Modeling of Piezo-Induced Ultrasonic Wave Propagation for Structural Health Monitoring

MODELING OF PIEZO-INDUCED ULTRASONIC WAVE PROPAGATION FOR STRUCTURAL HEALTH MONITORING

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND ASTRONAUTICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Kuldeep P. Lonkar August 2013

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/sm236hy1179

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Fu-Kuo Chang, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Richard Christensen

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Kincho Law

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

The process of implementing a damage detection and characterization strategy for engineering structures is referred to as structural health monitoring (SHM). Recently, damage detection using piezoelectric actuators and sensors has gained extensive at- traction. Piezoelectric actuators are used to induce elastic stress waves that can propagate for long distances in thin-walled structures with very little amplitude loss enabling inspection of large areas of a structure. These waves interact with damage and provide information on location, size, and type of damage; this information is extracted from sensor signals by diagnostic algorithms. However, these signals are sensitive to the operating conditions. Change in ambient temperature or loading conditions may affect the wave propagation and the sensor response leading to inaccurate diagnostics. Hence, fundamental understanding of the physics of wave propagation, their interaction with the structure, and the influence of varying operating conditions is crucial for developing appropriate diagnostic algorithms. Therefore, the prime objectives of this investigation are: (i) develop an efficient and accurate numerical model to simulate the sensor signals and the piezo-induced acoustoelastic wave propagation in prestressed homogeneous and layered media, (ii) study the effect of load on wave propagation (acoustoelastic effect) and varying ambient temperature on the sensor signals. A numerical model called Piezo-Enabled Spectral Element Analysis (PESEA), based on spectral element method is developed. PESEA can accurately and efficiently simulate ultrasonic wave propagation in complex structures with built-in piezoelectric sensor network. Experiments and simulations are performed on metallic and composite structures to verify and validate the accuracy of PESEA. Fatigue cracks in metallic

iv structures and debond/delamination is laminated composite structures are modeled by separating the nodes to create volume split. Simulations are carried out to show the wave-damage interaction and the scatter in the sensor signals due to damage. These simulation results show that PESEA can be used as a powerful tool to gain physical insights into the effect of different types of damage on wave propagation and sensor response. The influence of loading on ultrasonic waves actuated and sensed by piezoelectric sensors in aluminum plate is studied. A numerical and experimental study of axially stressed aluminum plates with surface-mounted piezoelectric sensors is carried out to investigate the dependence of wave velocity on applied load. Simi- larly, simulations and experiments are presented to understand how the piezoelectric sensor signal amplitude changes with adhesives of different thickness and material when the structure is exposed to elevated temperature. In SHM, piezo-induced ultrasonic waves are used for damage detection and localization. The accuracy of damage localization depends strongly depends on the a priori knowledge of the wave velocity. The estimation of the wave velocity for complex structures is challenging since analytical solutions only exist for simple structures. Hence, PESEA simulations are used to estimate the wave velocity profile for a given structures, which is then used for the offline training of the damage diagnostic imaging algorithm. PESEA simulations are carried out to validate the proposed model-assisted damage diagnostics. Accuracy of damage detection and localization also depends on the number of piezoelectric sensors and their placement. In this dissertation, a methodology is presented to utilize PESEA simulations to optimize the sensor network by maximizing the probability of damage detection. PESEA is used to understand the effect of crack and uncertainty in material properties on the sensor signal. This information is then used in a genetic algorithm based optimization code. This code maximizes the probability of detection and gives an optimized sensor network. The proposed methodology is used to optimize the piezoelectric sensor network for a stiffened aluminum panel. This demonstrates that PESEA can be used to optimize sensor placement for a given structure and improve the accuracy of the diagnostic algorithms.

v Acknowledgments

First of all, I would like to express my gratitude to Professor Fu-Kuo Chang for his guidance and support during the course of my PhD. I consider myself very fortunate that I got a chance to work with him. I would like to thank Professor Richard Christensen and Professor Kincho Law for their invaluable suggestions and careful review of this work. I also wish to thank Professor Debbie Senesky for serving on my oral examination committee and Professor Peter Pinsky for chairing the examination committee. I am also grateful to Jayanthi Subramaniam, Haruko Makitani, Barbara Briscoe, Ralph Levine, Robin Murphy, Liza Julian, and Patrick Ferguson for their administrative support. In addition, I gratefully acknowledge the financial support from the National Aero- nautics and Space Administration (NASA), the Air Force Office of Scientific Research (AFOSR), and Alcoa Inc. Finally, I would like to thank my friends and family. I thank all the current and past members of the Structures and Composites Laboratory (SACL). Specifically, I wish to thank Surajit Roy, Cecilia Larrosa, Zhiqiang (Steve) Guo, Nathan Salowitz, and Yu-Hung Li for their support, valuable discussions, and friendship. I am highly in- debted to my friends especially Mayank Agarwal, Shrey Kumar Shahi, Supreet Singh Bahga, Bhupesh Chandra, Uzma Hussain Barlaskar, Rohit Gupta, Manu Bansal, and Nitin Dua for their help, support, and making my stay at Stanford memorable. I will always be grateful to my wife, Amrita, for her unconditional love and care that helped me focus on my research. I wish to thank my sister, Dhanashree, for her encouragement. Finally, I dedicate this thesis to my parents who have always supported me in all my decisions.

vi Contents

Abstract iv

Acknowledgments vi

1 Introduction1 1.1 Structural Health Monitoring...... 1 1.2 Piezoelectric Materials...... 2 1.3 Wave Propagation in Thin-Walled Structures...... 3 1.4 Challenges in SHM based on Piezo-Induced Waves...... 5 1.4.1 Modeling of Piezo-Induced Ultrasonic Waves...... 5 1.4.2 Eﬀect of Operating Conditions...... 6 1.4.3 Diagnostic Algorithms and Optimal Sensor Placement.....8

2 Problem Statement 11

3 Method of Approach 13 3.1 Modeling of Piezo-Induced Ultrasonic Wave Propagation...... 13 3.2 Effect of Operating Conditions...... 15 3.2.1 Effect of Load...... 15 3.2.2 Effect of Temperature...... 15 3.3 Model-assisted Damage Diagnostics...... 16 3.4 Model-assisted Sensor Network Optimization...... 16

vii 4 Governing Equations for Wave Propagation 17 4.1 Introduction...... 17 4.2 Equations of Motion...... 17 4.3 Coupled Governing Equations...... 22 4.3.1 Stress-Free Initial State...... 24 4.4 Conclusions...... 25

5 Spectral Element Method 26 5.1 Introduction...... 26 5.2 Weak Formulation...... 27 5.3 Solid Spectral Element...... 27 5.3.1 Matrix Representation of Weak Form...... 28 5.3.2 Global System of Equations...... 32 5.3.3 Rayleigh Damping...... 33 5.3.4 Global System of Equations for Stress-Free Initial State.... 34 5.4 Numerical Integration in Spatial Domain...... 35 5.4.1 Nodal Quadrature...... 36 5.5 Time Integration...... 37 5.5.1 Central Diﬀerence Method...... 37 5.6 Layered Solid Spectral Element...... 39 5.6.1 Numerical Integration of Stiﬀness Matrix...... 40 5.7 Piezo-Enabled Spectral Element Analysis...... 42 5.7.1 Implementation of PESEA...... 42

6 Verification & Validation of PESEA 44 6.1 Validation of Solid Spectral Element...... 44 6.2 Verification of Layered Solid Spectral Element...... 47 6.2.1 Accuracy of LSSE...... 48 6.2.2 Efficiency of LSSE...... 56 6.3 Validation of Layered Solid Spectral Element...... 57 6.4 Validation of Acoustoelastic Formulation...... 59 6.4.1 Experimental Setup...... 59

viii 6.4.2 Simulations...... 63 6.4.3 Results...... 67 6.5 Conclusions...... 67

7 Wave Propagation in Structures 71 7.1 Introduction...... 71 7.2 Wave Propagation in Metallic Structures...... 72 7.3 Wave Propagation in Composite Structures...... 75 7.3.1 Modeling of Delamination & Debond...... 78 7.3.2 Multilayered Composite Plate with Delamination...... 78 7.3.3 Stiﬀened Composite Plate with Delamination/Debond.... 85 7.4 Conclusions...... 91

8 Eﬀect of Elevated Temperature on Sensor Signal 92 8.1 Introduction...... 92 8.2 Experiments and PESEA Simulations...... 94 8.2.1 Experimental Setup...... 94 8.2.2 Variation in Material Properties with Temperature...... 95 8.2.3 PESEA Simulations...... 97 8.3 Parametric Studies...... 100 8.3.1 Amplitude Variation with Adhesive CW2400...... 102 8.4 Conclusions...... 103

9 Model-Assisted Damage Diagnostics 105 9.1 Introduction...... 105 9.2 Method of Approach for Model-Assisted Damage Diagnostics..... 107 9.2.1 Damage Diagnostics Imaging Algorithm...... 107 9.2.2 PESEA Simulations...... 111 9.2.3 Temperature Compensation Model...... 113 9.3 Performance of Model-Assisted Damage Diagnostics...... 116 9.3.1 Oﬄine Training to Get ToF...... 117 9.3.2 Simulations for Panel in Pristine and Damaged State..... 117

ix 9.3.2.1 Training of Temperature Compensation Model.... 120 9.3.2.2 Compensation of Temperature Eﬀect...... 120 9.3.3 Results and Discussion...... 121 9.4 Conclusions...... 125

10 Model-Assisted Sensor Network Design 127 10.1 Introduction...... 127 10.2 Method of Approach for Model-Assisted Sensor Network Optimization 128

10.2.1 Probability of Detection of Sensor Network (PODnet)..... 131 10.3 PESEA Simulations to Compute DDD ...... 132 10.3.1 Stochastic Spectral Element Method...... 134 10.3.2 Monte Carlo Simulations...... 134 10.3.2.1 Uncertainty in Material Properties...... 135 10.3.2.2 Uncertainty in Material Properties...... 137 10.4 Optimal Sensor Network for a Stiﬀened Panel...... 142 10.5 Conclusions...... 144

11 Conclusions 147 11.1 Concluding Remarks...... 147 11.2 Contributions...... 149

A Piezoelectric Constitutive Equations 150

B High Order Shape Functions for Spectral Elements 151

C Material Properties 153 C.1 Mechanical Properties...... 153 C.2 Electrical Properties...... 154 C.3 Third Order Elastic Constants...... 154

D Error Quantiﬁcation 155

Bibliography 157

x List of Tables

5.1 GLL nodal and integration points and weights for interval [-1,1]... 36

6.1 Error in velocities of S0 and A0 modes in x-direction for layup [02/904]s 51

6.2 Error in velocities of S0 and A0 modes in x-direction for layup [902/02/902]s 51

6.3 Error in velocities (in m/s) of S0 and A0 modes x-direction for layup

[0/90/45/-45]s and [45/-45/0/90]s ...... 56 6.4 Comparison of computational costs for SSE and LSSE...... 56

8.1 Variation in the electrical properties of PZT...... 95 8.2 Variation in the mechanical properties of PZT and aluminum 2024-T3 96

8.3 Variation in the shear modulus of Hysol R EA 9696 and CW2400... 102

10.1 Uncertainty in material properties of PZT, aluminum, and adhesive. 137

B.1 GLL nodal and integration points for interval [-1,1]...... 152

C.1 Mechanical properties...... 153 C.2 Third order elastic constants...... 154

xi List of Figures

1.1 Structure with piezoelectric sensors for structural health monitoring.2 1.2 Piezoelectric material (adapted from [6])...... 3

1.3 Distribution of displacement for fundamental symmetric (S0) and an-

tisymmetric (A0) modes (adapted from [5])...... 4 1.4 Dispersion curve for aluminum (adapted from [5])...... 4 1.5 Eﬀect of Temperature on Sensor Signal (300 kHz)...... 7

2.1 Plate with surface-mounted piezoelectric sensors...... 12 2.2 Piezo-induced ultrasonic wave propagation in metallic or laminated composite structure...... 12

3.1 Convergence comparison between solid spectral element and linear ﬁ- nite element method (adapted from [6])...... 14

4.1 Coordinates of a material point at natural ξ, initial X, and ﬁnal x conﬁguration of a pre-deformed body...... 18 4.2 Plate with surface-mounted piezoelectric sensor...... 22

5.1 Solid spectral element (SSE) of order 4×4×4...... 28 5.2 Layered solid spectral element (LSSE) of order 4×4×4...... 40 5.3 Simpsons 1/3 rule in stacking direction...... 41 5.4 Schematic of how PESEA works...... 43 5.5 Implementation of PESEA to model acoustoelastic wave propagation 43

6.1 Aluminum plate with T-stiﬀener...... 45

xii 6.2 Mesh for the half geometry created in Abaqus/CAE R ...... 46 6.3 Five-cycle tone burst input signal (central frequency of 200 kHz)... 46 6.4 Comparison between experimental & simulated signal for sensor S1. 47 6.5 Geometric conﬁguration of the composite plate considered for veriﬁcation 48

6.6 Simulated signals for sensor S for composite plate with layup [02/904]s using: (a) 3 SSEs in thickness (blue curve), (b) 1 SSE with smeared material properties (red curve), and (c) 1 LSSE (black curve)..... 49

6.7 Veriﬁcation LSSE for layup [02/904]s ...... 50

6.8 Veriﬁcation LSSE for layup [02/902/02]s ...... 52 6.9 Comparison between simulated signals for quasi-isotropic CFRP plates

with layup (a) [0/90/45/-45]s and (b) [45/-45/0/90]s ...... 54 6.10 Surface plots of out-of-plane displacement at 36 µs for quasi-isotropic

CFRP for layup (a) [0/90/45/-45]s and [45/-45/0/90]s using 1 SSE in

thickness with smeared properties (b) [0/90/45/-45]s using 1 LSSE in

thickness and (c) [45/-45/0/90]s using 1 LSSE in thickness...... 55

6.11 CFRP T800S/3900-2 composite plate with layup [02/904]s ...... 57 6.12 Geometric conﬁguration of the composite plate considered for validation 58 6.13 Data acquisition hardware and software...... 58

6.14 Simulated signals for sensor S1 for composite plate with layup [02/904]s using: (a) experiment (blue curve), (b) simulation using 1 SSE with smeared material properties (red curve), and (c) simulation using 1 LSSE (black curve)...... 59

6.15 Simulated signals for sensor S2 for composite plate with layup [02/904]s using: (a) experiment (blue curve), (b) simulation using 1 SSE with smeared material properties (red curve), and (c) simulation using 1 LSSE (black curve)...... 60 6.16 Composite plate: comparison between experimental and simulated signal for sensor S1...... 61 6.17 Composite plate: comparison between experimental and simulated signal for sensor S2...... 62 6.18 Geometric conﬁguration and test setup for the experiments...... 64

xiii 6.19 Calculating velocity of S0 mode using zero crossing times...... 65 6.20 Mesh of aluminum plate created in Abaqus/CAE R ...... 65 6.21 Contours of displacement in (a) x, (b) y, and (c) z direction for 1 kip applied load...... 66 6.22 Relative change in group velocity at 200 kHz for path 16...... 68 6.23 Relative change in group velocity at 250 kHz for path 16...... 68 6.24 Relative change in group velocity at 200 kHz for path 45...... 69 6.25 Relative change in group velocity at 250 kHz for path 45...... 69 6.26 Relative change in sensor signal amplitude at 200 kHz for path 16.. 70

7.1 Geometric conﬁguration of a curved aluminum plate with a cutout and cracks (all dimensions are in mm)...... 72 7.2 A curved aluminum plate with a cutout and cracks...... 73 7.3 Sensor signals at S2 for pristine condition (baseline) and for damaged condition with cracks of various lengths...... 74 7.4 Surface plots of out-of-plane displacement for a curved aluminum plate with a cutout...... 75 7.5 Geometric conﬁguration of the composite plate...... 76

7.6 Surface plots of out-of-plane displacement for layup (a) [012] (b) [02/904]s 77 7.7 Surface plots of out-of-plane displacement for quasi-isotropic layup (a)

[0/30/60/90/-60/-30]s (b) [0/90/45/-45]s (c) [45/-45/0/90]s ...... 78 7.8 Modeling of debond by creating a volume split...... 79 7.9 Geometric conﬁguration of the composite plate with delamination.. 80 7.10 Surface plots of out-of-plane displacement for a ﬂat cross-ply composite

plate with layup [02/904]s ...... 81 7.11 Surface plots of out-of-plane displacement for a ﬂat quasi-isotropic

composite plate with layup [0/30/60/90/-60/-30]s ...... 82 7.12 Comparison of sensor signals for sensor S1 for the composite plate

with delamination at diﬀerent ply-interfaces for layup (a) [02/904]s (b)

[0/30/60/90/-60/-30]s ...... 83

xiv 7.13 Comparison of scatter signals for sensor S1 for the composite plate

with delamination at diﬀerent ply-interfaces for layup (a) [02/904]s (b)

[0/30/60/90/-60/-30]s ...... 84 7.14 Geometric configuration of the stiffened composite plate with debond 86 7.15 Surface plots of out-of-plane displacement for a stiffened cross-ply com-

posite plate with layup [02/904]s ...... 87 7.16 Surface plots of out-of-plane displacement for a stiﬀened quasi-isotropic

composite plate with layup [0/30/60/90/-60/-30]s ...... 88 7.17 Comparison of scatter signals for sensor S1 for the stiﬀened composite plate with delamination (interface 10-11) or debond for layup (a)

[02/904]s (b) [0/30/60/90/-60/-30]s ...... 89

8.1 Temperature eﬀect on sensor signal (300 kHz)...... 93 8.2 Geometric conﬁguration of the aluminum plate...... 95

8.3 An MTS test for measuring adhesive (Hysol R EA 9696) stiﬀness... 96 8.4 Frequency response with thin adhesive layer (40 µm)...... 98 8.5 Frequency response with thick adhesive layer (120 µm)...... 99 8.6 Parametric study with thin adhesive layer (40 µm)...... 100 8.7 Parametric study with thick adhesive layer (120 µm)...... 101 8.8 Frequency response with thin CW2400 adhesive (40 µm)...... 103 8.9 Frequency response with thick CW2400 adhesive (120 µm)...... 104

9.1 Method of approach for model-assisted damage diagnostics...... 108 9.2 (a) Pulse-echo and (b) pitch-catch conﬁgurations for SHM...... 109 9.3 Scatter signal corresponding to pixel P and actuator-sensor pair 2-3. 111 9.4 PESEA simulations to get ToF to all sensors...... 112 9.5 PESEA simulations to get ToF to all sensors...... 115 9.6 PESEA simulations to get ToF to all sensors...... 116 9.7 PESEA simulations to get ToF...... 118 9.8 Stiﬀened aluminum panel with 8 PZT sensors...... 119 9.9 Five-cycle tone burst input signal with central frequency of 250 kHz. 120 9.10 Diagnostic images for aluminum panel using sensor signals at 25 ◦C. 123

xv 9.11 Diagnostic images for aluminum panel using sensor signals at 60 ◦C. 124

10.1 Schematic of the approach for the sensor network optimization of an SHM system...... 129

10.2 Schematic of the aluminum plate and ScNRaX ratio...... 133 10.3 Parametric study (% change in the sensor signal amplitude...... 136 10.4 Parametric study (% change in the group velocity...... 136 10.5 Five-cycle tone burst input signal with central frequency of 250 kHz. 138 10.6 Variation in the ﬁrst wave packet of the sensor signal due to the variation in the material properties...... 138 10.7 PDF for the sensor signal amplitude with respect to the mean value. 139 10.8 PDF for the group velocity with respect to the mean value...... 140

10.9 Scatter to noise ratio (ScNRaX ) for different wave propagation distances taking into account the uncertainty in material properties... 141 10.10Stiffened aluminum panel with a cutout...... 142 10.11Top view of the stiffened aluminum panel. All dimensions are in mm 143 10.12Optimized sensor network for damage detection and localization... 145

D.1 Sensor signal and its Hilbert transform...... 156

xvi Chapter 1

Introduction

1.1 Structural Health Monitoring

Structural health monitoring (SHM) is the process of implementing a damage detection and characterization strategy for engineering structures. Four major tasks in an SHM system are: (i) damage detection, (ii) damage localization, (iii) damage classification and quantification, and (iv) estimation of the current and future impact on the structure. In the recent years, SHM based on acousto-ultrasound method has emerged as a promising technique for inspection of structural damage [1–4]. In this method ultrasonic elastic stress waves are generated in the structure under inspection. These waves can be used to detect damage at early stages of its development, before it can endanger the safety of the structure [5]. Elastic stress waves can propagate for long distances without significant decrease in amplitude. This allows inspection of large areas of a structure. A wedge method is commonly used, which uses wedges to deliver energy from ultrasonic transducers to an inspected structure. However, the inspection may require bulky equipment such as a laser vibrometer for sensing, and it needs to be manually conducted for locations of interest. Accordingly, the wedge method is difficult to use in hard-to-access areas without disassembling structural parts [6]. An SHM system based on built-in piezoelectric sensors is a promising technique

1 CHAPTER 1. INTRODUCTION 2

because it can automatically inspect and interrogate structural damage in hard-to- access areas [2,3,7,8]. In this method, diagnostic waves are generated by an adhesively bonded piezoelectric actuator, which converts an electric voltage input into mechanical strain. While the waves travel through the structure, they interact with it. As a result, structural information is delivered to neighboring piezoelectric sensors, which convert mechanical strain into electric voltage output. If there is damage in the structure, the diagnostic waves interact with both the structure and the damage (as shown in Figure 1.1), thus carrying information about location, size, and type of damage to the neighboring sensors. This information is extracted from the sensor signals by appropriate diagnostic algorithms.

Piezoelectric sensors

Input signal Output signal

Figure 1.1: Structure with piezoelectric sensors for structural health monitoring

1.2 Piezoelectric Materials

The most typical piezoelectric material is PZT (lead zirconate titanate). As illustrated in Figure 1.2 (adapted from [6]), a piezoelectric sensor produces strain, which generate elastic stress waves in the host structure when an electric ﬁeld is applied (actuator mode). Conversely, strain in the piezoelectric material induced by the elastic stress CHAPTER 1. INTRODUCTION 3

waves generates an electric ﬁeld which is proportional to the changes (sensor mode). Constitutive equations for piezoelectric materials are given in AppendixA.

- V + Vin out

strain - Vin + Vout poling direction poling direction

Figure 1.2: Piezoelectric material (adapted from [6])

1.3 Wave Propagation in Thin-Walled Structures

Elastic waves are mechanical waves propagating in an elastic medium as an eﬀect of forces associated with volume deformation (compression and extension) and shape deformation (shear) of medium elements. Depending on restrictions imposed on the elastic medium, wave propagation may vary in character [5]. Bulk waves propagate in inﬁnite media. A three-dimensional medium bounded by one surface allows for propagation of surface waves (Rayleigh waves and Love waves). Bounding a three- dimensional elastic medium with two surfaces (thin-walled structures), which is often the case for aircraft structures, results in the generation of Lamb waves. Depending on the distribution of displacement on the top and bottom surface, two forms of

Lamb waves modes appear: (i) symmetric mode, denoted as S0,S1,S2, ... , and (ii) antisymmetric mode, denoted as A0,A1,A2,... .S0 is the fundamental symmetric mode in which a crest of the top surface coincides with a crest of the bottom surface, as shown in Figure 1.3(a). A0 is the fundamental antisymmetric mode in which a crest of the top surface coincides with a trough of the bottom surface, as shown in Figure 1.3(b). CHAPTER 1. INTRODUCTION 4

(a) Fundamental symmetric (S0) mode (b) Fundamental symmetric (A0) mode

Figure 1.3: Distribution of displacement for fundamental symmetric (S0) and antisymmetric (A0) modes (adapted from [5])

Figure 1.3 shows the phase velocity of fundamental symmetric (S0) and antisymmetric (A0) modes as well as higher order modes ( S1,A1,S2,A2, etc.) as a function of frequency parameter. Frequency parameter is deﬁned as the product of frequency and thickness of the plate (distance between top and bottom surfaces).

10 S 2 A2 Symmetric A Antisymmetric 1 S 8 3 A3

(m/ms)

c 6 S0

A 2 0

Phase velocity 0 0 2 4 68 10 1214 16 18 20 Frequency parameter f×h (MHz mm)

Figure 1.4: Dispersion curve for aluminum (adapted from [5]) CHAPTER 1. INTRODUCTION 5

1.4 Challenges in SHM based on Piezo-Induced Waves

1.4.1 Modeling of Piezo-Induced Ultrasonic Waves

In SHM based on piezo-induced waves, piezoelectric sensor output contains the structural information in its waveform. Diagnostic algorithms are used to extract this information in order to detect, localize, classify, and quantify damage. Hence, fundamental understanding of the physics of these waves and their interaction with the structure is crucial for developing appropriate diagnostic algorithms. Giurgiutiu (2005) and Raghavan (2007) derived analytical solutions for 2-D and 3-D Lamb wave propagation in a structure, respectively [9, 10]. However, analytical solutions are only available for some regular and simple structures; numerical methods must be used for modeling wave propagation in complex structures. To accurately simulate ultrasonic wave propagation, it is important to model the dynamic interaction between piezoelectric actuators/sensors and the host structure. Conven- tional linear finite element method (FEM) can be used to solve coupled equations of motion and Gausss law to physically model piezoelectric actuators/sensors and simulate ultrasonic waves. However, it requires a very fine spatial discretization to handle high-frequency and short-wavelength nature of these waves, which results in high computational cost in terms of runtime and memory [6, 11–13]. The spectral element method (SEM), which was developed by Patera (1984) [14] has the potential to solve these problems and is well suited for this purpose. Though the formulation of SEM is very similar to that of FEM, it uses high order elements with nodes defined at Gauss-Lobatto-Legendre points and nodal quadrature for numerical integration of element matrices. This makes the simulations accurate and fast. SEM based on solid spectral elements was studied by Kim et al. (2008) [15] to simulate piezo-induced ultrasonic waves in isotropic material. It was further optimized by Ha et al. (2010) [13] to simulate waves in thin plates. Ostachowicz et al. (2009) [16] compared the performance of a 2D spectral element based on Mindlin plate theory and a 3D solid spectral element in modeling wave propagation in flat plates. It was found that, in general, CHAPTER 1. INTRODUCTION 6

3D solid spectral elements give more accurate results. However, modeling ultrasonic waves in composite structures is still very challenging. Dispersion, anisotropy, and attenuation make it very difficult to numerically simulate ultrasonic waves in composite plates. In contrast to metallic materials, attenuation due to viscous damping is an important effect in viscoelastic composite materials. Solid spectral elements cannot be used to efficiently model these waves in such anisotropic layered medium. For accurate results, one spectral element per layer can be used; though it quickly becomes computationally very expensive in terms of memory storage and computation time for large complex structures. Fewer elements in thickness with smeared material properties can be used to improve computational efficiency [17,18], however it may result in inaccurate group velocities and dispersion. Hence, an efficient spectral element needs to be developed to accurately model anisotropic layered composite structures.

1.4.2 Eﬀect of Operating Conditions

Another challenge in real-life applications of SHM based on ultrasonic waves is to carry out an accurate damage diagnosis in varying operating conditions. One has to pay close attention to conditions like change in ambient temperature, stress in the structure. Figure 1.5 shows the first symmetric mode in sensor signals collected at 25 C and 75 C. Although the same apparatus is used at both temperatures, the signals have significantly different amplitudes and arrival times. Therefore, a comprehensive understanding of temperature effect is essential before the practical implementation. Several studies have been conducted on the temperature effects on piezo-induced wave propagation. A temperature increase results in a decrease in the wave propagation velocity [19, 20]. Blaise and Chang (2001) [21] reported amplitude reduction and time delay of sensor signal at 90 C. Raghavan and Cesnik (2008) [22] studied the effects of elevated temperature on Lamb waves by theoretically simulating peak-to-peak amplitudes and arrival times of sensor signals. They compared these to experimental data and determined that the arrival times correlated reasonably well with the experimental data, but the peak-to-peak amplitudes did not. The authors CHAPTER 1. INTRODUCTION 7

suggested adhesive layer effects, which were not included in the simulations, as a possible reason for the discrepancies [22]. Recently Ha and Chang (2010) [23]] comprehensively studied adhesive layer effects on sensor signals using SEM simulations. Parameter studies with adhesive stiffness and thickness were conducted to investigate the physics of adhesive layers. An interesting finding was that the amplitude of the sensor signal may increase even with a lower shear modulus adhesive layer if the actuation frequency is near the resonant frequency of the surface-mounted sensor. Since elevated temperature causes reduction in the shear modulus of the adhesive layer, it was inferred that the adhesive layer at elevated temperature may introduce more complicated phenomenon than only signal amplitude reduction.

Figure 1.5: Eﬀect of Temperature on Sensor Signal (300 kHz)

Similarly, change in loading conditions causes changes in the piezo-induced wave velocity and sensor signals. However, load-induced changes in the wave velocities depend on the loading directions and the direction of wave propagation. The stress dependence of wave velocities is known as acoustoelasticity. Theory of acoustoelasticity was developed in 1953 by Hughes and Kelly (1953) [24] based on the Murnaghan theory of ﬁnite deformations for pre-deformed but initially isotropic solids. It has CHAPTER 1. INTRODUCTION 8

since been generalized by Toupin and Berstein (1961) [25] and Thurston and Brugger (1964) [26] to materials of arbitrary symmetry. The theory of acoustoelasticity has been extensively studied for bulk waves. There are a few papers in the literature on the theory of acoustoelasticity for Lamb waves. Lematre et al. (2006) [27] obtained dispersion curves along the loading direction. Gandhi and Michaels (2010) [28, 29] extended that methodology to obtain dispersion curves along an arbitrary direction of wave propagation under bi-axial loading. However, this method is limited to simple loading conditions and structures with simple geometry. Piezoelectric sensors are also not physically modeled in this method. Kang et al. (2011) [30] showed that the electromechanical coupling coefficients of piezoelectric materials change consid- erably under compressive stress. This may cause the amplitude of the sensor signal to change. Hence, physical modeling of piezoelectric sensors is required to study the effect of load on sensor signals. Numerical method has to be used to study the acoustoelastic effect along arbitrary propagation directions in large complex structures under different loading conditions.

1.4.3 Diagnostic Algorithms and Optimal Sensor Placement

As mentioned earlier, four major tasks in an SHM system are: (i) damage detection, (ii) damage localization, (iii) damage classification and quantification, and (iv) estimation of the current and future impact on the structure. The first two tasks are very important part of any SHM system, since estimation of extent of damage and remaining useful life depend on the accuracy of detection and localization of damage. Many Lamb wave based damage diagnostic imaging techniques have been developed to inter- pret the sensor signal in order to detect and localize the damage. Some baseline-free techniques use reflected signals from damage, such as cracks and holes in metallic plates to directly monitor the presence and location of damage [31, 32]. However, these techniques are only suitable for some simple structures [33], where the reflected wave from damage is not overlapped with reflections from other structural features, such as edges, or stiffeners. Hence, baseline-subtraction techniques [2,3,7,8, 33] are widely used for distinguishing the damage from other structural features. In these CHAPTER 1. INTRODUCTION 9

techniques sensor signals are recorded when the structure is in pristine state (baseline data) and damaged state (current data). The difference between the baseline and current sensor signals gives us the scatter in the signal due to the presence of the damage. The location of the damage is estimated using the scatter in the signal and ultrasonic Lamb wave velocities [2,3,7,8].Typically the accuracy of damage localization depends on a priori knowledge of the wave velocities. Estimation of the ultrasonic Lamb wave velocities for complex structures is very challenging since analytical relations only exist for structures with simple geometries. To overcome this challenge, SEM-based simulations can be used to model ultrasonic wave propagation in complex structures and generate an accurate velocity profile of ultrasonic Lamb waves, which can be used for offline training of the diagnostic algorithm. In addition to accurate diagnostic algorithms, optimal placement of the piezoelectric sensors is critical for desired performance of an SHM system in practical implementation. Markmiller and Chang (2010) [34] showed that for passive SHM systems, developed for impact monitoring, sensor network layout significantly affects the probability of detection (POD) of impacts on the structure. Therefore, optimal sensor placement is required to maximize the performance of an SHM system. How- ever, there have been limited studies on sensor network optimization. Gao and Rose (2006) [35] presented a genetic algorithm (GA) based technique which optimizes the sensor placement by minimizing miss-detection probability with the covariance matrix adaptation evolutionary strategy. Das et al. (2009) [36] estimated the sensor certainty region through experimental data and used it to optimize sensor network by minimizing the overlap region. However, it may not always be possible to per- form experiments to optimize a sensor network for a complex structure. SEM-based simulations can instead be used to model ultrasonic wave propagation in complex structures and their interaction with damage. Simulations can be used to gain physical insights into how damage and uncertainty in the material properties affect the sensor signals. And this information can be used to obtain an optimal sensor network that maximizes the probability of damage detection of an SHM system. Thus, the objective of the current research is to develop an SEM based simulation tool to accurately and efficiently simulate ultrasonic wave propagation and sensor CHAPTER 1. INTRODUCTION 10

response in structures with built-in piezoelectric sensor network. This tool will be used to gain physical insights into how these waves propagate in complex structures and to understand the wave-damage interaction and its effect on the sensor signals. It will also be used to study the effect of temperature on sensor signals and the effect of load on wave propagation; and to show how this simulation tool can be used to improve the accuracy of damage diagnostics and optimize the piezoelectric sensor network. Chapter 2

Problem Statement

In this dissertation, a distributed network of piezoelectric sensors and actuators mounted on the surface of a plate is considered (as depicted in Figure 2.1). It is desired to develop a computational tool to accurately simulate the sensor signals in response to an excitation from a neighboring piezoelectric actuator at ultrasonic frequencies (Figure 2.2). In this study, the plate can be a metallic plate with or without cracks or a laminated composite plate with or without delamination/debond. The plate can be subjected to static loads. In laminated composite plates, the ply ori- entation of the plate can be arbitrary. The geometry of the structure can also be arbitrary as long as its conﬁguration can be modeled through a traditional FE mesh generator such as Abaqus/CAE R . Finally, the thickness of the adhesive layer between the piezoelectric sensors/actuators and the plate must be given and all the material properties must be known to simulate the ultrasonic wave propagation and the sensors response. Tha main objectives of this dissertations are to:

• develop an eﬃcient and accurate simation tool to model the sensor signals and the piezo-induced wave propagation in prestressed homogeneous as well as layered media,

• study the eﬀect of load on wave propagation (acoustoelastic eﬀect) and varying ambient temperature on the sensor signals, and

11 CHAPTER 2. PROBLEM STATEMENT 12

• develop an integrate damage diagnostics and a sensor network optimization tool using the piezo-induced wave propagation simulations.

Piezoelectric sensors

Adhesive plate

Figure 2.1: Plate with surface-mounted piezoelectric sensors

0.03 0.02 0.01 0

Output (V) Output −0.01

−0.02

−0.03 0 20 40 60 80 100 120 Time (µ s) Actuation Signal Sensor Signal

0 Input (V)Input

-25

-50 0 10 20 30 40 50 Time (µ s)

Figure 2.2: Piezo-induced ultrasonic wave propagation in metallic or laminated composite structure Chapter 3

Method of Approach

To achieve the objectives of this dissertation, the method of approach involved the following key steps:

1. Modeling of piezo-induced ultrasonic wave propagation in prestressed structures,

2. Simulations to study the eﬀect of load on wave propagation and varying temperature on sensor signals,

3. Integration of simulations and damage diagnostic imaging algorithms, and

4. Simulations for sensor network optimization.

3.1 Modeling of Piezo-Induced Ultrasonic Wave Propagation

In this dissertation, spectral element method (SEM) was used to accurately and eﬃciently model sensor response and piezo-induced ultrasonic wave propagation in structures subjected to static loads. First coupled governing equations for wave propagation were derived based on the theory of acoustoelasticity. Acoustoelasticity refers to the stress dependence of wave velocities. Then SEM was used to solve these equations numerically. SEM uses high order Lagrangian shape functions to approximate

13 CHAPTER 3. METHOD OF APPROACH 14

the displacement and electric potential field inside an element. It exhibits a much faster convergence rate than a linear finite element method (FEM). Figure 3.1 compares the arrival time solution obtained using SEM and FEM with full/reduced integration for different mesh sizes [6,15]. SEM requires only 10 nodes per wavelength of the fundamental antisymmetric (A0) mode, while FEM requires about 100 nodes per wavelength. Hence, FEM becomes computationally prohibitive for large structures and higher frequencies (shorter wavelengths). Moreover, SEM achieves diagonal mass matrix by using nodal quadrature for numerical integration without mass lumping, which is used in FEM.

FEM: r-integ. 119 FEM: f-integ.

µ SEM Extrapolated solution 118

117

wave packet (

116

115

Arrival time of A Arrival of time 114 0 20 40 60 80 100

Node numbers per A 0 wavelength

Figure 3.1: Convergence comparison between solid spectral element and linear ﬁnite element method (adapted from [6])

A simulations tool based on SEM, called Piezo-Enabled Spectral Element Analysis (PESEA), was developed to simulate sensor response and piezo-induced wave propagation in prestressed structures. The initial stress could be due to an application of mechanical or thermal loads. In operating conditions, such loading conditions may change with time. However, in terms of short duration of ultrasonic Lamb waves they can be considered as static. Derivation of the coupled governing equations CHAPTER 3. METHOD OF APPROACH 15

in given in Chapter4 and the spectral element formulation is given in Chapter5. Experiments and simulations on stiﬀened metallic plate and a laminated composite plate were carried out to validate PESEA. The validation studies are presented in Chapter6. Chapter7 shows how PESEA can be used to visualize piezo-induced waves and their interaction with damage such as cracks in metallic structures and debonds/delamination in laminated composite structures.

3.2 Eﬀect of Operating Conditions

Varying operating conditions may affect the wave propagation and change the sensor response. It is very important to understand this effect to achieve accurate damage diagnostics and minimize false alarms. In this dissertation, the effect of load on wave propagation and the effect of varying temperature on the sensor response were studied.

3.2.1 Eﬀect of Load

In this study, the inﬂuence of loading on ultrasonic waves actuated and sensed by piezoelectric sensors in aluminum plate was investigated. Acoustoelasticity refers to the stress dependence of wave velocities. Chapter6 presents a numerical and experimental study of axially stressed aluminum plates with surface-mounted piezoelectric sensors. In numerical simulations, PESEA formulation presented in Chapter5 was used.

3.2.2 Eﬀect of Temperature

The main objective of this study was to understand how the piezoelectric sensor signal amplitude changes with adhesives of diﬀerent thickness and material when the structure is exposed to elevated temperature. Experiments and simulations were carried out for an aluminum (Al 2024-T3) plate with piezoelectric sensors (6.35 mm diameter, 250 m thickness and lead zirconate titanate (PZT) 5A material). To model the eﬀect of varying temperature on the sensor response in PESEA simulations, the CHAPTER 3. METHOD OF APPROACH 16

material properties of the plate, adhesive, and sensors were varied with temperature. The temperature distribution throughout the structure was assumed to be uniform. Thermal stresses and thermal expansion were not considered in simulations. The variation in sensor signal amplitude at diﬀerent actuation frequencies at diﬀerent temperatures is shown in Chapter8.

3.3 Model-assisted Damage Diagnostics

In SHM, piezo-induced ultrasonic waves are used for damage detection and localization [37]. The accuracy of damage localization depends strongly depends on the a priori knowledge of the wave velocity. The estimation of the wave velocity for complex structures is challenging since analytical solutions only exist for simple structures. Hence, in this dissertation, PESEA simulations were used to estimate the wave velocity profile for a given structures, which was then used for the offline training of the damage diagnostic imaging algorithm. In order to achieve accurate diagnostics in varying temperature environments, temperature compensation algorithm, which was proposed by Roy et al. (2011) [38], was used to compensate for the effect of temperature on the sensor response. PESEA simulations were carried out to validate the proposed model-assisted damage diagnostics. The simulation results are presented in Chapter9.

3.4 Model-assisted Sensor Network Optimization

In this dissertation, a methodology is presented to utilize PESEA simulations to optimize the sensor network by maximizing the probability of damage detection. PESEA was used to understand the eﬀect of crack and uncertainty in material properties on the sensor signal. This information was then used in a genetic algorithm based optimization code. This code maximizes the probability of detection and gives an optimized sensor network. A stiﬀened aluminum plate is considered as an example. The details of this methodology are given in Chapter 10. Chapter 4

Governing Equations for Wave Propagation

4.1 Introduction

This chapter presents the governing equations to model piezo-induced wave propagation in a prestressed structure. In operating conditions, the structure is subjected to variety of external mechanical or thermal loads that may change with time. However, in terms of short duration of ultrasonic Lamb waves such loading conditions can be considered as static. The following sections present the derivation of the coupled governing equations based on the theory of acoustoelasticity.

4.2 Equations of Motion

Theory of acoustoelasticity is used to describe the incremental stresses, strains and displacements due to wave propagation in a prestressed media as given in [24,39–41]. The configuration of a body along with position vectors ξ, X and x of a material point in the natural (unstressed), initial (stressed) and final state (wave motion) is shown in Figure 4.1). It is assumed that the deformation from natural to the initial state is static and the displacement of the particles is denoted by u0(ξ, t) = X − ξ. The displacements from the natural to the final state is represented by the vector

17 CHAPTER 4. GOVERNING EQUATIONS FOR WAVE PROPAGATION 18

uf (ξ, t) = x − ξ. The diﬀerence between these two vectors u(ξ, t) = uf − u0 is the dynamic displacement from the initial state to the ﬁnal state.

x X

ξ u

f Initial state u (stressed) Final state (wave motion)

Natural state (unstressed)

Figure 4.1: Coordinates of a material point at natural ξ, initial X, and ﬁnal x con- ﬁguration of a pre-deformed body

The Lagrangian strain, εij, in the initial and ﬁnal states is given by:

0 0 0 0 0 1 ∂ui ∂uj ∂uk ∂uk εij = + + (4.1a) 2 ∂ξj ∂ξi ∂ξi ∂ξj f f f f ! f 1 ∂ui ∂uj ∂uk ∂uk εij = + + (4.1b) 2 ∂ξj ∂ξi ∂ξi ∂ξj

If the superimposed dynamic strain is small, i.e., ku k kui k and εf − εi α α αβ αβ i εαβ , then the diﬀerence between the two strains is approximated as shown in Equa- tion (4.2). CHAPTER 4. GOVERNING EQUATIONS FOR WAVE PROPAGATION 19

    1 ∂uf ∂u0 ∂uf ∂u0 ∂uf ∂uf ∂u0 ∂u0  ε =  i − i + j − j + k k − k k  ij   2  ∂ξj ∂ξj ∂ξi ∂ξi ∂ξi ∂ξj ∂ξi ∂ξj  | {z } | {z }  ∂ui ∂uj ∂ξj ∂ξi 1 ∂u ∂u ∂u ∂u0 ∂u ∂u0 ∂u0 ∂u0 = i + j + k + k k + k − k k 2 ∂ξj ∂ξi ∂ξi ∂ξi ∂ξj ∂ξj ∂ξi ∂ξj 1 ∂u ∂u ∂u ∂u ∂u ∂u0 ∂u0 ∂u ∂u0 ∂u0 ∂u0 ∂u0 (4.2) = i + j + k k + k k + k k + k k − k k 2 ∂ξj ∂ξi ∂ξi ∂ξj ∂ξi ∂ξj ∂ξi ∂ξj ∂ξi ∂ξj ∂ξi ∂ξj   0 0 1 ∂ui ∂uj ∂uk ∂uk ∂uk ∂uk ∂uk ∂uk  =  + + + +  2 ∂ξj ∂ξi ∂ξi ∂ξj ∂ξi ∂ξj ∂ξi ∂ξj  | {z } small 1 ∂u ∂u ∂u ∂u0 ∂u0 ∂u ≈ i + j + k k + k k 2 ∂ξj ∂ξi ∂ξi ∂ξj ∂ξi ∂ξj

Stress-strain relationship for hyperelastic material can be approximated by:

1 σ = C ε + C(3) ε ε (4.3) ij ijkl kl 2 ijklmn kl mn

(3) where, σij is the stress, Cijkl are the second order constants, and Cijklmn are the third order elastic constants, respectively. These third (and higher) order elastic constants represent the nonlinearity in the stress-strain relationship. It is observed that the particle motion associated with the wave propagation is very sensitive to the elastic constants of medium. Hence, to accurately model the acoustoelastic eﬀect, it is important to consider the nonlinearity in the stress-strain relationship. The third order elastic constants for isotropic materials can be represented in terms of the Murnaghan constants l, m, and n [41]: CHAPTER 4. GOVERNING EQUATIONS FOR WAVE PROPAGATION 20

n C(3) = 2 l − m + δ δ δ ijklmn 2 ij kl mn n + 2 m − (δ I + δ I + δ I ) (4.4) 2 ij klmn kl mnij mn ijkl n + (δ I + δ I + δ I + δ I ) 2 ik jlmn il jkmn jk ilmn jl ikmn where, δ δ + δ δ I = ik jl il jk (4.5) ijkl 2 0 In order to obtain incremental stresses, the initial stress, σij, is subtracted from f the ﬁnal stress, σij, as shown below.

1 1 σ = σf − σ0 = C εf + C(3) εf εf − C ε0 − C(3) ε0 ε0 ij ij ij ijkl kl 2 ijklmn kl mn ijkl kl 2 ijklmn kl mn   f 0 1 (3) h f f 0 0 i = Cijkl εkl − εkl + Cijklmn εklεmn − εklεmn | {z } 2 εkl 1 (3) 0 0 0 0 = Cijklεkl + C εkl + ε εmn + ε − ε ε 2 ijklmn kl mn kl mn (4.6) 1 = C ε + C(3) ε ε + ε ε0 + ε0 ε + ε0 ε0 − ε0 ε0 ijkl kl 2 ijklmn kl mn kl mn kl mn kl mn kl mn   1 (3) 0 0 = Cijklεkl + Cijklmn εklεmn +εklεmn + εklεmn 2 | {z } small 0 ≈ Cijklεkl + Cijklmnεklεmn

Equation (4.6) can be further simpliﬁed approximating the Lagrangian strains in 0 the second term by inﬁnitesimal (or Cauchy) strain tensors kl and mn:

(3) 0 σij ≈ Cijkl εkl + Cijklmn kl mn (4.7) where,

0 0 1 ∂uk ∂uk ∂um ∂um ∂um ∂um εkl ≈ + + + (4.8a) 2 ∂ξl ∂ξl ∂ξk ∂ξl ∂ξk ∂ξl CHAPTER 4. GOVERNING EQUATIONS FOR WAVE PROPAGATION 21

1 ∂uk ∂ul kl = + (4.8b) 2 ∂ξl ∂ξk

Now the equation of equilibrium for the static pre-deformation is given by:

0 ∂ 0 0 ∂ui σji + σjk = 0 (4.9) ∂ξj ∂ξk

And that for the ﬁnal state can be expressed as:

f ! 2 f ∂ f f ∂ui ∂ ui σji + σjk = ρ 2 (4.10) ∂ξj ∂ξk ∂t where, ρ is the density in the natural state. Subtracting Equation (4.9) from Eqaua- tion (4.10),   ∂ ∂uf ∂u0 ∂2u σf − σ0 +σf i − σ0 i  = ρ i (4.11)  ji ji jk jk  2 ∂ξj | {z } ∂ξk ∂ξk ∂t σji Substituting,

f 0 σjk = σjk + σjk (4.12a) ∂uf ∂u ∂u0 i = i + i (4.12b) ∂ξk ∂ξk ∂ξk in Equation (4.11),

0 0 2 ∂ 0 ∂ui ∂ui 0 ∂ui ∂ ui σji + σjk + σjk + − σjk = ρ 2 ∂ξj ∂ξk ∂ξk ∂ξk ∂t ∂ ∂u ∂u0 ∂u ∂u0 ∂u0 ∂2u σ + σ i + σ i + σ0 i + σ0 i − σ0 i = ρ i ∂ξ ji jk ∂ξ jk ∂ξ jk ∂ξ jk ∂ξ jk ∂ξ ∂t2 j k k k k k (4.13)   0 2 ∂  ∂ui ∂ui 0 ∂ui  ∂ ui σji + σjk +σjk + σjk  = ρ 2 ∂ξj  ∂ξk ∂ξk ∂ξk  ∂t | {z } small

Finally, combining Equations (4.2), (4.7), and (4.13), equations of motion for the CHAPTER 4. GOVERNING EQUATIONS FOR WAVE PROPAGATION 22

incremental displacement u (ξ, t) in natural coordinates can be obtained as:

0 2 ∂ ∂ui 0 ∂ui ∂ ui σji + σjk + σjk = ρ 2 (4.14) ∂ξj ∂ξk ∂ξk ∂t where σij is given by Equation (4.7). After some simpliﬁcations, Equation (4.14) becomes: 2 ∂ ∂ui 0 ∂uk ∂ ui σkj + Γijkl = ρ 2 (4.15) ∂ξj ∂ξk ∂ξl ∂t where Γijkl are the eﬀective elastic moduli given by:

0 0 ∂ui ∂uk (3) 0 Γijkl = Cijkl + Cpjkl + Cijpl + Cijklmn mn (4.16) ∂ξp ∂ξp

4.3 Coupled Governing Equations

Figure 4.2: Plate with surface-mounted piezoelectric sensor

Equation (4.15) represents the modiﬁed equation of motion for wave propagation taking into account the acoustoelastic eﬀect. However, to physically model piezoelectric sensors, coupled equations have to be considered. The equations of motion are given by Equation (4.17) and Equation (4.18) is the Guasss law for electricity.

0 2 ∂ ∂ui 0 ∂ui ∂ ui S+P σij + σjk + σjk = ρ 2 , in Ω (4.17) ∂ξj ∂ξk ∂ξk ∂t CHAPTER 4. GOVERNING EQUATIONS FOR WAVE PROPAGATION 23

P Di,i = 0, in Ω (4.18) where Di is the electric displacement and the other variables are as described in the previous section. The boundary conditions and the initial conditions are as given by Equations (4.19).

S+P ui =u ¯i and/or σijnj = t¯i, on ∂Ω (4.19a) ¯ ¯ P φ = V and/or Dini = Q, on ∂Ω (4.19b) S+P u(t = 0) = u0 & ˙u(t = 0) = ˙u0 in Ω , (4.19c)

Here Ω means the physical domain and ∂Ω represents its boundary. Superscripts S and P represent the domain of structure + adhesive and piezoelectric material, ¯ ¯ respectively, as shown in Figure 4.2. φ is the electric potential.u ¯i, t¯i, V , and Q are prescribed displacement, traction, electric potential, and electric charge on the boundaries. u0 and ˙u0 are initial conditions i.e. displacement and velocity at time t = 0. The constitutive equations are given by Equations (4.20) and (4.21). More details about the piezoelectric constitutive equations are given in AppendixA.

(3) 0 σij = Cijklεkl + Cijklmn kl mn − ekijEk (4.20)

Di = eijkεjk + κijEj, (4.21) where

0 0 1 ∂uk ∂uk ∂um ∂um ∂um ∂um εkl ≈ + + + (4.22a) 2 ∂ξl ∂ξl ∂ξk ∂ξl ∂ξk ∂ξl 1 ∂uk ∂ul kl = + , (4.22b) 2 ∂ξl ∂ξk and the electric ﬁeld is given by the following relation.

∂φ Ei = − (4.23) ∂ξi CHAPTER 4. GOVERNING EQUATIONS FOR WAVE PROPAGATION 24

Substituting Equations (4.20)−(4.23) in Equations (4.17) and (4.18), the following equations are obtained.

2 ∂ 0 ∂ui ∂uk ∗ ∂φ ∂ ui S+P σkj + Γijkl + ekij = ρ 2 , in Ω (4.24) ∂ξj ∂ξk ∂ξl ∂ξk ∂t

∂ ∂φ P eijkεjk − κij = 0, in Ω (4.25) ∂ξi ∂ξj ∗ where Γijkl are the effective elastic moduli and ekij are the effective piezoelectric coefficients as given by the following equations.

0 0 ∂uk ∂ui (3) 0 Γijkl = Cijkl + Cijpl + Cpjkl + Cijklmn mn (4.26) ∂ξp ∂ξp

0 ∗ ∂ui ekij = ekij + ekjp (4.27) ∂ξp

4.3.1 Stress-Free Initial State

If the initial state is the same as the natural state, in other words if the initial state is stress-free, then the governing equations derived above can be simpliﬁed by ignoring the terms dependent on the initial stress and the initial strain. In this case, Equation

(4.24) can be written as Equation (4.28). Using the minor symmetry in Cijkl, i.e.

Cijkl = Cijlk, the ﬁrst term on the left hand side of Equation (4.28) can be further simpliﬁed as shown in Equation (4.29).

2 ∂ ∂uk ∂φ ∂ ui Cijkl + ekij = ρ 2 (4.28) ∂ξj ∂ξl ∂ξk ∂t

∂uk 1 ∂uk ∂uk 1 ∂uk ∂ul Cijkl = Cijkl + Cijkl = Cijkl + = Cijkl kl (4.29) ∂ξl 2 ∂ξl ∂ξl 2 ∂ξl ∂ξk Now the equations of motion and Gausss law can be written as:

2 ∂ ∂φ ∂ ui S+P Cijkl kl + ekij = ρ 2 , in Ω (4.30) ∂ξj ∂ξk ∂t CHAPTER 4. GOVERNING EQUATIONS FOR WAVE PROPAGATION 25

∂ ∂φ P eijkjk − κij = 0, in Ω (4.31) ∂ξi ∂ξj The constitutive equations are given by:

σij = Cijkl kl − ekijEk (4.32)

Di = eijkjk + κijEj, (4.33) and the strain, kl, and the electric ﬁeld, Ek, are as given by Equations (4.22) and (4.23). The boundary conditions and the initial conditions are as given by Equations (4.19).

4.4 Conclusions

In this chapter, the governing equations to model piezo-induced wave propagation in a prestressed structure were presented. The initial stress could be due to an application of mechanical or thermal loads. In operating conditions, such loading conditions may change with time. However, in terms of short duration of ultrasonic Lamb waves they can be considered as static. In this formulation, the equations of motion and Gausss law were considered to physically model the piezoelectric sensors. The coupled governing equations are given by Equations (4.24) and (4.25). In case of stress-free initial state, these equations were simpliﬁed to obtain Equations (4.30) and (4.31). The following chapter presents the spectral element formulation to numerically solve these coupled governing equations for an arbitrary voltage input to the piezoelectric sensors. Chapter 5

Spectral Element Method

5.1 Introduction

The coupled governing equations presented in Chapter 4 can be solved analytically only for simple structures, such as flat infinite plate [9,10]. Hence, numerical methods must be used for modeling piezo-induced wave propagation in complex structures. In this dissertation, spectral element method (SEM), developed by Patera (1984) [14], is used to solve the governing equations. Like the finite element method (FEM), the spectral element method (SEM) is a numerical method to find approximate solutions of partial differential equations. However, SEM uses high-order shape functions as opposed to the linear shape functions used in conventional FEM. Hence, SEM offers benefits over FEM with regards to the convergence of the solution. FEM needs a very fine mesh to achieve a given solution accuracy, whereas the same level of accuracy can be achieved with a coarser mesh using SEM. Moreover, SEM uses nodal quadrature for numerical integration which results in inherently diagonal mass matrix without a mass lumping approximation, which is used in FEM. Diagonal mass matrix enables the use of faster explicit time integration algorithms, such as central difference method [6]. This chapter presents the framework of an SEM-based numerical tool, called Piezo-Enabled Spectral Element Analysis (PESEA), to solve the coupled governing equations presented in the previous chapter.

26 CHAPTER 5. SPECTRAL ELEMENT METHOD 27

5.2 Weak Formulation

Weak form of the coupled governing equations is given below.

Z 2 ∂ 0 ∂ui ∂uk ∗ ∂φ ∂ ui σkj + Γijkl + ekij − ρ 2 δuj dΩ = 0 (5.1) ∂ξj ∂ξk ∂ξl ∂ξk ∂t ΩS+P

Z ∂ ∂φ eijkεjk − κij δφ dΩ = 0 (5.2) ∂ξi ∂ξj ΩP After integrating by parts and substituting the boundary conditions, Equations (5.3) are obtained. Z Z Z 0 ∗ ¯ Γijkl uk,l δui,j + σkj ui,k δui,j + ρ u¨i δui dΩ + ekij φ,k δui,jdΩ = ti δui d(∂Ω) ΩS+P ΩP ∂ΩS+P (5.3a) Z Z ¯ (eijk εjk δφ,i − κij φ,j δφ,i)dΩ = Q δφ d(∂Ω) (5.3b)

ΩP ∂ΩP 5.3 Solid Spectral Element

The weak form given above can be solved using the same procedure as in FEM, which divides the entire domain into a number of sub-domains called elements.

Ωe, e = 1, 2, ... Each element in the physical domain is mapped to a parent (ξ1, ξ2, ξ3) domain, [−1, 1] × [−1, 1] × [−1, 1]. However, SEM uses high-order Lagrange polynomials as shape functions (AppendixB). These shape functions are defined at the Gauss-Lobatto-Legendre (GLL) nodal points to approximate the field variables e.g. displacement and electric potential in this case. GLL points ξ1i ∈ [−1, 1] are N + 1 0 roots of Equation (5.4), where PN is the derivative of Legendre polynomial in ξ1 of order N. GLL points in ξ2 and ξ3 directions are defined in a similar fashion.

2 0 (1 − ξ1 )PN (ξ1) = 0 (5.4) CHAPTER 5. SPECTRAL ELEMENT METHOD 28

ξ3 ξ2

ξ1

Figure 5.1: Solid spectral element (SSE) of order 4×4×4

As an example, a solid spectral element (SSE) of order 4 × 4 × 4 is shown in Figure 5.1. Displacement ﬁeld ue within an element e is approximated as shown in Equation (5.5), where de is the nodal displacement vector for that element. Electrical potential φe within a piezoelectric element e is approximated as shown in Equation e (5.6), where V is the nodal electrical potential vector for that element. (ξ1, ξ2, ξ3) is e the coordinate system of element e in parent domain and hp (ξ1) is the shape function for element e at node p in ξ1 direction.

Nξ1 Nξ2 Nξ3 e X X X e e e e(pqr) ui (ξ1, ξ2, ξ3, t) = hp(ξ1)hq(ξ2)hr(ξ3) di (t) (5.5) p=0 q=0 r=0

Nξ1 Nξ2 Nξ3 e X X X e e e e(pqr) φ (ξ1, ξ2, ξ3, t) = hp(ξ1)hq(ξ2)hr(ξ3) V (t) (5.6) p=0 q=0 r=0

5.3.1 Matrix Representation of Weak Form

After substituting the approximate displacement and electric potential ﬁelds into Equations (5.3), a matrix representation of the weak form for element e is given by the following equations.

e¨e e e e e e M d (t) + [Kuu + KG] d (t) = −KuφV (t) (5.7a) CHAPTER 5. SPECTRAL ELEMENT METHOD 29

e e e e KφφV (t) = Kφud (t), (5.7b)

e e e e where, M , Kuu, KG, and Kφφ are element mass, mechanical stiffness, geometric stiff- e e ness, and electric stiffness matrices, respectively. Kuφ and Kφu are electromechanical coupling matrices for element e. Z e e T e M = Hu ρ HudΩ (5.8) Ωe Z e e T e Kuu = Gu ΓGudΩ (5.9) Ωe Z e e T 0 e KG = Gu S GudΩ (5.10) Ωe Z e e T ∗T e Kuφ = Gu e BφdΩ (5.11) Ωe Z e e T ∗e Kφu = Bφ e B udΩ (5.12) Ωe Z e e T e Kφφ = Bφ κ BφdΩ (5.13) Ωe Displacement, infinitesimal (Cauchy) strain, approximate Lagrangian strain, electric potential, and electric field in element e are given by Equations (5.14).

e e e u = Hud (5.14a) e e e = Bud (5.14b) e ∗e e ε = B ud (5.14c) e e e φ = HφV (5.14d) e e e e E = −∇φ = −BφV (5.14e)

In Equations (5.14), e Hu: matrix relating nodal displacements to displacement ﬁeld in element e CHAPTER 5. SPECTRAL ELEMENT METHOD 30

e Bu: strain-displacement matrix for infinitesimal (Cauchy) strain ∗e B u : strain-displacement matrix for approximate Lagrangian strain (Equation (4.22)) e Hφ: matrix relating nodal electric potential to electric potential field in element e e Bφ: matrix relating nodal electric potential to electric field in element e e In each element, Hu can be expressed as:

e e e e Hu = [Hu111 Hu112 ... Hupqr ... ]   he(ξ )he(ξ )he (ξ ) 0 0 i 1 j 2 k 3 (5.15) e  e e e  H =  0 h (ξ1)h (ξ2)h (ξ3) 0  uijk  i j k  e e e 0 0 hi (ξ1)hj(ξ2)hk(ξ3)

e and Hφ can be expressed as:

He = [He He ... He ... ] φ φ111 φ112 φpqr (5.16) He = he(ξ )he(ξ )he (ξ ) φijk i 1 j 2 k 3

e In both structural and piezoelectric elements, Bu is represented by:

 ∂  ∂ξ 0 0  1   0 ∂ 0   ∂ξ2   0 0 ∂  e  ∂ξ3  e Bu =   Hu, (5.17)  0 ∂ ∂   ∂ξ3 ∂ξ2     ∂ 0 ∂   ∂ξ3 ∂ξ1  ∂ ∂ 0 ∂ξ2 ∂ξ1

∗e and B u is given by: ∗e e e B u = B1u + B2u (5.18)

e e B1u = Bu (5.19) CHAPTER 5. SPECTRAL ELEMENT METHOD 31

 0 0 0  ∂u1 ∂ ∂u2 ∂ ∂u3 ∂ ∂ξ1 ∂ξ1 ∂ξ1 ∂ξ1 ∂ξ1 ∂ξ1  ∂u0 ∂u0 ∂u0   1 ∂ 2 ∂ 3 ∂  ∂ξ2 ∂ξ2 ∂ξ2 ∂ξ2 ∂ξ2 ∂ξ2  0 0 0   ∂u1 ∂ ∂u2 ∂ ∂u3 ∂  e  ∂ξ3 ∂ξ3 ∂ξ3 ∂ξ3 ∂ξ3 ∂ξ3  e B2u =  ∂u0 ∂u0 ∂u0 ∂u0 ∂u0 ∂u0  Hu (5.20)  1 ∂ + 1 ∂ 2 ∂ + 2 ∂ 3 ∂ + 3 ∂   ∂ξ3 ∂ξ2 ∂ξ2 ∂ξ3 ∂ξ3 ∂ξ2 ∂ξ2 ∂ξ3 ∂ξ3 ∂ξ2 ∂ξ2 ∂ξ3   ∂u0 ∂u0 ∂u0 ∂u0 ∂u0 ∂u0  1 ∂ + 1 ∂ 2 ∂ + 2 ∂ 3 ∂ + 3 ∂  ∂ξ3 ∂ξ1 ∂ξ1 ∂ξ3 ∂ξ3 ∂ξ1 ∂ξ1 ∂ξ3 ∂ξ3 ∂ξ1 ∂ξ1 ∂ξ3   ∂u0 ∂u0 ∂u0 ∂u0 ∂u0 ∂u0  1 ∂ + 1 ∂ 2 ∂ + 2 ∂ 3 ∂ + 3 ∂ ∂ξ2 ∂ξ1 ∂ξ1 ∂ξ2 ∂ξ2 ∂ξ1 ∂ξ1 ∂ξ2 ∂ξ2 ∂ξ1 ∂ξ1 ∂ξ2 e In piezoelectric elements, Bφ is represented by:

 ∂  ∂ξ1 e  ∂  e Bφ =   Hφ (5.21)  ∂ξ2  ∂ ∂ξ3

∗ In this formulation, Γijkl are the effective elastic moduli and ekij are the effective piezoelectric coefficients (Equations (4.26) and (4.27)). As it can be noticed from these equations, the minor symmetry in the elastic moduli tensor is lost because of the additional terms that depend on the initial strain. In other words, Γijkl 6= Γjikl and Γijkl 6= Γijlk. However, the major symmetry is still preserved i.e. Γijkl = Γklij. Therefore, Γ is now a 9×9 matrix, unlike the original elastic moduli matrix, C, which ∗ ∗ is 6×6. Similarly, ekij 6= ekji due to the additional term dependent on the initial strain, and therefore, e∗ is now a 3×9 matrix, unlike e, which was 3×6. For this e reason, a new 9×1 strain vector, G, is defined as given by the following equations.

∂u e = i (5.22) Gij ∂ξj

e e e G = Gud (5.23) CHAPTER 5. SPECTRAL ELEMENT METHOD 32

e where, Gu is:  ∂ 0 0  ∂ξ1  ∂   0 ∂ξ 0   1   0 0 ∂   ∂ξ1   ∂   0 0   ∂ξ2  Ge =  0 ∂ 0  He (5.24) u  ∂ξ2  u    0 0 ∂   ∂ξ2   ∂   0 0   ∂ξ3   0 ∂ 0   ∂ξ3  0 0 ∂ ∂ξ3 S0 is the matrix with initial stresses. It is required to compute the geometric e 0 stiﬀness matrix KG for element e as shown in Equations (5.10). S is given by Equation (5.25).

 0 0 0  σ11 0 0 σ21 0 0 σ31 0 0  0 0 0   0 σ11 0 0 σ21 0 0 σ31 0     0 0 σ0 0 0 σ0 0 0 σ0   11 21 31   σ0 0 0 σ0 0 0 σ0 0 0   12 22 32  S0 =  0 0 0  (5.25)  0 σ12 0 0 σ22 0 0 σ32 0     0 0 σ0 0 0 σ0 0 0 σ0   12 22 32  0 0 0  σ13 0 0 σ23 0 0 σ33 0 0     0 σ0 0 0 σ0 0 0 σ0 0   13 23 33  0 0 0 0 0 σ13 0 0 σ23 0 0 σ33

5.3.2 Global System of Equations

A global system of equations, given by Equations (5.26) and (5.27).

¨ Md + (Kuu + KG) d = Fu (V, t) (5.26)

KφφV = Fφ (d, t) (5.27) CHAPTER 5. SPECTRAL ELEMENT METHOD 33

where, global mass matrix (M), global mechanical stiffness matrix (Kuu), global geometric stiffness matrix (KG) global mechanical force vector (Fu(V, t)), global electrical stiffness matrix (Kφφ), and global electrical force vector (Fφ(d, t)) are obtained by assembling element matrices as shown in Equations (5.28), where nel is the total number of spectral elements in the structure.

nel M = A Me (5.28a) e=1 nel e Kuu = A Kuu (5.28b) e=1 nel e KG = A KG (5.28c) e=1 nel e e Fu(V, t) = A −KuφV (t) (5.28d) e=1 nel e Kφφ = A Kφφ (5.28e) e=1 nel e e Fφ(d, t) = A Kφud (t) (5.28f) e=1

5.3.3 Rayleigh Damping

In structural dynamics simulations, Rayleigh damping model is often used to include e viscous damping. In Rayleigh damping, the element damping matrix, Cuu is pro- e e portional to the element mass matrix, M , and the element stiﬀness matrix, Kuu, as shown in Equation (5.29). Global damping matrix can be obtained by assembling the element damping matrices as shown in Equations (5.30). The damping term can be added to Equation (5.26) to get Equation (5.31).

e e e Cuu = a0M + a1Kuu (5.29)

nel e Cuu = A Cuu (5.30) e=1 ¨ ˙ Md(t) + Cuud(t) + [Kuu + KG] d(t) = Fu(V, t) (5.31) CHAPTER 5. SPECTRAL ELEMENT METHOD 34

5.3.4 Global System of Equations for Stress-Free Initial State

In case of stress-free initial state, the weak form of the coupled governing equations can be written as given below.

Z 2 ∂ ∂uk ∂φ ∂ ui Cijkl + ekij − ρ 2 δuj dΩ = 0 (5.32) ∂ξj ∂ξl ∂ξk ∂t ΩS+P

Z ∂ ∂φ eijkεjk − κij δφ dΩ = 0 (5.33) ∂ξi ∂ξj ΩP After integrating by parts and substituting the boundary conditions, Equations (5.34) are obtained.

Z Z Z (Cijkl kl δij + ρ u¨i δui)dΩ + ekijφ,k δijdΩ = t¯i δui d(∂Ω) (5.34a)

ΩS+P ΩP ∂ΩS+P Z Z ¯ (eijk jk δφ,i − κij φ,j δφ,i)dΩ = Q δφ d(∂Ω) (5.34b)

ΩP ∂ΩP

After substituting the approximate displacement and electric potential ﬁelds into Equations (5.34), a matrix representation of the weak form for element e is given by the following equations.

e¨e e ˙ e e e e e M d (t) + Cuud (t) + Kuud (t) = −KuφV (t) (5.35a) e e e e KφφV (t) = Kφud (t), (5.35b) where, Z e e T e M = Hu ρ HudΩ (5.36a) Ωe Z e e T e Kuu = Bu CBudΩ (5.36b) Ωe e e e Cuu = a0M + a1Kuu (5.36c) CHAPTER 5. SPECTRAL ELEMENT METHOD 35

Z e e T T e Kuφ = Bu e BφdΩ (5.36d) Ωe Z e e T e Kφφ = Bφ κ BφdΩ (5.36e) Ωe Z e e T e Kφu = Bφ e BudΩ (5.36f) Ωe

e e e Matrices Hu, Bu, and Bφ are as given by Equations (5.15), (5.17), and (5.21), respectively. A global system of equations, given by Equations (5.37) and (5.38).

¨ ˙ Md(t) + Cuud(t) + Kuud(t) = Fu(V, t) (5.37)

KφφV(t) = Fφ(d, t), (5.38) where, global mass matrix (M), global mechanical stiﬀness matrix (Kuu), global mechanical force vector (Fu(V, t)), global electrical stiﬀness matrix (Kφφ), and global electrical force vector (Fφ(d, t)) are obtained by assembling element matrices shown in Equation (5.36).

5.4 Numerical Integration in Spatial Domain

In numerical methods such as FEM, SEM, direct evaluation of the exact value of defi- nite integrals is often ineffective and not desirable. Hence, numerical integration methods are used to compute element matrices given in Equations (5.8)−(5.13) and (5.36). The term numerical quadrature is more or less a synonym for numerical integration algorithm, especially as applied to one-dimensional integrals. There are various numerical integration algorithms available, such as trapezoidal rule, Simpson’s 1/3 rule, Newton-Cotes rules, Gaussian quadrature, nodal quadrature. Gaussian quadrature is preferred in FEM. In SEM, nodal quadrature, also called Gauss-Lobatto-Legendre quadrature, is used to evaluate integrals for both mass and stiffness matrices. Nodal quadrature is explained in detail in the following subsection. CHAPTER 5. SPECTRAL ELEMENT METHOD 36

5.4.1 Nodal Quadrature

In nodal quadrature the locations of the nodes and the integration points are identical. The nodal locations and weights are listed in Table 5.1. Accordingly, the values of the Lagrange interpolation functions (shape functions for spectral elements) evaluated at integration points become either 1 or 0. Consequently, matrices evaluated at each integration point become sparse, and computation time for matrix-vector multiplication and matrix assembly can be greatly reduced. Spectral elements, which utilize nodal quadrature, require significantly less computation time than the same order finite elements, which use Gaussian quadrature for numerical integration. Hence high-order spectral elements can be used in practice. Moreover, using nodal quadrature for computing mass matrix of a spectral element produces a diagonal mass matrix [6]. Diagonal mass matrix enables the use of explicit time integration methods, such as central difference method, to solve the differential equations efficiently.

Table 5.1: GLL nodal and integration points and weights for interval [-1,1]

Number of points (n) Points (ξi) Weights (wi) 2 ± 1.000000 1.000000 3 0.000000 1.333333 ± 1.000000 0.333333 4 ± 0.447214 0.833333 ± 1.000000 0.166667 5 0.000000 0.711111 ± 0.654654 0.544444 ± 1.000000 0.100000 6 ± 0.285232 0.554858 ± 0.765055 0.378475 ± 1.000000 0.066667 CHAPTER 5. SPECTRAL ELEMENT METHOD 37

5.5 Time Integration

The governing equations given by Equations (5.27) and (5.31) are semi-discrete equations, discretized in spatial domain. The initial value problem for these equations consists of ﬁnding a displacement, d(t), and electric potential, V(t), satisfying Equa- tions (5.27) and (5.31) and the given initial conditions,

d(0) = d0 (5.39a) ˙ d(0) = v0 (5.39b)

Perhaps the most widely used family of direct methods to solve Equation (5.31) with initial conditions given by Equations (5.39) is the Newmark family [42]. The Newmark family contains as special cases many well-known and widely used methods. One of these methods is the central diﬀerence method, which is explicit when M and

Cuu are diagonal. When the time step restriction is not too severe, as is often the case in elastic wave-propagation problems, the central diﬀerence method is generally the most economical direct time integration procedure and is thus widely used [42]. The time integration procedure using the central diﬀerence method is described in detail in the following subsection.

5.5.1 Central Diﬀerence Method

The central diﬀerence method is used for direct time integration of Equation (5.31). In this method, velocity and acceleration at a given time t are expressed as:

d(t + ∆t) − d(t − ∆t) v(t) = d˙ (t) = (5.40a) 2∆t d(t + ∆t) − 2d(t) + d(t − ∆t) a(t) = d¨(t) = (5.40b) ∆t2

The velocity at time t can also be represented in terms of the acceleration at time t and t − ∆t as follows:

∆t v(t) = v(t − ∆t) + (a(t − ∆t) + a(t)) (5.41) 2 CHAPTER 5. SPECTRAL ELEMENT METHOD 38

Equation (5.41) can be rewritten as Equation (5.42)

∆t v(t) = ˜v(t) + a(t), (5.42) 2 where ˜v(t)is the partially updated velocity given by:

∆t ˜v(t) = v(t − ∆t) + a(t − ∆t) (5.43) 2

After substituting Equations (5.40), Equation (5.31) reduces to a system of algebraic equations for each given time step, as given by the following equation.

1 1 2 M + C d(t + ∆t) = F (t) − K − M d(t) ∆t2 2∆t uu u uu ∆t2 (5.44) 1 1 − M − C d(t − ∆t) ∆t2 2∆t uu

However, for Equation (5.44) to be explicit, the damping matrix Cuu has to be diagonal. Since the stiﬀness matrix Kuu is not diagonal, this restricts the damping matrix to be only proportional to the diagonal mass matrix M and it cannot be proportional to the stiﬀness matrix. To overcome this problem, the damping term in Equation (5.31) is assumed to be dependent only on the partially updated velocity

˜v(t), which is known at time t. This damping term now becomes Cuu˜v(t) and can be moved to the RHS. After the modiﬁcations (5.44) becomes:

1 1 2 M d(t + ∆t) = F (t) − K + C − M d(t) ∆t2 u uu ∆t uu ∆t2 (5.45) 1 1 − M − C d(t − ∆t) ∆t2 ∆t uu

An easy way of implementing the algorithm given by Equation (5.45) is as given below: ¯ 1. Given initial conditions, d0 & v0, and boundary conditions, d & V¯ .

2. At t = 0:

Compute electrical force vector, Fφ(d0, 0). CHAPTER 5. SPECTRAL ELEMENT METHOD 39

Get electric potential vector, V(0), by solving KφφV(0) = Fφ(d0, 0).

Compute mechanical force vector, Fu(V, 0).

Get acceleration, a(0), by solving M a(0) = Fu(V, 0) − [Kuu + KG] d0 − Cuuv0. Set t := 0.

∆t 3. Partially update velocity vector, ˜v(t + ∆t) = v(t) + 2 a(t). Completely update displacement vector, d(t + ∆t) = d(t) + ∆t ˜v(t + ∆t).

4. At t + ∆t:

Compute electrical force vector, Fφ(d, t + ∆t).

Get electric potential vector, V(t+∆t), by solving KφφV(t+∆t) = Fφ(d, t+∆t).

Compute mechanical force vector, Fu(V, t + ∆t). Get acceleration, a(t + ∆t), by solving

M a(t + ∆t) = Fu(V, t + ∆t) − [Kuu + KG] d(t + ∆t) − Cuu˜v(t + ∆t).

∆t 5. Completely update velocity vector, v(t + ∆t) = ˜v(t + ∆t) + 2 a(t + ∆t).

6. Set t := t + ∆t.

7. Repeat steps 3−6 until t = tmax.

5.6 Layered Solid Spectral Element

Laminated composites are widely used in aerospace applications. Hence accurate modeling of piezo-induced wave propagation in composite structures is very important. Laminated composite is a layered medium, made by stacking and curing plies with different fiber orientations. Even though solid spectral element can be used to efficiently and accurately model ultrasonic waves in homogeneous media, it would be computationally expensive one solid spectral element per layer. To avoid too many elements in the stacking directions, smeared or average stiffness is generally used to model composites [17,18,43]. Smeared material properties may not always give accurate results for group velocity and dispersion of these waves. Hence, formulation of a layered solid spectral element (LSSE) has been developed that can efficiently model several plies per element, without using smeared material properties [44]. CHAPTER 5. SPECTRAL ELEMENT METHOD 40

Ply 4

Ply 3

Ply 2 ξ3 ξ2 Stacking Ply 1 direction ξ1

(a) Node locations and stacking direction (b) Integration points for computation of element stiﬀness matrix

Figure 5.2: Layered solid spectral element (LSSE) of order 4×4×4

As an example, a 4×4×4 order LSSE with 4 plies is shown in the parent domain in Figure 5.2(a). ξ3 is the stacking direction. Since the density of ply doesn’t depend on fiber direction and is the same for all the plies, the integrand in the element mass matrix, Me, is continuous in laminated composites, and nodal quadrature can still be used for numerical integrating of Me. Unlike Me, the integrand of the element e stiffness matrix, Kuu is continuous within a ply and discontinuous across plies. Hence nodal quadrature cannot be used in the stacking direction, unless smeared material properties are used. Therefore, a new combined integration rule has been proposed for numerical integration of the stiffness matrix.

5.6.1 Numerical Integration of Stiﬀness Matrix

To overcome the challenge mentioned above, LSSE uses Simpson’s 1/3 rule (also e known as 3-point closed Newton-Cotes rule) for numerical integration of Kuu in the stacking direction and nodal quadrature in the remaining directions. Integration points for this combined integration rule are shown in Figure 5.2(b). Simpson’s 1/3 rule is a simple method for numerical integration of a function a+b f(x) over interval [a, b]. It evaluates the integrand f(x) at a, b, and m = 2 ; and approximates it by a quadratic function P (x) as shown in Figure 5.3. CHAPTER 5. SPECTRAL ELEMENT METHOD 41

f(x): integrand f(x) P(x) P(x): quadratic fit

f(b)

f(a)

a m b

Figure 5.3: Simpsons 1/3 rule in stacking direction

The approximate integration of function f(x) over interval [a, b] is given by:

Z b b − a f(x)dx ≈ [f(a) + 4f(m) + f(b)] (5.46) a 6

With the hybrid integration rule, the modified computation of the element stiffness e matrix, Kuu, is given by Equation (5.47). That for the element geometric stiffness matrix is given by Equation (5.48). In case of stress-free initial state, Equation (5.49) is used for the computation of the element stiffness matrix.

Nξ1 Nξ2 3Nplies e X X X e T e nodal nodal Simpson0s Kuu = Gu ΓGu|J| i,j,k wi wj wk , (5.47) i=0 j=0 k=0

Nξ1 Nξ2 3Nplies e X X X e T 0 e nodal nodal Simpson0s KG = Gu S Gu|J| i,j,k wi wj wk , (5.48) i=0 j=0 k=0

Nξ1 Nξ2 3Nplies e X X X e T e nodal nodal Simpson0s Kuu = Bu CBu|J| i,j,k wi wj wk , (5.49) i=0 j=0 k=0 where |J| is the determinant of the Jacobian matrix, and w is the weight at integration e e point for numerical integration. Except for Kuu and KG, nodal quadrature is used for CHAPTER 5. SPECTRAL ELEMENT METHOD 42

e e e e numerical integration of the remaining matrices which are M , Kuφ, Kφφ, and Kφu. The Central diﬀerence method is used for time integration of Equation (5.31).

5.7 Piezo-Enabled Spectral Element Analysis

Based on the formulation of solid spectral element (SSE) and layered solid spectral element (LSSE), a numerical tool called Piezo-Enabled Spectral Element Analysis (PESEA) was developed. To summarize, PESEA uses solid spectral elements to model piezo-induced wave propagation in homogeneous media. Nodal quadrature is used for the numerical integration of element matrices and the central difference method is used for time integration to solve the equations of motion. To model ultrasonic waves in layered media, e.g. laminated composites, layered solid spectral elements are used to model several plies within one element in order to achieve accuracy and efficiency. LSSE uses combined integration rule for numerical integration of the element stiffness matrix, as discussed in the previous section.

5.7.1 Implementation of PESEA

It is worth noting that PESEA can interface with commercial ﬁnite element meth generators such as Abaqus/CAE R to model structures with complex geometries. Fig- ure 5.4 shows how PESEA works. First, a model with given geometry is created in

Abaqus/CAE R and a ﬁnite element (FE) mesh is generated. Piezoelectric sensors and thin adhesive layer between the sensors and the structure are also modeled. If the structure is under an external load, the static problem is solved using Abaqus/CAE R . PESEA consists of an interface program which uses the FE mesh to generate a spectral element (SE) mesh. This interface program also interpolates the initial displacements at the nodes of the SE mesh, as shown in Figure 5.5. This (SE) mesh along with the initial displacements, the input actuation signal and the boundary conditions is sent to the solver in PESEA as inputs. Based in the initial displacements, initial stresses and strains are computed at each node of all elements. The initial strains are used to get the eﬀective material properties using Equations (5.26) and (5.27), and the initial CHAPTER 5. SPECTRAL ELEMENT METHOD 43

Piezo Geometry of the Structure

Abaqus/CAE® Adhesive PESEA

Interface Program Voltage Input to Actuator Elastodynamics and Voltage Output Electric Potential at Sensors Material Properties, Boundary Conditions

Figure 5.4: Schematic of how PESEA works stresses are used to compute the geometric stiﬀness matrix given by Equation (5.10). PESEA then solves the coupled equations (Equations (5.27) and (5.31)) and outputs the sensor signal in volts.

Abaqus/CAE ® FE → SE PESEA

Wave Solve static Interpolate Calculate initial displacements for stresses & new propagation problem spectral elements material properties problem

Figure 5.5: Implementation of PESEA to model acoustoelastic wave propagation Chapter 6

Veriﬁcation & Validation of PESEA

This chapter presents validation of PESEA simulations using solid spectral element with experiments on an aluminum plate. The proposed layered solid spectral element is also veriﬁed and validated with experiments on cross-ply carbon ﬁber reinforced plastic (CFRP) plate. A numerical and experimental study of axially stressed aluminum plates with surface-mounted piezoelectric sensors is also presented in this chapter to validate the formulation of the acoustoelastic wave propagation.

6.1 Validation of Solid Spectral Element

In order to validate PESEA using solid spectral element, experiments were conducted on a 300 mm × 200 mm stiffened aluminum (Al 2024-T3) plate. The geometric configuration of the plate is shown in Figure 6.1(a) Two 1 mm thick aluminum plates were bonded together and a T-stiffener was bonded on one side using Hysol R EA 9696 thin film adhesive. Six single-PZT (PZT: lead zirconate titanate) SMART

Layers R (material PZT-5A) from Acellent Technologies were attached on the stiﬀ- ened plate as shown in Figure 6.1(b), using Hysol R EA 9696 adhesive. PZT actuator/sensor were 6.35 mm in diameter and 250 µm in thickness. ScanGenie R data acquisition hardware and SmartCompositeTM software were used to carry out the

44 CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 45

experiment in pitch-catch setup. Five-cycle tone burst actuation voltage signal with center frequency of 200 kHz was given to PZT A. Response was recorded at PZT sensors S1−S3.

L: ½”× ½”×⅛” 50 50 A S2 S1 S3 200 80 55 55 300

(a) Geometric conﬁguration (dimensions in mm)

(b) Laboratory sample

Figure 6.1: Aluminum plate with T-stiﬀener

In numerical simulations, solid spectral elements of order 5×5×3 were used. Ad- hesive thickness was assumed to be 50 µm. Taking advantage of the symmetry of the problem, only half of the plate with symmetric boundary conditions was modeled, CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 46

Figure 6.2: Mesh for the half geometry created in Abaqus/CAE R as shown in Figure 6.2. The total number of elements and nodes were 7,647 and 542,570 respectively. Traction free boundary conditions were assumed and the input excitation voltage, as shown in Figure 6.3, was applied to the top surface nodes of PZT A. The material properties used in the simulations are given in AppendixC.

0 Input (V)Input

-25

-50 0 10 20 30 40 50 Time (µs)

Figure 6.3: Five-cycle tone burst input signal (central frequency of 200 kHz)

Simulated signal for PZT sensor S1 is compared with the experimental signal in Figure 6.4. Simulated signal using PESEA matches very well with the experimentally CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 47

obtained signal. Error in the simulated sensor signal with respect to the experimental signal was quantiﬁed using the procedure described in AppendixD. The percentage error in the ﬁrst wave packet and the second wave packet is 4.07% and 3.22%, respectively

0.04 Experiment Simulation

0.02

0 Output (V) Output -0.02 error = 4.07% error = 3.22%

-0.04 30 40 50 60 70 80 Time ( µs)

Figure 6.4: Comparison between experimental & simulated signal for sensor S1

6.2 Veriﬁcation of Layered Solid Spectral Element

A CFRP composite plate as shown in Figure 6.5 was considered to test the accuracy and the eﬃciency of the layered solid spectral element (LSSE). Four types of laminates, namely, symmetric cross-ply [02/904]s and [902/02/902]s, and quasi-isotropic

[0/90/45/-45]s and [45/-45/0/90]s were considered in this investigation. Fiber orien- tation is deﬁned with respect to x-direction as shown in Figure 6.5. In all simulations and experiments, piezoelectric sensors made of lead zirconate titanate (PZT) type 5A were used. The polarization direction of the piezoelectric material is z and the excitation voltage was applied to the nodes on the top surface of PZT actuator, while bottom surface nodes were grounded. Actuator A was excited with a ﬁve-cycle tone burst signal with center frequency of 200 kHz [Figure 6.3]. Response was recorded CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 48

at sensors S. The material properties of the composite, adhesive, and piezoelectric material used in the simulations are given in AppendixC.

76.2 A S y

152.4 θ 76.2 x

76.2 152.4 76.2

Figure 6.5: Geometric conﬁguration of the composite plate considered for veriﬁcation

6.2.1 Accuracy of LSSE

As mentioned above, plies with different fiber orientations have significantly different material stiffness matrices. However, since it is computationally expensive to use multiple spectral elements in thickness, smeared or average properties are sometimes used to model composites for efficiency. Smeared material properties may not always give accurate results for group velocity and dispersion. To demonstrate this, first a cross-ply composite plate with layup [02/904]s i.e. [02/908/02] is considered. Compos- ite plate with this layup can be accurately modeled using three solid spectral elements in the thickness; one for bottom 02 plies, one for 908 plies, and one for top 02 plies. Figure 6.6 shows the simulated signals for sensor S for composite plate with layup

[02/904]s using 3 SSEs in thickness (blue curve), 1 SSE with smeared material properties (red curve), and 1 LSSE (black curve). Figure 6.7(a) compares the simulated voltage output at the sensor obtained by using three elements in thickness and using one element with smeared material properties. These smeared material properties were obtained as given in [17]. As it can be seen, there is significant mismatch in the simulated signals. Error in the sensor signal obtained using smeared properties was quantified as described in AppendixD. The error in the first wave packet (S 0 mode) CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 49

of the sensor signal is more than 30% and that in the second wave packet (S0 mode reflections) is more than 116%. However, PESEA can accurately simulate the sensor response using only one LSSE in thickness. Figure 6.7(b) compares the simulated sensor signals obtained by using three elements in thickness and using one LSSE with four layers. The error in the first wave packet (S0 mode) is only 0.55%, while that in the second wave packet (S0 mode reflections) is approximately 2%.

0.05 0

Output (V)Output -0.05 20 40 60 80 100 120

0.05 0

Output (V)Output -0.05 20 40 60 80 100 120

0.05 0

Output (V)Output -0.05 20 40 60 80 100 120 Time (µs)

Figure 6.6: Simulated signals for sensor S for composite plate with layup [02/904]s using: (a) 3 SSEs in thickness (blue curve), (b) 1 SSE with smeared material properties (red curve), and (c) 1 LSSE (black curve)

There is also signiﬁcant error in the group velocities of Lamb wave modes. Table

6.1 lists the errors in the group velocities of the fundamental symmetric (S0) and antisymmetric (A0) wave modes in x-direction. Smeared material properties give rise to 6.34% and 4.33% error in the group velocities of S0 and A0 modes, respectively. However, the error in the velocities of both modes is 0.44% and 0.93%, respectively, in case of LSSE. CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 50

3 SSE in thickness 0.06 1 SSE (smeared)

0.04 0.02 0

Output (V)Output -0.02 -0.04 -0.06 error = 30.41% error = 116.07%

20 40 60 80 Time (µs )

(a) Comparison between simulated signals using 3 SSEs and 1 SSE with smeared material properties

3 SSE in thickness 0.06 1 LSSE

0.04 0.02 0

Output (V)Output -0.02 -0.04 error = 0.55% error = 2.01% -0.06

20 40 60 80 Time (µs )

(b) Comparison between simulated signals using 3 SSEs and 1 LSSE

Figure 6.7: Veriﬁcation LSSE for layup [02/904]s CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 51

Table 6.1: Error in velocities of S0 and A0 modes in x-direction for layup [02/904]s

Model S0 mode A0 mode Velocity (m/s) Error (%) Velocity (m/s) Error (%) 3 SSE in thickness 5331.47 - 1562.28 - 1 SSE (smeared) 5669.64 6.34 1629.95 4.33 1 LSSE 5354.88 0.44 1576.82 0.93

This demonstrates that using smeared material properties to improve efficiency leads to inaccurate sensor signals. Another drawback of using smeared properties is that the same sensor signal is obtained even if the sequence of plies in a given layup ◦ is changed. For example, [02/904]s and [902/02/902]s have the same number of 0 and 90◦ plies, so they have the same smeared properties and the same sensor signal is obtained if only one element in the thickness direction is used with the smeared properties. However, wave propagation, in particular the antisymmetric wave mode, depends on the stacking sequence of the layup. Sensor signals for layup [902/02/902]s using five elements in thickness and 1 element with smeared properties are compared in Figure 6.8(a). The error in the first wave packet (S0 mode) is 1.87%. Figure 6.8(b) compares the simulated sensor signals obtained by using five elements in thickness and using one LSSE with five layers. The error in the first wave packet (S0 mode) is 0.96%.

Table 6.2: Error in velocities of S0 and A0 modes in x-direction for layup [902/02/902]s

Model S0 mode A0 mode Velocity (m/s) Error (%) Velocity (m/s) Error (%) 5 SSE in thickness 5515.74 - 1527.05 - 1 SSE (smeared) 5669.64 2.79 1629.95 6.74 1 LSSE 5530.76 0.27 1517.93 -0.60

Table 6.2 lists the errors in the group velocities of the S0 and A0 modes in x- direction. Smeared material properties give rise to 2.79% and 6.74% error in the group velocities of S0 and A0 modes, respectively. However, the error in the velocities of both CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 52

5 SSE in thickness 0.06 1 SSE (smeared)

0.04 0.02 0

Output (V)Output -0.02 -0.04 -0.06 error = 1.87% error = 6.16%

20 30 40 50 60 70 Time (µs )

(a) Comparison between simulated signals using 5 SSEs and 1 SSE with smeared material properties

5 SSE in thickness 0.06 1 LSSE

0.04 0.02 0

Output (V)Output -0.02 -0.04 -0.06 error = 0.96% error = 3.49%

20 30 40 50 60 70 Time (µs )

(b) Comparison between simulated signals using 5 SSEs and 1 LSSE

Figure 6.8: Veriﬁcation LSSE for layup [02/902/02]s CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 53

modes is 0.27% and 0.60% in case of LSSE. It can be noticed that the sensor signals for cross-ply layups [02/904]s and [902/02/902]s obtained by using multiple elements in thickness do not match. It can also be observed that the smeared properties do not give rise to large errors in terms of signal envelope and group velocity for S0 mode for layup [902/02/902]s. However, the error is signiﬁcant for A0 mode. This is expected since S0 (fundamental symmetric mode) is an extensional mode, while

A0 (fundamental antisymmetric mode) is a bending mode. And any change in the stacking sequence of the plies changes the bending properties of the composite plate. To further illustrate this point, simulations were carried out for a quasi-isotropic composite plate. Figure 6.9(a) and Figure 6.9(b) compare the simulated sensor signals obtained by using one LSSE and one SSE with smeared material properties for layups

[0/90/45/-45]s and [45/-45/0/90]s, respectively. Figure 6.10(a) shows the surface plots of the out-of-plane displacements at 36 µs from simulation using smeared material properties. Figure 6.10(b) and Figure 6.10(c) show the surface plots from simulations using the LSSE for layups [0/90/45/-45]s and [45/-45/0/90]s, respectively. As it can be observed, the error in the signal envelops and group velocities of the

first wave packet (S0 mode) and second wave packet (S0 mode reflections) is negligible. There is significant mismatch in the angular dependence of the group velocity of the

A0 mode [Figure 6.10]. Simulations using SSE with smeared material properties give a circular wave front for the A0 mode, as shown in Figure 6.10(a). However, from Figure 6.10(b) and Figure 6.10(c), it can be observed that the group velocity of the

A0 mode is greater in the direction of the ﬁbers in the outer layers. A0 mode is ◦ ◦ ◦ ◦ faster in the 0 and 180 directions in layup [0/90/45/-45]s, and in the 45 and 225 directions in layup [45/-45/0/90]s. This is because the outer layers dominate the bending properties related to the A0 mode.

Table 6.3 lists the errors in the group velocities of the S0 and A0 modes in x- direction. As expected, the smeared material properties do not give rise to large errors in the velocity of the S0 mode. However the errors are relatively larger for the

A0 mode, in particular for layup [0/90/45/-45]s. CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 54

0.04 1 LSSE 1 SSE (smeared)

0.02

0 Output (V)Output -0.02

error = 0.87% error = 1.20% -0.04 20 30 40 50 60 70 Time (µs )

(a) Using 1 SSE with smeared material properties and 1 LSSE

0.04 1 LSSE 1 SSE (smeared)

0.02

0 Output (V)Output -0.02 error = 0.07% error = 0.19% -0.04 20 30 40 50 60 70 Time (µs )

(b) Using 1 SSE with smeared material properties and 1 LSSE

Figure 6.9: Comparison between simulated signals for quasi-isotropic CFRP plates with layup (a) [0/90/45/-45]s and (b) [45/-45/0/90]s CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 55

(a)

(b)

(c)

Figure 6.10: Surface plots of out-of-plane displacement at 36 µs for quasi-isotropic CFRP for layup (a) [0/90/45/-45]s and [45/-45/0/90]s using 1 SSE in thickness with smeared properties (b) [0/90/45/-45]s using 1 LSSE in thickness and (c) [45/- 45/0/90]s using 1 LSSE in thickness CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 56

Table 6.3: Error in velocities (in m/s) of S0 and A0 modes x-direction for layup

[0/90/45/-45]s and [45/-45/0/90]s

Model S0 mode A0 mode Velocity (m/s) Error (%) Velocity (m/s) Error (%)

[0/90/45/-45]s, 1 LSSE 6476.84 - 1795.90 - 1 SSE (smeared) 6485.11 0.13 1690.52 -6.87

[45/-45/0/90]s, 1 LSSE 6496.16 - 1702.79 - 1 SSE (smeared) 6485.11 -0.17 1690.52 -0.72

6.2.2 Eﬃciency of LSSE

The details of the computational cost of the simulations are listed in Table 6.4. Mod- eling of the cross-ply [02/904]s composite plate with three solid spectral elements in thickness requires 5,544 elements with 389,093 degrees of freedom (DOF). One time step in the simulation takes approximately 0.6396 s. Smeared material properties can be used to reduce the memory storage and computational time in order to improve eﬃciency; however it yields inaccurate results as shown above. Modeling with the LSSE, on the other hand, gives accurate results while using less memory storage (1,880 elements and 151,709 DOF) and slightly less computational time per time step (0.6118 s).

Table 6.4: Comparison of computational costs for SSE and LSSE Layup Model Elements Total DOF Computational time per time step (s)

[02/904]s 3 SSE in thickness 5,544 389,093 0.6396 1 LSSE 1,880 151,709 0.6118

[902/02/902]s 5 SSE in thickness 9,208 626,477 0.9625 1 LSSE 1,880 151,709 0.7644

In this study, the simulation results obtained using layered solid spectral elements are in good agreement with those obtained using multiple solid spectral elements in CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 57

thickness. This demonstrates that the proposed LSSE is accurate and eﬃcient in modeling the wave propagation in layered composites. Hence it provides an eﬀective tool to gain insights into the ultrasonic wave propagation and the sensor signals in composite structures.

6.3 Validation of Layered Solid Spectral Element

To validate the proposed PESEA model with LSSE, experiments were conducted on a 609.6 mm × 228.6 mm ﬂat laminated CFRP cross-ply composite plate with layup [02/904]s and geometry as shown in Figure 6.11 and Figure 6.12. Carbon ﬁber prepreg (T800S/3900-2) was used to manufacture the plate. Single-PZT SMART

Layers R (PZT-5A) from Acellent Technologies were attached on the composite plate, using Hysol R EA 9696 thin ﬁlm adhesive. PZT actuator/sensors were 6.35 mm in diameter and 250 µm in thickness. ScanGenie R data acquisition hardware and SmartCompositeTM software were used [Figure 6.13] to carry out the experiment in pitch-catch setup. Actuator A was excited with a ﬁve-cycle tone burst signal with center frequency of 200 kHz [Figure 6.3]. Response was recorded at sensors S1 and S2.

Figure 6.11: CFRP T800S/3900-2 composite plate with layup [02/904]s CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 58

38.1

A S1 76.2

228.6 S2 y 50.8 θ

x 63.5

76.2 152.4 76.2 304.8

Figure 6.12: Geometric conﬁguration of the composite plate considered for validation

Figure 6.13: Data acquisition hardware and software

Figure 6.14 and Figure 6.15 show the signals for sensor S1 and S2, respectively, from experiments (blue curve), simulations using 1 SSE with smeared material properties (red curve), and simulations using 1 LSSE (black curve). Error in the ﬁrst two wave packets in the simulated signals for S1 and S2 with respect to the experimental signals is shown in Figure 6.16 and Figure 6.17, respectively. When smeared material properties are used, signiﬁcant mismatches in the group velocities of the wave modes as well as the waveform are observed. Simulated signal using PESEA with LSSE, on the other hand, correlates very well with the experimental signal with very small error. The error in the simulated signals for sensors S1 and S2 with respect to CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 59

0.05

0 Output (V)Output -0.05 20 40 60 80 100 120 0.05

0 Output (V)Output -0.05 20 40 60 80 100 120 Time (µs)

Figure 6.14: Simulated signals for sensor S1 for composite plate with layup [02/904]s using: (a) experiment (blue curve), (b) simulation using 1 SSE with smeared material properties (red curve), and (c) simulation using 1 LSSE (black curve) the experimental signals is shown in Figure 6.16 and Figure 6.17. Very small errors show that PESEA can accurately and eﬃciently model ultrasonic wave propagation in composite plates using fewer elements in thickness.

6.4 Validation of Acoustoelastic Formulation

6.4.1 Experimental Setup

Experiments were performed to quantify the change in wave velocity as a function of applied load as well as to validate the numerical model. The geometry of the aluminum plate and the test setup for loading experiments are shown in Figure 6.18.

Eight single-PZT SMART Layers R (PZT-5A) from Acellent Technologies were attached on the plate, using Hysol R EA 9696 thin ﬁlm adhesive. PZT sensors were CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 60

0.02

0 Output (V)Output -0.02 40 60 80 100 120 0.02

0 Output (V)Output -0.02 40 60 80 100 120 Time (µs)

Figure 6.15: Simulated signals for sensor S2 for composite plate with layup [02/904]s using: (a) experiment (blue curve), (b) simulation using 1 SSE with smeared material properties (red curve), and (c) simulation using 1 LSSE (black curve) CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 61

Experiment 0.06 Simulation (smeared)

0.04 0.02 0

Output (V)Output -0.02 -0.04 error = 26.45% -0.06 error = 20.04%

20 40 60 80 Time (µs)

(a) Simulations using smeared material properties and solid spectral element

Experiment 0.06 Simulation (LSSE) 0.04 0.02 0

Output (V)Output -0.02 -0.04 error = 0.78% error = 3.35% -0.06

20 40 60 80 Time (µs)

(b) Simulations using layered solid spectral element

Figure 6.16: Composite plate: comparison between experimental and simulated signal for sensor S1 CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 62

Experiment 0.02 Simulation (smeared)

0.01

0 Output (V)Output -0.01

error = 10.44% error = 187.25% -0.02

40 60 80 100 Time (µs)

(a) Simulations using smeared material properties and solid spectral element

0.02 Experiment Simulation (LSSE) 0.01

0 Output (V)Output -0.01

error = 6.32% error = 17.18% -0.02

40 60 80 100 Time (µs)

(b) Simulations using layered solid spectral element

Figure 6.17: Composite plate: comparison between experimental and simulated signal for sensor S2 CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 63

6.35 mm in diameter and 250 µm in thickness. ScanGenie R data acquisition hardware and SmartCompositeTM software were used. Sensors 1−4 were used as actuators and sensors 5−8 were used as sensors (Figure 6.18). The actuation signal was a ﬁve- cycle tone burst signal with 200 kHz and 250 kHz central frequencies. Sensor signals were recorded for all pairs of actuators-sensors under uniaxial loads from 0 kip (= 0 N) to 5 kips (≈ 22.24 kN) in steps of 1 kip (≈ 4.448 kN).

Velocity of the S0 mode is calculated from the time of arrival of the first wave packet in the sensor signals. Due to small size of the plate, there were a lot of reflections from the edges in the sensor signal. Hence a clean first wave packet was not always obtained. Therefore the time of arrival for the S0 mode was obtained from the zero crossing times as shown in Figure 6.19. The velocity was obtained by dividing the distance between the actuator and sensor by this time of arrival.

6.4.2 Simulations

Mesh of the aluminum plate created in Abaqus/CAE R is shown in Figure 6.20. Finite element mesh is used to solve the static problem for the applied load. Load is applied in x-direction at the nodes belonging to node set A, while the node set B is constrained to zero displacement (as shown in Figure 6.20). The same finite element mesh is used to create a spectral element mesh for PESEA simulations. The displacements at the nodes of the finite element mesh are used to interpolate the displacements for the spectral element mesh. The displacement field in x, y, and z directions for 1 kip (≈ 4.448 kN) is shown in Figure 6.21. The material properties used in the simulations are given in AppendixC. In PESEA simulations, solid spectral elements of order 5×5×3 were used. Fixed boundary conditions at nodes belonging to sets A and B (Figure 6.20) were assumed and the input excitation voltage was applied to the top surface nodes of the actuator. The material properties used in the simulations are given in AppendixC. The velocity of the S0 mode for different applied loads is calculated from the time of arrival. The time of arrival was obtained from the zero crossings of the actuation and sensor signals as shown in Figure 6.19. CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 64

31.7 17 17 17 31.7

Glass/epoxy tab 38.1

aluminum plate 1 2 3 4 50.8 waveform generator 177.8 355.6

data MTS acquisition

5 6 7 8 50.8

Glass/epoxy tab 38.1

114.4 (a) Simulations using (b) Simulations using layered solid spectral ele- smeared material prop- ment erties and solid spectral element

Figure 6.18: Geometric conﬁguration and test setup for the experiments CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 65

1 time of arrival Normalized input Normalized output

0.5

zero crossing Input or (V)Output Input -0.5

-1 0 10 20 30 40 50 Time (µs)

Figure 6.19: Calculating velocity of S0 mode using zero crossing times

Node set B Node set A

z x

Figure 6.20: Mesh of aluminum plate created in Abaqus/CAE R CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 66

U, Ux (in mm) +7.923e-2 +7.263e-2 +6.602e-2 +5.942e-2 +4.622e-2 +3.961e-2 +3.301e-2 +2.641e-2 +1.981e-2 +1.320e-2 +6.602e-3 +0.000e-0 (a)

U, Uy (in mm) +5.474e-3 +4.561e-3 +3.649e-3 +2.737e-3 +1.825e-3 +9.123e-4 +2.190e-8 −9.123e-4 −1.825e-3 −2.737e-3 −3.649e-3 −5.474e-3 (b)

U, Uz (in mm) +2.968e-3 +1.087e-3 −7.929e-4 −2.673e-3 −4.553e-3 −6.433e-3 −8.314e-3 −1.019e-2 −1.207e-2 −1.395e-2 −1.583e-2 −1.771e-2 −1.959e-2 (c)

Figure 6.21: Contours of displacement in (a) x, (b) y, and (c) z direction for 1 kip applied load CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 67

6.4.3 Results

The relative change in the group velocity, Vgr, at 200 kHz and 250 kHz for all paths at diﬀerent applied load levels was computed from the experimental and simulated sensor signals. Figure 6.22 and Figure 6.23 show the relative Vgr for 200 kHz and 250 kHz when sensor 1 was actuated and the signal was recorded at sensor 6 (path 16).

Whereas, Figure 6.24 and Figure 6.25 show the relative Vgr for 200 kHz and 250 kHz when sensor 5 was actuated and the signal was recorded at sensor 5 (path 45). As the figures show, the simulation results are in excellent agreement with the experimental results. In experiments, it was observed that the sensor signal amplitude increases significantly with load. The relative change in the amplitude for pat 16 at 200 kHz actuation frequency is shown in Figure 6.26. The amplitude increases by 58% at 5 kips. It is hypothesized that the increase in amplitude is caused by the change in piezoelectric coefficients and dielectric constants due to stress. These material parameters are very sensitive to stress [30] and the signal amplitude is very sensitive to these parameters.

Parametric study tells us that 1% increase in d31 causes the amplitude to increase by 2.3% and 1% increase in κ33 can decrease the amplitude by 1.2%. However, the required variation in d31 and κ33 is not available in literature. Hence no variation was assumed in simulations and the change in simulated signals is only about 2% (blank curve in Figure 6.26). If we assume that relative change in d31 is +0.35%/MPa, then we can capture the variation in signal amplitude as shown in Figure 6.26 (red curve). Here, it should be noted that the acoustoelastic formulation to model the eﬀect of load is complete, but experimental studies are required to characterize d31 and κ33 under stress.

6.5 Conclusions

In this chapter, validation of PESEA simulations using solid spectral element with experiments on an aluminum plate was presented. The proposed layered solid spectral element was also veriﬁed and validated with experiments on cross-ply carbon ﬁber CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 68

1 Experiments Simulations

0.995

Relative Vgr Relative 0.99

0.985 0 1 2 3 4 5 Load (kips)

Figure 6.22: Relative change in group velocity at 200 kHz for path 16

1 Experiments Simulations

0.995

Relative Vgr Relative 0.99

0.985 0 1 2 3 4 5 Load (kips)

Figure 6.23: Relative change in group velocity at 250 kHz for path 16 CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 69

1 Experiments Simulations

0.995

Relative Vgr Relative 0.99

0.985 0 1 2 3 4 5 Load (kips)

Figure 6.24: Relative change in group velocity at 200 kHz for path 45

1 Experiments Simulations

0.995

Relative Vgr Relative 0.99

0.985 0 1 2 3 4 5 Load (kips)

Figure 6.25: Relative change in group velocity at 250 kHz for path 45 CHAPTER 6. VERIFICATION & VALIDATION OF PESEA 70

1.6 Experiments

Simulations with no d31 variation

1.5 Simulations with d31 variation

1.4

1.3

Relative Amp Relative 1.2

1.1

1 0 1 2 3 4 5 Load (kips)

Figure 6.26: Relative change in sensor signal amplitude at 200 kHz for path 16 reinforced plastic (CFRP) plate. A numerical and experimental study of axially stressed aluminum plates with surface-mounted piezoelectric sensors was presented in this chapter to validate the formulation of the acoustoelastic wave propagation. This shows that PESEA can accurately and eﬃciently model piezo-induced ultrasonic wave propagation in homogeneous as well as layered media. Chapter 7

Wave Propagation in Structures

7.1 Introduction

Performance of any SHM system depends on the accuracy of damage detection and localization. In order to develop accurate diagnostic algorithms, it is very important to gain physical insights into how the ultrasonic waves induced by piezoelectric actuators interact with different types of damage. Fatigue cracks in metallic structures can cause significant reduction in the strength of the structure [45]. The damage forms in composite structures are quite complex, involving both intralaminar damage (e.g. matrix micro-cracking) and interlaminar damage (e.g. delamination) [46, 47]. Typically, transverse micro-cracking through the thickness of the ply occurs first, and then delamination follows [46, 47]. Delamination is one of the predominant forms of failure in laminated composites due to the lack of reinforcement in the thickness direction. Delamination and in case of stiffened structures skin-stiffener debond can occur as a result of impact or manufacturing defect. Delamination and debond can cause significant reduction in the strength and integrity of the structure which can lead to a catastrophic failure [46,47]. This chapter shows how PESEA simulations can be used for visualization of piezo- induced wave propagation as well as wave-damage interaction in metallic & composite structures. Cracks in metals and debond/delamination in laminated composite material are modeled by separating nodes in the damage area to create a volume split.

71 CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 72

This chapter presents simulation results for a curved aluminum plate with a cutout and fatigue cracks, flat composite plates with different layups and delaminations at different ply-interfaces, and a stiffened composite plate with a debond.

7.2 Wave Propagation in Metallic Structures

To demonstrate the capability of PESEA to model complex structures, numerical simulations were performed for a curved aluminum plate (Al 2024-T3) with a cutout and cracks. The geometry of the plate is as shown in Figure 7.1. Four PZTs (6.35 mm diameter and 250 µm thickness) were modeled, one as an actuator and three as sensors, along with 50 µm adhesive layer (Hysol R EA 9696). Since PZT is transversely isotropic material, local coordinate system was deﬁned at each PZT actuator/sensor and material properties were transformed into global coordinate while calculating element matrices/vectors.

152.4 76.2

S1 S2 114.3 Φ = 12.7 76.2 R=469.9 228.6 A S3

2 50.8

304.8 304.8

(a) Front view (b) Top view

Figure 7.1: Geometric conﬁguration of a curved aluminum plate with a cutout and cracks (all dimensions are in mm)

Fatigue cracks were modeled by separating nodes of corresponding spectral elements [45] as shown in Figure 7.2(a). PESEA model with solid spectral elements of CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 73

order 4×4×4 was used with mesh size 5 mm × 5 mm × 2 mm. The total number of elements was 2,084. The mesh is shown in Figure 7.2(b).

(a) Modeling of cracks by separating nodes

(b) Mesh for PESEA simulations

Figure 7.2: A curved aluminum plate with a cutout and cracks

Actuator A was excited with a ﬁve-cycle tone burst signal with center frequency of 200 kHz [Figure 6.3]. Sensor signals at S2 for pristine state and three diﬀerent CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 74

crack lengths are as shown in Figure 7.3. As expected, the time of arrival of the wave increases and amplitude of the signal decreases with the crack length.

0.02 Baseline 8 mm Crack 14 mm Crack 20 mm Crack 0.01

Sensor output (V) output Sensor -0.01

-0.02 30 40 50 60 70 Time ( µs)

Figure 7.3: Sensor signals at S2 for pristine condition (baseline) and for damaged condition with cracks of various lengths

Figure 7.4(a) and Figure 7.4(b) show the ultrasonic wave development in the curved aluminum plate in pristine and damaged state at various times. A MATLAB script was written transfer the PESEA output to MATLAB to plot displacement contours for visualization of wave generation and propagation. These figures show how the ultrasonic waves are reflected and scattered after the wave arrives at the hole and crack. The scattered waves further mix with the slower wave modes and reflections leading to complicated wave development as time progresses. Figure 7.4(c) shows only the propagating scattered waves at 36 µs, 48 µs, and 60 µs. Scattered waves are obtained by taking the difference in the displacement fields for the plate in the pristine state and the damaged state. Figure 7.4(c) clearly shows that the scattered waves are induced at the crack. This helps us observe the scattering at the crack and understand how ultrasonic waves interact with damage. CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 75

(a) Plate without cracks (b) Plate with 20 mm cracks (c) Waves scattered by cracks

Figure 7.4: Surface plots of out-of-plane displacement for a curved aluminum plate with a cutout

7.3 Wave Propagation in Composite Structures

In this section, wave propagation in a composite structure is presented. As shown earlier, PESEA can eﬃciently and accurately model piezo-induced ultrasonic wave propagation in multilayered media. The geometric conﬁguration of the CFRP composite plate considered in simulations is as shown in Figure 7.5(a). Five PZTs (6.35 mm diameter and 250 m thickness) were modeled, one as an actuator and three as sensors, along with 50 µm adhesive layer (Hysol R EA 9696). PESEA model with layered solid spectral elements of order 4×4×4 was used with in-plane mesh size of CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 76

304.8

S2 76.2

S3 A S1 76.2 304.8

S4 76.2 y θ

x 76.2

76.2 76.2 76.2 76.2

(a) Geometric conﬁguration (dimensions are in mm)

(b) Mesh created in Abaqus/CAE R

Figure 7.5: Geometric conﬁguration of the composite plate CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 77

5 mm × 5 mm, as shown in Figure 7.5(b). The time step was 4 ns. Actuator A was excited with a ﬁve-cycle tone burst signal with center frequency of 200 kHz [Figure 6.3]. Figure 7.6 shows out-of-plane displacement contour in at 56 µs in 2 mm thick

CFRP plates with layups [012] (unidirectional) and [02/904]s (cross-ply). In unidirectional composites, the S0 (fundamental symmetric) mode is continuously converted into the A0 (fundamental antisymmetric) mode. Figure 7.6(a) shows that PESEA can accurately capture this complex nature of ultrasonic wave propagation in the unidirectional CFRP plate.

(a) Unidirectional layup (b) Cross-ply layup

Figure 7.6: Surface plots of out-of-plane displacement for layup (a) [012] (b) [02/904]s

Figure 7.7 shows out-of-plane displacement contour in at 56 µs in 2 mm thick

CFRP plates with quasi-isotropic layups [0/30/60/90/-60/-30]s, [0/90/45/-45]s and

[45/-45/0/90]s. Even though all three layups are quasi-isotropic, the A0 (bending) mode is sensitive to the layup and the shape of the wavefront depends on the ﬁber direction of the outermost layers. CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 78

(a) Quasi-isotropic layup 1 (b) Quasi-isotropic layup 2 (c) Quasi-isotropic layup 3

Figure 7.7: Surface plots of out-of-plane displacement for quasi-isotropic layup (a)

[0/30/60/90/-60/-30]s (b) [0/90/45/-45]s (c) [45/-45/0/90]s

7.3.1 Modeling of Delamination & Debond

In order to simulate wave propagation in structures containing delamination or debond, ﬁrst, mesh is created without any delamination or debond. Then, delamination and debond are modeled by creating a volume split in the delaminated and debonded region. At the interface of adjacent elements that are aﬀected by debond/delamination, nodes are separated by small by small distance, such as 5-10 µm [48,49]. Volume split method to model debond is illustrated in Figure 7.8. It should be noted that, to model delamination in laminated composite plates, two elements are needed in the thickness direction, one element for the plies above the damaged ply interface, and one element for the plies below the damaged ply interface.

7.3.2 Multilayered Composite Plate with Delamination

PESEA was used to study the eﬀect of delamination on wave propagation in ﬂat composite plates with cross-ply layup [02/904]s and quasi-isotropic layup [0/30/60/90/-

60/-30]s. The geometry and the mesh of the plate are as shown in Figure 7.5. Actuator A was excited with a ﬁve-cycle tone burst signal with center frequency of 200 kHz [Figure 6.3]. Delamination of size 25.4 mm × 25.4 mm at diﬀerent ply interfaces was modeled by creating a volume split as described above. The area of delamination is CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 79

(a)

(b)

Figure 7.8: Modeling of debond by creating a volume split CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 80

highlighted in Figure 7.9.

Delamination (25.4×25.4) 304.8

S2 76.2

S3 A S1 76.2 304.8

S4 76.2 y θ

x 76.2

76.2 76.2 76.2 76.2

Figure 7.9: Geometric conﬁguration of the composite plate with delamination

Eﬀect of delamination on the out-of-plane displacement (z direction) is more prominent than that on the in-plane displacements (x and y directions). Surface plots of out-of-plane (z direction) displacements at 68 µs, 88 µs, and 108 µs for the cross-ply and quasi-isotropic composite plates are shown in Figure 7.10 and Figure 7.11, respectively. Figure 7.10(a) and Figure 7.11(a) show the displacement contours in the pristine plates, while subplot (b) shows the interaction of the antisymmetric

A0 mode with the delamination at the interface between plies 6 and 7 in both laminates. Scattered waves, as shown in Figure 7.10(c) and Figure 7.11(c), are obtained by subtracting the displacement field for the delaminated plate from that for the pristine plate. These figures show that the A0 mode is sensitive to delamination in both composite plates. The sensor signals for sensor S1 for cross-ply and quasi-isotropic composite plates in pristine state (no delamination) and delamination at different ply interfaces are shown in Figure 7.12. Scatter in the sensor signal at S1 for the cross-play composite plate with delaminations at various ply interfaces is plotted in Figure 7.13(a). Scatter in the signal is obtained by taking the difference between the sensor signal from the CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 81

(a) Plate with no delamina- (b) Plate with delamination (c) Waves scattered by delam- tion between plies 6 and 7 ination

Figure 7.10: Surface plots of out-of-plane displacement for a ﬂat cross-ply composite plate with layup [02/904]s CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 82

(a) Plate with no delamina- (b) Plate with delamination (c) Waves scattered by delam- tion between plies 6 and 7 ination

Figure 7.11: Surface plots of out-of-plane displacement for a ﬂat quasi-isotropic composite plate with layup [0/30/60/90/-60/-30]s CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 83

0.1 No delam 2-3

6-7 0.05 10-11

Output (V)Output -0.05

-0.1 0 20 40 60 80 100 120 Time (µs )

(a) Cross-ply composite plate

0.12 2-3No delam 2-3 6-7 6-7 0.051 10-11

Output (V)Output

-0.05-1 S catter S ignal (mV)

-0.1-2 200 2040 4060 60 80 80 100100 120120 TimeTime ((µµss ) )

(b) Quasi-isotropic composite plate

Figure 7.12: Comparison of sensor signals for sensor S1 for the composite plate with delamination at diﬀerent ply-interfaces for layup (a) [02/904]s (b) [0/30/60/90/-60/- 30]s CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 84

2 2-3 6-7

1 10-11

-1 S catter S ignal (mV)

-2 20 40 60 80 100 120 Time (µs )

(a) Cross-ply composite plate

2 2-3 6-7 10-11

-1 S catter S ignal (mV)

-2 20 40 60 80 100 120 Time (µs )

(b) Quasi-isotropic composite plate

Figure 7.13: Comparison of scatter signals for sensor S1 for the composite plate with delamination at diﬀerent ply-interfaces for layup (a) [02/904]s (b) [0/30/60/90/-60/- 30]s CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 85

pristine plate and the delaminated plate. Similarly, the scatter in the sensor signal at S1 for the quasi-isotropic composite plate is shown in Figure 7.13(b). It can be observed that the scatter depends on the depth location of the delamination. It is also evident that the scatter due to the delamination at the central plane of the plate (between plies 6-7) is smaller than that due to delaminations between plies 2-3 and

10-11. In particular, the scatter in the S0 mode is negligible in case of delamination between plies 6-7, while that in the A0 mode is dominant.

From the simulation results, it can be concluded that the S0 mode is less sensitive to delaminations at the central plane of the plate. These results are in good agreement with the observations in [48, 49]. Furthermore, the wavelength of the A0 mode is smaller than that of the S0 mode. Hence, the A0 mode can be used to detect delaminations of smaller sizes. Since scatter in the sensor signal varies with where the delamination is in thickness, PESEA can be used to gain more insight into how ultrasonic waves interact with delaminations at diﬀerent interfaces to improve the accuracy of damage localization.

7.3.3 Stiﬀened Composite Plate with Delamination/Debond

The advent of composites has lead to the replacement of fastened joints with bonded joints between the skin and the stiffeners of an aircraft structure. Depending on the energy and the velocity, a transverse impact can cause a debond between the skin and the stiffener or a delamination in the skin. Delamination and debond will affect the strength of the structure to different extent; and it is very important for an SHM system to classify between delamination and debond accurately. In this subsection, PESEA is used to model complex structures such as stiffened composite plate with delamination and debond. Numerical simulations were performed for a stiffened composite plate with delamination or debond of size 15 mm × 30 mm. Delamination was introduced by creating volume split at the interface between plies 10 and 11 and was at the same in-plane location as the debond. The geometry of the plate is as shown in Figure 7.14(a). Cross-ply layup [02/904]s and quasi-isotropic layup [0/30/60/90/-60/-30]s were considered for the plate. Layup of CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 86

1.33 50.8 114.3 t = 1.33 114.3 A S1 S2 S3 76.2 88.9 S4 50.8 88.9 76.2

(a) Geometric conﬁguration (dimensions are in mm)

(b) Mesh created in Abaqus/CAE R

Figure 7.14: Geometric conﬁguration of the stiﬀened composite plate with debond CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 87

(a) Plate with no debond (b) Plate with debond (c) Waves scattered by de- nond

Figure 7.15: Surface plots of out-of-plane displacement for a stiﬀened cross-ply composite plate with layup [02/904]s CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 88

(a) Plate with no debond (b) Plate with debond (c) Waves scattered by de- nond

Figure 7.16: Surface plots of out-of-plane displacement for a stiﬀened quasi-isotropic composite plate with layup [0/30/60/90/-60/-30]s CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 89

Delamination (10-11) 0.6 Debond

0.4 0.2 0 -0.2

-0.4 S catter S ignal (mV) -0.6 20 40 60 80 100 120 Time (µs )

(a) Cross-ply composite plate

1.5 Delamination (10-11) Debond

0.5

-0.5

S catter S ignal (mV) -1

-1.5 20 40 60 80 100 120 Time (µs )

(b) Quasi-isotropic composite plate

Figure 7.17: Comparison of scatter signals for sensor S1 for the stiﬀened composite plate with delamination (interface 10-11) or debond for layup (a) [02/904]s (b) [0/30/60/90/-60/-30]s CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 90

the stiffener was [02/902]s, respectively. Five PZTs (6.35 mm diameter and 250 µm thickness) were modeled, one as an actuator and three as sensors, along with 50 µm adhesive layer (Hysol R EA 9696). PESEA model with layered solid spectral elements of order 4×4×4 was used with in-plane mesh size of 5 mm × 5 mm as shown in Figure 7.14(b). The time step was 4 ns. Actuator A was excited with a five-cycle tone burst signal with center frequency of 200 kHz [Figure 6.3]. Mesh of the stiffened CFRP plate is shown in Figure 7.14(b). Area of debond is highlighted in the figure. Figure 7.15(a) and Figure 7.15(b) show the ultrasonic waves at 80 µs, 92 µs, and 104 µs in the cross-ply composite plate in pristine (no debond) state and with debond, respectively. Figure 7.15(b) shows how the waves are partially reflected and transmitted after the first arrival at the stiffener and debond. The scattered waves further mix with the slower wave modes and reflections leading to complicated wave development as time progresses. Figure 7.15(c) shows the waves scattered by the debond. It shows how the A0 mode waves interact with the damage. Similarly, Figure 7.16(a) and Figure 7.16(b) show the wave propagation at 80 µs, 92 µs, and 104 µs in the quasi-isotropic composite plate in pristine (no debond) state and with debond, respectively. Figure 7.16(c) shows the waves scattered by the debond. Figure 7.17(a) and Figure 7.17(b) show the scatter in the sensor signals at sensor S1 for the cross-ply and quasi-isotropic stiffened composite plates, respectively. The blue curve is the scatter signal due to the delamination at the interface of plies 10 and 11, and the red curve is the scatter signal due to the debond. These figures show that the change in the sensor signals due to delamination and debond is different. This offers the potential ability of PESEA to detect and classify delaminations and debonds. Furthermore, PESEA could be used to optimize key parameters such as sensor shape, size, locations, and input diagnostic waveform to achieve the maximum damage detection capability with a minimum number of sensors. CHAPTER 7. WAVE PROPAGATION IN STRUCTURES 91

7.4 Conclusions

This chapter shows that PESEA is a powerful tool to directly study the scattered waves and the sensitivity of sensor signals to damage in plate-like structures. It can be used to design the electrical excitation waveform to the piezoelectric actuator in order to maximize the sensitivity. PESEA can also be used to develop diagnostics algorithms to classify delamination and debond, and to study the actuator and sensor placement for a particular structure to maximize damage detection and localization capability of an SHM system. Chapter 8

Eﬀect of Elevated Temperature on Sensor Signal

8.1 Introduction

As mentioned earlier, SHM system based on acousto-ultrasound method uses built-in piezoelectric sensors. Diagnostic stress waves are generated by applying an electric voltage to the piezoelectric actuator, which is adhesively bonded to the structure. These waves travel and interact with the structure under inspection. As a result, structural information is delivered to neighboring adhesively-bonded piezoelectric sensors, which convert mechanical strain into electric voltage output. Damage detection algorithms extract damage information by comparing the sensor signal with the baseline signal for the same structure before damage. However, this technique is vulnerable to temperature variation in environment because varying ambient temperature can change the sensor signal significantly even without the presence of any damage. Figure 8.1 shows the first symmetric mode in sensor signals collected at 25 ◦C and 75 ◦C[50]. Although the same setups were used in this experiment, the signals have quite different amplitudes and arrival times. Therefore, comprehensive understanding of temperature effects is critical for practical implementation of the SHM systems based on acousto-ultrasound method using built-in piezoelectric sensors.

92 CHAPTER 8. EFFECT OF ELEVATED TEMPERATURE ON SENSOR SIGNAL93

Figure 8.1: Temperature eﬀect on sensor signal (300 kHz)

Several studies have been conducted on the temperature effects in piezo-induced Lamb wave propagation. Blaise and Chang (2001) reported amplitude reduction and time delay of sensor signal at -90 ◦C[21]. Lee et al. (2003) repeated experiments for damaged and undamaged aluminum plates to investigate hysteretic behavior, but it was not evidently shown in experimental data. The experiments showed amplitude ◦ decrease and arrival time delay of S0 mode as temperature increases from 35 C to 75 ◦C[51]. Raghavan and Cesnik (2007) studied effects of elevated temperature on Lamb wave. Peak-to-peak amplitudes and arrival times of sensor signal were theoretically simulated and compared to experimental data. The arrival times were reasonably correlated with experimental data, but the peak-to-peak amplitudes showed mismatches. The authors suggested adhesive layer effects, which were not included in the simulations, as a possible reason [22]. Recently, Ha (2009) comprehensively studied adhesive layer effects on PZT sensor signals with spectral element simulations [23]. Parameter studies with adhesive stiffness and thickness were conducted to investigate physics of adhesive layers. An interesting fact was found in his study; the amplitude of CHAPTER 8. EFFECT OF ELEVATED TEMPERATURE ON SENSOR SIGNAL94

a sensor signal may increase even with lower shear modulus adhesive layer around the resonant frequency of a surface-mounted piezoelectric sensor. Elevated temperature causes reduction in the shear modulus of the adhesive layer. Consequently, it is inferred that the adhesive layer at the elevated temperature may introduce more complicated phenomena than only the signal amplitude reduction. This chapter presents both experiments and PESEA simulations that were carried out to study the effects of adhesive layer on sensor signal under elevated temperature. Effect of varying ambient temperature was modeled by changing material properties of the sensors, adhesive, and the structure with temperature. Uniform temperature is assumed throughout the structure. As reported by Wilcox et al. (2007) [52], thermal expansion has negligible effect on the sensor signals. And since the aluminum plate considered in the experiments had free boundary conditions, the thermal stresses were negligible and only present where the sensors were bonded to the plate. Hence, thermal expansion and thermal stresses were ignored in the simulations. After the validations of the simulations, parametric studies were carried out to investigate the sensitivity of adhesive shear modulus compared to other material parameters.

8.2 Experiments and PESEA Simulations

8.2.1 Experimental Setup

In experiments for validation, an aluminum plate with 1.98 mm thickness was used. Dimensions of the plates are shown in Figure 8.2. Piezoelectric sensors (PZT-5A) with 250 µm thickness and 6.35 mm diameter were mounted to the surface using

Hysol R EA 9696 adhesive. The sensor signals were collected in pitch-catch mode. A ﬁve-cycle tone burst wave was used as actuation signal. The specimen was heated in a high-temperature-controlled oven to simulate the elevated environmental temperatures, and signals were measured from 25 ◦C to 75 ◦C. Experiments on temperature eﬀects were conducted with: (i) thin adhesive layers under both PZT actuator and sensor (thin adhesive), and (ii) thick adhesive layers under both PZT actuator and sensor (thick adhesive). Using the Hysol R EA 9696 CHAPTER 8. EFFECT OF ELEVATED TEMPERATURE ON SENSOR SIGNAL95

adhesive ﬁlm, thin and thick adhesive layers were controlled to be 40 µm and 120 µm, respectively.

225 mm

40 μm adhesive 120 μm adhesive

Plate dimension : 810 mm × 304 mm × 1.98 mm

Figure 8.2: Geometric conﬁguration of the aluminum plate

8.2.2 Variation in Material Properties with Temperature

In order to simulate piezo-induced ultrasonic wave propagation in an aluminum plate at elevated temperatures, material properties of PZT, aluminum, and adhesive were changed with temperature. Table 8.1 and Table 8.2 list the variation in the material properties used in the simulations.

Table 8.1: Variation in the electrical properties of PZT Electrical Properties

d31 d33 ε11 ε33 (×10−12 C/N) (×10−12 C/N) (Relative) (Relative) 25 ◦C 75 ◦C 25 ◦C 75 ◦C 25 ◦C 75 ◦C 25 ◦C 75 ◦C PZT [53,54] -171 -173.6 374 391 1730 1867 1700 1849

Unfortunately, the adhesive (Hysol R EA 9696) stiﬀness variation with temperature could not be found in literature. Hence, tension tests were conducted using an MTS machine with a specimen made of the adhesive material to measure its Young’s CHAPTER 8. EFFECT OF ELEVATED TEMPERATURE ON SENSOR SIGNAL96

Table 8.2: Variation in the mechanical properties of PZT and aluminum 2024-T3 Mechanical Properties E (GPa) G (GPa) ν 25 ◦C 75 ◦C 25 ◦C 75 ◦C 25 ◦C 75 ◦C PZT [53,54] 60.97 62.02 21.05 21.83 0.440 0.439 Aluminum [55] 69.00 67.69 25.94 25.15 0.330 0.346 modulus at diﬀerent temperatures as shown in Figure 8.3. The specimen was heated up to 100 ◦C and covered in non-woven polyester breather as an insulating material. A K-type thermocouple was attached to one side of the specimen to measure the current temperature. As the temperature of the specimen dropped from 75 ◦C to 25 ◦C, the Young’s modulus was measured at 8 discrete temperature points, and it was ﬁtted with a 2nd order polynomial to obtain its variation as a function of temperature. The entire test lasted for about 15-20 minutes.

Figure 8.3: An MTS test for measuring adhesive (Hysol R EA 9696) stiﬀness

The Young’s modulus of adhesive (Hysol R EA 9696) as a function of temperature CHAPTER 8. EFFECT OF ELEVATED TEMPERATURE ON SENSOR SIGNAL97

obtained experimentally is given by:

2 EAdh = 0.106 × T − 19.9 × T + 2500 (8.1)

◦ where, T is the temperature in C and EAdh is the Young’s modulus in GPa. Poisson’s ratio, νAdh, of the adhesive was assumed to be 0.3 for all temperatures considered.

8.2.3 PESEA Simulations

To validate PESEA simulations for temperature effects, simulations were conducted with the same configuration as the experimental setup in the previous section. In this validation, frequency responses of normalized maximum amplitudes at three different temperatures (25 ◦C, 50 ◦C, and 75 ◦C) were compared for thin and thick adhesive. In both experiments and simulations, the normalized maximum amplitudes of the

first wave packet (S0 mode) were determined from the sensor signals with five-cycle tone-burst inputs. The amplitudes were fitted with Gaussian curves in MicroCalTM Origin to observe the shift of a resonant frequency due to temperature change. Figure 8.4 shows variation of frequency response with thin adhesive at elevated temperatures. As temperature increases, the resonant frequency moves to a lower frequency in both experiment and simulation with thin adhesive. As a result, amplitude increases at 300 kHz, 350 kHz, and 400 kHz as temperature increases. The simulations are in a good agreement with the experiments. Frequency response with thick adhesive is shown in Figure 8.5. A resonant peak with the thick adhesive is located at a little lower frequency than the thin adhesive. Moreover, the shear lag effect is more pronounced in case of thick adhesive than thin adhesive. Consequently, the rate of amplitude increase at 300 kHz is much smaller than the thin adhesive. Although the simulation locates the resonant peak at a little lower frequency than experiment, it shows a reasonably correlated result in the trend of amplitude change at elevated temperatures. It is important to study the reason for the trends shown in Figure 8.4 and Figure 8.5. Parametric studies using PESEA simulations were conducted to investigate the effects of major parameters. CHAPTER 8. EFFECT OF ELEVATED TEMPERATURE ON SENSOR SIGNAL98

1.0 25 °C 50 °C 0.8 75 °C

0.6

0.4

0.2

0.0 Normalized Max. Amplitude (V) Amplitude Max. Normalized

100 150 200 250 300 350 400 450 500 550 Frequency (kHz)

(a) Experiments

1.0 25 °C 50 °C 0.8 75 °C

0.6

0.4

0.2

0.0 Normalized Max. Amplitude (V) Amplitude Max. Normalized

100 150 200 250 300 350 400 450 500 550 Frequency (kHz)

(b) Simulations

Figure 8.4: Frequency response with thin adhesive layer (40 µm) CHAPTER 8. EFFECT OF ELEVATED TEMPERATURE ON SENSOR SIGNAL99

1.0 25 °C 50 °C 0.8 75 °C

0.6

0.4

0.2

0.0 Normalized Max. Amplitude (V) Amplitude Max. Normalized

150 200 250 300 350 400 450 500 Frequency (kHz)

(a) Experiments

1.0 25 °C 50 °C 0.8 75 °C

0.6

0.4

0.2

0.0 Normalized Max. Amplitude (V) Amplitude Max. Normalized

100 150 200 250 300 350 400 450 500 550 Frequency (kHz)

(b) Simulations

Figure 8.5: Frequency response with thick adhesive layer (120 µm) CHAPTER 8. EFFECT OF ELEVATED TEMPERATURE ON SENSOR SIGNAL100

8.3 Parametric Studies

The objective of parametric studies is to understand the effects of major parameters at elevated temperatures. As the first step, material properties, which change with temperature, were selected as parameters. Sensor signals were simulated at two different temperatures, 25 ◦C and 75 ◦C. The major parameters, which significantly affect signal amplitude, are varied individually to study the effect of each parameter on the amplitude of the senor signal.

G Adh 30 E Al

d 31

20 E PZT

ε33 10 500 kHz 150 kHz 350 kHz

-10 Max. Amplitude Change (%) Change Amplitude Max.

-20

Figure 8.6: Parametric study with thin adhesive layer (40 µm)

Parametric studies for PZT sensors with thin (40 µm) and thick (120 µm) adhesive layers were conducted to understand the eﬀects of each parameter on sensor signals. Figure 8.6 and Figure 8.7 show percentage changes in maximum amplitudes at 150 kHz, 350 kHz, and 500 kHz excitation. The percentage changes in the maximum amplitudes are calculated with respect to those at 25 ◦C when temperature is elevated to 75 ◦C.

The ﬁve major parameters, Young’s modulus of aluminum (EAl), Young’s modulus of PZT (EPZT), shear modulus of adhesive (GAdh), PZT coupling coeﬃcient (d31), CHAPTER 8. EFFECT OF ELEVATED TEMPERATURE ON SENSOR SIGNAL101

15 500 kHz 10 5 350 kHz 150 kHz 0 -5 -10 -15 -20 G Adh -25 E Al -30 d 31 -35 E PZT Max. Amplitude Change (%) Change Amplitude Max. -40 ε33 -45

Figure 8.7: Parametric study with thick adhesive layer (120 µm)

PZT electric permittivity (ε33) were selected and their effects are below. Parametric study with a thin adhesive configuration is shown in Figure 8.6. The most influential parameter is the adhesive shear modulus (GAdh). The decrease in adhesive shear modulus at a higher temperature (75 ◦C) causes the amplitude increase at 350 kHz while it causes the amplitude to decrease at 150 kHz and 500 kHz. It is well correlated with the trend of amplitude variation shown in Figure 8.4. The shear modulus reduction moves the resonant frequency of surface-mounted PZTs closer to 350 kHz and thus increases the amplitude. The shift of the resonant frequency with the variation of the shear modulus was also reported in the previous study [23]. However, the amplitude at 500 kHz decreases as the resonant frequency recedes from 500 kHz. Figure 8.7 shows the result of parametric study with a thick adhesive configuration.

The most inﬂuential parameter is also the adhesive shear modulus (GAdh). However, the pattern of amplitude change is a quite diﬀerent from the thin adhesive case. De- crease in adhesive shear modulus at a higher temperature (75 ◦C) causes amplitude reductions at all frequencies. This phenomenon can be explained with shear lag effect, which causes reduction in signal amplitude with a thicker and softer adhesive layer. From the parametric studies, it is evident that the most important parameter CHAPTER 8. EFFECT OF ELEVATED TEMPERATURE ON SENSOR SIGNAL102

aﬀecting the amplitude of sensor signals is the adhesive shear modulus. Accordingly, more simulations were carried out to investigate variations of signal amplitudes with a diﬀerent type of adhesive.

8.3.1 Amplitude Variation with Adhesive CW2400

CW2400 (ITW CHEMTRONICS, Inc.), which is two-part conductive epoxy, was selected to investigate the pattern of signal change at elevated temperatures. To measure Young’s modulus (EAdh), a compression test was conducted using the MTS machine on a 22 mm × 5.6 mm × 45 mm small epoxy specimen. The measurement was conducted in the same way as mentioned earlier for Hysol R EA 9696. However, only an approximate value of the Young’s modulus at 25 ◦C could be obtained because the test coupon was small in dimensions. The Young’s modulus of adhesive (CW2400) as a function of temperature obtained experimentally is given by:

−5 2 EAdh = 5.6799 × 10 × T − 1.7534 × 10−2 × T + 1.6192 (8.2)

where, T is the temperature in C and EAdh is the Young’s modulus in GPa. Poisson’s ratio, νAdh, of the adhesive was assumed to be 0.3 for all temperatures considered. Shear moduli of Hysol R EA 9696 and CW2400 are summarized in Table 8.3.

Table 8.3: Variation in the shear modulus of Hysol R EA 9696 and CW2400 25 ◦C 75 ◦C Reduction (%)

Hysol R EA 9696 0.735 GPa 0.573 GPa 22.5 CW 2400 0.434 GPa 0.223 GPa 48.6

Frequency responses with thin CW2400 adhesive are plotted in Figure 8.8. The same conﬁguration as the previous simulations was used. Overall trends of CW2400 are diﬀerent from those of Hysol R EA 9696. The amplitude reduction rate of resonant peaks at higher temperatures is a lot higher than that of Hysol R EA 9696 in a CHAPTER 8. EFFECT OF ELEVATED TEMPERATURE ON SENSOR SIGNAL103

1.0 25 °C 0.8 50 °C 75 °C

0.6

0.4

0.2

Normalized Max. Amplitude (V) Amplitude Max. Normalized 0.0

150 200 250 300 350 400 450 Frequency (kHz)

Figure 8.8: Frequency response with thin CW2400 adhesive (40 µm) thin adhesive case. CW2400 adhesive has less stiffness and more temperature variation than Hysol R EA 9696, which was used in previous section. Consequently, the shear lag effect becomes more dominant, and it causes more reduction in the signal amplitude as temperature increases. Figure 8.9 shows simulated frequency responses with thick CW2400 adhesive. The amplitude reduction rate of resonant peaks at higher temperatures is a lot higher than that of Hysol R EA 9696, which is common with thin CW2400 adhesive. In particular, higher temperatures always cause lower signal amplitudes at all frequencies due to more shear lag effect.

8.4 Conclusions

In conclusions, the eﬀects of adhesive layer thickness on sensor signals with temperature variations, which have attracted little attention in the previous studies, were studied in this chapter. Experiments showed that the adhesive layer thickness can CHAPTER 8. EFFECT OF ELEVATED TEMPERATURE ON SENSOR SIGNAL104

1.0 25 °C 50 °C 0.8 75 °C

0.6

0.4

0.2

Normalized Max. Amplitude (V) Amplitude Max. Normalized 0.0

150 200 250 300 350 400 Frequency (kHz)

Figure 8.9: Frequency response with thick CW2400 adhesive (120 µm) affect sensor signals in different patterns. Numerical analysis with PESEA could simulate this phenomenon. From parametric studies with PESEA simulations, it was found that adhesive stiffness is very sensitive to temperature variation and significantly affects the sensor signal. The physical insights gained from the parametric studies can be used to develop a physics-based model to compensate the effect of temperature on the sensor signal for accurate damage diagnosis. Chapter 9

Model-Assisted Damage Diagnostics

9.1 Introduction

In SHM systems based on piezoelectric sensor network, ultrasonic waves are propa- gated through a structure induced and sensed by an active piezoelectric sensor network. The damage is detected by comparing current sensor signals (with damage) with baseline signals (without damage). Diagnostic algorithm interprets changes in the signals to detect and localize damage. An ultrasonic Lamb wave velocity based imaging algorithm was developed by Ihn and Chang (2003 & 2004) [2,3,7]. This algorithm utilizes the scatter in the sensor signal of each diagnostic actuator-sensor path. Lamb wave S0 mode velocities are calculated from the baseline data. Based on this, a damage diagnostic imaging algorithm was developed in Ref. [37] to account for material anisotropy and directional dependence of Lamb wave velocity to improve localization of damage. In this algorithm, the imaging domain is discretized into pixels which correspond to spatial points of the structure under inspection. Ultrasonic Lamb waves in induced in the structure and waves scattered from the damage are used to calculate the probability of damage at the pixels. In order to get accurate damage image, accurate estimation of time-of-ﬂight (ToF) from these pixels to all sensors is essential [56]. However, in existing algorithms, ToF for all pixels is calculated from

105 CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 106

Lamb wave velocity. Hence, the accuracy of this algorithm depends on the a priori knowledge of the Lamb wave velocity profile. Estimation of Lamb wave velocity profile for complex structures is very challenging since analytical relations only exist for simple structures. Sensor signals are typically used to estimate the velocity profile but since only a limited number of actuator-sensor paths are available, estimation of complete velocity profile for all parts of the structure is not possible. To overcome this challenge, advanced numerical simulations of ultrasonic wave propagation [44,57] can be used for accurate estimation of ToF from all pixels of the imaging domain to all sensors. Another challenge in damage diagnostics is compensation of the effect of environment (e.g. temperature change) on the sensor signals. Change in ambient temperature affects the sensor signal, even in the absence of damage [56]. If these effects are not compensated, we will get false alarms and inaccurate location of damage. To compensate for these effects, a physics-based strategy is proposed by Roy et al. (2011) [38] to improve damage detection and localization. The objective of this chapter is to present a novel model-assisted diagnostics for structural health monitoring of aerospace structures with complex geometries in varying temperature environments [56]. Given a complex structure with built-in piezoelectric sensor network, it is desired to develop a diagnostic system, for accurate damage detection and localization, which integrates

1. damage diagnostic imaging algorithm based on baseline-subtraction technique,

2. an advanced numerical tool to simulate acousto-ultrasonic Lamb wave propagation, and

3. physics-based temperature compensation model. CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 107

9.2 Method of Approach for Model-Assisted Dam- age Diagnostics

In this study a damage diagnostic imaging algorithm developed in Ref. [37] is adopted. PESEA is used to get accurate ToF from diﬀerent locations on the structure to the sensors. The damage diagnostics is further integrated with a temperature compensation model [38] to remove the eﬀect of ambient temperature change on the sensor signals for real-time application. The method of approach for the proposed model- assisted integrated damage diagnostics is as follows (Figure 9.1)[56]:

1. Validate the accuracy of SEM-based PESEA model on a smaller structure with distributed piezoelectric sensors.

2. Run oﬄine PESEA simulations for a structure with complex geometry (stiﬀeners/cut- outs/varying thickness etc.) to get ToF from some locations on the structure to the piezoelectric sensors. Interpolate to get ToF from the rest of the locations to the piezoelectric sensors and save in a table.

3. In real-time application, use the ToF from the table and use temperature compensation model to remove the eﬀect of change in ambient temperature on the sensor signals to get near real-time accurate damage image.

9.2.1 Damage Diagnostics Imaging Algorithm

It is well known that ultrasonic waves propagating in a structure can interact with structural features such as boundaries and material discontinuities (e.g. damage). The scatter in the sensor signal due to damage is obtained by subtracting the baseline signal from the current signal. Edges of damage in the structure act as a secondary source of wave generation which results in the form of scatter in the signal and can reflect part of the energy of the incoming ultrasonic wave [37], as shown in Figure 9.2(a) (pulse-echo configuration) and Figure 9.2(b) (pitch-catch configuration). Using scatter in the sensor signal from all actuator-sensor paths, damage diagnostic imaging CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 108

Input signal 2 3 1 4

(a) Step 1: Validation of numerical model

Force excitation

(b) Step 2: Oﬄine training to get ToF

Figure 9.1: Method of approach for model-assisted damage diagnostics CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 109

Induced stress Induced stress wave wave 2 1 2 1 Input Input signal Pulse-echo signal Scattered Scattered wave wave Pitch-catch

3 4 3 4

(a) Pulse-echo (b) Pitch-catch

Figure 9.2: (a) Pulse-echo and (b) pitch-catch configurations for SHM techniques have been developed. These techniques visualize a structural damage event intuitively in a two-dimensional image whose pixels correspond to spatial points of the structure under inspection [2,3,8,37,58]. In this study, a damage diagnostic imaging algorithm given in Ref. [37] is adopted to get a two-dimensional heat map image to indicate the probability of the presence of damage at a specific point of the structure. Steps followed in this algorithm are briefly explained below

1. Active diagnostic baseline and current data are preconditioned by passing through a band-pass ﬁlter to remove noise from the sensor signals.

2. The scatter sc (t) in the sensor signal for each actuator-sensor pair is calculated, using Equations (9.1) and short-time Fourier transform (STFT) is applied to

compute the spectral amplitudes, Sc (τ, fc), corresponding to the central fre-

quency fc of the actuation signal, using Equation (9.2), where, s (t) is the cur-

rent signal, sb (t) is the baseline signal.

sc (t) = s (t) − sb (t) (9.1) CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 110

∞ Z −i2πfct STFT {sc (t)} = Sc (τ, fc) = sc (t) w (t − τ) e dt (9.2) −∞

3. Velocity of S0 mode as a function of propagation angle is obtained from a polynomial least squares ﬁt through velocities computed from available actuator-sensor baseline signals, as explained in detail in Ref. [37]

4. The imaging domain is discretized into pixels of speciﬁed resolution.

5. For each pixel P and actuator-sensor pair i, the following are computed:

th (a) ToF, t1i, from actuator ai of i actuator-sensor pair to pixel P

th (b) ToF, t2i, from pixel P to sensor si of i actuator-sensor pair

LaiP t1i = (9.3) VaiP

LP si t2i = (9.4) VP si 6. ToF of the scatter signal corresponding to pixel P and ith actuator-sensor pair is calculated using Equation (9.5), as illustrated in Figure 9.3(a).

tiP = t1i + t2i (9.5)

7. Spectral amplitude of the scatter in the signal corresponding to pixel P and ith

actuator-sensor pair, S(tiP , fc), is obtained from STFT computed in step 2. As illustrated in Figure 9.3(b).

8. Intensity value, IP , for pixel P from N actuator-sensor pairs is obtained using Equation (9.6). N 1 X I = S(t , f ) (9.6) P N iP c i=1

9. Steps 5-8 are repeated for all pixels in the discretized domain. CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 111

2 t2P 1

P SC (t,f c)

tP3 STFTamplitude

t = t + t 3 4 2P P3 time

(a) Actuator-sensor pair 2-3 (b) STFT of signal from sensor 3

Figure 9.3: Scatter signal corresponding to pixel P and actuator-sensor pair 2-3

Pixels with higher intensity values have higher possibility of damage existing at those pixels [58]. Heat map of these intensity values (or log of intensity values) gives us the damage diagnostic image highlighting the area with high probability of damage. These steps are explained in detail in Ref. [37]. Accuracy of damage location in damage diagnostic image using this algorithm greatly depends on the accuracy of estimation of velocity of S0 mode in step 3. Since only a limited number of actuator-sensor paths are available, the estimation of S0 mode velocity at other propagation angles may not be very accurate. Especially in case of structures with complex geometries, where the wave interacts locally with the structural features such as stiﬀeners, cut-out, varying thickness, the ToF estimates based only on the S0 mode velocity ﬁt will be inaccurate. Error in velocity estimation gives erroneous ToF, tiP , which in turn gives erroneous intensity values for pixel P . Hence, in this work, numerical simulations of ultrasonic wave propagation using

PESEA are used to estimate ToF, tiP , as explained in the following section.

9.2.2 PESEA Simulations

In this study, piezoelectric sensors and adhesive layer are physically modeled in PE- SEA to get accurate sensor signals for the structure in pristine and damaged state. CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 112

2 1 t tP2 P1

tP4 tP3

3 4

(a) In-plane force at corner nodes

-25 Input (N) Force Nodal Input

-50 0 10 20 30 40 50 Time (µs)

(b) Five-cycle tone burst signal with central frequency of 250 kHz

Figure 9.4: PESEA simulations to get ToF to all sensors CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 113

However, to just get the ToF, physical modeling of piezoelectric sensors is not necessary. To verify this, simulations were carried out for a small aluminum plate to compare the ToF obtained by physical modeling of piezoelectric sensors and by applying in-plane nodal force [6, 59]. They matched very closely within 1 µs. Hence in this study, in-plane force is applied at the corner nodes of the top surface of an element to induce ultrasonic waves in the structure, as shown in red in Figure 9.4(a), in order to get ToF from that element to all sensors. Direction of the force at each node is along the line connecting the respective node and the centroid of the corner nodes and the magnitude for a ﬁve-cycle tone burst input signal with central frequency of 250 kHz is as shown in Figure 9.4(b). Displacements are recorded at all sensor locations and ToF to all sensors are calculated from the arrival of S0 mode.

9.2.3 Temperature Compensation Model

As mentioned earlier, another challenge in damage diagnostics is compensation of the effect of change in ambient temperature on the sensor signals. Change in temperature affects the sensor signal, and if these effects are not compensated, we will get false alarms and inaccurate location of damage. To compensate for these effects, a physics- based temperature compensation model developed in [38] is used. This model linearly relates the changes in the sensor signals to the changes in key physical properties using sensor signals taken at different ambient temperatures. Steps followed in this model are briefly explained below.

1. Sensor signals are decomposed in time-frequency domain using Gabor dictionary. Gaussian window function, given by Equation (9.7) are used to construct the elements of Gabor dictionary by scaling c, translating u, and modulating v Equation (9.7), as given by Equation (9.8).

g(t) = e−πt2 (9.7)

t − u g (t) = g cos(vt + w) (9.8) (c,u,v,w) c CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 114

2. After N iterations of decompositions, as given in [38], the sensor signal is rep- k resented as shown in Equation (9.9), where R s (t) is the residual and γk =

(ck, uk, vk, wk) are the parameters after k iterations.

N−1 X k N s(t) = R f (t) , gγk + R s (t) (9.9) k=0

3. After decomposition, coeﬃcients ak and bk given by Equations (9.10) and (9.11)

along with γk completely describes the signal [38]. k t − uk ak = R s (t) , g cos(vkt) (9.10) ck

k t − uk bk = R s (t) , g sin(vkt) (9.11) ck

4. Now, baseline sensor signals at a few diﬀerent ambient temperatures are decom-

posed and the relative change in coeﬃcients ak and bk are expressed as a linear combination of relative change in physical properties due to temperature with respect to a reference temperature (25 ◦C here) to obtain unknown constants

α1k and β1k as given by Equation (9.12).   ( ) " # ∆GAdhesive ∆ak α1k α2k α3k   ≈ ∆d31,P ZT (9.12) ∆bk β1k β2k β3k T ◦C    ∆ε33,P ZT  T ◦C

Once the constants αik and βik (i = 1, 2, 3) are obtained, they can be used to reconstruct a signal at any speciﬁed temperature or remove the eﬀect of ambient temperature change from the sensor signal as explained in [38]. CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 115

(a) Stiﬀened aluminum panel with a cutout

(b) An open crack of length 5 mm and width 200 µs

Figure 9.5: PESEA simulations to get ToF to all sensors CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 116

9.3 Performance of Model-Assisted Damage Diag- nostics

To test the performance of proposed model-assisted damage diagnostics, a flat aluminum panel with stiffeners and a cut-out was considered (Figure 9.5(a)), similar to an aircraft fuselage section with stiffeners and a cut-out for window. Simulations were carried out using PESEA for the quarter part of the panel in pristine state and in damaged state, with an open crack of 200 µm width and 5 mm length, as demonstrated in Figure 9.5(b). Crack was modeled by separating the nodes of the appropriate neighboring elements to create a volume split.

60 300×20×2 300 60 3 60 2 6 5 120 1 8 4 7 60 200 60 60 20

(a) Stiﬀened aluminum panel with a cutout

(b) An open crack of length 5 mm and width 200 µs

Figure 9.6: PESEA simulations to get ToF to all sensors

Conﬁguration of the panel is as shown in Figure 9.6(a). In numerical simulations, six 6.35 mm diameter and 250 µm thick piezoelectric sensors of type PZT-5A were CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 117

physically modeled at locations shown in Figure 9.6(a). Adhesive (Hysol R EA 9696) thickness was assumed to be 50 µm. The mesh created in Abaqus/CAE R is shown in Figure 9.6(b).

9.3.1 Oﬄine Training to Get ToF

The next step now is to carry out simulations for offline training in order to get ToF from different locations on the panel to the sensors. Nodal in-plane force was applied at the corner nodes of the top surface of an element. Magnitude of the force is five- peak tone burst signal with center frequency 250 kHz as shown in Figure 9.4(b); and direction of the force at a node is along the line joining the node and the centroid of top surface nodes of the element, as shown in Figure 9.7(a). Such nodal in-plane force was applied to 156 elements highlighted in Figure 9.7(b) one by one. Displacements were recorded at all sensor locations and ToF to all sensors were calculated from the arrival of S0 mode. ToF from the rest of the locations to the sensors was interpolated from the ToF obtained from in-plane force excitation simulations of the 156 elements highlighted in Figure 9.7(b). These ToF values were stored in a table and used in real-time application of the damage diagnostic imaging algorithm.

9.3.2 Simulations for Panel in Pristine and Damaged State

To test the performance of proposed model-assisted damage diagnostics, simulations were carried out for a ﬂat aluminum panel with stiﬀeners and a cut-out as shown in Figure 9.6. Top view of the panel is shown in Figure 9.8(a). For the damaged case, an open crack of 5 mm length and 200 µm width was modeled at the cut-out as shown in Figure 9.8(b). Damage in the form of crack was modeled by separating the nodes of the appropriate neighboring elements to create a volume split. Five-cycle tone burst voltage input signal given to the piezoelectric sensors is shown in Figure 9.9. Simulations for the baseline case were carried out for the pristine geometry at temperatures in the range of 25 ◦C - 75 ◦C with interval of 10 ◦C. Piezoelectric sensors 1-8 were actuated one at a time by applying actuation to the upper surface CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 118

F F

(a) In-plane bodal force at the corner nodes of an element

(b) In-plane force was applied at the highlighted elements

Figure 9.7: PESEA simulations to get ToF CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 119

60 60 180

60 1 2 3 60 200 4 5 6 60

60 7 8 100 R=15 60

60 60 60

(a) Geometric conﬁguration (dimensions in mm)

(b) Mesh created in Abaqus/CAE R

Figure 9.8: Stiﬀened aluminum panel with 8 PZT sensors CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 120

nodes and sensor response was recorded at the other sensors. Simulations for the damaged case were carried out for the aluminum panel with crack at temperatures 25 ◦C and 60 ◦C. To model the eﬀect of temperature, material properties of aluminum, adhesive, and piezoelectric material were changed with temperature.

0 Input (V)Input

-25

-50 0 10 20 30 40 50 Time (µs)

Figure 9.9: Five-cycle tone burst input signal with central frequency of 250 kHz

9.3.2.1 Training of Temperature Compensation Model

Baseline sensor signals simulated for temperatures in the range of 25 ◦C - 75 ◦C with interval of 10 ◦C were used to train the temperature compensation model to get constants αik and βik (i = 1, 2, 3 & k = 1, 2, 3, ..., N − 1). The parametric decomposition of each simulated sensor signals involves maximum of twenty iterations (N = 20), based on the amount of signal residue, for eﬃcient signal reconstruction.

9.3.2.2 Compensation of Temperature Eﬀect

Sensor signals simulated for damaged case at 60 ◦C include the effect of damage as well as the effect of change in ambient temperature. In order to eliminate the effect ◦ of later, the relative change in the sensor signal parameters ∆ak and ∆bk at 60 C CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 121

are obtained with respect to 25 ◦C, using Equation (9.13).

  ( ) " # ∆GAdhesive ∆ak α1k α2k α3k   ≈ ∆d31,P ZT (9.13) ∆bk β1k β2k β3k 60 ◦C    ∆ε33,P ZT  60 ◦C

Sensor signals at 60 ◦C for an undamaged structural state are reconstructed with ◦ the known model parameters ∆ak and ∆bk and the baseline sensor signal at 25 C. The eﬀect of the change in ambient temperature on simulated sensor signals for cracked aluminum panel is compensated by the following assumed principle of superposition (shown in Equation (9.14)), wherein the inﬂuence of change in temperature and the presence of damage does not interact with each other.

s(t)(damage,60 ◦C) ≈ sC (t)(damage) + s(t)(reconstructed,60 ◦C) (9.14)

s(t)compensated;(damage,25 ◦C) = sC (t)(phase shifted,damage) + s(t)(baseline,25 ◦C) (9.15)

The unknown eﬀect of structural damage on the sensor signals can be approximated by subtracting the reconstructed sensor signals from the simulated sensor signals with damaged condition at 60 ◦C. The contribution of the structural damage on the sensor signals needs to be phase compensated before being added to the baseline sensor signals at 25 ◦C to achieve the desired environmental compensation as shown in Equation (9.15).

9.3.3 Results and Discussion

Damage diagnostic images were constructed using the aforementioned simulated signals for stiﬀened panel in pristine and damaged state at diﬀerent ambient temperatures (25 ◦C and 60 ◦C). Simulated sensor signals for panel in pristine state at 25 ◦C were used as baseline data. For damage images, dB values of the intensity at each pixel normalized with respect to the maximum intensity, as given by Equation (9.16), CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 122

were used to create the heat map.

IP IdB (P ) = 20log10 (9.16) Imax

Damage image obtained using only imaging algorithm with no oﬄine training for ToF estimates is shown in Figure 9.10(a), while damage image obtained using the model-assisted damage diagnostics is shown in Figure 9.10(b). Simulated sensor signals for cracked panel at 25 ◦C were used to construct these images. Location of maximum intensity Imax is marked by x. As it can be seen, the damage location in

Figure 9.10(a) is far from the actual crack location since average S0 velocity obtained from the limited number of actuator-sensor pairs was used in the imaging algorithm. However, Figure 9.10(b) shows more accuracy in terms of the probable location of the crack. The mismatch between the actual location of crack and location of maximum intensity value (Imax) obtained using the damage diagnostics is due small damage size (5 mm crack length) and few number of sensors near the damage. It should also be noted that the actual crack location is outside the area covered by the eight piezoelectric sensors. Figure 9.11 highlights the importance of temperature compensation model in damage diagnostics. Damage image obtained using integrated damage diagnostic imaging algorithm with offline training for ToF estimates is shown in Figure 9.11(a). Simu- lated sensor signals for cracked panel at 60 ◦C were used to construct these images. In this case, sensor signals are affected by the crack as well as the change in ambient temperature. Since the effect of ambient temperature change was not removed from the signal, we get erroneous damage image (Figure 9.11(a)). On the other hand, damage image shown in Figure 9.11(b) is more accurate as it uses the model-assisted damage diagnostics with temperature compensation model. In principle, sensor signals at 25 ◦C and sensor signals at 60 ◦C after temperature compensation should give the same damage image, however, the mismatch between Figure 9.10(b) and Figure 9.11(b) is partly due to the inherent error in the reconstruction of sensor signals at 60 ◦C using the temperature compensation model and partly due to the assumption of independency of the underlying causes that affect the sensor response. Figure 9.10 CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 123

(a) Diagnostic algorithm given in [37]

(b) Model-assisted damage diagnostics

Figure 9.10: Diagnostic images for aluminum panel using sensor signals at 25 ◦C CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 124

(a) Without temperature compensation

(b) With temperature compensation

Figure 9.11: Diagnostic images for aluminum panel using sensor signals at 60 ◦C CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 125

and Figure 9.11 demonstrate that the proposed model-assisted damage diagnostics is more accurate than the existing algorithm in terms of the probable location of damage.

9.4 Conclusions

In this chapter, a damage diagnostics strategy that integrates acousto-ultrasonic wave propagation simulations, temperature compensation model and damage diagnostic imaging algorithm was presented for accurate damage localization in complex structures. The steps involved in this improved model-assisted damage diagnostics are as follows:

1. Validation of the SEM-based PESEA model on a smaller structure with distributed piezoelectric sensors.

2. Oﬄine PESEA simulations for the given structure to get ToF from some locations on the structure to the piezoelectric sensors. Interpolation of ToF from the rest of the locations to the piezoelectric sensors. All ToF values are then stored in a table.

3. In real-time application, ToF for all pixels are obtained from the table in step 2 and the temperature compensation model is used to remove the eﬀect of change in ambient temperature on the sensor signals to get near real-time accurate damage image.

Performance of model-assisted damage diagnostics was tested on simulated signals for a flat aluminum panel with stiffeners in pristine and damaged (crack) state at different ambient temperatures. Effect of temperature was modeled by varying material properties of aluminum, adhesive and piezoelectric material with temperature. The results showed that damage, as small as 5 mm in the form of an open crack, in aluminum stiffened panel could be identified and accurately localized with the integrated damage diagnostic imaging algorithm using the sensor signals at room temperature (25 ◦C) and elevated temperature (60 ◦C). The model-assisted damage CHAPTER 9. MODEL-ASSISTED DAMAGE DIAGNOSTICS 126

diagnostics is, therefore, demonstrated to be able to identify and localize damage in complex structures. Oﬄine training to get the ToF table also reduces computa- tions in the diagnostic algorithm making the integrated diagnostics faster in real-time applications. Chapter 10

Model-Assisted Sensor Network Design

10.1 Introduction

As mentioned earlier, structural health monitoring (SHM) based on acousto-ultrasound method has emerged as a promising technique for inspection of structural damage. The detection capability of a given SHM system strongly depends not only on its algorithm, but also on the sensor-actuator density, their distribution, the hardware sensitivities (signal to noise ratio), and uncertainties in the material properties of the sensors and the actuators. Moreover, quantification of the performance of an SHM system is critical for practical implementation on structures. Markmiller and Chang (2010) [34] showed that for passive SHM systems, developed for impact monitoring, sensor network layout significantly affects the probability of detection (POD) of impacts on the monitoring structure. Therefore, optimal sensor placement is required. Reliable sensor network optimizations procedures should consider (i) characteristics and material properties of structure, sensor and adhesive, (ii) performance of data acquisition hardware, and (iii) performance of diagnostic techniques used in the SHM system. However, most of the current sensor networks are designed through the experience obtained from trial and error procedures. Moreover, the deterministic

127 CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 128

approach in the existing technique does not address the key issue of the sensitivity of the sensor network to the variation in the material characteristics of a real structure. Recently, Janapati et al. (2012) [60] developed a sensor network optimization technique, called Model-Assisted Sensor Network Optimization Technique (ModSENOPT). ModSENOPT utilizes wave propagation simulations, a pre-selected diagnostic algorithm, and genetic algorithm to optimize the sensor network layout by maximizing the probability of damage detection (POD) for a given structure [60]. This chapter shows how PESEA simulations can be used to study the eﬀect of damage and uncertainty in material properties on the sensor signal and how this information can be used in ModSENOPT to optimize the sensor network for a given structure. A stiﬀened metallic panel was selected to demonstrate its performance. The results show that the proposed methodology can provide an optimal sensor network for complex structures.

10.2 Method of Approach for Model-Assisted Sen- sor Network Optimization

In this study, a sensor network optimization methodology, Model-Assisted Sensor Network Optimization Technique (ModSENOPT), introduced in [60] is adopted. An active sensing diagnostic algorithm (pitch-catch and pulse-echo) [56] is selected for detecting and localizing a damage in the structure. In this algorithm, detection is based on the difference between the sensor signal from a structure in damaged state and the baseline signal from the pristine structure. This difference is called the scatter signal. If the amplitude of the scatter signal exceeds a predefined threshold, then the damage is detected. For a given structure, the amplitude of the scatter signal depends on the distance of the damage from the actuator and the sensor. As this distance increases, the amplitude of the scatter signal decreases. The distance at which the scatter signal is almost equal to the predefined threshold is defined as the damage detection distance (DDD). The threshold depends on the SHM hardware and software. For a given structure with an SHM system, DDD depends on the critical CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 129

Figure 10.1: Schematic of the approach for the sensor network optimization of an SHM system CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 130

damage size 0a0, and how the amplitude of the scatter signal decreases with distance depends on the geometry of the structure, material properties, etc. Probability of detection (POD) of damage [56] depends on the DDD and the sensor layout. Hence, once the software and hardware for SHM is chosen, the value of DDD must be determined for a given structure in order to optimize the sensor layout that maximized the POD. In order to compute DDD for a given structure with an SHM system, a parameter

ScNRaX is defined. ScNRaX is the ratio of the amplitude of the scatter signal due to damage of size 0a0 for wave propagation distance 0X0 and the amplitude of the noise level 0N 0 in the data acquisition system. In this study, the threshold of the scatter amplitude for determining the DDD value is set to be 3.16 times the noise 0 0 level. Hence, the DDD corresponds to the distance X for which ScNRaX = 10 dB. ModSENOPT uses simulations of ultrasonic wave propagation in structures to estimate DDD profile for a critical damage size 0a0. PESEA is used to accurately and efficiently simulate piezo-induced ultrasonic waves in structures. Once the DDD is estimated, genetic algorithm (GA) based optimization is implemented to optimize the sensor network by maximizing the probability of detection of the network (PODnet). Detailed definition of PODnet is given in [60]. A schematic of how ModSENOPT works is shown in Figure 10.1. However, the deterministic approach of determining DDD presented in [60] does not address the key issue of the sensitivity of the sensor network to the variation in material characteristics of a real structure. This challenge is overcome by the application of nondeterministic analyses. Monte Carlo simulations using PESEA are carried out to quantify the effect of uncertainty in the material properties on the amplitude of the sensor signal. This uncertainty quantification is used to obtain a more conservative value of DDD value. Next subsection describes how the probability of detection of a sensor network

(PODnet) is determined for a given DDD. In the following sections present the formulation of SEM-based PESEA, computation of DDD using PESEA, and uncertainty quantiﬁcation using Monte Carlo simulations. CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 131

10.2.1 Probability of Detection of Sensor Network (PODnet)

PODnet is deﬁned as the overall network POD of the monitoring system in detecting damage anywhere in the structure. In general, PODnet should be 100% for a given monitoring system. In this work, total structure is discretized into a grid of points based on the required resolution in X and Y directions and it is assumed that the detectability of a point is either ’0’ or ’1’ based on the following criterion.

m n  P P  = 1 if Dmin ≤ D(a, s, P ) ≤ DDD POD(P ) a=1 s=1 (10.1)  = 0 otherwise where, P is index of a point in the grid, a, s are indices of actuator and sensor respectively, m, n are numbers of actuator and sensors respectively. Dmin is minimum distance between any actuator and sensor (this is required due to the signals’ crosstalk). D(a, s, P ) is the summation of distance from actuator ’a’ to point ’P ’ and from point ’P ’ to sensor ’s’. Evolutionary computation techniques such as GA [61] are search algorithms based on the mechanics of natural selection. They are robust, conceptually simple and very eﬃcient in ﬁnding a near global optimum. The GA based inverse method of reconstruction starts with a population of randomly guessed candidate solutions (three sensor locations) where each candidate solution corresponds to the location of sensors. For each candidate solution, PODnet is evaluated using the objective function given by Equation (10.2), where P is index of the point and N is total number of points in the grid. N P POD(P ) POD = P =1 (10.2) net N

Candidate solutions which have high PODnet value can be placed in the selection process to go to the next generation while the rest of them are discarded. Typi- cally, this process needs to be carried out several hundred times to achieve the maximum PODnet value required for the given application. Once the maximum PODnet CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 132

reaches/exceeds the required PODnet speciﬁed in the requirement, then the optimization process stops its evolution. If the maximum PODnet in a generation doesnt reach required PODnet, then one more sensor is added and the evolution process starts all over again and this process is repeated until required PODnet is achieved. In order to accurately localize damage, at least three diagnostic paths (actuator-sensor pairs) need to detect the damage. Hence, probability of damage localization for point P in the grid is said to be 1 if probability of detection at that point is 1 for three distinct actuator-sensor pairs (Equation (10.1)). The same optimization algorithm is then used for maximizing the probability of damage localization for the sensor network.

10.3 PESEA Simulations to Compute DDD

To demonstrate how DDD is computed using PESEA, simulations were carried out for an aluminum plate (254 mm × 609.6 mm × 2 mm) in pristine state and in damaged state with a 10 mm long crack at different orientations. Geometry of the 1 plate is as shown in Figure 10.2(a). 4 inch diameter and 10 mil thick lead zirconate titanate (PZT) actuator was given a voltage input of five-cycle tone burst signal with central frequency of 250 kHz and the output voltage signal was recorded at sensors 2 to 6 placed 76.2 mm apart as shown in Figure 10.2(b). Simulated sensor data were analysed to calculate the DDD profile through the estimation ScNRaX for different wave propagation distances (for a given noise level in the data acquisition system).

ScNRaX is computed using the following formula.

Sc ScNR = 20log (10.3) aX 10 N where, Sc and N are the amplitude of the scatter signal and noise level, respectively. From Figure 10.2(b), DDD can be estimated as 245 mm (≈ 9.65”) corresponding to

ScNRaX = 10 dB. CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 133

114.3 381 114.3

88.9

6 5 4 3 2 1

254 crack 76.2 Φ = 15.24

88.9

609.6

(a) Schematic of aluminum plate with PZT sensors (all dimensions are in mm)

(dB) aX

10 ScNR

0 100 150 200 250 300 350 400 450 Distance (mm)

(b) Scatter to noise ratio (ScNRaX ) for diﬀerent wave propagation distances

Figure 10.2: Schematic of the aluminum plate and ScNRaX ratio CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 134

10.3.1 Stochastic Spectral Element Method

PESEA simulations described above, assume no variations in the inputs. Hence it is a deterministic simulation tool. In order to examine the effect of uncertainty in the material properties on the sensor response and ScNRaX , uncertainty analysis was performed using stochastic spectral element method. Like stochastic FEM, stochastic SEM can be broadly classified into two categories: (a) analytical methods and (b) sampling-based methods. Analytical methods involve either the differentiation of governing equations and subsequent solution of a set of auxiliary sensitivity equations, or the reformulation of original equations using stochastic algebraic/differential equations. On the other hand, the sampling-based methods involve running the original deterministic simulation for a set of input/parameter combinations (sample points) and estimating the sensitivity/uncertainty using the output response at those points. Sampling-based methods do not require access to governing equations. These methods involve running the simulations at a set of sample points, and establishing a relationship between the inputs and the outputs at the sample points. In this study, Monte Carlo method was used to quantify the effect of the uncertainty in material properties on the sensor response.

10.3.2 Monte Carlo Simulations

Monte Carlo simulations are widely used for uncertainty analysis. Its implementation consists of random sampling from the distribution of inputs and successive deterministic simulation runs until a statistically significant distribution of outputs is obtained. Since it may require a large number of samples, its applicability is sometimes limited to small scale simulations due to time and computational resources required. The computational efficiency is enhanced by the use of a sampling technique that samples from the input distribution in an efficient manner, so that the number of necessary simulations is reduced [62]. Latin hypercube sampling (LHS) is one such widely used sampling technique. In this technique, the range of probable values for each uncertain input parameter is divided into segments of equal probability. The advantage of this approach is that the random samples are generated from all the ranges of possible CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 135

values, thus giving insight into the extremes of the probability distributions of the outputs. In this study, Monte Carlo method with Latin hypercube sampling was employed. For each input sample, a deterministic spectral element analysis was carried out using PESEA, giving an output sample for the sensor response. Finally, a response sampling is obtained, from which the mean and the standard deviation of the response can be derived. The estimator of the response,µ ˆ, is deﬁned by:

n 1 X µˆ = y(i) (10.4) n i=1

n 1 X 2 σˆ2 = y(i) − µˆ (10.5) n i=1 where,µ ˆ andσ ˆ denote the mean and variance of the response and n is the number of samples.

10.3.2.1 Uncertainty in Material Properties

First, parametric studies were carried out to understand the effect of each material property on the sensor response. The material properties were varied individually by 1% and sensor signal was simulated. Effect of each material property on the signal amplitude and the group velocity was studied. Figure 10.3 and Figure 10.4 show the major material properties and their effect on the signal amplitude and the group velocity, respectively. Based on the insights obtained from the parametric studies, only the material properties that significantly affect the sensor response were chosen as random input variables in the Monte Carlo simulations. It was assumed that all random input variables had normal distribution with corresponding mean values and standard deviation as given in Table 10.1[63]. Standard deviation in the mechanical properties of the adhesive and the piezoelectric material (PZT in this case) were assumed to be 2%, while that in the electromechanical properties of PZT was assumed to be 5%. CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 136

Since the material properties of aluminum are well-characterized, standard deviation for aluminum 2024-T3 was assumed to be 1%.

2.5 2 1.5 1 0.5

% % change amplitude in -0.5

-1 -1.5 adh al al al pzt pzt pzt pzt pzt pzt G ρ E ν ρ E11 E33 d31 d33 ε33

Figure 10.3: Parametric study (% change in the sensor signal amplitude

0.6

0.4

0.2

-0.2

-0.4 % change in group velocity % change group in -0.6

al al al pzt pzt ρ E ν d33 ε33

Figure 10.4: Parametric study (% change in the group velocity CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 137

Table 10.1: Uncertainty in material properties of PZT, aluminum, and adhesive Material Input property Mean value Standard deviation (%) PZT ρ (kg/m3) 7750 2

E11 (GPa) 60.97 2

E22 (GPa) 60.97 2

E33 (GPa) 53.19 2

d31 (C/N) -171e-12 5

d32 (C/N) -171e-12 5

d33 (C/N) 374e-12 5

ε33/ε0 1700 5 Aluminum ρ (kg/m3) 2700 1

E11 (GPa) 69.00 1 ν 0.33 1

Hysol R EA 9696 G (GPa) 1 2

10.3.2.2 Uncertainty in Material Properties

A 2 mm thick aluminum 2024-T3 plate was considered for the Monte Carlo simulations. PESEA was used to carry out the simulations for 100 different sets of material properties sampled from their probability distribution. Solid spectral elements of order 4×4×4 with mesh size of 5 mm were used. 6.35 mm diameter and 250 µm thick piezoelectric sensor and actuator made of PZT-5A were modeled on the plate. The distance between the actuator and the sensor was 6 inches. A 50 µm thick adhesive layer was modeled between the sensor/actuator and the aluminum plate. The actuator was excited with a five-cycle tone burst signal with central frequency of 250 kHz. The actuation signal is shown in Figure 10.5. Voltage output was recorded at the sensor. The variation in the first wave packet of the sensor signal due to the variations in the material properties is shown in Figure 10.6. The magenta curve is the sensor signal obtained using mean material properties as given in Table 10.1. Amplitude and the group velocity are two very important parameters of the sensor signal. The ratio of the amplitude of the scatter signal and the noise level in the data CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 138

0 Input (V)Input

-25

-50 0 10 20 30 40 50 Time (µs)

Figure 10.5: Five-cycle tone burst input signal with central frequency of 250 kHz

0.05

0.025

0 Output (V)Output -0.025

-0.05 30 35 40 45 50 Time (µs)

Figure 10.6: Variation in the ﬁrst wave packet of the sensor signal due to the variation in the material properties CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 139

acquisition system (ScNR) is used in optimal sensor network design, while the group velocity is used in the diagnostic algorithms for damage localization. Hence, the uncertainty in the sensor signal was quantified in terms of the probability distribution function (PDF) of the peak-to-peak amplitude and the group velocity of the first wave packet in the sensor signal. Figure 10.7 shows the histogram and the PDF for the sensor amplitude. Figure 10.8 shows the histogram and the PDF for the group velocity. Both amplitude and group velocity were normalized with respect to their mean values. From the parametric study, it can be observed that the sensor amplitude is more sensitive to the electromechanical properties of the sensors, whereas, the group velocity is more sensitive to the mechanical properties of aluminum. The variation in the electromechanical properties of the piezoelectric material is more than that in the mechanical properties of aluminum. Hence, as it can be seen in Figure 10.7 and Figure 10.8, the standard deviation for the sensor amplitude is very significant, while that for the group velocity is very small.

3.5 µˆ =1 σˆ = 0.1168 3 2.5

PDF 1.5

0.5

0 0.7 0.8 0.9 1 1.1 1.2 1.3 Amplitude with respect to mean value

Figure 10.7: PDF for the sensor signal amplitude with respect to the mean value CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 140

80 µˆ =1 σˆ = 0.0062

40 PDF

0 0.98 0.99 1 1.01 1.02 Group velocity with respect to mean value

Figure 10.8: PDF for the group velocity with respect to the mean value

Once the standard deviationσ ˆ is known, 2ˆσ upper and lower bounds on ScNRaX can be estimated as following:

Sc × (1 ± 2ˆσ) ScNR = 20log (10.6) aX 10 N

Sc ScNR = 20log + 20log (1 ± 2ˆσ) (10.7) aX 10 N 10

ScNRax withσ ˆ upper and lower bounds in shown in Figure 10.9(a), and with 2ˆσ upper and lower bounds is shown in Figure 10.9(b). Only the lower bound on ScNRax needs to be considered to obtain a conservative estimate of DDD corresponding to

ScNRax = 10 dB. From Figure 10.9 DDD can be estimated as 229 mm (≈ 9”) using σˆ lower bound, and 212 mm (≈ 8.4”) using 2ˆσ lower bound. It should be noted that, when damage diagnostic algorithms are used, the uncertainty in group velocity may introduce some error in damage localization. However, the variation in the group velocity with respect to its mean value was found to be very small (ˆσ = 0.0062), hence its eﬀect has been ignored in ModSENOPT. CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 141

25 Mean value ()1σˆ 20 Lower bound Upper bound ()1σˆ

(dB) aX

10 ScNR

0 100 150 200 250 300 350 400 450 Distance (mm) (a) One standard deviation upper and lower bounds

25 Mean value 2σˆ 20 Lower bound () Upper bound ()2σˆ

(dB) aX

10 ScNR

0 100 150 200 250 300 350 400 450 Distance (mm) (b) Two standard deviation upper and lower bounds

Figure 10.9: Scatter to noise ratio (ScNRaX ) for diﬀerent wave propagation distances taking into account the uncertainty in material properties CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 142

10.4 Optimal Sensor Network for a Stiﬀened Panel

To demonstrate the performance of the proposed methodology of piezoelectric sensor network optimization, ModSENOPT, a stiffened aluminum panel with a cutout, as shown in Figure 10.10, was considered. Dimensions of the panel were 375 mm × 375 mm × 2 mm. The dimensions of the cutout were 50 mm × 50 mm, while the dimensions of the stiffeners 375 mm × 50 mm × 2 mm. Top view of the structure is shown in Figure 10.11. PESEA was used to model the ultrasonic wave propagation in the structure with surface-mounted piezoelectric sensors. Several piezoelectric sensor elements were modeled on the structure and simulations were carried out for the structure in pristine state and structure with a crack at different locations. Simulated sensor signals from the structure with and without simulated damage were analyzed to estimate the DDD profile in different regions on the structure.

Figure 10.10: Stiﬀened aluminum panel with a cutout CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 143

125 50 100 100

175

50 375 50

100

375

Figure 10.11: Top view of the stiﬀened aluminum panel. All dimensions are in mm

In order to achieve a practical sensor network, along with the DDD profile, a few more parameters such as the minimum distance from the edges, the minimum and the maximum distance between any actuator-sensor pair were given as input to Mod- SENOPT. The minimum distance between the sensors and the edges of the structure was selected as 3 inches so that direct and boundary reflected sensor signal were well separated. In order to remove the effect of crosstalk in PC network configuration, the minimum distance between actuator and sensor was constrained to 4 inches and to improve the damage detectability maximum distance between any actuator-sensor pair was constrained to 90% of the DDD in that direction. The above parameters allowed the sensor network optimization tool to minimize the computational time in designing a more practical sensor network to satisfy the required performance levels set for the structure [60]. The estimated DDD profile and the above constraints on sensor locations were given as inputs to the GA-based optimization tool, ModSENOPT, to optimize the sensor network to achieve 100% detection and localization capability. GA was implemented with the following GA parameters. Number of generations was set as 250, crossover was set as 1.0, mutation chance and creep chance were set as 0.25, and creep CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 144

amount was set as a random amount within ±5% from the mean of candidate solution. To preserve top candidate solution from the GA-based operations, candidate solution that corresponds to the highest ﬁtness in a generation was directly placed into the next generation. The optimization process started as a two sensor optimization problem and was run for 250 generations. At the end of 250 generations, the maximum PODnet corresponding to the optimized sensor network was less than the required PODnet, hence, one more sensor was added to the parameter set and the optimization process was started all over again. This optimization process was repeated until the maximum PODnet was equal to 100%. The results of the sensor network optimization are given in the following subsection. Similarly, ModSENOPT was applied to optimize sensor network for maximizing the performance of the SHM system for damage localization. For accurate damage localization, at least three diagnostic paths need to detect damage. For the given structure, DDD corresponding to the 2ˆσ lower bound (212 mm) was used in ModSENOPT. The optimized piezoelectric sensor network for the given structure is shown in Figure 10.12. In this more conservative sensor network, eight piezoelectric sensors are needed for accurate damage detection, as shown in Figure 10.12(a); while 13 sensors are needed for accurate damage localization, as shown in Figure 10.12(b).

10.5 Conclusions

In this chapter, a probability of detection (POD) based sensor network optimization tool, ModSENOPT, was presented. ModSENOPT uses a physics-based wave propagation model, PESEA, and genetic algorithm evolution process. The steps involved in this proposed methodology are as follows:

1. For a given structure, damage detectable distance (DDD) is computed by running PESEA simulations.

2. Monte Carlo simulations are carried out using PESEA to understand the eﬀect of the uncertainty in the material properties on the sensor signals and DDD. CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 145

(a) Damage detection (TCL I)

(b) Damage localization (TCL II)

Figure 10.12: Optimized sensor network for damage detection and localization CHAPTER 10. MODEL-ASSISTED SENSOR NETWORK DESIGN 146

3. Genetic algorithm is used to maximize the probability of detection (POD) of a damage of size 0a0 by varying the number of sensors and their placements.

4. Finally, ModSENOPT gives an optimal sensor layout with minimum number of sensors that maximized the POD of a damage of given size for damage detection as well as localization.

Optimal sensor layout was obtained for a stiﬀened panel with a cutout. 100% probability of detection was achieved with eight sensors and that of localization was achieved with thirteen sensors. Chapter 11

Conclusions

11.1 Concluding Remarks

A simulation tool, Piezo-Enabled Spectral Element Analysis (PESEA), based on spectral element method (SEM) was developed to simulate sensor response and piezo- induced wave propagation in prestressed structures. PESEA solves coupled electromechanical governing equations for a given arbitrary voltage input to a piezoelectric actuator and outputs the voltage response of the piezoelectric sensors. In case of modeling piezo-induced waves in layered media such as laminated composites, one element per layer is computationally expensive. Smeared material properties may give inaccurate results; hence a layered solid spectral element (LSSE) was introduced. LSSE can model several layers per element. It is worth noting that PESEA can interface with commercial finite element (FE) mesh generators such as Abaqus/CAE R to model structures with complex geometries. Experiments were carried out for metallic and composite plates to validate PESEA. Simulations were carried out for different layups to verify the accuracy of LSSE. The results showed that LSSE can accurately and efficiently model ultrasonic wave propagation in layered media. Cracks in metals and delamination & debond in laminated composites were modeled by separating nodes to create a volume split. PESEA can be used to directly study the scattered waves and the sensitivity of sensor signals to damage in plate-like structures. It can be also used to design the electrical excitation waveform to the piezoelectric actuator

147 CHAPTER 11. CONCLUSIONS 148

in order to maximize the sensitivity. In practical implementation of SHM systems based on built-in piezoelectric sensor network, it is important to understand the effect of operating conditions (e.g. temperature, loading conditions) on sensor signals. The influence of loading on the velocity of ultrasonic waves was investigated. PESEA formulation takes into account the acoustoelastic effect. Simulations and experiments were performed for axially stressed aluminum plates with surface-mounted piezoelectric structures. It was found that PESEA can accurately capture the change in wave velocity, however accurate characterization of electromechanical properties of sensors under different loading conditions is required to capture the change in sensor signal amplitude. PESEA simulations and experiments were performed to study the effects of adhesive layer on sensor signal at elevated temperature. Effect of change in temperature was modeled by varying the material properties with temperature. Parametric studies were also performed to understand the effect of individual material properties on sensor signal. These studies revealed that the change in adhesive stiffness due to temperature is the most influential parameter for the change in sensor signals. The physical insights gained from this study were used by Roy et al. (2011) [38] to develop a temperature compensation model in order to improve the accuracy of damage diagnosis. A model-assisted integrated diagnostics methodology was presented. Typically, in SHM systems, the accuracy of damage localization depends on a priori knowledge of the wave velocities. Estimation of the ultrasonic wave velocities for complex structures is very challenging. Hence, PESEA was used to generate an accurate velocity profile of ultrasonic waves, which was used for offline training of the damage diagnostic imaging algorithm. Simulations were carried out to validate the proposed approach. Similarly, a model-assisted sensor network optimization tool (ModSENOPT) was presented. PESEA simulations were used to study the effect of damage and uncertainty in material properties on sensor output. This information was used in an optimization code to optimize the sensor network by maximizing the probability of damage detection. CHAPTER 11. CONCLUSIONS 149

11.2 Contributions

Key contributions of this dissertation are:

1. SEM-based simulation tool (PESEA) was developed to model ultrasonic waves induced by piezoelectric actuator. Layered solid spectral element (LSSE) was introduced to accurately and eﬃciently model piezo-induced ultrasonic waves in layered media.

2. Acoustoelastic eﬀect was modeled using PESEA. Simulations and experiments were carried out to investigate the change in ultrasonic wave velocities due to applied load.

3. PESEA was used to study the eﬀect of temperature on the amplitude of sensor signals. Parametric studies were performed to understand the eﬀect of individual material properties on sensor signal at elevated temperature.

4. A model-assisted integrated diagnostics was proposed. In this methodology, PESEA simulations were used to increase the accuracy of damage localization.

5. PESEA simulations were used in the model-assisted sensor network optimization (ModSENOPT) tool to optimize the sensor placement in order to maximize the probability of damage detection. Appendix A

Piezoelectric Constitutive Equations

Out of four diﬀerent forms, stress-charge form of piezoelectric constitutive equations has been used in the formulation. Linear constitutive equations for piezoelectric materials in stress-charge form are given by Equations (A.2).

σ = C − eT E (A.1a) D = e + κ E (A.1b)

In these equations, κ is the dielectric constant (electric permittivity) matrix and e is the piezoelectric coeﬃcient matrix in stress-charge form. Material properties for piezoelectric materials are usually given for strain-charge form. εσ is the dielectric constant (electric permittivity) matrix at constant stress, and d is the piezoelectric coeﬃcient matrix in strain-charge form. Therefore, the following transformations are required.

e = d C (A.2a)

T κ = εσ − d C d (A.2b)

150 Appendix B

High Order Shape Functions for Spectral Elements

In SEM, high order Lagrange polynomials are used as shape functions (interpolation functions). These polynomials are deﬁned at Gauss-Lobatto-Legendre (GLL) points. One-dimensional shape function of order N is shown below:

N Q (ξ − ξj) j=0,j6=i h (ξ) = (B.1) i N Q (ξi − ξj) j=0,j6=i where, ξi (i = 0, 1, ..., N) is a GLL point. GLL points for diﬀerent orders of shape functions are given in Table B.1.

151 APPENDIX B. HIGH ORDER SHAPE FUNCTIONS FOR SPECTRAL ELEMENTS152

Table B.1: GLL nodal and integration points for interval [-1,1]

Number of points (n) Points (ξi)) 2 −1.000000 +1.000000 3 −1.000000 0.000000 +1.000000 4 −1.000000 −0.447214 +0.447214 +1.000000 5 −1.000000 −0.654654 0.000000 +0.654654 +1.000000 6 −1.000000 −0.765055 −0.285232 +0.285232 +0.765055 +1.000000 Appendix C

Material Properties

C.1 Mechanical Properties

The mechanical properties used in the simulations are given in Table C.1.

Table C.1: Mechanical properties Property Unit Aluminum CFRP Adhesive Piezo

Al 2024-T3 T800S/3900-2 Hysol R EA 9696 PZT-5A

E11 GPa 69.00 156.00 2.60 60.97

E22 GPa 69.00 9.09 2.60 60.97

E33 GPa 69.00 9.09 2.60 53.19

G23 GPa 25.94 3.24 1.00 21.05

G31 GPa 25.94 6.96 1.00 21.05

G12 GPa 25.94 6.96 1.00 22.57

ν23 0.33 0.400 0.30 0.4402

ν13 0.33 0.228 0.30 0.4402

ν12 0.33 0.228 0.30 0.3500 ρ kg m−3 2700 1540 1100 7750

153 APPENDIX C. MATERIAL PROPERTIES 154

C.2 Electrical Properties

Electromechanical and electrical properties of PZT-5A used in this dissertation are given below in matrix form. ε0 is the permittivity of vacuum which is approximately 8.8542e-12 Fm−1.   0 0 0 0 584 0   −12 −1 d =  0 0 0 584 0 0 × 10 CN (C.1)   −171 −171 374 0 0 0

  1730 0 0   εσ =  0 1730 0  × ε0 (C.2)   0 0 1700

C.3 Third Order Elastic Constants

In order to model acoustoelastic eﬀect on the piezo-induced wave propagation, the (3) third order elastic constants (Cijklmn) are required. In the simulations, the third order elastic constants of aluminum were used, while those for the adhesive and (3) piezoelectric material were ignored. Cijklmn for isotropic materials can be represented in terms of the Murnaghan constants l, m, and n [28,29]. The Murnaghan constants for aluminum 2024-T3 used in the simulations are given below:

Table C.2: Third order elastic constants Property Unit Aluminum Al 2024-T3 l GPa -281.50 m GPa -375.00 n GPa -351.00 Appendix D

Error Quantiﬁcation

Quantifying errors in simulated sensor signal relative to experimental signal by comparing raw signals in time domain may result in signiﬁcant error with even small mismatch in the time of arrival. To overcome this problem, envelope of the signals is used to compute the error in simulated signals. Hilbert transform is used to compute the envelope of the sensor signals as given by Equations (D.1) and (D.2), as shown in Figure D.1. After the envelopes of simulated and experimental signals are computed, the error env is calculated by using Equation (D.3), where Ssim (t) is the envelope of the simulated env signal and Sexp (t) is the envelope of the experimental signal. ti is the initial time of the wave packet and tf is the ﬁnal time of the wave packet.

env ssim(t) = |H (ssim (t))| (D.1)

env sexp(t) = |H (sexp (t))| (D.2)

R tf env env 2 ssim (t) − sexp (t) dt error = ti × 100% (D.3) R tf senv (t)2 dt ti exp

155 APPENDIX D. ERROR QUANTIFICATION 156

0.1 Sensor signal Signal envelope

0.05

0 Output (V)Output -0.05

-0.1 20 40 60 80 Time (µs)

Figure D.1: Sensor signal and its Hilbert transform Bibliography

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