ASSESSMENT OF ULTRASONIC GUIDED
WAVE INSPECTION METHODS FOR
STRUCTURAL HEALTH MONITORING
By

Gerges Dib

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Electrical Engineering — Doctor of Philosophy

2014

ABSTRACT
ASSESSMENT OF ULTRASONIC GUIDED WAVE INSPECTION
METHODS FOR STRUCTURAL HEALTH MONITORING
By
Gerges Dib
Structural health monitoring (SHM) has the potential to significantly increase safety and
reduce manufacturing and maintenance costs of industrial structures. The use of piezoelectric
material such as Lead-Zirconate-Titanate (PZT) in exciting and sensing ultrasonic guided
waves for damage detection has become popular since its allows the rapid inspection of
large areas in a structure using non-intrusive sensors. Ultrasonic guided waves interact with
discontinuities in the structure, giving information about the potential presence of a damage,
its size and location.
The main concerns about using such methods is that PZT sensors and guided waves are
affected by environmental conditions. The performance of the PZT sensors, in terms of their
ability to detect damage, degrades over time and varies depending on current environmental
conditions and surrounding, resulting in inconsistent measurements. This work gives a novel
formulation of a model-based probability of detection method, which is able to quantify
the performance of guided wave inspection in a stochastically varying environment. An
analytically and experimentally verified finite element model is used to generate data that
represent the effects of varying environmental conditions. Then the stochastic approach
is used to evaluate the probability of detection of cracks in riveted aluminum plates and
delaminations in composite plates. Also, the performance of guided wave imaging algorithms
under degrading PZT conditions is examined.
Another concern is the ability to transfer data from the PZT sensors, which are per-

manently located on the structure, to a computer where the data could be processed in
real-time or near real-time. Wireless sensor networks (WSN) use low footprint smart sensor
nodes that are permanently mounted on the structure. The sensor nodes have their own
power supply and wireless communication devices to communicate with other sensor nodes
or a base station. Wireless sensor node for guided waves require an actuation interfaces
and high frequency sampling of the guided wave measurements. A proof-of-concept wireless
sensor node prototype is developed for the data acquisition and actuation using PZT sensors
and guided waves.

Copyright by
GERGES DIB
2014

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor and mentor, Prof. Lalita Udpa,
for her extraordinary support, guidance and encouragement during my studies. Prof. Lalita
has advised me not only on my research, but also on my career and life. I feel fortunate to
work with her and proud of being a member of the Non-destructive Evaluation Laboratory
(NDEL) research group.
I would to express my sincere thanks to my committee member Prof. Mahmood Haq for
his continuous support and guidance, and for giving me access to great experimental facilities.
I would like to thank my PhD studies committee members Prof. Shantanu Chakrabartty,
Prof. Nizar Lajnef, and Prof. Satish Udpa for their guidance and suggestions.
I would like to thank all the current and past members of NDEL. Special thanks goes to
Guang Yang for all her invaluable help. I would also like to thank all my friends who made
my stay at MSU memorable.
Finally, I would like to thank my parents Hana and Hanna, and my sister Haneen for
their constant support and encouragement. Their love and affection was the beacon that
lighted my way towards finishing my studies. Ofcourse, thank you Ali Deek for always being
entertaining, and Barbell for always being there when I needed to lift.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . .

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ix

LIST OF FIGURES

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Chapter 1 Introduction . . . . . . . . . . . . .
1.1 Motivation . . . . . . . . . . . . . . . . . . .
1.2 Structural Health Monitoring . . . . . . . .
1.3 Challenges in Guided Wave SHM . . . . . .
1.4 Problem Statement and Thesis Organization

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Chapter 2
2.1
2.2
2.3
2.4
2.5
2.6
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Background for GW-PZT SHM modeling, damage
reliability and instrumentation . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Piezoelectric Transducers . . . . . . . . . . . . . . . . . . . . . .
Ultrasonic Guided Waves . . . . . . . . . . . . . . . . . . . . . .
GW-PZT Modeling . . . . . . . . . . . . . . . . . . . . . . . . .
Damage Detection Methods . . . . . . . . . . . . . . . . . . . .
PZT Sensor Diagnostics and Effects on GW Measurement . . .
Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . .
2.7.1 Sensor nodes . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2 Distributed embedded software . . . . . . . . . . . . . .
2.7.3 Communication protocols . . . . . . . . . . . . . . . . .

Chapter 3 Finite Element Modeling of GW-PZT .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
3.2 Problem formulation . . . . . . . . . . . . . . . . .
3.3 FEM Mesh Validation . . . . . . . . . . . . . . . .
3.3.1 Analytical Validation using a 2-D Model . .
3.3.2 Validation of the 3-D Aluminum Plate Mesh
3.3.3 Validation of the 3-D Composite Plate Mesh
3.4 Experimental Validation . . . . . . . . . . . . . . .
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4 GW Propagation in Structures . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
4.2 GW Propagation in a Riveted Aluminum Plate . . .
4.3 Guided wave propagation in a Glass Fiber Reinforced

vi

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Chapter 5 Formulation of a Stochastic Method for Performance Evaluation
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 PZT degradation modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Signal Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Model-Based Performance Evaluation Approach . . . . . . . . . . . . . . . .
5.4.1 Deterministic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Probability Density Function of the Signal Feature . . . . . . . . . .
5.4.3 Decision Threshold Computation . . . . . . . . . . . . . . . . . . . .
5.4.4 Receiver Operating Characteristic Computation . . . . . . . . . . . .
5.5 POD Calculation for PZT Sparse Arrays . . . . . . . . . . . . . . . . . . . .
5.6 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 Confidence Bounds of POD Estimates . . . . . . . . . . . . . . . . .

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Chapter 6 Reliability of Crack Detection in Riveted Aluminum Plates .
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Baseline Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Bonding Young’s Modulus Variation Effects . . . . . . . . . . . . . . . . .
6.3.1 Effects of Short-term Variations . . . . . . . . . . . . . . . . . . . .
6.3.2 Effects of Long-term Variation . . . . . . . . . . . . . . . . . . . . .
6.4 Bonding Thickness Variation Effects . . . . . . . . . . . . . . . . . . . . .
6.4.1 Effects of Short-term Variations . . . . . . . . . . . . . . . . . . . .
6.4.2 Effects of Long-term Variation . . . . . . . . . . . . . . . . . . . . .
6.5 Sensor Debonding Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Effects of Short-term Variations . . . . . . . . . . . . . . . . . . . .
6.5.2 Effects of Long-term Variation . . . . . . . . . . . . . . . . . . . . .
6.6 Performance of Array Implementations . . . . . . . . . . . . . . . . . . . .
6.7 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 7 Damage Detection in GFRP Composite Plates
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Experimental Procedure for Impact Damage Detection . .
7.2.1 Impact Test and Measurements . . . . . . . . . . .
7.2.2 Monitoring the PZT Sensors Health . . . . . . . . .
7.2.3 Guided Wave Inspection . . . . . . . . . . . . . . .
7.3 Reliability of Impact Damage Detection using PZT Sensors
7.3.1 Effects of Short-term Variations . . . . . . . . . . .
7.3.2 Effects of Long-term Variation . . . . . . . . . . . .
7.4 Detection of Hidden Delamination . . . . . . . . . . . . . .
7.4.1 Effects of Short-term Variations . . . . . . . . . . .
7.4.2 Effects of Long-term Variation . . . . . . . . . . . .
7.5 Summary and Conclusions . . . . . . . . . . . . . . . . . .

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Chapter 8 Effects of PZT degradation on GW
8.1 Introduction . . . . . . . . . . . . . . . . . . .
8.2 Delay-and-Sum Imaging Algorithm . . . . . .
8.3 Results for Riveted Plate Crack Detection . .
8.3.1 Imaging with no PZT Degradation . .
8.3.2 Imaging under PZT Degradation . . .
8.4 Conclusions . . . . . . . . . . . . . . . . . . .

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Chapter 9 Sensor Node Prototype for Guided Wave SHM
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Hardware development . . . . . . . . . . . . . . . . . . . . .
9.2.1 Actuation circuit . . . . . . . . . . . . . . . . . . . .
9.2.2 Sensing Circuit . . . . . . . . . . . . . . . . . . . . .
9.3 Software development . . . . . . . . . . . . . . . . . . . . . .
9.3.1 Sensor node application . . . . . . . . . . . . . . . .
9.3.2 Base station application . . . . . . . . . . . . . . . .
9.3.3 Wireless communication . . . . . . . . . . . . . . . .
9.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.1 Actuator circuit validation . . . . . . . . . . . . . . .
9.4.2 Sensor circuit validation . . . . . . . . . . . . . . . .
9.4.3 Power consumption . . . . . . . . . . . . . . . . . . .

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Chapter 10 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . 152
10.1 Contributions and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 152
10.2 Looking Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Appendix A Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Appendix B FEM Simulation Signals . . . . . . . . . . . . . . . . . . . . . . . . 160
BIBLIOGRAPHY

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viii

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LIST OF TABLES

Table 3.1:

Indices for the bonding thickness and Young’s modulus (Y) sets used
in the FEM simulations for comparison with experiment. . . . . . .

47

Table 4.1:

Wave group velocities of the A0 and the S0 modes for different actuatorsensor paths, with PZT-5 as actuator. . . . . . . . . . . . . . . . . . 58

Table 9.1:

Properties of the Iris mote. . . . . . . . . . . . . . . . . . . . . . . . 137

Table 9.2:

Power consumption depending on the software state. . . . . . . . . . 150

Table A.1:

Mechanical properties of material. . . . . . . . . . . . . . . . . . . . 158

ix

LIST OF FIGURES

Figure 1.1:

Piezoelectric actuator excites a guided wave in a structure, and piezoelectric sensors measure the waveform. Reflections of the guided wave
from a structural feature and a crack could be seen. . . . . . . . . .

4

GW signals after a measurement is subtracted from the measurement
when the structure was in its pristine state. (a) Signal due to defect
scatter. (b) Signal due to difference in PZT sensor properties. . . . .

5

A rectangular PZT wafer with 200 µm thickness and 8x7 mm surface
area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

A PZT straining by expanding or contracting in the 1 and 2 directions
when a voltage is applied to it in the 3 direction and vice versa. . .

11

Guided wave particle displacement pattern (mode shape) for antisymmetric and symmetric modes. . . . . . . . . . . . . . . . . . . .

14

Figure 2.4:

Dispersion curves of guided waves in an aluminum plate. . . . . . .

15

Figure 2.5:

Effective surface traction τ (x, t) due to a surface bonded PZT. . . .

16

Figure 2.6:

Processing steps for typical damage detection scheme in GW-PZT
SHM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

The schematic of a centralized wireless sensor network consisting of
two entities: The sensor nodes and the base station. The sensor nodes
are mounted on the structure and connected to embedded or surface
bonded transducers. The base station is a remote PC that collects
data from the sensor nodes for processing. . . . . . . . . . . . . . . .

27

Mote platform versions supplied by Memsic Inc., from left to right:
Iris, micaZ, and lotus motes. . . . . . . . . . . . . . . . . . . . . . .

29

Domains for the multiphysics coupled model. The ΩS domain includes the structure and the bonding adhesive. The ΩP domain includes the PZT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

Figure 1.2:

Figure 2.1:

Figure 2.2:

Figure 2.3:

Figure 2.7:

Figure 2.8:

Figure 3.1:

x

Figure 3.2:

The configuration for the 2D aluminum plate mesh. . . . . . . . . .

Figure 3.3:

The actuation signal in (a) time domain, and (b) its Fourier transform. 37

Figure 3.4:

The resulting waveforms for different mesh element sizes, with 4nodes linear quadrilateral elements and for the analytical solution. .

38

The resulting waveforms for the analytical solution and for different
mesh element sizes, with 8-nodes biquadratic quadrilateral elements.

39

The resulting numerical mesh error with different mesh element sizes
for both linear and biquadratic elements. . . . . . . . . . . . . . . .

40

Cross-sectional view of the three-dimensional model, showing the colocated PZT sensors on the top and bottom surfaces of the plate. . .

40

(a) The three-dimensional FEM simulation for mesh verification of
an Aluminum plate, with two circular PZTs bonded on the top. (b)
Three-dimensional closeup of the PZT and the thin adhesive bonding
layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Figure 3.5:

Figure 3.6:

Figure 3.7:

Figure 3.8:

Figure 3.9:

36

The resulting waveforms for the 3-D mesh with different element sizes. 41

Figure 3.10: Numerical error due to increasing mesh element size. . . . . . . . . .

42

Figure 3.11: Simulation running time versus mesh element size. . . . . . . . . . .

43

Figure 3.12: Resulting waveforms with different mesh element size for a composite
plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Figure 3.13: Numerical error due to increasing mesh element size for a composite
plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Figure 3.14: (a) Experimental Setup. (b) Experimental setup showing a closeup
on the PZT sensors. . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Figure 3.15: Error as compared with experiment. . . . . . . . . . . . . . . . . . .

46

Figure 3.16: Experimental waveform comparison with FEM results. . . . . . . . .

48

Figure 4.1:

A three layered structure with a rivet, and a notch crack at the bottom
layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi

50

Figure 4.2:

The FEM model geometry for a riveted plate. The rivet hole is 10mm
in diameter. There are eight PZT sensors at each surface of the plate,
situated at an equal distance of 60mm from the center of the rivet. .

51

Figure 4.3:

A closeup of the (a) defect mesh, and (b) PZT and bonding mesh. .

51

Figure 4.4:

Wave propagation in a riveted aluminum plate at time 40 µs. . . . .

53

Figure 4.5:

The configuration for the FEM model geometry for a GFRP. There
are ten PZT sensors at each surface of the plate, distributed into two
rows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

(a) Cross-sectional view of the delamination area located midway
through the plate thickness, shown within the circled area. (b) Closeup of the delamination area. . . . . . . . . . . . . . . . . . . . . . .

54

Figure 4.7:

Wave propagation in a GFRP plate at time 70 µs. . . . . . . . . . .

56

Figure 4.8:

Calculating the time of flight (TOF) for obtaining the wave mode
velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

Figure 5.1:

The truncated variability probability density function. . . . . . . . .

62

Figure 5.2:

Representation of the waveforms due to direct path from ActuatorSensor and due to the scatter from a defect. . . . . . . . . . . . . . .

63

Block diagram of the different components of the received waveforms
passing through different channels. . . . . . . . . . . . . . . . . . . .

64

Example of a received waveform after baseline subtraction ∆r(t). It
is composed of the superposition of two waveforms: Distortion due
to sensor degradation ∆sa (t), and scatter from defect sd (t). . . . . .

67

The work flow for the calculation of the probability of detection of a
given defect size under known variability statistics, and given a false
alarm criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

Computing the threshold for a given false alarm rate criterion. The
false alarm rate is the area under the no defect likelihood function.
The probability of detection PD is the area under the defect likelihood
function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

Figure 4.6:

Figure 5.3:

Figure 5.4:

Figure 5.5:

Figure 5.6:

xii

Figure 5.7:

Design of an array receiver with total votes thresholding. . . . . . .

74

Figure 5.8:

An example of the upper and lower 99 % confidence bounds for a POD
calculation, where (a) 105 samples are used, and (b) 106 sampled are
used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Figure 6.1:

The baseline waveforms measured at PZT-7 when PZT-3 is actuated.

80

Figure 6.2:

Measured signals and their scatter after baseline subtraction for the
sensor path 3-7 for (a) A0 mode, (b) S0 mode, when defect length is
a = 4mm and the Young’s modulus ξ = 2080M P a. . . . . . . . . . .

81

The surface for the defect scatter features yA = fA (a, ξ) and yS =
fS (a, ξ), which are functions of the PZT adhesive bonding Young’s
modulus (Y ) and defect length. . . . . . . . . . . . . . . . . . . . .

82

The likelihood functions for different defect lengths, when the predicted standard deviation of the Young’s modulus σξ = 520M P a. . .

83

The probability of detection with short-term variations in the PZT
adhesive bonding Young’s modulus. . . . . . . . . . . . . . . . . . .

84

Figure 6.3:

Figure 6.4:

Figure 6.5:

Figure 6.6:

The probability of detection as the standard deviation σξ of the PZT
adhesive bonding Young’s modulus increases, for different defect sizes. 85

Figure 6.7:

The receiver operating characteristic curve for different defect sizes,
and at σξ = 520M P a. . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Figure 6.8:

The probability of detection for changing µξ with σξ = 260M P a. . .

86

Figure 6.9:

The probability of detection as the mean µξ changes due to the longterm variations, for different defect sizes. . . . . . . . . . . . . . . .

87

Figure 6.10: The surface of y variations due to varying bonding thickness and
defect length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Figure 6.11: The likelihood functions for different defect sizes, when the predicted
standard deviation of the bonding thickness σξ = 7.5µm. . . . . . .

89

Figure 6.12: The probability of detection with short-term variations in bonding
thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

xiii

Figure 6.13: The probability of detection as σξ increases, for different defect sizes
(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Figure 6.14: The receiver operating characteristic curve for different defect sizes,
and at σξ = 7.5µm. . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Figure 6.15: The probability of detection due to bonding thickness variations for
changing µξ with σξ = 7.5µm. . . . . . . . . . . . . . . . . . . . . .

91

Figure 6.16: The probability of detection due to bonding thickness variations as
the mean µξ changes, for different defect sizes. . . . . . . . . . . . .

91

Figure 6.17: The surface of y variations due to varying PZT diameter and defect
size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Figure 6.18: The likelihood functions for different defect sizes, when the predicted
standard deviation of the PZT diameter σξ = 0.5µm. . . . . . . . .

94

Figure 6.19: The probability of detection with short-term variations in bonding
coverage area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Figure 6.20: The probability of detection as σξ increases, for different defect sizes
(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Figure 6.21: The receiver operating characteristic curve for different defect sizes,
and at σξ = 0.25mm. . . . . . . . . . . . . . . . . . . . . . . . . . .

96

Figure 6.22: The probability of detection due to bonding coverage area variations
for changing µξ with σξ = 0.5mm. . . . . . . . . . . . . . . . . . . .

96

Figure 6.23: The probability of detection due to bonding coverage area variations
as the mean µξ changes, for different defect sizes. . . . . . . . . . . .

97

Figure 6.24: Received signal feature amplitude for PZT sensors with different wave
front incident angles. . . . . . . . . . . . . . . . . . . . . . . . . . .

98

Figure 6.25: Likelihood function for the PZT array elements, when there is no
defect, for the PZT adhesive Young’s modulus degradation mode.
The decision threshold for each PZT is shown as a dotted vertical
line. The standard deviation σξ = 260M P a. . . . . . . . . . . . . .

99

xiv

Figure 6.26: Likelihood function for the PZT array elements, when there is no
defect, for the PZT adhesive thickness degradation mode. The decision threshold for each PZT is shown as a dotted vertical line. The
standard deviation σξ = 7.5µm. . . . . . . . . . . . . . . . . . . . . 100
Figure 6.27: Likelihood function for the PZT array elements, when there is no defect, for the PZT adhesive bonding coverage area degradation mode.
The decision threshold for each PZT is shown as a dotted vertical
line. The standard deviation σξ = 0.5mm. . . . . . . . . . . . . . . 100
Figure 6.28: Probability of detection with PZT adhesive Young’s modulus degradation mode. The standard deviation σξ = 260M P a. . . . . . . . . 101
Figure 6.29: Probability of detection with PZT adhesive thickness degradation
mode. The standard deviation σξ = 7.5µm. . . . . . . . . . . . . . . 101
Figure 6.30: Probability of detection with PZT adhesive bonding coverage area
degradation mode. The standard deviation σξ = 0.5mm. . . . . . . 102
Figure 7.1:

Schematic of the experimental sample and the locations of the six
PZT sensors bonded to its surface. Three of the PZT sensors were
used as actuators and other three as sensors for guided wave inspection. The impact location is marked by ’X’. . . . . . . . . . . . . . . 104

Figure 7.2:

Experimental procedure for impact damage inspection which invloves
three steps: (a) A drop-weight impact. (b) Measurement of the PZT
impedance. (c) Guided wave inspection of the sample for detecting
impact damage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Figure 7.3:

Fixture for the drop-weight impact experiment. . . . . . . . . . . . . 105

Figure 7.4:

Picture of the sample after each impact, with varying energies. (a)
11 J, (b) 22 J, (c) 43 J, (d) 65 J, (e) 88 J, (f) 109 J. . . . . . . . . . 106

Figure 7.5:

(a) The total energy in the impactor object for all the six impact tests.
(b) The percentage absorbed energy by the sample during impact, for
the six different impacts. . . . . . . . . . . . . . . . . . . . . . . . . 107

Figure 7.6:

(a) Plate deflection due to applied load by the impactor, for six different impact energies. (b) Percentage of the stiffness change after
each impact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

xv

Figure 7.7:

The change in capacitance of the six PZTs bonded to the test sample
after each impact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 7.8:

The guided wave pitch-catch paths considered for impact damage
detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Figure 7.9:

Guided wave measurements of the baseline and after each of the six
impacts for the path 1-6. . . . . . . . . . . . . . . . . . . . . . . . . 111

Figure 7.10: Feature extraction of the measured waveform s(t) relative to the baseline signal sb (t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Figure 7.11: The signal features for the three different paths. (a) Amplitude ratio
change feature (y1 ); (b) Phase shift feature (y2 ); (c) Scatter envelop
energy (y3 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Figure 7.12: FE model geometry, with two PZT sensors on each surface of the
plate. The defect due to impact at the center of the plate is shown
in a lighter color to indicate a reduction in stiffness in that region. . 114
Figure 7.13: The surface variation of the feature (y) due to varying PZT diameter
and change in the stiffness of the defect region of the plate. . . . . . 115
Figure 7.14: The likelihood functions for different defect sizes, when the predicted
standard deviation of the PZT diameter σξ = 0.5µm. . . . . . . . . 116
Figure 7.15: The probability of detection of the change in stiffness with short-term
variations in bonding contact area of the PZT sensor. . . . . . . . . 116
Figure 7.16: The probability of detection as σξ increases. . . . . . . . . . . . . . 117
Figure 7.17: The probability of detection of the change in stiffness with short-term
variations in bonding contact area of the PZT sensor. . . . . . . . . 117
Figure 7.18: The probability of impact damage detection due to long-term PZT
bonding coverage area variation with σξ = 0.5mm. . . . . . . . . . . 118
Figure 7.19: The baseline waveforms measured at PZT-6 when PZT-5 is actuated. 119
Figure 7.20: The surface variation of the feature y due to varying PZT diameter
and delamination diameter. . . . . . . . . . . . . . . . . . . . . . . . 121

xvi

Figure 7.21: The likelihood functions for different defect sizes, when the predicted
standard deviation of the PZT diameter σξ = 0.5µm. . . . . . . . . 122
Figure 7.22: The probability of detection with short-term variations in bonding
contact area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Figure 7.23: The probability of detection as σξ increases, for different defect sizes
(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Figure 7.24: The receiver operating characteristic curve for different defect sizes
(a), and at σξ = 0.5mm. . . . . . . . . . . . . . . . . . . . . . . . . 124
Figure 7.25: The probability of detection due to long-term PZT bonding coverage
area variation with σξ = 0.5mm. . . . . . . . . . . . . . . . . . . . . 124
Figure 7.26: The probability of detection due to long-term PZT bonding coverage
area as the mean µξ changes, for different defect sizes. . . . . . . . . 125
Figure 8.1:

Results when there is no PZT degradation for a 4mm defect size
without thresholding. . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Figure 8.2:

Results when there is no PZT degradation for a 4mm defect size. . . 131

Figure 8.3:

Results when there is no PZT degradation for a 7mm defect size. . . 131

Figure 8.4:

Results when there is no PZT degradation for a 10mm defect size. . 131

Figure 8.5:

Results when there is no PZT degradation for a 10mm defect size,
and the signal detection thresholds are set to a high value. . . . . . 132

Figure 8.6:

Results when there is no PZT degradation for a 10mm defect size
after image thresholding, and the signal detection thresholds are set
to a high value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Figure 8.7:

The thresholded image when there is a PZT degradation and a 10mm
defect length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Figure 8.8:

The thresholded image when there is a PZT degradation and no defect
in the geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

xvii

Figure 9.1:

The extension board for iris mote. On the left is the picture of the
Iris mote connected to the developed extension circuit board. On the
right the block diagram of the extension circuit board components. . 139

Figure 9.2:

The actuation circuit schematic. . . . . . . . . . . . . . . . . . . . . 140

Figure 9.3:

The sensing circuit components. Figure 9.3a is the charge amplifier
schematic, Figure 9.3b is the threshold voltage schematic, and Figure
9.3c is the envelop detector schematic . . . . . . . . . . . . . . . . . 142

Figure 9.4:

The sensor node state machine.

Figure 9.5:

The actuation circuit validation. Figure (a) shows the square wave
as input and filtered output; (b) shows the frequency content of the
square wave input, the actuator output and a 7-cycle Hanning window
tone burst. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Figure 9.6:

An experimental setup with two sensor nodes, each connected to a
PZT piezoelectric wafer surface bonded to an aluminum plate. An
oscilloscope is connected to the outputs of the charge amplifier and
envelop detector of sensor node 2 so that the analog signals could be
investigated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Figure 9.7:

A analog output of the envelop detector is compared with the output of the charge amplifier. The digital samples are also shown for
comparison with the analog envelop signal. . . . . . . . . . . . . . . 149

Figure B.1:

S0 mode waveforms in a riveted aluminum plate, with defect effects.

Figure B.2:

S0 mode waveforms in a riveted aluminum plate, with sensor degradation effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Figure B.3:

A0 mode waveforms in a riveted aluminum plate, with defect effects. 163

Figure B.4:

A0 mode waveforms in a riveted aluminum plate, with sensor degradation effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Figure B.5:

S0 mode waveforms in a composite plate, with defect effects. . . . . 165

Figure B.6:

S0 mode waveforms in a composite plate, with sensor degradation
effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

xviii

. . . . . . . . . . . . . . . . . . . . 144

161

Figure B.7:

A0 mode waveforms in a composite plate, with defect effects. . . . . 167

Figure B.8:

A0 mode waveforms in a composite plate, with sensor degradation
effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

xix

Chapter 1
Introduction

1.1

Motivation

Materials and parts in complex systems can suffer from fatigue due to use, age and environmental stresses experienced during service. It is therefore essential to evaluate fatigue
damage and predict the remaining life, reliability and safety of structures. This is particularly important in the aerospace, oil and gas, and power generation industries, where a
structural failure could prove to be catastrophic. Nondestructive evaluation (NDE) is used
for assessing the integrity of critical structures. Various NDE methods are available, and
the selection of an inspection method depends on the material type and the kind of damage
that could occur in the structure.
Current industry practice requires inspection at scheduled intervals necessitating the
structure to be shutdown and go out of service throughout the period of inspection. NDE
methods adopted by the industry require specialized and expensive equipment, structures
often need to be dismantled to inspect inaccessible components, and the inspection needs
to be conducted by qualified trained personnel. All those requirements impose large costs
to the industry. As an example, in the commercial and military aerospace industry, 27%
of an aircraft life cycle cost is spent on inspection and repair, excluding the opportunity
cost due to the time the aircraft is out of service for scheduled inspection [1]. Moreover,

1

scheduled inspections may not be adequate or timely for detecting impending hazards due
to unanticipated loads and stresses. This safety drawback of current practices have led to
numerous commercial aircraft accidents due to structural failure, such as the crash of Japan
Airlines 123 due to rear pressure bulkhead failure, and the in-flight disintegration of China
Air flight 611 due to metal fatigue. Also similar causes for accidents in oil transportation
industry include the Pacific Gas and Electric pipeline explosion in San Bruno, California,
and the Enbridge crude oil pipeline rupture in Grand Marsh, Wisconsin.

1.2

Structural Health Monitoring

Structural health monitoring (SHM) is envisioned to improve safety and reduce maintenance
costs by providing real time information about the structure’s integrity and warnings about
impending hazards. The integration of SHM systems within industrial structures would
change current safety and inspection practices, eliminating the need for regularly scheduled
inspections and migrating towards condition-based inspections. An SHM system uses a network of sensors that are permanently surface mounted or embedded in the structure. Those
sensors measure physical quantities that are dependent on the structure’s properties (mass,
stiffness and damping). Detecting changes in the measured values indicates the possible
presence of damage. Sohn et al. [2] defined five different levels for damage identification in
an SHM system:
1. Existence: Is there damage in the system?
2. Location: Where is the damage in the structure?
3. Type: what kind of damage is present?

2

4. Extent: How severe is the damage?
5. Prognosis: How much useful life remains?
Identification of the type and extent of the damage requires knowledge of the failure modes of
the structure, and how the sensing modality interacts with each failure mode. For example,
in metallic structures, fatigue cracks and corrosion are of most concern. In fiber reinforced
plastic composites, delamination, fiber and matrix breaking are more of a concern. Also, the
type of the structural part can introduce critical damage areas. Stiffeners in a composite
structure can delaminate, fasteners in metallic structures can introduce stress concentrations
or crack initiation sites, and adhesively bonded parts can peel. Locating damage depends on
the type of sensing modality, sensor type, and the number of sensors. Prognosis relates the
estimated defect type and extent to the ultimate strength of the material in order to predict
the remaining useful life.
In general, most NDE techniques could be used as the sensing modality in an SHM system
(see Grandt [3] for a thorough reference about most common NDE techniques), however
vibration-based techniques have been the most popular. This is due to the availability of a
multitude of low profile vibration-based sensors that are convenient to adhesively bond to
the structure’s surface or embed in composite structures. These sensors include foil strain
gages, accelerometers, optical fibers, and piezoelectric sensors. Comprehensive reviews on
vibration-based SHM until the year 2006 have been reported by Doebling et al., [4], Sohn et
al. [2], and Montalvao et al. [5]. Vibration-based methods measure global-variations in the
structure such as changes in its dynamic natural frequency, damping ratio, and mode shapes.
A major drawback in those methods is the difficulty in locating damage, which requires a
dense network of sensors, and they are not effective in detecting small damages.

3

Sensors

Actuator

Structural Feature
Crack

Figure 1.1: Piezoelectric actuator excites a guided wave in a structure, and piezoelectric
sensors measure the waveform. Reflections of the guided wave from a structural feature and
a crack could be seen.
The ultrasonic guided wave method using built-in thin piezoelectric films has emerged
as a promising option for locating and characterizing damage in SHM. Guided waves are
elastic stress waves that are guided within the confines of a structure’s surface, when one
dimension of the structure is smaller than the propagating wavelength. Common industrial
substructures, such as an airplane wing, oil pipelines, and steam generator tubes in nuclear
power plants, are natural wave guides. A guided wave travels along the surface of the
structure with non-propagating perturbations (standing wave) along the thickness. The
propagating wave energy diffuses only in two dimensions, reducing attenuation and allowing
the wave to propagate for a longer distance. This allows the monitoring of a large structural
area using a sparse network of fixed sensors, which makes it attractive to SHM applications.

4

20
Voltage (mV)

Voltage (mV)

10
10
0
−10

5
0
−5
−10

0

20

40
time (µs)

60

80

0

20

(a)

40
time (µs)

60

80

(b)

Figure 1.2: GW signals after a measurement is subtracted from the measurement when
the structure was in its pristine state. (a) Signal due to defect scatter. (b) Signal due to
difference in PZT sensor properties.
In ultrasonic guided wave SHM, an adhesively bonded piezoelectric film transducer excites
a wave that propagates through the structure for inspection. This transducer that excites
a waveform is also called the actuator. Nearby piezoelectric transducers sense the actuated
waveform, which may contain the incident (line-of-sight) wave packet from the actuator and
its scattered and reflected wave packets due to the presence of defects or other discontinuities
in the structure. The main problem in guided wave SHM is the detection of the signals due
to defects and the ability to distinguish them from signals due to other structural features.
This problem inherently requires the comparison of the detected waveforms at inspection
time with the waveforms that were measured when the structure was known to be in a
pristine state (baseline waveform).

1.3

Challenges in Guided Wave SHM

One of the main challenges in using Guided Wave with PZT sensors (GW-PZT) for SHM applications, is that the PZT sensors themselves might degrade over time due to environmental

5

conditions, structural loading, and impacts that might break the PZT sensor or change its
properties. The PZT sensor could still operate, however its measurements would be different
from the baseline signals, even if there is no damage present. The problem is to be able to
distinguish changes due to sensor degradation from changes due to structural damage. This
problem is demonstrated in Figure 1.2. Two signals are shown after subtracting the baseline
waveform from the measurement. One of the figures is solely due to scatter from a defect
whereas the other figure is solely due to changes in the PZT properties, and no defect is
present. Could you tell which figure is due to defect and which is due to sensor degradation?
If this question is not be answered correctly, it could lead to missing the detection of critical
defects, or it could result in multiple false calls that would render the use of such a system
impractical and unreliable.
Conducting an experimental procedure that would reproduce all possible types of PZT
degradation and defect sizes in order to quantify the performance of an SHM system is prohibitively expensive in terms of time and cost. Moreover, the condition of the PZT sensor
is not always known, and it is difficult to obtain an accurate measure of their current properties and condition. As computational power becomes abundant and cheap, and advanced
numerical methods are available to simulate complex SHM system, a model-based approach
for performance assessment becomes an attractive alternative.
Another challenge in implementing an SHM system using GW is to transfer data from the
PZT sensor network to a base station. Connecting the sensors directly through cables and
wires has a high installation and material cost. Wireless sensor networks offer a promising
solution for continuous SHM. Wireless sensor networks are inherently highly scalable and
configurable and do not require high installation and maintenance cost. A sensing device is
connected to the PZT sensor to acquire and digitize the measured waveforms. Due to the lack
6

of physical wiring, these devices need to be self powered. This limits the amount of power
available to them, requiring the use of low power electronics which limits the computational
performance. This poses a challenge for the data acquisition of ultrasonic signals at high
frequencies, which requires sampling rates higher than those supported by current low power
wireless sensing devices.

1.4

Problem Statement and Thesis Organization

The two main problems that are addressed in this thesis are:
1. Formulate a stochastic model-based approach to quantify the performance of GWPZT SHM system. The system performance is evaluated in terms of the probability of
detection of different defect sizes, and the probability of false alarm.
2. Implement a low power wireless sensor node prototype that could be used as part of
a wireless sensor network for the data acquisition and actuation of guided waves using
PZT sensors.
The background and some literature review is discussed in Chapter 2. This includes
different modeling methods for GW-PZT, damage detection methods and algorithms, reliability of GW-PZT under varying environmental conditions, and wireless sensor networks for
data acquisition.
Chapter 3 describes the finite element model (FEM) used for modeling guided wave
propagation and the PZT sensors. The FEM model is verified analytically using a simple
2-D geometry, and then a full 3-D model is implemented and verified using experimental
measurements. Chapter 4 describes the two different structures that are modeled using
FEM: a plate with a rivet hole, and a fiber reinforced plastic composite plate.
7

The different modes for PZT sensor degradation are discussed in Chapter 5. The stochastic method for evaluating the performance of GW-PZT under degrading PZT sensors is then
formulated. GW-PZT performance results for detecting cracks in a riveted structure are
discussed in Chapters 6. Chapter 7 conducts an experimental investigation on the effects
of impacts on glass fiber composite plates, and the detection of such impact damage using
GW-PZT. A model-based study is performed for investigating the performance of GW-PZT
in detecting impact damage, and also for detecting small hidden delaminations in composite
plates. and The effects of PZT sensor degradation on the reliability of a GW imaging algorithm are described in Chapter 8. Chapter 9 describes the wireless sensor network prototype
implemented for data acquisition and actuation of guided wave signals. The conclusions
and prospects for continued work based on the results of this dissertation are discussed in
Chapter 10.

8

Chapter 2
Background for GW-PZT SHM
modeling, damage detection,
reliability and instrumentation

2.1

Introduction

A considerable amount of research has been done in SHM using guided waves and PZT
transducers. This is a truly multidisciplinary field which requires fundamental understanding
of the physical aspects of guided waves, modeling their excitation and sensing by PZT,
the interaction of waves with structural damage, and damage detection and classification
methods. Also, another important aspect is the data collection and instrumentation from
a network of PZT sensors. Raghavan and Cesnik [6] reviewed the state of the art and
technologies that have been used in GW-SHM, and the possible application areas. Lynch
and Loh [7] reviewed wireless sensor technologies for SHM applications. However, most of
the SHM applications were based on vibration methods. This chapter describes the most
notable works in GW-PZT modeling, damage detection, and wireless sensor networks for
SHM. The use of GW-PZT has gained interest mostly in monitoring airplane components,
such as riveted airplane wing splices and fiber reinforced plastic composites, especially with

9

recent introduction of composites in aircrafts.

2.2

Piezoelectric Transducers

Piezoelectricity is the two-way effect between stress/strain and electric field/voltage difference in a material. When a stress field is applied to a piezoelectric material, or when
they are strained, an electric potential difference is produced across the material. This is
called the piezoelectric effect. The converse is also true. When voltage difference is applied
across a piezoelectric material, it will expand and contract, producing a stress field. The
linear constitutive equation for the direct effect is given according to the IEEE Standard on
piezoelectricity [8] using the matrix indices notation:

εk = s E
km σm + djk Ej

(2.1)

and the linear constitutive equation for the converse effect:

Di = dim σm + σij Ej

(2.2)

where εk (k ∈ {1...6}) is the mechanical strain vector (m/m); σm (m ∈ {1...6}) is the stress
vector (N/m2 ); Ej (j ∈ {1...3}) is the electric field vector (V olt/m); sE
km is the mechanical
compliance matrix of size 6 × 6 at zero electric field (m2 /N ); djk , dim are the piezoelectric
coupling coefficient matrices of size 3 × 6 (m/V olt and V olt/m respectively); Di (i ∈ {1...3})
is the electric displacement vector (Coulomb/m2 ); σij is the dielectric constant matrix of
size 3 × 3 at zero stress (F arad/m). The constitutive equations combine the electric field
equation with the acoustic field equation through the coupling coefficient matrix d. This
10

Figure 2.1: A rectangular PZT wafer with 200 µm thickness and 8x7 mm surface area.

Figure 2.2: A PZT straining by expanding or contracting in the 1 and 2 directions when a
voltage is applied to it in the 3 direction and vice versa.
direct relation between mechanical and electrical fields make piezoelectric material most
commonly used as ultrasonic transducers. Also, the reciprocity of the piezoelectric effect
allows the same transducer to operate both as an actuator and sensor.
There are a multitude of material that exhibit piezoelectric properties, but Lead-ZirconateTitanate (PZT) has gained the most amount of popularity in guided wave SHM applications.
PZT has a high piezoelectric coupling coefficient and high stiffness, which allows it to efficiently actuate a structure by applying low voltage levels. It has a broadband frequency
response which makes it ideal for use with guided waves, whose frequencies range from 50
kHz up to 1 MHz. It is of low cost and is manufactured as thin films that are convenient to
bond to the surface of structures.
PZT transducers for guided wave applications come in different forms. Most commonly

11

used are the monolithic PZT ceramics cut as thin films as shown in Figure 2.1. Sirohi and
Chopra have investigated the use of PZT as sensors in [9], and as actuators in [10]. This
class of PZT ceramics is electrically polled across their thickness in the 3-direction, and they
are transversely isotropic in the 1-2 plane, according to the coordinate reference in Figure
2.2. When acting as sensors, the applied electric field is zero, and the PZT is stressed,
accumulating charge on its top and bottom surfaces. This charge is collected by electrodes
that cover the surfaces. The sensor equation described by the converse effect in equation
(2.2), could then be expressed as:








 D1 






 D2  = 






D3










0
0
0
0 d15 0  




0
0
0 d15 0 0 



d31 d31 d33 0
0 0 





σ1 


σ2 



σ3 



τ23 



τ13 


τ12

(2.3)

where the electric displacement is directly related to the stress by the piezoelectric coupling
matrix. The coefficients d31 , relates the normal stress in the 1 and 2 directions to the electric
fields along the 3-direction (polling direction). Similarly d33 relates the normal stress in the
3-direction to the electric field in the 3 direction, and d15 relates the shear stress in the 2-3
and 1-3 planes to the transverse electric fields (in PZT films, there are no shear stresses).
PZT sensors have electrodes at bottom and top surfaces (whose normal is in the 3-direction),
and only the electric displacement D3 is measured. Thus, only the normal stress components
(σ1 , σ2 , σ3 ) is measured. Moreover, since the thickness is much smaller than the area in
the 1-2 plane, the stress σ1 and σ2 would have the most effect on the measurement.
12

On the other hand, when there is no stress applied to the PZT, and a voltage is applied
across the PZT, it will strain mostly in the 1 and 2 directions since those dimensions are
significantly larger than the thickness in the 3-direction. The actuation equation is described
by the direct effect in equation (2.1), and it could be expressed as:


























ε1 
0 d31 
 0






 0


ε2 
0 d31 





 E1




 0


ε3 
0 d33 




 = 
  E2 




 0 d15 0  

γ23 





 E3



 d15 0
γ13 
0 






γ12
0
0
0

(2.4)

Again, as in the sensing case, only E3 could be excited, and thus according to equation (2.4),
only normal strains (ε1 , ε2 , ε3 ) could be actuated.

2.3

Ultrasonic Guided Waves

The first known study of waves propagating along an interface was by Lord Rayleigh in 1889
who derived the solutions for waves propagating along a free surface with infinite extent and
depth. In 1917, Horace Lamb explained the wave propagation in an isotropic plate with finite
thickness and infinite extent, however he did not attempt to generate them experimentally.
Consequently guided waves traveling along the boundaries of an elastic plate are commonly
known as Lamb waves. In 1967, Viktorov gave a thorough investigation of Lamb waves
by describing and evaluating their properties and their application for flaw detection [11].
However, his investigation was confined to isotropic materials and the two dimensional plane

13

(a) Antisymmetric mode shape

(b) Symmetric mode shape

Figure 2.3: Guided wave particle displacement pattern (mode shape) for antisymmetric and
symmetric modes.
strain assumption. Numerous texts have since dealt with this topic in three dimensions and
in anisotropic and layered media, most notable are the classical texts by Rose [12], Auld [13],
and Nayfeh [14].
Ultrasonic bulk waves travelling in an unbounded in elastic medium have three propagation modes, which are obtained by solving the wave equation: longitudinal mode where the
particle displacement is parallel to the wave propagation, shear horizontal mode where the
particle displacement is perpendicular to the wave propagation in the horizontal direction,
and shear vertical mode where the particle displacement is perpendicular to the wave propagation in the vertical direction. When the wave propagation is confined by two surfaces
of a structure, the longitudinal and shear vertical solutions of the wave equation become
coupled and produce infinite possible modes. Depending on the distribution of the particle
displacement across the thickness of the plate, the modes could be classified as antisymmetric modes (denoted A0 , A1 , ..., An ) and symmetric modes (denoted S0 , S1 , ..., Sn ). The
wave mode shapes are shown in Figure 2.3. There are at least two modes present at any
frequency, which are the fundamental antisymmetric A0 and symmetric S0 modes. As the
wave frequency gets higher, higher order modes are activated. Unlike bulk waves which have
a constant velocity at all frequencies, guided waves are dispersive, where their velocity depends on their frequency. For an isotropic material, with plate-like structure, the dispersion
equations could be obtained [12] as:

14

A1

S2
S1

5

A2

A

S3 A

4

3

S0
A0

0
0

6S

S4

0

Velocity (km/s)

Velocity (km/s)

10

2
4
6
8
10
Frequency.Thickness (MHz.mm)

S1
A1

4

2 A0
0
0

A3

S2

A2

S

3

A4

S4

2
4
6
8
10
Frequency.Thickness (MHz.mm)

(a) Phase Velocity

(b) Group Velocity

Figure 2.4: Dispersion curves of guided waves in an aluminum plate.
tan(qh) 4k 2 tan(ph)
= 0 for symmetric modes
+
2
q
q2 − k2

(2.5)

2

q 2 − k 2 tan(ph)
q tan(qh) +
= 0 for antisymmetric modes
4k 2 p

(2.6)

where 2h is the thickness of the plate, and

p2 =

ω 2
− k2
cL

and q 2 =

ω 2
− k2
cT

(2.7)

where cL and cT are the longitudinal and transverse bulk ultrasonic wave speeds, respectively.
Those speeds depend solely on the material properties. The angular frequency is ω, and k is
the wave number. In those equations, a solution for k could be found for every ω. Knowing
the relation k = ω/cp , the guided wave phase velocity could be calculated, and the dispersion
curves showing the relationship between frequency and velocity could be plotted (Figure
2.4(a)). In equations (2.6) and (2.5), as ω increases, more than one solution could satisfy the
equation, resulting in higher order symmetric and antisymmetric propagating wave modes.
In the presence of dispersion, the wave group velocity is different from the phase velocity,

15

tc

ts = 2h

Figure 2.5: Effective surface traction τ (x, t) due to a surface bonded PZT.
and it defined as:
cg =

dω
dk

(2.8)

The dispersion curves for the group velocity in an aluminum plate are shown in Figure 2.4(b).
For anisotropic material, such as fiber-reinforced composites, the wave velocity depends on
the direction of propagation. Obtaining the dispersion curves for such material is more involved than the isotropic case. The reader is referred to Nayfeh [14] and Wang and Yuan [15]
for detailed derivation of dispersion curves in composite lamina, and composite multilayered
structures.

2.4

GW-PZT Modeling

The two dimensional analytical solutions for guided wave excitation in isotropic material by
a PZT actuator have been derived by Giurgiutiu [16]. The 2-D geometry of this problem
is shown in Figure 2.5. The derivation is based on the simplification assumption that the
transducer mass is small relative to the total mass of the system, where its mechanical
dynamics could be ignored. A static model is used to derive the effective surface traction
force that the PZT actuator applies on the surface of a beam through a bonding layer.

16

This is called the shear-lag model, and have been first derived by de Luis and Crawley [17].
The wave equation could then be solved by applying the effective PZT traction force as a
boundary condition. This effective force model is valid only for low frequencies, when there
stress distributions across the thickness of the beam could be assumed linear or uniform.
Bartoli [18] derived the analytical solutions for high frequency excitation, with non-linear
and non-uniform Lamb wave mode shapes. Closed form solutions were found for for the
fundamental Lamb wave modes, but the closed-form solution for a finite number of modes
was difficult to achieve. Raghavan and Cesnik [19] derived the three dimensional analytical
solutions of GW excited by a PZT actuator, based on the PZT shear-lag effective force
model.
Although analytical solutions allows rapid computation of guided wave propagation, they
are only valid for regularly shaped and simple structures. They do not model reflections and
scattering from structural features, edges and defects. Moreover, the decoupled shear-lag
models are not accurate at higher frequencies. They may lead to accurate results when the
wavelength of Lamb wave is much larger than the dimension of the PZT, but may lead to
significant errors when the wavelength becomes smaller. The dynamic coupled equations
between the PZT and the substructure need to be considered for an accurate solution,
however it is difficult to obtained a closed form solution analytically.
Numerical methods based on the discretization of the space-time domain have been extensively studied for modeling GW-PZT. The domain in space is discretized into finite elements,
and the time is discretized into steps, and then the equation for each finite element at each
time step has to be solved, and the global solution is found by assembling all the solutions
for each finite element. This approach requires a lot of computational resources, especially
for large structures. Semi-analytical finite elements (SAFE) have been proposed to reduce
17

computational cost by combining analytical solutions in the direction of wave propagation
with linear finite elements in the other two dimensions. Other numerical methods have been
examined such as the finite element method (FEM), the finite difference method (FDM),
the boundary element method (BEM), the finite strip element method (FSM), the spectral
element method (SEM), and the mass spring lattice method (MSLM) [20]. FEM is the most
popular method due to its ability to model complex shapes and structures, and there are
numerous commercial FEM codes available. As computing power becomes more abundant
for modeling wave propagation.

2.5

Damage Detection Methods

The feasibility of using GW for inspecting complex structures have been well studied. Alleyne
and Cawley [21] studied the interaction of GW with notches in metallic plates using a two
dimensional FEM model and also experimental data. They have shown that the sensitivity
of GW is affected by operating frequency, mode type (symmetric and antisymmetric), mode
order, and the notch geometry. In [22], they propose guidelines for proper selection of
mode types, frequency, and excitation signal. In [23], they propose a signal processing
technique using 2D Fourier transform for obtaining information from multiple wave modes
present in the signal. This technique requires multiple measurements of the propagating
wave, at increasing distances from the excitation source. Rose [24] showed the potential
uses and benefits of GW inspection. The use of GW in railway, aerospace, and pipeline
inspection was demonstrated. Different types of transducers such as angle beam, comb, and
EMAT transducers that could be used for GW inspection were described. The application
of GW to fiber reinforced composite plastic have also been investigated by Chimenti [25] and

18

Figure 2.6: Processing steps for typical damage detection scheme in GW-PZT SHM.
Ramadas [26]. A review of the GW transducers and damage detection methods for composite
material is given by Su [27]. The damage detection methods and transducers described in
those works are intended for NDE. Bulky NDE sensors are used, and a trained operator
is present to collect data. The operator could change the sensor location and operating
frequencies to obtained clear indications about possible defect locations.
A major challenge in GW-PZT is that information about the structure’s health should
be obtained from an autonomous pre-installed sparse network of PZT sensors, and decisions
about the presence of defects should also be automated. Ideally, it is desired to obtain an
image representing the structure and its features, and indicate regions where possible damage
may have occurred. Figure 2.6 shows a general scheme to obtain an image, and then use
features in the image to detect defective regions, and classify the type and severity of defects.
The first step in damage detection is to obtain the data from the PZT sensors. Important

19

variables to consider are the excitation signal shape and bandwidth. Alleyne and Cawley [21]
studied the effects of those parameters on notch detection in metallic plates. Kessler [28]
describes the optimization of such parameters for damage detection in composite material.
Another important factor is the positions of the PZT sensors on the structure. It is
desired to minimize the number of sensors to minimize the total cost and weight of the SHM
system. On the other hand, enough sensors should be used for reliable detection of defects.
Kaiser [29] investigated the general problem of selecting a set of sensor locations from a
larger set of candidate locations. Genetic Algorithms have been used to search for optimal
locations of sensors for GW-PZT SHM. Janapati et al. [30] implemented a genetic algorithm
using model generated data for selecting the sensor placement that maximizes the probability
of detection of fatigue cracks in aluminum plates. Flynn and Todd [31] studied the optimal
placement of PZT sensors based on the framework of detection theory. The features used
for detection are assumed to be noisy, and the objective function for the genetic algorithm
is to find the sensor placement that maximizes the probability of detection or minimizes the
probability of false alarms.
The next step in damage detection is preprocessing, feature extraction, and time-of-flight
(TOF) extraction (Figure 2.6). The TOF is used to indicate the location of defect, and the
features are used to estimate the type and severity of damage. The type of preprocessing
depends on the type of features that are of interest. It is desired to obtain a signal feature
that is most sensitive to the defect type that needs to be detected. The features and TOF
could then be combined to construct an image. Once the regions of interest (ROI) where
damage might be present are identified, a report is issued to describe the predicted type of
defect in each ROI. Hu et al. [32,33] developed a theoretical framework to estimate the shape
of elliptical cracks and holes in plates by finding the sizes of the major and minor axes of the
20

elliptical defect. This method depends on obtaining the ratio of incident to reflected wave
packets. However, this approach will fail if the defect is near the sensor, since the incident
and reflected wave packet may be overlapping. Also, in a complex structure which might
include rivet holes and stiffeners, the received signals would be too complicated to isolate
wave packets reflected from defects.
Most methods for damage detection using GW-PZT involve comparison with waveforms
measured when the structure was in its healthy state. This waveform is called the baseline
signal. This comparison may be made directly using the time domain signals, or the signals
could be transformed into the frequency or time-frequency domains for comparison. The
simplest method for comparison is to subtract the baseline signal from the measurement
signal. If the remaining difference signal is not zero, this would indicate a change in the
structure, and a probable presence of damage. Several other features have been investigated
to obtain more information from the measured signals.
Michaels [34] described three different features extracted from the time domain signals for
detecting cracks and holes in an aluminum plate under varying temperature. These features
quantify the difference between the baseline signal x(t), and a measured signal y(t), and are
described as follows:
1. The normalized squared error between the signal and baseline, within a time window
T:

T

E1 =

(y(t) − x(t))2 dt

0

(2.9)

T

x(t)2 dt
0

2. The drop in the correlation coefficient between the signal and baseline, within a time

21

window T :

T

(x(t) − µx ) y(t) − µy dt
E2 = 1 −

0

σx σy

(2.10)

3. The loss of local temporal coherence between the signal and baseline, within a time
window T :
T

1
E3 = 1 −
max γxy (τ, t) dt
τ
T 0

(2.11)

where γxy (τ, t) is the local temporal coherence:
Rxy (τ, t)
Rxx (0, t)Ryy (0, t)

γxy (τ, t) =

(2.12)

and Rxy is the short time cross correlation, and Rxx (τ, t), Ryy (τ, t) are the short time
auto correlations:
t+ ∆T

2
1
x(s)ω(s − t)y(s + τ )ω(s + τ − t)ds
Rxy (τ, t) =
∆T t− ∆T
2

(2.13)

where ω(t) is a rectangular window function.
Zhao et al. [35] used the correlation coefficient feature E2 for detecting rivet cracks and
corrosion in a realistic aeroplane wing specimen. Monnier [36] used Fourier transform to
define a damage index feature such that:

DI =

n
i=0 |Fi − F Di |
n
i=0 |Fi |

(2.14)

where Fi and F Di are the Fourier transform coefficients for the baseline and the measured
signals, respectively. The damage index identifies changes in the signal amplitude and phase.

22

The use of damage index was examined for delamination detection in multi-layered carbon
fiber reinforced plastic (CFRP) composite plate and industrial structure with stiffeners.
Ihn and Chang [37, 38] used the short-time Fourier transform (STFT) to represent measured signals in the time-frequency domain. and defined a damage index feature:


tf

, t)|2 dt

α


 t |Ssc (ω0


DI =  i t



f
2
|Sb (ω0 , t)| dt

(2.15)

ti

where Ssc is the STFT of the scatter signal, which is the difference between the measured
and baseline time signals. Sb is the STFT if the baseline signal, ω0 is the selected excitation
frequency, tf and ti are the end and start of a time window, and α is a gain factor. They
applied this damage index for detecting cracks in a riveted aeroplane fuselage structure, and
cracks in a metallic plate repaired with a bonded composite patch.
There are two main methods for constructing an image using GW-PZT methods. In
general, an image is represented as a grid of pixels, and each pixel location is illuminated
based on some feature of the measured time signals from the PZT sparse array. The delayand-sum approach for constructing images uses a representation of the waveform in some
domain, usually either the time domain envelop or the time-frequency domain. Each time
sample of the signal represents a pixel location in the image, using a delay rule. For each
actuator-sensor path, the measured waveform is then transformed into an image using this
rule. Then, the images due to all the possible measured paths in the PZT network are added
to construct the final image. Ihn and Chang [39] used a delay-and-sum approach using
the short-time Fourier transform of the baseline subtracted time domain signals, to detect
damage in aluminum plates and in a stiffened composite panel. Michaels and Michaels [34,40]

23

used a broad band excitation and digital filtering of the time domain waveforms by convolving
with three cycle Hanning-windowed tone burst, to obtain a family of signals at different
frequencies. The time domain signals were subtracted from a baseline signal to obtain
damage information. An image was constructed using the delay-and-sum approach, and
tested for detecting defects in an aluminum plate.
Another imaging method is based on finding a single feature of the signal, that corresponds to a single time instance. If the velocity of wave propagation is known, this time
instance could be translated into an ellipse. By drawing the ellipses from all actuator-sensor
paths, the point where all the ellipses intersect indicates the damage location. Zhou et al. [41]
and Su et al. [42] used this method to detect holes in aluminum plates and delaminations in
carbon fiber reinforced epoxy composites.

2.6

PZT Sensor Diagnostics and Effects on GW Measurement

GW-PZT damage detection methods are based on information contained in the wave scatters
reflected or transmitted from defects. However changes in the measured signals due to faulty
PZTs might give false calls about the presence of damage, or they might fail to record data
properly and miss the detection of damage. Several methods have been suggested to enable
the PZT sensor to self diagnose and track any sensor fault that might give false indications
about the current health of the structure.
The adhesive (e.g. epoxy) between the PZT sensor and the structure has a significant
effect on the performance and the ability of actuation and sensing of guided waves. The
adhesive layer provides the necessary mechanical coupling to transfer the strains between
24

the PZT and the structure. It also keeps the PZT in place at the surface of the structure.
Qing et al. [43] conducted an experimental study on the effects of the adhesive layer thickness
and modulus on the piezoelectric impedance and sensed guided wave voltage amplitude. Ha
and Chang [44] studied the effects of the adhesive layer thickness on PZT-induced Lamb
waves.
The most common method to determine PZT integrity is measuring changes in its electric
impedance. At low frequencies, a PZT could be modeled as a capacitor [45]. The analytical
model of PZT impedance assuming perfect bonding between the PZT and the structure was
first proposed by Liang et al. [46]. Park et al. [47] used the analytical solution of a bonded
PZT impedance, supported by experimental results, to show that the imaginary part of the
PZT impedance (which is proportional to the capacitance), could be used to determine the
health of the PZT. Experimental results showed that a decrease in capacitance indicated
that the PZT sensor is broken, and an increase in capacitance indicates a bonding defect.
Then they extended their work [48] to show the effects of degrading sensors on guided wave
amplitude, phase, and time of flight. They argued that if signal processing methods based
on the wave attenuation or the time-of-flight information are used, changes due to bonding
defects could be mistakenly considered as structural damage.
Mulligan et al. [49, 50] investigated high frequency impedance metrics to asses and compensate degradation of adhesive layer of surface bonded piezoceramic transducers for SHM
applications. The investigated metrics include capacitance, resonant frequency, and damping ratio. To take into account the PZT resonance at high frequency, it was modeled as
resistor-capacitor (RC) circuit in series with inductor-resistor-capacitor (LRC) circuit. The
LRC circuit represents the first resonance of the PZT. The effects of PZT debonding and
changes in the adhesive properties were studied numerically using a FE model. Experimen25

tally, they investigated the effects of change of PZT bonding area by applying acetone to
the PZT, which reduces the bonding area by corroding the epoxy adhesive. They concluded
that the damping ratio gives the best sensitivity to changes in bonding area and adhesive
mechanical properties. A model based signal correction scheme for GW-PZT was then used
to modify the received signal amplitude and phase based on the PZT’s measured damping
ratio.
Overly et al. [51] proposed a method that tracks the capacitive value of PZT transducers,
which manifests in the imaginary part of the measured electrical admittance, to find an
outlier in a sensor array. The PZT sensors that causes the maximum change in the standard
deviation of admittance slope measurement of the combined average was assumed defective.
A different approach was proposed by Lee et al. [52], which does not require direct
comparison of measured signals with a baseline. The reciprocity of the guided wave signal
measurement, assuming the actuator and sensor have the same properties was used to detect
changes in the PZT. For each actuator-sensor path, another measurement is obtained with
the actuator and sensor flipped. The two measurements are symmetric if both actuator
and sensor have the same properties. A root square error quantifier was used to measure
the difference in the two signals. Then, in a network of PZT sensors, an outlier analysis
was conducted to find the defective PZT sensors. This approach does have limitations since
healthy PZT sensors might not always have the same properties, and hence make it difficult
to differentiate damage from PZT degradation.

26

Figure 2.7: The schematic of a centralized wireless sensor network consisting of two entities:
The sensor nodes and the base station. The sensor nodes are mounted on the structure and
connected to embedded or surface bonded transducers. The base station is a remote PC
that collects data from the sensor nodes for processing.

2.7

Wireless Sensor Networks

The simplest architecture of a Wireless Sensor Network (WSN) for SHM is the centralized
architecture, as shown in Figure Figure 2.7. There are two different entities in this architecture: the sensor nodes and the base station. The sensor nodes are smart low power
devices equipped with signal conditioning and data acquisition devices, microcontroller, digital memory, power supply, and a radio. Multiple sensor nodes are deployed on the structure
and connected to the PZTs, forming a wireless sensor network. The main power source in a
sensor node is a battery. Additional power supplies that could harvest energy from the environment or from other nearby devices could be used depending on the environment where the
27

sensors are deployed. Due to power scarcity, the computational speed, wireless transmission
power, and data acquisition speed are limited. The presence of a microcontroller enables
total control of the sensor node, and thus makes WSNs highly configurable and automated.
This is achieved by programming the microcontroller using software.
One base station communicates and controls the network of sensor nodes mounted on
the structure. The base station has a gateway that is responsible for transferring data from
the WSN to a PC and vice versa. The PC has a user interface where an administrator could
control and configure the network or individual sensor nodes. It also has visualization and
prognostic tools indicating the presence of potential hazards in the structure.

2.7.1

Sensor nodes

At the interface of the sensor node with the PZT, data acquisition circuitry are used to
transform signals from the transducer into a readable voltage signal that could be sampled
and digitized using an analog to digital (ADC) converter. The digital samples are then
stored in memory. The sensor node could then do further processing on the acquired data
or transmit them directly to the base station, depending on the application. If actuation is
needed, as is the case for guided wave SHM, an interface is used to change a digital signal
that is stored in the sensor node memory, or received from the base station, to an analog
signal that could drive the actuator.
A sensor node usually needs to have specialized hardware and software based on the
sensing application that it is used for. To overcome this issue, most commercial sensor nodes
are designed to be modular where a basic wireless module can be extended by connecting it
to external sensing modules depending on the application. The most common commercial
sensor node is the Mote platform initially developed at the University of California-Berkeley
28

Figure 2.8: Mote platform versions supplied by Memsic Inc., from left to right: Iris, micaZ,
and lotus motes.
[53], and commercialized by Memsic Inc. (www.memsic.com). Some examples of their
sensor nodes include the MicaZ mote, Iris mote, and the Lotus mote (Figure 2.8). The
review paper by Lynch and Loh [7] gives a general description of sensor node platforms
and specifically describes seventeen different sensor nodes hardware that were designed in
academic labs. The sensor nodes described were employed and tested for collecting vibrations
measurements mainly using accelerometers and strain gages. The use of sensor nodes for
global-based damage detection is becoming very popular, but there are few sensor nodes that
are compatible with guided wave SHM. Unlike global-based damage detection such as modal
analysis, guided wave SHM requires high data acquisition rates, where signal bandwidth
could be up to 1 MHz. Moreover, guided wave SHM is an active inspection method and it
requires an actuation interface.
Liu and Yuan [54] developed a sensor node that could be interfaced with PZT transducers.
They implemented a dual controller architecture, an FPGA directly controlling the ADC
and storing samples in memory, and a microcontroller unit controls other parts of the sensor
node. This allows a dedicated control of the sensing interface, increasing performance and
minimizing power consumption. However, their sensor node does not have an actuation
interface. Researchers at the University of California-San Diego [55] developed the Shimmer
sensor node, with the capability of actuating signals up to 1 MHz frequency, and sampling

29

frequency up to 10 MHz. The power of Shimmer is supplied solely by solar energy harvesting
cells. Pertsch et al. [56] developed an intelligent stand-alone ultrasonic device with and ADC
sampling rate around 8 MHz and actuation capability of signals with 1 MHz frequency. The
sensor node was tested by interfacing it with ultrasonic wedge transducers. However, to
provide the high performance needed, the sensor node requires large batteries and has an
overall large size compared to commercial sensor nodes.

2.7.2

Distributed embedded software

As shown in Figure 2.7, software needs to be installed in the computational core to control
the behavior of the sensor node. The software is usually divided into layers to simplify
the implementation, and separate the jobs of system designers from application designers.
Usually there are four different functional software layers: (1) Operating system; (2) Services;
(3) Application; (4) Wireless communication protocols.
The operating system is at the lowest level, and it has the implementation of hardware
drivers that allows upper layers to access the underlaying sensor node hardware without the
need to know their implementation details. Also the operating system can have a scheduler
that supports and organizes multiple application tasks being executed by the microprocessor.
The most widely used operating system for wireless sensor networks is TinyOS [57] developed
specifically for the mote platform, but has been extended to support multiple other platforms.
The advantage of TinyOS is that it is an open-source OS freely available for the public, and it
is intended to maximize the performance of low power applications. It needs only 256 bytes of
RAM to run, and it is written using NesC programming language [58], which is based on the
C programming language, but designed such that it is easier to write for applications that are
triggered by external environmental factors, which makes it easier to program applications
30

for wireless sensor networks.
Built upon the OS are the various network services that differ according to the application. Services enhance the performance of the overall network. Those services can run as a
distributed application in the network to coordinate the sensor nodes. For example in the
case of guided wave SHM, a data collection service would be needed to prevent multiple sensor nodes to transmit their data to the base station at the same time, reducing the network
throughput and increasing power consumption.

2.7.3

Communication protocols

The development of a reliable and energy-efficient protocol stack is important for WSN
applications. Sensor nodes use the protocol stack to communication with each other and the
base station. Various energy efficient protocols have been proposed for the transport layer,
network layer, data link layer, and physical layer. Yick et el. [59] have provided a review of
the various protocols for each layer.

31

Chapter 3
Finite Element Modeling of GW-PZT

3.1

Introduction

Model-based studies of the effects of PZT degradation on damage detection requires the use
of a validated finite element model. This chapter describes the FE model used for simulating
guided waves excited and sensed by PZT transducers. The Abaqus

FEM software is used

for this purpose. The coupled governing equations are described, and the implementation
of the model in Abaqus is verified analytically for a 2-D model and experimentally for the
3-D model. The effects of mesh size and the mesh element order is studied. The numerical
error using quadrilateral linear and biquadratic elements is compared for the 2-D model.
Quadratic brick elements are used in the 3-D model, and an optimal element density is
determined which is a compromise between computational time and solution accuracy. The
results of the 3-D model are then verified using experimental measurements.

3.2

Problem formulation

Physical modeling of PZT sensors adhesively bonded to a structure requires solving coupled
equations. Figure 3.1 shows the solution domain, which includes the adhesive and the plate
(ΩS ), and the PZT (ΩP ).
The equations of motion are solved over the domains ΩS+P . From the theory of linear
32

Ωp
ΩS

Figure 3.1: Domains for the multiphysics coupled model. The ΩS domain includes the
structure and the bonding adhesive. The ΩP domain includes the PZT.
elasticity, the equation of motion could be derived starting by Newton’s second law, expressed
in tensor form as:
∂σij
∂ 2u
= ρ 2i
∂xj
∂t

(3.1)

with the constitutive stress-strain relation:

σij = cijkl εkl

(3.2)

and the strain-displacement relation:

εkl =

1
2

∂ul
∂u
+ k
∂xk
∂xl

, where i, j, k, l = 1, 2, 3.

(3.3)

Here σij is the stress tensor, εkl is the strain tensor, ui is the displacement vector, ρ is
the material density, and cijkl is the stiffness tensor which represents the constant elastic
properties of the material. The stiffness tensor cijkl has symmetry properties, where the
following equalities should hold:

cijkl = cjikl = cijlk = cjilk

33

and
cijkl = cklij
To obtain the equation of motion in terms of the displacement ui entries only, substitute
(3.3) in (3.2), and then substitute the result in (3.1):

ρ

∂ 2 ul
∂ 2 ui
=
c
, in ΩS+P
ijkl
2
∂xj ∂xk
∂t

(3.4)

where the symmetry of the stiffness tensor cijkl has been used.
Gauss law for electric fields is solved over the domain ΩP , with no free charge present
within the domain:
Di,i = 0

(3.5)

The boundary conditions in the model are:

ui = ui

on ∂ΩS,P

σij ni = tj

on ∂ΩS,P
(3.6)
∂ΩP

V =V

on

Di ni = Q

on ∂ΩP

where ∂Ω represents the boundary of the domain, ui , ti , V , and Q are the prescribed
displacement, traction, voltage, and charge on the boundaries, respectively.
Equations 3.5 and 3.6 can be expressed by the weak form as:

ΩS,P

∂ 2 ui
∂ 2 ul
ρ 2 − cijkl
∂xj ∂xk
∂t

34

∂ui dΩ = 0

(3.7a)

ΩP

Di,i ∂V dΩ = 0

The weak form could then be discretized into finite elements. The Abaqus

(3.7b)
FEM software

is used for modeling GW-PZT, with dynamic implicit time integration. A zero voltage
boundary condition is applied on the PZTs ground plane at its nodes that are connected to
the adhesive bonding layer. A PZT is excited by applying an electric potential boundary
condition on the nodes on free surface of the PZT. For sensing measurements, the voltage
at the nodes of the top free surface of all PZTs in the model is recorded.
FEM methods for wave propagation problems have numerical dispersion errors due to the
accumulation of errors over time, as have been demonstrated by Mullen and Belytschko [60].
FEM studies have shown it is required for the maximum element size to be at least twenty
times smaller than the minimum wavelength of the propagating wave modes to obtain a
solution with a less than 1.0% dispersive error [61, 62].
Guided waves at frequencies around 200 kHz propagating in plate-like structures with
thickness around 4mm have a minimum wavelength in the order of several millimeters. The
fundamental antisymmetric mode (A0 ) has a lower velocity that the fundamental (S0 ) mode,
resulting in a smaller wavelength. Typical A0 wavelength ranges from 6 to 12 mm, and S0
mode from 12 to 50 mm. For linear quadrilateral mesh elements, this requires a maximum
element size of approximately 0.5 mm. The next section shows the effects of mesh size on
the error, for both linear and quadratic elements.

35

3.5mm

4mm

Z
X

Figure 3.2: The configuration for the 2D aluminum plate mesh.

3.3
3.3.1

FEM Mesh Validation
Analytical Validation using a 2-D Model

A two-dimensional finite element model using Abaqus

FEM software was firstly developed

to obtain a quick guideline for the required mesh size with an acceptable error. The 2D
analytical model proposed by Giurgiutiu et al. [16] was used to validate the FE model.
Figure 3.2 shows the schematic for the 2-D finite element model of an aluminum plate
used to investigate the effects of the mesh size on the accuracy. Utilizing the axisymmetric
nature of this problem, only half the PZT actuator and plate length were modeled. A zero
displacement boundary condition along the x-direction was imposed at the left boundary.
The 2D effective PZT force model [16] was utilized to model the effective force of a perfectly
bonded PZT actuator, which is given by:

τ (x, t) =

Es ts Ea d31 V0
[δ(x − a) − δ(x + a)] f (t)
Es ts + 4Ea ta

(3.8)

where Es and ts are the Young’s modulus and thickness of the plate, Ea , ta , d31 are the
Young’s modulus, thickness, and piezoelectric coupling coefficient of the PZT actuator, respectively. The properties used for aluminum and PZT are shown in Appendix A. V0 is

36

−4

x 10

1
0.5

|f˜(ω)|

4

0

2

−0.5
−1
0

5

10
15
time (µ s)

20

25

(a)

0
0

100

200
300
400
Frequency (kHz)

500

(b)

Figure 3.3: The actuation signal in (a) time domain, and (b) its Fourier transform.
the applied voltage across the PZT, and it was set to 10 volts. The modeled PZT actuator
has a length of 7mm (3.5mm half length). Equation (3.8) shows that the effective force is
concentrated at the tips of the PZT. For the axisymmetric model, this effective force is a
concentrated force at a single node as shown in Figure 3.2. The time signal f (t) was applied
as a 200 kHz modulated Gaussian tone-burst, with 50% frequency bandwidth. The actuation
waveform f (t) and its Fourier transform are shown in Figure 3.3. The maximum frequency
content in the actuation signal, as seen by Figure 3.3b is 300 kHz. From the dispersion
equations for the aluminum material used (see Figure 2.4 and equations (2.6) and (2.5)),
the minimum wavelength for the A0 mode is approximately λA0 = 8mm, and for the S0
mode is λS0 = 17mm.

The wave displacement at a distance of 20cm from the center of

the actuator was recorded for the nodes at the top and bottom surface of the plate. When
the measured waveforms at the two nodes are subtracted, only the antisymmetric modes
are obtained. When they are added, only the symmetric modes are obtained. This allows
investigating the mesh effect on each mode separately. The total length of the 2D plate was
such that the reflections from the right edge would not overlap with the incident waveforms
at the sensing points.
37

Displacement (nm)

Displacement (nm)

1

Analytical
0.5 0.1mm
0.5mm
1.0mm
0
2.0mm
−0.5
50

60

70
80
Time (µs)

Analytical
0.1mm
0.5
0.5mm
1.0mm
0
2.0mm
−0.5
−1
20

90

(a) A0 mode

30

40
50
Time (µs)

60

70

(b) S0 mode

Figure 3.4: The resulting waveforms for different mesh element sizes, with 4-nodes linear
quadrilateral elements and for the analytical solution.
A uniform mesh was used for the plate geometry. The mesh element size was varied
from 0.1 mm to 2 mm using 4 node plane strain linear elements. The resulting waveforms
when the element size is 0.1mm, 5mm, 1mm, and 2mm are shown in Figure 3.4(a) for the
A0 mode and in Figure 3.4(b) for the S0 mode. The analytical waveforms are also shown.
Since the A0 mode has a smaller wavelength than the S0 mode, it is affected more by the
element size, and numerical dispersion is more apparent at smaller mesh element sizes. The
same investigation was then conducted for 8 node plane strain biquadratic quadrilateral
mesh elements, with sizes varying from 0.1mm to 2mm. The resulting waveforms for some
of the mesh element sizes and the analytical solution are shown in Figure 3.5. Unlike the
linear elements, the biquadratic elements do not suffer much from distortion, even at large
mesh element size. At 2mm mesh element size, which is just 4 times smaller than the A0
wavelength and 8.5 times smaller than the S0 mode, the error is still negligible, compared
to the error in the linear mesh elements.

The mesh error is quantitatively evaluated by

comparing the FEM simulation results for different mesh sizes with the analytical result,
which is assumed to be the reference waveform. The error is quantified using the method

38

Displacement (nm)

Displacement (nm)

Analytical
0.5 0.1mm
0.5mm
1.0mm
0
2.0mm
−0.5
50

60

70
80
Time (µs)

Analytical
0.1mm
0.5
0.5mm
1.0mm
0
2.0mm
−0.5
−1
20

90

30

(a) A0 mode

40
50
Time (µs)

60

70

(b) S0 mode

Figure 3.5: The resulting waveforms for the analytical solution and for different mesh element
sizes, with 8-nodes biquadratic quadrilateral elements.
described by Lonkar [63]. The envelop of the waveforms is computed using the Hilbert
transform (H), and the percentage error could then be computed such that:
T

% error =

0

|H(ssim (t))| − H(sref (t))

dt
× 100

T
0

2

(3.9)

2

H(sref (t)) dt

where sref (t) is the reference waveform (the analytical result in this case), and ssim (t) is the
simulation waveform. The evaluated errors with the element sizes ranging from 0.1mm to
2mm for both linear elements and biquadratic elements are shown in Figure 3.6(a) for the A0
mode and in Figure 3.6(b) for the S0 mode. The meshing error becomes significant when the
linear mesh element size becomes larger than 0.5mm, however the errors for the biquadratic
mesh elements stays close to zero, even when the mesh size is 2mm, which corresponds to
only 4 elements per wavelength for the A0 mode, and 8.5 elements per wavelength for the S0
mode. It is apparent that using biquadratic elements is more suitable for wave propagation
problems, and this type of elements were then used for the 3-D FEM simulations.

39

% Error

% Error

60

Quadratic Elements
Linear Elements

40
20
0.5

1
1.5
Element size (mm)

2

(a) A0 mode

14
12
10
8
6
4
2

Quadratic Elements
Linear Elements

0.5

1
1.5
Element size (mm)

2

(b) S0 mode

Figure 3.6: The resulting numerical mesh error with different mesh element sizes for both
linear and biquadratic elements.

Figure 3.7: Cross-sectional view of the three-dimensional model, showing the co-located PZT
sensors on the top and bottom surfaces of the plate.

3.3.2

Validation of the 3-D Aluminum Plate Mesh

A three-dimensional finite element model using the coupled model that includes the actual
PZT sensor and the thin adhesive bonding layer that connects it to the plate was developed.
The model geometry is shown in Figure 3.7. The material properties for the PZT, the
adhesive bonding layer, and the aluminum plate are given in Appendix A. The PZT thickness

40

(a)

(b)

Figure 3.8: (a) The three-dimensional FEM simulation for mesh verification of an Aluminum
plate, with two circular PZTs bonded on the top. (b) Three-dimensional closeup of the PZT
and the thin adhesive bonding layer.
0.8mm
1.5mm
5
2.0mm
3.0mm
0

Voltage (mV)

Voltage (mV)

10

−5

4 0.8mm
1.5mm
2 2.0mm
3.0mm
0
−2
−4

−10

40

60
Time (µs)

0

80

(a) A0 mode

20

40
60
Time (µs)

80

(b) S0 mode

Figure 3.9: The resulting waveforms for the 3-D mesh with different element sizes.
is 0.2mm, with a diameter of 7mm. The epoxy adhesive layer thickness is modeled to be
60µm. Two co-located PZT sensors (see Figure 3.8) are bonded 15cm apart. PZT-A is
excited using the signal shown in Figure 3.3 at 200 kHz, and the responses at the top and
bottom PZT-B are recorded to obtain the A0 and S0 modes separately, as described in
Section 3.3.1.
To test the numerical error effects due to the mesh element size, 20-node biquadratic
elements were used and their sizes was varied between 0.8mm and 3mm. A total simulation
time of 100µs with 100ns time steps was used. The waveform results measured at PZTB are shown in Figure 3.9 for element sizes 0.8mm, 1.5mm, 2mm, and 3mm. There is
little numerical dispersion apparent in the S0 mode results (see Figure 3.9(b)). At the
largest element size used, with size 3mm, there are approximately 6 elements per wavelength.
41

0.4
% Error

% Error

60
40
20
0

0.3
0.2
0.1

1

1.5
2
2.5
Element size (mm)

3

(a) A0 mode

0

1

1.5
2
2.5
Element size (mm)

3

(b) S0 mode

Figure 3.10: Numerical error due to increasing mesh element size.
However for the A0 mode, there are only 2 elements per wavelength, and the numerical
dispersion is high when the element size is 3mm (see Figure 3.9(a)). To quantify the error
due to increasing mesh size, the error in the waveforms with respect to the waveform with the
smallest mesh element sizes was computed using equation (3.9). Thus the reference waveform
sref (t) in this case is the measured waveform using the mesh with 0.8mm element size. The
resulting errors are shown in Figure 3.10. When the mesh element size is more that 2mm, the
numerical error in the A0 mode is greater than 3%, and that of S0 is still very low at 0.15%.
To select the optimum mesh element size that could be used at the operating frequency, the
simulation running times for different mesh size are compared, as shown in Figure 3.11. The
FEM simulations are run using parallel computing, and are allowed to use 8 CPU cores on
a 16 core machine, with 500 GB of memory. The computing time increases exponentially
as the element size decreases. To balance between computing time and meshing numerical
errors, a mesh element size of 1.5mm are used for all subsequent simulations of aluminum
plates at 200kHz excitation frequency in this dissertation. This corresponds to 0.46% error
in the A0 mode, and 0.01% error in the S0 mode, which is virtually negligible.

42

Time (minutes)

500
400
300
200
100
1

1.5
2
2.5
Element size (mm)

3

Figure 3.11: Simulation running time versus mesh element size.

3.3.3

Validation of the 3-D Composite Plate Mesh

In fiber-reinforced composite material such as fiber-glass reinforced plastics, wave propagation is further complicated by the heterogeneity and anisotropy. Wave properties depend
on the direction of propagation, and the guided waves scatter at the interfaces of the fibers
and the matrix. Tauchert and Guzelsu [64] measured the ultrasonic wave scattering in
boron/epoxy composites at frequencies where the corresponding wavelengths were small
enough to produce scattering. For extensional waves, significant scattering occurred at a
wavelength on the order of fiber diameter. For flexural waves, scattering appeared when
the wavelength to fiber diameter ratio is 40. Considering that typical fiber diameter ranges
from 5-25 µm, and typical guided wave frequencies are below 1 MHz, where the wavelength
is not less than 5 mm for most material, it is safe to assume no scattering occurs. Then
each lamina can be treated as macroscopically homogeneous and orthotropic. Using this
assumption, a woven glass fiber reinforced plastic (GFRP) plate was modeled as a single
homogeneous layer, with mechanical properties given in Appendix A.
To verify that the mesh element size requirements that were obtained for the aluminum

43

1.5mm
2.0mm
5 2.5mm
3.0mm
0

Voltage (mV)

Voltage (mV)

10

−5
0

20

40
60
Time (µs)

5 1.5mm
2.0mm
2.5mm
3.0mm
0

−5
0

80

(a) Antisymmetric modes

20

40
60
Time (µs)

80

(b) Symmetric mode

Figure 3.12: Resulting waveforms with different mesh element size for a composite plate.

0.1
% Error

% Error

0.6
0.4

0.05

0.2
0
1.5

2
2.5
Element size (mm)

3

(a) Antisymmetric modes

0
1.5

2
2.5
Element size (mm)

3

(b) Symmetric modes

Figure 3.13: Numerical error due to increasing mesh element size for a composite plate.
plate are also acceptable for the composite material, a plate with 4mm thickness and two
PZTs with a distance of 10cm between them were modeled. Even though the composite
plate could be assumed to be homogeneous, wave attenuation due to the material is quite
considerable in composites. This viscous attenuation is modeled using proportional Rayleigh
damping, similar to that used by Ramadas et al. [65]. A mass proportional damping ratio
α = 80250 Np/s is used, and the stiffness damping ratio β = 0. The resulting waveforms for
element sizes varying from 1.5mm to 3.0mm are shown in Figure 3.12 for both symmetric
and antisymmetric modes. The numerical dispersion for both modes is not as apparent as

44

(a)

(b)

Figure 3.14: (a) Experimental Setup. (b) Experimental setup showing a closeup on the PZT
sensors.
that in an aluminum plate. The numerical errors with respect to a 1.5mm element sizes
are shown in Figure 3.13. The numerical error for the largest element sizes is still less than
1%. Consequently, a mesh element size of 2.5mm are used for all subsequent simulations of
GFRP plates at 200kHz excitation frequency in this dissertation.

3.4

Experimental Validation

The 3-D FEM model for the aluminum plate is verified experimentally. The experimental
setup is shown in Figure 3.14. Six circular PZTs with a radius of 7mm are bonded on the

45

% Error Average

25

S0 mode
A0 mode

20
15
10
5
0

1

2
3
4
5
6
7
8
FEM bonding (thickness, modulus) index

9

Figure 3.15: Error as compared with experiment.
top of an aluminum plate. A seventh circular PZT (the sensor) is bonded at a distance
of 15cm from all the other PZTs. Each of the six PZTs are actuated with the waveform
shown in Figure 3.3(a) at 200 kHz, with a peak-to-peak voltage of 6 volts, using a function
generator. The waveforms are recorded by the same PZT sensor using an oscilloscope, which
is connected directly to the PZT. No amplification on the sensed signal was used, so that
the electric potential difference across the PZT could be directly compared to the FEM
measurements without the need to scaling and normalization. The aluminum plate has 2
mm thickness. The mechanical properties of aluminum and and PZT are given in Appendix
A. As for the adhesive bonding layer, the PZTs were bonded using super-glue, but the
mechanical properties and thickness of the bonding layer are not known exactly, and it is
difficult to measure them. A range of FEM simulations using different values for the Young’s
modulus and the thickness for the bonding layer were used. A set of three values for the
Young’s modulus was selected (1.3 GPa, 2.0 GPa, 2.6 GPa), and a set of three bonding
thickness was selected (30 µm, 60 µm, 90 µm), for a total of 9 simulations.
Since there are a total of 9 simulation results, and 6 different experimental measurements,
46

Table 3.1: Indices for the bonding thickness and Young’s modulus (Y) sets used in the FEM
simulations for comparison with experiment.
Index

Thickness (µm)

Y (GPa)

1

30

1.3

2

60

1.3

3

90

1.3

4

30

2.0

5

60

2.0

6

90

2.0

7

30

2.6

8

60

2.6

9

90

2.6

the error for each FEM simulation is calculated by finding the error using equation (3.9)
with respect to each of the experimental waveforms and then the average error over all the
experimental measurements is computed, as shown in Figure 3.15. The bonding thickness
and Young’s modulus sets and their corresponding indices as used in the horizontal axis in
Figure 3.15 are shown in Table 3.1. The smallest error for both the A0 and S0 modes is
when the bonding thickness is 30 µm and the bonding Young’s modulus is 1.3 GPa, where
the error for both modes is approximately 1.75 %. The corresponding waveforms for the
experimental data and the FEM simulation corresponding to the bonding properties with
the least error are shown in Figure 3.16.

3.5

Conclusion

A 3-D FEM model for wave propagation in aluminum and GFRP plates, using adhesively
bonded PZT transducers was implemented using Abaqus

FEM software. An optimized

mesh density was found that minimizes meshing numerical errors, and has an acceptable

47

Voltage (mV)

10

FEM
Experiment

0
−10
20

40

60
Time (µ s)

80

Figure 3.16: Experimental waveform comparison with FEM results.
running time. The model is successfully validated experimentally for the aluminum plate
model, with an average error of less than 2%. These models are used in subsequent chapters
to conduct parametric studies to investigate the effects of PZT sensor variations on the
detection of damage, and the assessment of the performance of GW-PZT SHM methods.

48

Chapter 4
GW Propagation in Structures

4.1

Introduction

Aeronautical components such as aircraft wings and riveted fuselage lap joints undergo
fatigue damage that occurs around fastener holes due to mechanical stresses. Continued
stresses on the structure result in fatigue cracks emanating from the rivet hole. Typically,
these structures are multi-layered, as shown in Figure 4.1, and are fastened together by rivets.
Undetected cracks hidden at fastener sites in layered structures can lead to catastrophic
failures. Fastened multi-layered structures have tiny air gaps between the layers. PZT
sensors mounted on the top surface layer would not be able to propagate to the lower
layers, especially using the very low energy excitation which is typical for such GW-PZT
applications. However, since the thickness of the PZT sensor is on the same order of the
inter-layer air gaps (around 100 µm), a riveted structure could be instrumented with a spare
PZT network at each structural layer for the continuous monitoring of fatigue cracks growth.
This chapter describes the FEM model geometry used to simulate wave propagation in a
single layer riveted aluminum plate.
The high stiffness to mass ratio of fiber reinforced composites compared to metallic
materials makes them resistant to fatigue and corrosive damage. The use of composites in
aeronatical and naval industries have significantly increased in the past decade. However,

49

Figure 4.1: A three layered structure with a rivet, and a notch crack at the bottom layer.
typical laminated composites are susceptible to invisible delamination due to impacts and
fatigue. The FEM model geometry and wave propagation in a glass fiber reinforced epoxy,
and its interaction with delamination is described in this chapter.

4.2

GW Propagation in a Riveted Aluminum Plate

The geometry and dimensions of the modeled aluminum plate with a rivet hole is shown in
Figure 4.2. The modeling of the actual rivet is not required, since the small air gap between
the rivet and the plate will not allow the wave to be transmitted into the rivet. Eight circular
PZT sensors are placed at a distance 60 mm from the rivet hole center, and distributed
uniformly at 45◦ angle intervals around the rivet hole. A closeup of a PZT mesh and its
bonding layer is shown in Figure 4.3(b). Colocated sensors are placed on the top and bottom
surfaces of the plate (a total of 16 PZT sensors are present, sensing 8 locations). When the
measurements of the top and bottom sensors are added, only antisymmetric modes are
obtained, and if they are subtracted, only the symmetric modes are obtained. The material

50

3
2

400 mm

4
Rivet

5
6

1
8

7

y

x
400 mm
Figure 4.2: The FEM model geometry for a riveted plate. The rivet hole is 10mm in diameter.
There are eight PZT sensors at each surface of the plate, situated at an equal distance of
60mm from the center of the rivet.

(a)

(b)

Figure 4.3: A closeup of the (a) defect mesh, and (b) PZT and bonding mesh.
properties for the aluminum plate, epoxy adhesive bonding, and the PZT sensors are given
in Appendix A. The plate, PZT, and adhesive bonding thicknesses are 4mm, 0.2mm, and 60
µm, respectively. A crack emanating from the rivet hole, in the x−direction is modeled as
a thin detachment of the mesh nodes at that location, as shown in Figure 4.3(a), where the
crack width is 100 µm.

51

A 200 kHz modulated Gaussian tone-burst, with 50% frequency bandwidth is excited at
PZT-3 on the top surface of the plate. The actuation waveform and its Fourier transform
are shown in Figure 3.3. The FEM simulation was run for a total of 85 µs, with 100 ns time
steps. The magnitude of the particle displacement at the surface of the plate at a time 40
µs is shown in Figure 4.4(a) for the case when there is no crack near the rivet hole. Figure
4.4(b) shows the wave propagation at the same time, with a crack of length 6mm at the rivet
hole. Due to the presence of the rivet hole, the scatter of the incident wave at the rivet hole
would interfere with the scatter from the defect. The reflected/transmitted signal received
at the PZT sensors will contain overlapping waveforms due to the incident waveform, rivet
scatter, and defect scatter. This is the main motivation behind baseline subtraction.
At the excitation frequency of 200 KHz, only the fundamental symmetric and antisymmetric modes are present. The group velocity of the wave modes could be obtained using the dispersion equation given in equations (2.5), (2.6), and (2.8). The group velocity
of the fundamental symmetric mode S0 and the fundamental antisymmetric mode A0 are
vS0 = 5290 m/s and vA0 = 2764 m/s respectively. The wave group velocity of the S0 mode
is almost two times faster than the A0 mode, and it could be seen in Figure 4.4 that the wave
S0 wave packet leads the A0 . This difference in velocity also means that the wavelength of
the A0 is almost half that of the S0 mode. Thus, in the presence of defect (Figure 4.4(b)),
the defect dimension to wavelength ratio will be higher for the A0 mode, resulting in a larger
sensitivity to defects. The time history waveforms recorded at the PZT sensors due to the
excitation of PZT-3 are shown in Figure B.1 for the S0 mode and B.3 for the A0 mode, in
Appendix B.

52

(a) Plate without defect

(b) Plate with defect

Figure 4.4: Wave propagation in a riveted aluminum plate at time 40 µs.

53

80 mm

400 mm

25 mm
2 4 6 8 10

1 3 5 7 9

400 mm

Figure 4.5: The configuration for the FEM model geometry for a GFRP. There are ten PZT
sensors at each surface of the plate, distributed into two rows.

(a)

(b)

Figure 4.6: (a) Cross-sectional view of the delamination area located midway through the
plate thickness, shown within the circled area. (b) Close-up of the delamination area.

4.3

Guided wave propagation in a Glass Fiber Reinforced Plate

A 400×400 mm composite plate is modeled as shown in Figure 4.5. The plate material
properties are given in Appendix A. Those material properties are based on the measurements
of a homogenized 8 layer woven glass fiber fabric reinforced plastic (GFRP) composite plate.
Ten circular PZT sensors are placed on each surface of the plate, for a total of 20 sensors.
54

The PZT sensors are distributed in two rows 80mm apart and the distance between the
centers of PZT sensors in each row is 25mm. The PZT number labels are given in Figure
4.5. The material properties for the epoxy adhesive bonding and the PZT sensors are given
in Appendix A. The viscous attenuation is modeled using proportional Rayleigh damping.
A mass proportional damping ratio α = 80250N p/s is used, and the stiffness damping ratio
β = 0. The plate, PZT, and adhesive bonding thicknesses are 4.8mm, 0.2mm, and 60 µm,
respectively. A circular delamination area is modeled as a thin detachment of the mesh
nodes at the mid-point of the thickness of the plate, as shown in Figure 4.6. The maximum
thickness of the delamination area is 100 µm. The same procedure described in Section 4.2
is used to obtain only symmetric and antisymmetric modes from the measurement of the
PZT sensors.
A 100 kHz modulated Gaussian tone-burst, with 50% frequency bandwidth is excited at
PZT-5 on the top surface of the plate. The FEM simulation was run for a total of 200 µs, with
100 ns time steps. The magnitude of the particle displacement at the surface of the plate at
a time 70 µs is shown in Figure 4.4(a) for the case when there is no delamination. Due to the
anisotropy of the material, the wave velocity depends on the direction of propagation. From
Figure 4.7, it could be seen that the velocity of S0 mode is dependent on direction more than
the A0 mode, due to the non-circular shape of the wave front. The wave group velocities for
this composite structure is obtained using the FEM model. Since the wave speed depends on
propagation direction, the group velocity is calculated for each actuator-sensor path in the
geometry, for a total of nine paths considering only PZT-5 as an actuator. For a given path,
the difference between the peak time of the envelop of the actuation signal (ta ) and that of
the incident wave (ts ) gives the time of flight of the wave packet. An example waveform for

55

(a) No defect

(b) Delamination with 6 mm diameter

Figure 4.7: Wave propagation in a GFRP plate at time 70 µs.

56

1

ta

TOF

ts
Actuator
Sensor

0.5
0
0.5
1
0

50

100
time (µ s)

150

200

Figure 4.8: Calculating the time of flight (TOF) for obtaining the wave mode velocity.
this calculation is shown in Figure 4.8. Then the wave group velocity could be calculated as:

vi =

di
ts − ta

(4.1)

The obtained wave group velocity for the A0 and the S0 modes are shown in Table 4.1 for
each actuator-sensor path, with PZT-5 as actuator. The A0 mode has the same velocity for
all sensor paths, and the average velocity could be considered for all paths. However, for the
S0 mode, the velocity is direction dependent and each path has a different velocity.
Figure 4.7(b) shows the wave propagation at time 70 µs, with a small circular delamination diameter of 6 mm at 2.4 mm depth through the thickness of the plate. The center of
the delamination area is located such that it is halfway between PZT-5 and PZT-6. It could
be seen in Figure 4.7(b) that there is a small scatter of the A0 mode due to the presence
of the delamination. However, it seams that the S0 mode is not signifcantly affected by the
delamination. The time history waveforms recorded at the PZT sensors due to the excitation
of PZT-5 are shown in Figure B.5 for the S0 mode and B.7 for the A0 mode, in Appendix
B.
57

Table 4.1: Wave group velocities of the A0 and the S0 modes for different actuator-sensor
paths, with PZT-5 as actuator.
Sensor

vS (m/s)

vA (m/s)

1

3783

1177

2

3379

1184

3

4155

1157

4

3534

1189

6

3834

1157

7

4155

1157

8

3534

1189

9

3783

1177

10

3379

1184

Average

-

1179

58

Chapter 5
Formulation of a Stochastic Method
for Performance Evaluation

5.1

Introduction

GW-PZT SHM detects changes in a response measurement by comparing it to previous
response of the system in its pristine state. These changes indicate presence of damage in
the structure, or they could also be due to a variety of factors that would influence the
measurement. It is important to quantify the reliability of an SHM system in terms of its
ability to consistently detect defects of a critical size, and at the same time, minimize the
number of false positives. An SHM system is subjected to different variables that could affect
the measured signal, and the signals generated by the same flaw at different times could be
different. A baseline waveform or a set of baseline waveforms are recorded for each PZT
actuator-sensor path. A measured waveform that is different from the baseline would then
either indicate the presence of a defect in the structure that needs immediate attention, or
a fault in the PZT actuator and sensor.
In traditional non-destructive evaluation (NDE), the reliability of a certain technique
is assessed using probability of detection (POD) concepts, as prescribed in the military
handbook for NDE reliability assessment [66]. POD model uses statistical predictions of the

59

output decision based on variabilities that affect the measurements. In NDE inspections,
these variabilities are mostly due to human and instrument factors, such as variations in the
experiment setup, instrumentation noise, and the varying levels of experience and training
between individual testers. POD procedures are prescribed to analyze data from experiments
on real structures, requiring the fabrication of multiple costly samples that could replicate
the physical geometry and conditions of interest. An SHM system comprises of sensors that
are permanently mounted on a structure, and the whole SHM-structure system is unique.
Precisely reproducing such a system in experimental procedures could become prohibitively
costly and difficult. Moreover, the sources of variability are difficult to control and reproduce
experimentally. Model-assisted probability of detection, in which computer models are used
to simulate the variabilities and results for a given SHM system could significantly decrease
time and costs of assessing reliability of monitoring.
Variabilities that affect the performance of an SHM system are more diverse than in
NDE. Although the human factor effects are minimized since an SHM system is usually
autonomous, it is affected by the same environment that the structure encounters which
could result in performance degradation of the PZT sensors over time [67]. This chapter
identifies PZT degradation modes, and the consequent variabilities that affect the system’s
defect detection. A novel POD concept for quantifying the reliability in terms of sensor life
is described.

5.2

PZT degradation modes

A PZT sensor network surface mounted on a structure is composed of two components that
are prone to failure: (i) the adhesive bonding layer between the PZT and the structure, and

60

(ii) the PZT sensor itself.
Humidity, temperature variations, and loading rate influences the adhesive bonding layer
durability over long time periods. Banea et al. [68] had shown that the tensile strength and
Young’s modulus of epoxy adhesives vary with temperature and loading rate. For example,
the Young’s modulus of an XN1244 epoxy adhesive linearly decreased with temperature,
while it logarithmically increased with applied stress strain. Chemical changes result in
erosion and crack initiation in the adhesive, reducing its coverage area. Apart from the slow
degradation of the bonding layer, impacts could result in reduction of bonding layer coverage
due to structural deformation and delamination due to shearing motion [50]. Moreover,
impact could result in the breaking of the PZT sensors, reducing the active area of the PZT
sensor. Based on those observations, the effects of three parameters on the probability of
detection in GW-PZT SHM are studied:
1. Variation in the Young’s modulus of the adhesive bonding layer between the PZT and
the structure.
2. Variation in the thickness of the adhesive bonding layer between the PZT and the
structure.
3. Variations in the bonding layer coverage area.
Variabilities in the PZT and its adhesive layer is classified into two categories: (i) Shortterm variations, and (ii) long-term variations. Short-term variations are non-permanent
changes in the properties of the PZT due to changes in temperature, humidity and loading.
Long-term variations are permanent changes in the material properties to incidents such as
corrosion and impacts. Each of the variations in the PZT and adhesive properties mentioned
above are modeled as a stochastic random variable. Given a variability ξ which could vary
61

p ( )

μξ

ξmin

ξmax

ξ

Figure 5.1: The truncated variability probability density function.
over the range [ξmin , ξmax ], the probability density function of this variability is a truncated
Gaussian distribution, with mean µξ (T ), and variance σξ2 (T ) (Figure 5.1):

pξ (ξ) =

√
σξ (T ) 2π erf

(ξ−µξ (T ))2
2 exp −
2σ 2 (T )
ξ
ξmin −µξ (T )
√
− erf
σξ (T ) 2

ξmax −µξ (T )
√
σξ (T ) 2

(5.1)

and erf is the error function given by:
x
1
erf(x) = √
exp(−t2 ) dt
π −x

(5.2)

The long-term effects are modeled by the changes in the mean µξ (T ). Note that µξ (T )
is a function of time T , however capital T denotes that this change occurs over an extended
period of time, usually much longer than the time of a single GW-PZT inspection and data
acquisition. The variance σξ2 (T ) of the random variable reflect the short-term variations. The
variance also changes over the long-term time T , indicating that the short-term variations
are also dependent on the long-term variations. In the following sections, µξ (T ) is referred

62

Figure 5.2: Representation of the waveforms due to direct path from Actuator-Sensor and
due to the scatter from a defect.
to as simply µξ and σξ (T ) as σξ , and it is understood that these statistics are a function of
the long-term time T.

5.3

Signal Representation

In a sparse network of K PZT sensors surface bonded to a structure, consider a single path
(called path i hereafter) which involves two PZTs, one is the actuator and one is the sensor.
The actuator is excited with a voltage signal s(t), and a guided wave propagates through
the structure. The guided wave received at the sensor is composed of two super-positioned
waveforms: (i) The waveform received due to the direct path between the actuator and
sensor, and its corresponding echoes due to structural edges and features. This waveform is
denoted by sai (t) in Figure 5.2. (ii) The waveform due to the scatter of the incident wave at
the defect. This waveform is denoted by sdi (t) in Figure 5.2. The total received signal for
path i:
ri (t) = sdi (t) + sai (t)
=

cdi (t; ξ)

s(t) +
63

(5.3)
cai (t; ξ)

s(t)

Incident path

s(t)

Defect path

sai (t)
ri(t)
sd
i (t)

Figure 5.3: Block diagram of the different components of the received waveforms passing
through different channels.
where

is the convolution operation. Equation (5.3) models the different components of

the waveform as passing through a channel, which is illustrated by the block diagram in
Figure 5.3. The block diagram shows the direct path as a channel with an impulse response
function cai (t; ξ) which is a function of the PZT and bonding layer properties, denoted by
the vector ξ. Similarly, the scatter from defect is modeled as passing through a channel
with impulse response function cdi (t; ξ). Note that in practice there is a zero mean additive
white Gaussian noise due to thermal noise in measurement equipment and other variations
of the signal not related to the sensor degradation. However, in this study, this noise term
is neglected in order to focus on the effects of sensor degradation on detection.
Each of the K sensors in the network has a record of baseline waveforms for the path
that it is involved in. The baseline waveform for path i is written as:

sbi (t) = cai (t; ξ0 )

s(t)

(5.4)

where the baseline waveforms are taken when the PZT sensors properties are given by ξ0
which could be different from the properties ξ during an inspection. The baseline waveform
is the expected measurement when there is no flaw in the structure.

64

Sometimes, a dictionary of baseline waveforms is used, to take into account different
circumstances that can be encountered. For example, use of different baseline corresponding
to pristine structure measurements at different temperatures is reported in [34] . Then the
optimal baseline is selected based on a minimum square error criterion between the measured
signal and the baseline signal. However, this method does not always result in the optimal
baseline signal, due to possible phase differences in the scattered wave from the defect, and
the distortion signal due to nonideal baseline subtraction, as shown in Figure 5.4.
After applying baseline subtraction, where equation (5.4) is subtracted from equation
(5.3), the resulting difference waveform is:

∆ri (t) = cdi (t; ξ)
=

sdi (t)

+

s(t) + cai (t; ξ)

s(t) − cai (t; ξ0 )

s(t)
(5.5)

∆sai (t)

which includes the scatter from defect sdi (t), which is non-zero only if a defect is present.
Likewise the difference in the direct path waveforms ∆sai (t) is non-zero only if ξ0 = ξ,
indicating changes in the PZT and adhesive properties.
After baseline subtraction, some kind of signal feature needs to be examined, based
on which a decision is made regarding the presence or absence of damage. As described in
Section 2.5, different signal features have been examined in the literature, either based on the
time domain amplitude and temporal correlation with the baseline signal, or by examining
the spectral properties of the signal in the frequency domain. However, most of the methods
used for detection are deterministic, that are derived so as to minimize the contribution of
the term ∆sai (t) in equation (5.5).
Due to its simplicity, a modified version of the time domain feature given in equation (2.9)

65

is used in this formulation. By taking into account the stochastic nature of this problem,
the effectiveness of using such a simple feature is studied, and improved, making this single
feature sufficient for reliable defect detection under changing ξ. Instead of considering the
time domain signal directly as in equation (2.9), the signal envelops are taken. Since the exact
sinusoidal modulation components of equation (5.5) are not known, the Hilbert transform
(H) is utilized to obtain the signal envelop, and the signal feature that is used for detection
could then be represented as:
tf

yi =

0
tf

|H (∆ri (t))| dt
(5.6)
H

0

sbi (t)

dt

where tf is the total time for which the signal is integrated. The minimum value of yi is
zero, which is when no defect nor PZT degradation is present. The maximum value for yi is
1. Note that when no PZT degradation is present, the non-zero value of yi is solely due to
the presence of defect. Likewise, when no defect is present, the non-zero value of yi is solely
due to PZT degradation. When both defect and PZT degradation are present, then there
is a contribution due to both. Then the effect of PZT degradation is treated as a distortion
noise, and the signal feature is further simplified as:

yi = a
ˆ i + ni

(5.7)

where a
ˆi and ni represent the components of the obtained feature due to defect and sensor
degradation, respectively. Hence, yi is called the signal feature, a
ˆi is called the damage
feature, and ni is called the distortion feature, or distortion noise. Note that if there is
no damage then a
ˆi = 0, and if there is no change in sensor properties (there is no error), then
66

Voltage (mV)

2
0

−2
20

30

40
Time (µs)

50

60

Figure 5.4: Example of a received waveform after baseline subtraction ∆r(t). It is composed
of the superposition of two waveforms: Distortion due to sensor degradation ∆sa (t), and
scatter from defect sd (t).
ni = 0. As an example, Figure 5.4 shows the different components of the baseline subtracted
waveform, and how their superposition affects the feature amplitude. Moreover, as can be
seen from Figure 5.3 and in equation (5.5), the waveforms sdi (t) and sai (t) are a function of
the vector ξ, and thus the feature a
ˆi , and the distortion noise ni are also a function of ξ.
Since ξ is a random variable, the detection problem becomes that of detecting a random
signal a
ˆi in noise ni .

5.4

Model-Based Performance Evaluation Approach

The FEM model described in Section 3.2 is used to model the stochastic processes as described in Section 5.3, and thereby determine the reliability of GW-PZT using probability of
detection (POD) concepts. The approach consists of two parts as illustrated in Figure 5.5:
(i) Deterministic modeling, and (ii) stochastic modeling.

67

Variability (ξ)

Stochastic Model

Defect (a)

Geometry
Deterministic Model

pξ(ξ)

False Alarm
Criterion ( )

Figure 5.5: The work flow for the calculation of the probability of detection of a given defect
size under known variability statistics, and given a false alarm criterion.

5.4.1

Deterministic Modeling

Deterministic modeling uses FEM to generate a model according to a given geometry which
includes K number of PZT sensors. This model takes the defect parameter (a) as input.
The defect parameter denotes a failure mode for the structure geometry that is studied. For
example, the length of a fatigue crack in a riveted aluminum plate, or delamination area
in a multilayered fiber reinforced plastic composite plate. Another parameter that is given
as input to the FEM model is the variability property vector ξ. In the current studies, the
value of each variable property is studied independently of the other variations. Then the
vector ξ could be considered a scalar, since only one component is variable while all the
other components are held constant, and it is denoted by the non-bold symbol ξ.
Firstly, one FEM simulation is conducted to obtain the baseline waveforms. For this
simulation, there is no defect (a = 0), and ξ = ξ0 . After the baseline waveforms are
recorded for each path i, the FEM model is run for a given set of values {ξ} of size Mξ for
68

ξ ∈ [ξmin , ξmax ], and a set of {a} values of size Ma where a ∈ [0, amax ]. This results in
a total number of M = Ma × Mξ simulations. The signal feature yi is extracted from each
model according to the signal representation discussed in Section 5.3. Since yi depends on
the defect size and variability parameter, it is expressed as a surface function:

yi = fi (a, ξ)

(5.8)

where fi (a, ξ) are a set of functions corresponding to each path in the PZT sensor network.
Those functions are determined numerically by linear interpolation between the M sampled
simulated values. Thus fi () is a mapping function from the geometry features (a, ξ) with
a ∈ [0, amax ] and ξ ∈ [ξmin , ξmax ], to the signal features yi .

5.4.2

Probability Density Function of the Signal Feature

The probability density function of signal feature yi is computed for each possible defect
size a. The function fi () in equation (5.8) is reduced to a function of a single variable, by
considering a known defect size variable such as a = a0 so that yi = fi (a = a0 , ξ) = fi (ξ).
This results in a set of continuous conditional probability density functions for each given
defect size. Let the functions ξi = fi−1 (yi ) be the inverse of the functions fi (). Since fi (ξ) is
generally non-monotonic (not strictly increasing or decreasing) over the range [ξmin , ξmax ],
the inverse function can have multiple domains where it is defined and over which it is
monotonous. The equation yi = fi (ξ) has a unique solution over each monotonous domain,
(l)

and ξi

−1 (l)

= fi

(yi ) are those solutions, and L(yi ) is the number of those solutions. As

described in Section 5.2, ξ is modeled as a stochastic variable with a probability density
function pξ (ξ), The probability density function for each path signal feature yi has the
69

probability density function:
L(yi )

d ξ (l)
p (ξ (l) )
d yi ξ

py |a (yi | a) =
i

(5.9)

l=1

Substituting equation (5.1) into equation (5.9), the probability density function for yi could
be obtained:

L(yi )

2 exp −

d ξ (l)
d yi

py | a (yi | a) =
i

√

σξ 2π erf

l=1

(ξ (l) −µξ )2
2σ 2
ξ

ξmin −µξ
√
σξ 2

− erf

(5.10)
ξmax −µξ
√
σξ 2

In practice it is usually more informative to be able to represent signal features in terms of
their signal to noise ratio. According to the signal representation model, when the defect
size a = 0, then the defect feature aˆi = 0, and yi = ni . The signal feature is solely due
to the noise, and the energy could be computed using the variance of its probability density
function:
σn2 =

(yi − µn )2 py |a (yi | a = 0) d yi
i

(5.11)

yi py |a (yi | a = 0) d yi
i

(5.12)

where µn is the noise mean:

µn =

Then, for a given signal measurement yi , the SNR value could be computed as:
y
SN R = 20 log i
σn

70

(5.13)

5.4.3

Decision Threshold Computation

A detector has to make a decision between two available hypothesis: (i) There is a defect
(hypothesis H1 ), or (ii) there is no defect present (hypothesis H0 ). The detector makes a
series of binary decisions, corresponding to a known defect size. There are four possible
outcomes due to this hypothesis testing:
• The detector predicted hypothesis H0 , but H1 is true. This is called a miss.
• The detector predicted hypothesis H1 , and H1 is true. This is called a true positive,
or detection.
• The detector predicted hypothesis H1 , and H0 is true. This is called a false positive,
or false alarm.
• The detector predicted hypothesis H0 , and H0 is true. This is called a true negative.
Based on the formulations in Section 5.3, the two hypothesis of the detector are given:

H0 : yi ∼ ni

(5.14)

H1 : yi ∼ a
ˆ i + ni
and the likelihood function for each of the hypothesis is written as:

py |H (yi | H0 ) = py |a=0 (yi | a = 0)
i 0
i

(5.15a)

py |H (yi | H1 ) = py |a=0 (yi | a = 0)
i 1
i

(5.15b)

Ideally, the optimal receiver would minimize the probability of false alarms, while maximizing the probability of detection. This is analogous to minimizing the misses and maximizing
71

the true negatives. Typically, the probability of one criterion is constrained, and the other
is maximized (or minimized) [69]. The receiver should be designed such that it uses a decision rule that satisfies these criteria. Figure 5.6 illustrates the regions that represent the
probability of detection PD , and the probability of false alarm PF , for some given likelihood
functions py |H (yi | H0 ) and py |H (yi | H1 ). The optimal receiver problem is to select a
i 0
i 1
proper detection threshold γ. The false alarm probability is
∞

PF =

py |H (yi | H0 ) d yi
i 0

γ

(5.16)

and the detection probability for a give defect size (a) is then:
∞

PD (a) =

γ

py |H (yi | H1 ) d yi
i 1

(5.17)

The optimal threshold γ is found by by constraining PF = α, and the probability of detection
is kept as high as possible. The threshold γ is chosen such that the area under the probability
density function py |H (yi | H0 ) is equal to the false alarm rate, as shown in 5.6. There is
i 0
no closed form solution to equation (5.16) and it could be computed numerically to obtain
the optimum threshold. In order to compute the threshold, the following parameters and
statistics should be known:
• A false alarm criterion α,
• The probability density function and statistics for the variability parameter: its mean
µξ and variance σξ2 ,
• The fitted function: yi = fi (a = 0, ξ).

72

Figure 5.6: Computing the threshold for a given false alarm rate criterion. The false alarm
rate is the area under the no defect likelihood function. The probability of detection PD is
the area under the defect likelihood function.
Once the threshold γ is computed, the probability of detection for a given defect size parameter could be computed using equation (5.17).

5.4.4

Receiver Operating Characteristic Computation

Sometimes there is no hard criterion on the false alarm rate, but there is a limit on the
minimum probability of detecting a defect with a given size a = a0 . To find the required
threshold to satisfy this criterion, the receiver operating characteristic (ROC) curve is computed. The ROC curve plots the probability of detection versus the false alarm rate is varied
for a given defect size. This could also be implemented by increasing the threshold from them
minimum possible value to the maximum value (where a 0 % false alarm rate is obtained).
Then the needed false alarm rate is selected that gives the required probability of detection.

73

y1
y2
Decision
yk
Figure 5.7: Design of an array receiver with total votes thresholding.

5.5

POD Calculation for PZT Sparse Arrays

In a sparse network of K PZT sensors, the maximum number of possible actuator-sensor
paths is K 2 (this includes the self-sensing actuators). For simplicity denote the number of
total measured paths by J. For a given path i (for i ∈ {1...J}) , the probability of detection
can be calculated using equations (5.17) and (5.10). A total of J decisions of Yes/No can
be obtained from this sensor network. This receiver design is shown in Figure 5.7. At each
sensor, a comparator outputs “1” if yi is above a threshold, or a “0” if it is below the
threshold. The votes of all the sensors are added to obtain to the total number of sensors
with a “Yes” vote. Then another comparator is used to detect if a total of at least M sensors
have voted “Yes”. If this is true, then the final decision is that a defect is present (“Yes”
vote), otherwise no defect is present (“No” vote).
The next step is to quantify the performance of such receiver in terms of probability of
detection for a given defect size. If the probability of detection of actuator-sensor paths are
independent but not identical, then the final probability of detection for this type of array
receiver design follows a cumulative Poisson binomial distribution. That is, in the sequence
of J independent paths each with a probability of detection denoted pD (yi |a), the probability

74

of having at least (m) successful detections from the total of J:
J

1 − pD (yj | a)

pD (yi | a)

PD (y | a) =
k=m A ∈ Fk i ∈ A

j

(5.18)

∈ Ac

where Fk is the set of all subsets of k paths that can be selected from the J paths. Fk
J!
contains a total of (J−k)!
elements.
k!

5.6

Numerical Implementation

Direct calculation of the probability density function given in equation (5.10) is in general,
not numerically stable if the function fi (ξ) is not monotonic. Then there will be a value
dy

such that dξi =

df (ξ)
dξ

(l)

ξ=ξk

= 0. This means that the inverse function dξdy → ∞ in the
i

neighborhood of the value ξ = ξk .
Alternatively, one could use the Monte-Carlo resampling method. Samples of ξ are drawn
from its given distribution pξ (ξ) defined in equation (5.1), and then compute yi = fi (ξ). If
enough samples of ξ are drawn, then the probability density function of yi could be accurately
calculated. This method is more stable than the analytical/numerical computation method,
however it requires a large number of samples in order to be able to construct py |a (yi | a).
i

5.6.1

Confidence Bounds of POD Estimates

The calculation of the probability of detection requires integrating the probability density
function of the signal feature (equation (5.17)). Numerically, pyi (yi | a) is stored as a vector
of numerical values of size N , which is the number of drawn ξ values. The probability of
detection is computed by finding the ratio of number of samples with value that are greater

75

than the threshold Nd to the vector size:
N
PˆD = d
N

(5.19)

where PˆD is the estimate of the true POD value PD . The confidence interval for the POD
estimate is derived from Chebyshev’s inequality, which states that for any ε > 0, a bound
with a certain confidence on the accuracy of the estimate is given by:

E
P

PˆD − PD > ε

PˆD − PD

≤

2

ε2

(5.20)

Let x1 , x2 , . . ., xN denote the N samples of the distribution, then equation (5.19) could be
rewritten as:

N

Ik
PˆD = k=1
N
with





1 if xk > γ
Ik =



0 otherwise

(5.21)

(5.22)

where γ is the detection threshold. The expected value and variance of Ik are given by:

E(Ik ) = 1.P (Ik = 1) + 0.P (Ik = 0) = PD
Var(Ik ) = E(Ik2 ) − E 2 (Ik ) = PD (1 − PD )

76

(5.23a)
(5.23b)

1

1

PD

PD

0.8
0.5

0.6
0.4
0.2

0

2

4
6
8
Defect length (mm)

10

2

4
6
8
Defect length (mm)

(a)

10

(b)

Figure 5.8: An example of the upper and lower 99 % confidence bounds for a POD calculation, where (a) 105 samples are used, and (b) 106 sampled are used.
Using equations (5.21) and (5.23),

E

PˆD − PD

2

= Var(PˆD )
N

=
=

k=1

Var(Ik )

(5.24)

N2
PD (1−PD )
N

The maximum values for equation (5.24) is when PD = 0.5, in which case we have:

E

PˆD − PD

2

≤

1
4N

(5.25)

≤

1
4N ε2

(5.26)

Substituting (5.25) into (5.20):

P

PˆD − PD > ε

Thus, (5.26) shows that for N samples, the POD estimate lies within [−ε, ε], with the
confidence level of 1 − P

PˆD − PD > ε . For example, for N = 106 , and a required

77

confidence level of 99 %, we have:

0.01 ≤

1
4N ε2

ε

≤ √ 1

ε

≤ 0.005

(5.27)

0.04N

In the case when probability of detection is fairly high, and PD >> ε, which is the case
of most interest, the confidence level of 99% has a tight variation around the actual PD
value. Figure 5.8(b) shows the upper and lower 99% confidence bounds for a calculated
POD curve, and the bounds almost overlap the calculated POD curve. Figure 5.8(a) shows
the result for 105 samples, indicating a good confidence interval. Using modern computers,
the computation of 106 values typically takes less than a second, allowing estimates with
high accuracy, without significant computational time costs. All consequent computations
presented in this work used 106 samples for POD calculations.

78

Chapter 6
Reliability of Crack Detection in
Riveted Aluminum Plates

6.1

Introduction

The capability of GW-PZT for detecting cracks emanating from fastener holes in a riveted
aluminum plate is investigated following the procedures described in Section 5.4. The riveted
plate geometry and parameters described in Section 4.2 are used in this model-based study.
The plate configuration is shown in Figure 4.2. Eight collocated PZT sensors on the top
and bottom surfaces of the plate allow measuring the A0 and S0 modes separately. The
performance by considering each mode separately is investigated. The amplitude of the
excited incident mode depends on frequency. The operating excitation frequency in this
study is 200 kHz. The amplitude of the excited A0 mode is higher than that of S0 mode,
resulting in a higher signal to noise ratio for the A0 mode, and generally a better probability
of detection. Only the excitation of PZT-3 (Figure 4.2), and the crack emanating from the
rivet hole in the x-direction is considered in this chapter. Three degradation modes of the
PZTs are considered: variations in the PZT bonding Young’s modulus, variations in the PZT
bonding thickness, and PZT peeling. Each PZT degradation mode is studied separately, to
understand its effects on the detection and the measured waveforms.

79

20

Voltage (mV)

Voltage (mV)

10

0
−20

5
0
−5
−10

0

20

40
time (µs)

60

80

0

(a) A0 mode

20

40
time (µs)

60

80

(b) S0 mode

Figure 6.1: The baseline waveforms measured at PZT-7 when PZT-3 is actuated.

6.2

Baseline Waveform

Before studying the effects of degrading PZT sensors and adhesive bonding, a baseline signal
is recorded when there is no defect and under “ideal” conditions at all PZT sensors when
PZT-3 is actuated. The changing parameters in this model are the bonding Young’s modulus,
bonding thickness, and PZT diameter. Under ideal conditions, their values are set to:
• Bonding Young’s modulus E = 2.6 GP a,
• Bonding thickness = 60 µm,
• PZT diameter = 7 mm.
In sections 6.3, 6.4, and 6.5, only effects of sensor degradation on the actuator-sensor path 37 (see Figure 4.2) is studied. The implications of using other paths on detection, and the use
of array receiver configurations are discussed in section 6.6. The baseline waveform recorded
at PZT-7 is shown in Figure 6.1, for both the S0 symmetric mode, and A0 symmetric mode.
This baseline waveform is subtracted from waveforms measured in the presence of a defect
and/or changing PZT and bonding properties.

80

Voltage (mV)

Voltage (mV)

20
10
0
−10
−20
0

5
0
−5

20

40
time (µs)

60

0

80

(a)

20

40
time (µs)

60

80

(b)

Figure 6.2: Measured signals and their scatter after baseline subtraction for the sensor path
3-7 for (a) A0 mode, (b) S0 mode, when defect length is a = 4mm and the Young’s modulus
ξ = 2080M P a.

6.3

Bonding Young’s Modulus Variation Effects

The sensor degradation mode where the adhesive bonding Young’s modulus (Y ) varies is
studied in this section. According to the nomenclature in Section 5.3, the variability parameter ξ = Y . The limits on the Young’s modulus (ξ) and the defect length (a) are such
that:
ξ ∈ [260, 5200] M P a
(6.1)
a ∈ [0, 10] mm
Sixteen values are sampled from the range of ξ, and 11 values are sampled from the range
of a, resulting in a total of 176 FEM simulations. The feature yi , as given by equation (5.6)
is extracted from each of the measured A0 and S0 waveforms separately. Figure 6.2 shows
an example measured waveform when the defect size a = 4 mm and the bonding Young’s
modulus ξ = 2080 MPa. The scattered waveform after baseline subtraction is also shown.
The extracted feature is indicated as the maximum of the signal envelop. The variation
surfaces such that:
yA = fA (a, ξ)
81

(6.2a)

(a) A0 mode

(b) S0 mode

Figure 6.3: The surface for the defect scatter features yA = fA (a, ξ) and yS = fS (a, ξ),
which are functions of the PZT adhesive bonding Young’s modulus (Y ) and defect length.
yS = fS (a, ξ)

(6.2b)

are fitted using linear interpolation, where yA corresponds to the extracted feature of the A0
mode waveforms, and yS corresponds to the extracted feature of the S0 mode waveforms,
both for the path PZT-3 → PZT-7. The fitted surface functions after interpolation are
shown in Figure 6.3.
82

50

30
20

γA

10
0

γS

40
Py(y|a)

Py(y|a)

40

50
a = 0 mm
a = 2.5mm
a = 5.0mm
a = 7.5mm
a = 10mm

30
20
10

−10

0
10
SNR (dB)

0

20

−10

(a) A0 mode

0

10
SNR (dB)

20

(b) S0 mode

Figure 6.4: The likelihood functions for different defect lengths, when the predicted standard
deviation of the Young’s modulus σξ = 520M P a.

6.3.1

Effects of Short-term Variations

The probability density function due to short-term variations in the adhesive bonding Young’s
modulus, given by equation (5.1), has the following statistics:
• ξmin = 260M P a, and ξmax = 5200M P a,
• µξ = 2600M P a, which is the value during the baseline recording.
• The standard deviation σξ is varied between [130, 780]M P a, to study the effects of
increasing variations on the POD.
Figure 6.4 shows the likelihood functions for different defect lengths when σξ = 520 MPa.
An optimal decision threshold is selected by specifying a false alarm criterion and setting the
threshold by utilizing equation 5.16. The calculated threshold for a 1% false alarm rate is
shown as a vertical dotted line in Figure 6.4(a) for the A0 mode and in Figure 6.4(b) for the S0
mode. The probability of detection for defect sizes between 0 and 10mm is then computed
for different values of σξ , and is shown in Figure 6.5. For the S0 mode, there is 100%
detection of defect sizes larger than 4mm if σξ < 390M P a. The A0 mode is less sensitive
83

1

1
σξ = 130 MPa

0.8

σξ = 260 MPa

0.6

σξ = 390 MPa

0.4
0.2
2

PD

PD

0.8

0.6

σξ = 520 MPa

0.4

σξ = 650 MPa

0.2

4
6
8
Defect length (mm)

10

(a) A0 mode

2

4
6
8
Defect length (mm)

10

(b) S0 mode

Figure 6.5: The probability of detection with short-term variations in the PZT adhesive
bonding Young’s modulus.
to such variations and defects as small as 2mm could be detected with 100% probability if
σξ < 390M P a, with a false alarm rate less than 1%. The probability of detection decreases
as the variance of the Young’s modulus value increases.
The effects of the Young’s modulus variance are shown in Figure 6.6. If the variance is
small enough, then the short-term variations do not have a significant effect on detection.
The effects of changing the probability of false alarm on the POD is given by the receiver
operating characteristic (ROC) as shown in Figure 6.7. The ROC curves demonstrate that
the effect of variations in the Young’s modulus are not significant. Even for a high variance,
the ROC curve rapidly goes near the upper left corner (ideal detection), for a defect length
as small as 2mm.

6.3.2

Effects of Long-term Variation

As explained in Section 5.2, the standard deviation σξ and then mean µξ change slowly with
time. Figure 6.6 shows that increasing σξ decreases the probability of detection. Thus the
probability of detection is best in environments where the variations in the PZT adhesive

84

1

1
a = 1mm
a = 2mm
a = 3mm
a = 4mm
a = 5mm

0.4

0.8
PD

0.6

P

D

0.8

0.6
0.4

0.2

0.2
200

400
600
σξ (MPa)

800

1000

200

(a) A0 mode

400
600
σξ (MPa)

800

1000

(b) S0 mode

Figure 6.6: The probability of detection as the standard deviation σξ of the PZT adhesive
bonding Young’s modulus increases, for different defect sizes.

0.5

0
0

0.2

0.4

P

0.6

D

1
a = 0.5mm
a = 1.0mm
a = 1.5mm
a = 2.0mm

P

PD

1

0.5

0
0

0.8

F

0.2

0.4

P

0.6

0.8

F

(a) A0 mode

(b) S0 mode

Figure 6.7: The receiver operating characteristic curve for different defect sizes, and at
σξ = 520M P a.
bonding Young’s modulus is minimum. However, the mean value of the adhesive Young’s
modulus µξ could change with time, and this change could significantly affect the probability
of detection.
The baseline waveform corresponds to the value of the adhesive Young’s modulus 2600
MPa. Long-term changes in µξ between 1560 and 3120 MPa, with a fixed σξ = 260M P a are
studied in this section. The POD (given a 1% false alarm rate) for different mean values are
shown in Figure 6.8. If the mean value µξ decreases in the long term, then the probability of

85

1
0.8

0.8

µξ = 1820 MPa

0.6

µξ = 2080 MPa

0.4

µξ = 2340 MPa

0.2

µξ = 2600 MPa
2

4
6
8
Defect length (mm)

PD

PD

1

µξ = 1560 MPa

0.6
0.4
0.2

10

(a) A0 mode

2

4
6
8
Defect length (mm)

10

(b) S0 mode

Figure 6.8: The probability of detection for changing µξ with σξ = 260M P a.
detection also decreases. This change is more significant in the S0 mode than A0 mode. For
a decrease of up to 40 % in the mean value, the probability of detection for a 4mm defect
size stays at 100 % for the A0 mode. For the S0 mode, that probability goes down from 100
% to less than 20 %. Figure 6.9(a) shows the change in POD for the A0 mode as µξ varies
around its initial value. The probability of detection does not change for defects as small as
3mm , when the mean value of the adhesive modulus decreases by 40 %. Figure 6.9(b) shows
the effects of varying mean value of the adhesive Young’s modulus when using the S0 mode.
The POD decreases rapidly for a decreasing mean value of the adhesive Young’s modulus.
However, an increase in the mean value does not affect the probability of detection. This
could be seen from the feature surface function in Figure 6.3. For ξ values larger than the
baseline value, the feature changes slowly in contrast to when ξ falls down below the baseline
value.

86

0

0

−40
−60

−20
D

% ∆ PD

−20

%∆P

a = 3mm
a = 4mm
a = 5mm
a = 6mm

−80

−40
−60
−80

−60

−40

−20
% ∆µξ

0

−60

(a) A0 mode

−40

−20
% ∆µξ

0

(b) S0 mode

Figure 6.9: The probability of detection as the mean µξ changes due to the long-term
variations, for different defect sizes.

6.4

Bonding Thickness Variation Effects

The sensor degradation mode where the adhesive bonding thickness varies is studied in this
section. The limits on ξ and the defect length (a) are:

ξ ∈ [10, 210] µm
(6.3)
a ∈ [0, 10] mm
All the other parameters of the sensor are set to fixed values, the same as when the baseline
waveforms were obtained, as described in Section 6.2. Twelve values were sampled from the
range of ξ, and 11 values were sampled from the range of a, resulting in a total of 132 FEM
simulations. The interpolated surfaces of envelop peak feature for both the A0 mode and
the S0 mode is shown in Figure 6.10. The surface is at zero when there is no defect and the
adhesive thickness is the same as the baseline at 60 µm. It is non-zero for all other values of
bonding thickness, indicating that the variations will increase the probability of false alarms,
and decrease the probability of detection.

87

(a) A0 mode

(b) S0 mode

Figure 6.10: The surface of y variations due to varying bonding thickness and defect length.

6.4.1

Effects of Short-term Variations

The short-term variation in the adhesive thickness is modeled by equation (5.1), and has the
following statistics:
• ξmin = 10µm, and ξmax = 210µm,
• µξ = 60µm, which is the value during the baseline recording.
• The standard deviation σξ is varied between [5, 20]µm. The value of σξ represents the
extent of the short-term variation.
88

y

Py(y|a)

100

γ

150
P (y|a)

γA

200

200

a = 0 mm
a = 2.5mm
a = 5.0mm
a = 7.5mm
a = 10mm

S

100
50

0

15

20

25
SNR (dB)

0

30

(a) A0 mode

15

20
25
SNR (dB)
(b) S0 mode

Figure 6.11: The likelihood functions for different defect sizes, when the predicted standard
deviation of the bonding thickness σξ = 7.5µm.
Figure 6.11(a) shows the likelihood functions when σξ = 7.5 µm for both the A0 mode, and
Figure 6.11(b) for the S0 mode. A false alarm criterion is set to 1%, and the threshold value
is indicated by a vertical dotted line. The POD curves for different values of σξ are shown in
Figure 6.12. For the A0 mode, a standard deviation up to 15 µm in the adhesive thickness
still results in a 100 % detection for defects larger than 2mm. For the S0 mode, the POD
decreases significantly as the standard deviation increases. This is further demonstrated in
Figure 6.13, where the probability of detection is plotted versus the standard deviation in
the adhesive thickness. The receiver operating characteristic is shown in Figure 6.14. Shortterm variations in the adhesive thickness does not have a significant effect on detection if
the standard deviation is small enough. In the ROC curve, the A0 mode has near perfect
detection with no false alarms for defects larger than 1mm. Detection using the S0 mode
is close to 100 % for defect sizes larger than 2mm. This defect detection probabilities are
computed for σξ = 7.5µm, that is 12.5 % of the mean value, which is a large variation in
practice.

89

1

0.8

0.6

σξ = 10 µm

0.6

0.4

σξ = 15 µm

PD

σξ = 5 µm

0.8
PD

1

σξ = 2 µm

0.4
0.2

0.2
2

4
6
8
Defect length (mm)

10

2

(a) A0 mode

4
6
8
Defect length (mm)

10

(b) S0 mode

Figure 6.12: The probability of detection with short-term variations in bonding thickness.
1

1
a = 1mm
a = 2mm
a = 3mm
a = 4mm
a = 5mm

0.6
0.4

0.8
PD

PD

0.8

0.6
0.4
0.2

0.2
5

σξ (µm)

10

15

5

(a) A0 mode

σξ (µm)

10

15

(b) S0 mode

Figure 6.13: The probability of detection as σξ increases, for different defect sizes (a).
a = 0.5mm
a = 1.0mm
a = 1.5mm
a = 2.0mm

0.5

0.8
0.6
PD

PD

1

0.4
0.2

0
0

0.2

0.4

PF

0.6

0
0

0.8

(a) A0 mode

0.2

0.4

PF

0.6

0.8

1

(b) S0 mode

Figure 6.14: The receiver operating characteristic curve for different defect sizes, and at
σξ = 7.5µm.

90

1

µ =40µm
ξ

0.8

µξ =50µm

0.8

0.6

µξ =60µm

0.6

0.4

µξ =70µm

0.2

µξ =80µm
2

4
6
8
Defect length (mm)

PD

PD

1

0.4
0.2

10

2

(a) A0 mode

4
6
8
Defect length (mm)

10

(b) S0 mode

Figure 6.15: The probability of detection due to bonding thickness variations for changing
µξ with σξ = 7.5µm.
0

−60

D

% ∆ PD

−40

80
%∆P

a = 3mm
a = 4mm
a = 5mm
a = 6mm

−20

60
40
20

−80
−80

−60

−40

−20
% ∆µξ

0

0

20

(a) A0 mode

−60

−40

−20
% ∆µξ

0

20

(b) S0 mode

Figure 6.16: The probability of detection due to bonding thickness variations as the mean
µξ changes, for different defect sizes.

6.4.2

Effects of Long-term Variation

The effects of slow change in the mean value of bonding thickness is studied in this section.
Figure 6.15 shows the POD for different values of µξ , with a fixed value of σξ at 7.5 µm.
Figure 6.16 shows the change in POD as the mean value ∆µξ changes from its initial (baseline) value. Detection using the A0 mode (Figure 6.16(a)) is not affected by a decrease in
the adhesive bonding thickness up to 40%. For a decrease larger than 40%, the POD starts
to deteriorate rapidly. An increase in the bonding thickness does not affect the POD. The

91

detection with S0 mode degrades as the value of µξ decreases from the initial value of 60µm.
The S0 mode (Figure 6.16(b)) shows significant degradation in detection starting from a 10%
decrease in the mean value. A 20% decrease of the adhesive thickness mean value results
in nearly 100 % decrease in detection of defects up to 4mm in length. Increasing the mean
value does not significantly affect the detection up to some value, where the probability of
detection will start to decrease again. This could be seen in Figure 6.16(b) when defect
length is 3mm.

6.5

Sensor Debonding Effects

The sensor degradation mode where the adhesive bonding peels from the structure is studied
in this section. Adhesive peeling cause a reduction in the total coverage area that bonds the
PZT to the structure, which decreases the strain transfer efficiency of the PZT. This change
in adhesive coverage area is modeled as a change in the PZT and the underlying adhesive
diameter. The variable ξ in this case is the total diameter of the PZT sensor that is in
contact with the structure. The range of ξ and the defect length (a) are such that:

ξ ∈ [3, 8] mm
(6.4)
a ∈ [0, 10] mm
All the other parameters of the sensor are set to fixed values, the same as when the baseline
waveforms were obtained, as described in Section 6.2. Ten values were sampled from the
range of ξ, and 11 values were sampled from the range of a, resulting in a total of 110 FEM
simulations. The interpolated surfaces of the signal feature y for both the A0 mode and
the S0 mode are shown in Figure 6.17(a) and Figure 6.17(b), respectively. The surface is at
92

(a) A0 mode

(b) S0 mode

Figure 6.17: The surface of y variations due to varying PZT diameter and defect size.
zero when there is no defect and no peeling, which corresponds to a PZT diameter of 7mm.
It is non-zero for all other values for PZT diameter, indicating the presence of damage, or
variations due to non-ideal baseline subtraction.

93

40

60

Py(y|a)

30
20

γS

γA
Py(y|a)

a = 0 mm
a = 2.5mm
a = 5.0mm
a = 7.5mm
a = 10mm

10
0
−30

−20

−10
0
SNR (dB)

40

20

0
−30

10

(a) A0 mode

−20

−10
0
SNR (dB)

10

(b) S0 mode

Figure 6.18: The likelihood functions for different defect sizes, when the predicted standard
deviation of the PZT diameter σξ = 0.5µm.

6.5.1

Effects of Short-term Variations

The short-term variation in the PZT diameter which has contact area with the structure is
modeled according to equation (5.1), and has the following statistics:
• ξmin = 3mm, and ξmax = 8mm,
• µξ = 7mm, which is the value during the baseline recording.
• The standard deviation σξ is varied between [0.5, 1.5]mm, to study the effects of increasing variations on the POD.
Figures 6.18(a) and 6.18(b) show the likelihood functions when σξ = 0.5 mm for the A0
and S0 modes, respectively. A false alarm criterion is set to 1%, and the resulting decision
threshold is shown as a vertical dotted line. The resulting POD curves for different values of
σξ are shown in Figure 6.19. The reduction of adhesive contact area results in a significant
reduction in the probability of detection for both the A0 and S0 modes. Using the A0 mode,
even a small standard deviation less than 4 % of the mean value, the POD decreases rapidly
to zero for defect lengths less than 2mm. The POD is significantly lower as the value of the
94

1

0.8

0.8

σξ=0.5mm

0.6

0.6

σξ=0.75mm

0.4

0.4

σξ=1mm

0.2

0.2

PD

PD

1

2

4
6
8
Defect length (mm)

10

σξ=0.25mm

2

(a) A0 mode

4
6
8
Defect length (mm)

10

(b) S0 mode

1

1

0.8

0.8

0.6

0.6

PD

PD

Figure 6.19: The probability of detection with short-term variations in bonding coverage
area.

0.4

0.4

0.2

0.2
0.2

0.4
0.6
σξ (mm)

0.8

1

(a) A0 mode

a = 1mm
a = 2mm
a = 3mm
a = 4mm
a = 5mm

0.2

0.4
0.6
σξ (mm)

0.8

1

(b) S0 mode

Figure 6.20: The probability of detection as σξ increases, for different defect sizes (a).
standard deviation increases. For the S0 mode, a standard deviation of 4 % from the mean
value results in less than 15 % POD for defect lengths less than 4mm. Figure 6.20 shows the
effects of increased σξ versus the decrease in POD. Increasing σξ does have a significant effect
on the POD. The receiver operating characteristic is shown in Figure 6.21. The detection
performance degrades rapidly when the false alarm rate is required to be low.

95

1

D

0.5

0
0

P

PD

1

0.2

0.4

PF

0.6

0.8

0
0

1

a = 0.5mm
a = 1.0mm
a = 1.5mm
a = 2.0mm

0.5

0.2

(a) A0 mode

0.4

PF

0.6

0.8

(b) S0 mode

Figure 6.21: The receiver operating characteristic curve for different defect sizes, and at
σξ = 0.25mm.
1
0.3

µξ =5mm

0.6

PD

PD

0.8

µξ =4mm

0.2

µξ =6mm
µξ =7mm

0.4
0.1

0.2
2

4
6
8
Defect length (mm)

10

(a) A0 mode

2

4
6
8
Defect length (mm)

10

(b) S0 mode

Figure 6.22: The probability of detection due to bonding coverage area variations for changing µξ with σξ = 0.5mm.

6.5.2

Effects of Long-term Variation

The effects of the slow change of µξ of the bonding area is studied in this section. Figure
6.22 shows the POD for different values of µξ , with σξ fixed at 0.5mm and a false alarm rate
of 1 %. For both the A0 and S0 modes, it is apparent that the there is a large effect on the
probability of detection if the PZT sensors bonding starts peeling from the structure. For
A0 mode, the POD does not exceed 50 % for a defect size of 10mm if the bonding area is
reduced to 4mm. For the S0 mode, the corresponding POD is as low as 10 %. Figure 6.23

96

0

−40

2

a = 3mm
a = 4mm
a = 5mm
a = 6mm

% ∆ PD

% ∆ PD

−20

−60

0
−2
−4

−80
−40

−30

−20
% ∆µξ

−10

0

(a) A0 mode

−40

−30

−20
% ∆µξ

−10

0

(b) S0 mode

Figure 6.23: The probability of detection due to bonding coverage area variations as the
mean µξ changes, for different defect sizes.
shows the change in the POD due to the long-term changes in the mean value of the bonding
area. The A0 mode (Figure 6.23(a)) is significantly affected by PZT peeling, and the POD
decreases by 90 % for a bonding area decrease of 40 %. The S0 mode (Figure 6.23(b)) is
not significantly affected by changing the mean value of the bonding area, as there is only a
decrease of 5 % in POD for a decrease of 40 % in the bonding area. However, this is largely
due to the fact that the probability of detection is already quite low just due to short-term
variations (Figure 6.22(b)). The POD is as low as 5% for a defect size of 4mm, even without
any long-term variation.

6.6

Performance of Array Implementations

In the previous sections, the performance of detection by using only one actuator-sensor path
(3-7) is investigated. The wave front of the excited waveform is always parallel to the defect,
whose length is along the x-axis. The sensor (PZT-7) receives the waveform whose wave front
is also parallel to the defect direction. This path gives the highest sensitivity to the the defect,
and hence the best performance. To see this, the signal feature amplitude received at all the
97

0.5

0.4
0.3
Feature yi

Feature yi

0.4
0.3
0.2

0.1

0.1
0

0.2

1

2

4
5
PZT

6

7

0

8

(a) A0 mode

1

2

4
5
PZT

6

7

8

(b) S0 mode

Figure 6.24: Received signal feature amplitude for PZT sensors with different wave front
incident angles.
sensors of the geometry in Figure 4.2 is shown in Figure 6.24 for a defect length of 9mm,
using both the A0 and S0 modes. The corresponding time domain waveforms are shown
in Appendix B, Figures B.1 and B.3. The actuator is always PZT-3. As the sensed wave
front becomes perpendicular to the defect, the scatter amplitude from the defect becomes
smaller. From Figure 6.24, PZT-1 receives the scatter that has a perpendicular wave front
with respect to the defect, and it has the smallest amplitude. The distortion noise due to
degradation of the sensors does not depend on the defect signal, and thus the SNR is much
smaller for the PZT sensors that do not receive the waveform whose wave front is parallel to
the defect. Considering a PZT array with elements distributed in a circular fashion around
the rivet, should make the detection insensitive to defect orientation.
A threshold is obtained for each PZT sensor element, as shown in Figure 5.7. Since
the distortion noise does not depend on defect orientation with respect to the path, the
thresholds for a given false alarm criterion at each PZT sensor should be similar, but it will
depend on the distance between the sensor and actuator. The likelihood functions of all PZT
sensors when PZT-3 is actuated are shown in Figure 6.25 for the adhesive Young’s modulus
98

20

10

0
−30

−20

−10

0
10
SNR (dB)

20

Py(y | a=0)

Py(y | a=0)

PZT−1
PZT−2
PZT−4
PZT−5
PZT−6
PZT−7
PZT−8
30

(a) A0 mode

20

10

0
−30

−20

−10

0
10
SNR (dB)

20

30

(b) S0 mode

Figure 6.25: Likelihood function for the PZT array elements, when there is no defect, for the
PZT adhesive Young’s modulus degradation mode. The decision threshold for each PZT is
shown as a dotted vertical line. The standard deviation σξ = 260M P a.
degradation mode, Figure 6.26 for the adhesive thickness degradation mode, and in Figure
6.27 for the bonding coverage area degradation mode. The likelihood functions in all the
figures are for the case when no defect is present, and the decision threshold that results in
a 1% false alarm rate is marked by a vertical line for each PZT. The decision thresholds are
very similar for all the PZT sensors, in each degradation mode criterion. The main reason for
the difference in the likelihood functions for different PZTs is the different distances between
the actuator and sensor resulting in different wave attenuation and also due to dispersion
effect.
After the threshold at each PZT sensor is determined, the detection performance for
this type of receiver could be obtained by utilizing equation (5.18). The POD with adhesive
Young’s modulus degradation mode, thickness degradation mode, and bonding coverage area
degradation mode are shown in Figure 6.28, Figure 6.29, and Figure 6.30 respectively. The
POD is shown when at least 1, 2, 3, or 4 PZTs vote “Yes” for the presence of defect. It could
be seen that when more PZTs are required to have a “Yes” vote, the POD goes down. This
is due to the fact that not all PZTs have the same sensitivity to the defect, which can have

99

200

100

0

14

16

18
SNR (dB)

20

300
Py(y | a=0)

Py(y | a=0)

PZT−1
PZT−2
PZT−4
PZT−5
PZT−6
PZT−7
PZT−8

200
100
0

22

14

(a) A0 mode

16

18
SNR (dB)

20

22

(b) S0 mode

Figure 6.26: Likelihood function for the PZT array elements, when there is no defect, for the
PZT adhesive thickness degradation mode. The decision threshold for each PZT is shown
as a dotted vertical line. The standard deviation σξ = 7.5µm.

4
2
0

−20

−10

0
SNR (dB)

10

y

Py(y | a=0)

6

6
P (y | a=0)

PZT−1
PZT−2
PZT−4
PZT−5
PZT−6
PZT−7
PZT−8

4

2

0

20

(a) A0 mode

−20

−10
0
SNR (dB)

10

(b) S0 mode

Figure 6.27: Likelihood function for the PZT array elements, when there is no defect, for
the PZT adhesive bonding coverage area degradation mode. The decision threshold for each
PZT is shown as a dotted vertical line. The standard deviation σξ = 0.5mm.
different orientation with respect to each sensing path. Thus it is apparent that the best
detector should consider all PZT sensors in the array, and then require at least one PZT
sensor to vote “Yes” for the presence of defect. Also, by comparing the three different PZT
degradation modes, the detection performance is most significantly affected by the reduction
in the bonding coverage area. The POD for defects larger than 1mm in A0 mode, and 2mm
in S0 mode, is almost 100 % with adhesive bonding modulus and thickness degradation.

100

1

1
1 vote
2 votes
3 votes
4 votes

0.6

0.8
PD

PD

0.8

0.6

0.4

0.4

0.2

0.2
2

4
6
8
Defect length (mm)

10

2

(a) A0 mode

4
6
8
Defect length (mm)

10

(b) S0 mode

Figure 6.28: Probability of detection with PZT adhesive Young’s modulus degradation mode.
The standard deviation σξ = 260M P a.
1

0.6

0.6

0.4

0.4

0.2

0.2
2

4
6
8
Defect length (mm)

1 Vote
2 Votes
3 Votes
4 Votes

0.8
PD

0.8
PD

1

1 Vote
2 Votes
3 Votes
4 Votes

10

(a) A0 mode

2

4
6
8
Defect length (mm)

10

(b) S0 mode

Figure 6.29: Probability of detection with PZT adhesive thickness degradation mode. The
standard deviation σξ = 7.5µm.
However for the bonding coverage area degradation mode, 100 % detection is for defects of
length larger than 2.5 mm in A0 mode, and the POD does not get above 50 % for defects
as large as 10mm.

6.7

Summary and Conclusions

The effects of three PZT degradation modes on the probability of detection of cracks in
a riveted aluminum plate is investigated in this chapter. The three degradation modes
101

1

0.6

0.4

0.6

PD

PD

0.8

0.4

1 Vote
2 Votes
3 Votes
4 Votes

0.2

0.2
2

4
6
8
Defect length (mm)

10

(a) A0 mode

2

4
6
8
Defect length (mm)

10

(b) S0 mode

Figure 6.30: Probability of detection with PZT adhesive bonding coverage area degradation
mode. The standard deviation σξ = 0.5mm.
considered are variations in the PZT adhesive Young’s modulus, thickness, and coverage area.
The probability of detection for the actuator-sensor path that has the highest sensitivity to
the modeled defect orientation was investigated. It was shown that short-term variations in
the adhesive bonding did not have a significant effect on the probability of detection, as long
as the standard deviation of these variations were reasonably small. Long term variations
did show significant effect on the POD. The decrease in the adhesive coverage area has the
biggest effect on the POD. The A0 mode has a higher sensitivity for detecting defects than
the S0 mode, however usually S0 mode is preferred since it exhibits a more linear relation
between defect size and the feature amplitude.
The use of a sensor array evenly distributed around the rivet in a circular fashion removes
the orientation preference, and the probability of detection is decided by the path with
highest sensitivity, if a minimum of 1 sensor path is required to be above the decision
threshold.

102

Chapter 7
Damage Detection in GFRP
Composite Plates

7.1

Introduction

Fiber reinforced plastic composites are becoming widely used in vehicles and aerospace structures due to their high strength to weight ratio. However unlike metals, the multilayered
composite structures are more susceptible to damage mechanisms such as disbonds and delaminations due to impacts. It is often difficult to visually detect the damage. This chapter
investigates experimentally the effects of cumulative impacts on the health of plane woven
glass fiber composite plates. Guided wave measurements are conducted for detecting increasing damage in the composite plate. Then, based on the damage mode observed in the
experiments, a model-based study is conducted to investigate the probability of detection of
impact damage under degrading PZT sensor. Also, the case of a small hidden delamination
is investigated used the model-based POD procedure.

103

X
185 mm

3
2
1

Actuators

s
sor
Sen

mm
85

8
7
6

360 mm

Figure 7.1: Schematic of the experimental sample and the locations of the six PZT sensors
bonded to its surface. Three of the PZT sensors were used as actuators and other three as
sensors for guided wave inspection. The impact location is marked by ’X’.

(a)

(b)

(c)

Figure 7.2: Experimental procedure for impact damage inspection which invloves three steps:
(a) A drop-weight impact. (b) Measurement of the PZT impedance. (c) Guided wave
inspection of the sample for detecting impact damage.

7.2

Experimental Procedure for Impact Damage Detection

The effect of cumulative impacts on the health of glass fiber composite material is studied
experimentally in this section. A plane woven glass fiber composite plate was prepared as
shown in Figure 7.1. Six PZT sensors were bonded to the plate surface using super glue
adhesive. Three of the PZTs were used as actuators for exciting guided waves, and other
three PZTs were used as sensors (see Figure 7.1). The experimental procedure involved
three steps as demonstrated in Figure 7.2.

The first step is the actual impact test using

a drop-weight impact testing machine. The sample is hit by the impactor with a specified
energy. The impact location is at the center of the sample, as indicated in Figure 7.1.

104

Figure 7.3: Fixture for the drop-weight impact experiment.
The second step includes measuring the PZT impedance to monitor the health of PZT and
its bonding layer. The third step includes the guided wave inspection for monitoring the
growth of delamination and cracks in the vicinity of the impact location. This experimental
procedure is repeated six times with increasing impact energy levels. Detailed description
and the results of each experimental step are described in the following subsections.

7.2.1

Impact Test and Measurements

For each impact, the sample was placed in the drop-weight impact machine, where it was
impacted at the same location with six different energy levels from 11 J up to 109 J. The
fixture where the sample is placed for impact is shown in Figure 7.3. The sample was held
tightly in place, and the impactor hit the sample at its center location. A visual inspection
of the sample at the impact location is shown in Figure 7.4. After the first impact in
Figure 7.4(a), a slight delamination at the impacted area could be seen. Subsequently,
when higher impacts are applied, matrix cracking is observed. To further understand the
effects of the impact on the sample, three parameters are measured in real-time during

105

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7.4: Picture of the sample after each impact, with varying energies. (a) 11 J, (b) 22
J, (c) 43 J, (d) 65 J, (e) 88 J, (f) 109 J.
the impact: (i) the deflection of the sample due to impact, (ii) the load applied by the
impactor, (iii) and the total energy of the impact. Figure 7.5(a) shows the energy with
time for all of the six impact experiments. The peak value of the energy curve represents
the energy at the initial moment of impact. The asymptotic value of the curve represents
the remaining energy in the impactor. The difference between the the values represents the
total energy absorbed by the sample. The percentage absorbed energy is shown in Figure
7.5(b). The increasing percentage of absorbed energy with increasing total impact energy
indicates increasing damage in the sample plate, which is verified by the visual inspections
of increasing area of delamination and matrix/fiber breaking shown in Figure 7.4. After
the last impact of 109 J, the percentage of absorbed energy is approximately 95 % of the
106

Impact 1
Impact 2
Impact 3
Impact 4
Impact 5
Impact 6

50

0
0

% Absorbed energy

Energy (J)

100

5

10
time (ms)

15

90
80
70
60

20

20

(a)

40
60
80
Impact Energy (J)

100

(b)

Figure 7.5: (a) The total energy in the impactor object for all the six impact tests. (b) The
percentage absorbed energy by the sample during impact, for the six different impacts.
11J 22J 43J 65J 88J 109J
% Stiffness Change

Force (KN)

15
10
5
0
0

5
10
Displacement (mm)

40
30
20
10
0

15

20

40
60
80
Impact Energy (J)

100

(b)

(a)

Figure 7.6: (a) Plate deflection due to applied load by the impactor, for six different impact
energies. (b) Percentage of the stiffness change after each impact.
impact energy, indicating almost complete failure of the sample. To have a more quantitative
evaluation of the increasing damage, the applied load versus plate deflection is plotted in
Figure 7.6(a) for each of the six impacts. The rising slope of the curve is proportional to the
stiffness of the sample plate. This slope decreases with increasing impact energy, indicating
a decrease in the effective stiffness of the plate due to impact. The percentage change of
stiffness relative to the pristine state of the sample is plotted in Figure 7.6(b). A total change
in stiffness up to 50 % is observed.

107

% ∆ Capacitance

PZT−1 PZT−2 PZT−3 PZT−6 PZT−7 PZT−8
2

1

0
0

20

40
60
80
Impact Energy (J)

100

Figure 7.7: The change in capacitance of the six PZTs bonded to the test sample after each
impact.

7.2.2

Monitoring the PZT Sensors Health

As described in Section 2.6, the PZT could be modeled as a capacitor at low frequencies.
Park el al. [47] proposed measuring the PZT capacitance to monitor its health, where it
was shown that a change in the capacitance indicates damage of the PZT, or change of the
coverage area in the bonding layer. This method is used in this experiment to ensure that
the health of the PZT sensors bonded to the sample has not changed due to the impacts.
The PZT impedance is measured using an impedance analyzer (see Figure 7.2(b)) with a
frequency sweep from 1kHz to 20kHz.The imaginary part of the impedance (Z) is represented
as:
Z =

1
2πf C

(7.1)

where f is the frequency, and C is the capacitance of the PZT. Thus the capacitance could
be computed by taking the inverse of the slope of the measured imaginary part of impedance.
The capacitance of the six bonded PZT sensors are measured when the plate was in its

108

3
2
1

Actuators

Sen

sor
s

8
7
6

Figure 7.8: The guided wave pitch-catch paths considered for impact damage detection.
pristine state, and after each impact test. The change in the measured capacitance with
respect to the capacitance in the pristine state is then plotted in Figure 7.7. Most of the
PZT sensors have a maximum change of 1% in their capacitance value, PZT-3 had a change
of 1.5% in its capacitance value after the first impact, but its capacitance remained within 1
% during subsequent impact tests. This small change could be attributed to environmental
condition changes, and it could be assumed that the PZT sensors are not significantly affected
by the impact. Thus changes in the PZT sensors have a negligible effect on the guided wave
measurements. This result was expected and the impact was not anticipated to have a
significant effect on the PZT sensors since they are relatively far from the impact location,
where they are located approximately 92mm away from the impact location. Also, the PZT
sensors are located outside the impact machine fixture shown in Figure 7.3, thus the fixture
would absorb most of the impact energy propagating in the plate before it reaches the PZT
sensors. On the other hand, the PZT sensors in the vicinity of the impact location are
observed to have significant damage and debonding, as is seen in Figure 7.4. The two PZT
sensors in Figure 7.4 are only 20mm away from the impact location and on the inside of the
fixture. However these two PZT sensors are not used in the guided wave measurements.

109

7.2.3

Guided Wave Inspection

The experimental setup and instrumentation for the guided wave inspection is shown in
Figure 7.2(c). It is composed of a function generator for exciting the PZT actuators. A
50% bandwidth tone burst is used for the excitation (Figure 3.3), with center frequency
100 kHz. At the sensor side, a 4-channel charge amplifier is used for increasing the SNR
of the measured signals. The outputs of the charge amplifier are directly connected to a 4channel oscilloscope, and the measured signals are recorded on a PC computer connected to
the oscilloscope. A pitch-catch test configuration is used, where each of the PZT actuators
(PZT-1, PZT-2, and PZT-3) is excited separately, and the responses at the PZT sensors
(PZT-6, PZT-7, and PZT-8) are recorded. This results in a total of 9 GW signal paths,
but only the results of three paths are shown in this section. The three paths are shown
in Figure 7.8. This guided wave measurement procedure is conducted before any impact
to obtain the baseline data. Then it is repeated after each of the six impacts to monitor
the effect of impacts on the sample plate. The guided wave measurements for the PZT-1 to
PZT-6 path are shown in Figure 7.9. The first wave packet arrival is shown, which is solely
the S0 mode. It is observed that as the impact energy increases the S0 mode is affected by
wave attenuation and a phase shift. This is expected since the net effect of the cumulative
impacts is to reduce the stiffness of the composite in the vicinity of the impact location,
resulting in a ’soft’ region. This region is indicated by a light color in the schematic of
Figure 7.8. The propagating guided wave would then be reflected at the interface of the
reduced stiffness region, resulting in attenuation of the received signal amplitude. Also, the
change in material stiffness causes a change in the wave speed, resulting in a net phase shift
with respect to the baseline data.

110

Voltage (V)

baseline 11J 22J 43J 65J 88J 109J
0.4
0.2
0

−0.2
−0.4
190

200

210
220
time (µs)

230

Figure 7.9: Guided wave measurements of the baseline and after each of the six impacts for
the path 1-6.
Based on these observations, three features of the waveforms are used to correlate with
the measured change in material stiffness. Consider the baseline signal denoted by sb (t).
The peak value of the signal is denoted by Ab and the peak time is τb , as demonstrated in
Figure 7.10. The measured signal after impact is denoted by s(t), its peak value is A, and
the peak time is τ . The three signal features considered are:
1. The change in amplitude ratio:
y1 = 1 −

A
Ab

(7.2)

2. The change in signal phase:
y2 = 2πf (τ − τb )
where f is the signal frequency which is 100 kHz.

111

(7.3)

sb(t)

|H(sb(t))| s(t)
Ab

Voltage (V)

0.4

|H(s(t))|

0.2
0
−0.2
−0.4
190

200

210
220
time (µs)

230

Figure 7.10: Feature extraction of the measured waveform s(t) relative to the baseline signal
sb (t).
3. The scatter envelop energy:

y3 = 1 −

tf
0

|H (s(t) − sb (t))| dt
tf
0

(7.4)

|H (sb (t))| dt

where |H(sb (t))| is the envelop of the baseline signal, and |H (s(t) − sb (t))| is the
envelop of the signal scatter.
The three feature values versus the stiffness change are shown in Figure 7.11. There is a
generally monotonic relationship between the three feature values and the increasing damage
(represented by increasing stiffness change) in the sample. Also, the sensitivity of the S0
mode to impact damage is high, where there is a change up to 85 % in the amplitude ratio
and the scatter energy. Also the change in phase is up to 120 degrees. The sensitivity of the
different signal paths follow similar trends and are fairly close to each other, but they are
not identical, since they depend on the relative distance of the signal path and the damaged
area.
112

0.6

Path 1−6
Path 2−7
Path 3−8

Phase Shift (degrees)

Amplitude Ratio Change

0.8

0.4
0.2
0
0

200
400
600
800
Stiffness Change (N/mm)

120
100
80

Path 1−6
Path 2−7
Path 3−8

60
40
20
0

200
400
600
800
Stiffness Change (N/mm)

(a)

(b)

0.8

y3

0.6

Path 1−6
Path 2−7
Path 3−8

0.4
0.2
0

200
400
600
800
Stiffness Change (N/mm)
(c)

Figure 7.11: The signal features for the three different paths. (a) Amplitude ratio change
feature (y1 ); (b) Phase shift feature (y2 ); (c) Scatter envelop energy (y3 ).

7.3

Reliability of Impact Damage Detection using PZT
Sensors

The experimental procedure described in the previous section helped identify the mode of
degradation of composite plates under cumulative impacts, and assess the sensitivity of
guided waves in detecting such defects. The PZT sensor conditions were not affected by the
impacts. However this is not always the case, and the PZT sensors could suffer from damage
and debonding due to such impacts. A model-based study on the performance of damage
detection under such condition is conducted based on the procedure described in Chapter 5.

113

Figure 7.12: FE model geometry, with two PZT sensors on each surface of the plate. The
defect due to impact at the center of the plate is shown in a lighter color to indicate a
reduction in stiffness in that region.
The FE model uses the same geometry of the experimental sample in the previous section,
but with only two PZT sensors located at the top surface, and two PZT sensors at the bottom
surface of the plate, as shown in Figure 7.12. The bonding layer thickness is 60 µm, and
the PZT diameter is 7mm.The performance of the S0 mode only is investigated. The right
PZTs on the top and bottom of the plate in Figure 7.12 are excited at 100 kHz frequency
simultaneously with the same waveform used for the experimental procedure in the previous
section. The defect parameter is modeled as a change in the axial stifness of center region
of the plate which was a 60mm length. The stiffness constants (E11 , E22 ) are varied from
23.1 GPa to 14.1 GPa in that region. Thus the defect parameter (a) is the change in this
stiffness and its range is:
a ∈ [0, 9] GP a

(7.5)

The effect of PZT degradation due to debonding and breaking is investigated. The variable
ξ is the total diameter of the PZT sensor that is varied such that:

ξ ∈ [4, 8] mm

(7.6)

Ten values were sampled from the range of a, and ten values from the range of ξ, resulting in
a total of 100 FEM simulations. The interpolated surfaces of the feature given by equation
(5.6) for the S0 mode is shown in Figure 7.13.

114

Figure 7.13: The surface variation of the feature (y) due to varying PZT diameter and change
in the stiffness of the defect region of the plate.

7.3.1

Effects of Short-term Variations

The short-term variation in the PZT diameter which has contact area with the structure is
modeled according to equation (5.1), and has the following statistics:
• ξmin = 4mm, and ξmax = 8mm,
• µξ = 7mm, which is the value during the baseline recording.
• The standard deviation σξ is varied between [0.25, 1.0]mm, to study the effects of
increasing variations on the POD.
Figure 7.14 shows the likelihood functions when σξ = 0.5 mm for the S0 mode. A false alarm
criterion is set to 1%, and the resulting decision threshold is shown as a vertical dotted line.
The resulting POD curves for different values of σξ are shown in Figure 7.15. The POD is
almost 100% for any change of stiffness in the defect area that is larger in 15 %, and the
standard deviation is less than 1mm. This probability of detection decreases as the variance
115

5

5 x 10

γS

Py(y|a)

4
3
2
1
0
−10

0

10
SNR (dB)

20

Figure 7.14: The likelihood functions for different defect sizes, when the predicted standard
deviation of the PZT diameter σξ = 0.5µm.

1

σξ=0.25mm

0.8

σ =0.5mm

PD

ξ

0.6

σξ=1mm

0.4
0.2
10

20
a (% ∆ Y)

30

Figure 7.15: The probability of detection of the change in stiffness with short-term variations
in bonding contact area of the PZT sensor.
of the bonding area coverage increases. For a small change in the stiffness, which is less than
1.5 GPa, a small variance in the bonding coverage, less than 0.5mm, cause the POD to drop
to near zero, as shown in Figure 7.16.
The ROC curve due to the short-term variations is shown in Figure 7.17. For a small
false alarm rate, small changes in the stiffness due to impact damage is nearly impossible to
116

1

a = 0.5 GPa
a = 1.0 GPa
a = 1.5 GPa

PD

0.8
0.6
0.4
0.2
0.5

σξ (mm)

1

1.5

Figure 7.16: The probability of detection as σξ increases.

1

PD

a = 0.5 GPa
a = 1.0 GPa
a = 1.5 GPa
0.5

0
0

0.2

0.4

PF

0.6

0.8

Figure 7.17: The probability of detection of the change in stiffness with short-term variations
in bonding contact area of the PZT sensor.
detect. A false alarm rate of at least 15% is required to detect stiffness changes of 1.5 GPa
with a probability greater than 90%. However for a large stiffness change, about 3.5 GPa
(15%), it is possible to reliably detect the damage even with low false alarm rate, as shown
in Figure 7.15.

117

1

µξ =7mm

PD

0.8

µξ =6mm
µξ =5mm

0.6
0.4
0.2
10

20
a (% ∆ Y)

30

Figure 7.18: The probability of impact damage detection due to long-term PZT bonding
coverage area variation with σξ = 0.5mm.

7.3.2

Effects of Long-term Variation

The effects of the slow change in µξ of the bonding area is studied in this section. Figure
7.18 shows the POD for different values of µξ , with σξ fixed at 0.5mm and a false alarm rate
of 1%. The decrease in the mean PZT bonding area results in decreasing the POD. Near
100% detection rate is obtained for a mean value of 7mm if the change of stiffness is more
than 9%. However when the mean bonding coverage area decreases to 5mm, the stiffness
change should be larger than 18% for a 100% detection rate.

7.4

Detection of Hidden Delamination

Another class of defect that occurs in composite material are delaminations hidden in the
inner layers of composite plate structures. The performance of GW-PZT for detecting hidden
delamintations in GFRP composite plate is investigated following the procedures described
in Section 5.4. The composite plate geometry and parameters described in Section 4.3 are

118

Voltage (mV)

Voltage (mV)

2
0
−2
0

50

100
time (µs)

150

2
0
−2
0

(a) A0 mode

50

100
time (µs)

150

(b) S0 mode

Figure 7.19: The baseline waveforms measured at PZT-6 when PZT-5 is actuated.
used in this study. The plate configuration is shown in Figure 4.5. The performance of the
A0 and S0 modes is investigated separately. PZT-5 is excited at a 100 kHz frequency, and
the received waveforms at all other sensors are measured. The effect of PZT degradation
due to debonding on the detection probability is considered in this chapter.
Only effects of sensor degradation on the actuator-sensor path 5-6 (see Figure 4.5) is
studied. The baseline waveform recorded at PZT-6 is shown in Figure 7.19, for both the
S0 symmetric mode, and A0 antisymmetric mode. The baseline waveform considered represents the condition when all the PZT sensors and their adhesive bonding have the following
properties:
• Bonding Young’s modulus E = 2.6 GP a,
• Bonding thickness = 60 µm,
• PZT diameter = 7 mm.
This baseline waveform is then subtracted from the measurements and the signal feature
yi is extracted.

The sensor degradation mode where the adhesive bonding peels from the

structure is studied in this section. The variable ξ in this case is the total diameter of the
119

PZT sensor that is in contact with the structure. The defect parameter a is the delamination
diameter. The delamination has a circular shape, and is introduced at 2.4mm depth through
the plate thickness. The limits on ξ and the defect size (a) are given by:

ξ ∈ [3, 8] mm
(7.7)
a ∈ [0, 20] mm
All the other parameters of the sensor are set to fixed values, the same as when the baseline
waveforms were obtained. Eleven values were sampled from the range of ξ, and 11 values were
sampled from the range of a, resulting in a total of 121 FEM simulations. The interpolated
surfaces of the feature for both the A0 mode and the S0 mode are shown in Figure 7.20(a)
and Figure 7.20(b), respectively.
Examining the surface plots for A0 mode (Figure 7.20(a)), it could be seen that there is
a large change in the feature values as the delamination diameter increases. However, there
is not a significant change in the signal feature due to PZT diameter changes in comparison
to the effect of the defect. The effect of PZT debonding becomes more significant when the
PZT diameter is significantly reduced to less than 4 mm. In the case of S0 mode (Figure
7.20(b)), an opposite behaviour to that of the A0 mode is observed. It has a flat response
to changes in the delamination diameter, but it is significantly affected by the change of the
effective PZT diameter.

7.4.1

Effects of Short-term Variations

The short-term variation in the PZT diameter which has contact area with the structure is
modeled according to equation (5.1), and has the following statistics:

120

(a) A0 mode

(b) S0 mode

Figure 7.20: The surface variation of the feature y due to varying PZT diameter and delamination diameter.
• ξmin = 3mm, and ξmax = 8mm,
• µξ = 7mm, which is the value during the baseline recording.
• The standard deviation σξ is varied between [0.25, 1.0]mm, to study the effects of
increasing variations on the POD.

121

4

x 10
8
γA

y

P (y|a)

Py(y|a)

4

x 10
15 a = 0 mm
a = 2mm
10 a = 4mm
a = 6mm
5
0
−30

6

γS

4
2

−20

−10
0
SNR (dB)

10

20

(a) A0 mode

0

−20

−10
0
SNR (dB)

10

(b) S0 mode

Figure 7.21: The likelihood functions for different defect sizes, when the predicted standard
deviation of the PZT diameter σξ = 0.5µm.
Figure 7.21(a) and 7.21(b) show the likelihood functions when σξ = 0.5 mm for the A0 and S0
modes, respectively. A false alarm criterion is set to 1%, and the resulting decision threshold
is shown as a vertical dotted line. The resulting POD curves for different values of σξ are
shown in Figure 7.22. For the A0 mode, the POD is almost 100% for any delamination
diameter more than 1mm. On the other hand, the S0 mode is relatively insensitive to
delamination diameters less than 20mm, and the POD for delaminations up to 20mm is near
zero.
Figure 7.23 shows the effects of increased σξ versus the decrease in POD. The changes in
the standard deviation of the PZT diameter does not significantly affect the probability of
detection for the A0 mode, as long as σξ is less than 1.2 mm. For the S0 mode, any increase
in the variance about 0.15 mm would make the POD for delamination diameter as large as
20 mm to go near zero.
The ROC curves due to the short-term variations are shown in Figure 7.24. Again, it
is shown that the probability of detection using the A0 mode is near 100 % for defect sizes
larger than 2mm, and false alaram rate more than 1 %. For the S0 mode, the POD increases
122

1

σξ=0.25mm

0.05

σξ=0.5mm

0.04

σξ=1mm

0.6

PD

PD

0.8

0.4

0.03
0.02

0.2

0.01
5
10
15
Delamination diameter (mm)

20

5
10
15
Delamination diameter (mm)

(a) A0 mode

20

(b) S0 mode

Figure 7.22: The probability of detection with short-term variations in bonding contact area.
1

0.6

0.6

0.4

0.4

0.2

0.2
0.5

a = 5mm
a = 10mm
a = 20mm

0.8
PD

PD

0.8

1
a = 2mm
a = 3mm
a = 4mm
a = 5mm

σξ (mm)

1

1.5

(a) A0 mode

0.5

σξ (mm)

1

1.5

(b) S0 mode

Figure 7.23: The probability of detection as σξ increases, for different defect sizes (a).
as the false alarm rate criterion is increased. A 100 % detection is achieved for defect sizes
larger than 5mm , however the false alarm rate is more than 50%.

7.4.2

Effects of Long-term Variation

The effects of the slow change in µξ of the bonding area is studied in this section. Figure 7.25
shows the POD for different values of µξ , with σξ fixed at 0.5mm and a false alarm rate of
1%. For the A0 mode, the decrease in the mean PZT bonding area results in a small decrease
in POD. When the mean diameter value of the PZT sensor reaches 4mm, the performance is

123

1
a = 1mm
a = 2mm
a = 3mm
a = 5mm

0.5

0
0

0.2

0.4

PF

0.6

0.8

PD

PD

1

0.5

0
0

1

(a) A0 mode

0.2

0.4

PF

0.6

0.8

1

(b) S0 mode

Figure 7.24: The receiver operating characteristic curve for different defect sizes (a), and at
σξ = 0.5mm.
1
µξ =4mm

0.04

µξ =5mm

0.6

µξ =6mm

0.4

µξ =7mm

PD

PD

0.8

0.02

0.2
5
10
15
Delamination diameter (mm)

20

(a) A0 mode

0

5
10
15
Delamination diameter (mm)

20

(b) S0 mode

Figure 7.25: The probability of detection due to long-term PZT bonding coverage area
variation with σξ = 0.5mm.
affected significantly. Delamination diameters less than 12mm have less than 20 % detection
rate. For S0 modes, the POD is less than 2% for all values of µξ , and is practically zero as
the mean value decreases. For a given defect size, the effects of decreasing µξ on the POD
are shown in Figure 7.26. For the A0 mode, the POD is practically unaffected by a decrease
in µξ less than 30 %. For a reduction of more than 30 %, the POD starts decreasing rapidly.
For the S0 mode, there is not a significant change in the POD when there is a decrease in
µξ up to 40 %. This is due to the fact that the POD is already too low even without any

124

0

0
a = 5mm
a = 10mm
a = 20mm

−40

−0.2
% ∆ PD

% ∆ PD

−20

−60
−80

−0.4
−0.6

−40

−30

−20
% ∆µξ

−10

0

(a) A0 mode

−40

−30

−20
% ∆µξ

−10

0

(b) S0 mode

Figure 7.26: The probability of detection due to long-term PZT bonding coverage area as
the mean µξ changes, for different defect sizes.
long term variation.

7.5

Summary and Conclusions

The effects of cumulative impacts on the health of plane woven glass fiber composite plates
were investigated in this chapter. It was observed that this would result in a soft region near
the impact location due to loss in the effective stiffness. This impact damage resulted in
attenuation of the S0 mode GW waveforms, and also in a phase shift. A model based study
on the reliability of detecting impact using the S0 mode at 100 kHz is conducted. It was
shown that the S0 mode could successfully detect large changes in the stiffness (more than
15 %), however it is not very sensitive to small changes that are less than 5 %. Also the
detection of hidden delaminations in composite plate is investigated. It was found that the
A0 mode shows a good performance in detecting delaminations, even under large decrease
in the bonding coverage area. The S0 mode is highly insensitive to delaminations up to
20mm. This is due to the large wavelength of the S0 mode at 100 kHz, which is near 37mm.
Moreover, the location of the delamination at the midpoint of the plate thickness makes the
125

symmetric mode insensitive to it.

126

Chapter 8
Effects of PZT degradation on GW
Imaging

8.1

Introduction

The performance of imaging algorithms for identifying cracks in a riveted plate are discussed
in this chapter. An imaging algorithm fuses data from multiple PZT measurements to
triangulate the location of the damage. Thus, in theory, at least three PZT sensors are
needed to be able to construct an image that could properly indicate the damage location.
A slightly modified version of the delay-and-sum imaging algorithm is implemented and used
for detecting cracks, and identifying their locations. The effects of PZT sensor degradation
on the ability to detect cracks and on false alarms, using an image, is discussed.

8.2

Delay-and-Sum Imaging Algorithm

A two stage delay-and-sum imaging algorithm is implemented for detecting cracks in a
riveted plate. For each propagation path, a threshold is applied based on a given false alarm
criterion, and a known likelihood function of the distortion noise. The feature used is given
in equation (5.6). If the feature for the specific propagation path is above the threshold then

127

it is used in the imaging algorithm. If is it below the threshold, the data for that path is
discarded.
The imaging algorithm meshes the structural surface into an M × N grid. The objective
of the imaging algorithm is to illuminate each pixel in this grid appropriately so that the area
in the vicinity of a defect will have the highest illumination. Assume a given actuator-sensor
path i, from a total of J available paths. The baseline waveform is subtracted from the
time series waveform measured for the actuator-sensor path. The envelop of the difference
is scaled by the integral of the envelope of the baseline waveform, such that:

|H (∆ri (t))|

ui (t) =

tf

H

0

sbi (t)

(8.1)
dt

where H is the Hilbert transform, ∆ri (t) is the scatter signal, defined in equation (5.5), and
sbi (t) is the baseline signal. The delay-and-sum algorithm then maps the time series signal
ui (t) into the spatial coordinates represented by the mesh grid.
Consider a pixel p in the grid with a given coordinates (xp , yp ). The distance from the
PZT actuator with coordinates (xai , yia ) to the pixel, and the distance from the pixel p to
the PZT sensor with coordinates (xsi , yis ) are used to compute:

dpi =

xp − xai

2

+ yp − yia

2

+

xp − xsi

2

+ yp − yis

2

(8.2)

where dpi represents the distance travelled by a wave excited in path i, passing through pixel
p. If the wave group velocity V is known, then the time for the wave to propagate to the

128

pixel p, and then propagate to the sensor is given by:

τpi =

dpi
V

(8.3)

The group velocity of the wave mode could be calculated using the GW dispersion curves if
the material properties are known, or from measured waveforms as described in Section 4.3.
Then the illumination function for the meshed grid could be computed for each actuatorsensor path such that:
Ii (xp , yp ) = ui (τpi )

(8.4)

In other words, the value at the pixel p is illuminated by the corresponding time-delayed
value of the scaled scattered waveform of amplitude ui (τpi ). The calculated illumination
functions for each path are then added together by calculating their arithmetic mean:

I=

1
J

Ii (xp , yp )

(8.5)

i

A threshold is then applied on the final image, where any pixel value less than a given
threshold with respect to the maximum pixel value is set to zero.

8.3
8.3.1

Results for Riveted Plate Crack Detection
Imaging with no PZT Degradation

The data from the FEM model of the riveted plate geometry in Figure 4.2 is used for the
imaging algorithm. The S0 and A0 modes are both investigated separately, and also when
the signal paths due to both modes are added together. This would double the total available
129

(a) A0 mode

(b) S0 mode

(c) S0 + A0 modes

Figure 8.1: Results when there is no PZT degradation for a 4mm defect size without thresholding.
paths for constructing the image. Only PZT-3 is actuated, and the signals at all the other
PZT sensors are considered for the image construction. This results in 7 paths when each
mode is considered separately, and 14 paths when both A0 and S0 modes are considered to
construct a single image. A threshold is computed at each PZT sensor based on a 1 % false
alarm criterion with respect to variations in Young’s modulus of the adhesive. The standard
deviation of the adhesive modulus is set at σξ = 130M P a. Figure 8.1 shows the illumination
functions when there is a 4mm defect for the A0 mode, S0 mode, and the summation of both
modes. To see the indication of the defect clearly, the images in Figure 8.1 are thresholded
by setting any value less than 95 % of the maximum image value to zero. The thresholded
images for the case of 4mm, 7mm, and 10mm defects are shown in Figures 8.2, 8.3, and 8.4,
respectively. The rivet and the defect are indicated on the image. Also, the locations of the
eight PZTs are also shown for reference. In all cases, the imaging algorithm is able to detect
defects accurately when there is no variation in the PZT properties.

130

(a) A0 mode

(b) S0 mode

(c) S0 + A0 modes

Figure 8.2: Results when there is no PZT degradation for a 4mm defect size.

(a) A0 mode

(b) S0 mode

(c) S0 + A0 modes

Figure 8.3: Results when there is no PZT degradation for a 7mm defect size.

(a) A0 mode

(b) S0 mode

(c) S0 + A0 modes

Figure 8.4: Results when there is no PZT degradation for a 10mm defect size.

131

(a) A0 mode

(b) S0 mode

(c) S0 + A0 modes

Figure 8.5: Results when there is no PZT degradation for a 10mm defect size, and the signal
detection thresholds are set to a high value.

8.3.2

Imaging under PZT Degradation

To first understand the effects of the signal threshold selection on localizing defects, a threshold is computed at each PZT sensor based on a 1 % false alarm, and a high standard deviation of the adhesive modulus is set at σξ = 520M P a. This results in rejecting most of the
actuator-sensor signal paths. Only the paths with the highest sensitivity to defects would
be able to detect the defect. Figure 8.5 shows the constructed image using the A0 mode,
S0 mode, and the summation of the two modes. The defect length is 10mm, and there is
no variations in the PZT properties with respect to the baseline signal. Figure 8.6 shows
the resulting images after image thresholding where any pixel value less than 95% of the
maximum pixel is set to zero. Unlike the case when the signal thresholding value were set
to a low value, the image could not focus on the defect location due to insufficient number
of sensing paths available to construct the image. Thus, setting a high detection threshold
to take into account the high variances in the PZT sensor properties could result in missing
a defect even if the PZT properties did not change.
Next, the PZT adhesive Young’s modulus is varied, with a given signal detection threshold

132

(a) A0 mode

(b) S0 mode

(c) S0 + A0 modes

Figure 8.6: Results when there is no PZT degradation for a 10mm defect size after image
thresholding, and the signal detection thresholds are set to a high value.
set for an anticipated standard deviation of the adhesive modulus σξ = 130M P a. The case
when there is a defect of length 10mm, and the PZT adhesive Young’s modulus have degraded
to a value of 1560 MPa, from the baseline value of 2600 MPa is shown in Figure 8.7 for A0
mode, S0 mode, and the summation of both modes. The image threshold is set to 80% of
the maximum pixel value. In this case, there is a potential of missing the defect, since the
resulting image mostly focuses on the actuator (PZT-3). This is because all the sensing paths
will have a component ∆si (t), as given in equation (5.5). Since there is only one actuator
considered in this case, this will enhance the image focus on that actuator, possibly missing
the detection of the defect. The image when both A0 and S0 mode are both considered
(Figure 8.7(c) does show an indication near the defect, however the indication at PZT-3 is
much stronger. If the image threshold is set to a value higher than 80%, the defect would
be missed.
The case when there is no defect and the PZT adhesive Young’s modulus have degraded
to a value of 1560 MPa, from the baseline value of 2600 MPa is shown in Figure 8.8 for A0
mode, S0 mode, and the summation of both modes. Again, it is seen that using any wave

133

(a) A0 mode

(b) S0 mode

(c) S0 + A0 modes

Figure 8.7: The thresholded image when there is a PZT degradation and a 10mm defect
length.

(a) A0 mode

(b) S0 mode

(c) S0 + A0 modes

Figure 8.8: The thresholded image when there is a PZT degradation and no defect in the
geometry.
mode type, the image still focuses on the PZT actuator. It is difficult to tell the difference
between the case of no defect, and the presence of defect, when there is a signal contribution
due to the presence of PZT variations. Consequently the effect of PZT variation on the POD
would be significant.

134

8.4

Conclusions

The performance of a delay-and-sum GW imaging algorithm was shown to be affected by the
variations of the PZT properties. This could result in missing the defect or results in false
alarms. The detection and localization using imaging could be enhanced by utilizing regionbased processing, where the areas outside the region of interest could be discarded. The
algorithm considered in this chapter used a simple thresholding scheme to filter the image.
However to improve detection, only the region around the rivet could be considered, thus
removing the problem of focusing around the actuator due to PZT degradation. This ROIbased processing will increase the probability of detection, however it would still generate
false calls, depending on the type of processing, and the value of the signal detection threshold
used in the detection phase.

135

Chapter 9
Sensor Node Prototype for Guided
Wave SHM

9.1

Introduction

A prototype for data actuation and GW actuation, using a wireless module is built and
verified. As shown in Figure 2.7, the sensor node hardware includes the following modules:
(1) Data acquisition module, which should be designed to be able to acquire data depending
on the selected transducer; (2) Microcontroller coordinates the behavior of all the components
of the sensor node, stores acquired measurements and carry simple computations locally; (3)
The RF interface for wireless communication. Sensor nodes should be autonomous and have
their own energy supply. This provides a big challenge, and design should take into account
building a system that consumes the least amount of energy to increase the lifetime of the
network.

9.2

Hardware development

The two most common guided wave setups are the pulse-echo and pitch-catch methods. In
the pulse-echo method, an excitation signal is sent using a transducer, and then the same

136

Table 9.1: Properties of the Iris mote.
Component Properties
Sensing module
ADC
Resolution: 10 bit
Resolution: Sampling rate: 273 KHz
Processing module
µcontroller
Atmel ATmega1821, 7.37 MHz clock
Digital I/O pins
Storage
RAM: 8 KB
Flash memory: 512 KB
Wireless communication module
RF230 Radio IEEE 802.14.5 compliant
Bit Rate: 250 Kbps
Range: 50m (indoor), 300m (outdoor)

transducer listens to any echoes of wave scatter reflections from defects. In the pitch-catch
method, an excitation signal is sent through the structure using one transducer, and another
transducer catches the signal to determine if it contains any wave packets due to scatter
from a defect.
For actuation, it is usually desirable to use a narrowband excitation signal which would
minimize the effect of dispersion. The excitation signal that is most commonly used is the
Hanning window tone burst signal. By using a narrowband excitation signal, the sensed
signal has a similar bandwidth, and the wave reflections from defects are determined by the
peaks of the sensed signal envelop.
The sensor node used in this work is the Iris wireless module (also called Iris mote),
available from Memsic Corporation (www.memsic.com). The Iris mote hardware components
and properties are summarized in Table 9.1. It operates on two AA sized batteries, and it is
designed using hardware components with very low power consumption for maximizing the
batteries lifetime. The power limitation imposes constraints on the processing speed, since
increasing the microcontroller’s clock frequency will increase power consumption. For data

137

acquisition, the Iris mote has a 10 bit ADC and the sampling frequency is limited to 273 ksps,
which is not sufficient for sampling ultrasonic guided wave signals having bandwidth up to 500
kHz. Moreover, the Iris mote does not have an actuation interface that is needed for guided
wave excitation. However, it has a 51-pin connector that makes peripheral interfaces of the
microcontroller available for connection to external boards that could extend its functionality.
This connector was used to implement an extension circuit board for signal conditioning
that would allow the acquisition of narrowband ultrasonic signals with central frequency
up to 1 MHz from the piezoelectric transducer. The extension circuit board also provides
an actuation interface that could convert a digital square signal into an analog tone burst
narrowband signal for exciting the transducer. The extension circuit board is designed using
off-the-shelf circuit components and operational amplifiers.
The extension circuit board connected to the Iris mote and the block diagram of its circuit
components are shown in Figure 9.1. The extension circuit board is provided its own power
source using four AAA size batteries. The ADG1419 digital switch from analog devices
(www.analog.com) is used to select the actuation or sensing functionality of the transducer
by connecting it to the appropriate circuits. The switch position is controlled directly by
the Iris microcontroller using a single General Purpose I/O (GPIO) digital pin provided by
the 51-pin connector. The following subsections describe the sensing and actuation circuits
in more details.

9.2.1

Actuation circuit

When the switch is programmed to be in the actuator position, the actuation circuit will be
directly connected to the transducer. The input signal for the actuator circuit is provided
via a GPIO pin. The GPIO pin is programmed to behave as an output pin, and the micro138

Figure 9.1: The extension board for iris mote. On the left is the picture of the Iris mote
connected to the developed extension circuit board. On the right the block diagram of the
extension circuit board components.
controller can be programmed to provide a square wave of a given frequency and number of
cycles which is applied to a 2nd order bandpass filter to limit the bandwidth of the square
wave. The detailed circuit schematic for the bandpass filter is shown in Figure 9.2. The
filter is designed such that its center frequency can be varied to match the center frequency
of the square wave input from the GPIO. The filter’s center frequency is controlled by a
digital variable resistor R1 . The digital variable resistor used is the AD5272 chip from Analog Devices Inc. (www.analog.com). It has a 10-bit programmable wiper positions, and a
maximum resistance of 20 kΩ. The position of the rheostat wiper is controlled through a two
wire I2C serial interface with the microcontrller, allowing the microcontroller to have direct
control on the filter properties. The relationship between R1 and the filter center frequency
fc is found by solving the transfer function of the filter circuit, and it is given by:

R1 =

1
4.74 × 10−14 fc2 − 1.67 × 10−5

(9.1)

Since the maximum R1 value is 20 kΩ, the minimum excitation frequency that could
be supported is 38 kHz. Also, due to the limitation of the GPIO pin, resolution of the
139

Figure 9.2: The actuation circuit schematic.
digital rheostat, and bandwidth of the amplifiers used in the filter design, the maximum
supported excitation frequency is around 500 kHz. The second stage in the actuation circuit
is a noninverting amplifier that amplifies the filtered signal and drives the transducer. The
voltage gain of the amplifier is controlled by another digital rheostat R2 , whose wiper position
value is also programmed through the I2C serial interface with the microcontroller.

9.2.2

Sensing Circuit

When the switch is programmed to be in the sensor circuit position, the transducer is connected directly to the signal conditioning and sensing circuit. The first stage in the sensing
circuit is a charge amplifier. The piezoelectric wafer transducer behaves as a capacitor when
it vibrates, accumulating charge at its electrodes. A charge amplifier is used to transform
the charge into a voltage signal. The schematic for the charge amplifier is shown in Figure
9.3a. If the piezoelectric wafer transducer has a capacitance Cp , the relationship between

140

the output voltage of the charge amplifier and the piezoelectric wafer charge is given by:

H(ω) = −

jωRCp
1 + jωRC

(9.2)

Equation (9.2) shows that the charge amplifier behaves as a highpass filter with cutoff frequency ωn = 1/RC, and gain C/Cp . Thus by decreasing the value of the feedback capacitor
C relative to the piezoelectric wafer capacitance Cp , the amplifier gain can be increased. It
should be noted that R should be large to keep the cutoff frequency of the filter as low as
possible.
The second stage in the sensing circuit is the envelop detector, used to demodulate
narrowband ultrasonic signals to a baseband frequency so that the bandwidth is small enough
to be sampled by the ADC. The envelop detector implementation consists of two parts as
shown in Figure 9.3c. The first part is a full wave rectifier that extracts the absolute value
of the input signal. The second part is a 3rd order Butterworth lowpass filter with a cutoff
frequency of 60 KHz. The filter output is an envelop of the absolute value of the signal. The
filter also acts as an antialiasing filter before sampling. The sensor gain can be modified by
programming the wiper of the digital rheostat R3 , enabling the gain to vary between 0.5 dB
and 26 dB.
The output of the envelop detector is then connected to a multiplexer input. The multiplexer is part of the Atmega1281 microcontroller’s architecture and it has two possible
outputs: to the ADC or to one of the comparator’s inputs. The ADC and the comparator
are also part of the microcontroller’s architecture. If the multiplexer is programmed to connect the envelop detector output directly to the ADC input, the envelop signal would be
sampled directly when the ADC is enabled. If the multiplexer connects the envelop detector

141

(a)

(b)

(c)

Figure 9.3: The sensing circuit components. Figure 9.3a is the charge amplifier schematic,
Figure 9.3b is the threshold voltage schematic, and Figure 9.3c is the envelop detector
schematic
.
output to the input of the comparator, the envelop signal would be compared to a threshold
voltage applied to the comparator’s other input (Figure 9.1). When the envelop signal is
larger than the threshold voltage, the comparator’s output will go high, triggering a software interrupt, which is then used to enable the ADC to start sampling. This thresholding
method is useful for passive sensing applications. The programmable threshold voltage is
implemented using a voltage divider, shown in Figure 9.3b. A Zener diode (LM4041 from
Texas Instruments, www.ti.com) with a constant voltage of 1.225V is used to provide a stable

142

voltage to the voltage divider. A digital rheostat R3 enables changing the threshold voltage
in software. The relationship between the threshold voltage and R3 is given by:

R3 =

100Vthresh
1.225 − Vthresh

(9.3)

The value of R3 can vary between 60 Ω and 100 kΩ, using 1024 steps (10 bits). This allows
Vthresh to vary between 0.7mV to 610mV with steps around 0.6mV.

9.3

Software development

9.3.1

Sensor node application

The software for the Iris motes is written using TinyOS (see section 2.7.2). The mote
application programs the microcontroller and manages the interaction with its peripherals
including the extension circuit board and the wireless communication radio. The mote is
programmed so that its state would change depending on commands received wirelessly from
the base station. The state machine for the mote application is shown in Figure 9.4.
When the mote is powered up, it will attempt to read the values of the digital variable
resistors in the extension board through the I2C bus. There are four different variable
resistors controlling four different values described in section 9.2: Actuation bandpass filter
center frequency, actuation gain, sensing gain, and sensing threshold voltage. If reading
any of those values did not succeed, the mote will assume that the extension circuit board
is not connected and it will shutdown. If the read is successful, the mote will go to idle
listening mode, where it will wait to receive commands from the base station via the wireless
communication channel. There are four different commands from the base station that could

143

Figure 9.4: The sensor node state machine.
change the state of the mote, prompting it to take action.
1. Network Request command: The mote transmits the values of the variable resistors to
the base station, and then goes back to the Idle Listening state.
2. Actuate command: This command should also include data specifying the actuation
frequency, actuation gain, and the number of cycles required for the actuation signal.
The mote will then set the variable resistor values that correspond to the specified
frequency and gain requested. Then, the mote will output a square wave with the
given frequency and number of cycles at the actuation GPIO pin. If this is successful,
an acknowledgement is sent back to the base station, and the mote goes back to the
Idle Listening state.
144

3. Sense Enable command: This command prepares the sensor node for an active sensing
inspection. The command should include data specifying the sensing gain value. The
mote will start snooping the wireless channel for actuation commands transmitted to
other neighboring motes. When it detects an actuation command, it will enable its
sensing circuit and the ADC stores 100 data samples (corresponding to approximately
266 µs time window), and then transmits them to the base station. The mote will
then go back to the Idle Listening state.
4. Passive Sense command: This command puts the sensor node in passive sensing mode.
The command should also include data specifying the sensing gain and threshold voltage value. The mote will then enable its sensing and triggering circuit, and updates
the variable resistors with the received values. When the envelop signal level becomes
higher than the threshold voltage, the ADC is enabled and 100 data samples are stored
and transmitted to the base station. The mote will then go back to the Idle Listening
state.

9.3.2

Base station application

The base station consits of two parts: A gateway, and a PC. The gateway is an Iris mote
that is connected to the PC through a serial port. Its job is to relay data received from the
sensor nodes through the wireless channel to the serial port, and vice versa. The application
for the gateway uses TinyOS.
The PC has a user interface for viewing data received from the sensor nodes through
the gateway, or sending commands to the sensor nodes. The user interface is written in
MATLAB. It lists all the active sensor nodes in the network and their parameters. Then

145

commands could be sent to each node or broadcasted to all the nodes, allowing a practical way
to configure the WSN or the individual parameters of the motes. This design allows for both
active and passive sensing. For passive sensing, a Passive Sense command is broadcasted
to all the nodes in the network with a given threshold voltage and sensing gain, setting
the required sensitivity. This allows the passive monitoring of the structure for impact
damages or acoustic emissions from growing cracks. To put the network in active sensing
mode, a Sense Enable command is broadcast to the network. Then, an Actuate command it
transmitted to the desired sensor node. The user interface also plots the time measurements
received from the sensor nodes when an event is detected, allowing further signal processing
for damage detection and localization.

9.3.3

Wireless communication

The Iris mote uses the Atmel RF230 radio (Table 9.1), operating in the unlicensed ISM
band between 2.405 GHz and 2.480 GHz, with the ability to select between 11 different
communication channels. The radio is compliant to the IEEE 802.15.4 standard [70]. This
standard is very similar to Wi-Fi, but it is modified to suit low data rate and low power
applications. It has only a 250 kbps bit rate compared to 11 Mbps in Wi-Fi.
Due to the low data rate in the wireless communication physical layer, upper layer communication protocols that are common in Wi-Fi networks and the internet such as the
TCP/IP protocol are not a suitable option for wireless sensor networks. TinyOS has its own
implementation of a light weight and simple communication protocol for transmitting and
receiving data packets called Active Message. This protocol has very little overhead, hence
maximizing the data rate throughput. It is a best effort protocol, thus packet loss could
occur. To minimize the number of packets lost, the receiver is programmed so that it sends
146

4

2
Normalized Spectrum

GPIO pin
Actuator output
Volts

2

0

−2

0

5

10
15
Time (µs)

20

25

30

GPIO pin
Actuator Output
7 cycle Hanning

1.5
1
0.5
0

200

400
600
800
Frequency (kHz)

1000

(b)

(a)

Figure 9.5: The actuation circuit validation. Figure (a) shows the square wave as input
and filtered output; (b) shows the frequency content of the square wave input, the actuator
output and a 7-cycle Hanning window tone burst.
back an acknowledgment for every packet it receives. If the transmitter does not receive
an acknowledgement within a given deadline, it will transmit the same packet again until it
receives the acknowledgement. This helps decrease the amount of data loss when there is
substantial traffic in the wireless channel, but it does not guarantee it.

9.4
9.4.1

Validation
Actuator circuit validation

To investigate the performance of the actuator, one extension board was connected to one
mote, and the actuator circuit input (the same as the output of the GPIO pin from the mote)
and output were connected to an oscilloscope. From the base station, an Actuate command
was transmitted to the sensor node with the parameters: 320 KHz frequency, 3.5 dB gain,
and 2 cycles. The resulting output signal of the actuation circuit and the square wave used
as input are shown in Figure 9.5a. The frequency content of the two signals is shown in
Figure 9.5b. It could be seen that half the output signal power is within a bandwidth of
147

Figure 9.6: An experimental setup with two sensor nodes, each connected to a PZT piezoelectric wafer surface bonded to an aluminum plate. An oscilloscope is connected to the
outputs of the charge amplifier and envelop detector of sensor node 2 so that the analog
signals could be investigated.
75 KHz around a center frequency of 322 KHz. The output signal has a similar shape and
frequency content as a 7-cycle Hanning windowed tone burst, whose spectrum is also shown
in Figure 9.5b.

9.4.2

Sensor circuit validation

The experimental setup shown in Figure 9.6 was used to test the performance of the sensor
nodes for active guided wave inspection. Two PZT wafers are bonded to an aluminum plate
using epoxy, and each PZT wafer is connected to a sensor node. Sensor node 1 is used for
actuation and sensor node 2 is used for sensing. The outputs of the charge amplifier and
the envelop detector of sensor node 2 are connected to the oscilloscope so that the analog
signals could be investigated and compared to the digital samples transmitted to the base
station. From the base station, a Sense Enable command is transmitted to sensor node 2
with the sensing gain parameter specified as 6 dB. Then an Actuate command with the same
parameters used for the actuator validation is transmitted to sensor node 1. The resulting

148

Analytical envelop
Waveform
Interpolated data
Wireless data

1

Volts

0.5
0
−0.5
−1
0

50

100
Time (µs)

150

200

Figure 9.7: A analog output of the envelop detector is compared with the output of the
charge amplifier. The digital samples are also shown for comparison with the analog envelop
signal.
sensed signal and its envelop recorded on the oscilloscope and the envelop data samples
received at the base station are shown in Figure 9.7. The envelop signals have been scaled to
compensate for the sensing gain and attenuation of the low pass filter in the envelop detector.
The charge amplifier output is time shifted to compensate for the group delay of the low
pass filter. The envelop detector has good sensitivity to the data variation and is seen to
detect all the peaks of the wave packets. The interpolated digital signal samples transmitted
wirelessly to the base station closely match the analog envelop signal from the oscilloscope.

9.4.3

Power consumption

One of the major challenges in wireless sensor networks is the power constraint. A special
attention is needed to keep power consumption to a minimum in order to extend the lifetime
of the batteries provided. There are six batteries available in the sensor node. Two AA

149

Table 9.2: Power consumption depending on the software state.
Mode
Idle Listen
Sense Enable
Passive Sensing
ADC Enable
Actuate

Mote Current
17 mA
17 mA
17 mA
24 mA
24 mA

Ext. Board Current
0.5 mA
25 mA
25 mA
25 mA
11 mA

batteries solely powering the Iris mote, and four AAA batteries solely powering the extension
board. Table 9.2 shows the power consumption of the Iris mote and the extension circuit
board at each different mode.
• Idle Listening mode: All the amplifiers in the extension board are disabled, and the
current drawn is due to quiescent current of the amplifiers and the triggering circuit.
The current consumed by the Iris mote is mostly due to the wireless radio that is in
listening mode.
• Sensing Enabled and Passive Sensing modes: The amplifiers in the sensing
circuit are enabled and they consume most of the power. The Iris mote wireless radio
stays in listening mode and draws most of the 17 mA current, while the microprocessor
is in sleep mode and consumes negligible current.
• ADC enabled: The microprocessor wakes up consuming the extra current. However
the mote stays in this mode for just a few hundred microseconds, and then it will go
to the Idle Listening mode.
• Actuate mode: The actuation circuit is enabled, and the sensing circuit is disabled.
Since the actuation circuit has only two operational amplifiers, it consumes only 11
mA. The Iris mode should have its microprocessor enabled and thus consumes the
extra energy. The mote stays in the Actuate mode for just a few microseconds to send
150

the excitation signal and then goes to Idle Listening mode.
Using the values in Table 9.2, we could calculate the average lifetime of the batteries.
If a sensor node is used mainly for passive sensing, it would spend most of its time in the
Passive Sensing mode. The extension circuit board operates on four AAA batteries, each
typically has 1200 mAh of energy stored. The extension board continuously draws 25 mA,
and the energy stored in the batteries would then last up to 8 days. The mote operates on
two AA batteries, each typically stores 2700 mAh of energy. The mote batteries would then
last 13 days if the mote is continuously in the Passive Sensing mode.
If a sensor node is used mainly for active sensing, the mote would spend most of its
time in Idle listening mote. It could go to Actuate mode, Sensing Enabled mode or ADC
enabled mode for just a few microseconds but it will go back to the Idle Listening mode
once a test is over. Based on this, the extension circuit board batteries could last more than
a year, but the mote batteries could last 13 days since the radio should keep listening for
commands. The batteries life in the mote could be extended by using duty cycling in the
radio, decreasing its power consumption by 90% [71].

151

Chapter 10
Conclusions and Future Work

10.1

Contributions and Conclusion

An SHM system used for monitoring the health of a structure faces many challenges for
the use in practical applications. This dissertation addresses those challenges, specifically
the use of SHM systems for monitoring common aerospace structures which include riveted
aluminum plates and fiber reinforced composite plates. The main contributions of this
dissertation are:
• A finite element model for simulating guided waves excited and sensed by thin PZT
films is verified by comparison with existing analytical models and with experiments.
This FE model is then used for GW data generation that is used for investigating the
performance of GW-PZT SHM systems.
• An SHM system is subject to performance degradation due to changes in the PZT
sensors just the same as the structure itself could incur changes. A novel model-based
probability of detection approach for evaluating the performance of guided wave SHM
system was formulated. This method evaluates the probability of detecting defects
using GW-PZT with variations in the properties of the PZT sensors. Three degradation modes for the PZT sensors were identified and studied: (i) Adhesive Young’s
modulus degradation, (ii) adhesive thickness degradation, and (iii) adhesive coverage
152

area degradation. The variations in the PZT sensors due to changing environmental
conditions were modeled as stochastic processes. Permanent changes in the PZT sensors due to damage of the PZT or permanent change in its properties were modeled
as variations in the statistics of a stochastic process. The performance and reliability
of a GW-SHM system was then quantitatively evaluated by finding the probability of
detection for a given defect type, and also the probability of false alarms.
• The performance of GW-PZT for detecting fatigue cracks in riveted aluminum plates
was investigated. The use of both A0 and S0 modes for damage detection was investigated. The A0 mode had a higher sensitivity to fatigue cracks, and less sensitivity to
the variations in PZT sensors. Under small variations in PZT sensors, the A0 mode
could detect cracks larger than 2mm with nearly a 100 % probability. However the
S0 mode does not have such a good performance, and the probability of detection was
less than 40 % for cracks as large as 8mm. It was observed that the variations in the
adhesive coverage area had the most significant effect on the probability of detection.
• A two-stage modified approach for the delay-and-sum guided wave imaging algorithm
was evaluated to understand the effects of PZT degradation on damage detection and
localization. The PZT degradation focuses the image at the actuators. Using global
image thresholding methods results in decreasing the probability of detection since the
indication focused at the actuators mask the defect indication. Also it was demonstrated that the choice the detection threshold affects the detection. Region based
image processing is needed in order to minimize the false alarm rate, and maximize
the probability of detection.
• The effects of cumulative impacts on glass fiber composites is investigated experimen153

tally. Load-displacement measurements, and energy measurements showed that the
impacts resulted in a reduction of stiffness at the region in the vicinity of the impact
location. GW-PZT measurements using the S0 mode showed a good correlation between the impact damage growth and changes in the waveform amplitude and phase.
The model-based POD study was then used to assess the performance of GW-PZT for
detecting impact damage based on the experimental results. The effects of variation
in the PZT bonding coverage area is investigated. The S0 mode could detect stiffness
changes up to 15 % of the pristine structure’s stiffness with near 100 % probability
and less than 1 % false alarm rate. However, changes in stiffness less than 5 % could
be detected reliably only if the false alarm rate is higher than 20 %.
• The performance of GW-PZT for detecting a small hidden delamination in a laminated
composite structure is investigated. A circular delamination up to 20mm diameter is
modeled at the midpoint of the thickness of a glass fiber composite plate. The S0
mode was found to be completely insensitive to such a delamination, and it could not
be used for detecting such defects. However, the A0 mode could be used for detecting
delamination with high sensitivity. Delamination of 2mm diameter are detected with
100 % probability and 1 % false alarm, if the PZT variability is small.
• To address the issue of data delivery and communication in SHM systems, a proofof-concept wireless sensor node prototype was developed specifically for the data acquisition and actuation of guided waves using PZT sensors. The use of the wireless
sensor node with GW-PZT was verified experimentally. The challenge of using high
frequency guided wave signals on the power consumption of the sensor node was addressed by measuring the envelop of the GW signal, which reduces the requirement on

154

the sampling rate.

10.2

Looking Forward

The ability to quantify the performance of an SHM system for detecting imminent defects is
a crucial step toward their large-scale deployment in practical structures. A comprehensive
model-based method was proposed in this dissertation and applied to different scenarios to
investigate the feasibility of using GW-PZT. However, there remains a multitude of challenges
that need to be addressed to come closer to such realization.
The combined effects of multiple degradation modes of the PZT sensor, and also the variations in the structure itself that is not due to defects could affect the system performance.
Monte-Carlo simulations that take into account multiple degradation modes and their corresponding probability distributions need to be used for evaluating the system performance.
Additive noise due to measurement equipment could significantly affect the probability of
detection. If an assessment is required for the overall system, and not just for the degrading PZT sensors, this noise needs to be incorporated in the formulation of the POD. Also,
different signal features could be considered to compare their performance in terms of the
POD.
The statistics of the stochastic process that represents the PZT variation need to be
estimated using experiments. This is specific for the SHM system that needs to be designed,
since each system encounters its own environment, and that they will incur different stochastic variations. Moreover, apart from the effects of the adhesive bonding of the PZT, other
effects, such as the random breaking shapes of the PZT sensor due to impacts need to be
modeled and investigated. Another important factor in designing an SHM system is the

155

ability to use such POD tools to estimate the lifetime for the SHM system relative to the
structure itself. Obviously, the SHM system’s lifetime estimate should be similar to that
of the structure. The SHM system’s performance should be designed such that the minimum detectable defect should be smaller than the defect size that could result in complete
structural failure.
The separation of the multiple GW modes was implemented by bonding co-located PZT
sensors at the top and bottom surfaces of the structure. However, this is not possible in some
geometries due to the infeasibility of attaching sensors to both surfaces of the structure, or
the structure surfaces are not symmetrical. In this case, different methods needs to be used
for separating the guided wave modes. One methods is to use a tuned PZT array, similar
to the implementation of a comb transducer (see for example the work by Rose et al. [72]),
and optimize its spectral response such that it has a near flat response in a given frequency
bandwidth for one wave mode, and a zero response (or a very small response) to the other
wave modes.
Apart from the requirement of quantifying the reliability of SHM systems, the main
hindrance in their widespread deployment in industry is the data collection from the sensors.
This data collection is required to be in real-time or near real-time. The vast number of
sensors that might be required to monitor a structural area would make it prohibitively
expensive to use physical wiring to connect all sensors to main base station. The use of
wireless sensor technology helps solve this issue, but current technology for wireless sensor
nodes that could be used with guided wave are still too large for practical mounting and
permanent installation in structures. Advances in MEMS technology and energy harvesting
could enable the reduction of sensor node size to that comparable with PZT sensors, which
makes it practical for embedding and surface mounting in real structures.
156

APPENDICES

157

Appendix A
Material Properties

A.1

Mechanical Properties

The mechanical properties for the PZT piezoelectric materials, aluminum plate, and epoxy
adhesive used in this dissertation are shown in Table A.1.
Table A.1: Mechanical properties of material.
Property

Unit

Aluminum

CFRP

Epoxy

PZT-5A

E11

GPA

71.00

23.1

2.60

76.00

E22

GPA

71.00

23.1

2.60

76 .00

E33

GPA

71.00

6.9

2.60

56 .00

ν23

0.32

0.28

0.30

0.44

ν13

0.32

0.28

0.30

0.44

ν12

0.32

0.355

0.30

0.35

G23

GPA

26.90

2.54

1.00

21.05

G13

GPA

26.90

2.54

1.00

21.05

G12

GPA

26.90

3.2

1.00

22.57

ρ

kg/m3

2793

1850

1100

7800

158

A.2

PZT Electric Properties

The electromechanical coupling coefficient for the PZT-5A used in this dissertation is given
by:



0




d=



0

0

0

584 0 


× 10−12 m/V
0
0
0 584 0 0 


−190 −190 450 0
0 0

(A.1)

and the electrical permittivity of PZT-5A in matrix form is:



0
0
 1900


ε= 0
1900
0


0
0
1850




 × ε0



where ε0 is the permittivity in free space and is given by: ε0 = 8.8542 × 10−12 F/m.

159

(A.2)

Appendix B
FEM Simulation Signals

B.1

Riveted Plate Signals

The resulting waveform signals for the riveted aluminum plate geometry in Figure 4.2 for
each of the seven PZT sensors, with PZT-3 as the actuator, are shown in this section. Figures
B.1 and B.3 show the waveforms for the case of no defect and for the 9mm defect length
case, using the S0 mode, and the A0 mode respectively. No PZT sensor degradation or
variations are assumed, and perfect baseline subtraction is obtained, and is shown in the
figure. Figures B.2 and B.4 show the baseline signal the same as in Figures B.1 and B.1, and
a signal with a degrading PZT adhesive Young’s modulus. The baseline subtraction result
due to just sensor variation is also shown.

B.2

Composite Plate Signals

The resulting waveform signals for the riveted aluminum plate geometry in Figure 4.5 for each
of the nine PZT sensors, with PZT-1 as the actuator, are shown in this section. Figures B.5
and B.7 show the waveforms for the case of no delamination and for a 20mm delamination,
using the S0 and A0 modes, respectively. No PZT sensor degradation or variations are
assumed, and perfect baseline subtraction is obtained. Figures B.6 and B.8 show the baseline
signal the same as in Figures B.5 and B.7, and a signal with a peeling PZT adhesive. The
160

(a) PZT-1

0
−10
20

40
60
Time (µs)

10
0
−10
0

80

(d) PZT-5

80

10
0
−10
20

40
60
Time (µs)

0
−10
0

80

40
60
Time (µs)

0
−10
20

40
60
Time (µs)

80

10
0
−10
0

20

40
60
Time (µs)

(g) PZT-8
Voltage (mV)

20
10
0
−10
0

80

20

10

0

20

(f) PZT-7

20
Voltage (mV)

Voltage (mV)

40
60
Time (µs)

10

(e) PZT-6

20

0

20

Voltage (mV)

0

20
Voltage (mV)

10

(c) PZT-4

20
Voltage (mV)

No defect
10mm defect
defect scatter

20
Voltage (mV)

(b) PZT-2

20

40
60
Time (µs)

80

Figure B.1: S0 mode waveforms in a riveted aluminum plate, with defect effects.
baseline subtraction result due to just sensor variation is also shown.

161

80

(a) PZT-1

10
0
−10
20

40
60
Time (µs)

20

80

10
0
−10
0

(d) PZT-5

40
60
Time (µs)

80

10
0
−10
20

40
60
Time (µs)

0
−10
0

80

40
60
Time (µs)

80

20

10
0
−10
0

20

(f) PZT-7

20
Voltage (mV)

Voltage (mV)

20

10

(e) PZT-6

20

0

20

Voltage (mV)

0

(c) PZT-4
Voltage (mV)

Baseline
Degraded PZT
Subtraction

Voltage (mV)

Voltage (mV)

20

(b) PZT-2

20

40
60
Time (µs)

80

10
0
−10
0

20

40
60
Time (µs)

80

(g) PZT-8
Voltage (mV)

20
10
0
−10
0

20

40
60
Time (µs)

80

Figure B.2: S0 mode waveforms in a riveted aluminum plate, with sensor degradation effects.

162

(a) PZT-1

20

40
60
Time (µs)

0

−50
0

80

(d) PZT-5

−50
0

80

Voltage (mV)

0

20

40
60
Time (µs)

40
60
Time (µs)

20

40
60
Time (µs)

80

0

−50
0

20

40
60
Time (µs)

(g) PZT-8
50

0

−50
0

80

50

0

−50
0

80

20

(f) PZT-7

50

Voltage (mV)

Voltage (mV)

40
60
Time (µs)

0

(e) PZT-6

50

−50
0

20

Voltage (mV)

−50
0

50
Voltage (mV)

0

(c) PZT-4

50
Voltage (mV)

No defect
10mm defect
defect scatter

50
Voltage (mV)

(b) PZT-2

20

40
60
Time (µs)

80

Figure B.3: A0 mode waveforms in a riveted aluminum plate, with defect effects.

163

80

(a) PZT-1

0

20

40
60
Time (µs)

50

0

−50
0

80

(d) PZT-5

40
60
Time (µs)

−50
0

80

0

20

40
60
Time (µs)

40
60
Time (µs)

80

50

0

−50
0

80

20

(f) PZT-7

50
Voltage (mV)

Voltage (mV)

20

0

(e) PZT-6

50

−50
0

50

Voltage (mV)

−50
0

(c) PZT-4
Voltage (mV)

Baseline
Degraded PZT
Subtraction

Voltage (mV)

Voltage (mV)

50

(b) PZT-2

20

40
60
Time (µs)

80

0

−50
0

20

40
60
Time (µs)

80

(g) PZT-8
Voltage (mV)

50

0

−50
0

20

40
60
Time (µs)

80

Figure B.4: A0 mode waveforms in a riveted aluminum plate, with sensor degradation effects.

164

(a) PZT-1

(b) PZT-2

5
0
−5
−10

5
0
−5
−10

100
Time (µs)

150

0

5
0
−5
−10

−5

0

Voltage (mV)

−5
−10

100
Time (µs)

100
Time (µs)

5
0
−5

0

150

150

100
Time (µs)

150

10

5
0
−5

0

50

(i) PZT-10

−10
50

150

−10
50

10

0

100
Time (µs)

10

(h) PZT-9

5

50

(f) PZT-7

0

150

10
Voltage (mV)

0

−10

100
Time (µs)

−5

150

No defect
10mm defect
defect scatter

5

(g) PZT-8

0

100
Time (µs)

10
Voltage (mV)

Voltage (mV)

10

50

0

(e) PZT-6

(d) PZT-4

0

5

−10
50

Voltage (mV)

50

Voltage (mV)

0

10
Voltage (mV)

10
Voltage (mV)

Voltage (mV)

10

(c) PZT-3

5
0
−5
−10

50

100
Time (µs)

150

0

50

100
Time (µs)

Figure B.5: S0 mode waveforms in a composite plate, with defect effects.

165

150

(a) PZT-1

(b) PZT-2

5
0
−5

100
Time (µs)

0
−5
−10
0

150

5
0
−5

50

100
Time (µs)

5

−5
−10
0

150

100
Time (µs)

Voltage (mV)

0
−5

100
Time (µs)

150

0
−5

50

100
Time (µs)

150

(i) PZT-10
10

5
0
−5
−10
0

150

5

−10
0

150

10

5

100
Time (µs)

10

(h) PZT-9

10
Voltage (mV)

50

50

(f) PZT-7

0

(g) PZT-8

50

−5
−10
0

150

No defect
10mm defect
defect scatter

10
Voltage (mV)

Voltage (mV)

10

−10
0

100
Time (µs)

0

(e) PZT-6

(d) PZT-4

−10
0

50

5

Voltage (mV)

50

5

Voltage (mV)

−10
0

10
Voltage (mV)

10
Voltage (mV)

Voltage (mV)

10

(c) PZT-3

50

100
Time (µs)

150

5
0
−5
−10
0

50

100
Time (µs)

150

Figure B.6: S0 mode waveforms in a composite plate, with sensor degradation effects.

166

(b) PZT-2

0
−20

20
0
−20

100
Time (µs)

150

0

Voltage (mV)

Voltage (mV)

20
0
−20

150

0

50

100
Time (µs)

150

No defect
10mm defect
defect scatter

0

0

0
−20
100
Time (µs)

150

50

100
Time (µs)

0

0

150

50

100
Time (µs)

150

(i) PZT-10

20
0

0

150

−20

−20
50

100
Time (µs)

20

(h) PZT-9

20

50

(f) PZT-7

−20

Voltage (mV)

Voltage (mV)

100
Time (µs)

20

(g) PZT-8

0

50

(e) PZT-6

(d) PZT-4

0

0

Voltage (mV)

50

20

−20

Voltage (mV)

0

(c) PZT-3
Voltage (mV)

20

Voltage (mV)

Voltage (mV)

(a) PZT-1

20
0
−20

50

100
Time (µs)

150

0

50

100
Time (µs)

Figure B.7: A0 mode waveforms in a composite plate, with defect effects.

167

150

(b) PZT-2

0
−20

20
0
−20

100
Time (µs)

150

0

Voltage (mV)

Voltage (mV)

20
0
−20

150

0

50

100
Time (µs)

150

No defect
10mm defect
defect scatter

0

0

0
−20
100
Time (µs)

150

50

100
Time (µs)

0

0

150

50

100
Time (µs)

150

(i) PZT-10

20
0

0

150

−20

−20
50

100
Time (µs)

20

(h) PZT-9

20

50

(f) PZT-7

−20

Voltage (mV)

Voltage (mV)

100
Time (µs)

20

(g) PZT-8

0

50

(e) PZT-6

(d) PZT-4

0

0

Voltage (mV)

50

20

−20

Voltage (mV)

0

(c) PZT-3
Voltage (mV)

20

Voltage (mV)

Voltage (mV)

(a) PZT-1

20
0
−20

50

100
Time (µs)

150

0

50

100
Time (µs)

150

Figure B.8: A0 mode waveforms in a composite plate, with sensor degradation effects.

168

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