5.23) can be simplified as

 u&1( x = l ) + u&2 ( y = l )   T1( x = l ) hl + T2( y = l ) hl = − Z s, eff  2  

(5.24)

Making use of Eq.(5.13), we get

( S1( x = l ) + S 2( y = l ) − 2d31 E3 )Y E hl (1 − ν )

+ u&2( y = l )   u&  = − Z s , eff  1( x = l ) 2  

Substituting the expressions for ( u&1 + u&2 )x=l and (S1+S2)

x=l

(5.25)

from Eqs. (5.16) and

(5.17) respectively, and with E3 = (Vo/h)ejωt, we can derive

A1 + A2 =

2d 31V o Z a , eff (cos kl ) kh( Z s , eff + Z a , eff )

115

(5.26)

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

The electric displacement (or the charge density) over the surface of the PZT patch can then be determined from Eq. (5.10). Substituting Eq. (5.13) into Eq. (5.10) and with E3=(Vo/h)ejωt, we get

Vo jωt d 31Y E  V jωt  D3 = ε e +  S1 + S2 − 2d31 o e  h (1 − ν )  h  T 33

(5.27)

The instantaneous electric current, which is the time rate of change of charge, can be derived as

I = ∫∫ D& 3 dxdy = jω ∫∫ D3 dxdy A

(5.28)

A

Substituting D3 from Eq. (5.27) and S1 and S2 from Eq.(5.17), and integrating from ‘–l’ to ‘+l’ with respect to both ‘x’ and ‘y’, we obtain 2 E 2d312 Y E  Z a , eff l 2  T 2d 31Y  I = 4V ωj ε 33 − + (1 − ν ) (1 − ν )  Z s ,eff + Z a ,eff h  

 tan κl   κl 

   

(5.29)

where V = Voe jωt is the instantaneous voltage across the PZT patch. Hence, the complex electro-mechanical admittance of the PZT patch is given by 2 E 2d 312 Y E  Z a , eff I l 2  T 2d31Y  Y = = G + Bj = 4ωj ε 33 − + (1 − ν ) (1 − ν )  Z s ,eff + Z a ,eff V h  

 tan κl     κl  (5.30)  

which is the desired coupling equation for a square PZT patch. It should be noted that a factor of 4 is introduced in the final expression, since ‘l’ represents halflength of the patch. In the previous models (1D- Liang et al., 1994 and 2D- Zhou et al., 1996), only one half and one-quarter of the PZT patch (from the nodal point to the end of the patch) respectively were considered as the generic elements (See Fig. 5.2). The governing equations in those models (such as Eq. 5.5) correspond to onehalf and one-quarter of the patch only. The main advantage of the present approach is that a single complex term for Zs,eff accounts for the two dimensional interaction of the PZT patch with the host structure. This makes the equation simple enough to be utilized for extracting the mechanical impedance of the structure from Y , which can be measured at any desired frequency using commercially available impedance analyzers. The related computational procedure is presented in the sections to follow.

116

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

5.6 EXPERIMENTAL VERIFICATION 5.6.1 Details of Experimental Set-up PZT patch 10x10x0.3mm

10mm

Host structure

N2260 multiplexer and 3499A/B switching box

HP 4192A impedance analyzer

48mm

48mm Base Plate

Personal Computer

Fig. 5.5 Experimental set-up to verify effective impedance based new electro-

mechanical formulations. Fig. 5.5 shows the experimental test set-up to verify the new effective impedance based electro-mechanical formulations. The test structure was an aluminum block, 48x48x10mm in size, conforming to grade Al 6061-T6. Table 5.1 lists major physical properties of Al 6061-T6. The test block was bonded to a much larger and stiffer base plate to simulate base support. The test block was instrumented with a PZT patch, 10x10x0.3mm in size, conforming to grade PIC 151 (PI Ceramic, 2003). Table 4.1 (page 100) lists the key properties of PIC 151. The patch was bonded to the host structure using RS 850-940 epoxy adhesive (RS Components, 2003), and was wired to a HP 4192A impedance analyzer (Hewlett Packard, 1996) via a 3499B multiplexer module (Agilent Technologies, 2003). In this manner, the electro-mechanical admittance signature, consisting of the real part (conductance- G) and the imaginary part (susceptance- B), was acquired in the frequency range 0-200 kHz. Table 5.1 Physical Properties of Al 6061-T6. Physical Parameter

Value

Density (kg/m3)

2715

Young’s Modulus, Y11E (N/m2)

68.95 x 109

Poisson ratio

0.33

117

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

5.6.2 Determination of Structural EDP Impedance by FEM

Before using Eq. (5.30) to derive theoretical signatures for comparison with experimental signatures, we need to evaluate the effective mechanical impedance of the PZT patch (Za,eff) as well as the EDP impedance of the structure (Zs,eff). Though a closed form expression has been derived for Za,eff (Eq. 5.22), it is not possible to derive such closed-form expression for Zs,eff, especially for complex structural systems characterized by non-trivial 3D geometries. This holds true for most reallife structures and systems where NDE is of prime importance. Hence, in this research,

a method based on 3D dynamic finite element analysis has been

developed to determine the EDP impedance of the host-structure. The main strength of the FEM lies in its ability to accurately model real-life complex shapes and boundaries. It should be noted that FEM is solely employed for verifying the new impedance formulations derived above. In actual application of the formulations for SHM, no finite element analysis is required, as will be illustrated in the later part of this chapter. The excitation of this smart system by a harmonic electric field is a typical case of linear steady state forced vibrations. Investigations by Makkonen et al. (2001) showed that fairly accurate results can be obtained for dynamic harmonic problems by FEM, even for frequencies in the GHz range. In FEM, the physical domain (such as the aluminum block) is discretized into elementary volumes called elements. Fig. 5.6 shows the finitely discretized volume of the aluminum block. Because of symmetry about the x and y axes, it suffices to perform computations using only one quadrant of the actual structure. Appropriate boundary conditions were imposed on the planes of symmetry, that is, the x and y components of displacements were set to zero on the yz and the zx planes of symmetry respectively. In addition, displacements of the bottom of the block were set to zero to simulate bonding with the base plate. The finite element meshing was carried out using the preprocessor tool of ANSYS 5.6 (ANSYS, 2000), with 1.0 mm sized linear 3D brick elements (solid 45), possessing three degrees of freedom at each node. Since the stiffness and the damping of the PZT patch are separately lumped in the term Za,eff (Eqs. 5.22 and 5.30), we need not mesh the PZT element (Liang et al., 1993).

118

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

Displacement in x-direction = 0 A

24 mm

B

10 mm

Boundary of PZT patch

C

D

Origin of coordinate system

24 mm

Displacements in y-direction = 0 z y O

x

Fig. 5.6 Finite element model of one-quarter of test structure.

In general, for a forced harmonic structural excitation, as in the present case, Galerkin finite element discretization of the 3D domain leads to the following differential equation (Zienkiewicz, 1977) [M ][u&&] + [C ][u& ] + [ K ][u ] = [ F ]

(5.31)

where [K] is the stiffness matrix, [M] the mass matrix, [C] the damping matrix, [F] the force vector and [u] the displacement vector. The continuous field quantities i.e. the mechanical displacements are approximated in each element through linear sums of the interpolation functions or the shape functions (linear in the present case). The natural boundary conditions are included in the load vector, and the essential boundary conditions are imposed by adjusting the load vector and the stiffness matrix (Bathe, 1996). The simplest approach to determine the EDP impedance of the host structure is to apply an arbitrary harmonic force (at the desired frequency) on the surface of the structure (along the boundary of the PZT patch), perform dynamic harmonic analysis by FEM, and obtain the complex displacement response at those points. The applied mechanical load can be expressed as

119

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

[ F ] = [ F1 + F2 j ]e jωt

(5.32)

The resulting displacements, which are also harmonic functions of time (at same frequency as the loads) can be similarly expressed as [u ] = [u1 + u2 j ]e jωt

(5.33)

Complex notation is employed here to account for the phase lag caused by the ‘impedance’ of the system (Zienkiewicz, 1977). Substituting Eqs. (5.32) and (5.33) into Eq. (5.31) and noting that [u& ] = jω [u ] , [u&&] = −ω 2 [u ] , we obtain 2 { [ K ] + jω [C ] − ω [ M ] } [u1 + ju2 ] = [ F1 + jF2 ]

(5.34)

which can be written in a form similar to the static analysis as [ A*][u ] = [ F ]

(5.35)

The only difference from the static case being that all the terms are complex. Eq. (5.35) can be decomposed into two coupled equations involving real numbers only, and can be written as   u1   F1   − ω 2 [M ] + [ K ] − ω[C ]   =    2 ω[C ] − ω [ M ] + [ K ] u2   F2  

(5.36)

This set of equations can be solved to obtain the displacement components u1 and u2. This solution method is called the full solution method. Reduced solution method (Makkonen et al., 2001) is another approach but it is not as accurate as the full solution method employed presently. It should be noted that computing the frequency response requires the solution of the FEM equations at each desired frequency throughout the range of interest. If the boundary of the PZT patch consists of N equal divisions on each adjacent edge (N = 5 in the present case, as shown in Fig. 5.6) , we can obtain effective displacement as ueff =

δA po

(5.37)

Substituting expression for δA and po, we get u eff

l 1 l 1 l  1 l 1 l  1  2 (u1x + u 2 x ) N + 2 (u 2 x + u 3 x ) N + ... + 2 (u Nx + u ( N +1) x ) N  +  2 (u1 y + u 2 y ) N + ... + 2 (u Ny + u ( N +1) y ) N  = 2l

Solving,

u eff =

1 (u eff , x + u eff , y ) 2

120

(5.38) (5.39)

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

where u eff , x = u eff , y =

0.5(u1x + u ( N +1) x ) + (u 2 x + u3 x + .. + u Nx ) N 0.5(u1 y + u( N +1) y ) + (u 2 y + u3 y + .. + u Ny ) N

(5.40) (5.41)

Further, by splitting the real and the imaginary terms we can alternatively write, u eff =

1 (ueff , r + ueff , i j ) 2

(5.42)

We can then obtain the EDP structural impedance from Eq. (5.8), noting that u& eff = j ω u eff . If a uniformly distributed planar force, with an effective magnitude F = Fr + Fi j is applied, from Eqs. (5.8) and (5.42), the EDP impedance of the host

structure can be derived as Z s , eff =

 2(F u − Fr ueff , i )  Fr ueff , r + Fiueff , i  2(Fr + Fi j ) =  i eff2 , r −  j (5.43) 2 2 2 jω (ueff , r + ueff ,i j )  ω (ueff , r + ueff   ω (ueff , r + ueff ,i )  ,i ) 

We can simplify the computations by applying a purely real force (Fi = 0), in which case, the effective impedance will be given by

 2 Fr u eff ,i   2 Fr u eff ,r  Z s ,eff = − − j 2 2 2 2  ω (u eff , r + u eff ,i )  ω (u eff ,r + u eff ,i ) 

(5.44)

This procedure enables the determination of the EDP structural impedance using any commercial FEM software, without any adjustment or warranting the inclusion of electric degrees of freedom in the finite element model. 5.6.3 Modelling of Structural Damping

In most commercial FEM software, the damping matrix is determined from the stiffness and the mass matrices as [C ] = α [M ] + β [ K ]

(5.45)

where α is the mass damping factor and β is the stiffness-damping factor. This type of damping is called Rayleigh damping. Further simplification can be achieved by defining damping as a function of the stiffness alone η  [C ] =   [K] ω 

121

(5.46)

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

Then, after substituting in Eq. (5.34), this simplification renders the stiffness matrix complex, as defined by [ K ] = (1 + ηj )[ K ]

(5.47)

where η is called the mechanical loss factor of the material. Its equivalent Rayleigh damping coefficients are α = 0 and β = η / ω . This type of damping is frequency independent. The present analysis considered α = 0 and β = 3 x 10-9, resulting in η ≈ 0.002 on an average for the frequency range considered. 5.6.4 Wavelength Analysis and Convergence Test

In dynamic harmonic problems, in order to obtain accurate results, a sufficient number of nodal points (3 to 5) per half wavelength should be present in the finite element mesh (Makkonen et al., 2001). In order to ensure this requirement, modal analysis was additionally performed. The frequency range of 0-200 kHz was found to contain a total of 24 modes. The modal frequencies are listed in Table 5.2, computed for four different element sizes- 2mm, 1.5mm, 1mm and 0.8mm. It can be observed from the table that good convergence of the modal frequencies is achieved at an element size of 1mm (which is the element size used in the present analysis). Thus, fairly accurate results are expected from the present analysis using FEM. In addition, Figs. 5.7(a), 5.7(b) and 5.7(c) respectively show the plots of the displacements ux, uy and uz, corresponding to the 24th mode (the highest excited mode), over the top surface of the block (z = 10mm). Also, the displacements in the three principal directions are plotted for the edge AB (see Fig. 5.6) to illustrate that there are sufficient number of nodes per half wavelength so as to ensure adequate accuracy of the analysis. 5.6.5 Comparison Between Theoretical and Experimental Signatures

Using the EDP structural impedance obtained by FEM, as described in the preceding sub-sections, the admittance functions were derived using Eq. (5.30). The T values of ε 33 and δ for the PZT patch were determined experimentally. The

Poisson’s ratio of the patch was assumed as 0.3. A MATLAB program listed in Appendix C was used to perform computations. Fig. 5.8 shows a comparison

122

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

between the experimental and the theoretical signatures, based on the proposed approach as well as that based on the model of Zhou et al. (1995, 1996). The prediction by the present method is quite close to that by Zhou and coworkers’ model. However, the present formulations are much easier to apply than the approach of Zhou et al. (1995, 1996), as evident from the very complex nature of the governing equation (Eqs. 5.5-5.7) in the approach of Zhou et al. (1995, 1996). It is observed that reasonably good agreement exists between the experimental and the theoretical plots of the real part- the conductance, predicted by the proposed model (Fig. 5.8a). Major peaks are accurately predicted, though the experimental spectrum contains few unpredicted peaks (mainly due to edge roughness and due to the inability of FEM to accurately model solid-air interactions at the boundaries). However, in the plots of the susceptance (Fig. 5.8b), large discrepancy is clearly evident, especially the difference in slopes of the curves. This discrepancy is attributed to the deviation of the PZT behavior from the ideal Table 5.2 Details of modes of vibration of test structure. MODAL FREQUENCY (kHz)

DESCRIPTION OF MODE

MODE 2mm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

81.710 89.354 90.991 106.667 125.847 139.579 139.910 142.425 146.653 148.645 150.387 156.807 157.744 165.482 168.217 176.823 181.411 183.001 185.590 191.910 192.133 195.335 196.805 200.887

1.5mm 81.480 89.105 90.765 106.335 125.125 138.916 139.227 141.406 145.852 148.017 149.511 155.576 156.706 164.333 166.960 174.370 180.035 181.943 183.573 189.760 190.116 193.208 194.432 199.026

1mm 81.320 88.944 90.610 106.101 124.623 138.521 138.845 140.745 145.420 147.624 149.000 154.882 156.119 163.660 166.207 172.701 179.145 181.222 182.242 188.364 188.776 191.869 192.986 197.845

123

0.8mm 81.256 88.884 90.547 106.016 124.464 138.367 138.691 140.525 145.249 147.484 148.801 154.623 155.905 163.417 165.941 172.186 178.841 180.984 181.808 187.902 188.345 191.424 192.519 197.457

Thickness shear (diagonal) Face shear Thickness shear (diagonal) Face Shear + Flexure Thickness flexure Bending about diagonal Bending about diagonal + Rotation Thickness Flexure Flexure Flexure Flexure Flexure Flexure+Thickness extension Flexure Flexure Thickness flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure Flexure

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

uy

ux

y (mm)

y (mm)

x (mm)

x (mm)

(a)

(b)

uz

y (mm)

x (mm)

(c)

Normalized displacement

Displacement in z direction Displacement in y direction Displacement in x direction 25 20 15 10 5 0 -5 -10 -15 0

4

8

12

16

20

24

Distance along edge (mm)

(d) Fig. 5.7 Examination of mode 24 to check adequacy of mesh size of 1mm (a) Displacements in x direction on surface z = 10mm. (b) Displacements in y direction on surface z = 10mm. (c) Displacements in z direction on surface z = 10mm. (d) Displacements in principal directions along the line defined by the intersection of surfaces y = 24mm and z = 10mm (see Fig. 5.6). 124

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications 1.00E-02

G (S)

1.00E-03 1.00E-04 1.00E-05 1.00E-06 0

40

80

120

160

200

Frequency (kHz)

(a) 1.00E-02

B (S)

8.00E-03 6.00E-03 4.00E-03 2.00E-03 0.00E+00 0

40

80

120

160

200

Frequency (kHz)

(b) Theoretical (Proposed model) Theoretical (Model of Zhou et al) Experimental

Fig. 5.8 Comparison between experimental and theoretical signatures.

(a) Conductance plot. (b) Susceptance plot.

behavior predicted by Eq. (5.22). Besides, many parameters of the PZT patch could deviate from the values provided by the manufacturer. Fortunately, we had obtained the admittance signatures of the PZT patch in ‘free-free’ condition prior to bonding it on the structure. Hence, it was possible to investigate the behavior of free PZT

125

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

patch and use this information to obtain more accurate plots. The next section describes the investigations in detail. 5.7 REFINING THE MODEL OF PZT SENSOR-ACTUATOR PATCH

The properties of piezoceramics are strongly dependent upon the process route and exhibit statistical fluctuations within a given batch (Giurgiutiu and Zagrai, 2000). The fluctuations are caused by inhomogeneous chemical composition, mechanical differences in the forming process, chemical modification during sintering and the polarization method (Sensor Technology Ltd., 1995). A variance of the order of 5-20% in properties is not uncommon. In the EMI technique, we solely depend upon PZT patches to predict the mechanical impedance spectra of the structures. Hence, it is very important to accurately model the behavior of the PZT patches when using the formulations derived in the previous sections. For this purpose, it is recommended that the signatures of the PZT patches be recorded in the ‘free-free’ condition prior to their bonding on to the host structure. Looking back at Eq. (5.30), for a free (unbonded) PZT patch, the complex electro-mechanical admittance can be derived (by substituting Zs,eff = 0 and and simplifying) as

Y free

2 E l 2  T 2 d31Y  tan κl   ε 33 + = 4ωj − 1  h (1 − ν )  κl  

T T = ε 33 (1 − δj ) , Substituting Y E = Y E (1 + ηj ) , ε 33

(5.48)

tan κl = r + tj and ω = 2πf (‘f’ κl

being the frequency of vibrations in Hz), and simplifying we get

Y free = G f + B f j

(5.49)

where

 2d312 Y E 8πfl 2  T {η (r − 1) + t} Gf = ε 33δ − (1 − ν ) h  

(5.50)

2 E  8πfl 2  T 2d31Y {(r − 1) − ηt} Bf = ε 33 + h  (1 − ν ) 

(5.51)

126

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

Further, under very low frequencies (typically < one-fifth of the first resonance frequency of the PZT patch),

tan κl → 1 (i.e. r→1, t→0) (Liang et al., 1993, 1994), κl

thereby leading to quasi-static sensor approximation (Giurgiutiu and Zagrai, 2002)

G f ,qs =

T 8πfl 2ε 33 δ h

(5.52)

B f , qs =

T 8πfl 2ε 33 h

(5.53)

Rearranging the various terms, Eqs. (5.52) and (5.53) can be rewritten as G *f , qs =

B*f , qs =

G f , qs h

= δf

T 8πl 2ε 33

B f , qs h 8πl

2

(5.54)

T = ε 33 f

(5.55)

From Eqs. (5.54) and (5.55), we can determine the electrical constants

T ε 33 and δ as

* the slopes of the frequency plots of B f ,qs (unit S/m) and G *f , qs (unit S/F) for

sufficiently low frequencies (typically < 10 kHz for 10mm long PZT patches). Figs. 5.9 (a) and (b) respectively show the typical plots of these functions in the frequency range 0-10 kHz for two PZT patches labelled as S2002-5 and S2002-6. 400

0.0002

S2002-5

*

300

S2002-6

Gf,qs ( S/F )

Bf,qs ( S/m )

S2002-5 0.00015 0.0001

S2002-6

200

*

0.00005 0 0

2000

4000

6000

f (Hz)

8000

100 0

10000

0

2000

4000

6000

8000

f (Hz)

(b)

(a)

Fig. 5.9 Plots of quasi-static admittance functions of free PZT patches to

obtain electric permittivity and dielectric loss factor. (a) B *f ,qs vs frequency. (b) G *f ,qs vs frequency.

127

10000

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

Patch S2002-5 was used as piezo-impedance transducer in the experiment described T in the previous section. From these plots, ε 33 was worked out to be 1.7919x10-8 F/m

and 1.7328x10-8 F/m respectively for S2002-5 and S2002-6, against a value of 2.124x10-8 F/m supplied by the manufacturer). Similarly, δ was worked out to be 0.0238 and 0.0225 respectively, against a value of 0.015 supplied by the manufacturer. Using the values of the PZT parameters obtained above, free conductance and susceptance signatures of the PZT patches s2002-5 and s2002-6 were obtained in the ‘free-free’ condition in the frequency range 1-1000 kHz, using Eqs. (5.50) and (5.51) respectively. These are shown in Fig. 5.10 and compared with the experimental free PZT signatures. Although a quick look at the figures suggests reasonable agreement between the analytical and the experimental signatures, there are some underlying discrepancies, which need closer examination. A close look in frequency range 0-300kHz (Figs. 5.10a and 5.10c) shows an unpredicted mode at around 240kHz. In the case of S2002-5 (Fig. 5.10a), twin peaks are observed in the experimental spectra around each of the prominent resonance frequencies. Besides, a general observation is that the experimental resonance frequency is slightly higher than the theoretical frequency. The twin peaks are due to the deviation in the shape of the PZT patch from perfect square shape during manufacturing. This leads to somewhat partly independent resonance peaks corresponding to the two slightly unequal edge lengths. The unpredicted modes in the admittance spectra are due to edge roughness induced secondary vibrations. Somewhat higher experimental natural frequency suggests additional 2D stiffening, which is unaccounted for in the present model. A similar comparison was reported by Giurgiutiu and Zagrai (2000, 2002), but considering 1D vibrations only. They assumed the patch to possess widely separated values for length, width and thickness so that length, width and thickness vibrations are practically uncoupled. Their analytical predictions matched the experimental results only for aspect ratios higher than 2.0 only. Presently, the frequency range of interest is 0-200 kHz. The unpredicted mode does not come into picture in this frequency range. In order to further ‘update’ the model of the PZT patch with respect to peaks, a correction factor is

128

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

Twin peaks Unpredicted mode 8.00E-02

8.00E-02

6.00E-02

4.00E-02

B (S)

120 140 160 180 200

G (S)

6.00E-02

2.00E-02

4.00E-02 2.00E-02 0.00E+00 -2.00E-02 0

200

400

600

800

1000

-4.00E-02

0.00E+00 0

200

400

600

800

1000

Frequency (kHz)

Frequency (kHz)

(a)

(b)

Unpredicted mode 4.00E-02

120 140 160 180 200

8.00E-02

2.00E-02

B (S)

G (S)

6.00E-02 4.00E-02

0.00E+00 -2.00E-02

2.00E-02

-4.00E-02

0.00E+00 0

200

400

600

800

0

1000

200

400

600

800

Frequency (kHz)

Frequency (kHz)

(d)

(c) Analytical

Experimental

Fig. 5.10 Experimental and analytical plots of free PZT signatures.

(a) S2002-5: Conductance (G) vs Frequency. (b) S2002-5: Susceptance (B) vs Frequency. (c) S2002-6: Conductance (G) vs Frequency. (d) S2002-6: Susceptance (B) vs Frequency.

129

1000

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

introduced in the term ( tan κl / κl ). In the case of PZT patch S2002-5, where twin peaks are observed, this term may be replaced by 1  tan(C1κl ) tan(C2κl )  +   2  C1κl C2κl 

By trial and error, values of C1 = 0.94 and C2 = 0.883 were found to update the model of the PZT patch. Further, following values of the PZT parameters were determined from the experimental plot using the techniques of curve fitting.

K=

2d 312 Y E = 5.16 x10− 9 NV-2 and η = 0.03 (1 − ν )

The value of K based on data supplied by the manufacturer is determined as 8.4x10-9 NV-2. Using these values and the correction factors C1 and C2, the free PZT signatures were again worked out in the frequency range 0-200 kHz. Figs. 5.11(a) and 5.11(b) compare the updated signatures with the experimental signatures. A very good agreement is observed between the two. Similarly, for the PZT patch S2002-6, a coefficient C =0.885 was found, such that the term (tan κl / κl ) , when replaced by [ tan(Cκl ) / Cκl ] yielded a good agreement between the experimental and the analytical plots of free PZT signatures. Further, K was computed to be 4.63x10-9 NV-2 and η again worked out to be 0.03. Figs. 5.11(c) and 5.11(d) compare the analytical and the experimental plots. Again, a good agreement is observed between the experimental signatures and the signatures using the updated PZT model. Hence, considering the necessity of updating the model of the PZT patch, Eq. (5.30) can be modified as 2 E 2d 312 Y E  Z a ,eff l 2  T 2d 31Y  Y = G + Bj = 4ωj ε 33 − + h  (1 − ν ) (1 − ν )  Z s ,eff + Z a , eff 

  T     

(5.56)

where the term T is the complex tangent ratio (ideally tanκl/κl), which can be expressed as

130

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

tan(Cκl ) Cκl

for single-peak behaviour.

T =

(5.57)

0.1

6.00E-02

0.01

4.00E-02

0.001

2.00E-02

B (S)

G (S)

1  tan C1κl tan C2κl    for twin-peak behaviour. + C2κl  2  C1κl

0.0001 0.00001 0.000001 0

25

50

125

150

175

200

225

-4.00E-02

75 100 125 150 175 200 225 Frequency (kHz)

Frequency (kHz)

(a)

(b)

1.00E-01

4.00E-02

1.00E-02

2.00E-02

1.00E-03

B (S)

G (S)

0.00E+00 100 -2.00E-02

1.00E-04 1.00E-05

0.00E+00 -2.00E-02

1.00E-06 0

-4.00E-02

25 50 75 100 125 150 175 200 225

0

Frequency (kHz)

(c)

25

50

75

100 125 150 175 200 225

Frequency (kHz)

(d) Experimental

Analytical

Fig. 5.11 Plots of free-PZT admittance signatures using an updated PZT model.

(a) S2002-5: Conductance (G) vs Frequency. (b) S2002-5: Susceptance (B) vs Frequency. (c) S2002-6: Conductance (G) vs Frequency. (d) S2002-6: Susceptance (B) vs Frequency.

131

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

Further, the corrected actuator effective impedance (earlier expressed by Eq. 5.22) can be written as Z a , eff =

2hY E jω (1 − ν )T

(5.58)

As mentioned before, PZT patch S2002-5 was the one bonded to the host structure shown in Fig. 5.5. The theoretical signatures for this test structure (with the PZT patch bonded on the surface) were again worked out using the updated PZT model (Eqs. 5.56, 5.57 and 5.58). A MATLAB program listed in Appendix D was used to perform computations. Fig. 5.12 compares the theoretical signatures based on the proposed effective impedance based model (after updating PZT model) and the experimental signatures. This time, a much better agreement is found between the two. Fig. 5.13(a) compares the idealized and the corrected effective impedance for the PZT patch S2002-5. The influence of twin peaks is clearly reflected in the plot of the updated impedance. If we were to solely depend upon the idealized model of PZT patch to identify the structure, significant errors could have been introduced, as can be clearly observed in Fig. 5.13(b), which shows the plot of |Zs,eff|-1. Further, Fig. 5.13 shows the plots of |Zs,eff| and |Za,eff| derived experimentally to illustrate that the present system satisfies the criteria |Zs,eff| > |Za,eff|. Hence, this case falls in the category of Case II described in Chapter 3. It should be noted that Giurgiutiu and Zagrai (2002) also evaluated the electro-mechanical admittance across PZT terminals using analytical and numerical methods. However, they could only model very simple structures, such as thin beams, under extremely simple boundary conditions (such as ‘free-free’). There were orders of magnitude of error between the experimental and the analytical impedance spectra. The present work, on the other hand, is more general in nature and is valid for all types of structures, whether 2D or 3D. The agreement between the analytical and experimental results is also much better as compared to the results of the previous researchers. The present work, which is a semi-analytical approach (numerical + analytical) is the first attempt to compare theoretical modelling for 3D structures with experimental data for such high frequencies.

132

Chapter 5: Generalized Electro-Mechanical Impedance Formulations: Theoretical Development and SHM Applications

1.00E-02

G (S)

1.00E-03 1.00E-04 1.00E-05 1.00E-06 0

40

80

120

160

200

160

200

Frequency (kHz)

(a) 0.008

B (S)

0.006 0.004 0.002 0 0

40

80

120

Frequency (kHz)

(b) Theoretical

Experimental

Fig. 5.12 Comparison between experimental and analytical signatures based

on updated PZT model. (a) Conductance (G) vs frequency. (b) Susceptance (B) vs Frequency.

133

Based on idealized PZT model

10000

0.01

1000

Based on idealized PZT model

0.008 |Z|-1 (mN-1s -1)

Effective Impedance (Ns/m)

Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications

100

Based on updated PZT model

10 1 0

50

100 150 200 Frequency (kHz)

0.006

Based on updated PZT model

0.004 0.002 0

250

0

25

50

75 100 125 150 175 200 Frequency (kHz)

(a)

(b)

|Zs,ef f |, |Za,ef f | (Ns/m)

100000

Structure

10000 1000 100 10

PZT patch

1 0

50

100

150

200

Frequency (kHz)

(c) Fig. 5.13 (a) PZT effective impedance, based on idealized and updated models. (b) Error in extracted structural impedance in the absence of updated PZT model. (c) Relative magnitudes of structure and PZT impedances.

5.8 DECOMPOSITION

OF

COUPLED

ELECTRO-MECHANICAL

ADMITTANCE As in the case of 1D impedance model (covered in Chapters 2 to 4), the electromechanical admittance (given by Eq. 5.56) can be decomposed into two components as Y = 4ωj

2 E l 2  T 2d 31Y  ε 33 −  h (1 − ν )   

+

8ωd 312 Y E l 2  Z a , eff  h(1 − ν )  Z s , eff + Z a , eff

Part II

Part I 134

 T j  

(5.59)

Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications

It can be observed that the first part solely depends on the parameters of the PZT patch and is independent of the host structure. The structural parameters make their presence felt in part II only, in the form of the EDP structural impedance, Zs,eff. Therefore, Eq. (5.59) can be written as Y = YP + YA

(5.60)

where YA is the ‘active’ component and YP the ‘passive’ component. YP can be T T broken down into real and imaginary parts by expanding ε 33 = ε 33 (1 − δj ) and

Y E = Y E (1 + ηj ) and can be expressed as

YP = GP + BP j where

and

(5.61)

GP =

4ωl 2 T δε 33 + Kη h

BP =

4ωl 2 T ε 33 − K h

K=

2d312 Y E (1 − ν )

{

{

}

(5.62)

}

(5.63) (5.64)

We can predict GP and BP with reasonable accuracy if we record the conductance and the susceptance signatures of PZT patch in ‘free-free’ condition, prior to its bonding to the host structures, as demonstrated in section 5.7. Hence, the PZT contribution can be filtered off from the raw signatures and the active component deduced as

or

YA = Y − YP

(5.65)

YA = (G + Bj ) − (GP + BP j )

(5.66)

Thus, the active components (GA and BA) can be derived from the measured raw admittance signatures (G and B) as

and

G A = G − GP

(5.67)

BA = B − BP

(5.68)

In the complex form, we can express the active component as YA = G A + B A j =

8ωd 312 Y E l 2  Z a ,eff  h(1 − ν )  Z s ,eff + Z a ,eff

135

 T j  

(5.69)

Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications

It was demonstrated in Chapter 4, using 1D interaction model, that the elimination of the passive component renders the admittance signatures more sensitive to structural damages. The same holds true for the 2D PZT-structure interaction considered in this chapter. Therefore, it is more pragmatic to employ active components rather than raw signatures for SHM and NDE. 5.9 EXTRACTION OF STRUCTURAL MECHANICAL IMPEDANCE Chapter 4 outlined a computational procedure for extracting 1D drive point mechanical impedance of skeletal structures using the active admittance signatures of surface-bonded piezo-impedance transducers. This section outlines the corresponding procedure for the more general class of structures, based on the new electro-mechanical admittance formulations. Substituting Y E = Y E (1 + ηj ) and T = r + tj into Eq. (5.69), and rearranging the various terms, we obtain  Z a , eff M + Nj =  Z  S , eff + Z a , eff where

M =

BA h 4ωKl 2

R = r − ηt

 ( R + Sj )  

GAh 4ωKl 2

and

N =−

and

S = t + ηr

(5.70)

(5.71) (5.72)

Further, expanding Z S , eff = x + yj and Z a , eff = xa + ya j , and upon solving, we can obtain the real and imaginary components of the EDP structural impedance as x=

M ( xa R − ya S ) + N ( xa S + ya R) − xa M 2 + N2

(5.73)

y=

M ( xa S + ya R) − N ( xa R − ya S ) − ya M 2 + N2

(5.74)

In all these computations, the term T (which plays a significant role), depends upon (tanκl/κl) (see Eqs. 5.56-5.58), where κl is a complex number. It is essential to determine this quantity precisely, by the procedure outlined in Chapter 4. Further, it should again be noted that |x| > |xa| and |y| > |ya| in order to ensure smooth computations. Else, the extracted impedance spectra might exhibit false peaks. The simple computational procedure outlined above results in the determination of the drive point mechanical impedance of the structure,

136

Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications

Zs,eff = x + yj, at a particular frequency ω, from the active admittance signatures. Following this procedure, ‘x’ and ‘y’ can be determined for the entire frequency range of interest. This procedure was employed to extract the structural EDP impedance of the test aluminium block (used for validating the new impedance model in sections 5.6 and 5.7). A MATLAB program listed in Appendix E was used to perform the computations. In the present case, |Z| > |Za|, as apparent from Fig. 5.13(c). Fig. 5.14 shows a plot of |Zeff|-1, worked out by this procedure, comparing it with the plot determined using FEM, as discussed in the preceding sections. Reasonable agreement can be observed between the two. The main reason for plotting |Zs,eff|-1 (instead of Zs,eff) is that the resonant frequencies can be easily identified as peaks of the plot. As will be demonstrated in the forthcoming sections, this procedure enables us to ‘identify’ any unknown structure without demanding any a-priori information governing the phenomenological nature of the structure. The only requirement is an ‘updated’ model of the PZT patch, which can be derived from preliminary specifications of the PZT patch and by recording its admittance signatures in the ‘free-free’ condition, prior to bonding to the host structure. It was demonstrated in Chapter 4 that the utilization of ‘x’ and ‘y’ (rather than raw signatures) leads not only to higher damage sensitivity but also facilitates greater insight into the mechanism associated with structural damage. The next section will present a simple procedure to derive system parameters from the structural EDP impedance.

|Zef f |-1(m/Ns)

0.1

0.01

Experimental 0.001

Numerical 0.0001 0

40

80

120

160

200

Frequency (kHz)

Fig. 5.14 Comparison between |Zeff|-1 obtained experimentally and numerically.

137

Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications

5.10

SYSTEM PARAMETER IDENTIFICATION FROM EXTRACTED IMPEDANCE SPECTRA The structural EDP impedance, extracted by means of the procedure

outlined in the previous section, carries information about the dynamic characteristics of the host structure. In Chapter 4, the host structure (1D skeletal structure) was idealized as a parallel combination of a resistive element (damper) and a reactive element (stiffness-mass factor). The extracted structural parameters were employed in evaluating structural damages. This section presents a more general approach to ‘identify’ the equivalent structural system. Before considering any real-life structural system for this purpose, it would be a worthwhile exercise to observe the impedance pattern of few simple systems. Fig. 5.15 shows plots of the real and the imaginary components of the mechanical impedance of basic structural elements- the mass, the spring and the damper. These basic elements can be combined in a number of different ways (series, parallel or a mixture) to evolve complex mechanical systems. Table 5.3 shows the impedance plots (x, y vs frequency) for some possible combinations of the basic elements (Hixon, 1988). In general, for any real-life structure, the two components (real and imaginary) of the extracted EDP impedance may not display an ideal behavior, such as pure mass or pure stiffness or pure damper. Both the ‘resistive’ and the ‘reactive’ terms might vary with frequency, similar to a combination of the basic elements. The ‘unknown’ structure can thus be idealized as an ‘equivalent’ structure (series or parallel combination of basic elements), and the equivalent system parameters can thereby be determined.

Mass Damper

x

y

0

Damper Spring

0

Spring, mass Frequency

Frequency

(b)

(a)

Fig. 5.15 Impedance plots of basic structural elements- spring, damper and mass. (a) Real part (x) vs frequency. (b) Imaginary part (y) vs frequency. 138

Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications

Table 5.3 Mechanical impedance of combinations of spring, mass and damper. No.

COMBIN— ATION

x

y

c

1



x vs Freq.

k ω

Y vs Freq.

0 0

c

2



0

3

mω −

c

4

mω −

0

0

0

0

k

ω

k ω

0 0

5

c c−2 + (ωm)−2

0

6

7

(ωm) c − 2 + (ωm) − 2 −1

−1

−1 (ω / k ) − (ωm) −1

0

0

0

0

− (ω / k − 1/ ωm) c− 2 + (ω / k − 1/ ωm)2

c−1 c + (ω / k − 1/ ωm)2 −2

0 0

c

8

ωmk k − ω 2m

0

9

m −1 − k (c −2 + ω −2 m −2 ) ω c − 2 + (ωm) − 2

c −1 c + (ωm) − 2

[

−2

ω [m(c + ω k ) − k c − 2 + (ω / k ) 2

10

c c − 2 + (ω / k ) 2

11

cm2ω 2 2 c + (ωm − k / ω )2

0

]

−2

−1

2

−2

−1

]

0

0

12

[

c −1

c + ω /( k − mω ) −2

2

k   mω c2 − (ωm − k / ω ) ω   c2 + (ωm − k / ω )2

]

2

−ω /( k − mω 2 )

[

c + ω /( k − mω 2 ) −2

0

0 0

0

]

2

0 0

13

ck 2 / ω 2 c + (ωm − k / ω )2 2

 c2 k  − km(ωm − k / ω ) +  m ω  2 2 c + (ωm − k / ω )

139

0 0

Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications

To demonstrate this approach, let us consider another aluminum (grade Al 6061-T6) block, 50x48x10mm in size, representing an unknown structural system. The PZT patch S2002-6 (10x10x0.3mm in size), whose updated model was derived in section 5.7, was bonded to the surface of this specimen. Experimental set-up similar to that shown in Fig. 5.5 was employed to acquire the raw admittance signatures (conductance and susceptance) of this PZT patch. The passive components were filtered off from the raw signatures and the structural EDP impedance was extracted out, using the MATLAB program listed in Appendix E (considering the parameters of patch S 2002-6 derived experimentally). A close examination of the extracted impedance components in the frequency range 25-40 kHz suggested that the system behavior was similar to a parallel spring-damper (k-c) combination (system 1 in Table 5.3). For this system, x=c

y=−

and

k

(5.75)

ω

Using Eq. (5.75) and the actual impedance plots, the average “equivalent” system parameters were worked as: c = 36.54 Ns/m and k = 5.18x107 N/m. The analytical plots of ‘x’ and ‘y’ obtained by these equivalent parameters match well with their

200

-150

150

-200 y (Ns/m)

x (Ns/m)

experimental counterparts, as shown in Fig. 5.16.

100 50 0

-250 -300 -350

25

30

35

40

25

30

35

40

Frequency (kHz)

Frequency (kHz)

(a)

(b) Experimental

Equivalent system

Fig. 5.16 Mechanical impedance of aluminium block in 25-40 kHz frequency range. The equivalent system plots are obtained for a parallel spring-damper combination. (a) Real part (x) vs frequency.

(b) Imaginary part (y) vs frequency.

140

Chapter 5: Generalized Electro-Mechanical Impedance Formulation: Theoretical Development and SHM Applications

Similarly, in the frequency range 180-200 kHz, the system behavior was found to be similar to a parallel spring-damper (k-c) combination, in series with mass ‘m’ (system 11 in Table 5.3). For this combination,

x=

cm ω 2

2

k  c +  mω −  ω 

2

and

2

 k k  mω c 2 −  mω −  ω ω   y= 2 k  c 2 +  mω −  ω 

(5.76)

and the peak frequency of the x-plot is given by

ωo =

k

(5.77)

c2 m− k

If x = xo (the peak magnitude) at ω = ωo and x = x1 (somewhat less than the peak magnitude) at ω = ω1 ( Dc. As pointed out earlier, instead of defining a unique value of the critical damage variable Dc, we are employing a fuzzy definition to take uncertainties into account. Using the fuzzy set theory, a fuzzy region may be defined in the interval

(DL, DU) where DL and DU respectively represent the lower

and the upper limit of the fuzzy region (Valliappan and Pham, 1993; Wu et al., 1999). D > DU represents a failure region with 100% failure possibility and D < DL

178

Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete

LR = 0 LR = 0.268 LR = 0.402 LR = 0.536 LR = 0.670 LR = 0.804 LR = 1.000

5.00E+07 4.00E+07 3.00E+07 2.00E+07 1.00E+07 0.00E+00 60

70

80

90

LR = 0 LR = 0.311 LR = 0.519

5.50E+07

Equivalent stiffness (N/m)

Equivalent Stiffness (N/m)

6.00E+07

100

5.00E+07 4.50E+07

LR = 0.726

4.00E+07

LR = 0.830 LR = 0.882 LR = 1.000

3.50E+07 3.00E+07 60

Frequency (kHz)

70

80

(b)

3.00E+07

2.00E+07

1.00E+07

0.00E+00 80

90

LR = 0 LR = 0.148 LR = 0.296 LR = 0.444 LR = 0.592 LR = 0.741 LR = 0.888 LR = 0.963 LR = 1.000

4.00E+07

LR = 0 LR = 0.172 LR = 0.345 LR = 0.517 LR = 0.690 LR = 0.862 LR = 1.000

Equivalent Stiffness (N/m)

Equivalent stiffness (N/m)

4.00E+07

70

100

Frequency (kHz)

(a)

60

90

3.50E+07 3.00E+07 2.50E+07 2.00E+07 1.50E+07

100

80

84

88

92

96

100

Frequency (kHz)

Frequency (kHz)

(d)

(c)

LR = 0 LR = 0.206 LR = 0.413 LR = 0.619 LR = 0.774 LR = 0.929 LR = 1.000

Equivalent stiffness (N/m)

4.50E+07 4.00E+07 3.50E+07 3.00E+07 2.50E+07 2.00E+07 1.50E+07 60

70

80

90

100

Frequency (kHz)

(e)

Fig. 6.19 Effect of damage on equivalent spring stiffness (LR stands for ‘Load ratio’).

(a) C17 (b) C43 (c) C52 (d) C60 (e) C86

179

Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete

1 Probability Dis tribution

Empirical Theoretical

0.8 0.6 0.4 0.2

Empirical Theoretical

0.8 0.6 0.4 0.2

0

0

0.4

0.5

0.6

0.7

0.8

0.9

1

0.2

0.3

0.4

D

0.5

0.6

D

(b)

(a)

1 Probability Distribution

1 0.8 0.6 Empirical 0.4

Theoretical

0.2

0.8 Empirical

0.6

Theoretical 0.4 0.2 0

0 0.5

0.6

0.7

0.8

0.9

0.35

1

0.4

D

0.45 D

(d)

(c)

1 Probability Dis tribution

Probability Dis tribution

Probability Dis tribution

1

Empirical Theoretical

0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

D

(e)

Fig. 6.20 Theoretical and empirical probability density functions near failure.

(a) C17 (b) C43 (c) C52 (d) C60 (e) C86

180

0.5

Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete

represents a safe region with 0% failure possibility. Within the fuzzy or the transition region, that is DL< D < DU, the failure possibility could vary between 0% and 100%. A characteristic or a membership function fm could be defined (0< fm(D) 80%

Failure imminent Incipient Damage

Fuzzy Failure Probability (%)

Severe Damage (Failure Imminent) 100 80 60 40 20 0 C 17

C 43

C 52

C 60

C 86

Fig. 6.21 Fuzzy failure probabilities of concrete cubes at

incipient damage level and at failure stage.

Fuzzy Failure Probability %

100 C17

80

Large visible cracks

C43 C52 C60

60

Failure Imminent Severe Damage

Cracks opening up

Moderate Damage

C86 40 Micro cracks

20

Incipient Damage

0 0

0.2

0.4

0.6

0.8

1

Load ratio Fig. 6.22 Fuzzy failure probabilities of concrete cubes at

various load levels.

182

Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete

Thus, the fuzzy probabilistic approach quantifies the extent of damage on a uniform 0-100% scale. This can be employed to evaluate damage in real-life concrete structures. 6.7 DISCUSSIONS

All the PZT patches exhibited more or less a uniform behaviour with damage progression in concrete, although the strength of concrete cubes varied from as low as 17 MPa to as high as 86 MPa. Hence, the PZT patches were subjected to a wide range of mechanical stresses and strains during the tests. At a load ratio of 1.0, almost same order of FFP is observed, irrespective of the absolute load or stress level (for example 17 MPa for C17 and 86 MPa for C86). In general, the PZT material shows very high compressive strength, typically over 500 MPa and it essentially exhibits a linear stress-strain relation up to strains as high as 0.006. A typical experimental plot for the PZT material is shown in Fig. 6.23 (Cheng and Reece, 2001). In the experiments conducted on concrete cubes, the strain level never exceeded 0.003 (50% of the linear limit). Also, it was observed that in all the cubes tested, the damage typically initiated near the edges of the cube and migrated to regions near the PZT patch with increasing load ratios. After failure of the cubes, all the PZT patches were found intact. Fig. 6.24 shows close ups of the cubes after the tests.

Fig. 6.23 Typical stress-strain plot for PZT (Cheng and Reece, 2001). 183

Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete

C43 C17

(b)

(a)

C60

C52

(c)

(d)

C86

(e) Fig. 6.24 Cubes after the test. (a) C17 (b) C43 (c) C52 (d) C60 (e) C86 184

Chapter 6: Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment in Concrete

The results show that the sensor response reflected more the damage to the surrounding concrete rather than damage to the patches themselves. In general, we can expect such good performance in materials like concrete characterized by low strength as compared to the PZT patches. Hence, damage to concrete is likely to occur first, rather than the PZT patch. Further, though the cubes were tested in compression, the same fuzzy probabilistic damage model can be expected to hold good for tension also. 6.8 CONCLUDING REMARKS

This chapter has covered the development of a new experimental technique based on EMI technique for evaluating concrete strength non-destructively. Also, it has shown the feasibility of monitoring concrete curing using piezo-impedance transducers. It is found that the equivalent spring stiffness of concrete “identified” by a surface bonded PZT patch can serve as a damage sensitive structural parameter. It could be utilized for identifying and quantifying damages in concrete. A fuzzy probability based damage model is proposed based on the extracted equivalent stiffness to evaluate the extent of damage using the impedance data. This has facilitated the calibration of the piezo-impedance transducers in terms of damage severity and this can serve as a convenient empirical phenomenological damage model for quantitatively estimating damage in concrete in the real-life structures.

185

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

Chapter 7 INCLUSION OF INTERFACIAL SHEAR LAG EFFECT IN IMPEDANCE MODELS

7.1 INTRODUCTION The piezo-impedance transducers are bonded to the surface of the host structures using an adhesive mix (such as epoxy), which forms a permanent finite thickness interfacial layer between the structure and the patch. In the analysis presented so far in this thesis, the effects of this layer were neglected. The force transmission from the PZT patch to the host structure was assumed to occur at the ends of the patch (1D model of Liang et al., 1994) or along the continuous boundary edges of the patch (2D effective impedance model, Chapter 5). In reality, the force transfer takes place through the interfacial bond layer via shear mechanism. This chapter reviews the mechanism of force transfer through the bond layer and presents a step-by-step derivation to integrate this mechanism into impedance formulations, both 1D and 2D. The influence of various parameters (associated with the bond layer) on the electro-mechanical admittance response are also investigated. 7.2 SHEAR LAG EFFECT

dx Differential dx Element

y PZT patch Bond layer

x l

Tp+ ∂Tp dx ∂x

Tp tp ts

τ

l BEAM

Fig. 7.1 A PZT patch bonded to a beam using adhesive bond layer.

186

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

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 is shown in Fig. 7.1. The patch has a length 2l, width wp and thickness tp, while the bonding layer has a thickness equal to ts. The adhesive layer thickness has been shown exaggerated to facilitate visualization. The beam has depth tb and width wb. Let Tp denote the axial stress in the PZT patch and τ the interfacial shear stress. Following assumptions were made by Crawley and de Luis (1987) and Sirohi and Chopra (2000b) in their analysis: (i)

The system is under quasi-static equilibrium.

(ii)

The beam is actuated in pure bending mode and the bending strain is linearly distributed across any cross section.

(iii)

The PZT patch is in a state of pure 1D axial strain.

(iv)

The bonding layer is in a state of pure shear and the shear stress is independent of ‘y’.

(v)

The ends of the segmented PZT actuator/ sensor are stress free, implying a uniform strain distribution across the thickness of the patch. A more detailed deformation profile is shown in Fig. 7.2, which shows the

symmetrical right half of the system of Fig. 7.1. Let ‘up’ be the displacement at the interface between the PZT patch and the bonding layer and ‘u’ the corresponding displacement at the interface between the bonding layer and the beam.

y

upo up

PZT patch Bonding layer x

B

B A

After deformation



A’ uo

u

x

Beam

Fig. 7.2 Deformation in bonding layer and PZT patch.

187

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

7.2.1

PZT Patch as Sensor Let the PZT patch be instrumented only to sense strain on the beam surface

and hence no external electric 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. 7.1(a), we can derive τ =

∂Tp ∂x

tp

(7.1)

At any cross section of the beam, within the portion containing the PZT patch, the bending moment is given by M = Tp wpt p (0.5tb + ts + 0.5t p )

(7.2)

Also, from Euler-Bernoulli’s beam theory,  I   M = −σ b   0.5tb 

(7.3)

where σb is the bending stress at the extreme fibre of the beam and ‘I’ the second moment of inertia of the beam cross-section. The negative sign signifies that sagging moment and tensile stresses are considered positive. Comparing Eqs. (7.2) and (7.3) and with I = wb t b / 12 , we get 3

 3Tp wpt p  (tb + t p + 2ts ) = 0 σ b +  2  wbtb 

(7.4)

Assuming (tp+2ts ) 93%, suggesting that shear lag effect can be ignored for relatively high (> 30 cm-1) values of Γ.

1 0.8

leff /l

0.6 0.4 0.2 0 0

20

40

60

80

100

Γ (cm-1)

Fig. 7.4 Variation of effective length with shear lag factor.

191

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

7.2.2

PZT Patch as Actuator If the PZT patch is employed as an actuator for a beam structure, it can be

shown (Crawley and de Luis, 1987) that the strains Sp and Sb will be as given by

Sp =

Λψ cosh Γx 3Λ + (3 + ψ ) (3 + ψ ) cosh Γl

(7.23)

Sb =

3Λ 3Λ cosh Γx − (3 + ψ ) (3 + ψ ) cosh Γl

(7.24)

where Λ = d31E3 is the free piezoelectric strain and ψ = (Ybtb/YEtp) is the product of modulus and thickness ratios of the beam and the PZT patch. Fig. 7.5 shows the plots of (Sp / Λ) and (Sb / Λ) along the length of the PZT patch (l = 5mm) for ψ = 15. It is observed that like in the case of sensor, as Γ increases, the shear is effectively transferred over small zone near the two ends of the patch. As Γ → ∞, the strain is transferred over an infinitesimal distance near the ends of the PZT patch. For the limiting case, as apparent from Fig. 7.5, Sb = S p =

3Λ (3 + ψ )

(7.25)

which sets the maximum fraction of the piezoelectric free strain Λ that can be induced into the beam. Further, as ψ → 0, Sb → Λ.Typically, for Γ > 30cm-1, the strain energy induced in the substructure by PZT actuator is within 5% of the perfectly bonded case. Therefore, for Γ > 30 cm-1, ignoring the effect of the bond layer will provide sufficiently accurate results for most engineering models. It should be noted here that the analysis carried out by Crawley and de Luis (1987) as well as Sirohi and Chopra (2000b) is valid for static conditions only. These researchers extended their formulations to dynamic problems under the assumption that the operating frequency is small enough to ensure that the PZT patch acts ‘quasi-statically’. However, in the EMI technique, the operational frequencies are of the order of the resonant frequency of the PZT patch, warranting that the actuator dynamics should not be neglected.

192

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

1 Γ = 10

0.8

20

0.6

30

(Sp/Λ)

0.4

40

50

0.2 60

0 -1

-0.5

0

0.5

1

(x/l)

(a)

0.2 60

0.15

40 30

(Sb/Λ) 0.1

50

20 Γ = 10

0.05

0 -1

-0.5

0

0.5

1

(x/l)

(b)

Fig. 7.5 Distribution of piezoelectric and beam strains for various values of Γ. (a) Strain in PZT patch. (b) Beam surface strain.

193

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

7.3 INTEGRATION OF SHEAR LAG EFFECT INTO IMPEDANCE MODELS It was observed in the previous section that both as an actuator as well as a sensor, shear lag effect is associated with the mechanism of force transmission between the PZT patch and the host structure via the adhesive bond layer. However, till date, this aspect has not been thoroughly investigated with respect to the EMI technique, where the same patch serves as a sensor as well as an actuator concurrently. Abe et al. (2002) encountered large errors in their stress prediction methodology using EMI technique. This error was attributed to imprecise modelling of the interfacial bonding layer. Xu and Liu (2002) proposed a modified impedance model in which the bonding layer was modelled as a SDOF system, connected in between the PZT patch and the host structure, as shown in Fig. 7.6. The bonding layer was assumed to possess a dynamic stiffness K b (or mechanical impedance K b /jω) and the structure a dynamic stiffness K s (or mechanical impedance, Zs = K s /jω). Hence, the resultant mechanical impedance for this series system can be determined as (using Eq. 3.5)

Z res

 K b  K s      jω  jω   K    = b  =  K + K Z s Kb K s b s   + j ω jω

PZT patch

Dynamic stiffness = Kb k

(7.26)

Structure

Z

m c Bonding layer

Fig. 7.6 Modified impedance model of Xu and Liu (2002) including bond layer.

194

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

or

Z res = ζZ s

where

ζ =

1 1 + K s / Kb

(

(7.27)

)

(7.28)

Hence, the coupled electromechanical admittance, as measured across the terminals of the PZT patch and expressed earlier by Eq. (2.24), can be corrected as Y = 2ωj

 Z a  2 E  tan κl  wl  T 2 E d 31Y   (ε 33 − d 31Y ) +  h   κl   Z a + ζZ s 

(7.29)

The term ζ in this equation modifies the dynamic interaction between the PZT patch and the host structure, taking into consideration the effect of the bonding layer. ζ = 1 implies a very stiff bonding layer where as ζ = 0 implies free PZT patch. Xu and Liu (2002) demonstrated numerically that for a SDOF system, as ζ decreases (i.e bond quality degrades), the PZT system would show an increase in the associated resonant frequencies. The investigators further stated that K b depends on the bonding process and the thickness of the bond layer. However, no closed form solution was presented to quantitatively determine K b and hence ζ (From Eq. 7.28). Besides, no experimental verification was provided. The fundamental mechanism of force transfer was therefore nowhere reflected in their analysis. Ong et al (2002) integrated the shear lag effect into impedance modelling using the analysis presented by Sirohi and Chopra (2000b). The PZT patch was assumed to possess a length equal to leff (Eq. 7.22) instead of the actual length. However, since the effective length was determined by considering sensor effect only, the method took care of the associated shear lag only partially. Also, the formulation was valid for beam type structures only and not general in nature. Besides, since the frequencies of the order of 100-150kHz are involved, quasi-static approximation (for calculating leff) is strictly not valid. This chapter presents a detailed step-by-step analysis for including the shear lag effect, first into 1D model (Liang et al., 1994) and then its extension into 2D effective impedance based model (covered in Chapter 5).

195

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

7.4 INCLUSION OF SHEAR LAG EFFECT IN 1D IMPEDANCE MODEL Consider the PZT patch, shown in Figs. 7.1 and 7.2, to be driven by an alternating voltage source and let it be attached to any host structure (not necessarily beam). All the assumptions of Sirohi and Chopra (2000) and Crawley and de Luis (1987) (except for static condition) still hold good. In addition, we assume that the PZT patch is infinitesimally small as compared to the host structure. This means that the host structure has constant mechanical impedance all along the points of attachment of the patch. By D’Alembert’s principle, we can write following equation for dynamic equilibrium of an infinitesimal element of the patch τwp dx + (dm)

∂ 2u p ∂t

2

=

∂Tp ∂x

t p wp dx

(7.30)

where ‘dm’ is the infinitesimal mass and up the displacement in the PZT patch at the location of the infinitesimal element. Because of the dominance of shear stress term, we can neglect the inertial term in Eq. (7.30). The inertial force term has been separately considered in impedance model (Chapter 2 and 5), where as a matter of fact, the shear lag effect was ignored. Hence, the two effects are individually considered and then combined. With this assumption, Eq. (7.30) can be reduced to or

τ =

∂Tp ∂x

tp

(7.31)

Further, assuming pure shear in the bonding layer, τ =

G s (u p − u ) ts

(7.32)

where G s = Gs(1+η ′ j) is the complex shear modulus of the bonding layer and η ′ the mechanical loss factor associated with the bond layer. The axial stress in the PZT patch is given by (from PZT constitutive relation, Eq. 2.14 )

or

Tp = Y E ( S p − ∧)

(7.33)

Tp = Y E (u′p − ∧)

(7.34)

where Y E is the complex Young’s modulus of the PZT patch, S p = u′p is the PZT strain and ∧ = E3d31 is the free piezoelectric strain. Substituting Eqs. (7.32) and (7.34) into Eq. (7.31) and simplifying, we get

196

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

 Y Et t  p s  up − u =  u′′  G  p s  

(7.35)

At any vertical section through the host structure (which includes PZT patch), the force transmitted to the host structure is related to the drive point impedance Z of the host structure by F = − Zu&

(7.36)

where u& is the drive point velocity at the point in question on the surface of the host structure. Since the PZT patch is infinitesimally small, Z is practically same along the entire length of the PZT patch. Eq. (7.36) can be further simplified as (noting that u& = jωu ) Tp w pt p = − Zujω

(7.37)

Substituting Eq. (7.34) and differentiating with respect to x (noting that Z is constant), and rearranging, we get  Zjω   u′ ′ ′ = − up w t YE   pp 

(7.38)

By rearranging various terms, Eq. (7.35) can be written as  G  s (u − u ) u′p′ =   Y Et t  p p s  

(7.39)

Substituting Eq. (7.39) into Eq. (7.38) and solving, we get  Zt jω  u p − u = − s u ′ G w   s p 

(7.40)

Eqs. (7.35) and (7.40) are the fundamental equations governing the shear transfer mechanism via the adhesive bonding layer. Differentiating Eq. (7.40) twice with respect to x and rearranging, we can obtain  Zt jω  u ′p′ = u ′′ −  s u ′′′ w G   p s

(7.41)

Substituting Eqs. (7.40) and (7.41) into Eq. (7.35), differentiating with respect to x, and rearranging, we get

197

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

  w p Gs  u ′′′ −  G s u ′′′′+  −  Y Et t  Zt s jω  s p   

(7.42)

wp Gs

p=−

Let

 u ′′ = 0  

(7.43)

Zt s jω

Substituting Z = x + yj , Gs = (1 + η ′j )Gs and simplifying, we get p = a + bj where

a=

wpGs ( y − η′x)

wpGs ( x + η′y )

b=

and

ωt s ( x 2 + y 2 )

(7.44) (7.45)

ωt s ( x 2 + y 2 )

Since η and η ′ are very small in magnitude, the coefficient of u′′ can be written as q=

Gs E

Y tst p



Gs Y E tst p

(7.46)

It should be noted that p is a complex term whereas the term ‘q’ is approximated as a pure real term. Hence, the resulting differential equation (Eq. 7.42) can be written as u′′′′+ pu′′′ − qu′′ = 0

(7.47)

The characteristic equation for this differential equation is λ4 + pλ3 − qλ2 = 0

(7.48)

Solving, we get roots of the characteristic equation as λ1 = 0, λ2 = 0, λ3 =

− p+

2

2

p + 4q − p − p + 4q , λ4 = 2 2

(7.49)

Hence, the solution of the differential equation Eq.(7.19) can be written as

u = A1 + A 2 x + Be

λ3x

+ Ce

λ4x

(7.50)

The constants A, B, C and D are to be evaluated from the boundary conditions. Differentiating with respect to x, we get u ′ = A2 + Bλ3 e λ3 x + Cλ 4 e λ4 x

(7.51)

Substituting Eqs. (7.50) and (7.51) into Eq.(7.40), we get  Zt jω  u p = ( A1 + A2 x + Beλ3 x + Ce λ 4 x ) −  s  A2 + Bλ3eλ3 x + Cλ4eλ 4 x w G   p s 

(

198

)

(7.52)

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

 Z t jω  Denoting  − s  = 1 / p by n , and simplifying, we get  w G  p s   u p = ( A1 + nA2 ) + A2 x + B(1 + nλ3 )e λ3 x + C (1 + nλ 4 )e λ4 x

(7.53)

Differentiating with respect to x, we can obtain the strain in the PZT patch as S p = A2 + Bλ3 (1 + nλ3 )e λ3 x + Cλ 4 (1 + nλ 4 )e λ4 x

(7.54)

At x = 0 (the mid point of the PZT patch, Figs. 7.1 and 7.2), u = 0, which leads to following condition from Eq. (7.50) A1 = -(B + C)

(7.55)

Further, the boundary condition that at x = 0 up = 0 leads to (from Eq. 7.53) A2 = -(Bλ3 + Cλ4)

(7.56)

Making substitution for A2 from Eq. (7.56) into Eq. (7.54), we get S p = B[λ3 (1 + nλ3 )e λ3 x − λ3 ] + C[λ 4 (1 + nλ 4 )e λ4 x − λ 4 ]

(7.57)

The third and the fourth boundary conditions are imposed by the stress free ends of the PZT patch. At x = -l and at x = +l, the axial strain in the PZT patch is equal to the free piezoelectric strain or Λ (Crawley and de Luis, 1987). The application of these two boundary conditions in Eq. (7.57) result in following equations

[ B[λ (1 + nλ )e

] [ − λ ] + C [λ (1 + nλ )e

]

B λ3 (1 + nλ3 )e − λ3l − λ3 + C λ 4 (1 + nλ 4 )e − λ4l − λ 4 = Λ 3

3

λ3l

3

4

4

λ4l

]

− λ4 = Λ

(7.58) (7.59)

After solving these equations, the constants B and C can be determined as k 4 − k 2  B Λ C  = (k k − k k )  k − k    3 1 4 2 3  1

(7.60)

k1 = λ3 (1 + nλ3 )e − λ3l − λ3

(7.61)

k 2 = λ 4 (1 + nλ 4 )e − λ4l − λ 4

(7.62)

k 3 = λ3 (1 + nλ3 )e λ3l − λ3

(7.63)

k 4 = λ 4 (1 + nλ 4 )e λ4l − λ 4

(7.64)

where

In general, the force transmitted to the host structure can be expressed as F = − Zjωu ( x =l ) , where u(x=l) 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

199

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

coworkers) assume perfect bonding between the PZT patch and the host structure, i.e. the displacement compatibility u(x=l) = up(x=l), thereby approximating the transmitted force as F = − Z jωu p ( x =l ) . However, due to the shear lag phenomenon associated with finitely thick bond layer, u(x=l) ≠ up(x=l). Based on the analysis presented in this section, we can obtain following relationship between u(x=l) and up(x=)l using Eq. (7.40) u( x = l ) u p ( x =l )

=

1  Zt jω  u′ 1 −  s  ( x =l )  w G  u( x = l )  p s

=

1  1 uo′  1 +  p uo  

(7.65)

where uo is as shown in Fig. 7.2. The term u (′x =l ) / u ( x =l ) can be determined by using Eqs. (7.50) and (7.51). Making use of this relationship, the force transmitted to the structure can be written as F = − Zu ( x =l ) −Z

or

F=

where

Z eq =

 1 u o′  1 +  p u o  

(7.66) jωu p ( x =l ) = Z eq jωu p ( x =l )

Z

(7.67)

(7.68)

 1 u o′  1 +  p u o  

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), Zeq = Z. On comparing with the result of Xu and Liu (2002) (Eq. 7.27), we find that ζ =

1 1 uo′ 1+ p uo

(7.69)

The interfacial shear stress can be calculated by using Eq. (7.32). Substituting Eq. (7.40) into Eq. (7.32), we get  − Zjω τ =  w p  From Eqs.(7.51), (7.56) and (7.70) we get

200

 u′  

(7.70)

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

τ=

[ (

)

(

)]

− Zjω Bλ3 e λ3 x − 1 + Cλ4 eλ 4 x − 1 wp

(7.71)

7.5 EXTENSION TO 2D-EFFECTIVE IMPEDANCE BASED MODEL The formulations derived above can be easily extended to the effective impedance based electro-mechanical model developed in Chapter 5. For this derivation, it is assumed that the PZT is square in shape with a length equal to 2l. The strain distribution and the associated shear lag are determined along each principal direction and the two effects are assumed independent, which means that the effects at the corners are neglected. Consider an infinitesimal element of the PZT patch in dynamic equilibrium, as shown in Fig. 7.7. Since this shows planar view, the shear stresses τxz and τyz are not visible in the figure. Considering equilibrium along x-direction we can write (De Faria, 2003), ∂T1 ∂τ xy τ xz + − =0 ∂x ∂y tp

(7.72)

Ignoring the terms involving rate of change of shear strains (consistent with the observation by Zhou et al., 1996), we get

∂T1 τ xz = tp ∂x

(7.73)

Further, using Eqs. (5.11) and (5.12), we can derive

YE T1 = [(S1 + νS2 ) − Λ(1 + ν )] (1 − ν 2 )

 ∂T  T2 +  2 dy  ∂y  y dx T1 x

dy τxy

 ∂τ  τ xy +  xy dy  ∂y   ∂T  T1 +  1 dx  ∂x 

T2

Fig. 7.7 Stresses acting on an infinitesimal PZT element. 201

(7.74)

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

where ν is the Poisson’s ratio of the PZT patch. or

T1 =

[

YE (u′px + υu′py ) − Λ(1 + ν ) (1 − ν 2 )

]

(7.75)

Differentiating with respect to x and ignoring the second order terms involving both x and y (Zhou et al., 1996), we get ∂T1 YE = u 'px' 2 ∂x (1 − ν )

(7.76)

Substituting Eq. (7.76) into Eq. (7.73) and expanding τxz, we get G (u − u ) YE u 'px' = s px 2 (1 −ν ) tst p

(7.77)

On rearranging, we get u px − u x =

Y E tst p Gs (1 − ν ) 2

u 'px'

(7.78)

u 'py'

(7.79)

Similarly, we can write, for the other direction u py − u y =

Y E tst p Gs (1 − ν 2 )

Adding Eqs. (7.78) and (7.79) and dividing by 2, we get

(u

px

+ u py ) 2



(u

x

+ uy ) 2

 Y E t t  (u′′ + u′′ ) p s py  px =  G (1 − ν 2 )  2  s 

(7.80)

From Eq. (5.19), based on the definition of ‘effective displacement’, we can write u p , eff − ueff

or

 Y Et t  p s u′′ =  G (1 − ν 2 )  p , eff  s 

u p , eff − uh , eff ≈

1 u′p′ , eff qeff

(7.81)

(7.82)

where qeff has been approximated as pure real number, as in the 1D case. Here, u p , eff , by definition, is the effective displacement at the interface between the PZT patch and the bond layer and ueff is the corresponding effective displacement at the interface between the structure and the bonding layer. Further, from the definition of effective impedance, introduced in Chapter 5, we can write, for the host structure

202

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

F1 + F2 = − Z eff ueff jω or

(7.83)

T1lt p + T2lt p = − Z eff ueff jω

(7.84)

From Eq. (5.13), we get Y E lt p ( S p1 + S p 2 − 2Λ) (1 − ν )

= − Z eff ueff jω

(7.85)

Substituting for S p1 = u′px and S p 2 = u′py , making use the definition of effective displacement, and differentiating, we can derive

 Z (1 − ν ) jω  u′p , eff u′p′, eff = −  eff  2Y E lt p 

(7.86)

Substituting for u′p′ , eff from Eq. (7.81), we get

or

 Z eff t s jω  ueff ′ u p , eff − ueff = −  2 G ( 1 + ν ) l  s 

(7.87)

 1  u′ u p , eff − ueff =   p  eff eff  

(7.88)

Eqs. (7.82) and (7.88) are the governing equations for 2D case. The parameters p eff and qeff are thus given by  2G (1 + ν )l   p eff = − s  Z t jω   eff s 

qeff ≈

Gs (1 − ν 2 ) Y E t pt s

(7.89)

The rest of the procedure is identical to the one outlined in the previous section for 1D case. The equivalent effective impedance can then be derived as Z eff , eq =

Z eff  1 ueff ′ ,( x =l )   1+   p ueff , ( x = l )   eff 

= ζ Z eff

(7.90)

7.6 EXPERIMENTAL VERIFICATION In order to verify the derivations outlined above, two PZT patches, 10x10x0.3mm and 10x10x0.15mm, conforming to grade PIC 151 (PI Ceramic,

203

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

2003), were bonded to two aluminium blocks, each 48x48x10mm in size. The experimental set-up shown in Fig. 5.5 (page 117) was employed. The PZT patches were bonded to the blocks using RS 850-940 two-part epoxy adhesive (RS Components, 2003). The adhesive layer thickness was maintained at 0.125 mm for both the specimens using two optical fibre pieces of this diameter, by the procedure outlined earlier in Chapter 6. The two specimens have (ts/tp) ratio equal to 0.417 and 0.833 respectively. For obtaining the effective mechanical impedance of the host structure, the numerical approach based on FEM, outlined earlier in chapter 5, was employed. The shear modulus of elasticity of the epoxy adhesive was assumed as 1.0 GPa in accordance with Adams and Wake (1984). The mechanical loss factor of commercial adhesives shows a wide variation and is strongly dependent on temperature. It might vary from 5% to 30% at room temperature, depending upon the type of adhesive (Adams and Wake, 1984). For this study, a value of 10% has been considered. A MATLAB program listed in Appendix G was used to perform the computations automatically. Fig. 7.8 shows the plot of normalized conductance (Gh/L2) worked out using the integrated 2D model developed in this chapter for the two specimens. The plot for perfectly bonded condition is also shown. It is observed that with increasing thickness of the adhesive layer, the sharpness of peaks in the conductance plot tends to diminish. This fact is confirmed by the experimental plots shown in Fig. 7.9 for the two specimens. Fig. 7.10 shows the plot of normalized susceptance (Bh/L2), worked out using the new model for three cases- no bond layer, (ts/tp) = 0.417 and (ts/tp) = 0.833.

Perfect bonding

0.02 0.015 0.01 0.005 0 0

50

0.03 0.025

ts/tp = 0.417

0.02 0.015 0.01 0.005

100 150 200 250

Frequency (kHz)

0.03

0 0

50

G, Normalized (S/m)

0.03 0.025

G, Normalized (S/m)

G, Normalized (S/m)

Again, it is observed that an increase in thickness tends to flatten the peaks. Besides,

0.015 0.01 0.005 0

100 150 200 250

Frequency (kHz)

(a)

ts/tp = 0.834

0.025 0.02

0

50

150 200 250

Frequency (kHz)

(b) Fig. 7.8 Theoretical normalized conductance. (a) Perfect bonding. (b) ts/tp = 0.417. (c) ts/tp = 0.834. 204

100

(c)

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

average slope of the curve also reduces marginally. This is confirmed by Fig. 7.11, which shows the curves determined experimentally for the two specimens. Thus, the shear lag model has made reasonably accurate predictions.

G, normalized (S/m)

0.025 0.02

ts/tp =0.417

0.015

ts/tp =0.834

0.01 0.005 0 0

50

100

150

200

250

Frequency (kHz)

Fig. 7.9 Experimental normalized conductance for ts/tp = 0.417 and ts/tp = 0.834. Perfect bonding

0.06 0.04 0.02 0 0

50

100

150

200

0.1

0.08

B, Normalized (S/m)

B, Normalized (S/m)

0.1

ts/tp = 0.417

0.06 0.04 0.02 0

250

0

50

Frequency (kHz)

100

150

200

0.08

ts/tp = 0.834

0.06 0.04 0.02 0

250

0

Frequency (kHz)

(a)

50

100

150

200

Frequency (kHz)

(b)

(c)

Fig. 7.10 Theoretical normalized susceptance. (a) Perfect bonding. (b) ts/tp = 0.417. (c) ts/tp = 0.834. 0.1 B, Normalized (S/m)

B, Normalized (S/m)

0.1 0.08

0.08

ts/tp =0.417

0.06 0.04 ts/tp =0.834

0.02 0 0

50

100

150

200

250

Frequency (kHz)

Fig. 7.11 Experimental normalized susceptance for ts/tp = 0.417 and ts/tp = 0.834.

205

250

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

That excessive bond layer thickness coupled with poor bond quality can adversely affect the signatures can be investigated by considering the case the case ts/tp = 1.5. An aluminum specimen, again 48x48x10mm, was instrumented with a PZT patch 10x10x0.3mm, but with a bond layer thickness of 0.45mm, implying a thickness ratio of 1.5. To achieve poor bond quality, many pieces of fisherman’s net cord (0.45 mm thickness) were laid on the surface of the host structure prior to applying the adhesive layer. Fig. 7.12 (a) and (b) show the plots of G and B, worked out using the shear lag model developed in this chapter, considering a value of Gs = 0.2GPa. It is clearly evident that free PZT behaviour tends to dominate itself over the structural characteristics. The peak around 150 kHz corresponds to the first PZT resonance. This finding is confirmed by the experimental plots shown in Fig. 7.12(c) and (d). Hence, the newly developed model can accurately predict shear lag

1.00E-01

6.00E-02

8.00E-02

4.00E-02

6.00E-02

2.00E-02

B (S)

G (S)

effect from small thickness to large thickness of the adhesive bond layer.

4.00E-02

0.00E+00 -2.00E-02

2.00E-02

-4.00E-02

0.00E+00 0

50

100

150

200

0

250

50

100

200

250

Frequency (kHz)

Frequency (kHz)

(b)

(a) 1.20E-02

1.20E-02

8.00E-03

8.00E-03 B (S)

G (S)

150

4.00E-03 0.00E+00

4.00E-03 0.00E+00

0

50

100

150

200

250

0

Frequency (kHz)

50

100

150

200

250

Frequency (kHz)

(c)

(d)

Fig. 7.12 Analytical and experimental plots for ts/tp equal to 1.5. (a) Analytical G. (b) Analytical B. (c) Experimental G. (d) Experimental B.

206

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

7.7 PARAMETRIC

STUDY

ON

ADHESIVE

LAYER

INDUCED

ADMITTANCE SIGNATURES From the derivations in the preceding sections, it can be observed that the extent to which the electro-mechanical admittance signatures are influenced by bond layer depends on following parameters  2G (1 + υ )l   p eff = − s  Z t jω  eff s  

q eff =

G s (1 − υ 2 ) Y Et pts

(7.91)

For this parametric study, we considered a PZT patch 10x10x0.3mm (grade PIC 151) bonded to an aluminum block (grade Al 6061T6), 48x48x10mm in size. The various factors affecting shear lag are Gs (or the ratio YE/G), thickness of adhesive layer (or the ratio ts/tp) and sensor length (l). The influence of all these parameters is studied in depth using the 2D shear lag based effective impedance formulations. The PZT parameters are considered as listed in Table 6.1 (page 165). For the bond layer, it is assumed that ts = 0.125mm, G = 1.0GPa, η ′ = 0.1 (i.e. 10%). The MATLAB program listed in Appendix G was employed to perform all the computations. 7.7.1

Influence of Bond Layer Shear Modulus (Gs) Fig. 7.13 shows the influence of bond layer shear modulus on the

conductance and susceptance signatures. It is observed that as Gs decreases, the peaks of conductance subside down and shift rightwards (i.e. the apparent resonant frequencies undergo an increase). That the peaks shift rightwards was also observed by Xu and Liu (2002). In the susceptance plot, it is observed that the average slope of the curve falls down slightly, besides peaks subsiding down. The worst results are observed for G = 0.05 GPa, where the PZT patch behaves more or less independent of the host structure, as marked by a peak at its resonance frequency, rather than identifying the host structure. In this regard, it should be noted that the imaginary part undergoes more identifiable change. Hence, it could be utilized in detecting problems related to the bond layer. From these observations, it is recommended that for best structural identification, an adhesive with high shear modulus should be used for bonding the PZT patch with the structure.

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Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

193 kHz

198 kHz

0.0025

0.0025

Gs = 1.0 GPa

0.002

Perfect bonding

0.0015

G (S)

G (S)

0.002 0.001 0.0005

0.0015 0.001 0.0005

0 0

50

100

150

200

0

250

0

50

Frequency (kHz)

100

200

250

Frequency (kHz)

(b)

(a) 0.0025

0.1

0.002

0.08

Gs = 0.5 GPa G (S)

G (S)

150

0.0015 0.001 0.0005

Gs = 0.05 GPa

0.06 0.04 0.02

0

0 0

50

100

150

200

250

0

Frequency (kHz)

50

100

150

200

250

Frequency (kHz)

(d)

(c) Perfect bonding 0.01

Gs = 0.05 GPa

B (S)

0.008

Gs = 1.0 GPa

Gs = 0.5 GPa

0.006 0.004 0.002 0 0

50

100

150

200

250

Frequency (kHz)

(e) Fig. 7.13 Influence of shear modulus of elasticity of bond layer. (a) Conductance vs frequency (perfect bonding).(b) Conductance vs frequency (Gs = 1.0GPa). (c) Conductance vs frequency (Gs = 0.5GPa). (e) Susceptance vs frequency.

208

(d) Conductance vs frequency (Gs = 0.05GPa).

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

7.7.2

Influence of Bond Layer Thickness (ts) Fig. 7.14 shows the plots of conductance and susceptance corresponding to

ts = 0.05mm (thickness ratio, ts/tp = 0.17) and 0.1mm (thickness ratio, ts/tp = 0.33). It is apparent that as bond layer thickness increases, the peaks subside down and shift rightwards. Besides, the average slope of the susceptance curve falls down. Hence, the overall effect is similar to that of reducing Gs. Exceptionally thick bond layer (thickness ratio > 1.0) may lead to highly erroneous structural identification, as illustrated in the preceding section. Hence, it is recommended that the bond layer thickness be maintained minimum possible, preferably less than 1/3rd of the patch thickness. 193 kHz 0.002

0.0025

0.002

Perfect bonding

0.0015

0.002

ts/tp = 0.17

0.0015

0.0015

0.001

0.001

0.001

0.0005

0.0005

0.0005

0 0

50

100

150

200

0 250 0

50

100

(a)

150

200

0 250 0

ts/tp = 0.33

50

100

198 kHz

150

200

250

(c)

(b)

0.01

Perfect bonding

0.008

B (S)

G (S)

198 kHz 0.0025

0.0025

ts/tp = 0.17

0.006 0.004 0.002

ts/tp = 0.33

0 0

50

100

150

200

250

(d)

Fig. 7.14 Influence of bond layer thickness. (b) Conductance vs frequency (perfect bonding). (b) Conductance vs frequency (ts/tp = 0.17) (c) Conductance vs frequency (ts/tp = 0.33).

209

(d) Susceptance vs frequency.

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

7.7.3

Influence of Damping of Bond Layer (η ′ ) Fig. 7.15 shows the influence of the damping of the bonding layer on

conductance and susceptance signatures. It is observed from Fig. 7.15(a) that as the damping increases, the slope of the baseline conductance tends to fall down. However, susceptance, on the other hand, remains largely insensitive to damping variations, as can be observed from Fig. 7.15(b). 0.002 Mech. Loss factor = 20% 0.0016

Mech. Loss factor = 10%

G (S)

Mech. Loss factor = 5% 0.0012 0.0008 0.0004 0 0

50

100

150

200

250

200

250

Frequency (kHz)

(a) 6.00E-03 Mech. Loss factor = 20%

5.00E-03

Mech. Loss factor = 10% Mech. Loss factor = 5%

B (S)

4.00E-03 3.00E-03 2.00E-03 1.00E-03 0.00E+00 0

50

100

150

Frequency (kHz)

(b) Fig. 7.15 Influence of damping of bond layer. (a) Conductance vs frequency.

210

(b) Susceptance vs frequency.

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

7.7.4

Overall Influence of Parameter peff Fig. 7.16 shows the influence of the parameter peff on conductance and

susceptance plots. This is achieved by multiplying peff by a constant factor. It can be observed that as peff increases, the sensor response tends to reach the ideal condition corresponding to perfect bonding. From Eq. (7.91), it can be observed that it is the shear modulus Gs and bond layer thickness, which govern the value of parameter peff . Higher peff implies higher Gs and lower ts, which, as observed earlier, are beneficial in getting better admittance response from the PZT patch. 0.0025

peff = 0.5 times

0.002

0.002

0.0015

G (S)

G (S)

0.0025

0.001 0.0005

peff = 1 times

0.0015 0.001 0.0005

0

0 0

50

100

150

200

250

0

100

(b)

(a) 0.0025

0.0025

0.002

0.002

0.0015

G (S)

peff = 2.0 times

0.001

0.001 0.0005

0

0 0

100

Perfect bonding

0.0015

0.0005

0

200

50

100

150

200

(d)

(c) 0.01 0.008

B (S)

G (S)

200

0.006

peff

Perfect bonding peff = 2.0 times peff = 1.0 times = 0.5 times

0.004 0.002 0 0

50

100

150

200

250

(e) Fig. 7.16 Influence of peff . (a) Conductance vs frequency (peff = 0.5 times). (b) Conductance vs frequency (peff = 1.0 times). (c) Conductance vs frequency (peff = 2.0 times). (d) Conductance vs frequency (perfect bonding). (e) Susceptance vs frequency. 211

250

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

7.7.5

Overall Influence of Parameter qeff Fig. 7.17 shows the influence of the parameter qeff on the admittance

signatures. On comparing Figs. 7.17(a), (b) and (c), it is apparent that the influence of qeff alone is not sufficient to improve the quality of conductance signatures. Rather, in the susceptance plots, an increase of qeff alone might marginally degrade the quality of signatures, as can be observed from Fig. 7.17(e).

0.0025

0.0025

qeff = 0.5 times

G (S)

G (S)

0.002 0.0015

0.002

0.001

0.001

0.0005

0.0005

0

0 0

50

100

150

200

250

0

100

0.0025

0.0025

0.002

0.002

G (S)

qeff = 2.0 times

0.0015

200

(b)

(a)

0.001

Perfect bonding

0.0015 0.001 0.0005

0.0005

0

0 0

100

0

200

50

100

150

(d)

(c) 0.01

Perfect bonding qeff = 0.5 times

0.008

B (S)

G (S)

qeff = 1 times

0.0015

qeff = 1.0 times

0.006 0.004 0.002

qeff = 2.0 times

0 0

50

100

150

200

250

(e) Fig. 7.17 Influence of qeff . (a) Conductance vs frequency (qeff = 0.5 times). (b) Conductance vs frequency (qeff = 1.0 times). (c) Conductance vs frequency (qeff = 2.0 times). (d) Conductance vs frequency (perfect bonding). (e) Susceptance vs frequency. 212

200

250

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

7.7.6

Influence of Sensor Length (l) Fig. 7.17 shows the influence of sensor length for two typical sizes of PZT

patch, l = 5mm and l = 20mm. It is observed that for small sensor lengths, the presence of bond layer does not affect the signature as adversely as for long PZT patches. Hence, small lengths of PZT patches are recommended for better structural identification. 0.04

0.0025

0.03

l = 5 mm G (S)

0.0015 0.001

l = 20 mm

0.02 0.01

0.0005

0

0 0

50

100

150

200

0

250

50

100

150

200

250

Frequency (kHz)

Frequency (kHz)

(a)

(b)

0.0084

0.12 0.1

l = 5 mm B (S)

0.0063

B (S)

G (S)

0.002

0.0042 0.0021

l = 20 mm

0.08 0.06 0.04 0.02

0

0 0

50

100

150

200

250

0

Frequency (kHz)

50

100

150

Frequency (kHz)

(c)

(d)

With bond layer

Perfect bonding

Fig. 7.17 Influence of sensor length. (a) Conductance vs frequency (l = 5 mm). (b) Conductance vs frequency (l = 20 mm). (c) Susceptance vs frequency (l = 5 mm). (d) Susceptance vs frequency (l = 20 mm).

213

200

250

Chapter 7: Inclusion of Interfacial Shear Lag Effect in Impedance Models

7.7.7

Quantification of Overall Influence of Bond Layer The parametric study described in the previous subsections showed the

influence of various parameters related to the bond layer on the admittance response. The overall influence of the bond layer can be quantified using the parameter ζ defined by Eq. (7.90). For best results, this factor should preferably be as close as possible to unity. It is important to include the shear lag effect into the analysis if ζ < 0.8. 7.8 SUMMARY AND CONCLUDING REMARKS This chapter has rigorously addressed the problem of incorporating the influence of adhesive layer in the electro-mechanical impedance modelling. The treatment presented is generic in nature and not restricted to beam structures alone, as in the case of Crawley and de Luis (1987) and Sirohi and Chopra (2000). Besides, dynamic equilibrium of the system has been considered rather than relying on equivalent length static coefficients. The formulations have been extended to 2D effective impedance based model and have been experimentally verified. Hence, the treatment is more general, rigorous and accurate. The study covered in this chapter showed that the bond layer can significantly influence structural identification if not carefully accounted for. Useful parametric study was also carried out to consider the influence of the various parameters related to adhesive bond layer. It is found 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. Too low shear modulus of elasticity or too large thickness of the bond layer can produce erroneous or misleading results, such as overestimation of peak frequencies or the dominance of PZT patch’s own frequencies. Further, in order to minimize the influence of the bond layer, small sized PZT patches should be employed for structural identification. In addition, the imaginary part of the admittance signature, so far considered redundant, can play meaningful part in detecting any deterioration of the bond layer since it is more sensitive to damages to the bond layer. It is therefore recommended to pay careful attention to the imaginary component of the admittance signature while applying the EMI technique for NDE or structural identification.

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Chapter 8: Practical Issues Related to EMI Technique

Chapter 8 PRACTICAL ISSUES RELATED TO EMI TECHNIQUE

8.1 INTRODUCTION In spite of key advantages such as cost-effectiveness and high sensitivity, there are several impediments to the practical implementation of the EMI technique for NDE of real-life structures, such as aerospace components, machine parts, buildings and bridges. The main challenge lies in achieving a consistent behavior from the surface bonded piezo-impedance transducers over sufficiently long periods, typically of the order of few years, under ‘harsh’ environmental conditions. Hence, protecting PZT patches from unfriendly environmental effects is very crucial in ensuring reliability of the patches for SHM. This chapter reports a dedicated investigation stretched over several months, carried out to ascertain long-term consistency of the electro-mechanical admittance signatures. Possible means of protecting the patches by suitable covering layer and the effects of such layer on the sensitivity of the patch are also investigated. The chapter also investigates on the possible use of multiplexing to optimize sensor interrogation time. 8.2 EVALUATION OF LONG TERM REPEATIBILITY OF SIGNATURES The PZT transducers are relatively new for SHM engineers, who are more accustomed to using conventional sensors such as strain gauges and accelerometers. They are often skeptical about the reliability of the signature based EMI technique. It is often argued that if the signatures are not repeatable enough over long periods of time, it could be very confusing for the maintenance engineers to make any meaningful interpretation about damage. No study has so far been reported to investigate this vital practical issue.

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Chapter 8: Practical Issues Related to EMI Technique

In this research, an experimental investigation, spanning over two months, was carried out in order to ascertain the repeatability of the admittance signatures. Fig. 8.1 shows the details of the specimen employed for this purpose. It was an aluminum plate, 200x160x2mm in size, instrumented with two PZT patches, which were periodically scanned for over two months. Very often, the wires from the patches to the impedance analyzer were detached and reconnected during the experiments. Fig. 8.2 shows the conductance signatures of patch #1 over the twomonth duration. Very good repeatability is clearly evident from this figure. Standard deviation was determined for this set of signatures at each frequency step. Average standard deviation worked out to be 4.36x10-6S (Seimens) against a mean value of 2.68x10-4 S. Hence, the normalized standard deviation (average standard deviation divided by mean) worked out to be 1.5% only, which shows that the repeatability of the signatures was excellent over the period of experiments. Fig. 8.3 similarly shows the susceptance plots of patch #1 over the same period. From this figure, it is observed that susceptance plots also exhibit good repeatability. Similar repeatability was also observed for the signatures acquired from the PZT patch #2. 8.3 PROTECTION OF PZT TRANSDUCERS AGAINST ENVIRONMENT If piezo-impedance transducers are to be employed for the NDE of real-life structures, they are bound to be influenced by environmental effects, such as temperature fluctuations and humidity. Temperature effects have been studied by many researchers in the past (e.g. Sun et al., 1995; Park et al., 1999) and algorithms for compensating these have already been developed. However, no study has so far been undertaken to investigate the influence of humidity on the signatures. In this research, an experiment was conducted on the specimen shown in Fig. 8.1. The PZT patch #2, which was not protected by any layer, was soaked in water for 24 hours and its signatures were recorded before as well as after this exercise (excess water was wiped off the surface before recording the signature). Figs. 8.4(a) and (b) compare the conductance and susceptance signatures respectively for two conditions. That humidity has exercised adverse effect on the signatures is clearly evident by the substantial vertical shift in the conductance signature (Fig. 8.4a). From Eq. (2.24), it is

most

probable

significantly increased the

216

that

the

presence of humidity has

Chapter 8: Practical Issues Related to EMI Technique

PZT patch #2

100 mm

100 mm = =

Hole (damage)

50 mm

= =

110 mm

PZT patch #1

Fig. 8.1 Test specimen for evaluating repeatability of admittance signatures.

0.0006

Day1

0.0005

Day 9

Day 20

Day 26

Day 40

Day 49

Day 64

G (S)

0.0004 0.0003 0.0002 0.0001 100

110

120

130

140

150

Frequency (kHz)

Fig. 8.2 A set of conductance signatures of PZT patch #1 spanning over two months.

0.006 0.005

Day1

Day 9

Day 20

Day 26

Day 40

Day 49

Day 64

B (S)

0.004 0.003 0.002 0.001 0 100

110

120

130

140

150

Frequency (kHz)

Fig. 8.3 A set of susceptance signatures of PZT patch #1 spanning over two months. 217

Chapter 8: Practical Issues Related to EMI Technique

electric permittivity of the patch. This experiment suggests that a protection layer is necessary to protect the PZT patches against humidity in the actual field applications. It should be mentioned here that upon drying by a stream of hot air, the signatures subsided down, although still not recovering the original condition completely. Silicon rubber was chosen as a candidate protective material since it is known to be a good water proofing material, chemically inert and at the same time very good electric insulator. Besides, it is commercially available as paste which can be solidified by curing at room temperature. To evaluate the protective strength of silicon rubber, PZT patch #1 (see Fig. 8.1) was covered with silicon rubber coating (grade 3140, Dow Corning Corporation, 2003). The previous experiment carried out on patch #2 (i.e. soaking with water for 24 hours) was repeated on patch #1. Figs. 8.4 (c) and (d) compare the signatures recorded from this patch in the dry state as well as humid state. It is found that there is very negligible change in the signatures even after long exposure to humid conditions. Hence, silicon rubber is capable of protecting PZT patches against humidity. Although this experiment clearly establishes the suitability of silicon rubber in providing protection against humidity, it is however likely that its presence could reduce the damage sensitivity of the PZT patch. In order to ascertain this doubt, damage was induced in the plate by drilling a 5mm-diameter hole equidistant from the two PZT patches (damage location is shown in Fig. 8.1). Fig. 8.5 shows the effect of this damage on the signatures of the two PZT patches- the protected patch (patch #1) and the unprotected patch (patch #2). The damage was quantified using the root mean square deviation of the signature from its baseline position using Eq. (2.28). The RMSD index was worked out to be 5.6% for the unprotected patch (patch #2) and 5% for the protected patch (patch #1). This shows that the silicon rubber covering layer has only marginal effect on the damage sensitivity of the PZT patches. Hence, silicon rubber is very suitable in protecting the PZT patches against environmental hazards, without significantly diminishing their sensitivity. It should be mentioned here that commercially available packaged QuickPack® actuators (Mide Technology Corporation, 2004) are also likely to be robust against humidity, though no study has been reported so far. However, the packaging itself enhances the cost by at least 10 times. The proposed protection, using silicon rubber, on the other hand, offers a simple and an economical solution to the problem of humidity.

218

Chapter 8: Practical Issues Related to EMI Technique 0.00065

0.005

Unprotected patch

Unprotected patch

0.004

0.00045

0.003

B (S)

G (S)

0.00055

0.00035

0.002

0.00025

0.001

0.00015 100

110

120

130

140

0 100

150

110

Frequency (kHz)

120

130

150

Frequency (kHz)

(a)

(b)

0.00045

0.005

Protected patch

Protected patch

0.004

0.00035

0.003 B (S)

G (S)

140

0.00025

0.002 0.001 0 100

0.00015

100

110

120

130

140

150

110

120

130

140

150

Frequency (kHz)

Frequency (kHz)

(c)

(d) Humid condition

Dry condition

Fig. 8.4 Effect of humidity on signature. (a) Unprotected patch: G-plot. (b) Unprotected patch: B-plot. (c) Protected patch: G-plot.

Pristine state

0.0004

Pristine state

0.00032

After damage Conductance (S)

Conductance (S)

(d) Protected patch: B-plot.

0.00035

0.0003

0.00025 128

130

132

134

136

138

0.00028

0.00024

0.0002 112

140

Frequency (kHz)

After damage 114

116

118

120

Frequency (kHz)

(b)

(a)

Fig. 8.5 Effect of damage on conductance signatures. (a) Unprotected PZT patch (patch #2). (b) PZT patch protected by silicon rubber (patch #1). 219

122

124

Chapter 8: Practical Issues Related to EMI Technique

8.4 MULTIPLEXING OF SIGNALS FROM PZT ARRAYS The EMI technique employs PZT arrays, which, for NDE, must be scanned on one-to one basis. In real-life applications, this could turn out to be a very time consuming operation. For example, if a structure has been instrumented with 50 PZT patches which are intended to be scanned in the frequency range 100-120 kHz at an interval of 100Hz, the entire operation would consume approximately one hour on the standard HP 4192A impedance analyzer, operating in the normal mode using PC interface. However, such a thorough scan may not be warranted most of the time. In this research, the feasibility of reducing PZT scanning time using a multiplexing device was investigated. The test specimen was an aluminium plate, 600x500x10mm in size, instrumented with 20 PZT patches, as shown in Fig. 8.6. The patches were not connected directly to the impedance analyzer. Rather, they were first wired to the 40 channel N2260A multiplexer module housed inside 3499B switch control system (Agilent Technologies, 2001), that was in-turn connected to the HP 4192A impedance analyzer. The entire set-up is shown in Fig. 8.7. With this system, any number of PZT patches (from one PZT patch to all) can be activated simultaneously for interrogation. Park et al. (2001) also reported connecting multiple patches to the impedance analyzer simultaneously. But his arrangement lacked the flexibility of scanning the patches individually should the need arise, since the patches were connected permanently. However, in the present system, the advantage is that both the options (individually or group or subgroup)

5 x 100 mm

6 x 100 mm

PZT patches Damage (10mm φ hole)

Fig. 8.6 Test specimen for evaluating signature multiplexing.

220

Chapter 8: Practical Issues Related to EMI Technique Controlling personal computer

3499B switch control system housing N2260A multiplexer module

HP 4192A impedance analyzer

Fig. 8.7 Experimental set-up consisting of impedance analyzer, controller PC and multiplexer.

are available at a button’s press. Any number of patches can be activated simply be switching, thus offering great optimization flexibility. The multiplexer module is especially manufactured for low-current applications, as in the present case. With this arrangement, there is no necessity to scan the patches on one-toone basis in the routine checks. All the patches can be simultaneously scanned regularly. In the case of an unusual observation from the collective signature (which would be the case at the onset of damage), one-to-one basis (or scanning small groups of PZT patches collectively) can be resorted back so as to localize the damage location. This idea of multiplexing PZT signatures was tested on the twenty PZT patches instrumented on the plate shown in Fig. 8.6. Fig. 8.8 shows the effect of damage (a 10mm diameter hole, shown in Fig. 8.6) on the collective signature of 20 PZT patches. Presence of damage can be easily inferred from the conductance plot shown in the figure. Hence, the multiplexing of PZT signals can enable the user to reduce the interrogation time substantially. Moreover, the presence of the

221

Chapter 8: Practical Issues Related to EMI Technique

0.007

0.08

Pristine state

0.07

B (S)

G (S)

0.006 0.005 0.004 0.003 120

121

122

123

124

0.06 0.05

After damage

0.04 120

125

Frequency (kHz)

Pristine state

After damage 121

122

123

124

125

Frequency (kHz)

(b)

(a)

Fig. 8.8 Effect of damage on collective signature of 20 PZT patches. (a) Conductance. (b) Susceptance. multiplexer module ensures much more stable and repeatable signatures due to more secure connections.

8.5 CONCLUDING REMARKS This chapter has addressed key practical issues related to the implementation of the EMI technique for NDE of real-life structures. The results of the repeatability study, which extended over a period of two months, demonstrated that PZT patches exhibit excellent repeatable performance and are reliable enough to be used for monitoring real-life structures. However, are at the same time, the signatures are highly sensitive to humidity. Silicon rubber has been experimentally found to be a good covering material to impart sound protection against humidity. Hence the presence of silicon rubber layer can enable the application of the method on realworld civil-structures, where it is necessary that the transducer should serve for long periods. The striking feature of the silicon rubber is that it does not adversely affect the damage sensitivity of the PZT patch. In addition, the feasibility of reducing the scanning time and effort using commercially available multiplexing system has also been demonstrated in this chapter.

222

Chapter 9: Conclusions and Recommendations

Chapter 9 CONCLUSIONS AND RECOMMENDATIONS

9.1 INTRODUCTION This thesis embodies findings from the research carried out for structural identification, health monitoring and non-destructive evaluation using structural impedance parameters extracted using surface bonded piezo-impedance transducers. Specifically, a major objective of the research was to upgrade the EMI technique from its present state-of-the art of relying on statistical non-parametric damage evaluation using raw signatures. This conventional approach lacked not only an understanding of the inherent damage mechanism but also a rigorous calibration to realistically estimate damage severity in real-life situations. The major novelty in the present research is that for the first time, extraction of mechanical impedance parameters has been attempted using piezo-impedance transducers. This approach has been shown more realistic as well as more sensitive to damage. Any ‘unknown’ structure can be ‘identified’ by the proposed method, without warranting any a priori information governing the phenomenological nature of the structure. The following sections outline the major contributions, conclusions and recommendations stemming out from this research. 9.2 RESEARCH CONCLUSIONS AND CONTRIBUTIONS Major research conclusions and contributions can be summarized as follows (i)

The raw conductance signature (real component), which is conventionally employed for SHM in the EMI technique, is mixed with a ‘passive’ component, arising out of the capacitance of the PZT patch. This passive

223

Chapter 9: Conclusions and Recommendations

component, which is ‘inert’ to structural damages, tends to lower down the damage sensitivity of the conductance signature. The raw susceptance signature (imaginary component) is similarly ‘camouflaged’, somewhat more heavily, thereby diminishing its usefulness for SHM to the extent of redundancy. In this research, a new concept of active-signatures has been introduced to extract damage sensitive ‘active’ components by carrying out signature decomposition. This filtering process is found to substantially improve the sensitivity of the both the real as well as the imaginary component. Rather, it has been found to raise the level of sensitivity of the imaginary component as high as its real counterpart. Hence, together, these can be employed to derive more pertinent information governing the phenomenological nature of the host structure. (ii)

A new method of analyzing the electro-mechanical admittance signatures has been developed for diagnosing damages in skeletal structures. This method involves extracting the ‘apparent’ drive point structural impedance from the active conductance and active susceptance signatures. Hence, both real and imaginary components are utilized for damage assessment. A complex damage metric has been proposed for quantifying damages using the extracted structural parameters. The real part of the damage metric indicates changes in the equivalent SDOF damping, whereas the imaginary part indicates the changes in the equivalent SDOF stiffness-mass factor resulting from damages. As proof-of-concept, the new methodology was applied on a model RC frame subjected to base vibrations on a shaking table. The proposed methodology was found to perform much better than the existing damage quantification approaches i.e. the low frequency vibration methods as well as the traditional raw-signature based damage assessment using the EMI technique. The instrumented PZT patches were also found to provide meaningful insight into the changes taking place in the structural parameters due to damages.

224

Chapter 9: Conclusions and Recommendations

(iii)

In order to extend the impedance based damage diagnosis method to the general class of structures, a new PZT-structure interaction model has been developed based on the concept of ‘effective impedance’. As opposed to the previous impedance-based models, the new model condenses the twodirectional mechanical coupling between the PZT patch and the host structure into a single impedance term. The model has been verified on a representative aerospace structural component over a frequency range of 0200 kHz. To the author’s best knowledge, this has been the first ever attempt to compare theoretical and experimental admittance signatures relevant to EMI technique for such high frequencies. As a byproduct, a new method has been developed to drive the EDP mechanical impedance of any complex real-life structure by 3D dynamic harmonic analysis using any commercial finite element software. The new model bridges gap between the 1D impedance model of Liang et al. (1993, 1994) and the 2D model proposed by Zhou et al. (1995, 1996). The new impedance formulations can be conveniently employed to extract the 2D mechanical impedance of any ‘unknown’ structure from the admittance signatures of a surface-bonded PZT patch. Besides NDE, the proposed model can be employed in numerous other applications, such as predicting system’s response, energy conversion efficiency and system power consumption.

(iv)

A new experimental technique has been developed to ‘update’ the model of the piezo-impedance transducer before it could be surface bonded for ‘identifying’ the host structure. This updating has been found to facilitate a more accurate identification of structural system’s parameters. The new impedance formulations, in conjunction with the ‘updated’ PZT model, can be employed to ‘identify’ the host structure and to carry out parametric damage assessment. Proof-of-concept applications of the proposed structural identification and health monitoring methodology have been undertaken on structures ranging from precision machine and aerospace components to large civil-structures. Since the dynamic characteristics of the host structure are not altered by small sized PZT patches, a very accurate structural

225

Chapter 9: Conclusions and Recommendations

identification is therefore possible by the proposed method. The piezoimpedance transducers can be installed on the inaccessible parts of crucial machine components, aircraft main landing gear fitting, turbo-engine blades, RCC panels of space shuttles and civil-structures to perform continuous real-time SHM. The equivalent system is identified from the experimental data alone. No analytical/ numerical model is required as a prerequisite. The proposed NDE method has also demonstrated ability to detect damages resulting from loss of mass, such as in the RCC panels of space shuttles due to oxidation. (v)

After identifying the impedance parameters, it is equally important to relate them with physical parameters such as strength/ stiffness and to calibrate the changes in the parameters with damage progression in the component. Towards this end, comprehensive tests were performed on concrete specimens up to failure to empirically calibrate the ‘identified’ system parameters with damage severity. It has been found that in the frequency range 60-100 kHz, concrete essentially behaves as a parallel spring damper combination. The equivalent spring stiffness has been found to reduce and the damping found to increase with damage progression. However, in most tests, the damping was found to undergo major changes towards specimen failure only. The equivalent stiffness, on the contrary, showed a uniform and consistent trend and was found more suitable for diagnosing damages ranging from incipient types to very severe types. A fuzzy probability based damage model has been proposed based on the extracted equivalent stiffness from the tests conducted on concrete. This has enabled the calibration of the piezo-impedance transducers in terms of damage severity and can serve as a practical empirical phenomenological damage model for quantitatively estimating damage severity in concrete.

(vi)

A new experimental technique has been developed to determine in situ concrete strength non-destructively using the EMI principle. The new technique is much superior than the existing strength prediction techniques such as ultrasonic methods. The new method demands only one free surface of the specimen only whereas the ultrasonic methods (Chapter 6) warrant

226

Chapter 9: Conclusions and Recommendations

two opposite surfaces. In addition, this research has shown the feasibility of monitoring curing of concrete using the EMI technique, demonstrating much higher sensitivity than the conventional methods. This method can be applied in the construction industry to decide the appropriate time of removal of the formwork and the time of commencement of prestressing operations in the prestressed concrete members. (vii)

This research has minutely investigated the mechanism of force transfer between the PZT patch and the host structure through the interfacial adhesive bond layer and has presented a step-by-step derivation to integrate its effects into impedance formulations, both 1D and 2D. The treatment presented in this research is of general nature and not restricted to beam structures alone as in the case of the analysis presented by Crawley and de Luis (1987) and Sirohi and Chopra (2000b). Useful parametric study has also been carried out to investigate the influence of the various parameters related to the adhesive bond layer. It has been found that a high shear modulus of elasticity and a small thickness of bond layer is imperative in ensuring accurate structural identification. Preferably, the bond layer should not be thicker than one-third of the PZT patch. Also, the length of the PZT patch should be kept as small as possible to minimize inaccuracies due to shear lag.

(viii) Finally, this research has addressed key practical issues related to the implementation of the EMI technique for NDE of real-life structures. The results of the repeatability study, which extended over a period of two months, demonstrated that the PZT patches exhibit excellent repeatable performance and are reliable enough for monitoring real-life structures. However, the signatures are at the same time highly prone to contamination by humidity. Silicon rubber has been experimentally shown to be a good material to impart sound protection against humidity. The striking feature of the silicon rubber is that it does not adversely affect the damage sensitivity of the PZT patch. The feasibility of reducing the scanning time and effort using commercially available multiplexing system has also been demonstrated.

227

Chapter 9: Conclusions and Recommendations

9.3 RECOMMENDATIONS FOR FUTURE WORK From the experience of carrying out research in the field of EMI technique, the author believes that the present research work can be further extended as follows (i)

In this research, methods have been developed to filter off the damage insensitive inert components from the admittance signatures. However, theoretically, it is possible, by adjusting the PZT properties (namely Y E , T ε 33 and d31) that the passive component is automatically nullified (Eq. 3.28).

This means that the raw signature itself will be as good as the active signature, thereby eliminating the requirements of filtering. Hence, research should be directed so as to obtain a material composition where this could be achieved. Besides, research could be focused on developing temperature tolerant PZT material so that the requirements of temperature compensation could also be eliminated. (ii)

The effective impedance based electro-mechanical formulations derived in Chapter 5 should be further extended to embedded PZT patches, such as in laminated beams. In this case, it could be necessary to consider vibrations in the thickness direction also in the analysis.

(iii)

This research has demonstrated the possibility of non-destructive concrete strength assessment using PZT patches. However, more tests should be conducted in order to take into consideration the effects of variables like type of cement, type and size of the aggregates, type and size of the PZT patches and their mechanical and electrical properties. All these issues should be addressed before the technique could be standardized and commercialized. Further experiments should also be performed so that the material strength estimation technique can be extended to other materials. A universal calibration chart could also be developed.

(iv)

The fuzzy probabilistic damage severity calibration methodology presented in this thesis for concrete can be extended to other materials. It adequacy should be tested for concrete subjected to tension and bending also.

228

Chapter 9: Conclusions and Recommendations

(v)

In reality, for an adhesively bonded piezo-impedance transducer, the governing differential equation is

τwp dx + (dm)

∂ 2u p ∂t 2

=

∂Tp ∂x

t p wp dx

(9.1)

In the present analysis, the inertial force term and the shear force term have been considered separately and the two effects are superimposed. However, this is only an approximation. It is recommended that ways and means should be developed to solve this differential equation by considering the two effects concurrently. The resulting impedance model would be, truly speaking, the most realistic one. The author strongly believes that there is great potential in developing the EMI technique as a universal cost-effective NDE technique.

229

Author’s Publications

AUTHOR’S PUBLICATIONS JOURNAL 1. Soh, C. K., Tseng, K. K.-H., Bhalla, S. and Gupta, A. (2000), “Performance of Smart Piezoceramic Patches in Health Monitoring of a RC Bridge”, Smart Materials and Structures, Vol. 9, No. 4, pp. 533-542. (Based on author’s M. Eng. thesis). 2. Bhalla, S. and Soh, C. K. (2003), “Structural Impedance Based Damage Diagnosis by Piezo-Transducers”, Earthquake Engineering and Structural Dynamics, Vol. 32, No. 12, pp. 1897-1916. (Based on Chapter 4 of thesis). 3. Bhalla, S. and Soh, C.K. (2004), “High Frequency Piezoelectric Signatures for Diagnosis of Seismic/ Blast Induced Structural Damages”, NDT&E International, Vol. 37, No. 1, pp. 23-33. (Based on Chapter 4 of thesis). 4. Bhalla, S. and Soh, C.K. (2004), “Structural Health monitoring by PiezoImpedance Transducers:

Modeling”,

Journal of Aerospace Engineering,

ASCE, Vol. 17, No. 4, pp. 154-165. (Based on Chapter 5 of thesis). 5. Bhalla, S. and Soh, C.K. (2004), “Structural Health monitoring by PiezoImpedance Transducers: Applications”, Journal of Aerospace Engineering, ASCE, Vol. 17, No. 4, pp. 166-175. (Based on Chapter 5 of thesis). 6. Bhalla, S. and Soh, C.K. and Liu, Z. (2005), “Wave Propagation Approach for NDE Using Surface Bonded Piezoceramics”, NDT&E International, Vol. 38, No. 2, pp. 143-150. 7. 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.

230

Author’s Publications

8. Soh, C. K. and Bhalla, S. (2004), “Calibration of Piezo-Impedance Transducers for Strength Prediction and Damage Assessment of Concrete”, Smart Materials and Structures, tentatively accepted (Based on Chapter 6 of thesis).

CONFERENCE: 1. Tseng, K. K.-H., Soh, C. K., Gupta, A. and Bhalla, S. (2000), “Health Monitoring of Civil Infrastructure Using Smart Piezoceramic Transducer Patches”, Proceedings of 2nd International Conference on Computational Methods for Smart Structures and Materials, edited by C. A. Brebbia and A. Samartin, 19-20 June, Madrid, WIT Press (Southampton), pp.153-162. (Based on author’s M. Eng. thesis). 2. Bhalla, S., Soh, C. K., Tseng, K. K.-H and Naidu, A. S. K. (2001), “Diagnosis of Incipient Damage in Steel Structures by Means of Piezoceramic Patches”, Proceedings of 8th East Asia-Pacific Conference on Structural Engineering and Construction, 5-7 December, Singapore, paper no. 1598. (Based on author’s M. Eng. thesis). 3. Bhalla, S., Naidu, A. S. K. and Soh, C. K. (2002), “Influence of StructureActuator Interactions and Temperature on Piezoelectric Mechatronic Signatures for NDE”, Proceedings of ISSS-SPIE International Conference on Smart Materials, Structures and Systems, edited by B. Dattaguru, S. Gopalakrishnan and S. Mohan, 12-14 December, Bangalore, Microart Multimedia Solutions (Bangalore), pp. 213-219. (Based on Chapter 3 of thesis).

4. Naidu, A. S. K. and Bhalla, S. (2002), “Damage Detection in Concrete Structures with Smart Piezoceramic Transducers”, Proceedings of ISSS-SPIE International Conference on Smart Materials, Structures and Systems, edited by B. Dattaguru, S. Gopalakrishnan and S. Mohan, 12-14 December, Bangalore, Microart Multimedia Solutions (Bangalore), pp. 639-645. 231

Author’s Publications

5. Ong, C. W. , Yang Y., Wong, Y. T., Bhalla, S., Lu, Y. and Soh, C. K. (2002), “The Effects of Adhesive on the Electro-Mechanical Response of a Piezoceramic Transducer Coupled Smart System”, Proceedings of ISSS-SPIE International Conference on Smart Materials, Structures and Systems, edited by B. Dattaguru, S. Gopalakrishnan and S. Mohan, 12-14 December, Bangalore, Microart Multimedia Solutions (Bangalore), pp. 191-197. 6. Bhalla, S., Naidu, A. S. K., Ong, C. W. and Soh, C. K. (2002), “Practical Issues in the Implementation of Electro-Mechanical Impedance Technique for NDE”, in Smart Structures, Devices and Systems, edited by E. C. Harvey, D. Abbott and

V. K. Varadan, SPIE’s International Symposium on Smart Materials,

Nano-, and Micro-Smart Systems, 16- 18 December, Melbourne, Proceedings of SPIE Vol. 4935, pp. 484-494. (Based on Chapter 8 of thesis) 7. Naidu, A. S. K., Bhalla, S. and Soh, C. K. (2002), “Incipient Damage Localization in Structures Using Smart Piezoceramic Patches” in Smart Structures, Devices and Systems, edited by E. C. Harvey, D. Abbott and V. K. Varadan, SPIE’s International Symposium on Smart Materials, Nano-, and Micro-Smart Systems, 16- 18 December, Melbourne, Proceedings of SPIE Vol. 4935, pp. 495-502. 8. Naidu, A. S. K., Bhalla, S. and Soh, C. K. (2002), “Damage Location Identification in Smart Structures Using Modal Parameters”, Proceedings of the 2nd International Conference on Structural Stability and Dynamics, edited by C. M. Wang, G. R. Liu and K. K. Ang, 16- 19 December, Singapore, World scientific Publishing Co. Pte. Ltd., pp. 737-742.

9. Bhalla, S., Naidu, A. S. K., Yang, Y. W. and Soh, C. K. (2003),

“An

Impedance-Based Piezoelectric-Structure Interaction Model for Smart Structure Applications”, Proceedings of Second MIT Conference on Computational Fluid and Solid Mechanics, edited by K. J. Bathe, June 17-20, Cambridge, pp. 107-

232

Author’s Publications

110. (Based on Chapter 5 of thesis. This paper was awarded Young Researcher Fellowship Award). 10. Naidu, A. S. K., Bhalla, S. and Soh, C. K. (2004), “Recent Developments in Smart Systems Based Structural Health Monitoring”, National Conference on Materials and Structures, 23-24 January, NIT-Warangal, pp. 273-278.

233

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251

Appendix A

APPENDIX (A) Visual Basic program to derive conductance and susceptance plots from ANSYS output. This program is based on 1D impedance model of Liang et al. (1994), Eq. 2.24. All units in the SI system ‘Inputs: Frequency (Hz) Fr (N) Fi (N) Ur (m/s) Ui (m/s)’ ‘Declaration of variables’ Public Const LA As Double = 0.005 ‘Length of PZT patch’ Public Const WA As Double = 1# ‘Width of PZT patch’ Public Const HA As Double = 0.0002 ‘Thickness of PZT patch’ Public Const RHO As Double = 7650# ‘Density of PZT’ Public Const D31 As Double = -0.000000000166 ‘Piezoelectric strain coefficient’ Public Const Y11E As Double = 63000000000# ‘Young’s modulus of PZT’ Public Const E33T As Double = 0.000000015 ‘Electric permittivity of PZT’ Public Const ETA As Double = 0.001 ‘Mechanical loss factor’ Public Const DELTA As Double = 0.012 ‘Electric loss factor’ Dim f As Double ‘Frequency in Hz’ Dim k_real As Double ‘Real component of wave number’ Dim k_imag As Double ‘Imaginary component of wave number’ Dim x, y As Double ‘Real and imaginary components of structural mechanical impedance’ Dim xa, ya As Double ‘Real and imaginary components of PZT mechanical impedance’ Dim r, t As Double ‘Real and imaginary components of tankl/kl’ Dim G, B As Double ‘Real and imaginary components of admittane’ Dim Fr, Fi, Ur, Ui As Double ‘Real and imaginary components of force and displacement’ ‘Main program’ Sub main() Dim index As Integer Dim kl_real, kl_imag As Double For index = 8 To 508 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) = G Cells(index, 14) = B Next index End Sub

‘Real and imaginary components of kl’

‘Subroutine to calculate (tankl/kl)’ Sub tanz_by_z(rl, im As Double)

252

Appendix A 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 r = (a * u - b * v) / q t = (-1#) * (a * v + b * u) / q End Sub ‘Subroutine to calculate kl’ 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 ‘Subroutine to calculate complex admittance’ 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 G=2* multi * (DELTA * E33T + t_r * D31 * D31 * Y11E) B =2* multi * (E33T + t_i * D31 * D31 * Y11E) End Sub ‘Subroutine to calculate actuator impedance’ Sub Z_actuator() Dim multi As Double multia = (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 ‘Subroutine to calculate structure impedance’ 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

253

‘Temporary variables’ ‘Temporary variables’

Appendix B

APPENDIX (B) Visual Basic program to derive real and imaginary components of structural impedance from admittance signatures. This program is based on 1D impedance model of Liang et al. (1994), Eq. 2.24. All units in the SI system ‘Inputs: Frequency (kHz) G (S) B (S)’ ‘Declaration of variables’ ‘Half-length of PZT patch’ Public Const LA As Double = 0.005 ‘Width of PZT patch’ Public Const WA As Double = 0.01 ‘Thickness of PZT patch’ Public Const HA As Double = 0.0002 ‘Density of PZT’ Public Const RHO As Double = 7800# ‘Piezoelectric strain coefficient’ Public Const D31 As Double = -0.00000000021 ‘Young’s modulus of PZT’ Public Const Y11E As Double = 66700000000# ‘Electric permittivity of PZT’ Public Const E33T As Double = 0.00000002124 ‘Mechanical loss factor’ Public Const ETA As Double = 0.001 ‘Electric loss factor’ Public Const DELTA As Double = 0.015 ‘Frequency in Hz’ Dim f As Double ‘Real and imaginary components of wave number’ Dim k_real, k_imag As Double ‘Real and imaginary components of kl’ Dim kl_real, kl_imag As Double ‘Active conductance' Dim Ga As Double ‘Passive susceptance' Dim Ba As Double Dim rgx, k, c As Double ‘Temporary variables’ ‘Real and imaginary components of structural mechanical impedance’ Dim x, y As Double Dim xa, ya As Double ‘Real and imaginary components of actuator mechanical impedance’ Dim xt, yt As Double ‘xt = x + xa and yt = y + ya’ Dim r, t As Double ‘Real and imaginary components of tankl/kl’ Dim Big_R, Big_T As Double ‘Temporary variables’ Dim multi As Double ‘Temporary variable’

‘Main program’

Sub main() Dim Index As Integer ‘For loop index’ For Index = 7 To 507 f = Cells(Index, 1) * 1000 ‘Conversion to Hz’ multi = (WA * LA * 2# * 3.14 * f) / HA G = 0.5*Cells(index, 2) Gp = multi * (E33T * DELTA + D31 ^ 2 * Y11E * ETA) Ga = G - Gp Cells(index, 4) = Ga B = 0.5*Cells(index, 3) Bp = multi * (E33T - D31 ^ 2 * Y11E) Ba = B - Bp Cells(index, 5) = Ba rgx = Ga / Ba Call calc_k(f) kl_real = k_real * LA kl_imag = k_imag * LA

254

Appendix B Call tanz_by_z(kl_real, kl_imag) Call Z_actuator k = D31 * D31 * Y11E * (WA * LA / HA) Big_R = r - ETA * t Big_T = t + ETA * r c = (Big_T + rgx * Big_R) / (rgx * Big_T - Big_R) ct = (ya - c * xa) / (c * ya + xa) xt = (-1#) * (2 * 3.14 * f) k * (ya * ct + xa) * (Big_T + Big_R * c) / (Ga * (1 + ct * ct)) yt = ct * xt x = xt - xa y = yt - ya Cells(Index, 6) = x Cells(Index, 7) = y Next row Next n End Sub ‘Subroutine to calculate wave number’ 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 ‘Subroutine to calculate (tankl/kl)’ 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 r = (a * u - b * v) / q t = (-1#) * (a * v + b * u) / q End Sub ‘Subroutine to calculate mechanical impedance of actuator’ Sub Z_actuator() Dim multia As Double multia = (WA * HA * Y11E) / (2 * 3.14 * LA * f) Big_rt = r * r + t * t xa = multia * (ETA * r - t) / Big_rt ya = multia * (-1#) * (r + ETA * t) / Big_rt End Sub

255

Appendix C

APPENDIX (C) MATLAB program to derive elecro-mechanical admittance signatures from ANSYS output. The program is based on the new 2D model based on effective impedance, covered in Chapter 5 (Eq. 5.30) All units in the SI system %Inputs: Frequency (Hz) Fr (N) Fi (N) Ur (m/s) Ui (m/s) data=dlmread('output250.txt','\t'); %Data-matrix, stores ANSYS output % The symbols declared below carry same meaning as in Appendices A, B LA=0.005; HA= 0.0003; RHO=7800; D31= -0.00000000021;mu=0.3; Y11E= 66700000000; E33T=1.7919e-8; ETA= 0.035; DELTA= 0.0238; f = data(:,1); Fr = data(:,2); Fi = data(:,3); Ur = data(:,4); Ui = data(:,5);

%Frequency in Hz %Real component of effective force %Imaginary component of effective force %Real component of effective displacement %Imaginary component of effective %displacement

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) = 2*(Fi(I) * Ur(I) - Fr(I) * Ui(I)) / (omega(I) * Big_U(I)); y(I) = 2*(-1.0) * (Fr(I)*Ur(I)+Fi(I)*Ui(I))/(omega(I) * Big_U(I)); %Calculation of wave number cons = (RHO *(1-mu*mu)/ (Y11E * (1 + ETA * ETA)))^0.5; k_real(I) = cons * omega(I); k_imag(I) = cons * omega(I) * (-0.5 * ETA); rl(I) = k_real(I) * LA; im(I) = k_imag(I) * LA; %Calculation of tan(kl)/kl 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) = (HA * Y11E) / (pi * (1-mu)* f(I)); Big_rt(I) = r(I) * r(I) + t(I) * t(I);

256

Appendix C 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) = (LA * LA * omega(I)) / HA; K = 2.0 * D31 * D31 * Y11E /(1 - mu); G(I) = 4*multi(I) * (DELTA * E33T + K * t_r(I)); B(I) = 4*multi(I) * (E33T + K * t_i(I)); end subplot(2,1,1); plot(f,G); subplot(2,1,2); plot(f,B);

257

Appendix D

APPENDIX (D) MATLAB program to derive PZT signatures from ANSYS output, using updated PZT model (twin-peak). The program is based on the new 2D model based on effective impedance, covered in Chapter 5. (Eq. 5.56) NOTE: Single peak case can also be dealt with by using cf1 = cf2 All units in the SI system %Inputs: Frequency (Hz), Fr (N), Fi (N), Ur (m), Ui (m) %PZT parameters- based on measurement. data=dlmread('output250.txt','\t'); %Data-matrix, stores the ANSYS output %PZT parameters based on updated model derived by experiment %Symbols for following variables carry same meaning as Appendices A,B LA=0.005; HA= 0.0003; RHO=7800; D31= -2.1e-10;mu=0.3; Y11E= 6.67e10; E33T=1.7919e-8; ETA= 0.03; DELTA= 0.0238; K =5.16e-9; f = data(:,1); Fr = data(:,2); Fi = data(:,3); Ur = data(:,4); Ui = data(:,5); N=size(f); cf1 = 0.94; cf2 = 0.883;

%Frequency in Hz %Real component of effective force %Imaginary component of effective force %Real component of effective displacement %Imaginary component of effective displacement %No of data points %Correction factors for PZT peaks %For single peak case, Cf1 = cf2

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) = 2*(Fi(I) * Ur(I) - Fr(I) * Ui(I)) / (omega(I) * Big_U(I)); y(I) = 2*(-1.0) * (Fr(I)*Ur(I) + Fi(I)*Ui(I))/(omega(I)* Big_U(I)); %Calculation of wave number cons = (RHO *(1-mu*mu)/ (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 * cf1; im(I) = k_imag(I) * LA * cf1; 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);

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Appendix D h(I) = u(I)^2 + v(I)^2; r1(I) = (a(I) * u(I) - b(I) * v(I)) / h(I); t1(I) = (-1.0) * (a(I) * v(I) + b(I) * u(I)) / h(I); rl(I) = k_real(I) * LA * cf2; im(I) = k_imag(I) * LA * cf2; 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; r2(I) = (a(I) * u(I) - b(I) * v(I)) / h(I); t2(I) = (-1.0) * (a(I) * v(I) + b(I) * u(I)) / h(I); r(I) = 0.5 * (r1(I)+r2(I)); t(I) = 0.5 * (t1(I)+t2(I)); %Calculation of actuator impedance multia(I) = (HA * Y11E) / (pi * (1-mu)* 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); %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; multia(I) = (LA * LA * omega(I)) / HA; G(I) = 4*multia(I) * (DELTA * E33T + K *t_r(I)); B(I) = 4*multia(I) * (E33T + K *t_i(I)); end subplot(2,1,1); plot(f,G); subplot(2,1,2); plot(f,B);

259

Appendix E

APPENDIX (E) MATLAB program to derive structural mechanical impedance from experimental admittance signatures, using updated PZT model (twin-peak). The program is based on the new 2D model based on effective impedance, covered in Chapter 5 (Eq. 5.56). NOTE: For single peak case, cf1 = cf2 All units in the SI system %Inputs: Frequency (kHz), G (S), B (S) %PZT parameters- based on measurement. data=dlmread('gb.txt','\t'); %Data-matrix, %The symbols for variables carry same meaning as in Appendices A,B LA=0.005; HA= 0.0003; RHO=7800; D31= -2.1e-10;mu=0.3; Y11E= 6.67e10; E33T=1.7919e-8; ETA= 0.03; DELTA= 0.0238; cf1 = 0.94; cf2 = 0.883; %Correction factors for PZT peaks %For single peak case, cf1 = cf2 f = 1000*data(:,1); %Frequency in Hz G = data(:,2); %Conductance B = data(:,3); %Susceptance K = 5.16e-9; no=size(f);

%K = 2*D31*D31*Y11E/(1-mu); %No of data points

for I = 1:no, %Calculation of active signatures omega(I) = 2*pi*f(I); multi(I) = 4*(LA * LA * omega(I)) / HA; Gp(I) = multi(I) * (E33T * DELTA + K * ETA); GA(I) = G(I)- Gp(I); Bp(I) = multi(I) * (E33T - K); BA(I) = B(I) - Bp(I); %Calculation of M and N M(I) = (BA(I)*HA)/(4*omega(I)*K*LA*LA); N(I) = (-GA(I)*HA)/(4*omega(I)*K*LA*LA); %Calculation of wave number cons = (RHO * (1-mu*mu)/ (Y11E * (1 + ETA * ETA)))^0.5; k_real(I) = cons * omega(I); k_imag(I) = cons * omega(I) * (-0.5 * ETA); rl(I) = k_real(I) * LA; im(I) = k_imag(I) * LA; %Calculation of tan(kl)/kl rl(I) = k_real(I) * LA * cf1; im(I) = k_imag(I) * LA * cf1; a(I) = (exp(-im(I)) + exp(im(I))) * sin(rl(I));

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Appendix E 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; r1(I) = (a(I) * u(I) - b(I) * v(I)) / h(I); t1(I) = (-1.0) * (a(I) * v(I) + b(I) * u(I)) / h(I); rl(I) = k_real(I) * LA * cf2; im(I) = k_imag(I) * LA * cf2; 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; r2(I) = (a(I) * u(I) - b(I) * v(I)) / h(I); t2(I) = (-1.0) * (a(I) * v(I) + b(I) * u(I)) / h(I); r(I) = 0.5 * (r1(I)+r2(I)); t(I) = 0.5 * (t1(I)+t2(I)); %Calculation of actuator impedance multia(I) = (HA * Y11E) / (pi * (1-mu)* 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); %Calculation of structural impedance R(I) = r(I) - ETA * t(I); S(I) = ETA * r(I) + t(I); P(I) = xa(I) * R(I) - ya(I) * S(I); Q(I) = xa(I) * S(I) + ya(I) * R(I); MN(I)= M(I)^2+N(I)^2; x(I) = (P(I)*M(I)+Q(I)*N(I))/MN(I) - xa(I); y(I) = (Q(I)*M(I)-P(I)*N(I))/MN(I) - ya(I); end dlmwrite('f.txt',f,'\t'); dlmwrite('x.txt',x,'\t'); dlmwrite('y.txt',y,'\t');

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Appendix F

APPENDIX (F) MATLAB program to compute fuzzy failure probability All units in the SI system x=sym('x'); mu = 0.3314; sigma= 0.0466;

% Mean damage variable % Standard deviation of damage variable

Dl = 0; Du = 0.4;

% Lower limit of fuzzy interval % Upper limit of fuzzy interval

fuzzy = 0.5 + 0.5*sin((pi/(Du-Dl))*(x-0.5*Du-0.5*Dl)); pow = (-0.5)*(x-mu)^2/(sigma^2); f = exp(pow)/(sqrt(2*pi)*sigma); ans = double(int(f*fuzzy,0,0.4)+ int(f,0.4,1))

262

Appendix G

APPENDIX (G) MATLAB program to derive electro-mechanical admittance signatures from ANSYS output, taking shear lag in the adhesive layer into account. The program is based on the new 2D model based on effective impedance, covered in Chapter 5 (Eqs. 5.56 and 7.90). NOTE: Single peak case can also be dealt with by using cf1 = cf2 ‘All units in the SI system’ %Inputs: Frequency (Hz), Fr (N), Fi (N), Ur (m), Ui (m) %PZT parameters- based on measurement. data=dlmread('output.txt','\t'); %Data-matrix, stores the ANSYS output %Parameters of PZT (averaged, as in Chapter 6)

%Symbols for following variables carry same meaning as Appendices A,B LA=0.005; HA= 0.0003; RHO=7800; D31= -0.00000000021;mu=0.3; Y11E= 66700000000; E33T=1.7785e-8; ETA= 0.0325; DELTA= 0.0224;K =5.35e-9; GE = 1.0e9; HE = 0.000125; BETA = 0.1;V=1.4;cf=0.898; %BETA represents mechanical loss factor of bonding material f = data(:,1); Fr = data(:,2); Fi = data(:,3); Ur = data(:,4); Ui = data(:,5); N=size(f);

%Frequency in Hz %Real component of effective force %Imaginary component of effective force %Real component of effective displacement %Imaginary component of effective displacement %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) = 2*(Fi(I) * Ur(I) - Fr(I) * Ui(I)) / (omega(I) * Big_U(I)); y(I) = 2*(-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*(1-mu*mu)/(Y11E*HA*HE); % qeff ae = 2*LA*GE*(1+mu)*(y(I)-BETA*x(I))/(omega(I)*HE*(x(I)^2+y(I)^2)); be = 2*LA*GE*(1+mu)*(x(I)+BETA*y(I))/(omega(I)*HE*(x(I)^2+y(I)^2)); pe=complex(ae,be); 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);

%peff

%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;

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Appendix G Be=PS*(k4-k2)/(k1*k4-k2*k3); Ce=PS*(k1-k3)/(k1*k4-k2*k3); A1=-Be-Ce; A2=-Be*root3-Ce*root4; ue=A1+A2*LA+Be*E3+Ce*E4; 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);

%End displacement on surface of host structure %End strain surface of host structure

%Calculation of wave number cons = (RHO *(1-mu*mu)/ (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 * cf; im(I) = k_imag(I) * LA * cf; 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) = (HA * Y11E) / (pi * (1-mu)* 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); %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) = (LA * LA * omega(I)) / HA; G(I) = 4*multi(I) * (DELTA * E33T + K *t_r(I)); B(I) = 4*multi(I) * (E33T + K *t_i(I)); Gnor(I)=G(I)*HA/(LA*LA); %Normalized conductance Bnor(I)=B(I)*HA/(LA*LA); %Normalized susceptance end plot(f,G); figure; plot(f,B);

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