Computational and Experimental

Studies on Aircraft Structural

Health Monitoring Systems

Joana Roque Capinha

Dissertação para obtenção do Grau de Mestre em

Engenharia Aeroespacial

Júri

Presidente: Professor Afzal SulemanCo-orientador: Professor Agostinho Rui Alves da FonsecaCo-orientador: Professor Afzal SulemanVogais: Professor Horácio Cláudio de Campos Neto

Outubro 2007

Abstract

Structural health monitoring (SHM) is a research �eld that has been growing in thelast years. It has the ultimate goal of guaranty the safety of the aircrafts but withthe minimal costs as possible. To achieve that, the aircrafts will be subjected tonondestructive tests (NDT) that don't need them to stop their activity. The sensorsused in the tests will be ideally embedded in the structure and will be interrogatedwhen the aircrafts stop to regular inspection between �ights or even when in the air.

Structural �aws represent changes in e�ective thickness and local material prop-erties. The Lamb waves can be used to detect those changes. Therefore mea-surements of variations in Lamb wave propagation can be employed to assess theintegrity of structures.

This thesis will focus on the generation and reception of Lamb waves to detectdamage in a aluminum plate.

Piezoelectric wafer active sensors (PWAS) were bonded on the plate to evaluateits capability in Lamb wave generation and detection. The advantages of PWAS aretheir simplicity, small size and potentially low cost.

The Lamb waves group velocity was predicted theoretically and compared withthe experimental results. The response to changes in the excitation frequency wasalso studied. Cuts were made in the plate in order to see if the Lamb waves coulddetect them.

Numerical simulations of the wave propagation process and interaction withdamage were also performed using the commercially available �nite-element codeANSYS.

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Resumo

A monitorização da integridade estrutural de aeronaves é um campo de pesquisaque tem vindo a crescer nos ultimos anos. O seu objectivo é garantir a segurançadas aeronaves com o minimo de custos possivel. Para atingir esse objectivo, asaeronaves serão sujeitas a testes não destrutivos que não impliquem a sua paragem deactividade. Os sensores usados nesses testes serão idealmente integrados na estruturae são interrogados quando a aeronave parar em actividades regulares de manutençãoentre voos ou até em pleno voo.

As falhas estruturais levam a alterações de espessura e de propriedades locaisdo material. As ondas de Lamb podem ser usadas para detectar essas alterações.Assim, medições de variação na propagação das ondas de Lamb podem ser usadaspara garantir a integridade de estruturas.

Esta tese vai focar-se na geração e recepção de ondas de Lamb para detectardanos numa placa de aluminio.

Como sensores e actuadores serão usados piezoelectricos para testar a sua ca-pacidade de gerar e receber ondas de Lamb. Este tipo de sensores/actuadores têmcomo vantagem a sua simplicidade, pequenas dimensões e baixo custo.

A velocidade das ondas de Lamb foi calculada teoricamente e comparada comos resultados experimentais. A resposta a variações na frequência de excitação foitambém estudada. Foram feitos cortes na placa para ver se eram detectaveis comas ondas de Lamb.

São também efectuadas simulações numericas do processo de propagação dasondas de Lamb e sua interação com danos utilizando o programa de elementos�nitos comercialmente disponivel, ANSYS.

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Acknowledgments

I would like to express my gratitude to my supervisors, professor Agostinho Fonsecaand professor Afzal Suleman, whose expertise, understanding, and patience, addedconsiderably to my graduate experience.

I have a special thank to Bruno Rocha who introduced me to the Lamb wavesconcept. His guidance and support was very important along this all process. Hewas able to pass me all his enthusiasm about this subject.

I would also have to thank Carlos Silva for his patience clarifying some doubtsand ideas changed.

Finally, I have to thank Pedro who always had faith in me and help me wheneverI needed.

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Contents

1 Introduction 3

1.1 Structural Health Monitoring . . . . . . . . . . . . . . . . . . . . . . 31.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Nondestructive Tests and Evaluation 7

2.1 Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 NDT/NDE Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Visual and Optical Testing (VT) . . . . . . . . . . . . . . . . 92.2.2 Liquid Penetrant Inspection (LPI) . . . . . . . . . . . . . . . . 92.2.3 Magnetic Particle Testing (MT) . . . . . . . . . . . . . . . . . 112.2.4 Electromagnetic Testing (ET) or Eddy Current Testing . . . . 132.2.5 Radiography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.6 Ultrasonic Testing (UT) . . . . . . . . . . . . . . . . . . . . . 18

3 Lamb Waves 22

3.1 Dispersion Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Selective excitation of Lamb wave modes . . . . . . . . . . . . . . . . 273.3 Damage Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Piezoelectrics Sensors and Actuators 31

4.1 Piezoelectric e�ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 PZT structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Piezoelectric as a sensor . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Piezoelectric Wafer Active Sensors . . . . . . . . . . . . . . . . . . . . 33

4.4.1 PWAS Generation of Lamb waves . . . . . . . . . . . . . . . . 364.5 Wave Propagation in Electromechanical Structures . . . . . . . . . . 36

4.5.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 374.5.2 Wave Propagation Due to a Single Actuator . . . . . . . . . . 38

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5 Experimental Setup 43

5.1 Lamb Waves Generation . . . . . . . . . . . . . . . . . . . . . . . . . 445.1.1 Limitations of the equipment . . . . . . . . . . . . . . . . . . 455.1.2 Actuation Signal . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 Dispersion Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.3 Time of Flight (TOF) . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3.1 Direct Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.3.2 Re�ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.4.1 Variation with the frequency . . . . . . . . . . . . . . . . . . . 575.4.2 2D Visualization of the Results . . . . . . . . . . . . . . . . . 59

6 Damage Detection 61

6.1 First Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Second cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.3 Increase of the damage . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7 Numerical Simulations 66

7.1 Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667.2 Natural Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.3 Lamb Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 687.4 Damage Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.4.1 1D Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 707.4.2 2D Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 71

8 Conclusions and Further Research 73

8.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738.2 Recommended Future Work . . . . . . . . . . . . . . . . . . . . . . . 73

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List of Figures

2.1 Liquid applied to the surface of the part [17] . . . . . . . . . . . . . . 102.2 Removal of the excess liquid from the surface [17] . . . . . . . . . . . 102.3 Penetrant out of the defect on the surface where it can be seen [17] . 102.4 Visual inspection under UV light[17] . . . . . . . . . . . . . . . . . . 102.5 A bar magnet with 2 poles [17] . . . . . . . . . . . . . . . . . . . . . 122.6 Flux leakage �eld [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Longitudinal magnetic �eld [17] . . . . . . . . . . . . . . . . . . . . . 122.8 Circular magnetic �eld [17] . . . . . . . . . . . . . . . . . . . . . . . . 122.9 The importance of magnetic �eld orientation [17] . . . . . . . . . . . 132.10 Detectability of �aws according to direction of magnetic �eld [17] . . 132.11 Lights for magnetic particle inspection [17] . . . . . . . . . . . . . . . 132.12 Lights for magnetic particle inspection [17] . . . . . . . . . . . . . . . 132.13 Eddy current principle [17] . . . . . . . . . . . . . . . . . . . . . . . . 142.14 Here a small surface probe is scanned over the part surface in an

attempt to detect a crack [17] . . . . . . . . . . . . . . . . . . . . . . 142.15 The electromagnetic spectrum [17] . . . . . . . . . . . . . . . . . . . 152.16 Radiation source [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.17 Radiographic images [17] . . . . . . . . . . . . . . . . . . . . . . . . . 172.18 Radiographic images [17] . . . . . . . . . . . . . . . . . . . . . . . . . 172.19 Typical ultrasonic system [17] . . . . . . . . . . . . . . . . . . . . . . 182.20 Longitudinal and shear waves [17] . . . . . . . . . . . . . . . . . . . . 192.21 Snell's Law [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 Symmetric mode, S0 [8] . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Anti-symmetric mode, A0 [8] . . . . . . . . . . . . . . . . . . . . . . . 223.3 Lamb waves movement [17] . . . . . . . . . . . . . . . . . . . . . . . 233.4 Wave speed vs frequency [4] . . . . . . . . . . . . . . . . . . . . . . . 253.5 Group velocity vs frequency [4] . . . . . . . . . . . . . . . . . . . . . 253.6 Example of application of TOFD method for damage location [1] . . 29

4.1 Internal Structure of an electret [23] . . . . . . . . . . . . . . . . . . . 324.2 A sensor based on the piezoelectric e�ect [23] . . . . . . . . . . . . . 33

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4.3 Piezoelectric wafer transducer a�xed to the host structure [4] . . . . 344.4 Interaction forces and moments [4] . . . . . . . . . . . . . . . . . . . 344.5 PWAS on a 1D structure [4] . . . . . . . . . . . . . . . . . . . . . . . 354.6 PWAS on a 2D structure [4] . . . . . . . . . . . . . . . . . . . . . . . 354.7 Actuators surface-bonded to an elastic medium [10] . . . . . . . . . . 37

5.1 Experimental setup used in the laboratory . . . . . . . . . . . . . . . 435.2 Aluminum plate used in the experiments . . . . . . . . . . . . . . . . 445.3 Function generator, oscilloscope and board . . . . . . . . . . . . . . . 445.4 Square wave (left) and respective power spectrum (right) . . . . . . . 465.5 Ramp signal (left) and respective power spectrum (right) . . . . . . . 465.6 Sine signal (left) and respective power spectrum (right) . . . . . . . . 465.7 Cosine signal (left) and respective power spectrum (right) . . . . . . . 475.8 Actuation wave (left) and respective power spectrum (right) . . . . . 475.9 Actuation wave (left) and respective power spectrum (right) . . . . . 485.10 Dispersion curves for an aluminum plate of 2 mm thickness (Fre-

quency vs Phase velocity) . . . . . . . . . . . . . . . . . . . . . . . . 495.11 Dispersion curves for an aluminum plate of 2 mm thickness (Fre-

quency vs Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . 495.12 Variation of S and A wavelength with frequency . . . . . . . . . . . . 505.13 Coordinates of the piezoeletric actuatiors/sensors in the plate . . . . 515.14 Distances between the piezoelectrics actuators/sensors . . . . . . . . 515.15 Re�ections of the waves in the boundaries . . . . . . . . . . . . . . . 525.16 Lamb Wave in the case of a in�nite plate . . . . . . . . . . . . . . . . 535.17 Re�ections arriving at sensor 1 . . . . . . . . . . . . . . . . . . . . . 535.18 Actuation wave (blue) and received wave (orange) from 1 to sensor 2 555.19 Actuation wave (blue) and received wave (orange) from 1 to sensor 2

with time shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.20 Wave generated by actuator 1 and received by sensor 2 (blue) and

sensor 3 (orange) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.21 In�uence of the frequency in the received waves (actuator 2 to sensor

1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.22 In�uence of the frequency in the received waves (actuator 2 to sensor

3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.23 Received wave for an actuation of 250kHz . . . . . . . . . . . . . . . 585.24 Actuation wave passing through sensor 1 . . . . . . . . . . . . . . . . 595.25 Actuation wave passing through sensor 3 . . . . . . . . . . . . . . . . 60

6.1 Location of the �rst cut in the plate . . . . . . . . . . . . . . . . . . . 616.2 Location of the �rst cut . . . . . . . . . . . . . . . . . . . . . . . . . 62

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6.3 Zoom of the �rst cut . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.4 Actuator 1 to Sensor 2 with the �rst damage(orange) and without

(blue) any damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.5 Actuator 1 to Sensor 3 with (orange) and without (blue) damage . . . 626.6 Location of the second cut in the plate . . . . . . . . . . . . . . . . . 636.7 Actuator 1 to Sensor 2 with the second cut . . . . . . . . . . . . . . . 636.8 Actuator 1 to Sensor 3 with the second cut . . . . . . . . . . . . . . . 646.9 Plate without damage . . . . . . . . . . . . . . . . . . . . . . . . . . 646.10 Plate with a 20mm cut . . . . . . . . . . . . . . . . . . . . . . . . . . 646.11 Plate with a 25 mm cut . . . . . . . . . . . . . . . . . . . . . . . . . 646.12 Plate with a 30mm cut . . . . . . . . . . . . . . . . . . . . . . . . . . 646.13 Plate with a 35mm cut . . . . . . . . . . . . . . . . . . . . . . . . . . 656.14 Plate with a 40mm cut . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7.1 SHELL63 geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667.2 First mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.3 Second mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.4 Third mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.5 Fourth mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.6 Fifth mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.7 Sixth mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.8 Convergence of the natural frequency . . . . . . . . . . . . . . . . . . 687.9 Convergence of the maximum displacement . . . . . . . . . . . . . . . 687.10 Grid and boundary conditions . . . . . . . . . . . . . . . . . . . . . . 697.11 Actuator 1 to Sensor 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 697.12 Actuator 1 to Sensor 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 707.13 Results for the three cases (1 nodes, 3 nodes and 5 nodes) . . . . . . 707.14 Case damaged vs undamaged . . . . . . . . . . . . . . . . . . . . . . 717.15 Actuator 1 to Sensor 2 (�rst cut)) . . . . . . . . . . . . . . . . . . . . 717.16 Actuator 1 to Sensor 3 (�rst cut) . . . . . . . . . . . . . . . . . . . . 717.17 Actuator 1 to Sensor 2 (second cut) . . . . . . . . . . . . . . . . . . . 727.18 Actuator 1 to Sensor 3 (second cut) . . . . . . . . . . . . . . . . . . . 72

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Acronyms

FFT Fast Fourier Transform

LPI Liquid Penetrant Inspection

MPI Magnetic Particle Inspection

NDE Nondestructive Evaluation

NDT Nondestructive Tests

PWAS Piezoelectric Wafer Active Sensor

SHM Structure Health Monitoring

TOF Time of Flight

UT Ultrasonic Testing

VT Visual Testing

S0 First Lamb wave symmetric mode

A0 First Lamb wave assymetric mode

cS S Waves Velocity

cP P Waves Velocity

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Chapter 1

Introduction

1.1 Structural Health Monitoring

Structure Health Monitoring (SHM) is an emerging technology leading to the devel-opment of integrated systems capable of continuously monitoring structures. Theultimate goal of SHM is to increase reliability, improve safety, enable light-weightdesign and reduce maintenance costs for all kinds of structures.

SHM involves integration of Nondestructive Tests (NDT) methods into a vehiclein order to improve damage detection and minimize the human intervention. ActualNDT systems are stationary and ground based which imply that the aircraft isfully stopped whenever it needs to be inspected. Aircrafts have regular mandatoryinspections and every minute that an aircraft is on the ground is an extra cost tothe company due to the non operation. With the new SHM systems the mandatoryperiodic procedures will be reduce which will result in a reduction in the maintenancecosts.

The development of integrated vehicle health monitoring could also reduce oreliminate a number of present design constrains which will make possible new struc-tures with less redundancy and therefore lighter.

Damage detection using Lamb waves is a new promising research �eld. This the-sis will focus on that area. Structural �aws represent changes in e�ective thicknessand local material properties, and therefore measurements of variations in Lambwave propagation can be employed to assess the integrity of these structures. Inthis work piezoelectric wafers are used to transmit and received the Lamb waves.

1.2 State of the Art

Damage in a structure would result in shifts in natural frequencies and changes inthe vibration modes. There are damage detection techniques that allow the mass

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and sti�ness matrices to be updated based on theses changes in natural frequenciesand modes. However, that changes only happens signi�cantly when the damage hasa certain size and not due to small incipient damages [1].

For damage detection using Lamb waves it is important to select the optimalwave mode and to know the propagation and the interaction with defects. Thereforeresearch �elds are: sensors, excitation, bonding/embedded of sensors, propagation[2], interaction with defects [2], signal processing and signal evaluation.

Experiments and simulations with Lamb waves to monitoring plates have beenperformed [3] such as experiments to detect �aws with the pulse-echo method [3].

Some �nite element analysis of wave propagation in a beam specimen have beensuccessfully done [4].

Edalati, Kermani, Seiedi, Movafeghi had investigated the numerical method fordrawing the dispersion and displacement curves of ultrasonic Lamb wave propa-gated in Aluminum thin plate. Two ultrasonic lamb wave techniques, pulse-echoand emission, were used for interpretation of notch defects.It was observed thatthese techniques are sensitive to evaluate defects, especially in short probe to defectdistances [5].

Lamb waves generally have multiple modes highly dispersive and in consequencepulse dispersion can become pronounced and can make di�cult of impossible theinterpretation of pulse-echo responses. Studies show that selective generation of onlyone mode will overcome that di�culties [3].

Ullate and Espinosa (2006) obtained the experimental dispersion curves applyinga 2D Fast Fourier Transform (FFT) algorithm to the data collected from an opticalvibrometer [6]. A �nite element model, using a commercial simulation program,PZFlex, was also developed to calculate the dispersion curves and to compare themwith the analytical and experiment ones.

Develop an e�cient method to model numerically elastic waves propagation fordamage detection [7]

Current work in SHM has focused on damage detection methods and sensoroptimization [2]. In order to put that systems in service there has to be test stan-dards and certi�cations. Wardle and Kessler [18] present experimental results fromdurability testing of piezoelectric Lamb wave transducers and o�er a framework fordeveloping SHM test standards. Lamb wave sensors have been tested in a variety ofenvironments (including high temperature and large strain) so that their operationalenvelop can be characterized.

The investigation of the fundamental aspects of using piezoelectric wafer ac-tive sensors (PWAS) to achieve embedded ultrasonics in thin-gage beam and platestructures opens the path for systematic application of PWAS for in situ healthmonitoring [4,8,9]. One of the aspects that was object of study was the optimal size

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and location of the piezoelectric transducers [13], [11].Nieuwenhuis, Neumann, Greve, and Oppenheim report the use of �nite element

simulation and experiments to further explore the operation of the wafer transducer[21].

Wang and Huang (2004) provided an analytical and numerical study to simu-late the wave propagation in an elastic half plane with surface-bonded piezoceramicactuators under high-frequency electric loads [10]. They had also provided a the-oretical study of elastic wave propagation in a cracked elastic medium induced byand embedded piezoelectric actuator [14].

Greve, Oppenheim, Sohn and Yue (2006) have been exploring an inductivelycoupled Lamb wave transducer that eliminates the need for wired contact. Anadvantage of this type of transducer is the absence of any electrical connections,which eliminates a major point of failure. In their paper [33] they present the resultsof experimental demonstrations using two di�erent transducer design, a ferrite potcore transducer and a planar coil transducer.

There are a lot of international research programs and activities which are de-voted to structural health monitoring. Van der Auweraer and Peeters [15] discussthe EU-cooperative and the main US and Far-East activities.

Farrar and Lieven [16] has discussed the concept of damage prognosis. It at-tempts to forecast system performance by assessing the current damage state of thesystem estimating the future loading environments for that system and predictingthrough simulation and past experience the remaining useful life of the system.

1.3 Thesis Outline

In the chapter 2 there is an introduction to the Nondestructive Tests (NDT). Afterthat is a summarized description of the NDTs that are actually used in the structureshealth monitoring. There is also a brief historical perspective of the NDTs.

The Lamb waves concept is introduced in chapter 3. The existence of multiplemodes is explained such as the dispersion curves. This chapter also refers to theirused in damage detection.

Chapter 4 starts with the general properties of the piezoelectric materials andhow are they used as sensors. Then it talks about the piezoelectric wafer active sen-sor (PWAS) and how can they generate Lamb waves. Finally, is presented an ana-lytical and numerical study to simulate the wave propagation in an elastic half planewith surface-bonded piezoceramic actuators under high-frequency electric loads.

In chapter 5 is shown the laboratory setup. The instruments used, sensors andplate. Some excitation signs are evaluate and the Lamb waves dispersion curves arecalculated. The theoretical time of �ight (TOF) is calculated and then compared

5

with the experimental one. There are also the results of tests where the frequencywas changed.

In chapter 6 are presented the results of the damage detection.Numerical simulations using ANSYS are presented in chapter 7. The natural

frequencies and modal vibrations of the plate are calculated. There are results forthe case of the undamaged and damaged plate. The cracks in the simulation weresimilar to the ones in the laboratory in order to compare the two results.

Chapter 8 presents the conclusions and recommended further work.

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Chapter 2

Nondestructive Tests and Evaluation

The �eld of Nondestructive Tests (NDT) is a very broad, interdisciplinary �eld thatplays a critical role in assuring that structural components and systems performtheir function in a reliable and cost e�ective fashion. NDT technicians and engineersde�ne and implement tests that locate and characterize material conditions and �awsthat might otherwise cause planes to crash, reactors to fail, trains to derail, pipelinesto burst, and a variety of less visible, but equally troubling events. These tests areperformed in a manner that does not a�ect the future usefulness of the object ormaterial. In other words, NDT allows parts and materials to be inspected andmeasured without damaging them. Because it allows inspection without interferingwith a product's �nal use, NDT provides an excellent balance between quality controland cost-e�ectiveness. Generally speaking, NDT applies to industrial inspections.While technologies are used in NDT that are similar to those used in the medicalindustry, typically nonliving objects are the subjects of the inspections.

Nondestructive Evaluation (NDE) is a term that is often used interchangeablywith NDT. However, technically, NDE is used to describe measurements that aremore quantitative in nature. For example, a NDE method would not only locate adefect, but it would also be used to measure something about that defect such asits size, shape, and orientation. NDE may be used to determine material propertiessuch as fracture toughness, formability, and other physical characteristics.

2.1 Historical Perspective

Nondestructive testing has been practiced for many decades. One of the earliestapplications was the detection of surface cracks in railcar wheels and axles. Theparts were dipped in oil, then cleaned and dusted with a powder. When a crackwas present, the oil would seep from the defect and wet the powder providing visualindicating that the component was �awed. This eventually led to oils that were

7

speci�cally formulated for performing these and other inspections and this inspectiontechnique is now called penetrant testing.

X-rays were discovered in 1895 by Wilhelm Conrad Roentgen(1845-1923) whowas a Professor at Wuerzburg University in Germany. Soon after his discovery,Roentgen produced the �rst industrial radiography when he imaged a set of weightsin a box to show his colleagues. Other electronic inspection techniques such as ul-trasonic and eddy current testing started with the initial rapid developments in in-strumentation spurred by technological advances, and subsequent defense and spacee�orts following World War II. In the early days, the primary purpose was the de-tection of defects. Critical parts were produced with a "safe life" design, and wereintended to be defect free during their useful life. The detection of a defects wasautomatically a cause for removal of the component from service.

In the early 1970's, two events occurred which caused a major change in the wayinspections were viewed. The continued improvement of inspection technology, inparticular the ability to detect smaller and smaller �aws, led to more and more partsbeing rejected (even though the probability of part failure had not changed). Atthis time the discipline of fracture mechanics emerged, which enabled one to predictwhether a crack of a given size would fail under a particular load if a particularmaterial property or fracture toughness were known. Other laws were developed topredict the rate of growth of cracks under cyclic loading (fatigue). With the adventof these tools, it became possible to accept structures containing defects if the sizesof those defects were known. This formed the basis for a new design philosophycalled "damage tolerant designs." Components having known defects could continueto be used as long as it could be established that those defects would not grow to acritical size that would result in catastrophic failure.

A new challenge was thus presented to the nondestructive testing community.Mere detection of �aws was not enough. One needed to also obtain quantitativeinformation about �aw size to serve as an input to fracture mechanics calculationsto predict the remaining life of a component. These needs, which were particularlystrong in the defense and nuclear power industries, led to the creation of a number ofresearch programs around the world and the emergence of nondestructive evaluation(NDE) as a new discipline.

2.2 NDT/NDE Methods

The number of NDT methods that can be used to inspect components and makemeasurements is large and continues to grow. Researchers continue to �nd new waysof applying physics and other scienti�c disciplines to develop better NDT methods.However, there are six NDT methods that are used most often. These methods are:

8

• Visual and Optical Testing (VT)

• Penetrant Testing (PT)

• Magnetic Particle Testing (MT)

• Electromagnetic Testing (ET) or Eddy Current Testing

• Radiography (RT)

• Ultrasonic Testing (UT)

2.2.1 Visual and Optical Testing (VT)

Visual inspection involves using an inspector's eyes to look for defects. The inspectormay also use special tools such as magnifying glasses, mirrors, or borescopes to gainaccess and more closely inspect the subject area.

2.2.2 Liquid Penetrant Inspection (LPI)

Liquid penetrant inspection is a method that is used to reveal surface breaking �awsby bleedout of a colored or �uorescent dye from the �aw. The technique is based onthe ability of a liquid to be drawn into a "clean" surface breaking �aw by capillaryaction. After a period of time called the "dwell," excess surface penetrant is removedand a developer applied. This acts as a blotter. It draws the penetrant from the �awto reveal its presence. Colored (contrast) penetrants require good white light while�uorescent penetrants need to be used in darkened conditions with an ultraviolet"black light".

The advantage that a liquid penetrant inspection (LPI) o�ers over an unaidedvisual inspection is that it makes defects easier to see for the inspector. It producesa �aw indication that is much larger and easier for the eye to detect than the �awitself and with a high level of contrast between the indication and the backgroundalso helping to make the indication more easily seen. When a �uorescent penetrantinspection is performed, the penetrant materials are formulated to glow brightly andto give o� light at a wavelength that the eye is most sensitive to under dim lightingconditions.

Liquid penetrant inspection (LPI) is one of the most widely used nondestruc-tive evaluation (NDE) due to its relative ease of use and its �exibility. LPI canbe used to inspect almost any material provided that its surface is not extremelyrough or porous. Materials that are commonly inspected using LPI include metals(aluminum, copper, steel, titanium, etc.), glass, many ceramic materials, rubber andplastics.

9

Figure 2.1: Liquid applied to the sur-face of the part [17]

Figure 2.2: Removal of the excess liq-uid from the surface [17]

Liquid penetrant inspection is used to inspect for �aws that break the surfaceof the sample. For example fatigue cracks, quench cracks, grinding cracks, overloadand impact fractures, porosity, laps, seams, pin holes in welds and lack of fusion orbraising along the edge of the bond line.

Figure 2.3: Penetrant out of the de-fect on the surface where it can beseen [17]

Figure 2.4: Visual inspection underUV light[17]

One of LPI primary advantages is its high sensitivity to small surface discon-tinuities. It has few material limitations, i.e. metallic and nonmetallic, magneticand nonmagnetic, and conductive and nonconductive materials may be inspected.Large areas and large volumes of parts/materials can be inspected rapidly and atlow cost as also parts with complex geometric shapes. Indications are produceddirectly on the surface of the part and constitute a visual representation of the �awand aerosol spray cans make penetrant materials very portable. Penetrant materialsand associated equipment are relatively inexpensive.

LPI primary disadvantages are that only surface breaking defects can be detectedand only materials with a relatively nonporous surface can be inspected. Surface�nish and roughness can a�ect inspection sensitivity. If the part is not very clean,the contaminants can mask the defects. Also the metal smearing from machining,

10

grinding, and grit or vapor blasting must be removed prior to LPI. Post cleaningof acceptable parts or materials is required. The inspector must have direct accessto the surface being inspected. Multiple process operations must be performed andcontrolled. Chemical handling and proper disposal is required.

2.2.3 Magnetic Particle Testing (MT)

Magnetic particle inspection (MPI) is a nondestructive testing method used for de-fect detection. MPI is fast and relatively easy to apply, and part surface preparationis not as critical as it is for some other NDT methods. These characteristics makeMPI one of the most widely utilized nondestructive testing methods.

MPI uses magnetic �elds and small magnetic particles (i.e.iron �lings) to detect�aws in components. The only requirement from an inspectability standpoint is thatthe component being inspected must be made of a ferromagnetic material such asiron, nickel, cobalt, or some of their alloys. Ferromagnetic materials are materialsthat can be magnetized to a level that will allow the inspection to be e�ective.

In theory, magnetic particle inspection (MPI) is a relatively simple concept. Itcan be considered as a combination of two nondestructive testing methods: magnetic�ux leakage testing and visual testing. Consider the case of a bar magnet. It has amagnetic �eld in and around the magnet. Any place that a magnetic line of forceexits or enters the magnet is called a pole. A pole where a magnetic line of forceexits the magnet is called a north pole and a pole where a line of force enters themagnet is called a south pole.

When a bar magnet is broken in the center of its length, two complete barmagnets with magnetic poles on each end of each piece will result. If the magnetis just cracked but not broken completely in two, a north and south pole will format each edge of the crack. The magnetic �eld exits the north pole and reenters atthe south pole. The magnetic �eld spreads out when it encounters the small air gapcreated by the crack because the air cannot support as much magnetic �eld per unitvolume as the magnet can. When the �eld spreads out, it appears to leak out of thematerial and, thus is called a �ux leakage �eld.

If iron particles are sprinkled on a cracked magnet, the particles will be attractedto and cluster not only at the poles at the ends of the magnet, but also at the polesat the edges of the crack. This cluster of particles is much easier to see than theactual crack and this is the basis for magnetic particle inspection.

To properly inspect a component for cracks or other defects, it is important tounderstand that the orientation between the magnetic lines of force and the �awis very important. There are two general types of magnetic �elds that can beestablished within a component: longitudinal magnetic �eld and circular magnetic

11

Figure 2.5: A bar magnet with 2 poles[17]

Figure 2.6: Flux leakage �eld [17]

�eld (�gures 2.7 and 2.8).

Figure 2.7: Longitudinal magnetic�eld [17]

Figure 2.8: Circular magnetic �eld[17]

The type of magnetic �eld established is determined by the method used tomagnetize the specimen. Being able to magnetize the part in two directions isimportant because the best detection of defects occurs when the lines of magneticforce are established at right angles to the longest dimension of the defect. Thisorientation creates the largest disruption of the magnetic �eld within the part andthe greatest �ux leakage at the surface of the part. As one can see in the �gure 2.9,if the magnetic �eld is parallel to the defect, the �eld will see little disruption andno �ux leakage �eld will be produced.

An orientation of 45 to 90 degrees between the magnetic �eld and the defect isnecessary to form an indication. Since defects may occur in various and unknowndirections, each part is normally magnetized in two directions at right angles to eachother. If the component below is considered, it is known that passing current throughthe part from end to end will establish a circular magnetic �eld that will be 90 degreesto the direction of the current. Therefore, defects that have a signi�cant dimension inthe direction of the current (longitudinal defects) should be detectable. Alternately,transverse-type defects will not be detectable with circular magnetization.

To properly inspect a part for cracks or other defects, it is important to be-come familiar with the di�erent types of magnetic �elds and the equipment used togenerate them.

For proper inspection of a component, it is important to be able to establish amagnetic �eld in at least two directions. A variety of equipment exists to establish

12

Figure 2.9: The importance of mag-netic �eld orientation [17]

Figure 2.10: Detectability of �aws ac-cording to direction of magnetic �eld[17]

the magnetic �eld for MPI. One way to classify equipment is based on its portability.Some equipment is designed to be portable so that inspections can be made in the�eld and some is designed to be stationary for ease of inspection in the laboratoryor manufacturing facility. As portable equipment there are permanent magnets,electromagnets, prods, portable coils and conductive cables and portable powersupplies. There are also stationary equipment for magnetic particle inspection.

Magnetic particle inspection can be performed using particles that are highly vis-ible under white light conditions or particles that are highly visible under ultravioletlight conditions.

Figure 2.11: Lights for magnetic par-ticle inspection [17]

Figure 2.12: Lights for magnetic par-ticle inspection [17]

2.2.4 Electromagnetic Testing (ET) or Eddy Current Testing

Electrical currents are generated in a conductive material by an induced alternatingmagnetic �eld. Interruptions in the �ow of eddy currents, caused by imperfections,dimensional changes, or changes in the material's conductive and permeability prop-erties, can be detected with the proper equipment.

Eddy currents are created through a process called electromagnetic induction.When alternating current is applied to the conductor, such as copper wire, a mag-netic �eld develops in and around the conductor. This magnetic �eld expands as

13

Figure 2.13: Eddy current principle [17]

the alternating current rises to maximum and collapses as the current is reducedto zero. If another electrical conductor is brought into the close proximity to thischanging magnetic �eld, current will be induced in this second conductor. Eddycurrents are induced electrical currents that �ow in a circular path. They get theirname from �eddies� that are formed when a liquid or gas �ows in a circular patharound obstacles when conditions are right.

Figure 2.14: Here a small surface probe is scanned over the part surface in anattempt to detect a crack [17]

One of the major advantages of eddy current as an NDT tool is the variety ofinspections and measurements that can be performed. In the proper circumstances,eddy currents can be used for crack detection, material thickness measurements,coating thickness measurements and conductivity measurements. The conductivitymeasurements can be used to material identi�cation, heat damage detection, casedepth determination and heat treatment monitoring.

Some of the advantages of eddy current inspection include its sensitive to smallcracks and other defects. The ability to detect surface and near surface defects. Theinspection gives immediate results and uses a very portable equipment. This method

14

can be used for much more than �aw detection and a minimum part preparationis required. Test probe does not need to contact the part and it inspects complexshapes and sizes of conductive materials.

Some of the disadvantages of eddy current inspection include the limitation toconductive materials and the accessibility of the surface. Skill and training requiredis more extensive than other techniques. Surface �nish and and roughness mayinterfere. The depth of penetration is limited and �aws such as delaminations thatlie parallel to the probe coil winding and probe scan direction are undetectable

2.2.5 Radiography

Radiography involves the use of penetrating gamma or X-radiation to examine partsand products for imperfections. An X-ray generator or radioactive isotope is usedas a source of radiation. Radiation is directed through a part and onto �lm or otherimaging media. The resulting shadowgraph shows the dimensional features of thepart. Possible imperfections are indicated as density changes on the �lm in the samemanner as a medical X-ray shows broken bones.

Figure 2.15: The electromagneticspectrum [17]

Figure 2.16: Radiation source [17]

When x-rays or gamma rays are directed into an object, some of the photonsinteract with the particles of the matter and their energy can be absorbed or scat-tered. This absorption and scattering is called attenuation. Other photons travelcompletely through the object without interacting with any of the material's parti-cles. The number of photons transmitted through a material depends on the thick-ness, density and atomic number of the material, and the energy of the individualphotons.

Even when they have the same energy, photons travel di�erent distances withina material simply based on the probability of their encounter with one or more of theparticles of the matter and the type of encounter that occurs. Since the probabilityof an encounter increases with the distance traveled, the number of photons reaching

15

a speci�c point within the matter decreases exponentially with distance traveled. If1000 photons are aimed at ten 1 cm layers of a material and there is a 10% chance ofa photon being attenuated in this layer, then there will be 100 photons attenuated.This leave 900 photos to travel into the next layer where 10% of these photos willbe attenuated. By continuing this progression, the exponential shape of the curvebecomes apparent.

The formula that describes this curve is:

I = I0e−µx (2.1)

Where,I is the intensity of photons transmitted across some distance x

I0 is the initial intensity of photonss is a proportionality constant that re�ects the total probability of a photon beingscattered or absorbedm is the linear attenuation coe�cientx is the distance traveled

The factor that indicates how much attenuation will take place per cm (10%in this example) is known as the linear attenuation coe�cient, m. It describes thefraction of a beam of x-rays or gamma rays that is absorbed or scattered per unitthickness of the absorber. This value basically accounts for the number of atoms ina cubic cm volume of material and the probability of a photon being scattered orabsorbed from the nucleus or an electron of one of these atoms.

When the incident x-ray photon is de�ected from its original path by an inter-action with an electron compton scattering occurs. The electron is ejected fromits orbital position and the x-ray photon loses energy because of the interactionbut continues to travel through the material along an altered path. Energy andmomentum are conserved in this process. The energy shift depends on the angleof scattering and not on the nature of the scattering medium. Since the scatteredx-ray photon has less energy, it has a longer wavelength and less penetrating thanthe incident photon.

The change in wavelength of the scattered photon is given by:

λ′ − λ =h

mec(1− cosθ) (2.2)

Where,λ is the wavelength of incident x-ray photonλ′ is the wavelength of scattered x-ray photon

16

h is the Planck's Constant (the fundamental constant equal to the ratio of the en-ergy E of a quantum of energy to its frequency υ: E = hυ)me is the mass of an electron at restc is the speed of lightq is the scattering angle of the scattered photon

Figure 2.17: Radiographic images [17]

Figure 2.18: Radiographic images [17]

Radiographic �lm interpretation is an acquired skill combining visual acuity withknowledge of materials, manufacturing processes, and their associated discontinu-ities. If the component is inspected while in service, an understanding of appliedloads and history of the component is helpful. A process for viewing radiographs(e.g. left to right, top to bottom, etc.) is helpful and will prevent overlooking anarea on the radiograph. This process is often developed over time and individual-ized. One part of the interpretation process, sometimes overlooked, is rest. Themind as well as the eyes need to occasionally rest when interpreting radiographs.

When viewing a particular region of interest, techniques such as using a smalllight source and moving the radiograph over the small light source, or changing theintensity of the light source will help the radiographer identify relevant indications.Magnifying tools should also be used when appropriate to help identify and evaluateindications. Viewing the actual component being inspected is very often helpful indeveloping an understanding of the details seen in a radiograph.

Interpretation of radiographs is an acquired skill that is perfected over time.By using the proper equipment and developing consistent evaluation processes, the

17

interpreter will increase his or her probability of detecting defects.

2.2.6 Ultrasonic Testing (UT)

Ultrasonics use transmission of high-frequency sound waves into a material to detectimperfections or to locate changes in material properties. The most commonly usedultrasonic testing technique is pulse echo, where in sound is introduced into a testobject and re�ections (echoes) are returned to a receiver from internal imperfectionsor from the part's geometrical surfaces. To illustrate the general inspection principle,a typical pulse/echo inspection con�guration is illustrated in �gure 2.19.

Figure 2.19: Typical ultrasonic system [17]

A typical UT inspection system consists of several functional units, such as thepulser/receiver, transducer, and display devices. A pulser/receiver is an electronicdevice that can produce high voltage electrical pulses. Driven by the pulser, thetransducer generates high frequency ultrasonic energy. The sound energy is intro-duced and propagates through the materials in the form of waves. When there is adiscontinuity (such as a crack) in the wave path, part of the energy will be re�ectedback from the �aw surface. The re�ected wave signal is transformed into an elec-trical signal by the transducer and is displayed on a screen. In the �gure 2.19, there�ected signal strength is displayed versus the time from signal generation to whena echo was received. Signal travel time can be directly related to the distance thatthe signal traveled. From the signal, information about the re�ector location, size,orientation and other features can sometimes be gained.

In solids, sound waves can propagate in four principle modes that are based onthe way the particles oscillate. Sound can propagate as longitudinal waves, shearwaves, surface waves, and in thin materials as plate waves. Longitudinal and shear

18

waves are the two modes of propagation most widely used in ultrasonic testing. Theparticle movement responsible for the propagation of longitudinal and shear wavesis illustrated in �gure 2.20.

Figure 2.20: Longitudinal and shear waves [17]

In ultrasonic testing, the inspector must make a decision about the frequency ofthe transducer that will be used. As we learned on the previous page, changing thefrequency when the sound velocity is �xed will result in a change in the wavelengthof the sound. The wavelength of the ultrasound used has a signi�cant e�ect onthe probability of detecting a discontinuity. A general rule of thumb is that adiscontinuity must be larger than one-half the wavelength to stand a reasonablechance of being detected.

When sound travels through a medium, its intensity diminishes with distance.When sound waves pass through an interface between materials having di�erent

acoustic velocities, refraction takes place at the interface. The larger the di�erencein acoustic velocities between the two materials, the more the sound is refracted.Notice that the shear wave is not refracted as much as the longitudinal wave. Thisoccurs because shear waves travel slower than longitudinal waves. Therefore, thevelocity di�erence between the incident longitudinal wave and the shear wave is notas great as it is between the incident and refracted longitudinal waves. Also notethat when a longitudinal wave is re�ected inside the material, the re�ected shearwave is re�ected at a smaller angle than the re�ected longitudinal wave. This is alsodue to the fact that the shear velocity is less than the longitudinal velocity withina given material.

sinθ1

VL1

=sinθ2

VL2

=sinθ3

VS1

=sinθ4

VS2

(2.3)

where,VL1 is the longitudinal wave velocity in material 1

19

Figure 2.21: Snell's Law [17]

VL2 is the longitudinal wave velocity in material 2VS1 is the shear wave velocity in material 1VS2 is the shear wave velocity in material 2

When a wave moves from a slower to a faster material, there is an incident anglewhich makes the angle of refraction for the longitudinal wave 90 degrees.

this is know as the �rst critical angle and all of the energy from the refractedlongitudinal wave is now converted to a surface following longitudinal wave. Thissurface following wave is sometime referred to as a creep wave and it is not veryuseful in NDT because it dampens out very rapidly.

Beyond the �rst critical angle, only the shear wave propagates into the material.For this reason, most angle beam transducers use a shear wave so that the signal isnot complicated by having two waves present. In may cases there is also an incidentangle that makes the angle of refraction for the shear wave 90 degrees. This is knowas the second critical angle and at this point, all of the wave energy is re�ected orrefracted into a surface following shear wave or shear creep wave. Slightly beyondthe second critical angle, surface waves will be generated.

Ultrasonic Inspection is a very useful and versatile NDT method. Some of theadvantages of ultrasonic inspection include its sensitive to both surface and subsur-face discontinuities and the depth of penetration for �aw detection or measurementthat is superior to other NDT methods. Only single-sided access is needed when thepulse-echo technique is used. It is highly accurate in determining re�ector positionand estimating size and shape. Minimal part preparation is required. Electronicequipment provides instantaneous results. Detailed images can be produced withautomated systems. In addition to �aw detection it has other uses, such as thicknessmeasurement.

As with all NDT methods, ultrasonic inspection also has its limitations, whichinclude skill and training more extensive than with some other methods. The surface

20

must be accessible to transmit ultrasound. It normally requires a coupling mediumto promote the transfer of sound energy into the test specimen. Materials that arerough, irregular in shape, very small, exceptionally thin or not homogeneous aredi�cult to inspect. Cast iron and other coarse grained materials are di�cult toinspect due to low sound transmission and high signal noise. Linear defects orientedparallel to the sound beam may go undetected. Reference standards are requiredfor both equipment calibration and the characterization of �aws.

21

Chapter 3

Lamb Waves

Discovered by Horace Lamb in 1910 [34], Lamb waves are elastic waves that prop-agate across thickness of thin wall structures with free boundaries parallel to themid-surface. They are guided waves and can also travel inside curved walls with theshallow curvature. Lamb waves can travel at large distances with very little ampli-tude loss even in materials with a high attenuation ratio and thus a broad area canbe quickly examined. With a high susceptibility to interference on a propagationpath like damage or boundaries Lamb waves are able to provide fast in-service in-spections without time consuming scanning. The entire thickness of the plate canalso be interrogated by various Lamb modes, a�ording the possibility of detectinginternal damage as well as that on surface.

Lamb waves techniques are emerging as one of the most e�ective methods fordamage detection in aeronautic structures.

Across the material thickness, Lamb waves present stationary wave patterns.Lamb waves can be either symmetric or anti-symmetric across the material thickness(Sn and An, respectively, where n represents the number of in�ection points acrossthe material thickness).

Figure 3.1: Symmetric mode, S0 [8] Figure 3.2: Anti-symmetric mode, A0

[8]

Symmetrical Lamb waves move in a symmetrical fashion about the median planeof the plate. This is sometimes called the extensional mode because the wave is

22

�stretching and compressing� the plate in the wave motion direction. Wave motionin the symmetrical mode is most e�ciently produced when the exciting force isparallel to the plate. The asymmetrical Lamb wave mode is often called the ��exuralmode� because a large portion of the motion moves in a normal direction to theplate, and a little motion occurs in the direction parallel to the plate. In this mode,the body of the plate bends as the two surfaces move in the same direction.

Figure 3.3: Lamb waves movement [17]

3.1 Dispersion Curves

The Lamb wave phase velocity, cL, depends on the product between frequency andthe material thickness, h. Since the wave speed varies with frequency, the propaga-tion of Lamb waves is essentially dispersive. For a given frequency multiple modescan exist and therefore the received signals are a complex mixture from di�erentmodes and di�cult to evaluate.

The analytical dispersion curves give an idea of the various existing modes andits velocities for each frequency of excitation. Therefore, it becomes necessary toplot the dispersion curves for this case to choose the optimal frequency excitation.ξ, ζ and d are de�ned as

ξ =

√c2Sc2P

ζ =

√c2Sc2L

d = 2π·f ·dcS

(3.1)

where f is the frequency and d is the half-thickness of the plate, d = h/2.To calculate S Waves Velocity (cS) and P Waves Velocity (cP ) one use the Lame

Constants:

cS =√

E2ρ(1+ν)

and cP =√

νE(1+ν)(1−2ν)ρ

+ Eρ(1+ν) (3.2)

The dispersion curves can now be obtained by solving the Rayleigh-Lamb fre-quency equation. For symmetrical motion the equation becomes:

23

tan√

1− ζ2d

tan√

ξ2 − ζ2+

4ζ2√

1− ζ2√

ξ2 − ζ2

(2ζ2 − 1)2 = 0 (3.3)

Hence, the two components of the displacement can be expressed as:

U(x, y, t) = Re[AkL(cosh(qz)

sinh(qd)− 2qs

k2L + s2

cosh(sz)

sinh(sd))ei(kLx−ωt π

2 ] (3.4)

W (x, y, t) = Re[Aq(sinh(qz)

sinh(qd)− 2k2

L

k2L + s2

cosh(sz)

sinh(sd))ei(kLx−ωt] (3.5)

And for anti-symmetrical motion:

tan√

1− ζ2d

tan√

ξ2 − ζ2+

(2ζ2 − 1)2

4ζ2√

1− ζ2√

ξ2 − ζ2= 0 (3.6)

and the two components of the displacement can be expressed as

U(x, y, t) = Re[AkL(cosh(qz)

cosh(qd)− 2qs

k2L + s2

cosh(sz)

cosh(sd))ei(kLx−ωt π

2 ] (3.7)

W (x, y, t) = Re[Aq(sinh(qz)

cosh(qd)− 2k2

L

k2L + s2

cosh(sz)

cosh(sd))ei(kLx−ωt] (3.8)

The Lamb waves group velocity represents the speed with which Lamb-wavepacks are sent and received along the thin-wall plate.

cg = c2[c− (fd)dc

d(fd)]−1 (3.9)

where cg is the Lamb wave group velocity, and c is the Lamb wave phase velocity.Equation 3.3 e equation 3.6 admits several roots, corresponding to several

symmetrical Lamb wave modes, called S0, S1, etc. A plot of the S0 Lamb wavespeed versus frequency is given in �gure 3.4. The calculation was made for analuminum plate with a thickness value 2d = 1.6mm. The roots of the equation werefound numerically using mathematical software. The �gures 3.4 and 3.5 show thedispersion curves for Lamb waves in a 1.6mm aluminum alloy plate

Examination of �gure 3.4 indicates that at low frequencies (f < 500kHz) thespeed of the symmetrical Lamb wave approaches the speed of axial waves, cP . Athigh frequencies (2500kHz), the dispersion curves for the S0 and A0 modes coalesce.

24

Figure 3.4: Wave speed vs frequency [4]

Figure 3.5: Group velocity vs frequency [4]

25

An analytical model of the Lamb dispersion curves was develop using the varia-tion of the Rayleigh-Lamb frequency equations made by Rose (1999). When plottingthe dispersion curves, one is only interesting in the real solutions of the equations,which present the (undamped) propagating modes of the structure. By collectingthe terms α and β, the equations take only real values for real or imaginary ξ. Thetwo equations become:

tan(βd)

β+

4ξ2αtan(αd)

(ξ2 − β2)2= 0 (3.10)

βtan(βd) +(ξ2 − β2)2tan()αd

4ξ2α= 0 (3.11)

The roots of the Rayleigh-Lamb equation are commonly plotted on fd vs. c/cS

coordinates. The fd is the product of frequency and half thickness can use samedispersion curves.

The procedure used to obtain the plot of the dispersion curves is the following[27]:

1. De�ne the desire range of the fd and c/cS, such that the fd vs. c/cS plane isde�ned.

2. Partition the fd and c/cS axes into small steps to create a mesh. Each nodeon the mesh has a pair of values (fd,c/cS)

3. For each of the nodes, calculate the corresponding c; plug into equation 3.10to obtain α and β. Then evaluate 3.10

4. For each fd value, sweep to vertical line fd = constant and �nd sign-changepoints. These points are close to the roots of the equations. Record the(fd,c/cS) values of these points.

5. Eliminate the singular points of 3.10 i.e. the points where a jump from +/-in�nity to -/+ in�nity is observed.

6. Use the sign-change points as guess value to �nd the approximated roots of3.10. For this step, various iterative root �nding algorithms can be used.For example, bisection, Newton-Raphson, etc. Bisection algorithm is usedhere, because it guaranties convergence. A tolerance is required to de�ne theprecision to be achieved.

7. After �ning the roots, �t the points with a spline and plot the dispersioncurves.

26

A �ne fd vs. c/cS grid is used to increase numerical resolution.Another important property of the Lamb waves is the group velocity dispersion

curves. By using the relation

cg = c− λ∂c

∂λ(3.12)

the group velocity, cg, can be derived from the phase velocity c. To reduce theprogramming e�orts, some manipulation to 3.12 are useful. Following Rose (1999),and using the de�nition wavelength as being

λ =c

f(3.13)

one can write

∂c

∂λ=

∂c

∂(c/f)=

∂c1f∂c− c

f∂f

=f 2∂c

(f∂c− c∂f)(3.14)

Hence,

cg = c− cf

f2∂c(f∂c−c∂f)

= c(1− f∂cf∂c−c∂f

) =

c(−f∂c−c∂ff∂c

)−1 = c2(c− fd ∂c∂fd

)−1

(3.15)

The derivative ∂c/∂fd is calculated from the phase velocity dispersion curve.The numerical derivation is done in a simple way by the �nite di�erence formula:

∂c

∂(fd)∼=

∆c

∆(fd)(3.16)

3.2 Selective excitation of Lamb wave modes

When plotting the received signal as a function of frequency is visible that the �rstpulse is due to the S0 emitted mode which is received at a time that is almostindependent of the frequency. The second pulse is from the A0 mode. It shows moredispersion and exhibits a frequency-dependent group velocity [3].

From simulations of Lamb wave excitation [3] we clearly see that the highestgroup velocity belongs to the S0 mode and this mode shows particle displacementmostly in the x direction (i.e. along the direction of propagation). Due to thePoisson e�ect there is a small dilation and expansion in the y direction. The A0 is

27

the slower wave mode (propagation velocity depends signi�cantly on frequency) andshows particle displacement mostly in the y direction. (symmetric about the centerof the plate).

Selective mode excitation can be achieved by bonding two piezoceramic on op-posite sides of a thin aluminum plate [6]. At frequencies below 1MHz (aluminum)two Lamb modes are present, A0 and S0 (�rst anti-symmetric mode and �rst sym-metric mode). When the second actuator is excited in phase with the �rst one,the symmetric mode (S0) is reinforced whereas the anti-symmetric mode (A0) ispartially canceled [3,6]. When the two actuators are excited out of phase, is theanti-symmetric mode that remained unaltered whereas the symmetric mode is can-celed. In this way, �ltering the appropriated Lamb modes, defects identi�cation canbe easily achieved.

There are many types of Lamb wave transducers, but the majority of them arebuilt based on using angled incident wave or applying force on the surface [27]. The�rst one often uses a wedge to make an incident pressure wave go into the target in anangle that is calculated by using Snell's law with the incident wave velocity and thedesired Lamb wave mode velocity. The Lamb wave mode can be selected by changingthis angle. The second method directly uses the surface stress/strain distribution ofLamb wave modes, and generates a surface load similar to this distribution, henceselectively generates a desired Lamb wave mode. Comb transducer and interdigitaltransducer are examples of this method.

3.3 Damage Detection

Structural �aws represent changes in e�ective thickness and local material proper-ties, and therefore measurements of variations in Lamb wave propagation can beemployed to assess the integrity of these structures.

During occurrence of a damage Lamb waves are generated, which can be receivedby PZT transducers (passive method). Lamb waves can also be generated andreceived at distinct positions (active method).

The interaction of Lamb waves with bolted joint boundary conditions is shownto be sensitive to the torque loading of the mounting bolts [5]. A case was shownwhere the re�ection of a Lamb wave from a bolted joint boundary changed when abolt was loosened. When a single sensor was used, the change in the re�ection fromthe bolted joint was not apparent. However a sensor array was able to distinguishbetween waves arriving from di�erent directions and isolate components arrivingfrom each boundary separately. The re�ection from the boundary corresponding tothe loosened bolt was shown to change, using the directionally �ltered response.

The principle to locate a damage is then very simple. If multiple sensors, in

28

di�erent locations, are utilized to detect the original generated Lamb waves and theirre�ections by the damage, knowing their velocity of propagation and the di�erencein time of detection from those di�erent sensors, by triangulation is possible todetect their source location, i.e., the damage. This method is presently referred asTOFD (time of �ight di�raction, or di�erence, or delay). After obtaining the sensorssignals, their analysis consist of an inverse problem that may be solved by a seriesof di�erent proposed methods, as an example Fourier transforms and numericalmethods, expanding potentials (logarithmic, etc), etc.

Figure 3.6: Example of application of TOFD method for damage location [1]

The two di�erent �rst modes of Lamb waves, S0 and A0, are prone to detectdi�erent types of damages. Cuts imitating cracks are detectable using symmetri-cal Lamb modes while surface damages (ex: �xed masses) a�ect mainly the anti-symmetrical mode of Lamb wave propagation [22]. Besides, the amplitude of re-�ected waves is proportional to damage dimension and dependent on damage ori-entation. Damages oriented perpendicular to the local propagation direction of theLamb waves (parallel to wave front, perpendicular to a radius centered in the originof Lamb waves) are better detected than those angled, being the extreme and moredi�cult case to detect when the damage is oriented along the propagation directionof the Lamb waves.

The Lamb wave mode utilized in SHM methods is also highly dependent onthe structural con�guration to inspect. For instance, if the structure possesses asti�ener, this component will attenuate the propagation of Lamb waves, decreasingtheir energy and amplitude, while generating strong re�ections. At the same time,it is known that re�ections are, at least, one magnitude lower than original incidentLamb waves, already of very low energy and amplitude.

This fact makes the task of detecting a re�ection very di�cult. Together with the

29

existence of a sti�ener with the implications referred before, one may conclude thatit will be very hard to detect a failure beyond the sti�ener. On this particular case,knowing that A0 Lamb waves have a higher energy than S0 Lamb waves, they wouldbe the better choice, although they present the disadvantage of their propagationvelocity being strongly dependent on the frequency.

The characteristics of the re�ected waves described before, namely low elasticdeformation energy and high frequency (as a consequence of the low energy of inci-dent waves), make them very prone to noise interference and also noise interferencein sensor readings. PZT sensors are very sensitive and since the waves that they areintended to detect have very low energy, obvious conclusion is that noise will playan important role in measurements. One attenuating factor is that the frequenciesof the Lamb waves applied in the inspection method are pre-determined and so thesensor signal can be �ltered. One other important aspect to consider is that whenthe re�ections from boundaries are sensed (summed to all Lamb waves and re�ec-tions travelling in the host medium) the sensor signal is useless. This, together withthe high propagation velocities of Lamb waves and the fact that on certain SHMmethods it is desirable to use an actuator also as a sensor, explains the need touse high frequency actuation. This way the actuation signal will be over and willnot interfere with re�ections. As mentioned earlier, methods for damage locationdepend on wave time of �ight calculations, that depend on propagation velocity ofthe waves in the host structure, that is calculated with base in the dispersion curves.Time of �ight is very small and propagation velocities are very high, so any disconti-nuity of the material, Young�s Modulus and/or density variation is a problem whendetermining damage position.

30

Chapter 4

Piezoelectrics Sensors and Actuators

4.1 Piezoelectric e�ect

The piezoelectric e�ect describes the relation between a mechanical stress and anelectrical voltage in solids. When a mechanical stress is applied it will generate avoltage and an applied voltage will generate a mechanical stress that will producea mechanical displacement (it will change the shape of the solid up to a 4% changein volume)[23].

The piezoelectric e�ect was discovered in 1880 by the Jacques and Pierre Curiebrothers. They found out that when a mechanical stress was applied on some crystalselectrical charges appeared, and this voltage was proportional to the stress. Itremained a mere curiosity until the 1940s. The property of certain crystals to exhibitelectrical charges under mechanical loading was of no practical use until very highinput impedance ampli�ers enabled engineers to amplify their signals. In the 1950s,electrometer tubes of su�cient quality became available and the piezoelectric e�ectwas commercialized.

The piezoelectric e�ect occurs only in non conductive materials. Piezoelectricmaterials can be divided in 2 main groups: crystals and ceramics. The most well-known piezoelectric material is quartz (SiO2).

4.2 PZT structure

The atoms are arranged in a cubical structure. At temperatures below the Curietemperature (depending on the material between 150°C and 200°C) the titaniumatom moves from its central position and the electrically neutral lattice becomesa dipole. This dipole lattice presents now piezoelectric characteristics and is con-sidered as one of the most economical piezoelectric material. By doping the PZTmaterial, its piezoelectric characteristics can be modi�ed: especially the hardness or

31

softness of the material [23].

Figure 4.1: Internal Structure of an electret [23]

Electrets are solids which have a permanent electrical polarization. Figure ??

shows a diagram of the internal structure of a electret. In general, the alignmentof the internal electric dipoles would result in a charge which would be observableon the surface of the solid. In practice, this small charge is quickly dissipated byfree charges from the surrounding atmosphere which are attracted by the surfacecharges. Electrets are commonly used in microphones.

Permanent polarization as in the case of the electrets is also observed in crystals.In these structures, each cell of the crystal has an electric dipole, and the cells areoriented such that the electric dipoles are aligned. Again, this results in excesssurface charge which attracts free charges from the surrounding atmosphere makingthe crystal electrically neutral. If a su�cient force is applied to the piezoelectriccrystal, a deformation will take place. This deformation disrupts the orientation ofthe electrical dipoles and creates a situation in which the charge is not completelycanceled. This results in a temporary excess of surface charge, which subsequentlyis manifested as a voltage which is developed across the crystal.

4.3 Piezoelectric as a sensor

In order to utilize this physical principle to make a sensor to measure force, we mustbe able to measure the surface charge on the crystal.

Figure 4.2 shows a common method of using a piezoelectric crystal to make aforce sensor. Two metal plates are used to sandwich the crystal making a capacitor.As mentioned previously, an external force cause a deformation of the crystal resultsin a charge which is a function of the applied force. In its operating region, a greaterforce will result in more surface charge. This charge results in a voltage V = Qf/C

, where Qf is the charge resulting from a force f, and C is the capacitance of thedevice.

32

Figure 4.2: A sensor based on the piezoelectric e�ect [23]

In the manner described above, piezoelectric crystals act as transducers whichturn force, or mechanical stress into electrical charge which in turn can be convertedinto a voltage. Alternatively, if one was to apply a voltage to the plates of thesystem described above, the resultant electric �eld would cause the internal electricdipoles to re-align which would cause a deformation of the material.

When piezoelectric ceramics were introduced, they soon became the dominantmaterial for transducers due to their good piezoelectric properties and their ease ofmanufacture into a variety of shapes and sizes. They also operate at low voltageand are usable up to about 300ºC. The �rst piezoceramic in general use was bariumtitanate, and that was followed during the 1960's by lead zirconate titanate composi-tions, which are now the most commonly employed ceramic for making transducers.New materials such as piezo-polymers and composites are also being used in someapplications.

The thickness of the active element is determined by the desired frequency ofthe transducer. A thin wafer element vibrates with a wavelength that is twiceits thickness. Therefore, piezoelectric crystals are cut to a thickness that is 1/2 thedesired radiated wavelength. The higher the frequency of the transducer, the thinnerthe active element. The primary reason that high frequency contact transducers arenot produced is because the element is very thin and too fragile.

4.4 Piezoelectric Wafer Active Sensors

Piezoelectric Wafer Active Sensors (PWAS) are small, inexpensive, unobtrusive andnon-invasive permanently attached piezoelectric wafers that can conceivably be de-ployed in health monitoring arrays without producing prohibitive weight and costpenalties [4]. Commonly they are manufactured from thin wafers of the piezoce-ramic Pb(Zr − Ti)O3 (a.k.a. PZT). The general constitutive equations of linearpiezoelectric material behavior describe a tensorial relation between mechanical andelectrical variables (mechanical strain Sij, mechanical stress Tkl, electrical �eld Ek,and electrical displacement Dj) in the form

33

Sij = sEijklTkl + dkijEk (4.1)

where sEijkl is the mechanical compliance of the material measured at zero electric

�eld (E=0) and dkij. This equation refers to the converse piezoelectric e�ect. Thedirect piezoelectric e�ect is

Dj = djklTkl + εEjkEk (4.2)

where εEjk is the dielectric permitivity measured at zero mechanical stress (T=0).

Figure 4.3: Piezoelectric wafer transducer a�xed to the host structure [4]

Figure 4.3 shows an active sensor consisting of a lead zirconate titanate (PZT)piezoceramic wafer a�xed to the structural surface. The PWAS is directly connectedto the source of electrical excitation through the connecting wires. The piezoelectricwafer is also intimately bonded to the structure, such that the strain/displacementcompatibility and stress/force equilibrium principles apply. As the PZT materialis electrically activated, strain is induced in the piezoelectric wafer, and interactionforces and moments appear at the interface between the sensor and the structure.

Figure 4.4: Interaction forces and moments [4]

In the pin-force model, the interaction force, FPZT , is assumed to act at thesensor boundary only. Induced by FPZT are activation forces and moments (Na and

34

Ma), which apply a pinching action to the structural surface and generate structuralwaves ( 4.4):

FPZT = ˆFPZT eiωt, Na = FPZT , Ma = FPZTh

2(4.3)

Conversely, when an elastic wave travels through the structure, the sensor be-comes activated through the strain/displacement compatibility condition. The straininduced in the sensor generates an electric �eld that is captured as voltage at thesensor terminals. In the pin-force model, the sensor strain is proportional with thedi�erence in displacement between its extremes. This observation underpins theconcept of 'sensor tuning', i.e., optimal coupling between sensor and structure isachieved when the sensor e�ective length equals the half wavelength of the elasticwave in the structure (a = λ/2).

Figure 4.5: PWAS on a 1D structure [4]

For a PWAS a�xed to a one-dimensional (1D) structure, e.g., a beam (as shownin �gure 4.5), the wave propagation is mainly 1D. In this case, the dominantelectromechanical coupling constant is d13. If the active sensor is placed on a two-dimensional (2D) structure, the wave propagation is, in principle 2D. Since theelectromechanical coupling constants d31 and d32 have essentially same value, radialsimmetry can be applied and the analysis can be reduced to a 1D case in the radialcoordinate, r.

Figure 4.6: PWAS on a 2D structure [4]

35

4.4.1 PWAS Generation of Lamb waves

PWAS (piezoelectric-wafer active sensors) are small wafers of piezoelectric materialthat are permanently bonded to the material surface, and can simultaneously actas elastic-waves transmitters and receptors [8]. PWAS are strain transducers thatcouple directly with the surface strains of the thin-wall structure. Due to the in-plane surface coupling, PWAS are ideally suited for the generation of guided platewaves (Lamb waves). A surface mounted PWAS can simultaneously excite bothaxial (S0) and �exural (A0) Lamb waves. For e�cient Lamb wave excitation, thePWAS length, la, must be an integer multiple of the Lamb wave half-length, λ/2,i.e.

la = kλ

2k = 1, 2, ... (4.4)

When a time-varying voltage, V (t), is applied to the PWAS electrodes, the PWASexpands and contracts in accordance with the laws of piezoelectricity. Thus, thePWAS acts as a Lamb wave generator. Conversely, when a Lamb wave is present inthe material under a PWAS, the surface expansions and contractions are felt by thePWAS and transformed in time varying electric signals. In this case, the PWAS actsas a wave sensor. Of particular importance is the fact that PWAS are coupled withthe material strains parallel to the material surface. Thus, the transmission andreception of Lamb waves in thin-wall structures is greatly facilitated. This type ofcoupling, which is parallel to the material surface, is signi�cantly more e�cient forthe excitation and reception of Lamb waves than that of the conventional ultrasonictransducers, which can only impinge normal to the material surface (or at an angle,when using wedge couplers). This observation, in additional to the much lowercost of PWAS transducers, highlights their advantage over conventional ultrasonictransducers in the transmission and reception of Lamb waves.

4.5 Wave Propagation in Electromechanical Struc-

tures

In this section is presented an analytical and numerical study to simulate the wavepropagation in an elastic half plane with surface-bonded piezoceramic actuatorsunder high-frequency electric loads. The study presented has been obtained fromresearch papers published by Wang and Huang (2001) [10]. Based on a one dimen-sional actuator model, the wave propagation induced by a single actuator is studiedby using integral transform method and solving the resulting integral equations.

36

Because of their advantages of quick response, low power consumption and highlinearity, piezoelectric actuators may also be used to induce high frequency elasticwave propagation in di�erent engineering structures for their health monitoring. Theinduced wave propagation will carry the information on the properties of existingdamage and, therefore, can be used to identify the location and nature of the damageby using properly arranged networks of sensors. The most fundamental issue sur-rounding the e�ective use of piezoelectric actuators in this type of applications is theevaluation of the generated wave propagation for di�erent actuator designs and ar-rangements. The possible mechanical failure of surface-bonded actuators is anotherconcern, which degrades the mechanical integrity of the structure. For example, thestress concentration near the ends of an actuator may result in undesired peeling-o�of the actuator from the host structure. An accurate assessment of the coupledelectromechanical behavior of piezoelectric structure would, therefore, necessitatethe detailed understanding of the local mechanical �eld around actuators.

In comparison with embedded ones, surface-bonded actuators have the advan-tages that they can be attached to existing structures to form an online monitoringsystem. They can also minimize the adverse e�ects on structures by avoiding inter-nal weak points induced by embedded actuators for cases where the e�ects of theseinclusions are signi�cant.

4.5.1 Statement of the Problem

Consider the plane strain problem of M thin piezoceramic actuators surface-bondedto a homogeneous and isotropic elastic insulator, as illustrated in Figure 4.5.1. Thehalf length and the thickness of actuator n are denoted as an and hn, respectively.The position of the centre of actuator n is described by its coordinate in the globalcoordinate system, (y0

n,0). A local coordinate system (yn,zn) will be used to describeactuator n with its origin at the centre of the actuator. It is assumed that the polingdirection of the actuators is along the z-axis.

Figure 4.7: Actuators surface-bonded to an elastic medium [10]

A voltage between the upper and the lower electrodes of actuator n is applied,which results in an electric �eld En

z of frequency ω along the poling direction of

37

the actuator, Enz = (V −

n − V +n )/hn . To study the resulting wave propagation,

only the steady state response of the system will be considered. In this case, thedisplacement,strain, stress and electric �elds of the system will generally involve atime factor exp(−iωt). For the sake of convenience, this factor will be suppressedand only the amplitude of the �eld variables will be considered.

4.5.2 Wave Propagation Due to a Single Actuator

Let us �rst consider the case where only one actuator is attached to the host medium.The actuator will extend (contract) when an electric �eld is applied and consequentlyresult in the deformation of the host elastic medium. Detailed description of thisprocess involves the analysis of complicated local stress distribution around theactuator. Because the thickness of the actuator used is very small in comparisonwith its length, the applied electric �eld will mainly result in a deformation along theaxial direction. Accordingly, the actuator can be modelled as an electroelastic linesubjected to the applied electric �eld and a distributed axial force, τ , as shown inFigure 4.5.1, where τ is the interfacial shear stress transferred between the actuatorand the host structure.

The attention will be focused on cases where high frequency electric �eld isapplied, which results in a wave propagation with the typical wavelength comparableto the length of the actuator. In this case, the inertia e�ect of the actuator must beconsidered. According to the actuator model, the equation of motion of the actuatorcan be expressed as

dσay

dy+ τ(y)/h + ρaω

2uay = 0 (4.5)

where ρa is the mass density of the actuator. The axial stress in the actuatorcan be expressed in terms of the axial displacement (ua

y) and the electric �eld (Ez)

as

σay = Ea

∂uay

∂y− eaEz (4.6)

where Ea and ea are e�ective material constants. The two ends of the surface-bonded actuator can be assumed to be traction free, i.e.

σay = 0, |y| = a. (4.7)

Using this boundary condition, the axial strain obtained by solving equation 4.5as

38

εay(y) = εE(y) +

sin(ka(a + y)

hEasin2kaa

∫ a

−acoska(ξ − a)τ(ξ)dξ −

∫ y

−acoska(ξ − y)

τ(ξ)

hEa

dξ(4.8)

where

εE(y) = ε0coskay

coskaa(4.9)

is the axial strain of a free actuator caused by Ez and

ε0 =Ezea

Ea

, ka =ω

ca

, ca =

√Ea

ρa

(4.10)

with ka and ca being the wave number and the axial wave speed of the actuator,respectively. The dynamic deformation of the actuator will be transformed to thehost medium trough interfacial shear stress. The boundary condition along thesurface (interface) of the host medium (z=0) can be expressed as

τ zy =

−τ(y) |y| < a

0 otherwise(4.11)

The resulting dynamic strain along the surface can be determined by solving theplane strain elastodynamic problem (Achenbach, 1973) as

εy(y, 0)host =λ0

2πµ[∫ a

−a

τ(ξ)

y − ξdxi−

∫ a

−aτ(ξ)m1(y − ξ)dxi] (4.12)

where µ is the shear modulus of the matrix, λ0 = 2(1 − υ) with υ being thePoisson's ratio, and

m1(y − ξ) =∫ ∞

0(

2k2sβ

λ0[(2s2 − k2)2 − 4s2αβ]+ 1)× sins(y − ξ)ds (4.13)

The kernel of the integration in equation 4.13 becomes singular when (2s2 −k2)2 − 4s2αβ approaches zero, which corresponds to the well-known Rayleigh wavespeed. This singular property will be used in the following discussion to determinethe behaviour of Rayleigh wave propagation. In general cases, an actuator will besubjected to an incident mechanical wave induced by applied load or other actuators.The continuity of deformation between the actuator and the host structure indicatesthat

39

εay(y) = εy(y) + εI

y(y) |y| < a, z = 0 (4.14)

where εy is the outgoing wave and the superscript 'a' and 'I' represent the actu-ator and the incident wave, respectively. By substituting equations 4.8 and 4.12into equation 4.14, the following integral equation can be obtained

−qv sinka(a+y)sin2kaa

∫ a−a coska(ξ − a)τ(ξ)dξ + qv

∫ y−a coska(ξ − a)τ(ξ)dξ+

∫ a−a

τ(ξ)y−ξ

dξ −∫ a−a τ(ξ)m1(y − ξ)dξ = εE(y)− εI

y, |y| < a

(4.15)

where

q =πE

2Ea

, E =E

1− υ2, υ =

a

h(4.16)

with E being the Young's modulus of the matrix. εIy is the strain of the incident

�eld and εE is the electric load given by 4.9 . Equation 4.15 is a �rst kind ofsingular integral equation, which involves a square-root singularity of τ at the endsof the actuator. The general solution of τ can be expressed in terms of Chebyshevpolynomial, such that

τ(y) =∑∞

j=0cjTj(y/a)/√

1− y2/a2 (4.17)

If the expansions in equation 4.17 are truncated to the Nth term and 4.15 issatis�ed at the following collocation points along the actuator

yl = acos[l − 1

N − 1π], l = 1, 2, ..., N (4.18)

N linear algebraic equations in terms of {c} = {c1, c2, ..., cN}T an be obtained as

[A]{c} = {F} (4.19)

where [A] is a known matrix and {F} is the applied load. From these equations,the unknown coe�cients in {c}, which represent the interfacial shear stress τ , canbe determined.

Based on the solution of interfacial in the host medium induced by the actuatorcan be obtained, using the general solution as

40

σyy(y, z) =∑

Nj=1cj

(−1)n

∫∞0 H1(s, z)Jj(sa)cos(sy)ds

j = 2n + 1

(−1)(n+1)∫∞0 H1(s, z)Jj(sa)sin(sy)ds

j = 2n

(4.20)

σzz(y, z) =∑

Nj=1cj

(−1)n

∫∞0 H2(s, z)Jj(sa)cos(sy)ds

j = 2n + 1

(−1)(n+1)∫∞0 H2(s, z)Jj(sa)sin(sy)ds

j = 2n

(4.21)

σyz(y, z) =∑

Nj=1cj

(−1)n

∫∞0 H3(s, z)Jj(sa)cos(sy)ds

j = 2n + 1

(−1)(n+1)∫∞0 H3(s, z)Jj(sa)sin(sy)ds

j = 2n

(4.22)

where H1(s, z), H2(s, z), H3(s, z) are given by

H1(s, z) =2sβ[(k2 + 2α2)eαz − (2s2 − k2)eβz]

(2s2 − β2)2 − 4s2αβ(4.23)

H2(s, z) =2sβ(2s2 − k2)(eβz − eαz)

(2s2 − β2)2 − 4s2αβ(4.24)

H3(s, z) =4s2αβeαz − (2s2 − k2)2eβz

(2s2 − β2)2 − 4s2αβ(4.25)

The stress �eld is singular near the tips of the actuator. This singular behaviorcan be characterized by a shear stress singularity factor (SSSF), S, de�ned by

Sr = limy→a[√

2π(a− y)τ(y)] (4.26)

Sl = limy→−a[√

2π(a + y)τ(y)] (4.27)

with the subscript ′r′ and ′l′ representing right and left tips, respectively.According to this de�nition, the SSSF can be expressed in terms of cj as

41

Sl =√

aπ∑

Nj=1(−1)jcj, Sr =

√aπ

∑Nj=1cj (4.28)

The singular stress �eld distribution near the right tip of the actuator can beobtained by concluding an asymptotic analysis as:

σrr = − Sr

a√

2πr(5cos

θ

2+ 3cos

3θ

2) (4.29)

σθθ = − 3Sr

a√

2πr(cos

θ

2− cos

3θ

2) (4.30)

σrθ = − Sr

a√

2πr(sin

θ

2− 3sin

3θ

2) (4.31)

42

Chapter 5

Experimental Setup

To perform the experimental work it was used the following setup:

Figure 5.1: Experimental setup used in the laboratory

This setup is composed by an aluminum plate, 3 piezoelectrics wafers, a functiongenerator, an oscilloscope and a GPIB board to connect the computer to the functiongenerator and the oscilloscope.

The experimental work was performed in the laboratory of robotics in IST, whichis a very large laboratory that has a lot of people working there in di�erent projects.All the experiments were performed without a controlled environment to simulate anenvironment as close as the reality as possible. There were no worries with soundsfrom machines, people talking, or every other noise, even knowing that all thatsounds can be felt by the piezoelectric sensors. This method of detecting damageswith Lamb waves with an integrated system of sensors and actuators in a controlledenvironment is already validated so the next step is to validate the method in anon-controlled environment.

The aluminum plate is a square of 1.5mx1.5m and 2mm thickness.

43

Figure 5.2: Aluminum plate used in the experiments

The function generator used is a Philips PM5138. To connect the function gener-ator and the oscilloscope to the computer is was used a GPIB board. In the computerthe "`Fluke AnyWave"' program was used to send the signal to the generator andto receive the response signals from the oscilloscope.

Figure 5.3: Function generator, oscilloscope and board

The oscilloscope is a Philips PM3335.This experimental setup was a preliminary one using the available material in

the laboratory. The objective was to do some tests and experiments in order toachieve the ideal characteristics of the new equipment to be bought.

5.1 Lamb Waves Generation

The piezoelectric wafers are used to transmit and received the Lamb waves. In eachtest, one of them is used as actuator and the other two as sensors. They are allconnected by wires to the function generator or to the oscilloscope depending ontheir function in the test.

44

5.1.1 Limitations of the equipment

Function Generator

This equipment only allows to create a discrete actuation wave with 1000 pointsequally spaced. That wave is sent continually and the function generator doesn'tallow to sent only a unique wave or to create a sequence of a actuation wave followedby a zero signal. Due to this fact the only results that one can use to analyze isthe results that arrive to the sensor before the second actuation wave arrives. Thewindow of the actuation signal has to be as long as possible so the waves have timeto arrive to the sensors and the �rsts re�ections. The problem of this is that the timeinterval between the points that de�ne the wave is increased and the wave becomespoor de�ned. One have to �nd a commitment between the number of points thatthe wave need to be well de�ned and the maximum time of window that is possible.Another limitation of the function generator is the amplitude of the actuation signal.The maximum amplitude that is allowed is 10V i.e. 20V peak to peak.

Oscilloscope

The ANYWAVE software saves the results for a time interval that is double of theinterval that is show in the oscilloscope window. The results are saved in a �lewith 'comma separated value' (csv) format. That is a microsoft Excel format whichsimpli�es the analysis of the results. The other option is to create a routine inMATLAB to read the �le and plot the results. The results are save in a discreteway with 4096 points. Since the number of points is constant, if one saves a shortperiod of time the results will be well de�ned. As explained before the window oftime that is interesting to analyse is until 0.1ms. Due to all this the x-scale chosento work with in the oscilloscope was 50µs. That results in a window of time showedin the oscilloscope of 0.05ms and a window saved in the �le of 0.1ms. The nextx-scale available in the oscilloscope is 20µs, which will result in a 0.04ms window ofthe saved �le. That time is not enough to all the re�ection arrive.

5.1.2 Actuation Signal

Due to the dispersive nature of Lamb waves it is important that the actuation signalexcites only one frequency. If more than one frequency is excited there would beLamb waves with various velocities that would turn the results obtained very di�cultto interpret. To choose the appropriate signal to excite the Lamb waves in the plateone have study several types of signals calculating the spectrum power of which one.

The wave used to excite the Lamb waves was:

45

Figure 5.4: Square wave (left) and respective power spectrum (right)

Figure 5.5: Ramp signal (left) and respective power spectrum (right)

Figure 5.6: Sine signal (left) and respective power spectrum (right)

46

Figure 5.7: Cosine signal (left) and respective power spectrum (right)

F = Asen(2πft) ∗ sen(2πf

10t) (5.1)

where, A is the amplitude, f is the frequency and t is the time.

Figure 5.8: Actuation wave (left) and respective power spectrum (right)

The power spectrum of the hanning windowed tone burst (excitation signal used)shows a wider main frequency peak, and there is almost no side band peak. In thiscase, the frequency component is concentrated around the main peak, hence reducethe frequency spread range.

47

The actuation wave sent by the function generator is not exactly the wave feltby the sensor and saved by the oscilloscope. For that reason it was also calculatedthe power spectrum of the actuation wave felt by the sensor.

Figure 5.9: Actuation wave (left) and respective power spectrum (right)

5.2 Dispersion Curves

As explained before, the Lamb waves have a dispersive nature, i.e., their velocitydepends on the frequency. To know their velocity in this case one have to obtainthe dispersion curves for this plate. The dispersion curves will give the informa-tion about how the Lamb waves velocity varies with the frequency. The dispersioncurves depends on the Young modulus, density, Poisson ratio and thickness of thepropagation medium, which in this case is the aluminum plate.

Property Valueρ(Kg/m3) 2700E (GPa) 70

ν 0.35Thickness h(m) 0.002

With the plate properties one can calculate the velocity of the P-waves and theS-waves in this plate.

vs =

√E

2ρ(1 + ν)= 3098.74m/s (5.2)

vs =

√Eν

(1− 2ν)(1 + ν)ρ+

E

ρ(1 + ν)= 6450.54m/s (5.3)

48

Knowing the vs,vp and the thickness of the plate the only two incognits in theRayleigh-Lamb equations are now the frequency and the Lamb waves velocity. UsingMatlab one obtains the dispersion curves for this plate.

Figure 5.10: Dispersion curves for an aluminum plate of 2 mm thickness (Frequencyvs Phase velocity)

Figure 5.11: Dispersion curves for an aluminum plate of 2 mm thickness (Frequencyvs Group Velocity

The group velocity represents the speed with which Lamb-wave packs are sentand received along the thin-wall plate.

As one can sees in �gure 5.10 for frequencies higher than 1.5MHz there aremultiple modes of Lamb waves. Below that frequency only exist the �rst symmetrical

49

mode (S0) and the �rst anti-symmetrical mode (A0). For frequencies higher than1MHz the velocity of the �rst mode of the anti-symmetric wave is approaching the�rst symmetric mode velocity. The goal in this experience is to have only the two�rst modes (A0 and S0) and that that modes have a velocity as di�erent as possible.If the velocity of the two modes is to close the waves will arrive at the same timeand it will be di�cult to distinguish the two. The arrival time of the waves is a veryimportant parameter of these method if not the most important. So one decides tohave excitation frequencies below 1MHz.

Figure 5.12: Variation of S and A wavelength with frequency

5.3 Time of Flight (TOF)

With the dispersion curves presented in the previous section one can know thevelocity of the waves for the actuation signal in question i.e. 100kHz. The valuestaken from 5.11 are presented in the next table.

Group velocity of S0 (100kHz) 5440 m/sGroup velocity of A0 (100kHz) 2300 m/s

Calculating the path of the waves from the actuator to the sensor, one can knowthe distance. With the distance and the velocity the theoretical time of �ight of thewaves can be calculated.

Figure 5.13 show the position in the plate of the 3 piezoelectrics actuators/sensorsbonded. It also shows the numbering used to refer to each one of them and the co-ordinate system.

50

Figure 5.13: Coordinates of the piezoeletric actuatiors/sensors in the plate

5.3.1 Direct Waves

The sensor will receive the direct wave from the actuator and then the it will re-ceive the re�ections from the boundaries or from damages. In the �rst part of theexperimental work the plate was undamage so the only re�ections that the sensorsreceived were the re�ections from the boundaries.

Figure 5.14: Distances between the piezoelectrics actuators/sensors

In the next table one had the results for the case of a 100kHz actuation wave.

51

S0 wavesActuator/Sensor distance (m) time (s)

2 to 1 0.533 9.80e-52 to 3 0.839 1.54e-4

A0 wavesActuator/Sensor distance (m) time (s)

2 to 1 0.533 2.31e-42 to 3 0.839 3.65e-4

5.3.2 Re�ections

The calculation of the distance of the re�ections it's not so linear. MathematicallyLamb waves are a circle with center in the actuator growing with time. When apart of that circle arrives to a boundary it is re�ected symmetrically in the oppositedirection. The next scheme shows that phenomenon and the cardinal directionsused to refer to the boundary. For example to refer to the re�ection arriving fromthe west (W) boundary one will say the 'W re�ection'. The waves that are re�ectedone time from the boundaries will be called primary re�ections, if it is already are�ection that hit the boundary it will be called secondary re�ection.

Figure 5.15: Re�ections of the waves in the boundaries

In the above picture one can see in green the instant where the actuation wave(from actuator 2) arrives to the sensor 1. It can be seen in the south boundaryalready a re�ection going up. In orange it is represented the instant where the samewave arrives to the sensor 3, there can now be seen re�ections from boundariessouth, east and west.

52

Figure 5.16: Lamb Wave in the caseof a in�nite plate

Figure 5.17: Re�ections arriving atsensor 1

Figure 5.16 shows the diameter of the Lamb wave in the case of a in�nite plate.However, the plate is �nite and the wave is re�ected in the boundaries. Picture 5.17now shows the instant where the re�ections from south and east arrive to sensor 1.To calculate that distance one have to sum the distance from the actuator to theboundary with the distance back to the sensor. This distance is not the distancebetween the boundary and the sensor but the distance showed in the picture as 'dE'.That is the radius of the circle. An easy way to calculate that radius is making are�ection of the sensor instead of re�ecting the wave. Having the sensor at the samedistance from the boundary but on the other side of it as showed in the followingpicture, one only have to calculate the distance between the actuator and that point.

To calculate all the combinations actuator/sensors it was done a little routinein matlab. The input of that routine is the coordinates of the three piezoelectricsand the velocity of the waves for the frequency used. It creates a vector with thecoordinates of the 3 possible actuators and a matrix with the coordinates of the 2possible sensors for which actuator. For all that combinations it �rst calculate thedirect distance between the actuator and the sensor as

di,j =√

(xai− xsi,j

)2 + (yai− ysi,j

)2 (5.4)

where (xa, ya) are the coordinates of the actuator and (xs, ys) are the coordinatesof the sensor.

To calculate the primary re�ections the routine calculates the coordinates of thepoint symmetric to the boundary in question. And then is calculates the distancebetween the actuator and that point with the same formula showed above. In thecase of south boundary:

53

x = xsi,j

y = ysi,j− 2ysi,j

(5.5)

North:

x = xsi,j

y = ysi,j+ 2(1.5− ysi,j

)(5.6)

East:

x = xsi,j+ 2(1.5− xsi,j

)

y = ysi,j

(5.7)

West:

x = xsi,j− 2xsi,j

y = ysi,j

(5.8)

Knowing the coordinates of this points one then calculates dN ,dS,dE and dW asbeing respectively the distances of the North, South, East and West re�ections.

Actuator 2 to Sensor 1Distance (m) S0 time (ms) A0 time (ms)dS = 1.1859 0.218 0.515dN = 1.9121 0.351 0.831dE = 1.1859 0.218 0.515dW = 1.9121 0.351 0.831

Actuator 2 to Sensor 3Distance (m) S0 time (ms) A0 time (ms)dS = 2.0194 0.371 0.878dN = 1.3521 0.248 0.588dE = 1.5462 0.284 0.672dW = 1.5462 0.284 0.672

5.4 Results

In this section are presented the plots of the received waves by the sensors.In �gure 5.18 there is a typical result obtain with the laboratory equipment. The

amplitude of the actuation wave was divided by 100 to �t the scale of the received

54

Figure 5.18: Actuation wave (blue) and received wave (orange) from 1 to sensor 2

wave. The actuation wave starts before the zero seconds in the negative part. Thisis due to the trigger level of the oscilloscope. The oscilloscope only does the triggerafter a while of the start of the actuation wave. The time of �ight of the receivedwaves is the di�erence between the time of arrival of the wave and the time whenthe actuation wave start. If the actuation wave doesn't start at zero the time ofarrival of the waves cannot be read directly from the x-axis. In order to simplifythat point all the signals were shift along the positive direction of the x-axis untilthe start of the actuation wave is on origin of the x-axis. This is shown in 5.19.

Figure 5.19: Actuation wave (blue) and received wave (orange) from 1 to sensor 2with time shift

The actuation wave that is received is similar for all the cases. Its primaryimportance is to quantify the shift of time that should be applied to each wave.That will be done for all the cases mas only the received waves will be plot with therespective shift.

With the Time of Flight (TOF) previous calculated one can identify the packsof waves that appear in the results. That is shown in �gure 5.20.

where,S0 - S0 waveS0(R) - Primary re�ection of the S0 waveS0(r) - Secondary re�ection of the S0 wave

55

Figure 5.20: Wave generated by actuator 1 and received by sensor 2 (blue) andsensor 3 (orange)

A0 - A0 waveA0(R) - Primary re�ection of the A0 wave

As de�ned before, primary re�ection is when the wave re�ects in one boundary.Secondary re�ection is when the wave re�ects in two boundaries before it arrives tothe sensor.

With the previously calculated time of �ight one can identify each pack thatarrives at the sensor.

In �gure 5.20 one can see that between the packs (sensor 2) it is visible somenoise, but between the second and the third pack the noise is higher that betweenthe other ones. The time of the occurrence is consistent with the position of thesensor 3.

Actuator/Sensor Distance Time of arrival Group Velocity Error(m) (s) (m/s)

1-2 0.530 0.95e-4 5353 1.6%1-3 0.839 1.55e-4 5412 0.51%

The results show a good accuracy between the theoretical group velocity (5440m/s)and the experimental values. The group velocity is very high when compare withthe distances which will give time of �ights in the order of 10−4 seconds. Any littlechange in the times readings will change signi�cantly the value of the calculated

56

velocity. So the calculation of the error is also dependent on the accuracy of thatreading which in this case were direct from the plots.

5.4.1 Variation with the frequency

In this section one will analyze the variation of the Lamb waves with the frequencies.

Frequency (kHz) A0 group velocity (m/s) 2-1 time (ms) 2-3 time (ms)100 2300 0.231 0.364125 2450 0.216 0.342150 2600 0.204 0.323175 2700 0.196 0.310200 2800 0.189 0.299

Frequency (kHz) S0 group velocity (m/s) 2-1 time (ms) 2-3 time (ms)100 5440 0.097 0.154125 5410 0.098 0.155150 5400 0.098 0.155175 5400 0.098 0.155200 5400 0.098 0.155

Figure 5.21: In�uence of the frequency in the received waves (actuator 2 to sensor1)

The S0 mode is received at a time that is almost independent of frequency whichis consistent with the calculated dispersion curves for this plate. The amplitude is

57

Figure 5.22: In�uence of the frequency in the received waves (actuator 2 to sensor3)

also dependent on frequency. Although, for more than one test done at the samefrequency the sensors used showed variations in the amplitude of the results. Dueto this, one cannot say for sure that the variation in the amplitude is a result offrequency variation or a defect of the sensors. As the frequency increases the wavesshow less dispersion.

They were also performed tests with higher frequencies than 200kHz. But theamplitude of the results is in the order of the noise amplitude and it is impossibleto distinguish anything. An example is shown in �gure 5.23.

Figure 5.23: Received wave for an actuation of 250kHz

58

5.4.2 2D Visualization of the Results

Lamb waves propagates as a circle, across thickness. In the results saved by thesensor with the oscilloscope one can only see a 1D plot of the waves arriving at thatsensor. To give some visibility to the results, that 1D results felt by the sensor wererotated 360º around the actuator. A little C program was developed to take theresults saved by the Fluke Anywave software and create an input �le to TECPLOT.The �le saved by the FlukeAnywave software has two columns, the �rst one withthe time and the second one with the amplitude. The �rst thing that the programdoes is multiply the time by the wave velocity in order to achieve the distance. Inthis way one gets the points in the form of a matrix with two columns with thedistance in the �rst and the amplitude in the second one. The next step is to createa regular square grid in the plate. Then one calculates the distance between everypoints in the grid and the actuator. When that distance is equal to a distance valuepresent in the matrix, the correspondent amplitude value is attributed to that point.After the grid is completely swept all the point have an amplitude attributed and iscreated an output �le. That �le has three columns (x, y, amp) and is written in away that could be read by TECPLOT.

Figure 5.24: Actuation wave passing through sensor 1

59

Figure 5.25: Actuation wave passing through sensor 3

60

Chapter 6

Damage Detection

One of the objectives of this work is to detect damages in the plate using Lambwaves. With the plate undamage a lot of results were obtained and that results willbe the reference to those with the damaged plate. One will compare the two andtry to �nd the di�erences in the waves obtained by the sensors. As explained beforea cut or a mass in the plate will re�ect the Lamb waves. That re�ection will bereceived by the sensors such as the re�ection from the boundaries are.

6.1 First Cut

It was made a cut through all the thickness of the plate with 20mm by 1mm. Thecut was made with a saw to be as clean as possible. This cut was made parallel tothe north and south boundaries. The location of the cut can be seen in �gure 6.1.

Figure 6.1: Location of the �rst cut in the plate

Comparing the case damaged with the case undamaged one can see by �gures6.4 and 6.5 that at visible eye there cannot be seen any di�erence. These can bedue to the orientation of the cut. Generated Lamb waves arrive at the cut andare re�ected but that re�ections have some directionality. In some directions the

61

Figure 6.2: Location of the �rst cut Figure 6.3: Zoom of the �rst cut

Figure 6.4: Actuator 1 to Sensor 2 with the �rst damage(orange) and without (blue)any damage

Figure 6.5: Actuator 1 to Sensor 3 with (orange) and without (blue) damage

62

amplitude of the re�ections is higher than in other directions. The re�ections thatarrive the sensors are too small to be distinguish from noise.

6.2 Second cut

The second cut was made perpendicular to the direction sensor 1 to sensor 2, i.e.,with an angle of 45º with the horizontal. This was done in order to be sure that there�ections from the damage will be received by these two sensors.

Figure 6.6: Location of the second cut in the plate

Figure 6.7: Actuator 1 to Sensor 2 with the second cut

Here it is visible the re�ection from the damage. It is also visible the re�ectionfrom the third sensor as mentioned in the previous chapter. Each sensor is actinglike a strange presence that is felt by the other sensors. In the case of using the 1-2ou 2-1 combination for actuator/sensor that re�ection occurs at a time where thereis nothing in the signal but noise which make it visible.

In the case of �gure 6.8 the re�ection from the damage is also visible mas witha minor amplitude. This could be due again to the orientation of the cut.

63

Figure 6.8: Actuator 1 to Sensor 3 with the second cut

6.3 Increase of the damage

To analyse the in�uence of the increase of the damage, the cut in the plate wasincreased slowly to verify if the re�ections of the cut also increase or not. All theresults presented in this section refer to actuator 1 and the sensor 2.

Figure 6.9: Plate without damage Figure 6.10: Plate with a 20mm cut

Figure 6.11: Plate with a 25 mm cut Figure 6.12: Plate with a 30mm cut

The results in the �gure 6.9 refer to the undamaged case. In the �gure 6.10the plate has a 20mm cut and there is visible the re�ection from that cut. It wasdrawn a reference pink line in the plots to be easier to compare them. In �gure6.10 the re�ection from the damage is below that pink line. As the cut is increased

64

Figure 6.13: Plate with a 35mm cut Figure 6.14: Plate with a 40mm cut

the re�ection from the damage is also slowly increasing until the case in �gure 6.14(40mm) where the re�ection from the damage pass the pink line.

One can conclude that the re�ection from the damage increases with the increaseof the damage.

65

Chapter 7

Numerical Simulations

In order to validate the experimental results were developed some numerical modelsof the experiments. Numerical simulations of the wave propagation process wereperformed using the commercially available �nite-element code ANSYS.

7.1 Element

The element chosen to use in the numerical simulations was SHELL63. This elementhas both bending and membrane capabilities. Both in-plane and normal loads arepermitted. The element has six degrees of freedom at each node: translations in thenodal x, y, and z directions and rotations about the nodal x, y, and z-axes. Stresssti�ening and large de�ection capabilities are included.

Figure 7.1: SHELL63 geometry

The geometry, node locations, and the coordinate system for this element areshown in �gure 7.1. The element is de�ned by four nodes, four thicknesses, anelastic foundation sti�ness, and the orthotropic material properties. Orthotropicmaterial directions correspond to the element coordinate directions.

66

7.2 Natural Frequencies

The �rst thing calculated in ANSYS was the natural frequencies of the plate andthe vibration modes.

Figure 7.2: First mode Figure 7.3: Second mode

Figure 7.4: Third mode Figure 7.5: Fourth mode

Figure 7.6: Fifth mode Figure 7.7: Sixth mode

Natural Frequencies:

Mode Freq (Hz)1 7.98752 16.2903 16.2904 24.0175 29.2056 29.343

67

Figure 7.8: Convergence of the natural frequency

Figure 7.9: Convergence of the maximum displacement

Maximum Displacement:

Mode Displacement (m)1 0.7062482 0.6973493 0.6973324 0.6409845 0.6933576 0.86885

7.3 Lamb Wave Propagation

To simulate the generation of the Lamb waves by the piezoelectric actuators a dis-placement was applied to the correspondent nodes. The function applied to thenodes was similar to the one applied in the laboratory experiments. The only dif-ference is the amplitude. In the experimental part the amplitude was 10V and inthe ANSYS simulations the amplitude was 10e-6 m.

F = Asen(2πft) ∗ sen(2πf

10t) (7.1)

where, A is the amplitude, f is the frequency and t is the time.

68

The grid chosen was an agreement between the computational e�ort and goodresults. The grid used has 100x100 elements.

Figure 7.10: Grid and boundary conditions

The results obtained from ANSYS were saved in a txt �le that has the value ofdisplacement on each node. A little routine was made in Matlab to receive theresults from ANSYS and make the plots.

Figure 7.11: Actuator 1 to Sensor 2

There is a good agreement between the time of arrival of the �rst pack of wavesand the arrival of the �rst boundary re�ection. In the laboratory the re�ection fromthe south and east boundaries arrive at the same time. However, in the numericalsimulations the wave that is created is not exactly circular and so, the re�ectionsfrom the two boundaries don't arrive at the same time. It is visible in �gure ?? thetwo re�ections arriving one after the other.

In the case of the sensor 3 the times of �ight of the �rst pack and the �rstre�ection also agree with the theoretical calculated times.

69

Figure 7.12: Actuator 1 to Sensor 3

7.4 Damage Detection

To experiment the options to how simulate a damage in ANSYS simpler tests wereperformed �rst. With simpler tests being tests not to heavy to the computer. Inthis case simpler tests cannot mean larger grid. The grid has to be small enough tothe wave to propagate and one be able to see it. So the solution was to perform 1Dtests �rst where the actuator only actuates the y direction and the sensor and thedamage are in the same x coordinate of the actuator. The three are in the samevertical line.

7.4.1 1D Simulations

In the �rst test only one node was clamped, then three and �nally �ve nodes wereclamped to simulate a damage. The results obtained were the following:

Figure 7.13: Results for the three cases (1 nodes, 3 nodes and 5 nodes)

It can be seen that the amplitude of the �rst pack to arrive is constant in all

70

the three cases but the amplitude of the second pack in slowly increasing as thenumber of clamped nodes increases. That is a sign that the clamped nodes aredoing anything.

As the crack length increases, the amplitude of the re�ection increases.To con�rm that the increase of amplitude is due to the damage another case was

tested. In this case the cut was placed in a way that will arrive at the sensor soonerin order not to interfere with another waves or re�ections.

Figure 7.14: Case damaged vs undamaged

It is visible in �gure 7.14 near 0.2ms the re�ection from the damage.

7.4.2 2D Simulations

Figure 7.15: Actuator 1 to Sensor 2 (�rst cut))

Figure 7.16: Actuator 1 to Sensor 3 (�rst cut)

71

The amplitude of the cut re�ection is in the order of the signal noise of thelaboratory. In the simulations there is no noise and one can see the re�ections, butit the experimental results it's very di�cult to distinguish the re�ection from thenoise.

Figure 7.17: Actuator 1 to Sensor 2 (second cut)

Figure 7.18: Actuator 1 to Sensor 3 (second cut)

In the case of the second cut the amplitudes from the re�ections are bigger thanin the case of the �rst one. As observed in the experimental results.

72

Chapter 8

Conclusions and Further Research

8.1 Conclusion

From the performed experiments it can be concluded that the piezoelectric sen-sors/actuators are e�cient in the generation and detection of Lamb waves. Theomnidirectionality of the waves was also proved because the sensor has received thedirect wave and re�ections from all the four boundaries.

There is a good agreement between the theoretical time of �ight and the resultsobtain in the lab (in the order of 99%).

Even with the material available good results were obtained and a damage wasdetected. The �rst cut was more di�cult to detect but with the right positioning ofthe sensors in order to eliminate that "dead zones" it could be more e�ective.

The pulse dispersion of the Lamb waves in combination with the presence ofmultiple modes can make the results di�cult to interpret but it can be passed overwith the right excitation signal.

With the increase of frequency the Lamb waves show less dispersion.The results from the numerical simulations had a good agreement with the ex-

perimental ones in the �rst waves arriving the sensors. The other ones were a�ectedby simulation errors.

8.2 Recommended Future Work

The bene�ts of Structure Health Monitoring (SHM) could result in cost savings andsafer aircrafts. It is a research area that deserves to be worked and studied.

For future works are recommended tests with more sensors and with variouscon�gurations. Tests with higher frequencies should already be done because theywill be more e�ective in the damage detection. That cannot be done with the actualequipment.

73

The use of phased arrays (an array of sensors) can be a good solution to the"dead zones" problem. Using phased arrays one can determine the direction of theincoming waves. This is very useful because when a re�ection is received one cantell from which direction it came. With single actuators/sensors one can only tellthat the source is at a speci�ed distance mas the direction is unknown.

Fiber Grating Sensor (FBG) is also a possibility. They are known to showhigh directivity, therefore signal amplitude will also depend on the direction of theincoming wave.

74

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[31] AGFA - "Ultrasonic Array Probes For Nondestructive Testing Online or in theField"

[32] Betz, D. C.; Thursby, G.; Culshaw, B.; Staszewski, W. J.; "Identi�cation ofstructural damage using multifunctional Bragg grating sensors: I. Theory andimplementation" Institute of Physics Publishing 2006

[33] Greve, D.W.; Oppenheim I.J.; Sohn H.; Yue C.P. "Structural health monitoringwith an inductively coupled (wireless) Lamb wave transducer" Paper presentedat the Third European Workshop on Structural Health Monitoring, Granada,

Spain, July 2006

[34] Lamb, Horace; "The Dynamical Theory of Sound" London, Edward Arnold,

1910

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