Condition Assessment of GFRP-Retrofitted Concrete

Condition Assessment of GFRP-Retrofitted Concrete Cylinders Using Electromagnetic Waves by Tzu-Yang Yu Submitted to the Department of Civil and Environmental Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the field of Structures and Materials at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2008 c Massachusetts Institute of Technology 2008. All rights reserved.

Author...... Department of Civil and Environmental Engineering May 8, 2008

Certiﬁed by...... Oral Buyukozturk Professor of Civil and Environmental Engineering Thesis Supervisor

Accepted by...... Daniele Veneziano Chairman, Departmental Committee for Graduate Students 2 Condition Assessment of GFRP-Retroﬁtted Concrete Cylinders Using Electromagnetic Waves by Tzu-Yang Yu

Submitted to the Department of Civil and Environmental Engineering on May 8, 2008, in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the ﬁeld of Structures and Materials

Abstract The objective of this study is to develop an integrated nondestructive testing (NDT) capability, termed FAR NDT (Far-field Airborne Radar NDT), for the detection of defects, damages, and rebars in the near-surface region of glass fiber reinforced polymer (GFRP)-retrofitted concrete cylinders through the use of far-field radar measurements (electromagnetic or EM waves). In this development, two far-field monostatic ISAR (inverse synthetic aperture radar) measurement schemes are identified for collecting radar measurements, and the backprojection algorithm is applied for processing radar measurements into spatial images for visualization and condition assessment. Reconstructed images are further analyzed by mathematical morphology to extract a numerical index representing the feature of the image as a basis for quantitative evaluation. The components of the development include dielectric modeling of materials, laboratory radar measurements, numerical simulation, and image reconstruction. It is found that using the developed technique the presence of near-surface defects can be detected by the oblique incidence measurements. Radar signals in the frequency range of 8 GHz to 18 GHz are found effective for damage detection in the near-surface region of the specimens. Numerical simulation using the finite-difference time-domain (FDTD) method is conducted to understand the propagation and scattering of EM waves from the defects and inclusions in two-dimensional and three-dimensional GFRP-concrete models. The FDTD simulation is capable of predicting the far-field response of GFRP-concrete cylinders and beneficial to better understanding the pattern of field measurements in the application of the FAR NDT technique. Dielectric properties of materials are investigated for their use in numerical simulation and for improving the precision of reconstructed images. Re- constructed images of GFRP-concrete cylinders with and without artificial features (rebar and defect) clearly indicate the presence of these features. Normal incidence scheme is found to be effective for rebar detection, and the oblique incidence scheme can discover near-surface defects such as GFRP debonding and delamination. The proposed FAR NDT technique is found to be capable of detecting near-surface defects in GFRP-concrete cylinders and potentially applicable for the field condition as-

3 sessment of GFRP-retroﬁtted reinforced concrete and other reinforced concrete civil infrastructure systems.

Thesis Supervisor: Oral Buyukozturk Title: Professor of Civil and Environmental Engineering

4 Acknowledgments

It is Professor Oral Buyukozturk (Course I) who led me into the research ﬁeld of condition assessment of concrete structures using electromagnetic waves through working on a research project several years ago, on which my doctoral dissertation is essentially based. It is also through working on this project, my research interests on various topics were subsequently developed/discovered. His guidance, encouragements, and supports are indispensable to me for making this dissertation possible. I am deeply indebted to the time he spent with me at nights and on weekends, and to his tolerance in the process of forging my research attitude and enhancing my research capabilities. It would not have been possible for me to accomplish this work without his training on many aspects. For that, I truly appreciate this precious opportunity he gave me at MIT.

Professor Jerome J. Connor (Course I) was kind enough for supervising my Mas- ter’s thesis when I came to MIT in 2001 and for joining my thesis committee in 2005. It is very diﬃcult not to be encouraged and inspired by him in every discussion we have made through his infectious passion on research and teaching.

I am also indebted to the late Professor Jin Au Kong (Course VI) for his leading me into the intriguing world of electromagnetism and for his valuable suggestions made in my committee meetings. His extraordinary sense of humor proportionally reﬂects the magnitude of his knowledge. His research philosophy inspires me and has made me a good friend of ”SAM” ever since. His sudden decease on March 13, 2008, is an unmeasured loss to me and everyone who knows him, while his lecturing and words still vividly survive in our memories.

It is my pleasure to have Dr. Tomasz M. Grezgorczyk (Course VI) serving on my thesis committee. His insightful suggestions to the electromagnetic problems I have encountered in conducting this research are most helpful and valuable. Dr. Grezgorczyk has also been very supportive to the completion of this research in many aspects. I am deeply grateful for his willing to guide me in exploring the world of electromagnetism.

5 I should like to take this opportunity to express my gratitude to Professor Michael C. Forde (University of Edinburgh, Scotland) serving as a member on the thesis committee. His constructive suggestions and questions I have received during his stay at MIT in 2004 and during my thesis defense in 2008 are valuable to the further improvement of this thesis. I am grateful for his supporting this research on many aspects and for his sharing his perspectives and thoughts on many critical problems in civil engineering. Special thanks go to Dr. Antonis Giannopoulos for the use of GPRMax2D/3D and his suggestions in the numerical simulation. Many productive and interesting discussions with Professors E. Kausel and D. Veneziano (both in Course I) are greatly appreciated. I would like to extend my thanks to a number of people for their help of various kinds; to Dennis Blejer and Alex Eapen (MIT Lincoln Laboratory) for their help on laboratory radar measurements and data interpretation; to Patricia Dixon and Cynthia Stewart when I was in need of help in 2004; to Donna Hudson for her help on proposal budgets; to Patricia (Patty) Glidden, Kris Kipp, Jeanette Marchocki, and Donna Beaudry for their everyday relentlessly greetings on the aisles in Building 1. This journey would have been much more colder without their warm smiles. I have been lucky enough to make many friends in Course I; O. Gunes, C. Au, E. Karaca, R. Sudarshan, J. Pei, A.E. Sew, M.A. Nikolinakou, J.A. Ortega, K. Ishimaru, S. Cheekiralla, P. Dohnalek, C. Tuakta, S. Lin, I. Tsai, J. Park, Y. Moriyama, I.(Aki) Choo, D. Lau, as well as in other Departments including K. Lee (Course VIII), J. Chen (Course VI), and M. Nikku (Course VIII). The time with my classmates including Marc, Bora, Carmen, Jason, Tashan, Vimal, Luca, Chinghuei, and Sakda was also memorisable. Last but not least, I will always have special gratitude and love for my grandma, Shun Jen, my father, Jr-Shen Yu, my mother, Kuei-Yin Shu, my brother, Shun-Hwa Yu. Their endless, unconditional supports warm my heart as always. Finally, I want to dedicate this thesis to my wife, Kaiwen Chen, who has been my Muse on many aspects ever since she walked into my life.

6 Contents

1 Introduction and Research Motivation 23 1.1 Research Objective ...... 29 1.2 Research Approach ...... 29 1.3 Organization of the Dissertation ...... 31

2 Literature Review 35 2.1 Nondestructive Testing (NDT) Techniques ...... 36 2.1.1 Optical Methods ...... 42 2.1.2 Acoustic Methods ...... 44 2.1.3 Thermal Methods ...... 48 2.1.4 Radiographic Methods ...... 51 2.1.5 Magnetic and Electrical Methods ...... 53 2.1.6 Microwave and Radar Methods ...... 56 2.2 Summary ...... 61

3 Numerical Simulation 63 3.1 Maxwell’s Curl Equations and Linearly Polarized EM Waves . . . . . 65 3.2 Finite Diﬀerence Time Domain Solution and Yee’s Algorithm . . . . . 68 3.3 Absorbing Boundary Condition – Perfectly Matched Layer ...... 73 3.4 Stability Criteria in Discretization ...... 74 3.4.1 Discretization in Space ...... 75 3.4.2 Discretization in Time ...... 76 3.5 Two-Dimensional and Three-Dimensional Simulations ...... 77

7 3.5.1 Validation of the Code ...... 77 3.5.2 Actual Far-Field Simulation ...... 82 3.6 Simulation Results ...... 85 3.6.1 Damage Detection in Normal Incidence ...... 86 3.6.2 Damage Detection in Oblique Incidence ...... 86 3.6.3 Eﬀect of Defect Width in Normal Incidence ...... 88 3.6.4 Eﬀect of Defect Depth in Normal Incidence ...... 89 3.6.5 Rebar Detection in Normal Incidence ...... 89 3.6.6 2D and 3D Responses ...... 107 3.7 Summary ...... 107

4 Laboratory Radar Measurements 111 4.1 Experimental Program ...... 112 4.2 Manufacturing of the Specimens ...... 116 4.3 Experimental Conﬁguration and Parameters ...... 118 4.3.1 Monostatic ISAR Normal Incidence Scheme ...... 120 4.3.2 Monostatic ISAR Oblique Incidence Scheme ...... 121 4.4 Calibration of Laboratory Radar Measurements ...... 121 4.4.1 PEC Specimen ...... 122 4.4.2 Lossy Dielectric Specimen and Its Optical Model ...... 124 4.5 Frequency-Angle Measurements ...... 127 4.5.1 Monostatic ISAR Normal Incidence Scheme ...... 133 4.5.2 Monostatic ISAR Oblique Incidence Scheme ...... 152 4.6 Summary ...... 155

5 Image Reconstruction 159 5.1 Single Scattering and Synthetic Aperture Radar ...... 160 5.2 Inverse Synthetic Aperture Radar ...... 172 5.3 Backprojection Algorithms ...... 175 5.3.1 Range Compression ...... 175 5.3.2 Backprojection Processing ...... 178

8 5.3.3 Support Band Analysis – Method of Stationary Phase . . . . . 179 5.3.4 Advantages of Backprojection Algorithms ...... 187 5.4 Implementation and Coding Procedure ...... 187 5.5 Eﬀects of Aperture Size and Bandwidth ...... 193 5.5.1 Aperture Size ...... 193 5.5.2 Bandwidth ...... 194 5.6 Summary ...... 195

6 Dielectric Modeling of GFRP-concrete Systems 201 6.1 Background ...... 202 6.1.1 Deﬁnition and Physics of Dielectric Properties ...... 202 6.1.2 Dielectric Spectroscopy and Dielectric Dispersion ...... 211 6.1.3 Energy Storage and Dissipation Mechanisms ...... 212 6.1.4 Dielectric Properties in Microwave and Radar NDT ...... 216 6.2 Approaches for the Determination of Dielectric Properties ...... 216 6.3 Integrated Methodology for the Determination of Dielectric Properties 219 6.3.1 Time Diﬀerence of Arrival (TDOA) ...... 220 6.3.2 Root-searching Optimization Scheme ...... 222 6.3.3 Validation of the Methodology ...... 227 6.4 Modeling Approach for the Dielectric Properties of Materials . . . . . 233 6.4.1 Internal Field Approach versus External Field Approach . . . 233 6.4.2 Geometrical Analysis ...... 242 6.5 Dielectric Properties of Water ...... 247 6.5.1 Free Water ...... 251 6.5.2 Bound Water ...... 253 6.6 Dielectric Properties of GFRP ...... 263 6.6.1 Epoxy Resin ...... 265 6.6.2 E-glass Fabric ...... 266 6.6.3 GFRP Layer/Sheet ...... 267 6.7 Dielectric Properties of Concrete ...... 272

9 6.7.1 Determination of Volumetric Fractions ...... 275 6.7.2 Dielectric Modeling of Oven-Dried Hydrated Cement Paste . . 277 6.7.3 Challenges in the dielectric modeling of concrete ...... 280 6.8 Summary ...... 281

7 Condition Assessment of GFRP-concrete Systems – FAR NDT 285 7.1 Components of FAR NDT ...... 286 7.2 Physical Inspection – Far-ﬁeld ISAR Measurements ...... 289 7.3 Numerical Processing – Image Reconstruction ...... 294 7.3.1 Physical Meaning of the Scattering Signals in the Images . . . 294 7.3.2 Progressive Image Focusing ...... 295 7.4 Image Resolutions and Damage Detectability ...... 299 7.5 Pattern Recognition – Damage Detection ...... 300 7.5.1 Local Index - Maximum Amplitude ...... 301 7.5.2 Global Index - Mathematical Morphology ...... 303 7.6 Summary ...... 307

8 Conclusions 319

A Phase Velocity of Love Waves in A Layer Underlain by A Half Space Medium 325

B Analytical Approach to Several Plane Wave Incidence Problems 331 B.1 Reflection Coefficient and Reflectivity ...... 331 B.2 A Two-dimensional Three-layer Model ...... 334 B.3 A Three-dimensional Infinite Dielectric Cylinder Model ...... 336 B.4 Summary ...... 341

10 List of Figures

1-1 Several FRP strengthening/repair scenarios of concrete structures . . 25 1-2 Intact multi-layer cementitious composite systems ...... 26 1-3 Damaged multi-layer cementitious composite systems ...... 26 1-4 Detected air voids at various locations in a bridge box-girder wall and their repair ...... 27 1-5 Several failure modes in GFRP-concrete systems ...... 28 1-6 Modeling of construction defects and structural damages using artiﬁcial anomaly ...... 31 1-7 Organization of the dissertation ...... 32

2-1 Procedure of NDT Techniques ...... 37

3-1 Configuration grid in a two-dimensional domain ...... 70 3-2 Derivation of global and local ABCs ...... 74 3-3 Quantization error Er (k∆x) vs. k∆x ...... 76 3-4 Geometry of a 2D model for validation ...... 79 3-5 Gaussian current source and the reflection from PEC ...... 80 3-6 Sinusoidal current source and the reflection from PEC ...... 81 3-7 Gaussian current source and the reflection from PEC ...... 82 3-8 Gaussian current source and the reflection from PEC – Close-Up . . . 83 3-9 Theoretical curve and the FDTD solution of the amplitude reflection coefficient of a 2D dielectric plate ...... 84 3-10 Relative difference/error between the theoretical curve and the FDTD solution ...... 85

11 3-11 Sinusoidal current source and the reflection from a 2D dielectric plate 86 3-12 Numerical domain for simulating actual far-field radar measurements 87 3-13 Incident field, total field, and net reflection field of a 9 GHz signal . . 88

3-14 Scattered (net reﬂection) ﬁeld of Ez – Intact, lossless concrete cylinder 91

3-15 Total ﬁeld of Ez – Intact, lossless concrete cylinder ...... 92

3-16 Total ﬁeld of Ez – Intact, lossy concrete cylinder ...... 93

◦ 3-17 Scattered ﬁeld of Ez, φ = 0 – Damaged, lossless concrete cylinder . . 94

◦ 3-18 Scattered ﬁeld of Ez, φ = 0 – Damaged, lossy concrete cylinder . . . 95

◦ 3-19 Scattered ﬁeld of Ez, φ = 30 – Intact, lossless concrete cylinder . . . 96

◦ 3-20 Scattered field of Ez, φ = 30 – Damaged, lossless concrete cylinder . 97 3-21 Thirteen 2D models for studying the effect of defect width in normal incidence ...... 98 3-22 Variation of reflected electric field with respect to different defect widths in normal incidence ...... 98 3-23 Sixteen 2D models for studying the effect of defect depth in normal incidence ...... 99 3-24 Variation of reflected electric field with respect to different defect depths in normal incidence ...... 99

3-25 Total ﬁeld of Hx – Plain concrete cylinder with a center rebar . . . . 100

3-26 Total ﬁeld of Hy – Plain concrete cylinder with a center rebar . . . . 101

3-27 Total ﬁeld of Ez – Plain concrete cylinder with a center rebar . . . . 102

3-28 Scattered ﬁeld of Hx – Plain concrete cylinder with a center rebar . . 103

3-29 Scattered ﬁeld of Hy – Plain concrete cylinder with a center rebar . . 104

3-30 Scattered ﬁeld of Ez – Plain concrete cylinder with a center rebar . . 105

3-31 Scattered ﬁeld of Ez – Plain concrete cylinder with a center rebar . . 106 3-32 A 3D lossless dielectric cylinder model ...... 107 3-33 Comparison between the 2D and 3D responses of a PEC cylinder model108

4-1 Specimens CON and GFRP ...... 114 4-2 Specimens CRE and 4RE ...... 114

12 4-3 Specimens AD1 and AD2 ...... 115 4-4 Specimen AD3 ...... 115 4-5 Compact RCS antenna range facility in the MIT LL [Courtesy of the MIT LL] ...... 119 4-6 Schematic of the compact RCS/antenna range facility in the MIT LL 125 4-7 Far-field monostatic ISAR normal incidence scheme. Note that the angular zero is referred to a selected point on the cylinder...... 126 4-8 Far-field monostatic ISAR oblique incidence scheme. Note that the angular zero is in alignment with the axis of the cylinder...... 127 4-9 PEC specimen – Aluminum tube [Courtesy of the MIT Lincoln Labo- ratory] ...... 128 4-10 Frequency-angle response of the Aluminum tube – Amplitude (dBsm), X-band ...... 128 4-11 Frequency-angle response of the Aluminum tube – Amplitude (dBsm), Ku-band ...... 129 4-12 Mean amplitudes and their standard deviations of the reflection response at different frequencies – Aluminum tube, X-band ...... 129 4-13 Mean amplitudes and their standard deviations of the reflection response at different incident angles – Aluminum tube, X-band . . . . . 130 4-14 Mean amplitudes and their standard deviations of the reflection response at different frequencies – Aluminum tube, Ku-band ...... 130 4-15 Mean amplitudes and their standard deviations of the reflection response at different incident angles – Aluminum tube, Ku-band . . . . 131 4-16 Correlation coefficients of the reflection response at different frequencies – Aluminum tube, X-band ...... 131 4-17 Correlation coefficients of the reflection response at different frequencies – Aluminum tube, Ku-band ...... 132 4-18 Lossy dielectric specimen – Plexiglass rod ...... 132 4-19 Frequency-angle response of the plexiglass specimen – Amplitude (dBsm), Ku-band ...... 133

13 4-20 Simulated and measured responses — Plexiglass rod, HH (upper level) and VV (lower level) polarizations [Courtesy of the MIT LL] . . . . . 134 4-21 Frequency-angle response of specimen CON (plain concrete) – Ampli- tude (dBsm), X-band ...... 136 4-22 Frequency-angle response of specimen CON (plain concrete) – Ampli- tude (dBsm), Ku-band ...... 136 4-23 Frequency-angle response of the specimen GFRP (plain concrete with GFRP) – Amplitude (dBsm), X-band ...... 137 4-24 Comparison between the power responses of specimens CON and GFRP, X-band ...... 139 4-25 Photos of the specimen CON showing surface roughness ...... 140 4-26 Misalignment between the cylinder specimen and the Styrofoam tower in the normal incidence scheme ...... 142 4-27 Frequency-angle response of specimen AD1B – Amplitude (dBsm), X- band ...... 142 4-28 Frequency-angle response of specimen AD1F – Amplitude (dBsm), X- band ...... 143 4-29 Comparison between the power responses of the intact and damaged sides of the specimen AD1, X-band ...... 144 4-30 Photos showing the concave region in the specimen ...... 145 4-31 Frequency-angle response of specimen AD2 – Amplitude (dBsm), X-band147 4-32 Frequency-angle response of specimen AD3 – Amplitude (dBsm), X-band147 4-33 Frequency-angle response of specimen CRE – Amplitude (dBsm), X-band148 4-34 Comparison between the power responses of specimens CON and CRE, X-band ...... 149 4-35 Frequency-angle response of specimen 4RE (rebar 1) – Amplitude (dBsm), X-band ...... 150 4-36 Frequency-angle response of specimen 4RE (rebar 2) – Amplitude (dBsm), X-band ...... 150

14 4-37 Frequency-angle response of specimen 4RE (rebar 3) – Amplitude (dBsm), X-band ...... 151 4-38 Frequency-angle response of specimen 4RE (rebar 4) – Amplitude (dBsm), X-band ...... 151 4-39 Frequency-angle response of the intact side of the specimen AD1 – Amplitude (dBsm), X-band ...... 153 4-40 Frequency-angle response of the damaged side of the specimen AD1 – Amplitude (dBsm), X-band ...... 154 4-41 Frequency-angle response of the damaged side of the specimen AD2 – Amplitude (dBsm), X-band ...... 154

5-1 Conﬁguration of SAR measurements ...... 170 5-2 Monostatic radar and its footprint ...... 171

5-3r ¯s,r ¯0, andr ¯j in the far-field region ...... 171 5-4 Configuration of ISAR measurements ...... 184 5-5 Two cases of the reflection of radar signals ...... 185 5-6 Time shifting error ...... 185 ˜ 5-7 Variables in I(kx, ky) and their relationship ...... 186 5-8 Far-field ISAR measurement of the specimen AD1 – HH polarization,

◦ φi = −15 ...... 188 5-9 Complex form of the far-field ISAR measurement ...... 189 5-10 Desired sidelobe pattern with different sidelobe levels (SLLs) ranging from 30dB to 90dB ...... 190 5-11 Weighted complex form of the far-field ISAR measurement ...... 191 5-12 Shifted 1D DFT of the weighted complex signal ...... 192 5-13 Poynting vector of the radar ...... 192 5-14 Projected image of the transformed far-field ISAR measurement . . . 193 5-15 Full-bandwidth far-field ISAR measurements with an azimuth vector of (−14.8◦, −15◦, −15.2◦) ...... 194

15 5-16 Full-bandwidth far-field ISAR measurements with an azimuth vector of (−14.8◦, −15◦, −15.2◦) ...... 195 5-17 Full-bandwidth far-field ISAR measurements with an azimuth vector of (−14.6◦, −14.8◦, −15◦, −15.2◦, −15.4◦) ...... 196 5-18 Full-bandwidth far-field ISAR measurements with an azimuth vector of (−14.4◦, −14.6◦, −14.8◦, −15◦, −15.2◦, −15.4◦, −15.6◦) ...... 197

5-19 Full-bandwidth half-aperture backprojection image – ∆φi × [−150 : 1 : ◦ 1] (∆φi = 0.2 ) ...... 197

5-20 Full-bandwidth full-aperture backprojection image – ∆φi × [−150 : 1 : ◦ 150] (∆φi = 0.2 )...... 198 5-21 Sub-bandwidth full-aperture backprojection image – [8, 11] (GHz) . . 198 5-22 Sub-bandwidth full-aperture backprojection image – [8, 10] (GHz) . . 199 5-23 Sub-bandwidth full-aperture backprojection image – [8, 9] (GHz) . . . 199

6-1 Dielectric dispersion of several types of polarization (Modified after Knight and Nur (1987) [135]) ...... 213 6-2 A two-dimensional free-space transmission model ...... 220 6-3 Overview of the proposed methodology ...... 221 6-4 Transmitting horn and the network analyzer for the free-space measurement [Courtesy of the MIT Lincoln Laboratory (MIT LL)] . . . . 223 6-5 Transmitting and receiving horns for the free-space measurement [Cour- tesy of the MIT LL] ...... 224 6-6 Error surfaces of the dielectric measurements collected from Teflon, Lexan, Bakelite, and Portland cement concrete ...... 225 6-7 Error surfaces of the dielectric measurements collected from Teflon, Lexan, Bakelite, and Portland cement concrete ...... 226

0 6-8 Convergence of estimates of r at diﬀerent frequency bandwidths using TDOA...... 232

00 6-9 Normalized phase velocity vs. loss factor r ...... 233

6-10 Behavior of the Cole-Cole model (s = 2, ∞ = 1) ...... 240

16 6-11 Behavior of the Davidson-Cole model (s = 2, ∞ = 1) ...... 241 6-12 The Cole-Cole diagram of Cole-Cole’s model ...... 243 6-13 The Cole-Cole diagram of Davidson-Cole’s model ...... 244 6-14 The Cole-Cole diagram and the equivalent circuit of Fröhlich’s model 244 6-15 The Cole-Cole diagram and the equivalent circuit of Cole-Cole’s model 247 6-16 The Cole-Cole diagram and the equivalent circuit of Davidson-Cole’s model ...... 248 6-17 Performance of the Cole-Cole model for free water ...... 253 6-18 Considered sorption model in the pore structure of hcp ...... 254 6-19 Jellium distance in the sorption model ...... 257 6-20 Used chemical potential accounting for the formation of the bound water on the surface of hcp ...... 258 6-21 Measurements and model prediction of non-porous and porous specimens of hydration products ...... 259 6-22 Calculated bonding potential of free water molecules ...... 260 6-23 Calculated relaxation time distribution over the bound water later . . 261 6-24 Performance of the curve-fitted model for the specific surface area of hcp by nitrogen adsorption ...... 264 6-25 Performance of the Cole-Cole model for epoxy ...... 266 6-26 Performance of the Cole-Cole model for E-glass fabric ...... 268 6-27 Performance of the Cole-Cole model for E-glass fabric – Real part and imaginary part ...... 269 6-28 Unidirectional GFRP layer ...... 269 6-29 Performance of six mixing models for GFRP-epoxy ...... 271 6-30 Relationship between the w/c ratio and product of the w/c ratio and dielectric constant of oven-dried cement paste specimens ...... 279 6-31 Curve-fitting results of the oven-dried hcp in the frequency range of 3 GHz to 24 GHz ...... 284

7-1 Overview of the FAR NDT ...... 287

17 7-2 Normal and oblique incidence inspection schemes of the FAR NDT technique ...... 288 7-3 Defect and rebar detection ...... 289 7-4 Specular dominant circumstance ...... 290 7-5 Specular recessive circumstance ...... 290 7-6 Computed far-field distances at various frequencies and two antenna apertures ...... 291 7-7 Inspection procedure of the FAR NDT technique ...... 292 7-8 Bridge inspection – Beam and column ...... 293 7-9 Two types of frequency bandwidth integration ...... 296 7-10 Improvement of image resolutions Progressive image focusing – Fre- quency integration, HH polarization ...... 297 7-11 Prediction error of cross-range resolution formulae ...... 298 7-12 Progressive image focusing – Frequency integration using shifting center frequency, HH polarization, θ = 15◦ ...... 309 7-13 Improvement of image resolutions progressive image focusing – Angular integration, HH polarization ...... 310 7-14 Comparison of images of the intact and damaged surfaces of the specimen AD1 at different incident angles (30◦ ∼ 10◦) ...... 311 7-15 Comparison of images of the intact and damaged surfaces of the specimen AD1 at different incident angles (−10◦ ∼ −30◦) ...... 312 7-16 Description of the used far-field ISAR measurements and specimens for damage detection ...... 313 7-17 Maximum amplitudes of the backprojection images of the specimen AD1 – Full bandwidth (8GHz∼12GHz), HH polarization ...... 313 7-18 Differential maximum amplitudes of the backprojection images of the specimen AD1 – Full bandwidth (8GHz∼12GHz), HH polarization . . 314 7-19 An eight-node element for morphological operations ...... 314 7-20 Backprojection images of the specimen AD1 – HH polarization, θ = −15◦ 315 7-21 Backprojection images of the specimen AD1 – VV polarization, θ = −15◦ 315

18 7-22 Variation of nE with respect to nthv of the intact-side of the specimen AD1 ...... 316

7-23 Variation of nE with respect to nthv of the damaged-side of the specimen AD1 ...... 316 7-24 Backprojection images and their feature-extracted version of the specimen AD1 (HH polarization, full bandwidth, θ = −15◦) ...... 317

7-25 Original nE(θ) curves of the intact and damaged surfaces of the specimen AD1 (HH polarization, full bandwidth) ...... 317

7-26 Filtered nE(θ) curves of the intact and damaged surfaces of the specimen AD1 (HH polarization, full bandwidth, L = 3) ...... 318

A-1 A layer underlain a solid half space ...... 326

B-1 A two-dimensional two-layer model with inﬁnite boundary (TE waves) 332 B-2 TE and TM waves and incident wave vector ...... 334 B-3 A two-dimensional three-layer model (TE waves) ...... 335 B-4 A three-dimensional inﬁnite dielectric cylinder impinged by plane waves337

19 20 List of Tables

2.1 NDT techniques utilizing mechanical waves ...... 38 2.2 Electromagnetic spectrum ...... 39 2.3 NDT Techniques ...... 41 2.4 Types of mechanical wave ...... 45 2.5 Microwave and radar NDT techniques ...... 57 2.6 Microwave and radar NDT applications in civil engineering ...... 59

4.1 Designed specimens ...... 113 4.2 Used materials and their suppliers ...... 117 4.3 Manufacturing standards ...... 117 4.4 Statistical parameters of the frequency-angle amplitude measurement of the Aluminum tube (PEC) ...... 123 4.5 Statistical parameters of the frequency-angle response of the plexiglass rod specimen ...... 125 4.6 Peak RCS of simulated and measured responses of the plexiglass rod specimen ...... 126 4.7 Signal contents of the radar measurements of laboratory specimens . 135 4.8 Maximum and minimum powers (dBsm) of specimens CON and GFRP, X-band ...... 137 4.9 Maximum and minimum RCS (dBsm) of the specimens AD1 . . . . . 141 4.10 Conﬁguration of the specimen 4RE ...... 148

5.1 Comparison of several SAR modes ...... 161

21 6.1 Types of magnetic materials ...... 206 6.2 Types of dielectric materials ...... 208 6.3 Types of materials deﬁned by conductivity ...... 210 6.4 Civil engineering applications of dielectric properties ...... 216 6.5 Thicknesses of the specimens ...... 227

0 6.6 TDOA measurements and dielectric constants r of the specimens . . 229 00 6.7 Loss factors r of the specimens ...... 229 0 6.8 Estimated r by TDOA using different frequency bandwidths (GHz) . 231 6.9 Comparison of internal field approach and external field approach . . 242 6.10 Measured complex permittivity of water at 20◦C ...... 252 6.11 Fitted parameters in Debye’s and Cole-Cole’s models of free water . . 253 6.12 Averaged statistical thickness, t (A˚), of adsorbed water layer on hcp . 259

6.13 Relaxation time, τw (ps), of free water molecules ...... 260 6.14 Speciﬁc surface area of hcp [83] ...... 263 6.15 Comparison of two GFRP-epoxy systems ...... 271 6.16 Performance of six mixing models for a GFRP-epoxy system . . . . . 272 6.17 Volumetric fractions of cementitious composites ...... 277 6.18 Density of cement paste specimens [83] ...... 277 6.19 Parameters in the oven-dried hcp model ...... 280

7.1 Image resolution formulae ...... 297

22 Chapter 1

Introduction and Research Motivation

”All diﬃcult things have their origin in that which is easy, and great things in that which is small.” —– DaoDeJing, Lao Tzu (∼ 600 B.C.)

Deterioration of manmade structures is an inevitable and non-stopping process, no matter how carefully structures are maintained or reserved. Among all manmade structures, civil infrastructures such as buildings, bridges, tunnels, dams, pipelines, roads, airports suﬀer from the severe attacks from the environment during their design lifespan which may range from 30 to 50 years (bridges designed for 120 years). In order to meet the expected lifespan and performance for civil infrastructures, strengthening and repair of concrete structures has become an important issue for public safety and for eﬀective infrastructure management. Engineering technologies are developed and introduced for extending the service life of concrete structures by means of restoring their design capacity for continuous use and/or upgrading them for possible future challenges from the environment, and for meeting the demand for increased service load conditions. Strengthening and repairing of concrete structures can be conducted either internally or externally. Internal strengthening techniques such as injection techniques use

23 adhesive materials to fill in interconnecting cracks and voids in concrete. In this type of techniques, targeted infiltration/penetration depth of injected adhesive may not be easy to achieve, and the effectiveness of construction sometimes remains question- able. External strengthening techniques, on the other hand, are more effective than internal techniques and can achieve a significant level of strengthening, especially for column-type structures. At present time, the use of fiber reinforced polymer (FRP) composites as an externally bonded element to confine the concrete in order to secure the integrity of concrete structures has been proven, both theoretically and practically, to be an effective strengthening/repair approach. FRP composite jacketing systems have emerged as an alternative to traditional construction, strengthening, and repair of reinforced concrete columns and bridge piers. A large number of projects, both public and private, have used this technology and escalating deployment is expected, especially in seismically active regions. Integration of the new FRP composite with the existing concrete substrate results in the formation of a new structural system. Differences in the material properties of the two structural components (FRP and concrete) pose challenging problems of predicting the behavior of the integrated structural system. Extensive research effort has been devoted to this active field as reported in the literature on structural engineering, and composite materials and construction.

Compared with the traditional materials such as steel, general advantages oﬀered by FRP composites include high strength-to-mass ratio, high stiﬀness-to-mass ratio, ease in handling, and resistance to corrosion. Ample research activities and applications of FRP strengthening in civil engineering have been reported in research papers [44] and reports [193, 49] in conjunction with developed design codes [53, 87, 198] and industrial manuals [253, 55, 152].

Typical FRP composites used in civil engineering applications are carbon FRP (CFRP), glass FRP (GFRP) and aramid FRP (AFRP). Among these strengthening materials, GFRP composites have been widely adopted since the 1990s [228, 202, 145, 225], particularly in retroﬁtting of reinforced concrete (R/C or RC) members, such as slabs [145], beams [45], walls [132] and columns [258] [160] to increase/restore

24 their mechanical capabilities (flexural, shear and compressive). GFRP composites can also be applied to masonry [250], metal [56] and wood structures [62]. Figure 1-1 demonstrates several strengthening/repair scenarios on concrete structures. In this dissertation, GFRP strengthened and retrofitted concrete columns are considered and modeled by GFRP-retrofitted concrete cylinder specimens. Figure 1-2 and Figure 1-3 show the schematics of intact and damaged GFRP-concrete column systems.

FRP sheet/layer for axial & flexure for shear

for flexure

(a) Beam strengthening (b) Column strengthening

for flexure

for shear

Figure 1-1: Several FRP strengthening/repair scenarios of concrete structures

Prior to the strengthening/repair construction, it is important to know (1) the location(s) for strengthening and (2) the level of strengthening at a specific location in order to properly and effectively strengthen the structure. Therefore, appropriate and field applicable nondestructive testing (NDT) or evaluation techniques need to be introduced to assess the current condition (level of damage) of the deteriorated/damaged structure.

25 FRP-epoxy

Epoxy

Steel reinforcement

Concrete

(a) FRP-wrapped concrete (b) FRP-wrapped reinforced column concrete column

Figure 1-2: Intact multi-layer cementitious composite systems

FRP-epoxy Epoxy

Steel reinforcement

Concrete Concrete cracking

Delamination (a) FRP-retrofitted concrete (b) FRP-retrofitted reinforced column concrete column

Figure 1-3: Damaged multi-layer cementitious composite systems

After the strengthening/repair construction, the integrated concrete structure with the externally bonded GFRP composite forms a multi-layer composite system. Construction defects and structural/environmental damages may occur within the GFRP-retroﬁtted concrete structures, and especially, in the vicinity of FRP- concrete interfaces. construction defects such as air voids/pockets being trapped between the GFRP wrap and the concrete substrate may be encountered. The presence of these air voids creates a region at which the shear stresses are discontinuous. The stress discontinuity will further encourage the formation and development of FRP debonding at and in the vicinity of the interface of FRP and concrete under associ-

26 ated loading conditions. Therefore, in practice, the defected areas must be identiﬁed and repaired. Figure 1-4 shows the detected air voids and their repair in a bridge rehabilitation project (the Jamestown Bridge, Rhode Island).

(a) Found air void defect with the removal of GFRP sheet

(b) Air void found near the corner

Figure 1-4: Detected air voids at various locations in a bridge box-girder wall and their repair

FRP-concrete interface and concrete conditions cannot be fully revealed until physical removal of the FRP composite layer unless the member has already been subjected to apparent substantive damage. Partial or complete removal of the FRP composite layer for observation of the damage may pose a danger of structural collapse. A FRP-retroﬁtted concrete beam or column could appear safe without showing any sign of substantial damage underneath the FRP composite while containing a severely deteriorated region. Such scenario could happen when the structure has

27 undergone a modest seismic event that has significantly damaged the FRP-concrete system while the system has not reached the failure stage. Failures of damaged FRP- concrete systems are often brittle, involving delamination of the FRP, debonding of concrete layers, and shear collapse, which can occur at load levels lower than the predicted theoretical strength of the retrofit system. Research results from large- scale retrofitted RC (reinforced concrete) beam tests [260, 34, 196] have also shown that failures of these systems may take place through various possible mechanisms, depending on the concrete grade, rebar provision, and properties of FRP. Identified failure modes include: (1) concrete crushing before steel yielding; (2) steel yielding followed by concrete crushing; (3) steel yielding followed by FRP rupture; (4) shear failure; (5) concrete cover delamination; and (6) debonding in the vicinity of the FRP/epoxy/concrete bond interface. Brittle debonding has been particularly observed [227, 34, 9]. Similar failure modes can occur to FRP-strengthened RC columns. Figure 1-5 illustrates several failure modes in two GFRP-concrete specimens and one GFRP-RC column. Gradual debonding of the FRP composite (structural damage)

(a) Global shear cracking in (b) Local concrete crumbling in a GFRP- (c) GFRP rupture and a CFRP-wrapped wrapped concrete specimen concrete crumbling in concrete specimen (Au and Buyukozturk, 2005) a GFRP-wrapped (Au and Buyukozturk, 2005) concrete column (Sheikh and Yau, 2002)

Figure 1-5: Several failure modes in GFRP-concrete systems under service load conditions may result in premature failures of the retroﬁtted system, leading to the total collapse of the structure. Thus, there is a need for inspection

28 of debonding in such multi-layer systems using NDT techniques in field conditions. This dissertation deals with the development of a nondestructive testing/evaluation (NDT/E) technique for concrete columns/brideg piers retrofitted by externally wrapped glass fiber reinforced polymer (GFRP) composites using far-field radar measurements. In this development, research components including analytical approach, experimental measurements, numerical simulation, and image reconstruction are described in detail in the following chapters. Research objectives, approach and the organization of the dissertation are given in the following.

1.1 Research Objective

The objective of this research is to develop a microwave-based NDT/E capability for the distant assessment of the physical condition of GFRP-retrofitted concrete structures with emphasis on the detection of anomalies and delaminations in the GFRP-concrete interface region and in the concrete cover areas. The research aims at the development of new knowledge on the interaction of GFRP-concrete systems with electromagnetic waves and an image reconstruction capability for physical imagery. A knowledge-based interpretation algorithm for an effective NDT/E technique as a basis for the condition assessment of GFRP-retrofitted concrete structures is developed.

1.2 Research Approach

The research approach to achieve the objective consists of four components:

1. Numerical simulation – Transmission and reflection of radar signals can be numerically simulated by the propagation and scattering of EM waves in a digital environment. Special models of the GFRP-concrete cylinders are constructed and impinged by plane EM waves using finite difference time domain (FDTD) methods. The purpose of the numerical simulation is to identify and study major system design parameters (measurement scheme, measurement frequency, and incident angle) in the development of a distant microwave-based NDT/E

29 technique. Parametric study on the effects of artificial defect and dielectric properties in the reflected radar signals is conducted.

2. Laboratory experimentation – Physical radar measurements were made from laboratory GFRP-wrapped concrete cylinders manufactured with and without artificial defects and rebars. Artificial defects including a cube, a thin-plate, and a strip (made of Styrofoam whose electromagnetic properties are same as air) were embedded at the interface between the GFRP layer and the concrete core. The selection of regular shapes for artificial defects represents the simplification to the complex shapes of real defects. Figure 1-6 shows this concept. The purpose of conducting radar measurements on laboratory specimens is twofold: First, a forward study can be performed using laboratory specimens whose defects are known. Radar measurements made on intact (without defect) and damaged (with defect) specimens can be evaluated based on their raw measurements and finally processed by image reconstruction as a basis for comparison with the real specimens. Secondly, the laboratory configuration provides a noise-free environment for the reflection radar measurements of the specimens. The removal of background noise is advantageous for better distinguishing the signal due to the presence of defect at a higher signal-to-noise ratio (SNR). Such measurements are insightful for damage detection and convenient to deal with in terms of the need of denoising.

3. Image reconstruction – In this research, collected distant radar signals (measurements) were processed by tomographic reconstruction (TR) methods in order to reconstruct the spatial profile of GFRP-wrapped concrete cylinders for condition assessment. Fast backprojection algorithm was applied for implementing TR methods and for developing numerical codes. The purpose of signal processing is to establish an effective and efficient transformation to visualize radar measurements.

4. Modeling of Dielectric Properties – For the use of radar signals (electromagnetic (EM) waves) on probing materials, knowledge is needed regarding the dielectric

30 GFRP-epoxy layer Styrofoam anomaly GFRP-epoxy layer

Concrete

Concrete Delamination between GFRP- epoxy and concrete A detected air void width

depth

The multi-layer cylinder structure

Structural cracks Artificial defects

Figure 1-6: Modeling of construction defects and structural damages using artiﬁcial anomaly

properties of the medium in which EM waves travel. Wave-medium interactions including transmission, reﬂection, and scattering of EM waves can not be understood and simulated without the knowledge of the dielectric properties of materials. For the materials considered in this research, dielectric properties are modeled in the frequency range of 8 GHz to 18 GHz for water, epoxy resin, E-glass fabric, GFRP, and oven-dried hydrated cement paste. This modeling eﬀort is important to an accurate numerical simulation, as well as in the image reconstruction for better locating defects (to be further explained in the chapters).

1.3 Organization of the Dissertation

This dissertation is organized in the following manner. Figure 1-7 illustrates the relations among the chapters in this dissertation. Chapter 2 reviews the current development of NDT techniques for the condition assessment of FRP-retrofitted concrete structures. Different NDT methods are compared for their current or potential use on GFRP-retrofitted concrete structures.

31 Literature review

Far-field Airborne Radar NDT

Far-field monostatic Numerical Laboratory ISAR measurements of simulation using radar FDTD GFRP-retrofitted concrete columns measurements Dielectric modeling of materials

Backprojection processing and image Image reconstruction using far- reconstruction field ISAR measurements

Condition assessment

Figure 1-7: Organization of the dissertation

Chapter 3 illustrates the numerical simulation results of electromagnetic waves propagation and scattering using finite difference time domain (FDTD) methods. Principles and implementation guidelines of FDTD methods are also introduced. Effects of system design parameters including measurement scheme, measurement frequency, and incident angle are studied.

Chapter 4 reports the radar measurements of GFRP-retroﬁtted concrete cylinders made in the MIT Lincoln Laboratory. Far-ﬁeld ISAR (inverse synthetic aperture radar) measurements of GFRP-concrete cylinders were collected by a horn antenna operating in monostatic mode. Linearly polarized continuous wave (CW) radar signals in the frequency range of 8 GHz to 18 GHz were used in probing the GFRP- concrete cylinders. Azimuthal (angular) range of 60 degrees were explored. Collected ISAR measurements were represented in the frequency-angle format.

32 Chapter 5 introduces the data processing approach for image reconstruction. Backprojection algorithms are used for processing far-field ISAR measurements into spatial images. Reconstructed range–cross-range images are also provided in this chapter. Chapter 6 addresses the dielectric modeling of the materials encountered in GFRP-concrete systems, including water, epoxy, GFRP, and oven-dried hydrated cement paste. Dielectric models applicable in the frequency range of 8 GHz to 18 GHz are developed based on the dielectric measurements of the materials reported in the literature. Chapter 7 addresses the condition assessment methodologies for inspecting the near-surface defects/damages in GFRP-retrofitted concrete cylinders using reconstructed images. Chapter 8 summarizes the research findings and discusses possible research topics for future work. Appendix A provides the derivation of the phase velocity of Love waves in a layer underlain by a half space medium, as the basis for condition assessment of multi-layered systems using the phase velocity of Love waves. Appendix B addresses the analytical investigation of several two-dimensional EM scattering problems in which a multi-layered dielectric medium is impinged by plane waves.

33 34 Chapter 2

Literature Review

”Read not to contradict and confute, not to believe and take for granted, but to weight and consider.” —– Essay on Of Studies, Francis Bacon (1561 ∼ 1626)

Condition assessment of materials and structures in a noninvasive manner is called nondestructive testing (NDT) or nondestructive evaluation (NDE). While the information regarding the condition of structural systems (e.g., bridge, buildings, dam, tunnel) designed for public use is not only crucial to the operation of the structure but also vital to public safety. Therefore, various NDT techniques have been developed for obtaining such information. Reported NDT techniques demonstrate the research results and ﬁndings based on diﬀerent approaches for assessing the condition of a material/structure system. The purpose of this chapter is to review these techniques.

In this chapter, general description of NDT is ﬁrst given. Current NDT techniques including optical NDT, acoustic NDT, thermal NDT, magnetic/electrical NDT, radiographic NDT, and microwave/radar NDT are reviewed with emphases on (1) physical principles regarding each NDT technique for understanding the characteristics (advantages and constraints) of each technique, and (2) reported or potential application of these techniques on GFRP-wrapped concrete systems.

35 2.1 Nondestructive Testing (NDT) Techniques

American Society for Nondestructive Testing (ASNT) deﬁnes NDT as

The testing of a specimen that determines its serviceability without damage that could prevent its intended use.

As indicated in the deﬁnition, NDT aims at providing information regarding the serviceability of a specimen (or material or structure) without damaging the specimen. Such a non-invasive scheme is favored for pertaining not only to the serviceability but also to the sustainability of the specimen under investigation. The specimen inspected by NDT techniques is sometimes termed the material under testing (MUT) which includes laboratory specimens and actual structures. General description of NDT is described in the following.

1. Purpose and Effectiveness of NDT Inspection — The purpose of NDT inspection is to understand the MUT via obtaining the information regarding its material properties through the knowledge of wave-medium or field-medium interaction. Serviceability of the MUT is based on the interpretation of material properties. Therefore, the effectiveness of NDT inspection in a particular application theoretically depends on the significance of such interaction and practically depends on the design of instrumentation. Should there be minor or no interaction between the chosen wave or field and the target MUT, the selected NDT method is theoretically ineffective. Should the signal-to-noise ratio (SNR) obtained through certain instrumentation be too small to be detectable, the selected NDT method is practically ineffective.

2. Procedure of NDT techniques — Figure 2-1 illustrates the general inspection procedure of NDT techniques.

3. Classiﬁcation of NDT Techniques — All NDT techniques are based on certain types of physical law for probing and manifesting the characteristics (material properties or geometrical properties) of the MUT. Current NDT techniques are distinguished by the following features.

36 Selection of NDT methods

Experimental configuration (Instrumentation) and installation

Man-made source Natural or man-made source for manifestation

Generation of manifesting agent for probing the MUT Natural source

Measuring the (Manifestation-medium response of the MUT interaction)

Interpretation of the measurement (Signal processing)

Condition assessment

Figure 2-1: Procedure of NDT Techniques

(a) Underlying physical principle — The physical principle behind each NDT technique is the relationship between the transmitted/produced waves (dynamically) or ﬁelds (statically) and the material properties of the MUT. The essential concept is to interpret the change in the MUT through the change in the received waves/ﬁelds.

(b) Type of wave or field — Selected waves or fields for probing the MUT can be mechanical, thermal, electric, magnetic, or electromagnetic. These waves/fields can be produced by man-made devices (e.g., electronic transducer and radar antenna) or by natural sources (e.g., thermal radiation and radiative decay) (Figure 2-1). It is obviously that one NDT technique utilizing mechanical waves is different from another technique utilizing electromagnetic waves. Table 2.1 shows several NDT methods utilizing mechanical waves. Table 2.2 lists the content of the electromagnetic spec-

37 trum.

Table 2.1: NDT techniques utilizing mechanical waves

Technique Frequency (Hz) Impact-echo and pulse-echo 5 × 104 ∼ 5 × 105 Acoustic emission (AE) 105 ∼ 106 Ultrasound — Pulse velocity 2×104 ∼ 2.5×105 Spectral analysis of surface waves ∼ 108

(c) Measurement device — Information resulting from wave-medium interactions is the response of a MUT and can be collected by electronic devices. Typically, an electronic device capable of manifesting a MUT (producing the manifesting agent) are equally capable of collecting the response of a MUT (reciprocal theorem). However, recent advances on novel measurement technology encourages the development of hybrid NDT whose manifesting agent is produced by one device and response is measured by another. Therefore, the use of a measurement device other than the one producing manifesting agent is considered a diﬀerent NDT technique.

(d) Instrumentation type — Information must be collected and transformed by devices through instrumentation configuration (equipment). Several instrumentation types such as single-input-single-output and single-input- multiple-output are possible for different interpretation schemes leading to the materialization of different NDT techniques.

For generality the classiﬁcation of NDT techniques in this thesis is made based on the underlying physical principles behind each NDT technique, rather than elaborating the details of each NDT technique.

4. Content of NDT techniques — While the NDT techniques are rapidly developing and expanding, their physical principles can be generally classiﬁed into

38 Table 2.2: Electromagnetic spectrum

Class Frequency Wavelength, λ Energy (eV) Type of inhomo- (Hz) (m) geneity Gamma 3 × 1019 ∼ 10−11 ∼ 10−14 > 1.24 × 105 rays 3 × 1022 X rays 3 × 1016 ∼ 10−8 ∼ 10−11 1.24×102 ∼ 1.24× Dislocations, pre- 3 × 1019 105 cipitates Ultraviolet 7.5×1014 ∼ 7 × 10−7 ∼ 10−8 3.1 ∼ 1.24 × 102 (UV) 3 × 1016 Visible 4.3×1014 ∼ (7 ∼ 4) × 10−7 1.77 ∼ 3.1 light 7.5 × 1014 Infrared 3 × 1011 ∼ 10−3 ∼ 4 × 10−7 1.24 × 10−3 ∼ 1.77 Texture, residual (IR) 4.3 × 1014 stresses, cracking, grain size, inclusions, ﬁber fracture, delaminations, porosity Microwaves 3 × 109 ∼ 10−3 ∼ 10−1 1.24 × 10−3 ∼ Texture, residual 3 × 1011 1.24 × 10−5 stresses, cracking, delaminations Radio freq. 3 × 10 ∼ 107 ∼ 10−1 1.24 × 10−13 ∼ Cracking, delami- (RF) 3 × 109 1.24 × 10−5 nations

39 the following categories. Reported examples are provided under each category.

• Optical methods — Including visual inspection [13], surface coating (e.g., photoelastic coating [41] and brittle coating), , fluorescent penetrants (e.g., dye penetrant [213], volatile liquid, and filtered particle), Moire interferometry [284], optical holography [100], shearography [97], laser speckle metrology [218], optical heterodyne interferometry [19], borescope [204], fiber optical sensors [109], and machine vision [110].

• Acoustic methods — Including impact-echo [231], sonics [144], ultrasonics [59, 247], acoustic emission [182], acousto-ultrasonics [281], electromagnetic acoustic transducer (EMAT) [161, 187], laser-ultrasonics [275], ultrasonic holography [136], acoustic microscopy [162], acoustography and vibro-acoustography [230, 189], and sonic signature analysis [112].

• Thermal methods — Including thermoelectric probe [71], impulse thermography [167], infrared thermography (IRT) [14], ultraviolet ﬂuorescence (UVF) [165], and emission spectroscopy [92].

• Magnetic and electrical methods — Including magnetic ﬁeld [170], magnetic particle [249], eddy current [222], nuclear magnetic resonance (NMR) [146], Kirlian photography [172], Barkhausen eﬀect [264], scanning electron microscopy (SEM) [98], and impedance spectroscopy [128].

• Radiographic methods — Including X-ray radiography [200, 94], gamma- ray radiography [101], neutron radiography [116], proton radiography [127], synchrotron radiation [86], and Compton backscatter [46].

• Microwave and radar methods — Including microwave radiometry [273], waveguides, coaxial probes, radar horn antenna [35, 195].

• Miscellaneous — Including mechanical impedance (hardness testing [61]), embedded ﬁber-optic strain gauge [232], chemical, liquid penetrant testing, replication microscopy, SQUID (Superconducting Quantum Interference Device) [276], and spark testing [131].

40 Table 2.3 summaries these methods classified by the elements mentioned previously in this section. These methods are briefly described in the following sections. While some NDT methods are applied in fluid dynamics and aerody- namics, this thesis only concerns solid medium as the MUT. Review of NDT methods in civil engineering can also be found at [133] and [175].

Table 2.3: NDT Techniques

Technique Type of Measurement Material wave/field device properties Optical EM waves in Human eye, lasers, optical spectrum and optical cameras Acoustic Mechanical Electronic Mechanical waves transducers properties Thermal EM waves in Ultraviolet and in- Thermal thermal radia- frared cameras properties tion spectrum Magnetic/ Magnetic field Magnetic electrical and electric field and electrical properties Radiographic X-rays, Gamma- Photographic films Radiographic rays, neutrons properties Microwave EM waves in Waveguides, coax- Dielectric and radar microwave special probes, radar properties trum and radio antennas frequency range

41 2.1.1 Optical Methods

Principles

Optical methods includes the earliest NDT method (visual inspection) and some mod- ern methods (e.g, laser speckle metrology and machine/computer vision), all relying on the electromagnetic radiation in the optical spectrum range, which can be detected by the human eye. The optical spectrum contains visible lights of wavelengths in air from approximately 400 nm to 700 nm (corresponding to frequencies from 750 THz to 428 THz) as the manifestation agent. Geometrical optics and physical optics can be applied for analyzing the propagation of visible light, while the former approach as- sumes zero-wavelength of light and considers no diﬀraction. However, the ray tracing technique in geometrical optics still serves as a convenient tool for providing insights in some problems. In physical optics, the scalar wave equation governs the motion of light waves in an isotropic medium [31].

1 ∂2φ(z, t) ∇2φ(z, t) = (2.1) c2 ∂t2

where z is the traveling distance of light waves, t is time, φ(z, t) is the space-time function of light waves (or wave function), and c is the speed of light waves in free space (c= 3 × 108 m/s). To account for the interference and diﬀraction phenomena of light waves during propagation, the Fresnel transform is used for determining the light intensity distribution on an observation plane (ξ, η) from a source plane (x, y).

∞ −ie−iωt Z Z 1 + cosθ ˆ g(ξ, η) = f(x, y)eik · rˆdxdy (2.2) λr 2 −∞

where i is the imaginary number, λ is the wavelength, ω is the radian frequency,r ˆ is the observation vector between the source point (x, y) and the observation point (ξ, η) with length r, θ is the angle between normal vector, and kˆ is the wave vector with magnitude k (wave number). The Fresnel transform of light intensity distribution in the source plane f(x, y) is the observed g(ξ, η). Fresnel transform is also the basis for

42 synthetic aperture radar (SAR) processing. Further investigation of optical analysis can be found at [31] and [265]. The ﬁrst appearance of qualitative visual inspection technique may be dated long back in history without a record, while the practice develops and matures with time. Visual inspection is still by far the predominant technique for the assessment of transportation infrastructures such as bridges, although its reliability for routine and in-depth inspections can be doubtful [163]. Results of evaluation using human eye are qualitative in nature and could be very subjective. The quality of such evaluation strongly relies on inspectors’ experience and judgement. As such, machine/computer/digital vision-based evaluation is introduced to eliminate the factor of subjectiveness and provides quantitative analysis. Signal analysis of recorded optical images using artiﬁcial intelligence approach (e.g., fuzzy set theory, expert system [110], neural network, and genetic algorithm) and computer vision approach provides alternatives to physical approach.

Optical NDT for GFRP-concrete Systems

At the present time and to the author’s knowledge, there is no literature of optical NDT for GFRP-concrete systems, although field engineers do use visual inspection for the detection of significant GFRP debonding and delamination in practice. How- ever, visual inspection is inherently not applicable to the condition assessment of GFRP-concrete systems for the ineffectiveness of optical lights in penetrating GFRP sheets/layers. In some circumstances, significant GFRP debonding, GFRP delamination, and concrete cracking may incur discoloring on the surface of GFRP, which is detectable to optical NDT. However, at this stage of failure, severe damage levels are reached without precaution. Additionally, similar discoloring of FRP could also due to the on-site variation in the curing of epoxy, causing potential false-alarm detection for optical NDT. Some optical NDT utilizes computer/mechine vision analysis incorporating artificial intelligence approach offers in-depth information through extensive simulation. But the approach is not physically sound, and the solution may not be unique.

43 In summary, constraints of optical NDT methods for GFRP-concrete systems include: (1) physical limitation of visual lights in penetrating the opaque GFRP layer, and (2) insuﬃcient credibility of optical signal (e.g., discoloring) for damage indication. These constraints hinder the use of optical NDT on GFRP-wrapped concrete systems for condition assessment.

2.1.2 Acoustic Methods

Principles

Acoustic methods are classiﬁed as the NDT techniques using mechanical waves as the manifesting agent to investigate MUTs. The use of mechanical waves, including body wave and surface wave, leads to the dynamic/vibrational response of MUTs. The governing equation of all acoustic methods from a microscopic point of view is the Navier equation of motion. For homogeneous and isotropic materials [171],

∂2u (λ + µ) ∇ (∇ · u) + µ∇2u + ρb = ρ (2.3) ∂t2

νE E where λ = and µ = are the Lame constants, u = Σu xˆ (1 + ν)(1 − 2ν) 2(1 + ν) i i is the displacement field, and b is the body force field per unit mass. The traction boundary condition is needed for solving the observable displacement field at the boundary. ∂u ∂u λ (∇ · u)n ˆ + µ i + j · nˆ = f (2.4) ∂xj ∂xi wheren ˆ is the normal vector at the boundary and f is the prescribed function acting on the boundary. Displacement fields at the boundary in different conditions (different mechanical properties; λ, µ) are measured in order to retrieve the variation in the ∂u ∂2u mechanical properties of materials. Velocity fields and acceleration fields can ∂t ∂t2 be determined either analytically when displacement fields u are obtained by evaluating the Navier’s equation, or numerically when displacement fields are measured over a period of time. Table 2.4 provides the relationship between wave velocity and mechanical properties of materials. It is clear that changes in the mechanical properties

44 (E, ν, ρ) of a MUT are reﬂected in the variation of these wave velocities. In Table 2.4, the direction of particle movement is determined with respect to the direction of wave propagation. Also, ρ denotes the density of the MUT, E the Young’s modulus, and ν the Poisson’s ratio. Another dispersive wave mode existing between body and surface waves is the Lamb wave whose wave velocities (phase and group) can only be numerically evaluated from the Rayleigh-Lamb equations [32].

Table 2.4: Types of mechanical wave

Class Particle Wave velocity Name movement E(1 − ν) 1/2 Parallel v = [255] P-wave, com- p ρ(1 + ν)(1 − 2ν) pressional wave, dilational wave Body E 1/2 wave Orthogonal v = [255] S-wave, shear wave, s 2ρ(1 + ν) distortional wave, equivolumnial wave, rotational wave 0.87 + 1.12ν E 1/2 Elliptical v ∼= Rayleigh wave R 1 + ν 2ρ(1 + ν) orbit (sym- [267] metrical Surface mode) wave Horizontally vL (See Appendix A) Love wave polarized shear mode

Mechanical responses of the MUT need to be collected by electronic devices (transducers) which translate mechanical responses into electrical signals for further analysis. Practical issues such as coupling between transducers and the surface of materials, multiple propagation paths, frequency-dependent characteristics of materials compli- cate the interpretation result.

45 Acoustic NDT for GFRP-concrete Systems

In principle, acoustic NDT such as acoustic emission is applicable to brittle materials like concrete [26]. Since the integration of concrete with GFRP layers also forms a brittle system, acoustic NDT is theoretically applicable to GFRP-concrete systems.

Mirmiran et al. (1999) [1] applied acoustic emission (AE) technique to GFRP- confined concrete cylinder specimens to study the correlation between AE signals and the stress state in concrete. AE transducers were mounted on the surface of the specimens by a highly viscous coupling agent in order to ensure the close contact condition. The used dominant peak frequency of AE transducers was 150 kHz. GFRP-confined concrete column specimens with different lengths, cross sections, jacket types, and jacket thickness were manufactured and subjected to cyclic compressive loadings. The Felicity and Kaiser effects of AE signals were also discussed. They found that the AE activity can be correlated to the extent of damage within the specimen. Higher AE activities were observed on specimens with longer dimensions and thicker jackets. Although the frequency content of AE signals is a function of AE transducers’ frequency response, they considered spectral analysis ineffective for evaluating the condition of GFRP-concrete columns.

Kundu et al. (1999) [247] studied the scanned ultrasonic images of concrete plate specimens attached by GFRP and CFRP composites. The GFRP-concrete system was formed by gluing a GFRP sheet with epoxy and mounting it onto a concrete plate. A circular delamination region of about 50 mm diameter was introduced between the GFRP layer and the concrete substrate for inspection. The ultrasonic signals were generated with frequencies from 200 kHz to 800 kHz. Both monostatic and bistatic conﬁgurations of ultrasonic transducers were used in the applied two scanning modes; Lamb wave scanning (L-scan) and longitudinal wave scanning (pulse-echo and C-scan). It was found that the longitudinal scanning mode is not eﬀective for distinguishing intact regions from damaged ones, while the Lamb wave scanning mode showed the damaged region (with delamination) as a bright spot in the produced image. They further concluded that the insensitivity of Lamb wave scanning mode to

46 the small variations in the epoxy and concrete properties has made the Lamb wave scanning mode superior for GFRP-concrete systems.

Mirmiran and Wei (2001) [183] used ultrasonic pulse velocity (UPV) to investigate the extent and progression of damage in concrete cylinders with and without E- glass FRP tube, also used as a tool for damage indication. The GFRP-concrete system was formed by filling concrete into GFRP tubes. Cyclic compressive loadings were applied on the specimens whose mid-section was attached by two ultrasonic sensors on two opposite sides of the mid-section. They found that the UPV in GFRP- concrete systems were sensitive not only at lower stress levels but also after concrete had significantly cracked, when compared with plain concrete specimens. Unlike the generally increasing trend of the AE signal in a GFRP-concrete system, the UPV signal exhibited a fluctuating pattern at different loading stages. They considered such difference as an opportunity to complement UPV with AE for the condition assessment of GFRP-concrete systems.

Bastianini et al. (2001) [80] used ultrasonic pulse amplitude (UPA) to locate the debonding defects in a polyurethane slab reinforced on both sides with GFRP, a concrete cylinder specimen wrapped with CFRP composites, and masonry columns wrapped with CFRP. An ultrasonic transducer was required in close contact condition with the surface of MUTs for eﬀective measurements of UPA. They found that the use of UPA is rather independent from the MUT and from the defect nature.

In summary, the features of acoustic NDT methods for GFRP-concrete systems include: (1) mechanical waves can penetrate through the GFRP layer in GFRP- concrete systems. Surface waves such as Lamb waves are found effective on detecting the presence of unseen delamination, due to the change of GFRP thickness [247]; (2) properties of ultrasound waves traveling within GFRP-concrete systems can be used for locating defects within the systems [183, 80]; (3) the use of coupling agent is required to assure the contact condition between transducers and the surface of MUTs, suggesting that acoustic NDT is essentially a contact technique; (4) limited to the size of acoustic transducers, inspection must be conducted on a point-by-point basis; and (5) interpretation of results from reflection measurement is difficult, especially

47 for heterogeneous materials like concrete. These constraints pose diﬃculties for the ﬁeld application of acoustic NDT on GFRP-wrapped concrete systems for condition assessment.

2.1.3 Thermal Methods

Principles

Thermal methods measure the thermal radiation (radiometry) or temperature (ther- mometry) emitted/reﬂected from the surface of the MUT, as well as from the ambient surroundings. Thermal radiation is the inherent electromagnetic radiation of materials above absolute zero temperature, which is a small fraction of the entire electromagnetic spectrum, containing electromagnetic waves ranging from ultraviolet light (3nm ∼ 400 nm) to far infrared radiation (300µm ∼ 1000µm). The radiant energy ﬂux at a given temperature and at a given wavelength is determined by [122]

Z λ e(λ, T ) = eλ(λ, T )dλ (2.5) 0

where eλ(λ, T ) is the distribution function of radiative ﬂux at wavelength λ (monochromatic emissive power), and T is the Kelvin (absolute) temperature. Since a MUT emits a unique distribution of energy in wavelength at a given temperature, as the result of the thermal properties of MUT, such distribution (at a given temperature) can be used for determining the thermal properties of the MUT and, furthermore,

for distinguishing one material from another. The relationship among eλ, λ, and T can be demonstrated by the Plank law of emission for a black body which yields the maximum value of eλ that a material can attain.

2π c2 e = ~ (2.6) λb h ~c i λ e kB T λ − 1

−34 where eλb is the eλ of a black body, ~ is Planck’s constant (~ = 6.62606876×10 J ·s), −23 c is the speed of light, and kB is Boltzmann’s constant (kB = 1.3806503×10 J/K).

48 Experimentally, the Stefan-Boltzmann law states the strong dependency of the radiant energy ﬂux on T as

4 eb(T ) = σT (2.7)

where σ is the Stefan-Boltzmann constant (σ = 5.6704 × 10−8W/m2 · K4)). Since the radiant energy flux of a MUT is a unique signature at a given temperature and significantly changes with varying temperatures, it is used for characterizing the MUT. As a NDT method, thermal radiation is practically more applicable for field application over other heat transfer modes including heat conduction (Fourier’s equation) and heat convection (Newton’s law of cooling). The evaluation of heat transfer process is governed by the heat diffusion equation. The heat diffusion equation for incompress- ible medium without convection is [122]

∂q ∂T ∇ · k∇T + = ρc (2.8) ∂t ∂t

watts where k is the thermal conductivity ( m·K ), T is the Kelvin temperature (K), q is the watts kg heat ﬂux ( m2 ), t is time (sec), ρ is the density ( m3 ), c is the speciﬁc heat capacity kJ ( kg·K ) [122]. Techniques for solving inverse heat transfer problems can be found in textbooks such as [201]. The applicability of this technique is based on the availability of heat (cold) source, thermal gradient, and induced heating [257]. Anomalies such as air voids between two material layers will obstruct the thermal radiation from the bottom layer and alter the thermal image.

Thermal NDT for GFRP-concrete Systems

Thermal NDT has been a popular inspection technique for FRP-concrete systems. Many of the applications are CFRP-wrapped systems [210, 166, 197] and some GFRP- wrapped systems [262]. Halabe et al. (2002) [262] used IR cameras on several laboratory specimens and composite bridge decks for delamination detection. The GFRP-concrete system in their study was a concrete bridge column wrapped with GFRP. Their laboratory tests suggested that delaminations in a composite deck with a depth of the order 0.5 in (1.3

49 cm) and thickness of the order of 0.05 in (0.13 cm) could be detected. However, issues aﬀecting the thermal contrast between delaminations and the background, including surface defects and oil spills, shadow from adjacent objects, and inspection angle, could interfere with the results.

Brown and Hamilton III (2003) [33] applied IR cameras and heating lamps on concrete block specimens mounted with GFRP/vinylester, CFRP/epoxy, and CFRP/ polyurethane composites for detecting the delamination within the FRP-concrete systems. The heating lamps for pre-warming purpose were placed 12 in from the surface of the specimen, and the IR camera was located 24 in from the specimen. A certain amount of time (30 sec or 60 sec) was needed for recording the cooling process on the surface of the specimen. They found that defects residing under single-layer CFRP are well detected, however, the IR detectability decreases when the thickness of the FRP increases. They also found that GFRP layers with a thickness of 0.3 in ∼ 0.4 in are diﬃcult to detect in their experiments.

In summary, the features of thermal NDT methods for GFRP-concrete systems include: (1) the presence of unseen delamination within GFRP layers or in the interface between GFRP and concrete may not be detectable from the surface measurement by thermal sensors (e.g., IR camera) as reported by some researchers [33]; (2) thermal NDT relies on the thermal contrast between damaged and intact regions, triggered by either natural (sun) or man-made (heating blanket or cooling liquid) sources. Al- though thermal NDT is basically a distant inspection technique, surface preparation may require direct access to the surface of MUTs; and (3) variations in the ambient thermal condition and surface emissivity of materials aﬀect the resolution of the imagery and the interpretation. Controlling ambient temperature could be a trouble- some and diﬃcult task in some practical situations.

50 2.1.4 Radiographic Methods

Principles

Radiographic methods are the technique of obtaining a shadow image of a MUT using penetrating radiation (radiation particles) including X-rays (or X-ray photons), gamma-rays (or gamma-ray photons), neutrons, positrons, protons, or electrons as manifesting agent. Among these radiation particles, gamma-ray photons are emitted by the disintegration of a radioactive isotope (radioactive decay), while other man- made particles are generated by accelerator or circuit, radioactive, reactor neutron sources. The images produced by radiographic methods are known as radiographs, and the contrast in a radiograph indicates the diﬀerential absorption of penetrating radiation in the MUT. In this section, characteristics of penetrating radiation are illuminated using X-rays as an example, since it is widely used in medical and engineering applications. Information regarding the use of penetrating radiation in NDT applications can be found at [25] and [43]. A typical X-ray instrumentation scheme consists of (1) X-ray tube (to generate X-rays), (2) collimators (to restrict X-rays within desired area), and (3) image receptor (to capture transmitted X-rays). Further information about X-rays can be found in [54].

Generation and Features of X-rays

X-rays are generated when an incident electron beam impinges on a solid matter. The electrons of the matter are ejected (photoelectric absorption), scattered (coherent scattering), recoiled (Compton scattering), released (pair and triplet production), and absorbed (photointegration) by the incident electron beam, depending on the energy the electron beam carries [42]. X-rays or X-ray photons are produced during the process of these interactions. Discovered by W.C. R¨ontgen in 1895, X-rays are characterized by (1) They are invisible, highly penetrating rays traveling in straight lines. Access to opposite sides of the MUT is required in order to capture the transmitted rays; (2) They are electrically neutral and immune to electromagnetic ﬁelds; (3) They are polyenergetic rays which can be generated at various energy levels; and (4) They

51 can ionize the MUT through penetration and excitation. Such ionization produces chemical and biological changes in the MUT, meaning that protecting personnel from X-ray radiation is vital.

X-ray Interaction with Matter

The absorption and scattering of X-rays in matter are responsible for the attenuation of X-rays in matter. Quality radiographs are usually obtained with minimum X-ray scattering. The attenuation (absorption) coefficient of X-rays for a narrow, well- collimated beam is defined by dI = −µdz (2.9) I where dI/I is the relative decrease in intensity, µ is the linear attenuation (or absorption) coefficient, and dz is the unit traveling distance of X-rays. Integrating Eq. 2.9 provides Lambert’s law (also known as Beer’s law) given by

−µz Iz = I0e (2.10)

where Iz is the intensity at a distance z and I0 is the intensity at the surface of the MUT. For a broad diverging X-ray beam, the linear absorption coeﬃcient µ is deﬁned by

µ = µT h + µph + µj + µξ + µη (2.11)

where µT h is the absorption coeﬃcient accounting for the Thomson scattering, µph for

the photoelectric eﬀect, µj for Compton scattering (absorption and scattering), µξ for

pair and triplet production, and µη for photodisintegration [103]. These coefficients represent the radiographic properties of the MUT. X-ray diffraction is one of the two phenomena (with x-ray fluorescence) associated with the onset of x-radiation. It is the elastic scattering of x-ray photons by atoms in a periodic lattice structure of materials. The phase information of the scattered monochromatic x-rays provides meaningful information about the structure of materials and can be further used to identify the material.

52 Radiographic NDT for GFRP-concrete Systems

At present time, only a limited amount of research has been devoted to the use of radiographic NDT for concrete (cementitious) materials. Masad et. al. (2002) [72] applied x-ray diffraction on identifying the distribution (sizes and locations) of air voids in asphalt mix specimens. Daigle et. al. (2005) [159] compared the performance of x-ray diffraction and ultrasound on concrete specimens based on several factors including cost and image resolution. Their research results suggest that the cost, safety concern, and experiment preparation of x-ray diffraction may limit its applicability in field, although it provides high resolution images.

To the author’s knowledge, there is no literature of radiographic NDT application for GFRP-concrete systems. The reason is simply because that radiographic NDT is essentially a laboratory method, and GFRP-concrete systems are situated in an open environment, making the application of the method in field and the protection of inspectors from excessive radiographic exposure (contamination) extremely difficult. In addition, the requirement to access on both sides of the MUT and the safety concern make radiographic NDT a near-contact technique, although radiographic rays are theoretically suitable for distant inspection. It is the operational and safety issues that limit the use of radiographic NDT for GFRP-concrete systems. The future application of radiographic NDT for GFRP-concrete systems may be conducted in laboratory, but it will not be possible to make radiographic NDT a field technique unless the previously addressed concerns can be resolved.

2.1.5 Magnetic and Electrical Methods

Principles — Magnetic Methods

Magnetic methods are described by the magnetic particle inspection (MPI) in this thesis as an example. In MPI methods, the constitutive relation connecting magnetic ﬂux density B(webers/m2), magnetic permeability µ, and magnetic ﬁeld strength H

53 (amperes/m) for homogeneous, isotropic, non-frequency dispersive media is read [138]

B = µH (2.12)

In an imperfect MUT with a crack, a discontinuity is created between two surfaces of a crack, resulting in a distorted magnetic ﬁeld termed the magnetic leakage ﬁeld. The constitutive relation becomes

B = µ0 H + M (2.13) where magnetization M (amperes/m) accounts for the contribution of the magnetic leakage and is used for detecting defects. For ferromagnetic materials, M H. Therefore, ∼ B = µ0M (2.14)

Such variation in magnetic ﬁelds can be related to the surface current density kM within an area ∆A which is deﬁned by

P qivi kM = lim (2.15) ∆A→0 ∆A

where qi is the moving charge on the surface, and vi is the velocity vector of qi. Con- sidering a Gaussian surface of negligible thickness with area ∆A across the boundary of two regions (i and j) consisting the crack, Gauss’s law provides

Z kM ∆A = jM dV (2.16) V

where jM = ∇ × M is the equivalent current distribution, and dV is the unit volume. Also, Z kM ∆A = ∇ × MdV =n ˆ × M j − M i ∆A V wheren ˆ is the normal vector pointing from region i to region j. It is seen that the surface current accounts for the total diﬀerence in magnetization between two regions

54 creating a crack if kM is measured. MPI is performed by (1) cleaning the surface of the MUT in order to allow magnetic particles capture the geometry of the MUT after magnetization, (2) magnetizing the MUT, (3) spreading magnetic particles over the surface of the MUT, (4)illuminating the surface for inspection based on the distribution of magnetic ﬁelds indicated by sprayed magnetic particles, and (5) demagnetizing the MUT [43].

Magnetic NDT for GFRP-concrete Systems

In order to successfully detect defects by MPI, the surface of the MUT must be cleaned and the MUT magnetized. Since neither GFRP-epoxy composites nor concrete can be magnetized, magnetic NDT methods are inapplicable to the condition assessment of GFRP-concrete systems. Although steel reinforcements are magnetizable, they are buried under concrete cover, and their cracking is not concerned in this research. Additionally, magnetic NDT methods provide only the surface information of the MUT, thus, not relevant for the purpose of in-depth investigation.

Principles — Electrical Methods

Electric methods are characterized by eddy current testing (ECT) which in this thesis. This section introduces the physical principles of ECT as an example for electrical NDT methods. When a coil carrying alternating currents approaches the surface of a MUT, an eddy current ﬂowing in a closed loop is formed on the surface of the MUT, causing a back electromotive force (EMF), f, opposing the in the coil. The magnitude of f is changed when defects are presented in the MUT, and is used for indicating the presence of defects. Faraday’s law provides [138]

I v f = E + × B · dl (2.17) C c

where C is the closed loop containing the considered magnetic flux density B, E is the electric field strength (volts/m) flowing through the closed loop C, v is the velocity

55 of a given point in the middle of an unit length dl on the loop, and c is the speed of light. The ﬁrst part in the R.H.S. (right hand side) of the above equation results is the result of the time-varying vector potential, and the second part is the result of the closed loop’s motion. The second part diminishes if the loop is stationary. With the constitutive relation E = −1 · D, where D is the electric displacement (coulombs/m) and the electrical permittivity, f is related to the variation of the MUT’s electrical properties. ECT is conducted by the use of inspection devices typically containing an induction coil circuit and an oscilloscope for illuminating the result. Contact inspection of ECT is not required, but the depth of penetration is usually restricted. Distance between the ECT device and the surface of a MUT, the frequency of the alternating current inside the loop, and design of the coil can be changed so as to achieve desired results [102].

Electrical NDT for GFRP-concrete Systems

As an example of electrical NDT methods, ECT is only applicable to electrically conductive materials. Variation in the electrical properties of GFRP-concrete systems due to the presence of defects (GFRP delamination, GFRP debonding, or concrete cracking) is not significantly detectable since GFRP and concrete are not highly conductive materials. The contrast between intact region and cracked region in a GFRP-concrete system due to the inclusion of air gaps can be amplified if the presence of water within the crack is found. Although the near-surface condition of GFRP- concrete systems can be evaluated by ECT, the restricted lift-off distance classifies ECT as a near-contact method which is not feasible for distant inspection in practice.

2.1.6 Microwave and Radar Methods

Principles

Microwave and radar (Radio Detection and Ranging) NDT methods are based on Maxwell’s equations which govern the spatial and temporal relationships between

56 electric field and magnetic field. The coupling between electric field and magnetic field is the kernel to the propagation of electromagnetic (EM) fields which are used as the manifesting agent in microwave and radar NDT methods. Although the terms ”microwave NDT” [8] and ”radar NDT” [4] are both used by various researchers, differences can be distinguished between them as listed in Table 2.5. By this definition, radar NDT also includes those used in geophysical and geotechnical applications (subsurface radar [69], georadar, ground penetrating radar or GPR), although those applications are not the focus of this thesis. The formulation of EM fields in microwave

Table 2.5: Microwave and radar NDT techniques

Microwave Radar Spectrum 0.3 GHz ∼ 300 GHz 0.003 GHz∼110 GHz Wavelength 1 m∼ 10−2m 3×102m ∼ 2.72 × 10−2m Measurement Waveguides, coaxial probes, Bow-tie antennas, horn an- device cavity resonators tennas

and radar NDT applications is typically carried out by excluding the source (EM ﬁeld radiator or radar antenna) within the domain of interest, unless the radiation pattern of the antenna is an issue. For this reason, the source-free Maxwell’s equations are used to derive the EM wave propagation equation as follows [138].

∂B ∇ × E = − (2.18) ∂t ∂D ∇ × H = (2.19) ∂t ∇ · B = 0 (2.20)

∇ · D = 0 (2.21)

Rearranging the above with the constitution relationships provided in the Section of Magnetic and Electrical Methods and using the vector identity ∇ × (∇ × X) =

57 ∇(∇·X)−∇2X where X is a vector ﬁeld, two Helmholtz wave equations are obtained.

∂2E ∇2E − µ = 0 (2.22) ∂t2 ∂2H ∇2H − µ = 0 (2.23) ∂t2

The use of microwave and radar NDT covers a wide variety of applications in civil engineering, which is summarized in the Table 2.6. As can be seen in Table 2.6, EM signals used for probing underground and underwater objects are generally of frequencies lower than 1 GHz ∼2 GHz for the sake of better penetration into lossy dielectric media such as wet soils, while higher frequencies are usually adopted for detecting anomalies (air voids, rebars, debonding, delamination) to achieve better resolution. It should be noted that Table 2.6 only demonstrates the explored frequency ranges in various applications. Frequencies outside the range listed in Table 2.6 are not excluded from their potential use in each application.

Microwave and Radar NDT for GFRP-concrete Systems

Several radar NDT techniques for assessing the condition of FRP-retrofitted/wrapped concrete structures are reported in the literature. Li and Liu (2001)[149] applied a bistatic radar NDT system to detect air voids in the interface region between GFRP-epoxy layer/jacket and concrete surfaces. Radar measurements were conducted at 10 GHz using a pair of dielectric lenses for signal focusing. The focusing distance between the dielectric lenses and the specimen was 10 in. (25 cm) at 10 GHz. The dielectric focus lenses enabled revealing of localized response from the GFRP-concrete specimen, with a spatial resolution as small as 4 in. (10.16 cm) in diameter. Artificial defects of size 0.07 in2 (0.43 cm2) were detected at 10 GHz. The defined imagery was constructed by assembling the reflection coefficient at each spatial point, and used for condition assessment. Feng et al. (2002)[185] used a horn antenna and a waveguide reflectometer for detecting air voids in the interface region of GFRP-wrapped concrete specimens. Dielectric lenses were also introduced to mechanically focus plane waves on a local-

58 Table 2.6: Microwave and radar NDT applications in civil engineering Application Frequency range References Material characterization 0.1 MHz ∼ 18 GHz [5] [168] [29] [176] [2] [248] [73] [29] [141] [30] [99] [237] [3] [8] Bridge pier scour detection 0.1 GHz ∼ 1 GHz [192] [181] [88] Bridge deck assessment 0.05 kHz ∼ 16 GHz [64] [60] [17] [164] Void and crack detection in 0.5 GHz ∼ 1.5 GHz [39] [27] [169] [38] stone and masonry structures Damage detection in sluices 0.5 GHz ∼ 0.9 GHz [168] Pavement thickness detection 10 MHz ∼ 2.5 GHz [246] [173] [229] [40] [6] Underground object detection 0 Hz ∼ 0.6 GHz [199] [93] Tunnel linings 0 Hz ∼ 1 kHz [4] Channel walls 0.2 GHz ∼ 0.9 GHz [216] Railway tracks 0.5 GHz ∼ 1.5GHz [37] [259] [158] Fatigue cracks detection in 24.13 GHz ∼ 33.6 [211] steel structures GHz Debonding in composites 8.5 GHz ∼ 40 GHz [283] [190] [149] [11] [48] Concrete cover thickness detec- 1.5 GHz [134] tion Concrete and cement cracking 1.5 GHz ∼ 10 GHz [35] [157] [217] [167] [191] Post-tensioned concrete beam 1.5 GHz [274] Rebar detection 1 GHz ∼ 12 GHz [217] [226] [233] Structural testing and remote 5.75 GHz ∼ 10 GHz [57] [234] [205] sensing

59 ized spot in order to enhance the strength of reflected signals. The focusing distance between the dielectric lenses and the specimen was 2.5 in. (6.4 cm) in X-band. Reflec- tion coefficients were used for evaluating the difference between intact and damaged responses. Artificial defects as small as 0.105 cm in concrete were detected at the frequency 12.4 GHz. The imagery was the assembling of reflection coefficients at different points on the surface of the specimen. Later, Kim et al. (2004)[279] proposed the use of planar slot antenna arrays for detecting air voids in concrete panel and block specimens. The slot antenna consisted of two arrays: transmitting array and receiving array, operating in 5.2 GHz with a resolution of 1 in. (2.5 cm) and penetration depth of 10 in. (25 cm). The focusing distance between the slot antenna and the specimen was 1.2 in. (3 cm). The imagery for damage detection was generated by integrating reflection responses of the receiving slots to obtain an equivalent electric current distribution within the target. This distribution was decomposed by the Hankel function of the second kind in the case of cylindrical waveguide problems. The coefficients operating with the Hankel function were considered as the focusing operator. This development was based on the use of cylindrical slot antenna arrays applicable for circular column structures.

Akuthota et al. (2004) [11] utilized an open-ended rectangular waveguide probe and near-field radar measurements for detecting disbonds and delaminations between CFRP (carbon FRP) laminate and the concrete substrate. Transmitted EM waves were linearly polarized with orientation orthogonal to the unidirectional carbon fibers in order to achieve strong penetration through the CFRP laminate. Artificial defects with sizes in the millimeter range were inserted between the CFRP laminate and the concrete substrate. Inspection distances were 0.06 in.∼0.12 in. (0.15∼0.3 cm) at 10 GHz and 0.12 in.∼0.27 in. (0.3∼0.7 m) at 24 GHz as the authors reported in their work.

Although EM waves can be used in distant ranges (far-field), most of the work on this particular problem adopted contact or near-contact (near-field) measurements. Tradeoffs between near-field and far-field techniques were discussed in the literature [142, 215]. FRP debonding, delamination, and air voids in the interface regions of

60 FRP-concrete systems were investigated by near-field inspections using different hard- ware systems including horn antenna, slot antenna arrays, and waveguide probe. High frequencies ranging from 5 GHz to 24 GHz were used in order to capture small defects. Reflection coefficients were calculated and used for damage assessment. Visualization of the results was generally performed by assembling the localized response at each inspection location to form a spatial profile in the reported techniques.

2.2 Summary

This chapter reviews current nondestructive testing (NDT) methods at large and their applicability to the particular structural system (GFRP-concrete system) considered in this thesis. Features of these NDT techniques are summarized as follows.

1. Optical NDT — Physical limitation of visual lights on penetrating the opaque GFRP layer, as well as the insuﬃcient credibility of optical signal for damage indication, constrain the use of optical NDT for GFRP-concrete systems.

2. Acoustic NDT — Acoustic waves (Lamb wave) and ultrasounds are found effective on detecting the presence of unseen delamination, owning to the change of GFRP thickness in affecting the propagation of waves. However, acoustic NDT is basically a near-contact, point-by-point inspection technique which is not ready for the field inspection of GFRP-concrete systems.

3. Thermal NDT — The presence of unseen delamination within GFRP layers or in the interface between GFRP and concrete may not be detectable from the surface measurement by thermal cameras as reported. Variations in the ambient thermal condition and surface emissivity of materials affect the resolution of the imagery and the interpretation. Controlling ambient temperature could be difficult in some practical situations. Although thermal NDT is inherently a distant inspection technique, surface preparation may require direct access to the surface of the system, making it not very preferable for the field application of GFRP-concrete systems.

61 4. Radiographic NDT — The cost and safety concerns of radiographic NDT made it rather a laboratory technique but a ﬁeld technique. These concerns also unfavor the use of radiographic NDT for the ﬁeld inspection of GFRP- concrete systems.

5. Magnetic and Electrical NDT — The non-magnetic and non-conductive nature of GFRP layers and concrete makes the use of magnetic and electrical NDT on GFRP-concrete systems inherently less eﬀective, especially for target- ing GFRP debonding and GFRP delamination.

6. Microwave and radar NDT — Microwave and radar NDT is in nature a field applicable method since EM waves attenuate sightly when traveling in free space and are capable of penetrating into non-metallic, lossy media such as concrete. Current microwave and radar NDT techniques for GFRP-concrete systems are either distant methods for surface inspection or near-field methods for in-depth inspection. While the in-depth inspection capability can be offered by some near-field methods, the need for mechanical focusing devices (e.g., lens, antenna arrays) is not practical for civil engineering application. On the other hand, surface information provided by distant methods is not critical for GFRP-concrete systems.

From the results of literature review, it is found that a distant NDT technique that is capable of conducting in-depth inspection of GFRP-concrete systems in field conditions is currently lacking. Microwave and radar NDT is identified to be applicable for developing such technique, based on the following reasons: (1) its distant inspection capability when compared with acoustic, magnetic, and electrical NDT, (2) its capability of penetrating into non-metallic, lossy media such as concrete when compared with optical and thermal NDT, (3) it is less vulnerable to temperature variation in field conditions when compared with thermal NDT, and (4) it is not limited by safety issues in field conditions when compared with radiographic NDT. According to these findings, it is concluded that microwave and radar NDT is potentially applicable for the distant, in-depth condition assessment of GFRP-concrete systems.

62 Chapter 3

Numerical Simulation

”Whatever philosophical standpoint one may adopt today, from every point of view the erroneousness of the world in which we think we live is the surest and ﬁrmest fact that we can lay eyes on: we ﬁnd reasons upon reasons for it which would like to lure us to hypothesis concerning a deceptive principle in ’the essence of things.’” —– Beyond Good and Evil, Friedrich Nietzsche (1844∼1900)

Understanding the transmission and reflection of radar signals in GFRP-retrofitted concrete columns is equivalent to studying the propagation and scattering of electromagnetic (EM) waves in a dielectric medium. In such investigation, it is beneficial to develop the knowledge of radar response from GFRP-concrete columns through numerical simulation such that a close link between radar response and structural integrity can be established. On the other hand, complexities of the dielectric properties of concrete can also be studied by the forward modeling in numerical simulation. The main interest of this simulation work is to investigate the time-domain far-field response of scattered EM fields from GFRP-concrete systems. Among many numerical simulation techniques, the finite difference time domain (FDTD) method using Yee’s algorithm [277] was chosen for its superior performance and capability of generating the space-time response of EM wave propagation and scattering. GprMax, a

63 GNU/GPL code based on the FDTD method, was used in producing the simulation results [95].

Conceptually, the radar can collect measurements reflected from the GFRP-concrete column (cylinder) at any angle. Should we define a plane containing the longitudinal axis of the cylinder, the profile of the cylinder will be rectangular in shape (oblique incidence); should we define another plane perpendicular to the longitudinal axis of the cylinder, the profile of the cylinder will be circular (normal incidence). These two scenarios represent two different measurement schemes. The coinciding use of a radar for transmitting and receiving signals leads to the operation mode of monostatic radar. It can be expected that the reflection response of GFRP-concrete cylinders depends on the selection of measurement scheme, measurement (incident) frequency, and incident angle.

The purpose of this chapter is to investigate the effects of system parameters (measurement scheme, measurement frequency, and incident angle) in the reflection response of GFRP-concrete cylinders with artificial features (near-surface defect and rebars). In the two-dimensional simulation of this work, GFRP-concrete cylinders are consequently modeled as rectangles and circles. The knowledge developed in numerical simulation can benefit the design of an efficient and effective experimental program (Chapter 4 Laboratory Radar Measurements).

In the time-domain far-ﬁeld region the incident EM waves are approximately planar. The use of linearly polarized EM waves leads to the decoupling of Maxwell’s curl equations into transverse electric (TE) and transverse magnetic (TM) wave modes [138]. In this chapter, the FDTD solutions to both TE and TM waves are provided, as the basis for the explicit evaluation to Maxwell’s curl equations and for numerical pro- gramming. Computational issues including stability criteria and absorbing boundary conditions are also addressed. Description to numerical models is provided, followed by the results of parametric study and their discussion.

64 3.1 Maxwell’s Curl Equations and Linearly Polar- ized EM Waves

In linear, isotropic, nondispersive, lossy, source-free problems, Maxwell’s curl equations are written as [138]

∂H¯ 1 = − ∇ × E¯ + σ∗H¯ (3.1) ∂t µ ∂E¯ 1 = ∇ × H¯ − σE¯ (3.2) ∂t where H¯ = magnetic field strength (amperes/m or A/m), µ = magnetic permeability of the medium (henrys/m or H/m), E¯ = electric field strength (volts/m or V/m), σ∗ = equivalent magnetic loss (ohms/m or Ω/m), = electric permittivity of the medium (farads/m or F/m), and σ = electric conductivity of the medium (1/ohms/m = siemens/m or S/m). In construction materials which are in general non-magnetically lossy, contribution of σ∗ can be neglected. In such problems, the medium in which electromagnetic waves propagate is fully characterized by its dielectric properties (µ, , and σ). Meanwhile, H¯ = H¯ (¯r, t) and E¯ = E¯(¯r, t), wherer ¯ is the three- dimensional spatial vector at the point where the EM fields (H¯ and E¯) are concerned, and t is the one-dimensional temporal variable at the time instant when the EM fields are evaluated. Note that Maxwell’s equations can be expressed in integral or differential form and in different coordinate systems (e.g., Cartesian, cylindrical, and spherical). In brief, the four-dimensional space-time behavior of EM waves is governed by Maxwell’s equations, regardless of their representation.

65 In Cartesian coordinate systems, Eqs.(3.1) and (3.2) are written as

∂H 1 ∂E ∂E x = − z − y (3.3) ∂t µ ∂y ∂z ∂H 1 ∂E ∂E y = − x − z (3.4) ∂t µ ∂z ∂x ∂H 1 ∂E ∂E z = − y − x (3.5) ∂t µ ∂x ∂y ∂E 1 ∂H ∂H x = z − y + σE (3.6) ∂t ∂y ∂z x ∂E 1 ∂H ∂H y = x − z + σE (3.7) ∂t ∂z ∂x y ∂E 1 ∂H ∂H z = y − x + σE (3.8) ∂t ∂x ∂y z

In the far-ﬁeld region where EM waves are planar, variation of the EM wave components in the direction normal to the plane on which EM waves propagate diminishes. In other words, the EM wave components in the direction perpendicular to the plane on which the wave vector (k¯) is deﬁned are constant. Suppose we choose the x-y plane ∂ as the wave propagation plane (k¯ =xk ˆ +yk ˆ ), we have = 0. This plane-wave x y ∂z ∂ condition eliminates the terms in Eqs.(3.3), (3.4), (3.6), and (3.7), leading to ∂z

∂H 1 ∂E x = − z (3.9) ∂t µ ∂y ∂H 1 ∂E y = − − z (3.10) ∂t µ ∂x ∂E 1 ∂H x = z + σE (3.11) ∂t ∂y x ∂E 1 ∂H y = − z + σE (3.12) ∂t ∂x y

Taking the time derivative of Eqs.(3.19) and (3.22) yields

∂2H 1 ∂ ∂E ∂ ∂E z = − y − x (3.13) ∂t2 µ ∂x ∂t ∂y ∂t ∂2E 1 ∂ ∂H ∂ ∂H ∂E z = y − x − σ z (3.14) ∂t2 ∂x ∂t ∂y ∂t ∂t

66 With Eqs.(3.9)∼(3.12), Eqs.(3.13) and (3.14) become

∂2H 1 ∂2H ∂2H ∂E ∂E z = − − z − z + σ y + x ∂t2 µ ∂x2 ∂y2 ∂x ∂y ∂2 ∂2 ∂2 ∂E ∂E ⇒ + − µ H − σ y + x = 0 (3.15) ∂x2 ∂y2 ∂t2 z ∂x ∂y ∂2E 1 ∂2E ∂2E ∂E z = y − x − σ z ∂t2 µ ∂x2 ∂y2 ∂t ∂2 ∂2 ∂2 ∂ ⇒ + − µ − σ E = 0 (3.16) ∂x2 ∂y2 ∂t2 ∂t z

It can be shown that two EM wave propagation modes can be deﬁned by properly grouping the six governing equations (Eqs.(3.9), (3.10), (3.19), (3.11), (3.12), and (3.22)) into two groups. Eqs.(3.17), (3.18), and (3.13) form the propagation mode of transverse electric (TEz) waves of waves or TE waves, governing equations are

∂E 1 ∂H x = z + σE (3.17) ∂t ∂y x ∂E 1 ∂H y = − z + σE (3.18) ∂t ∂x y ∂H 1 ∂E ∂E z = − y − x (3.19) ∂t µ ∂x ∂y

It is clear that TE waves can be fully described by Ex, Ey, and Hz. In radar terminology, transmitted EM waves with an electric field horizontally polarized are called H-polarized waves. Orientation of the electric field is defined with respect to the plane formed by the wave vector k¯ and the characteristic axis (e.g., the longitudinal axis of a cylinder). When the received electric field is also horizontally polarized, the radar response and its associated images are called HH-polarized. In this case, TE waves are HH-polarized signals. Similarly, Eqs.(3.20), (3.21), and (3.14) form the propagation mode of transverse magnetic (TM) waves since it is the magnetic field ¯ Hz normal/transverse to the wave vector k. The governing equations of TM waves

67 are

∂H 1 ∂E x = − z (3.20) ∂t µ ∂y ∂H 1 ∂E y = − − z (3.21) ∂t µ ∂x ∂E 1 ∂H ∂H z = y − x + σE (3.22) ∂t ∂x ∂y z

TM waves can be fully described by Hx, Hy, and Ez. When TM waves are transmitted and received in vertical polarization, VV-polarized signals are obtained. In this case, TM waves are identical to VV-polarized signals. In this dissertation, elliptically polarized EM signals are not used, nor are the switched transmission and receiving modes (HV- and VH-polarized) considered. Both TE and TM waves are linearly polarized since the variation of the electric ﬁeld is linear. HH- and VV-polarized signals are used in the laboratory radar measurements.

3.2 Finite Diﬀerence Time Domain Solution and Yee’s Algorithm

While the governing equations of TE and TM waves are sufficient for analytically studying the EM wave propagation and scattering problems, they are not directly applicable for numerical application. Spatial and temporal discretization is necessary for implementing Maxwell’s curl equations in the discrete domain established by computing machines. Among current discretization methods in numerical analysis, the finite difference method in time domain or the finite difference time domain (FDTD) method is chosen for simulating the propagation and scattering of radar signals in space and time. The main reason for choosing the FDTD method are (1) its explicit evaluation of Maxwell’s curl equations, which avoids potential numerical instability, and (2) its accuracy of approximation. Details of the FDTD method can be found in [251]. The central difference scheme is selected for its higher order of accuracy over

68 the forward and backward difference schemes. It can be shown that, with Taylor’s theorem, for a one-dimensional function f(x), the first-order derivative of f(x) by several difference schemes is

∆f(x) • Forward difference: ∆f(x) = f(x + h) − f(x) ⇒ − f 0(x) = O(h). h ∇f(x) • Backward difference: ∇f(x) = f(x) − f(x − h) ⇒ − f 0(x) = O(h). h f(x + h) − f(x − h) δf(x) • Central difference: δf(x) = ⇒ − f 0(x) = O(h2). 2 h

When the spacing h is small enough, the error of the central difference scheme decreases faster than the forward and backward difference schemes. Should we list the table of Stirling numbers [104], it is obvious that, given sufficient continuity of f(x), the central difference is superior to the other two since the forward and backward differences use only one side of the Stirling numbers in approximating f(x), while the central difference uses both sides. Second and third orders of derivative of f(x) by the central difference are

• Second derivative – d2f d df δ2f f(x + h) − 2f(x) + f(x − h) = ≈ = dx2 dx dx h2 h2 ⇒ δ2f = f(x + h) − 2f(x) + f(x − h)

• Third derivative – d3f d d2f δ3f f(x + 2h) − 2f(x + h) + 2f(x − h) − f(x − 2h) = ≈ = dx3 dx dx2 h3 2h3 f(x + 2h) − 2f(x + h) + 2f(x − h) − f(x − 2h) ⇒ δ3f = 2

Replacing x with t leads to the expression of discretization in time domain. As for two-dimensional functions, consider f = f(x, y) in a grid as shown in Figure 3-1 and its Taylor’s series expanding at node 0 in the x direction:

∂f 1 ∂2f 1 ∂3f 2 3 f(x, y) = f(x0) + (x − x0) + 2 (x − x0) + 3 (x − x0) + ... ∂x x0 2! ∂x x0 3! ∂x x0 (3.23)

Using a uniform spacing h in both x and y directions, ∆x = ∆y = h. Since x3 = x0−h

69 y 10

7 2 6

11 3 0 1 9

8 4 5

Figure 3-1: Conﬁguration grid in a two-dimensional domain

and x1 = x0 + h, we have

∂f ∂2f h2 ∂3f h3 f(x3, y3) = f3 = f0 − h + − + ... (3.24) ∂x 0 ∂x2 0 2 ∂x3 0 6 ∂f ∂2f h2 ∂3f h3 f(x1, y1) = f1 = f0 + h + + + ... (3.25) ∂x 0 ∂x2 0 2 ∂x3 0 6

When h is small, higher order terms become insigniﬁcant.

∂f ∂2f h2 f3 = f0 − h + (3.26) ∂x 0 ∂x2 0 2 ∂f ∂2f h2 f1 = f0 + h + (3.27) ∂x 0 ∂x2 0 2

Then we have

∂f f − f = 1 3 (3.28) ∂x 0 2h ∂2f f − 2f + f = 1 0 3 (3.29) ∂x2 0 h2 (3.30)

70 ∂f ∂2f Similarly, and can be obtained. ∂y 0 ∂y2 0

∂f f − f = 2 4 (3.31) ∂y 0 2h ∂2f f − 2f + f = 2 0 4 (3.32) ∂y2 0 h2

Replacing f(x, y) with E¯ and H¯ in the governing equations of TE and TM waves, the FDTD solution to Maxwell’s curl equations are reached.

As for three-dimensional problems, Yee’s algorithm (grid and notation) is widely used in most FDTD solutions and is adopted in this dissertation [277]. In Yee’s

∆x ∆y ∆z ∆t notation, unit spacings in space and time are chosen to be 2 , 2 , 2 , and 2 . We use dummy indices i, j, and k for denoting the spatial location of E¯ and H¯ and a dummy index n for the time variable in E¯ and H¯ . Note that i and j are not the imaginary number, nor is k the wave vector here. Following Yee’s notation, (xi, 1 1 ¯ 1 1 yj + ∆y, zk + ∆z, tn) is written as (i, j + 2 , k + 2 ,n), and the Ex at (i, j + 2 , k + 2 ,n) n is Ex 1 1 . i,j+ 2 ,k+ 2

Considering Eq.(3.17) as an example, its FDTD expression is

∂E 1 ∂H x = z + σE ∂t ∂y x n+ 1 n− 1 E 2 − E 2 x i,j+ 1 ,k+ 1 x i,j+ 1 ,k+ 1 ⇒ 2 2 2 2 = ∆t " n n # Hz i,j+1,k+ 1 − Hz i,j,k+ 1 1 2 2 n 1 1 + σi,j+ ,k+ Ex i,j+ 1 ,k+ 1 (3.33) 1 1 ∆y 2 2 2 2 i,j+ 2 ,k+ 2

n Linear interpolation (semi-implicit approximation) is used for Ex 1 1 . i,j+ 2 ,k+ 2

n+ 1 n− 1 E 2 + E 2 x i,j+ 1 ,k+ 1 x i,j+ 1 ,k+ 1 n 2 2 2 2 Ex i,j+ 1 ,k+ 1 = (3.34) 2 2 2

71 1 1 n+ 2 n− 2 Rearranging Eq.(3.33) with respect to Ex 1 1 and Ex 1 1 gives i,j+ 2 ,k+ 2 i,j+ 2 ,k+ 2

! σi,j+ 1 ,k+ 1 · ∆t n+ 1 1 + 2 2 E 2 = x i,j+ 1 ,k+ 1 2 1 1 2 2 i,j+ 2 ,k+ 2  n n  ! H − H σi,j+ 1 ,k+ 1 · ∆t n− 1 ∆t z i,j+1,k+ 1 z i,j,k+ 1 1 − 2 2 E 2 + 2 2 (3.35) x i,j+ 1 ,k+ 1   2 1 1 2 2 1 1 ∆y i,j+ 2 ,k+ 2 i,j+ 2 ,k+ 2

n+ 1 The explicit expression of a time-stepping E 2 is found after dividing both x i,j+ 1 ,k+ 1 ! 2 2 σi,j+ 1 ,k+ 1 · ∆t sides in Eq.(3.35) by 1 + 2 2 . Finally, the used FDTD expression of 2 1 1 i,j+ 2 ,k+ 2 1 n+ 2 Ex 1 1 is obtained. i,j+ 2 ,k+ 2

  σi,j+ 1 ,k+ 1 · ∆t 1 − 2 2 n+ 1  2i,j+ 1 ,k+ 1  n− 1 E 2 =  2 2  · E 2 + x i,j+ 1 ,k+ 1 x i,j+ 1 ,k+ 1 2 2  σi,j+ 1 ,k+ 1 · ∆t  2 2 1 + 2 2  2 1 1 i,j+ 2 ,k+ 2   ∆t  n n   i,j+ 1 ,k+ 1  Hz i,j+1,k+ 1 − Hz i,j,k+ 1  2 2  2 2   ·   (3.36)  σi,j+ 1 ,k+ 1 · ∆t  ∆y 1 + 2 2  2 1 1 i,j+ 2 ,k+ 2

Other electric and magnetic ﬁelds can be obtained following same procedure. Only the results, as used in this numerical simulation work, are presented here.

  σi− 1 ,j+1,k+ 1 · ∆t 1 − 2 2   n+ 1 2i− 1 ,j+1,k+ 1 n− 1 E 2 =  2 2  · E 2 y i− 1 ,j+1,k+ 1   y i− 1 ,j+1,k+ 1 2 2  σi− 1 ,j+1,k+ 1 · ∆t  2 2 1 + 2 2  2 1 1 i− 2 ,j+1,k+ 2   ∆t  n n   i− 1 ,j+1,k+ 1  Hz i,j+1,k+ 1 − Hz i−1,j+1,k+ 1  2 2  2 2 −   ·   (3.37)  σi− 1 ,j+1,k+ 1 · ∆t  ∆y 1 + 2 2  2 1 1 i− 2 ,j+1,k+ 2

72   σi− 1 ,j+ 1 ,k+1 · ∆t   1 − 2 2 ∆t   n+ 1 2i− 1 ,j+ 1 ,k+1 n− 1  i− 1 ,j+ 1 ,k+1  E 2 =  2 2  · E 2 −  2 2  · z i− 1 ,j+ 1 ,k+1   y i− 1 ,j+ 1 ,k+1 2 2  σi− 1 ,j+ 1 ,k+1 · ∆t  2 2  σi− 1 ,j+ 1 ,k+1 · ∆t  1 + 2 2  1 + 2 2  2 1 1 2 1 1 i− 2 ,j+ 2 ,k+1 i− 2 ,j+ 2 ,k+1  n n n n  Hy i,j+ 1 ,k+1 − Hy i−1,j+ 1 ,k+1 Hx i− 1 ,j+1,k+1 − Hz i− 1 ,j,k+1 2 2 − 2 2  ∆x ∆y  (3.38)

 n+ 1 n+ 1  ! E 2 − E 2 ∆t y i− 1 ,j+ 3 ,k+1 z i− 1 ,j+ 1 ,k+1 n+1 n  2 2 2 2  Hx i− 1 ,j+1,k+1 = Hx i− 1 ,j+1,k+1 + · 2 2 µ 1  ∆y  i− 2 ,j+1,k+1 (3.39)  n+ 1 n+ 1  ! E 2 − E 2 ∆t z i+ 1 ,j+ 1 ,k+1 z i− 1 ,j+ 1 ,k+1 n+1 n  2 2 2 2  Hy i,j+ 1 ,k+1 = Hy i,j+ 1 ,k+1 + · 2 2 µ 1  ∆x  i,j+ 2 ,k+1 (3.40) ! n+1 n ∆t Hz i,j+1,k+ 1 = Hz i,j+1,k+ 1 + · 2 2 µ 1 i,j+1,k+ 2  n+ 1 n+ 1 n+ 1 n+ 1  E 2 − E 2 E 2 − E 2 x i,j+ 3 ,k+ 1 x i,j+ 1 ,k+ 1 y i+ 1 ,j+1,k+ 1 y i− 1 ,j+1,k+ 1  2 2 2 2 − 2 2 2 2   ∆y ∆x 

(3.41)

The leapfrog scheme in Yee’s algorithm proves to be eﬃcient and reliable in the space-time simulation of EM waves.

3.3 Absorbing Boundary Condition – Perfectly Matched Layer

In the numerical simulation of EM wave propagation and scattering in free space, the use of reflectionless boundaries is necessary in order to simulate the infinite physical environment (open space) in a finite numerical environment (computational domain).

73 Such boundary conditions are termed absorbing boundary condition (ABC). ABC can be derived from either the integral form or the differential form of the reflectionless condition, leading to the global and local ABCs (Figure 3-2). The computational burden of global ABCs on storing and processing massive data makes local ABCs attractive and efficient in most applications. Types of local ABCs include (1) predictor-based ABC (radiation boundary condition): Merewether (1971) [115], (2) operator-based ABC [78] [186] [18] [111], and (3) material-based ABC (perfectly matched layer or PML) [24]. In this simulation work, PML is chosen for its demonstrated efficiency in various applications.

Maxwell’s equations

Assumed outgoing wave field solutions (E, H)

Reflectionless condition of EM fields at the artificial boundary

Integral form of the Differential form of the reflectionless condition reflectionless condition

Global ABCs Local ABCs

Figure 3-2: Derivation of global and local ABCs

3.4 Stability Criteria in Discretization

In the discretization process of continuous functions spatial and temporal intervals must be chosen with care to avoid numerical dispersion and divergence throughout the computation. To do so, stability criteria are applied for determining the spatial and temporal intervals. Considered stability criteria in the simulation work are introduced in this section.

74 3.4.1 Discretization in Space

Quantization criterion for the chosen spatial interval of sinusoidal functions (EM waves) can be evaluated in the following manner. Consider a one-dimensional E(x, t) as E(x, t) = sin (kx − ωx) (3.42) where k = the wave vector (rad/m), x = spatial variable (m), ω = radian frequency (rad/sec), and t = temporal variable (sec). Following Yee’s notation for discretization, we have n E(x, t) = E(i, n) = E i = sin (k · i∆x − ω · n∆t) (3.43)

The second-order space derivative of E(i, n) and its ﬁnite diﬀerence approximation are

∂2E = −k2 sin (kx − ωt) (3.44) ∂x2 2 ⇒ [cos (k∆x) − 1] · sin (k · i∆x − ω · n∆t) (3.45) ∆x2

Eq.(3.45) will be identical to Eq.(3.44) if

(k∆x)2 − = cos (k∆x) − 1 (3.46) 2 is valid. Should we deﬁne the approximation error as

2 cos (k∆x) − 1 + (k∆x) Er (k∆x) = 2 · 100% (3.47) cos (k∆x) a relationship between k∆x and Er (k∆x) can be obtained to evaluate the accuracy of discretization in space. Figure 3-3 shows the k∆x-Er (k∆x) curves. For example, to achieve Er (k∆x) ≤ 5%, the criterion for ∆x is

0.9 9c ∆x < = (3.48) k 20πf

75 where ∆x is in m/rad, c is the speed of light (m/s), and f is the temporal frequency

8 0.07

7 0.06 6 0.05 5 0.04

x), % 4 x), % ∆ ∆ 0.03

Er(k 3 Er(k 0.02 2

1 0.01

0 0 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 k∆x k∆x Figure 3-3: Quantization error Er (k∆x) vs. k∆x

(1/s or Hertz or Hz). Note that the factor 0.9 in Eq.(3.48) is determined in Figure 3-3. For EM waves of frequency f= 10 GHz, ∆x must be less than 0.043 m (1.69 in). The approximation error for E(x, t) will be 5% if ∆x= 0.043 m (1.69 in) at f= 10 GHz.

3.4.2 Discretization in Time

According to the sampling theorem which is deﬁned by the folding or Nyquist frequency, the necessary sampling rate in time, ∆t, is determined by

1 ∆t < (3.49) 2fmax

where fmax is the maximum frequency content (Hz) (Nyquist frequency) that is al- lowed to ensure the signal can be fully reconstructed at sampling rate ∆t [Hamming 1989]. For instance, given a maximum frequency f= 10 GHz, the sampling rate must be less than 5×10−11 sec. or 5×10−2 ns. Another criterion on the coupling between ∆t and spatial increments (∆x, ∆y, ∆z) is the Courant-Friedirchs-Levy (CFL) or the Courant stability criterion which is

76 deﬁned by [214] 1 ∆t < (3.50) q 1 1 1 c ∆x + ∆y + ∆z

For uniform discretization in three-dimensional problems we have ∆t < ∆√x , and c 3 ∆t < ∆√x for two-dimensional problems. Derivation of the CFL stability criterion c 2 can be found at [252]. For example, ∆t must be less than 3.538×10−3 ns. for ∆x= 0.0015 m. The quantization approach applied on ∆x can be used on ∆t. The second-order time derivative of E(x, t) and its ﬁnite diﬀerence approximation are

∂2E = −ω2 sin (kx − ωt) ∂t2 2 ⇒ [cos (ω∆t) − 1] · sin (k · i∆x − ω · n∆t) (3.51) ∆t2 which means 0.9 ∆t = (3.52) 2πf for an approximation error Er(ω∆t) ≤ 5% at f = 10 GHz. This gives a ∆t = 2.865×10−11 sec or 2.865×10−2 ns.

3.5 Two-Dimensional and Three-Dimensional Sim- ulations

In this research, two-dimensional and three-dimensional simulations of EM wave propagation and scattering are conducted using GprMax [132]. The code was developed based on the principles previously introduced in this chapter.

3.5.1 Validation of the Code

The code is validated by four examples whose theoretical solutions are available; (1) reﬂection response from a two-dimensional (2D) perfectly electric conductor (PEC) plate due to a Gaussian current source, (2) reﬂection response from the 2D PEC

77 plate due to a continuous/sinusoidal current source, (3) reflection response from a 2D dielectric plate due to a Gaussian current source, and (4) reflection response from the 2D dielectric plate due to a continuous current source. Geometrical configuration of the four examples is provided in Figure 3-4.

0 0 In Figure 3-4, a 0.3m-by-0.3m computational domain ﬁlled with air (r= 1, µr= 1) was ﬁrst built with uniform spatial discretization of ∆x = ∆y = 0.0025 m (0.098

0 in.). At the right boundary a target plate was inserted, which is made of PEC (r= 0 0 0 3000(>> 1), µr= (>> 1)) or dielectric(r= 5∼25, µr=1). Eight layers of PML ABC, whose total thickness is 0.02 m (0.79 in.), were deployed on four boundaries (top, bottom, left, and right) of the square domain to simulate the reflectionless effect in an infinite space. A current source (transmitter) was placed at the location (0.075 m, 0.15 m). A target plate made of PEC or dielectric was located at the right boundary of the domain, simulating an infinite surface (x ∈ [0.25m, 0.3m], y ∈ [0m, 0.3m]). The reflection response was collected at the receiver’s location (0.15 m, 0.15 m) between the current source and the target plate. The receiver was 0.1 m (3.94 in.) from the target plate to the right and 0.075 m (2.95 in.) from the source to the left. Total response collected at the receiver was comprised of the incident wave from the transmitter and the reflected wave from the target plate. Later, the net reflection response was extracted from the total response, considering the time-delay between the transmitter and the receiver, and compared with theoretical values. The wave ¯ vector was k =xk ˆ x +yk ˆ y, and the current source was in the z direction. This led to the use of TM waves (Hx, Hy, and Ez) in the examples. Time interval was determined by the CFL stability criterion to be 5.89×10−12 sec or 5.89×10−3 ns, and the total duration of simulation was 1×10−8 sec or 10 ns, leading to a time series response with 1,696 points collected at the receiver location. In these examples, the impulse current source is a Gaussian function of the following form.

" # 1 2 I(t, f) = exp −ζ t − (3.53) f where ζ = 2(πf)2. The continuous/sinusoidal current source is a sinusoidal function

78 y PML layer PEC or dielectric

free space 0.3 m

Hertzian current dipole 0.15 m 0.02 m

0.075 m x 0.15 m 0.25 m Transmitter Receiver 0.3 m

Figure 3-4: Geometry of a 2D model for validation of the following form. tf I(t, f) = sin (2πft) (3.54) 4 The theoretical response of EM waves reflected from an infinite interface can be evaluated by the amplitude reflection coefficient, r. √ 0 µ0 − 1 √ r r r = 0 0 (3.55) rµr − 1

0 For non-magnetic media, µr = 1. In the considered examples, the 2D target plate extends to the boundaries of the square domain, making it equivalently an infinite plate to any sources within the domain. The amplitude reflection coefficient of a PEC plate is √ √ 0 >> 1 − 1 0 √ r √ r r = 0 ∼ 0 = 1 (3.56) r >> 1 − 1 r

79 0 with a phase diﬀerence of π. As for dielectric plates with a r ranging from 5 to 25, r ranges from 0.38 to 0.67. Comparison of impulse response is performed at the peak value of the impulse and continuous responses. These results are provided in the following.

1. 2D PEC plate subjected to a Gaussian current source – The current source (solid line) and the net reflection (dashed line) are shown in Figure 3-5. In Figure 3-5, the difference in the peak amplitudes is around 0.06%, an insignificant difference. The amplitude reflection coefficient is 0.9968 by FDTD, and the theoretical value is 1. The time delay between the two peaks is 114∆t = 0.2017 m ∼ 0.2 m which is twice the distance between the receiver and the plate. In this example, the FDTD solution is in excellent agreement with the theoretical value.

1.5 1.2165 Source Reflection from PEC 1

0.5

0 (V/m) z E -0.5

-1 1.2172 -1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec) -8 215 t x 10 329 t

Figure 3-5: Gaussian current source and the reﬂection from PEC

2. 2D PEC plate subjected to a sinusoidal current source – Figure 3-6 shows the sinusoidal current source and the reﬂection response from the PEC plate. A constant time delay of 114∆t is observed between two curves, representing twice the traveling distance between the receiver and the plate. The

80 diﬀerence in the peak values of incident and reﬂected amplitudes is less than 0.1%, also suggesting an excellent agreement between the FDTD solution and theoretical prediction.

1.5

0.5

0 (V/m) z E -0.5

-1

-1.5 Source Reflection from PEC -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec) -8 x 10

Figure 3-6: Sinusoidal current source and the reﬂection from PEC

3. 2D dielectric plate subjected to a Gaussian current source – Figure 3-7 shows the impulse current source and the reﬂection responses from a 2D dielec-

0 tric plate with five values of the dielectric constant r of the plate; 5, 10, 15, 20, and 25. In Figure 3-7, the time delay representing twice the traveling distance between the receiver and the plate (0.2017 m) is almost identical to the actual value (0.2 m). Figure 3-8 shows the close-up of the reflection responses. It is found that the greater the dielectric constant the higher the reflection amplitude, as Eq.(3.55) predicts. As for the amplitude reflection coefficient, Figure 3-9 shows the theoretical curve and the FDTD solution. Error or relative difference is calculated and provided in Figure 3-10. It is found that the maximum error (∼2.4%) is less than 3%.

4. 2D dielectric plate subjected to a sinusoidal current source – Figure 3-11 shows the sinusoidal current source and the reﬂection response from a 2D

81 1

0.5

Source (V/m) z ε' =5 E -0.5 r ' =10 ε r ' =15 ε r -1 ' =20 ε r ' =25 ε r -1.5 0 1 2 3 4 5 -9 215 t Time (sec) x 10 329 t

Figure 3-7: Gaussian current source and the reﬂection from PEC

0 dielectric plate with a dielectric constant r= 25. The error is also less than 3%.

From the four examples provided in this section, the accuracy of the FDTD simulations is validated, both qualitatively and quantitatively. Hence, it is believed that the used FDTD code is reliable.

3.5.2 Actual Far-Field Simulation

An incident wave impinging on a finite specimen with boundaries modeled within the domain of consideration produces multiple reflections and transmissions. When analyzing such time domain response as a forward study, one can choose to investigate the content of the signal with respect to certain reflection/transmission by windowing the time series. In the considered problem, it is the first complete reflection (reflected from the backside of the specimen) that is concerned. While multiple reflections could be also advantageous in the analysis, it is not convenient to distinguish/determine the amplitude and phase differences attributed to presence of a local defect in the specimen from the total response. Therefore, the windowed time domain response is

82 0.8 ' =5 ε r 0.6 ' =10 ε r ε' =15 0.4 r ' =20 ε r ' =25 0.2 ε r (V/m) z 0 E

-0.2

-0.4

-0.6 1.5 2 2.5 3 Time (sec) -9 x 10

Figure 3-8: Gaussian current source and the reflection from PEC – Close-Up analyzed. The monostatic radar operation defined in the developed NDT technique and in laboratory radar measurements indicates a coincidental location for both the source (transmitter) and the receiver. As the result of it, difficulties including angular coupling between the source and the receiver and a required computational domain in order to contain the source and the receiver are avoided. In laboratory radar measurements the monostatic radar is placed in alignment with the center of cylinder specimens and perpendicular to the longitudinal axis of cylinder specimens. Such configuration is also followed in numerical simulation. While the option of an ideal equivalent Huygen’s surface (surface equivalence theorem) is not available in GprMax, the infinite range distance for perfect plane waves is not possible to be modeled in the simulation using GprMax. Rather, considering that in the proposed NDT technique the radar is configured beyond the far-field distance but located at infinite range from the structure, it is decided to place the radar in the far-field region (approximate plane waves) in the simulation. This way, the simulation will be honestly reproducing actual far-field measurements. Figure 3-12 shows the computational configuration. The size of the numerical

83 0.65

0.6

0.55

0.5

0.45

0.4 Reflection coefficient Reflection

0.35 Theoretical FDTD solution 0.3 5 10 15 20 25 ε' r

Figure 3-9: Theoretical curve and the FDTD solution of the amplitude reflection coefficient of a 2D dielectric plate domain is determined so as to satisfy the far-field distance. Snapshots of the incident fields are shown in Figure 3-13 using a 9 GHz frequency sinusoidal signal as example.

The purpose of performing two-dimensional (2D) simulations is twofold; first, 2D models are convenient to built and results are efficient to obtain. Secondly, the negligence of the third dimension in 2D models removes the influence from the third dimension in the simulation results and, thus, purifies the parametric study by considering only the parameters within the 2D plane. In other words, scattering/reflection responses, owing to the incidence of plane waves, from the third dimension are eliminated. In the simulation, the normal and oblique incidence schemes are both considered, where in the normal incidence scheme the concrete cylinder specimens with/without defect are defined as a circle and in the oblique incidence scheme they are described as a rectangle. All models are situated in free space (air), as the physical condition suggested in laboratory radar measurements. The defect is placed at the surface of the concrete core, which is characterized by its width, depth, and length. System parameters include incident frequency, incident angle, defect width, defect depth, defect length, and concrete permittivity. Frequency range is from 8 GHz to 18 GHz,

84 2.5

2.4

2.3

2.2

2.1

Relative difference (%) difference Relative 2

1.9 5 10 15 20 25 ε' r

Figure 3-10: Relative diﬀerence/error between the theoretical curve and the FDTD solution and the models are constructed in analogy to the GFRP-concrete specimens AD1 and AD2 whose dimensions are previously provided in the chapter on laboratory radar measurements.

In this section, time-domain responses in the neighborhood of the 2D models in the normal and oblique incidence schemes are produced to illustrate the scattering response due to the presence of a near-surface defect and rebar(s) in concrete (dielectric) and metallic (PEC) specimens.

3.6 Simulation Results

In the following cases, only scattered Ez is provided and discussed in order to save the space. The use of monostatic radar suggests that only the reﬂection response in the line-of-sight of radar can be perceived. In the following results, incident waves are introduced from the left boundary of the numerical domain, implying that the monostatic radar is located on the left boundary. Additionally, the line-of-sight of the radar intersects with the left boundary at Y = 0.2 m, the center of left boundary.

85 2

1.5

0.5

0 (V/m) z E -0.5

-1

-1.5 Source Reflection from dielectric -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec) -8 x 10

Figure 3-11: Sinusoidal current source and the reﬂection from a 2D dielectric plate

3.6.1 Damage Detection in Normal Incidence

In the numerical simulation of normal incidence responses for damage detection, the interest is to know whether the scattered signal due to the presence of a near-surface defect can be detected. Figure 3-14 shows the net reflection field of Ez from an intact, lossless concrete cylinder in the normal incidence scheme. It is noticed that, in Figure 3-14, a strong reflection (specular return) from the surface of the cylinder is observed. This specular return will present at all incident angles in the normal incident scheme due to the circular geometry of the cylinder in such scheme, suggesting a permanent influence of specular return on the normal incidence responses. Scattering signals due to the features (defect or rebar) of the cylinder will be mixed with the specular return in the observation of the radar. A low signal-to-noise ratio (SNR) can be expected.

3.6.2 Damage Detection in Oblique Incidence

In the oblique incidence scheme, it is of interest to know if the near-surface defects can be revealed by the reduction/removal of specular returns in such scheme. In this section, the FDTD simulation explains and demonstrates this behavior using two incident angles 0◦ and 30◦.

86 source receiver PEC or dielectric y free space 0.6 m 0.3 m

0.075 m x 0.15 m 5.7 m

6 m

Figure 3-12: Numerical domain for simulating actual far-ﬁeld radar measurements

The concrete cylinder in the oblique incidence scheme was modeled as a rectangle in 2D space, and the cubic defect AD1 was used. Four cases were considered at the incident angle of φ = 0◦; (1) intact, lossless concrete cylinder, (2) intact, lossy concrete cylinder, (3) damaged, lossless concrete cylinder, and (4) damaged, lossy concrete cylinder. Among them, cases (1) and (2) were used as the background signals for cases (3) and (4) for extracting the scattered ﬁelds due to defect. Figures 3-15

and 3-16 show the total ﬁelds of Ez of cases (1) and (2), respectively. The extracted scattered ﬁelds of Ez of cases (3) and (4) are provided in Figures 3-17 and 3-18. In

Figures 3-17 and 3-18, the scattered Ez due to defect was found in the line-of-sight direction of the monostatic radar. However, given the intensity of specular returns in

Figures 3-15 and 3-16, the existence of such scattered Ez is difficult to assert. Two cases were considered at the incident angle of φ = 30◦; (1) intact, lossless concrete cylinder and(2) damaged, lossless concrete cylinder. It is found that, in Figure 3-19, the specular return from the cylinder is no longer perceivable by the monostatic radar. Meanwhile, the scattered field becomes detectable by the radar as shown in Figure 3-20 since the background noise (specular return) has been removed. It is understood that the geometry of near-surface defects is related to the selection of incident angle even after the removal of specular returns. While greater incident angles (>30◦) may demonstrate satisfactory results, they pose a difficulty when they are applied in field applications; greater incident angles need shorter inspection dis-

87 1

(V/m) 0.5 z 0 -0.5 -1 Incident E Incident 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (ns)

2 (V/m) z 1

-1 -2

Total reflected E reflected Total 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (ns)

0.5 (V/m) z

-0.5

Net reflectedNet E 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time (ns)

Figure 3-13: Incident field, total field, and net reflection field of a 9 GHz signal tances than what smaller ones need. Therefore, it is of interest to explore the use of smaller incident angles. In the cases considered here, the angular range [0◦, 30◦] is to be explored.

3.6.3 Eﬀect of Defect Width in Normal Incidence

In studying the effect of defect width in normal incidence thirteen 2D models are constructed as shown in Figure 3-21. As one can observe in the pattern of variation among these models, the width (opening) of the artificial defect decreases from the benchmark defect (physically embedded in the specimen AD1) (1.5 in.) to the extreme case of 0.12 in. The purpose of this part of parametric study is to learn how the variation of defect width affects the reflection amplitude in a monostatic radar measurement scenario. In Figure 3-21, two frequencies, 8 GHz and 9 GHz, were considered. It is found that, in general, the use of higher frequencies leads to a greater

88 reflected electric field, while the measurement frequencies have their own region of defect width in which the reflected electric field is suddenly decreased.

3.6.4 Eﬀect of Defect Depth in Normal Incidence

For the study of defect depth another sixteen 2D models are prepared as shown in Figure 3-23. In this group of models it is the depth of the defect decreasing from 1 in. to 0.06 in. The purpose of this study is to learn how the variation of defect depth affects the reflected electric field. In Figure 3-24, the reflection coefficient is the ratio between the reflected electric field and the incident electric field. In Figure 3-24, two frequencies of signals were used; 8 GHz and 9 GHz. It is found that these two frequencies demonstrate different reflection patterns with respect to the variation of defect depth. In other words, detectability of near-surface defects of this kind depends on the use of incident frequency. Variation of the reflected electric field displays a sinusoidal pattern with respect to the changes of defect depth.

3.6.5 Rebar Detection in Normal Incidence

Three simulation cases were prepared; (1) plain concrete cylinder, (2) plain concrete cylinder with a center rebar, and (3) plain concrete cylinder with four rebars con- ﬁgured as the rebars in the specimen 4RE. In these cases, the concrete cylinder was modeled as a circle in 2D space, and the rebar was a small circle in the center of the cylinder. Case (1) was used as a background signal for extracting the scattering signal due to rebar(s). Dielectric constant of concrete was taken to be 5, and it is considered lossless (nonconductive)). Dimensions of the cylinders and rebars were same as the ones used in physical laboratory specimens when described in a 2D domain. The rebars were modeled as perfectly electric conductor. Simulation time was determined in order to incorporate the instants right before the impinging of incident waves.

The total fields (Hx, Hy, and Ez) of the time-domain response of case (2) are first provided as shown in Figures 3-25, 3-26, and 3-27. Scattered fields due to the presence of rebar(s) in cases (2) and (3) are extracted from the total fields using the case (1)

89 response as the background signal. Figures 3-28, 3-29, and 3-30 show the scattered fields of case (2). Figures 3-25, 3-26, and 3-27 demonstrate different patterns of wave scattering due to the presence of a point PEC inside a dielectric circle, as the FDTD simulation result suggested. The scattered fields reveal the scattering response of the point PEC. It is observed that the amplitude of scattered Ez (due to the rebar) in the line-of-sight direction of the radar is not significant. This confirms the finding on the laboratory measurements on specimens PC and CRE; difference between the reflection responses with and without a rebar embedded at a depth of 3 inches inside a concrete cylinder is not strong in the normal incident scheme, when compared with the amplitude of specular returns. Nonetheless, this difference leads to a greater reflection response of the concrete specimen with rebar, also to be consistent with the laboratory measurement results. When four rebars are present at different depths inside a concrete cylinder, it is found that the amplitude of scattered Ez (due to the rebars) in the line-of-sight direction of the radar is increased. This is attributed to the decrease of embedment depth (from 3 in. to 0.25 in.) and the presence of adjacent rebars in the concrete cylinder. Figure 3-31 shows the scattered Ez of case (3). It is noticed that strong reflections are present not only in the line-of-sight direction of the radar, but also in other directions. The snapshot images at time instants 21.4 ns, 21.7 ns, and 22 ns in Figure 3-31 clearly demonstrate this phenomenon. The strong reflections in other directions are caused by adjacent rebars next to the rebar on the line-of-sight direction. These reflections are the reason why the normal incidence scheme is capable of detecting rebars outside the line-of-sight direction.

90 iue31:Satrd(e eeto)fil of field reflection) (net Scattered 3-14: Figure Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E z z z z at time =18.7 ns at time =21.4 ns at time =20.5 ns at time =19.6 ns 0.1 0.1 0.1 0.1 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E z z z at time =19.9 ns z at time =21.7 ns at time =20.8 ns 0.1 0.1 0.1 0.1 at time 19 = ns X (m) 91 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 E z nat osescnrt cylinder concrete lossless Intact, –

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E z z z at time =20.2 ns at time =19.3 ns z at time =21.1 ns 0.1 0.1 0.1 0.1 at time =22 ns X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E z z z z at time =18.7 ns at time =21.4 ns at time =20.5 ns at time =19.6 ns 0.1 0.1 0.1 0.1 iue31:Ttlﬁl of ﬁeld Total 3-15: Figure X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E z z z at time =19.9 ns z at time =21.7 ns at time =20.8 ns 0.1 0.1 0.1 0.1 E at time 19 = ns z X (m) nat osescnrt cylinder concrete lossless Intact, – 92 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 E E E E z z z at time =20.2 ns at time =19.3 ns z at time =21.1 ns 0.1 0.1 0.1 0.1 at time =22 ns X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E z z z z at time =18.7 ns at time =21.4 ns at time =20.5 ns at time =19.6 ns 0.1 0.1 0.1 0.1 iue31:Ttlﬁl of ﬁeld Total 3-16: Figure X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E z z z at time =19.9 ns z at time =21.7 ns at time =20.8 ns 0.1 0.1 0.1 0.1 at time 19 = ns E z X (m) 93 nat os oceecylinder concrete lossy Intact, – 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) iue31:Satrdﬁl of ﬁeld Scattered 3-17: Figure 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E z z z z at time =18.7 ns at time =21.4 ns at time =20.5 ns at time =19.6 ns 0.1 0.1 0.1 0.1 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E z E z z z , at time =19.9 ns z at time =21.7 ns at time =20.8 ns 0.1 0.1 0.1 0.1 at time 19 = ns φ 0 = X (m) 94 0.2 0.2 0.2 0.2 ◦ aae,lsls oceecylinder concrete lossless Damaged, – 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 iue31:Satrdﬁl of ﬁeld Scattered 3-18: Figure E E E E z z z z at time =18.7 ns at time =21.4 ns at time =20.5 ns at time =19.6 ns 0.1 0.1 0.1 0.1 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E E z z z z at time =19.9 ns z at time =21.7 ns at time =20.8 ns 0.1 0.1 0.1 0.1 at time 19 = ns , φ X (m) 0 = 95 0.2 0.2 0.2 0.2 ◦ aae,lsycnrt cylinder concrete lossy Damaged, – 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 iue31:Satrdﬁl of ﬁeld Scattered 3-19: Figure E E E E z z z z at time =18.7 ns at time =21.4 ns at time =20.5 ns at time =19.6 ns 0.1 0.1 0.1 0.1 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E E z z z z at time =19.9 ns z at time =21.7 ns at time =20.8 ns 0.1 0.1 0.1 0.1 at time 19 = ns , φ X (m) 30 = 96 0.2 0.2 0.2 0.2 ◦ 0.3 0.3 0.3 0.3 nat osescnrt cylinder concrete lossless Intact, –

Y (m) Y (m) Y (m) Y (m) iue32:Satrdﬁl of ﬁeld Scattered 3-20: Figure 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E z z z z at time =18.7 ns at time =21.4 ns at time =20.5 ns at time =19.6 ns 0.1 0.1 0.1 0.1 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E z E z z z , at time =19.9 ns z at time =21.7 ns at time =20.8 ns φ 0.1 0.1 0.1 0.1 at time 19 = ns 30 = X (m) 97 0.2 0.2 0.2 0.2 ◦ aae,lsls oceecylinder concrete lossless Damaged, – 0.3 0.3 0.3 0.3

width

(1.42, 1) (1.30, 1) (1.18, 1)

depth

(width, depth) = (1.5 in., 1 in.) (1.06, 1) (0.94, 1) (0.83, 1)

(0.71, 1) (0.59, 1) (0.47, 1)

(0.35, 1) (0.24, 1) (0.12, 1)

Figure 3-21: Thirteen 2D models for studying the eﬀect of defect width in normal incidence

0.3

0.25

0.2

0.15 Reflection coefficient f = 8 GHz f = 9 GHz 0.1 0.017 0.018 0.019 0.02 0.021 0.022 0.023 0.024 0.025 Defect width (m)

Figure 3-22: Variation of reflected electric field with respect to different defect widths in normal incidence

98 (1.5, 0.94) (1.5, 0.86) (1.5, 0.83) (1.5, 0.77)

(1.5, 0.71) (1.5, 0.65) (1.5, 0.59) (1.5, 0.53)

(1.5, 0.47) (1.5, 0.41) (1.5, 0.35) (1.5, 0.3)

(1.5, 0.24) (1.5, 0. 18) (1.5, 0.12) (1.5, 0.06)

Figure 3-23: Sixteen 2D models for studying the eﬀect of defect depth in normal incidence

0.3

0.25

0.2

0.15 Reflection coefficient f = 8 GHz f = 9 GHz 0.1 0.005 0.01 0.015 0.02 0.025 Defect depth (m)

Figure 3-24: Variation of reflected electric field with respect to different defect depths in normal incidence

99 Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 iue32:Ttlﬁl of ﬁeld Total 3-25: Figure H H H H x x x x at time =18.7 ns at time =21.4 ns at time =20.5 ns at time =19.6 ns 0.1 0.1 0.1 0.1 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) H 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 x li oceeclne ihacne rebar center a with cylinder concrete Plain – H H H H x x x x at time =19.9 ns at time =21.7 ns at time =20.8 ns 0.1 0.1 0.1 0.1 at time 19 = ns 100 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 H H H H x x x x at time = 20.2 ns 20.2 = time at ns 19.3 = time at at time = 21.1 ns 21.1 = time at 0.1 0.1 0.1 0.1 at time =22 ns X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 iue32:Ttlﬁl of ﬁeld Total 3-26: Figure H H H H y y y y at time =21.4 ns at time =20.5 ns at time =19.6 ns at time =18.7 ns 0.1 0.1 0.1 0.1 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 H y H H H H li oceeclne ihacne rebar center a with cylinder concrete Plain – y y y y at time =19.9 ns at time =21.7 ns at time =20.8 ns 0.1 0.1 0.1 0.1 at time 19 = ns X (m) 101 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 H H H H y y y y at time = 20.2 ns 20.2 = time at ns 19.3 = time at at time = 21.1 ns 21.1 = time at 0.1 0.1 0.1 0.1 at time =22 ns X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 iue32:Ttlﬁl of ﬁeld Total 3-27: Figure E E E E z z z z at time =18.7 ns at time =21.4 ns at time =20.5 ns at time =19.6 ns 0.1 0.1 0.1 0.1 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E z E E E li oceeclne ihacne rebar center a with cylinder concrete Plain – E z z z at time =19.9 ns z at time =21.7 ns at time =20.8 ns 0.1 0.1 0.1 0.1 at time 19 = ns X (m) 102 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 iue32:Satrdﬁl of ﬁeld Scattered 3-28: Figure H H H H x x x x at time =21.4 ns at time =20.5 ns at time =19.6 ns at time =18.7 ns 0.1 0.1 0.1 0.1 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 H H H H H x x x x x at time =19.9 ns at time =21.7 ns at time =20.8 ns 0.1 0.1 0.1 0.1 at time 19 = ns li oceeclne ihacne rebar center a with cylinder concrete Plain – X (m) 103 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 iue32:Satrdﬁl of ﬁeld Scattered 3-29: Figure H H H H y y y y at time =21.4 ns at time =20.5 ns at time =19.6 ns at time =18.7 ns 0.1 0.1 0.1 0.1 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 H H H H H y y y y y at time =19.9 ns at time =21.7 ns at time =20.8 ns 0.1 0.1 0.1 0.1 at time 19 = ns li oceeclne ihacne rebar center a with cylinder concrete Plain – X (m) 104 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 iue33:Satrdﬁl of ﬁeld Scattered 3-30: Figure E E E E z z z z at time =18.7 ns at time =21.4 ns at time =20.5 ns at time =19.6 ns 0.1 0.1 0.1 0.1 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 E E E E E z z z z at time =19.9 ns z at time =21.7 ns at time =20.8 ns 0.1 0.1 0.1 0.1 li oceeclne ihacne rebar center a with cylinder concrete Plain – at time 19 = ns X (m) 105 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

Y (m) Y (m) Y (m) Y (m) 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.1 iue33:Satrdﬁl of ﬁeld Scattered 3-31: Figure E E E E z z z z at time =18.7 ns at time =21.4 ns at time =20.5 ns at time =19.6 ns 0.1 0.1 0.1 0.1 X (m) 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3

3.6.6 2D and 3D Responses

A 3D PEC cylinder model is shown in Figure 3-32, whose height is 12 in. and diameter is 6 in. A plane wave source is initiated by the introduction of Hyguen’s surface such that the longitudinal axis of the cylinder is parallel to the wavefront of the plane wave. The reflected electric field of this model is compared with the one from the 2D version of the model with identical dielectric and geometrical properties. Figure 3-33 shows the comparison between the 2D and 3D responses of the PEC cylinder model. It is found that the 3D response is weaker than the 2D response since the 3D model has less area for reflection. With more variables (e.g., dielectric properties, local defects) involved, the difference between 2D and 3D response can also be expected.

Figure 3-32: A 3D lossless dielectric cylinder model

3.7 Summary

Findings in this chapter are summarized and discussed in the following.

• In the FDTD simulation of normal incidence measurements, direct reﬂection (specular return) is dominant, leading to a low signal-to-noise ratio (SNR) (Fig- ure 3-14). Near-surface defects are unlikely to be detected in such measurement scheme.

107 2

1.5

0.5

0 (V/m) z E -0.5

-1 Source -1.5 2D FDTD -- PEC 3D FDTD -- PEC -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time (sec) -8 x 10

Figure 3-33: Comparison between the 2D and 3D responses of a PEC cylinder model

• In the FDTD simulation of oblique incidence measurements, specular returns are dramatically reduced (Figures 3-19 and 3-20). This suggests a high SNR for detecting the presence of near-surface defects in the far-ﬁeld region.

• When four rebars are present at diﬀerent depths inside a concrete cylinder, it

is found that the amplitude of scattered Ez (due to the rebars) in the line-of- sight direction of the radar is increased. This is attributed to the decrease of embedment depth (from 3 in. to 0.25 in.) and the presence of adjacent rebars in the concrete cylinder. Strong reﬂections are present not only in the line-of-sight direction of the radar, but also in other directions. These reﬂections are the reason why the normal incidence scheme is capable of detecting rebars outside the line-of-sight direction.

• 3D reflection response is different from 2D reflection response in a way that 3D reflection is weaker than 2D reflection. While 2D models offer certain computational advantages (less computational cost), 3D models are needed for better understanding the radar response (electromagnetic reflection) of the considered GFRP-concrete systems.

108 With these ﬁndings, laboratory radar measurements (Chapter 4) are planned to further investigate the system design parameters such as measurement scheme, measurement frequency, and the range of incident angles and to understand how the actual radar response of GFRP-concrete systems in a laboratory environment should be.

109 110 Chapter 4

Laboratory Radar Measurements

”Every eﬀect obviously has its cause, which can be retracted from cause to cause into the abyss of eternity; but every cause does not have its eﬀect to the end of time.” —– Philosophical Dictionary, Fran¸cis-Marie Arouet (Voltaire) (1694∼1778) ”A crucial experiment is one which gives a result consistent with one hypothesis and inconsistent with another.” —– Anonymous

This chapter reports the laboratory radar measurements collected on concrete and GFRP-retrofitted concrete cylinders with artificial features (defect and rebar). As found in Chapter 3 Numerical Simulation, measurement scheme, measurement frequency, and incident angle play a key role in the reflection response of GFRP- concrete cylinders. The purposes of conducting radar measurements on laboratory specimens are multiple; (1) a forward study can be performed using laboratory specimens whose defects are known. Radar measurements made on intact (without defect) and damaged (with defect) specimens can be evaluated based on their raw measurements and finally compared after being processed into images. (2) the laboratory configuration provides a noise-free environment for the reflection radar measurements of the specimens. The removal of background noise is advantageous for better distinguishing the signal due to the presence of defect at a higher signal-to-noise ratio

111 (SNR). Such measurements are insightful for damage detection and convenient to deal with in terms of the need of denoising; and (3) the radar measurement on a plain concrete cylinder can provide insights to the variation of dielectric properties of concrete and is advantageous for the dielectric modeling of concrete (Chapter 6). GFRP-retrofitted concrete cylinder specimens were manufactured in the Concrete Laboratory of the Department of Civil and Environmental Engineering at MIT. Phys- ical radar measurements were performed under the supervision of Dennis Blejer at the MIT Lincoln Laboratory (MIT LL). In this chapter, the design of experimental program, manufacturing of laboratory specimens, and experimental configuration and parameters are first described. Selected radar measurements are reported in this chapter for the discussion of damage detection and rebar detection.

4.1 Experimental Program

An experimental program aiming at investigating the eﬀect of unseen near-surface anomaly in the far-ﬁeld monostatic ISAR measurement of GFRP-wrapped concrete specimens is prepared. In this work, there are three types of anomaly that are targeted for detection.

1. GFRP Delamination and Debonding – GFRP delamination occurs when wrapped multi-layer GFRP sheets deteriorate due to the application of service loads and/or environmental effects. GFRP debonding happens either during construction (trapped air voids between GFRP sheets and irregular surface of concrete during the wrapping of GFRP sheets), or due to the application of service loads and/or environmental effects. This type of anomaly (Artificial Defect, AD) is simulated by the introduction of Styrofoam pieces between GFRP sheets and the surface of concrete.

2. Reinforcement – Detection of the location of reinforcement is important to the rehabilitation and repair of corroded R/C structures. Steel rebars are embedded in GFRP-wrapped concrete specimens to simulate the detection of re-

112 inforcement in such structure.

Table 4.1 lists all laboratory specimens in the experimental program in this research. All specimens are casted in cylinder forms.

Table 4.1: Designed specimens

Spec. Rad. H Freq. Azi. Scheme Description Figure (in) (in) band (deg.) CON 3 12 X, Ku ±30◦ Normal Concrete cylinder 4-1 CRE 3 12 X, Ku ±30◦ Normal Concrete cylinder 4-2 with a rebar in the center GFRP 3 12 X ±30◦ Normal Concrete cylinder 4-1 wrapped with GFRP 4RE 3 12 X ±15◦ Normal Concrete cylinder 4-2 with four rebars AD1B 3 12 X, Ku ±30◦ Normal, GFRP-con. cylinder 4-3 Oblique with AD1, intact side AD1F 3 12 X, Ku ±30◦ Normal, GFRP-con. cylinder 4-3 Oblique with AD1, damaged side AD2 3 15 X ±45◦ Normal, GFRP-con. cylinder 4-3 Oblique with AD2, damaged side AD3 3 15 X ±45◦ Normal GFRP-con. cylinder 4-4 with AD3, damaged side

It is noteworthy to point out that scaled GFRP-wrapped concrete specimens diﬀer from GFRP-wrapped concrete columns in the following aspects:

1. Size (radius and height) – The radius of the GFRP-concrete specimens is approximately 3 inches/in. (or 7.62 centimeters/cm), and the specimens are designed in two heights; 12 in. and 15 in. The radius and height of actual concrete columns vary in a wide range of values, depending on the level of

113 12 in 12 in (30.4 cm) (30.4 cm)

6.2 in 6 in 6 in (15.2 cm) (15.2 cm)

CON GFRP

Figure 4-1: Specimens CON and GFRP

12 in 15 in (30.4 cm) (38.1 cm)

1.5 in 6 in −135° (15.2 cm) (3.8 cm) 0.1 in (0.25 cm) −45° −135°

0.5 in (1.27 cm) 1 in 45° (2.5 cm)

CRE 4RE

Figure 4-2: Specimens CRE and 4RE

114 1.5 in (3.8 cm) 3 in 12 in (7.6 cm) (30.4 cm) 15 in 1 in (38.1 cm) (2.5 cm) 0.2 in (0.5 cm) 6.2 in 6 in (15.2 cm)

1.5 in 3 in (3.8 cm) (7.6 cm)

AD1 AD2

Figure 4-3: Specimens AD1 and AD2

15 in (38.1 cm)

6.2 in 6 in (15.2 cm)

1 in 1 in (2.5 cm) (2.5 cm) AD3

Figure 4-4: Specimen AD3

115 needed bearing capacity. In any case, they are much greater than the values of the specimens used in this study.

2. Number of GFRP sheets/layers – Only one sheet/layer of GFRP composite is used in this work. In practice, the use of more than one layer of GFRP composite is usual.

3. Reinforcement – Steel reinforcement is considered separate from the artificial defect in this study in order to distinguish their contributions in the far-field monostatic ISAR measurements. Such design allows us to simplify the problem by investigating their individual effects.

4.2 Manufacturing of the Specimens

The concrete and GFRP-wrapped concrete specimens were manufactured in the Con- crete Laboratory of the Department of Civil and Environmental Engineering at MIT. In this section, descriptions of the materials used in the experimental work are provided. Material suppliers are also indicated. Materials used in the manufacturing of GFRP-wrapped concrete specimens and their suppliers are listed in Table 4.2. Stan- dards by ASTM (American Society for Testing and Materials) and material suppliers for the quality control of the materials used in the manufacturing of concrete and GFRP-concrete specimens are indicated in Table 4.3. Equipments used in the manufacturing of concrete specimens and GFRP-epoxy sheets include a concrete mixer for rotationally blending material components, an iron plate for casting concrete into cylinder forms, a plastic rectangular plate for molding E-glass fabric with epoxy, and other auxiliary accessories such as handheld shovels. The manufacturing of GFRP- wrapped concrete specimens involves the following steps.

1. Calculation of the needed fractions of each material component in concrete (water, cement, ﬁne aggregate, and coarse aggregate) and GFRP sheets (E- glass fabric and epoxy resin)

116 Table 4.2: Used materials and their suppliers

Material Brand Supplier

Cement Dragon r Portland Cement Dragon Products Company, Type I Thomaston, Maine

Fine Walbro r Mortar Sand B Vitalini Inc., Milford, aggregate Massachusetts

Coarse Walbro r PEA Gravel B Vitalini Inc., Milford, aggregate Massachusetts Water Batch water M.I.T.

Styrofoam Foamular 250 r Extruded Owens Corning Corp., Polystyrene Insulation Toledo, Ohio

E-glass Tyfo r SEH-51 Composite Fyfe Co. LLC, San Diego, fabric California

Epoxy Tyfo r S Epoxy Fyfe Co. LLC, San Diego, resin California

Table 4.3: Manufacturing standards

Material Standard General ASTM C125-03 Standard Terminology Relating to Concrete and Concrete Aggregates Aggregates ASTM C33-03 Standard Specification for Concrete Aggregates ASTM C136-04 Standard Test Method for Sieve Analysis of Fine and Coarse Aggregates ASTM C127-04 Standard Test Method for Density, Rela- tive Density (Specific Gravity), and Absorption of Coarse Aggregate Water ASTM C1602/C1602M-05 Standard Specification for Mixing Water Used in the Production of Hydraulic Cement Concrete Concrete ASTM C192/C192M-05 Standard Practice for Making and Curing Concrete Test Specimens in the Laboratory

GFRP sheet Fyfe 01-05 Quality Control Manual for the Tyfo r FIBR- WRAP System

117 2. Preparation of the needed amount (in weight) of materials for each batch in the mixing of concrete and GFRP

3. Preparation of the equipments and concrete cylinder forms (two types of forms are used; paperboard and iron)

4. For concrete specimens with anomaly:

• Attaching artiﬁcial defects onto the inside surface of the form at a prescribed location for artiﬁcially damaged concrete specimens

• Attaching and ﬁxing rebar(s) in the prescribed location(s) inside the cylinder form for concrete specimens with rebar(s)

5. Brushing the inside surface of concrete cylinder forms with lubricant (oil) in order to ensure a clean separation between the form and the concrete

6. Mixing concrete patch, casting and consolidating concrete cylinder specimens

7. Removal of the forms of concrete specimens after one day

8. Curing concrete cylinder specimens in water tank for 28 days

9. Mixing E-glass fabrics with epoxy resin in a rectangular plate to form GFRP sheets

10. Brushing the surface of concrete specimens using epoxy resin

11. Wrapping one layer of GFRP sheet onto the concrete specimen

12. Curing the GFRP sheet in room temperature for one day

4.3 Experimental Conﬁguration and Parameters

Laboratory radar measurements were performed in the MIT Lincoln Laboratory using the compact RCS (Radar Cross Section)/antenna range facility. The experimental set-up consists of a horn antenna, stepped-frequency radar, network analyzer systems,

118 and a Harris dual-shaped reflection system (Model 1606) designed for conducting far- field measurements which are made at a range beyond the far-field distance. The far-field distance, dff , is generally defined by [243, 137]

2D2 d = (4.1) ff λ where D is the maximum dimension of radar antenna apertures, and λ is the minimum wavelength of radar signals. The reflection system is used for transforming cylindrical/spherical waves (by the radar antenna) into plane waves in the limited space of laboratory, which is not needed in field measurements where the far-field distance requirement is typically satisfied. Design of reflection systems can be found at [16]. The picture and schematic of the facility are provided in Figures 4-5 and 4-6. The compact RCS/antenna range facility can achieve a 6-ft (20-meter) quite

Figure 4-5: Compact RCS antenna range facility in the MIT LL [Courtesy of the MIT LL]

119 zone with the installation of Chebyshev absorbers (shown in Figure 4-5) covering the ﬂoors, walls, and the support tower of the laboratory. The facility can make fully polarimetric RCS measurements and achieve high signal-to-noise ratio (SNR) for a large frequency bandwidth ranging from 0.7 GHz (Ultra High Frequency, UHF) to 100 GHz (Extremely High Frequency, EHF). As found in Chapter 3 Numerical Simulation, a time-harmonic/continuous wave (CW) radar signal at 8 GHz frequency whose wavelength is 0.0375 m (1.48 in) can produce scattering signals due to the presence of a near-surface cubic-like defect in a dielectric medium. Since the use of higher frequency CW signals can better detect defects than lower frequency CW signals, CW radar signals in the frequency range of 8 GHz to 18 GHz (8GHz 12GHz: X-band, and 12GHz 18GHz: Ku-band) at an interval of 0.02 GHz were used in the laboratory radar measurements. Azimuth angle ranging from −30◦ to 30◦ with 0◦ corresponding to perpendicular incidence (θ = 90◦ were explored at an angular interval of 0.2◦. Linearly polarized radar signals were transmitted and received (HH- and VV-polarized). The time needed for collecting the steady-state response per frequency per angle was 0.14 second [254]. As also found in Chapter 3 Numerical Simulation, two measurement schemes were considered in this research; normal incidence scheme and oblique incidence scheme, which are to be further explained in the following sections.

4.3.1 Monostatic ISAR Normal Incidence Scheme

Deﬁned monostatic ISAR normal incidence scheme is illustrated in Figure 4-7. In Figure 4-7, the synthetic aperture plane is always perpendicular to the longitudinal axis of the cylinder. For cylinders of perfect geometry the measurements should be independent of azimuth angle, suggesting a theoretically circumferential symmetry of ISAR normal incidence measurements. Additionally, the monostatic ISAR normal incidence measurements capture the specular returns from the cylinder at all azimuth/incident angles since the radar is aligned with the center of the cylinder.

◦ ◦ π 2π The default range of azimuth angle in the measurements is θ ∈ [60 , 120 ][ 3 , 3 ], or ◦ ◦ equivalently θn ∈ [−30 , 30 ] where θn is normalized to the center of the aperture.

120 4.3.2 Monostatic ISAR Oblique Incidence Scheme

Deﬁned monostatic ISAR oblique incidence scheme is sketched in Figure 4-8. In Fig- ure 4-8, the longitudinal axis of the cylinder is on the synthetic aperture plane. For cylinders of perfect geometry the measurements are not circumferentially symmetrical. Unlike in the normal incidence scheme, the monostatic ISAR oblique incidence measurements capture the specular returns from the cylinder only when φ equals

◦ π 90 ( 2 ) which is the case of normal/perpendicular incidence. The diﬀerence in the pattern of specular returns is believed to favor the revealing of scattering signals due to defects. The default range of azimuth angle in the measurements is φ ∈ [60◦, 120◦]

◦ ◦ ◦ ◦ , which is equivalent to a range of [−30 , 30 ], or equivalently φn ∈ [−30 , 30 ] where ◦ φn is normalized to the angle at perpendicular incidence (φ = 90 ).

4.4 Calibration of Laboratory Radar Measurements

Laboratory radar measurements were calibrated using a perfect electric conductor (PEC) tube specimen made of Aluminum and a lossy dielectric rod specimen made of plexiglass in the normal incidence scheme. Two approaches applied in this calibration are described as follows.

1. Second-order statistics – In the normal incidence scheme, the circumferential symmetry in the measurements can be used to evaluate the measurement quality and the geometrical perfectness of a cylinder specimen. For this purpose,

second-order statistics (the use of mean µ, standard deviation σs, and correlation coeﬃcient ρ) is applied. In the frequency-angle measurement A(f,¯ θ¯) of a specimen featuring angular symmetry, its statistical properties are:

• The mean amplitude at one angle over all frequencies should be identical or close to the mean amplitude at another angle over all frequencies.

¯ ¯ µ[A(f, θi)] = µ[A(f, θj)] (4.2)

where A denotes the amplitude response.

121 • The standard deviation of amplitude and phase responses at one frequency over all angles equals or close to zero.

¯ σs[A(fi, θ)] = 0 (4.3) ¯ σs[Φ(fi, θ)] = 0 (4.4)

where Φ denotes the phase response.

• The correlation coeﬃcient of amplitude and phase responses evaluated between any two angles over all frequencies equals or close to unity.

ρ[A(¯ω, θi),A(¯ω, θj)] = 1 (4.5) ¯ ¯ ρ[Φ(f, θi),A(f, θj)] = 1 (4.6)

These parameters quantitatively evaluate the circumferential/angular symmetry in the frequency-angle measurements of cylinder specimens in the normal incidence scheme.

2. Optical backscattering analogy – The far-ﬁeld radar reﬂection response of a dielectric circular/spherical scatterer can be analogically simulated by its optical model. The optical backscattering model by Inada (1974) [118] and Stephens et al. (1975) [124] is used in this analogy. This calibration was conducted by Dennis Blejer at the MIT Lincoln Laboratory, and the result is provided here at his courtesy.

4.4.1 PEC Specimen

An Aluminum tube specimen of 12 in. (15.24 cm) height, 3 in. (7.62 cm) radius , and 0.25 in. (0.635 cm) thickness used as an example of PEC was prepared by the laboratory staﬀ in the MIT Lincoln Laboratory for the purpose of calibrating the compact RCS/antenna range facility. Figure 4-9 (a) shows a picture of the tube specimen. The scattering signals showing in Figure 4-9 (b) indicates a strong specular return

122 from the PEC specimen. The far-field monostatic ISAR normal incidence responses (amplitude) of this PEC specimen were collected in X- and Ku-band for both HH (TE) and VV (TM) polarizations, as shown in Figures 4-10 and 4-11. Figure 4-12 (a) and (b) show the mean and standard deviation of amplitudes at X-band. Figure 4-13 (a) and (b) show the mean and standard deviation of phases at X-band. Figure 4-14 (a) and (b) show the mean and standard deviation of amplitudes at Ku-band. Figure 4-15 (a) and (b) show the mean and standard deviation of phases at Ku-band. Correlation coefficients of the radar measurements in X- and Ku-band are computed and shown in Figures 4-16 and 4-17. Table 4.4 lists the statistical parameters of the far-field ISAR normal incidence frequency-angle amplitude measurements of the ¯ PEC tube. It is found that (1) the fluctuation of µ[A(f, θi)] indicates the changes of ¯ ¯ the surface properties of the PEC tube; (2) both σs[A(fi, θ] and σs[Φ(fi, θ] are close to zero; and (3) both ρ[A(¯ω, θi),A(¯ω, θj)] and ρ[Φ(¯ω, θi),A(¯ω, θj)] are close to unity. It is found that, although a PEC specimen was used, there exists some differences

Table 4.4: Statistical parameters of the frequency-angle amplitude measurement of the Aluminum tube (PEC)

Parameter Theo. Measured value value Mean — X-band: −2.48 ∼ −1.8(HH) / −2.55 ∼ −1.9(VV) ¯ µ[A(f, θi)] Ku-band: −1.85 ∼ −0.75 (HH) / -2.18∼-1 (VV) Std. dev. 0 X-band: 0.124 ∼ 0.277(HH) / 0.052 ∼ 0.396(VV) ¯ σs[A(fi, θ)] Ku-band: 0.239 ∼ 0.549(HH) / 0.156 ∼ 0.658(VV) Corr. coef. 1 X-band: 0.99838(HH) / 0.99497(VV) ¯ ¯ ρ[A(f, θi),A(f, θj)] Ku-band: 0.98606(HH) / 0.96593(VV)

between theoretical values and measured values in Table 4.4. The diﬀerences are attributed to the following factors:

• Geometrical imperfectness of the specimen – The manufactured tube specimen may not be a perfect cylinder in shape, owing to the limited precision of manufacturing. Local surface roughness of the tube specimen may also contribute to the reﬂection response in steady-state.

123 • Alignment error – The center of the cylinder specimen is supposed to coincide with the center of the Styrofoam tower in order to satisfy the condition of normal incidence scheme in Figure 4-7. Should this alignment condition of centers is not met, the specular return from the cylinder will be reduced.

• Background noise – Measurements made in the steady-state could amplify the background noise from the imperfect reﬂectionless boundary condition. While this type of error cannot be completely eliminated, its contribution is considered to be secondary when compared with other factors.

4.4.2 Lossy Dielectric Specimen and Its Optical Model

A lossy dielectric cylinder specimen made of plexiglass of 7.335 in. (18.63 cm) height and 1 in. (2.54 cm) radius, as shown in Figure 4-18 (a), was prepared by the MIT LL staff for calibration purpose. The laser scattering picture shows not only the specular return, but also the back reflection and surface wave from the lossy dielectric specimen in Figure 4-18 (b). The plexiglass rod were manufactured with fibers ori- ented in parallel to the axis of the rod. The measured Ku-band amplitude response is shown in Figure 4-19. In this measurement set, heterogeneity of the lossy dielectric specimen is due to the nonuniform distribution of fibers within the specimen, resulting in the fluctuating reflection response shown in Figure 4-19 The second-order statistical parameters of the amplitude response are listed in Table 4.5 in which the ¯ circumferential heterogeneity of the specimen is suggested by an increased σs[A(fi, θ)]. An optical model for estimating the backscattering waves and the surface creeping waves of a lossy dielectric cylinder was developed by D. Blejer (MIT LL) and used for the far-field ISAR normal incidence measurements of the plexiglass rod [118, 124]. This model produced the simulated far-field response of the plexiglass rod with an dielectric constant (r=2.6) and an electrical conductivity (σ=0.03 mho/m) for the plexiglass rod. Both simulated and measured (Figure 4-19) responses were processed by a backprojection algorithm (to be further explained in Chapter 5 Image Re- construction) to render the range-crossrange imagery as shown in Figure 4-20. In

124 Reflector 2 GFRP-concrete specimen Plane wave

Network Reflector 1 analyzer Stepped-freq. radar and horn Styrofoam antenna tower

Figure 4-6: Schematic of the compact RCS/antenna range facility in the MIT LL

Table 4.5: Statistical parameters of the frequency-angle response of the plexiglass rod specimen

Parameter Theo. Measured value value ¯ Mean µ[A(f, θi)] — Ku-band: −8.5 ∼ −4.7 (HH) / −6.8 ∼ −3.25 (VV) Std. dev. 0 Ku-band: 0.683 ∼ 1.808(HH) / 0.609 ∼ 1.64(VV) ¯ σs[A(fi, θ)] Corr. coef. 1 Ku-band: 0.98937(HH) / 0.98204(VV) ¯ ¯ ρ[A(f, θi),A(f, θj)]

125 GFRP-concrete Horn antenna specimen ξ θ = 60°∼120°

θint/2 GFRP-concrete Styrofoam r specimen tower

Top view Front view

Figure 4-7: Far-ﬁeld monostatic ISAR normal incidence scheme. Note that the angular zero is referred to a selected point on the cylinder.

Figure 4.6, the images produced by measured radar response using HH- and VV- polarized signals approximate the ones by simulated far-ﬁeld optical response. This indicates a good agreement between theory (optics) and measurement (the compact RCS/antenna range facility). Table 4.6 also lists the peak RCS of simulated and measured responses. It is found that the measured response demonstrates satisfactory approximation to the theoretical values predicted by an optical model, both qualitatively (Figure 4-20) and quantitatively (Table 4.6). This example, along with the PEC specimen, validates the reliability of laboratory radar measurements used in this dissertation.

Table 4.6: Peak RCS of simulated and measured responses of the plexiglass rod specimen

Polarization Simulated RCS (dBsm) Measured RCS (dBsm) HH -11.7 -11.2 VV -7.2 -7.1

126 Horn antenna ξ φ = 60°∼120° GFRP-concrete specimen

θint/2 Styrofoam r GFRP-concrete tower specimen

Top view Front view

Figure 4-8: Far-ﬁeld monostatic ISAR oblique incidence scheme. Note that the angular zero is in alignment with the axis of the cylinder. 4.5 Frequency-Angle Measurements

Selected laboratory radar measurements of the GFRP-concrete cylinder specimens are reported in this section. Only the amplitude response is provided. The measured reﬂection responses of GFRP-concrete cylinders are the combined result of the following factors:

• Variation in the dielectric properties of the cylinder – The magnitude of reflected radar signals is a function of the dielectric properties of the medium (cylinder). Variation in the dielectric properties in different parts of the cylinder can result in the change of reflected radar signals.

• Variation in the surface roughness of the cylinder – Theoretically, the incident angle should be the diﬀerential angle between the incident wave vector and the normal vector perpendicular to the surface of the cylinder on the line- of-sight of the radar. Should there be any change in the surface geometry (roughness or smoothness) of the cylinder, such condition is not preserved, leading to the change of reﬂected radar signals.

• Magnitude of misalignment – In the normal incidence scheme, the incident wave vector is supposed to be perpendicular to the normal vector of the surface

127

Specular

(a) Picture of the Aluminum tube (b) Light scattering from the Aluminum tube

fname:Figure CYL03 4-9:, HH PEC Pol., specimenmax = 4.77 –dBsm Aluminum tube [Courtesy of the MIT Lincoln Labora- 12 tory] 0 12 0

11.5 11.5 -2 -2 11 11

10.5 10.5 -4 -4

10 10

-6 -6 9.5 9.5 Frequency (GHz) Frequency Frequency (GHz) Frequency 9 9 -8 -8 8.5 8.5

8 -10 8 -10 60 70 80 90 100 110 120 60 70 80 90 100 110 120 θφ ((deg)deg) θ φ( deg(deg))

(a) HH (TE) (b) VV (TM)

Figure 4-10: Frequency-angle response of the Aluminum tube – Amplitude (dBsm), X-band

128 fname: CYL03K, HH Pol., max = 5.92 dBsm fname: CYL03K, VV Pol., max 5.76 dBsm 18 0 18 0

17 17 -2 -2

16 16 -4 -4

15 15

-6 -6 14 14 Frequency (GHz) Frequency Frequency (GHz) Frequency

-8 -8 13 13

12 -10 12 -10 60 70 80 90 100 110 120 60 70 80 90 100 110 120 θφ ((deg)deg) θφ ((deg)deg)

(a) HH (TE) (b) VV (TM)

Figure 4-11: Frequency-angle response of the Aluminum tube – Amplitude (dBsm), Ku-band

-0.5 HH (TE) HH (TE) VV (TM) 0.35 VV (TM) -1

σ 0.3 -1.5

0.25 -2 (dBsm) µ

-2.5 0.2 Mean, Mean,

-3 Standard deviation, 0.15

-3.5 0.1

8 9 10 11 12 8 9 10 11 12 Frequency (GHz) Frequency (GHz) (a) µ[A] (b) σ [A] s Figure 4-12: Mean amplitudes and their standard deviations of the reﬂection response at diﬀerent frequencies – Aluminum tube, X-band

129 HH (TE) -1.9 0.96 VV (TM) 0.95 -2 σ 0.94 -2.1 0.93 (dBsm)

µ -2.2 0.92 -2.3 Mean, 0.91 Standard deviation, -2.4 0.9 HH (TE) -2.5 0.89 VV (TM) -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 θ (deg) θ (deg) (a) µ[A] (b) σ [A] s Figure 4-13: Mean amplitudes and their standard deviations of the reﬂection response at diﬀerent incident angles – Aluminum tube, X-band

0.65 HH (TE) VV (TM) 0.6 -1 0.55

σ 0.5

-1.5 0.45 (dBsm)

µ 0.4

0.35 -2 Mean, Mean, 0.3 Standard deviation, 0.25 -2.5 HH (TE) 0.2 VV (TM)

12 13 14 15 16 17 18 12 13 14 15 16 17 18 Frequency (GHz) Frequency (GHz) (a) µ[A] (b) σ [A] s Figure 4-14: Mean amplitudes and their standard deviations of the reﬂection response at diﬀerent frequencies – Aluminum tube, Ku-band

130 -0.8 HH (TE) VV (TM) 0.55 -1

σ -1.2 0.5

-1.4 (dBsm) µ 0.45 -1.6 Mean, Mean,

-1.8 Standard deviation, 0.4

-2 HH (TE) VV (TM) 0.35 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 θ (deg) θ (deg) (a) µ[A] (b) σ [A] s Figure 4-15: Mean amplitudes and their standard deviations of the reﬂection response at diﬀerent incident angles – Aluminum tube, Ku-band

(a) ρ[A] – HH (b) ρ[A] – VV Figure 4-16: Correlation coefficients of the reflection response at different frequencies – Aluminum tube, X-band

131

(a) ρ[A] – HH (b) ρ[A] – VV Figure 4-17: Correlation coefficients of the reflection response at different frequencies – Aluminum tube, Ku-band

Specular Back reflection

7.335 in Surface wave

2 in

(a) Schematic of the plexiglass rod (b) Light scattering from the plexiglass rod [Courtesy of the MIT LL]

Figure 4-18: Lossy dielectric specimen – Plexiglass rod

132 fname: CYL08K, HH Pol., max 4.33 dBsm fname: CYL08K, VV Pol., max 1.81 dBsm 18 0 18 0

17 17 -2 -2

16 16 -4 -4

15 15

-6 -6 14 14 Frequency (GHz) Frequency (GHz) Frequency

-8 -8 13 13

12 -10 12 -10 60 70 80 90 100 110 120 60 70 80 90 100 110 120 θφ ((deg)deg) θ φ( deg(deg))

(a) HH (TE) (b) VV (TM) Figure 4-19: Frequency-angle response of the plexiglass specimen – Amplitude (dBsm), Ku-band

of the cylinder. This condition will be violated if the center of the cylinder does not coincide with the center of the measurement tower in laboratory radar measurements. Misalignment will be further explained later in this chapter.

Analysis and discussions are provided on the topics of damage detection and rebar detection in both normal and oblique incidence schemes.

4.5.1 Monostatic ISAR Normal Incidence Scheme

Amplitude responses of the specimens measured in the far-field monostatic ISAR normal incidence scheme are provided in this section, including the measurements of specimens CON, GFRP, CRE, 4RE, AD1B, AD1F, AD2, and AD3. Table 4.7 describes the signal content of each specimen, as well as the use of each specimen in the discussion. In Table 4.7, DD denotes ”Damage Detection” and RD denotes ”Rebar Detection”. In what follows, effects of the presence of GFRP layer and the heterogeneity of concrete in the far-field monostatic ISAR normal incidence measurements (or normal incidence measurements) are discussed. The use of normal incidence measurements

133 Simulated: ε = 2.6, σ = 0.03 = 22.9183 , [f ,f ] = [12,18] GHz, lo rhi Ψ Measuredlo hi 1 1

0.8 -10 0.8 -10

0.6 -20 0.6 -20

0.4 -30 0.4 -30

0.2 -40 0.2 -40

0 -50 0 -50 meters meters -0.2 -60 -60

Range (m) Range (m) Range -0.2

-70 -0.4 -0.4 -70

-80 -0.6 -0.6 -80

-90 -90 -0.8 -0.8

-1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 meters meters Cross-range (m) Cross-range (m)

Simulated: εr = 2.6, σ = 0.03 Measured 1 1

0.8 -10 0.8 -10

0.6 -20 0.6 -20

0.4 -30 0.4 -30

0.2 -40 0.2 -40

0 -50 0 -50 meters meters -0.2 -60 -0.2 -60 Range (m) Range Range (m) Range -70 -0.4 -70 -0.4

-80 -0.6 -80 -0.6

-90 -90 -0.8 -0.8

-1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 meters meters Cross-range (m) Cross-range (m)

Figure 4-20: Simulated and measured responses — Plexiglass rod, HH (upper level) and VV (lower level) polarizations [Courtesy of the MIT LL]

134 for damage and rebar detections is discussed. Measurements on the mechanically- damaged are also reported and discussed.

Table 4.7: Signal contents of the radar measurements of laboratory specimens

Specimen Signal content Usage

CON Splain concrete DD, RD

GFRP Splain concrete + SGFRP DD

CRE Splain concrete + Sone center rebar RD

4RE Splain concrete + Sfour near-surface rebars RD

AD1B Splain concrete + SGFRP DD

AD1F Splain concrete + SGFRP + Sartiﬁcial defect AD1 DD

AD2 Splain concrete + SGFRP + Sartiﬁcial defect AD2 DD

AD3 Splain concrete + SGFRP + Sartiﬁcial defect AD3 DD

Eﬀects of GFRP and Concrete Heterogeneity

For the understanding of eﬀects of GFRP and concrete heterogeneity in normal incidence measurements, two specimens (CON, GFRP) are considered. Specimen CON is a plain concrete cylinder, and the specimen GFRP is a plain concrete cylinder wrapped with one layer of GFRP sheet. Radar measurements made on the specimen CON are shown in Figures 4-21 and 4-22. Figure 4-23 shows the radar measurements of the specimen GFRP.

135 fname: CYL05, HH Pol., max = -3.34 dBsm fname: CYL05, VV Pol., max 3.77 dBsm 12 0 12 0

11.5 11.5 -2 -2 11 11

10.5 10.5 -4 -4

10 10

-6 -6 9.5 9.5 Frequency (GHz) Frequency (GHz) Frequency 9 9 -8 -8 8.5 8.5

8 -10 8 -10 60 70 80 90 100 110 120 60 70 80 90 100 110 120 θφ ((deg)deg) θ φ( deg(deg))

(a) HH (TE) (b) VV (TM)

Figure 4-21: Frequency-angle response of specimen CON (plain concrete) – Amplitude (dBsm), X-band

fname: CYL05K, HH Pol., max 1.77 dBsm fname: CYL05K, VV Pol., max 2.37 dBsm 18 0 18 0

17 17 -2 -2

16 16 -4 -4

15 15

-6 -6 14 14 Frequency (GHz) Frequency (GHz) Frequency

-8 -8 13 13

12 -10 12 -10 60 70 80 90 100 110 120 60 70 80 90 100 110 120 θφ ((deg)deg) θ φ( deg(deg))

(a) HH (TE) (b) VV (TM)

Figure 4-22: Frequency-angle response of specimen CON (plain concrete) – Amplitude (dBsm), Ku-band

136 fname: CYL06, HH Pol., max = -3.35 dBsm fname: CYL06, VV Pol., max 3.79 dBsm 12 0 12 0

11.5 11.5 -2 -2 11 11

10.5 10.5 -4 -4

10 10

-6 -6 9.5 9.5 Frequency (GHz) Frequency (GHz) Frequency 9 9 -8 -8 8.5 8.5

8 -10 8 -10 60 70 80 90 100 110 120 60 70 80 90 100 110 120 θφ ((deg)deg) θ φ( deg(deg))

(a) HH (TE) (b) VV (TM)

Figure 4-23: Frequency-angle response of the specimen GFRP (plain concrete with GFRP) – Amplitude (dBsm), X-band

In Figures 4-21 and 4-22, the circumferentially symmetry of the reﬂection response from the plain concrete cylinder is observed. Unlike other homogeneous media, the heterogeneity of concrete as a cementitious composite material produces a location- dependent radar response which also varies with respect to the incident angle. This heterogeneity is sensible by the chosen radar signal in the X- and Ku-band whose half-wavelength ranges from 1.87 cm (0.74 in, 8GHz) to 0.83 cm (0.32 in, 18GHz) and are comparable to the size of aggregates used in concrete. Table 4.8 lists the maximum and minimum power response (in dBsm) values of specimens CON and GFRP for both HH and VV polarizations.

Table 4.8: Maximum and minimum powers (dBsm) of specimens CON and GFRP, X-band

Polar. CON(plain concrete) GFRP(plain concrete with GFRP) HH Max.=0.6808 / Min.=0.2503 Max.=0.68 / Min.=0.2286 VV Max.=0.6479 / Min.=0.216 Max.=0.6464 / Min.=0.2101

The eﬀect of the GFRP wrap in the normal incidence scheme can also be illustrated

137 by comparing the radar measurements of specimens CON and GFRP. Figure 4-24 shows the selected power responses (dBsm) of specimens CON and GFRP in HH polarization. It can be seen that the presence of the GFRP wrap slightly reduces the level of radar reflection. Same result is observed in VV polarization. In Figure 4-24, local fluctuations in the power response of specimens CON and GFRP is the result of not only the surface roughness of the cylinder but also the material heterogeneity of the cylinder, while global variation may be caused by the misalignment of cylinder on the measurement tower. Figure 4-25 shows the surface roughness (porous surface) of the specimen CON. Figure 4-26 illustrates the weakening of reflected power response due to misalignment; the misaligned specimen does not produce a perfectly normal reflection after the rotation of the tower. It is noteworthy to point out that the measured power response is the combined result of all factors.

Damage Detection

In the damage detection using far-field monostatic ISAR normal incidence measurements six specimens (CON, GFRP, AD1B, AD1F, AD2, AD3) are considered. Figures 4-27 and 4-28 are the far-field radar responses of the intact and damaged (with an artificial defect) sides of the specimen AD1 in X-band. In Figure 4-27, while no defect is presented, variation of the power response in the circumferential direction is still observed. Table 4.9 lists the extreme values of the measurements on the specimen AD1. Although the presence of a local defect globally increases the power level of far-field ISAR normal incidence measurements as the experimental data suggest, it should be noted that these extreme values do not occur at same locations. The angular pattern of the power fluctuation is better illustrated in Figure 4-29. In Figure 4-29, fluctuation of power response curve representing the specimen GFRP-con. (intact side of AD1) is attributed to the surface roughness and the material heterogeneity of the specimen. There is a descending region around 20◦ and 30◦ in the GFRP-con. curve. A careful investigation on the specimen AD1 reveals a concave concave on the surface of the specimen as shown in Figure 4-30. The distribution of this concave region matches the descending region in the curve around

138 [ 8 GHz ] 1 Con.w.GFRP Con. 0.5

0 -30 -20 -10 0 10 20 30 Ref. power (dBsm) [ 9 GHz ] 1

0.5

0 -30 -20 -10 0 10 20 30 Ref. power (dBsm) [ 10 GHz ] 1

0.5

0 -30 -20 -10 0 10 20 30 Ref. power (dBsm) [ 11 GHz ] 1

0.5

0 -30 -20 -10 0 10 20 30 Ref. power (dBsm)

[ 12 GHz ] 1

0.5

0 -30 -20 -10 0 10 20 30

Ref. power (dBsm) Incident angle (deg.)

Figure 4-24: Comparison between the power responses of specimens CON and GFRP, X-band

139

θ = –30° θ = 0° θ = 0° θ = 30°

Figure 4-25: Photos of the specimen CON showing surface roughness

140 30◦. On the other hand, the curve representing the damaged side of the specimen AD1 has a descending region around 0◦, while there is no any surface irregularity at the location corresponding to 0◦ except the embedded artiﬁcial defect underneath the GFRP wrapping. The descending becomes most obvious at frequencies around 10 GHz to 11 GHz whose wavelengths (1.18 in. to 1.07 in.) are approximate to the depth of the defect (1 in) (the half-wavelength of radar signals between 8 GHz and 12 GHz ranges from 0.73 in. to 0.49 in.). This phenomenon suggests a possibility of discovering local near-surface defect in GFRP-retroﬁtted concrete columns, although other factors leading to a false alarm result must be excluded, such as surface irregularity. Radar measurements collected in Ku-band are reported in the Appendix.

Table 4.9: Maximum and minimum RCS (dBsm) of the specimens AD1

Intact side of AD1 Damaged side of AD1 HH Max.=0.5439 / Min.=0.3327 Max.=0.6615 / Min.=0.3908 VV Max.=0.5321 / Min.=0.3479 Max.=0.6412 / Min.=0.4027

141 Monostatic radar

Center of the Styrofoam specimen tower Misalignment

Misalignment Cylinder Center of the specimen tower (a) Before rotation (b) After rotation

Figure 4-26: Misalignment between the cylinder specimen and the Styrofoam tower in the normal incidence scheme

fname: CYLAD1BV, HH Pol., max = -5.29 dBsm fname: CYLAD1BV, VV Pol., max = -5.48 dBsm 12 0 12 0

11.5 11.5 -2 -2 11 11

10.5 10.5 -4 -4

10 10

-6 -6 9.5 9.5 Frequency (GHz) Frequency (GHz) Frequency 9 9 -8 -8 8.5 8.5

8 -10 8 -10 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 θ φ( deg(deg)) θφ ( (deg)deg)

(a) HH (TE) (b) VV (TM)

Figure 4-27: Frequency-angle response of specimen AD1B – Amplitude (dBsm), X- band

142 fname: CYLAD1FV, HH Pol., max = -3.59 dBsm fname: CYLAD1FV, VV Pol., max 3.86 dBsm 12 0 12 0

11.5 11.5 -2 -2 11 11

10.5 10.5 -4 -4

10 10

-6 -6 9.5 9.5 Frequency (GHz) Frequency (GHz) Frequency 9 9 -8 -8 8.5 8.5

8 -10 8 -10 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 θ φ( deg(deg)) θ (φdeg (deg))

(a) HH (TE) (b) VV (TM)

Figure 4-28: Frequency-angle response of specimen AD1F – Amplitude (dBsm), X- band

Far-field monostatic ISAR normal incidence measurements of specimens AD2 and AD3 are also reported here, as shown in Figures 4-31 and 4-32, both in X-band. Since the intact side of the specimen AD2 was not measured, there is no available baseline to be used as the intact response. The intact response from the specimen AD1 is not relevant here because the size of AD2 is different from AD1. Nonetheless, in Figure 4-31, a weaker power response is found, centering at the 0◦ where the delamination defect is embedded. A reasonable postulation suggests that this weakening in power response is due to the scattering of the defect, after excluding the possibility of surface irregularity via carefully checking the surface of the specimen AD2. Firm conclusion can be obtained should the intact side of AD2 be measured for comparison. Similar constraint occurs to the measurement on the specimen AD3. In Figure 4-32, a significant weaker region is observed at 0◦ where a strip defect is placed at the interface between GFRP and concrete along the longitudinal axis of the cylinder. Surface irregularity has also been excluded after inspecting the specimen. No baseline data can be used for classifying the fluctuations of power response into different causes, although it seems obvious that the weaker region is caused by the presence of a strip defect. In this case it is suggested that a reliable assessment should be conducted

143 [ 8 GHz ] 0.5

0.4 GFRP-con.w.AD1 GFRP-con. 0.3 -30 -20 -10 0 10 20 30 Ref. power (dBsm) [ 9 GHz ]

0.5

0.4

0.3 -30 -20 -10 0 10 20 30 Ref. power (dBsm) [ 10 GHz ]

0.4

-30 -20 -10 0 10 20 30 Ref. power (dBsm) [ 11 GHz ]

0.4

-30 -20 -10 0 10 20 30 Ref. power (dBsm)

[ 12 GHz ]

0.6

0.4

-30 -20 -10 0 10 20 30

Ref. power (dBsm) Incident angle (deg.)

Figure 4-29: Comparison between the power responses of the intact and damaged sides of the specimen AD1, X-band

144 θ = 0° [Max. depth of the concave region is θ = 30° about 0.25 in. (0.635 cm).]

Figure 4-30: Photos showing the concave region in the specimen

145 via other supportive evidences, rather than solely based on the power reflection response. It is noted that the damage detection using far-field monostatic ISAR normal incidence measurements relies not only on the half-wavelength of CW radar signals (whether the signal can be interfered by the defect) but also on how the detectability is defined. While the half-wavelength of a 8 GHz signal, 0.73 in., seems to be capable of detecting a 1-in.-depth defect (specimen AD1), there is no clear indication of the presence of defect in the reflection response at 8 GHz. Rather, clear indication is observed in the neighborhood of 10 GHz and 11 GHz (Figure 4-27). However, care must be taken when defining the ”indication” using far-field monostatic ISAR normal incidence measurements. Questions such as ”how much difference should be used as a positive sign for the presence of defect?” need to be answered in advance. Another issue further complicating the detectability in field conditions is the presence of ambient noise. Without field measurements, it is not clear how the ambient, background electromagnetic noise affects far-field monostatic ISAR normal incidence measurements. Performance of de-noising techniques using compact (wavelets) and non-compact (e.g., Fourier series) basis functions remains unknown in this particular application.

Rebar Detection

In the rebar detection using far-field monostatic ISAR normal incidence measurements three specimens (CON, CRE, and 4RE) are considered. Figure 4-33 shows the frequency-angle power reflection response/measurement of the specimen CRE (plain concrete cylinder with a rebar at the center), and Figure 4-34 provides a comparison between the power responses of specimens CON and CRE. Knowing that the curve of the specimen CON is not perfectly ideal in Figure 4-34, the power response of the specimen CRE is in average greater than the one of the specimen CON. As expected, this is due to the presence of a PEC rebar in the center of the cylinder. The difference is not significant since the rebar is embedded approximately 3 inches from the surface. In the case of the specimen 4RE (four rebars buried at different depths from the surface of a plain concrete cylinder), significant fluctuations of the power response are

146 fname: CYLAD2V, HH Pol., max = -3.71 dBsm fname: CYLAD2V, VV Pol., max 3.14 dBsm 12 0 12 0

11.5 11.5 -2 -2 11 11

10.5 10.5 -4 -4

10 10

-6 -6 9.5 9.5 Frequency (GHz) Frequency (GHz) Frequency 9 9 -8 -8 8.5 8.5

8 -10 8 -10 -40 -20 0 20 40 -40 -20 0 20 40 θ φ( deg(deg)) θ φ(deg (deg))

(a) HH (TE) (b) VV (TM)

Figure 4-31: Frequency-angle response of specimen AD2 – Amplitude (dBsm), X- band

fname: CYLAD3, HH Pol., max = -2.43 dBsm fname: CYLAD3, VV Pol., max 2.34 dBsm 12 0 12 0

11.5 11.5 -2 -2 11 11

10.5 10.5 -4 -4

10 10

-6 -6 9.5 9.5 Frequency (GHz) Frequency (GHz) Frequency 9 9 -8 -8 8.5 8.5

8 -10 8 -10 -40 -20 0 20 40 -40 -20 0 20 40 θ φ( deg(deg)) θ φ(deg (deg))

(a) HH (TE) (b) VV (TM)

Figure 4-32: Frequency-angle response of specimen AD3 – Amplitude (dBsm), X- band

147 fname: CYL04, VV Pol., max 2.96 dBsm 12 0 12 0

11.5 11.5 -2 -2 11 11

10.5 10.5 -4 -4

10 10

-6 -6 9.5 9.5 Frequency (GHz) Frequency Frequency (GHz) Frequency 9 9 -8 -8 8.5 8.5

8 -10 8 -10 60 70 80 90 100 110 120 60 70 80 90 100 110 120 θφ ((deg)deg) θ (φdeg (deg))

(a) HH (TE) (b) VV (TM)

Figure 4-33: Frequency-angle response of specimen CRE – Amplitude (dBsm), X- band

expected due to the configuration of local PEC scatterers (the four rebars). Configu- ration of the rebars is illustrated in Figure 4-2 and summarized in Table 4.10. Figures 4-35, 4-36, 4-37, and 4-38 show the power reflection responses of the specimen 4RE at four angular regions; namely, RE1∼RE4. Four data ranges are consequently defined, centering at the location where one rebar is buried. Mean RCS or power reflection response at each data range is also reported. It is found that the power reflection

Table 4.10: Conﬁguration of the specimen 4RE

Rebar Angle Date range Embed. Mean RCS (dBsm) depth (in) 1 −135◦ [−150◦ ∼ −120◦] 0.25 HH=0.6375/VV=0.6265 2 −45◦ [−60◦ ∼ −30◦] 1.5 HH=0.4830/VV=0.5388 3 45◦ [30◦ ∼ 60◦] 1 HH=0.4833/VV=0.4801 4 135◦ [120◦ ∼ 150◦] 0.5 HH=0.5281/VV=0.527

response exhibits a clear two-dimensional sinusoidal pattern of change when power attenuation in concrete is small (the rebar is buried very close to the surface, RE1).

148 [ 8 GHz ]

0.45 0.4 Con.w.1 rebar 0.35 Con.

-30 -20 -10 0 10 20 30 Ref. power (dBsm) [ 9 GHz ] 1

0.5

0 -30 -20 -10 0 10 20 30 Ref. power (dBsm) [ 10 GHz ] 1

0.5

0 -30 -20 -10 0 10 20 30 Ref. power (dBsm) [ 11 GHz ] 1

0.5

0 -30 -20 -10 0 10 20 30 Ref. power (dBsm)

[ 12 GHz ] 1

0.5

0 -30 -20 -10 0 10 20 30

Ref. power (dBsm) Incident angle (deg.)

Figure 4-34: Comparison between the power responses of specimens CON and CRE, X-band

149 fname: CYLRE1, HH Pol., max = -1.51 dBsm fname: CYLRE1, VV Pol., max 2.11 dBsm 12 0 12 0

11.5 11.5 -2 -2 11 11

10.5 10.5 -4 -4

10 10

-6 -6 9.5 9.5 Frequency (GHz) Frequency (GHz) Frequency 9 9 -8 -8 8.5 8.5

8 -10 8 -10 -150 -145 -140 -135 -130 -125 -120 -150 -145 -140 -135 -130 -125 -120 θφ ( (deg)deg) θ φ( deg(deg))

(a) HH (TE) (b) VV (TM)

Figure 4-35: Frequency-angle response of specimen 4RE (rebar 1) – Amplitude (dBsm), X-band

fname: CYLRE2, HH Pol., max = -1.52 dBsm fname: CYLRE2, VV Pol., max 1.55 dBsm 12 0 12 0

11.5 11.5 -2 -2 11 11

10.5 10.5 -4 -4

10 10

-6 -6 9.5 9.5 Frequency (GHz) Frequency (GHz) Frequency 9 9 -8 -8 8.5 8.5

8 -10 8 -10 -60 -55 -50 -45 -40 -35 -30 -60 -55 -50 -45 -40 -35 -30 θ φ( deg(deg)) θ (φdeg (deg))

(a) HH (TE) (b) VV (TM)

Figure 4-36: Frequency-angle response of specimen 4RE (rebar 2) – Amplitude (dBsm), X-band

150 fname: CYLRE3, HH Pol., max = -3.72 dBsm fname: CYLRE3, VV Pol., max 4.26 dBsm 12 0 12 0

11.5 11.5 -2 -2 11 11

10.5 10.5 -4 -4

10 10

-6 -6 9.5 9.5 Frequency (GHz) Frequency (GHz) Frequency 9 9 -8 -8 8.5 8.5

8 -10 8 -10 30 35 40 45 50 55 60 30 35 40 45 50 55 60 θ φ( deg(deg)) θ φ( deg(deg))

(a) HH (TE) (b) VV (TM)

Figure 4-37: Frequency-angle response of specimen 4RE (rebar 3) – Amplitude (dBsm), X-band

fname: CYLRE4, HH Pol., max = -3.07 dBsm fname: CYLRE4, VV Pol., max 3.35 dBsm 12 0 12 0

11.5 11.5 -2 -2 11 11

10.5 10.5 -4 -4

10 10

-6 -6 9.5 9.5 Frequency (GHz) Frequency (GHz) Frequency 9 9 -8 -8 8.5 8.5

8 -10 8 -10 120 125 130 135 140 145 150 120 125 130 135 140 145 150 θφ ( (deg)deg) θ φ( deg(deg))

(a) HH (TE) (b) VV (TM)

Figure 4-38: Frequency-angle response of specimen 4RE (rebar 4) – Amplitude (dBsm), X-band

151 When the power attenuation is signiﬁcant (the rebar is buried at a depth), this pattern becomes blur and unclear. This attenuation issue determines whether such pattern can be used for rebar detection or not. The specimen representing the most extreme case of power attenuation for rebar detection is the specimen CRE in which the rebar is buried at a depth of 3 inches from the surface. There is almost no indication of any sinusoidal pattern in the power response of the specimen CRE in Figure 4-33, as can be observed in Figures 4-35∼4-38. In Table 4.10, the mean power response of the frequency-angle measurement in each data range is also provided. It is expected that the deeper the rebar is buried from the surface of concrete, the weaker the power response should be. This understanding is in good agreement with the power response in HH polarization and generally correct with the power response in VV polarization.

4.5.2 Monostatic ISAR Oblique Incidence Scheme

The amplitude and phase responses of the specimens measured in far-field monostatic ISAR oblique incidence scheme are reported in this section. The original purpose for using this measurement scheme is to alleviate the effect of specular returns in the reflection response. Considered specimens include AD1 (GFRP-concrete cylinder with a cubic defect) and AD2 (GFRP-concrete cylinder with a delamination defect) (refer to Table 4.1 and previous figures for a detailed description to these specimens). Both the intact surface (without artificial defect) and the damaged surface of the specimen AD1 were measured, while only the damaged surface of the specimen AD2 was measured. Only X-band measurements are available. Specimens with rebars are not measured in the oblique incidence scheme. Figures 4-39 and 4-40 show the oblique incidence measurements of the intact and damaged sides of the specimen AD1. It is seen that, the specular return dominates the reflection response at azimuth angles around φ = 0◦ (−2◦ ∼ 2◦). The specular return is reduced to a significant degree when the incident angle φ deviates from 0◦, and the alleviation of specular return reveals the hidden scattering signals due to the presence of defect. The spreading signals in Figure 4-40 indicates the presence of the

152 near-surface artificial defect in the specimen AD1 for both HH and VV polarizations. Figure 4-40 demonstrates the effectiveness of the use of the oblique incidence scheme in eliminating the specular return at different incident angles. The radar measurements on the specimen AD2 in Figure 4-41 also shows scattering signals similar to those obtained for the specimen AD1, suggesting the presence of the artificial defect in the specimen AD2. It should be noted that, the scattering signals due to defect are not continuously distributed outside the specular-dominant zone in the oblique incident measurements. Rather, variation of power amplitude with respect to the incident angle φ is found. This is attributed to the sensitivity of monostatic ISAR measurements to the geometry of a scatterer (the specimen or the defect) in the oblique incidence scheme; not only the specular return but also the scattering signals varies with φ. This suggests the existence of a set of incident angles for better inspection.

fname: CYLAD1FH, HH Pol., max = -5.09 dBsm 12 0 12 0

11.5 11.5

11 -5 11 -5

10.5 10.5

10 -10 10 -10

9.5 9.5 Frequency (GHz) Frequency Frequency (GHz) Frequency 9 -15 9 -15

8.5 8.5

8 -20 8 -20 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 φ (deg) φ (deg) Figure 4-39: Frequency-angle response of the intact side of the specimen AD1 – Amplitude (dBsm), X-band

153 fname: CYLAD1BH, HH Pol., max = -6.41 dBsm ,, 12 0 12 0

11.5 11.5

11 -5 11 -5

10.5 10.5

10 -10 10 -10

9.5 9.5 Frequency (GHz) Frequency (GHz) Frequency 9 -15 9 -15

8.5 8.5

8 -20 8 -20 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 φ (deg) φ (deg) Figure 4-40: Frequency-angle response of the damaged side of the specimen AD1 – Amplitude (dBsm), X-band

12 0 12 0

11.5 11.5

11 -5 11 -5

10.5 10.5

10 -10 10 -10

9.5 9.5 Frequency (GHz) Frequency Frequency (GHz) Frequency 9 -15 9 -15

8.5 8.5

8 -20 8 -20 -40 -20 0 20 40 -40 -20 0 20 40 φ φ( deg(deg)) φ φ( deg(deg))

(a) HH (TE) (b) VV (TM)

Figure 4-41: Frequency-angle response of the damaged side of the specimen AD2 – Amplitude (dBsm), X-band

154 4.6 Summary

This chapter reports the laboratory radar measurements conducted in far-ﬁeld monostatic ISAR normal and oblique incidence schemes. Steady-state, complex responses in terms of amplitude and phase were collected for each harmonic frequency (f) and azimuth angle (θ in normal incidence or φ in oblique incidence). The frequency-angle measurements were obtained. Research ﬁndings are summarized as follows.

• Far-ﬁeld monostatic ISAR normal incidence measurements

1. Variations in the material composition and the dielectric properties of the components of concrete lead to the fluctuation of measured reflection power responses at differen incident angles. The measured power responses essentially collect the specular returns from the specimens in the normal incidence scheme. Additionally, changes in the geometry of the cylinder specimen can also result in the variation of power response.

2. Presence of the externally bonded GFRP sheet slightly reduces the total power response of the GFRP-wrapped concrete cylinder (Figure 4-24). This is attributed to the lossness of GFRP in the steady-state response, which consumes a small amount of incident energy.

3. Presence of a local, artificial defect (cavity) in the GFRP-retrofitted concrete cylinder may produce a weakened power response at the right incident angle (Figure 4-27), due to the local scattering of signals at the edges of the rectangular defect. However, considering other possible causes such as the geometry of the cylinder, it is not recommended to use power response as the only information for asserting the presence of defect. In other words, changes in the reflected power response can be attributed to the presence of defect or to other causes, considering the strong influence of specular returns in the normal incidence scheme.

4. Alignment accuracy is important in the far-ﬁeld monostatic ISAR norma incidence measurements. Misalignment (Figure 4-26) can cause global,

155 signiﬁcant change of the power response. This is mainly due to the use of monostatic radar.

5. From the results on specimens AD1, AD2, and AD3, it is believed that a reliable assessment should be achieved via the use of power reﬂection response (frequency-angle measurement) and other supportive evidences; power reﬂection response in the normal incidence scheme may be used as an indicator, but not as a conclusive tool.

6. A sinusoidal pattern of change is observed in the frequency-angle measurements of the specimen 4RE, accounting for the scattering from the rebars. This pattern decays when the depth of rebar location increases.

7. Based on the findings in the normal incidence scheme, it is believed that the far-field ISAR normal incidence measurements can not be exclusively used for determining the presence of near-surface defects, due to the strong influence of specular returns in the measurements. On the other hand, normal incidence measurements of the specimen 4RE (concrete cylinder with four rebars) demonstrate a clear pattern different from the measurements of specimens without rebar. This suggests the feasibility of using the normal incidence scheme for rebar detection.

• Far-ﬁeld monostatic ISAR oblique incidence measurements

1. In Figures 4-39 and 4-40, the specular return dominates the reﬂection response at azimuth angles around φ = 0◦ (−2◦ ∼ 2◦). The specular return is reduced to a signiﬁcant degree when azimuth angle φ deviates from 0◦, and the alleviation of specular return reveals the hidden scattering signals due to the presence of defect.

2. Since the monostatic ISAR measurements are sensitive to to the geometry of a scatterer (the specimen or the defect) in the oblique incidence scheme, not only the specular return but also the scattering signals varies with the incident angle φ. This suggests the existence of a set of incident angles for better inspection.

156 3. From the Figures 4-39 and 4-40, it is found that the importance of alignment has been reduced in the oblique incidence measurements. It is because that the measurement surface is no longer always normal to the line-of-sight of the radar;. Although misalignment in the neighborhood of perpendicular incidence (φ ∼ 0◦) can reduce the intensity of scattering signals due to defect, it is not the angular range which the oblique incidence scheme rely on for damage detection. Other than the perpendicular incidence region, misalignment will not produce a significant impact to the intensity of defect signals since the background specular return has been removed. This feature offers the oblique incidence scheme another benefit as it is more robust than the normal incidence scheme.

4. Based on the ﬁndings in the oblique incidence scheme, it is believed that the far-ﬁeld ISAR oblique incidence measurements can be used for detecting near-surface defects such as GFRP debonding and delamination. The alleviation of specular returns in such measurement scheme encourages the revealing of scattering signals due to defect.

For the purpose of condition assessment, collected frequency-angle measurements (far-ﬁeld monostatic ISAR measurements) of GFRP-concrete cylinders are further processed by the image reconstruction algorithm in order to render a spatial proﬁle of the cylinders. Principles of signal processing are described in Chapter 5 Image Reconstruction.

157 158 Chapter 5

Image Reconstruction

”Between the matter of the physical image and its object there is a very diﬀerent relation: they resemble each other. What should we understand by this?” —– The Imaginary, Jean-Paul Sartre (1905∼1980)

The purpose of this chapter is to provide a theoretical background and principles of the image reconstruction method used for processing the collected far-field ISAR measurements (Chapter 4 Laboratory Radar Measurements) in this research. The image reconstruction method is based on tomographic reconstruction (TR) methods [203, 129] and implemented by the fast backprojection algorithm [176, 194, 278]. In this chapter, classical formulation of synthetic aperture radar (SAR) and the backprojection algorithm in the fields of radar engineering and remote sensing is provided for demonstrating the process of transforming inverse SAR (ISAR) signals to backprojection images from a theoretical perspective. This image reconstruction processing is the basis for producing the images used in Chapter 7 Condition Assessment of GFRP-concrete Systems – FAR NDT. Credits for developing fundamental equations in SAR [261, 47, 121]and the backprojection algorithm [68, 239, 67, 7, 194, 240] should go to the researchers in both fields. The author’s contributions lay in (1) deriving the form of backprojection images using the ISAR-processed signal with a support band analysis, and (2) identifying

159 the importance of dielectric properties in the formulation regarding image resolution.

In this chapter, a brief introduction to traditional radar and SAR/ISAR measurements is ﬁrst provided, followed by the principle of backprojection algorithms. A support band analysis using the method of stationary phase for determining a band region in which the spatial spectrum of backprojection images is non-trivial. Cod- ing procedure for the implementation of backprojection algorithms is also provided. Eﬀects of aperture size and frequency bandwidth in the performance of the image reconstruction method are discussed.

5.1 Single Scattering and Synthetic Aperture Radar

Consider the planar scattering problem in a domain Ωs containing N point scatterers and an incident wave with unit amplitude to be [261]

1 ψ (¯r) = exp ik¯ · r¯ (5.1) inc r

¯ where ki = kixxˆ − kiyyˆ is the incident wave vector,r ¯ is the relative position vector √ from the radar to any observation point, and i = −1 is the imaginary number. The scattered ﬁeld from scatterer j atr ¯j and observed atr ¯ is [261]

ˆ ˆ sj ks, ki ψscat,j (¯r, r¯j) = ψs,j (¯r, r¯j) = exp (ik |r¯ − r¯j|) · ψinc (¯r) (5.2) |r¯ − r¯j|

¯ where ki = kixxˆ−kiyyˆ is the incident wave vector,r ¯ is the relative position vector from √ ˆ ˆ the radar to any observation point, i = −1 is the imaginary number, sj = sj ks, ki ˆ is the scattered amplitude at scatterer j due to an incident wave of ki, observed at r¯. When N scatterers are presented, the total scattered ﬁeld observed at r is the summation of the scattered ﬁelds from all scatterers. It can be approximated to be

N ˆ ˆ X sj ks, ki ψ (¯r) = ψ (¯r) = exp (ik |r¯ − r¯ |) · ψ (¯r) (5.3) scat s |r¯ − r¯ | j inc j=1 j

160 ¯ ¯ ¯ where ks = ksxxˆ + ksyyˆ is the scattering direction vector. ks = −ki when the radar operates in monostatic mode. Note that in Eq.(5.3) interactions among scatterers are neglected as the result of approximation (independent scattering approximation).

The spatial resolution in traditional radar imaging is limited to its frequency and bandwidth, while the amplitude and phase of scattered waves can both be measured. To remedy this disadvantage, the concept of synthetic aperture radars (SAR) is introduced in which the spatial resolution is improved by phase compensation (focusing). Several major SAR modes are listed and compared in Table 5.1. In Table 5.1, ”ﬂight path” is the shape of synthetic aperture, and ”footprint” is the area of radar investigation. Other SAR modes and their derivatives can be found in the radar literature. [180, 58, 240] The conﬁguration of SAR measurements is illustrated in Figure 5-1. For

Table 5.1: Comparison of several SAR modes

SAR mode Flight path Footprint Spotlight Straight line Circular Strip Straight line Strip Inverse Circular Circular

SAR measurements using monostatic radar located atr ¯s = (xs, ys, zs), conducting at constant zs and at constant ﬂight speed v Figure 5-1, we have

zs = Rs cos θi = h (5.4)

ys = Rs sin θi = h tan θi (5.5)

xs = xs0 + vt (5.6)

where Rs = |r¯s| is the range distance, θi is the inclination angle with respect to the z

axis, h is the height of the radar, and xs0 = xs(t = 0) is the initial x position of the monostatic radar. For ISAR measurements conducting at constant zs and at constant

161 ﬂight speed v, we have

zs = Rs cos θi = h (5.7) vt ys = Rs sin θi sin = Rs sin θi sin(ωt) (5.8) Rs sin θi vt xs = Rs sin θi cos = Rs sin θi cos(ωt) (5.9) Rs sin θi

v where ω = is the angular velocity of the radar. Assuming a constant prop- Rs sin θi agation time of radar signals in the measurements, the distance variables (e.g.,r ¯s, r¯j) are exchangeable with propagation time. Therefore, ψinc (¯rs, r¯j) = ψinc(ω, t) and

ψs (¯rs, r¯j) = ψs (ω, t). Assuming that G contains N scatterers, the received scattered wave due to the presence of N scatterers is

N X ψs(ω, t) = ψs,j(ω, t) (5.10) j=1 where

ˆ ˆ fj(ks, ki) ψs,j(ω, t) = exp [ikRs,j] · ψinc(¯rj) (5.11) Rs,j

Rs,j = |r¯s − r¯j| (5.12)

ˆ ˆ ˆ ˆ and fj(ks, ki) = fj(−ki, ki) is the backscattering amplitude of the nth scatter. With- out losing generality, consider the case of single scatterer and its response in the far-

ﬁeld region (|r¯ − r¯j| ≈ |r¯| = r for amplitude approximation and |r¯ − r¯j| ≈ r + rj cos θ for phase approximation). The size of the circular footprint is assumed to be Gaussian amplitude with radius g (Figure 5-4(b)). The incident wave on the footprint surface is w(¯rj) ψinc (¯rs, r¯j) = exp [ikiRs,j] (5.13) Rs,j where w(¯rj) is a Gaussian amplitude distribution function at the scattererr ¯n =

(xn, yn, zn) ∈ G , Rs,j is the distance betweenr ¯j and the radar locationr ¯s and

p 2 2 2 Rs,j = |r¯s − r¯j| = (xj − xs) + (yj − ys) + (zj − zs) . The projected G on x − y

162 plane is an ellipse with characteristic dimensions gx and gy as shown in Figure 5-2.

λ g = R (5.14) x L s λ g = sec θ R (5.15) y L i s

Considering the footprint domain located at zj = 0, the Gaussian amplitude distribution function is

2 2 (xj − xs) (yj − ys) w(xj, yj, 0) = w(xj, yj) = exp − − (5.16) gx gy

Therefore,

N ˆ ˆ 2 2 X fj(−ki, ki) (xj − xs) (yj − ys) ψ (ω, t) = exp [2ikR ] exp − − (5.17) s R2 s,j g2 g2 j=1 s,j x y

The signal can be focused at a locationr ¯0 = (x0, y0, z0) by multiplying ψs (ω, t) by a reference signal

ψ0(¯r0, ω, t) = exp [2ikRs,0] (5.18)

where Rs,0 = |r¯s − r¯0|. The focused signal becomes

F (¯r0, ω, t) = ψs(ω, t)ψ0(¯r0, ω, t) (5.19)

Integrating F (¯r0, ω, t) over ω ∈ [ωc − Bπ, ωc + Bπ] and t ∈ [t0, tf ] provides the ﬁnal

focused signal C(¯r0) atr ¯0.

tf ω + Bπ tf ω + Bπ Z c Z Z c Z C(¯r0) = dt dωF (¯r0, ω, t) = dt dωψs(ω, t)ψ0(¯r0, ω, t) (5.20)

t0 ωc − Bπ t0 ωc − Bπ

where [to, tf ] and [ωc −Bπ, ωc +Bπ] are the time window and frequency window of the

ﬂight path in which ψs(ω, t) is collected and processed, respectively. In the far-ﬁeld

163 region, Rs,j is approximately Rs. C(¯r0) becomes

N 1 ZZ X C(¯r ) = dtdω f (−kˆ , kˆ ) exp [2ik(R − R )] 0 R2 j i i s,j s,0 s j=1 2 2 (xj − xs) (yj − ys) · exp − 2 − 2 (5.21) gx gy

The reason ψ0(¯r0, ω, t) can focus C(¯r0) atr ¯0 is due to the phase 2k(Rs,j − Rs,0). ∼ ∼ (Rs,j −Rs,0) = 0 whenr ¯n = r¯0. This way C(¯r0) projects the scattering from a scatterer nearr ¯0 and creates an image centered atr ¯0. The integral of C(¯r0) is inherently computationally-expensive since a two-dimensional integral over ω and t is required for eachr ¯0. To alleviate this disadvantage, approximation on the phase is typically performed in order to utilize computationally-eﬃcient transforms like fast Fourier ∼ ∼ transform (FFT). Considering in the region wherer ¯n = r¯0, Rs,j = Rs,0. Also, in the ∼ ∼ far-ﬁeld region, Rs,0 = Rs since Rs,j = Rs (Figure 5-3).

R2 − R2 R2 − R2 R2 − R2 s,j s,0 ∼ s,j s,0 ∼ s,j s,0 Rs,j − Rs,0 = = = (5.22) Rs,j + Rs,0 2Rs,0 2Rs

The phase in C(¯r0) becomes

R2 − R2 ∼ s,j s,0 Φ(¯r0, ω, t) = 2k(Rs,j − Rs,0) = (5.23) Rs k 2 2 2 2 2 2 = [ xj − x0 + yj − y0 + zj − z0 Rs

+2xs(x0 − xj) + 2ys(y0 − yj) + 2zs(z0 − zj)] (5.24) k = [−∆x (xj + x0) − ∆y (yj + y0) − ∆z (zj + z0) Rs

+2xs (x0 − xj) + 2ys (y0 − yj) + 2zs (z0 − zj)] (5.25) where

∆x = x0 − xj (5.26)

∆y = y0 − yj (5.27)

∆z = z0 − zj (5.28)

164 Considering the conﬁguration of SAR measurements, the phase Φ(¯r0, ω, t) can be represented by Φ(¯r0, ω, t) = k aj + bjtˆ (5.29) where

1 aj = [∆x [2xs0 − (xj + x0)] Rs

+∆y [2ys − (yj + y0)] + ∆z [2zs − (zj + z0)]] (5.30)

bj = 2∆x (5.31) v tˆ= t (5.32) Rs

With these relations, the focused signal C(¯r0) becomes

ωc + πB vt

c Rs N c Z Z X C(¯r ) = dk dtˆ f (−kˆ , kˆ ) exp ik a + b tˆ 0 vR j i i j j s j=1 ωc − πB 0 c 2 2 (xj − xs0 − tRˆ s) yj · exp − 2 − 2 (5.33) gx gy

The ﬂight speed v of the monostatic radar is constrained by the size of the footprint

G (gx and gy) and the Gaussian amplitude distribution function. In the conﬁguration of SAR measurements (Figure 5-1), the maximum movement of radar is related to

the size of the footprint gx by

|xs0 + vt| ≤ gx −g − x vt g − x ⇒ x s0 ≤ ≤ x s0 (5.34) Rs Rs Rs

165 λ Since g = R , x L s

λ x vt λ x − − s0 ≤ ≤ − s0 L Rs Rs L Rs λ x λ x ⇒ − − s0 ≤ tˆ≤ − s0 (5.35) L Rs L Rs

∼ Assuming that xj = xs0 + tRˆ s, the integrand in C(¯r0) is then

ω + πB λ xs0 c − c L Rs Z Z N c X ˆ ˆ C(¯r0) = dk dtˆ fj(−ki, ki) vRs ω − πB λ x j=1 c − − s0 c L Rs y2 ˆ j · exp ik aj + bjt exp − 2 (5.36) gy

ω + πB λ xs0 c − c L Rs Z N y2 Z c X ˆ ˆ j ˆ ˆ = dk fj(−ki, ki) exp − 2 · dt exp ik aj + bjt (5.37) vRs gy ω − πB j=1 λ x c − − s0 c L Rs

First, we evaluate the integrand over tˆ.

λ x λ x − s0 − s0 L R L R Z s Z s dtˆexp ik aj + bjtˆ = exp[ikaj] · dtˆexp[ikbjtˆ] λ x λ x − − s0 − − s0 L Rs L Rs

166 λ x − s0 exp[ikaj] L Rs = exp[bjtˆ]| ikb λ xs0 j − − L Rs exp[ikaj] λ λ xs0 = exp ikbj − exp −ikbj · exp −ikbj ikbj L L Rs exp[ikaj] λ xs0 = · 2i sin kbj exp −ikbj ikbj L Rs 2 exp[ikaj] λ xs0 = sin kbj exp −ikbj (5.38) kbj L Rs

Therefore,

ωc + πB c Z N 2 c X yj C(¯r ) = dk f (−kˆ , kˆ ) exp − 0 vR j i i g2 s j=1 y ωc − πB c λ 2 sin kbj L xs0 · exp[ikaj] exp −ikbj (5.39) kbj Rs

Since C(¯r0) is a correlation integral of F (¯r0, ω, t), it indicates the likeliness of seeing a scatterer within the footprint. The correlation is deﬁned by aj and bj. The aj dependence is a exponentially decaying as a function of ∆x, ∆y, and ∆z. The bj sin x dependence is 2∆x = 2 (x − x ) and has a form. To positively assert the 0 j x presence of a scatterer, we need

λ sin kb ≤ 1 (5.40) j L and kb λ L j ≤ π ⇒ b ≤ π (5.41) L j kλ

Since bj = 2∆x = 2(x0 − xj), xj should be within the neighborhood of

L Lπ L 2 |x − x | ≤ π = = (5.42) 0 j kλ 2π 2

167 L The azimuth or cross-range resolution (ρ ) of SAR measurements is now where L is xr 2 λ L the antenna size, improved from g = R to . For the radar antenna operating in x L s 2 the range of X-band (8 GHz∼12 GHz) and Ku-band (12 GHz∼18 GHz), L is roughly 1 0.3 m. Take f = 12 GHz and λ = m as an example, the far-ﬁeld distance is 40 10 approximately 10 m. The cross-range resolution ρ is improved from =0.83 m to xr 12 L =0.15 m, a 5.53 times improvement. This example demonstrates the eﬀectiveness 2 of SAR measurements on improving cross-range resolution ρxr.

ωc Should we use the center frequency fc (or ωc = 2πfc or kc = ) for the wave ω c number variable (k = ) and have the radar antenna start from x = 0, we have c s0 1 λ 1 2π 2 sin kbj = 2 sin bj since kλ = 2π. The correlation integrand C(¯r0) kbj L kcbj L can be written to be

πB kc + 2π c 2 sin b Z N y2 j c X ˆ ˆ j L C(¯r0) = dk fj(−ki, ki) exp − 2 · exp[ikaj] vRs gy kcbj πB j=1 k − c c 2π πBaj N 2 2 sin bj 2 sin c X yj L c = f (−kˆ , kˆ ) exp − · · (5.43) vR j i i g2 k b a s j=1 y c j j

The aj dependence is also constrained by

πBa πBa c sin j ≤ 1 ⇒ j ≤ π ⇒ a ≤ (5.44) c c j B

Knowing that

1 aj = [∆x [−(xj + x0)] + ∆y [2ys − (yj + y0)] + ∆z [2zs − (zj + z0)]] (5.45) Rs

Assuming that the radar inspection is performed on a similar level with the footprint, zs is then much smaller than ys. Consequently, ∆x << ys and ∆z << ys. aj is

168 dominated by ys.

∼ 1 aj = [∆y [2ys − (yj + y0)]] Rs ∼ 1 = 2ys∆y (5.46) Rs

The aj dependence is then

1 c 2ys∆y ≤ (5.47) Rs B c R ⇒ ∆y ≤ · s (5.48) 2B ys

ys Since = sin θi ≤ 1, Rs

c Rs c 1 ∆y = |y0 − yj| ≤ · = · (5.49) 2B ys 2B sin θi c ⇒ |y − y | sin θ ≤ (5.50) 0 j i 2B

This is the limit of the range resolution of SAR measurements. In view of the form of C(¯r0), the SAR point response S(x, y sin θi) is usually written as

πy sin θi πx S(x, y sin θi) = sinc sinc (5.51) ρr ρxr

sin πx where sinc(x) = . πx

For traditional radar imaging, the range resolution ρr is

λ λRs ρr ≤ gy = sec θiRs = (5.52) L L cos θi

1 ∼ ∼ 10 Use λ = m, L =0.3 m, Rs =10 m, and θi = 0 (therefore zs << ys), gy = = 40 c 12 0.83 m. For B= 4 GHz, the range resolution for SAR imaging is =0.0375 m, 2B showing a 22 times improvement. Additional improvement on ρr can be expected, providing more bandwidth B. This example demonstrates the eﬀectiveness of SAR measurements on improving range resolution ρr.

169 z

Radar position

rxyzs = ()sss,,

Flight path

θ ˆ i ki

Footprint domain G

x (a) Synthetic aperture radar (SAR)

Footprint domain G g Radar position x y

g y

Flight path

x (b) SAR – Plane view

Figure 5-1: Conﬁguration of SAR measurements

170 z

rxyzs = ()sss,, Monostatic radar

θ kˆ i i Footprint domain G

rxyzjjjj= ( ,,) y gx

g y

Figure 5-2: Monostatic radar and its footprint

rxyzs = ()sss,, Monostatic radar

rrs − j

rrs − 0

Footprint domain G y nth scatter

focused location rxyzjjjj= ( ,,) rxyz= ( ,,) x 0000

Figure 5-3:r ¯s,r ¯0, andr ¯j in the far-ﬁeld region

171 5.2 Inverse Synthetic Aperture Radar

In this research, far-ﬁeld ISAR measurements are adopted. ISAR is a circular mode of SAR. Conﬁgurations of SAR measurements is illustrated in Figure 5-4. In ISAR measurements, an incident wave vector is considered.

ˆ ki = − sin θi cos φixˆ − sin θi sin φiyˆ − cos θizˆ (5.53)

ˆ where θi is the angle between ki and the z axis (Figure 5-4(a)) and φi the angle ˆ between ki and the x axis (Figure 5-4(b)). The radar is located at the range distance of Rs and denoted by a position vectorr ¯s = (xs, ys, zs) where

xs = Rs sin θi cos φi (5.54)

ys = Rs sin θi sin φi (5.55)

zs = Rs cos θi = h (5.56)

and Rs = |r¯s|. Assuming a Gaussian amplitude distribution for the circular footprint with radius g containing N scatterers (Figure 5-4(b)), the scattered ﬁeld ψs(ω) is

N ˆ ˆ 2 2 X fj(−ki, ki) (xj − xs) (yj − ys) ψ (ω) = exp [2ikR ] exp − − (5.57) s R2 s g2 g2 j=1 s x y

and the phase Φ(¯r0, ω) is

∼ k Φ(¯r0, ω) = [−∆x (xj + x0) − ∆y (yj + y0) − ∆z (zj + z0) Rs

+2xs (x0 − xj) + 2ys (y0 − yj) + 2zs (z0 − zj)] (5.58)

In the far-ﬁeld region, Rs sin θi is usually much greater than the radius of G, g (Figure

5-4). Therefore, the phase Φ(¯r0, ω) is approximated by

∼ −2k Φ(¯r0, ω) = [xs∆x + ys∆y + zs∆z] (5.59) Rs

172 The focused signal C(¯r0) can be written as

N 1 ZZ X 2ik C(¯r ) = dtdω f (−kˆ , kˆ ) · exp (x ∆x + y ∆y + z ∆z) (5.60) 0 R2 j i i R s s s s j=1 s

vt in which t ∈ 0, , ω ∈ [ωc − πB, ωc + πB], and ωc is the radian center frequency. Rs Knowing that

rφi = vt ⇒ rdφi = vdt (5.61) and r = Rs sin θi, dt is transformed to dφi by

R sin θ dt = s i dφ (5.62) v i

φi ∈ [0, 2π] (5.63)

Also,

dω = cdk (5.64) πB πB k ∈ [k − , k + ] (5.65) c c c c

where kc is the center wavenumber or radian period (c = λf where λ is the time c period, f is the time frequency. By analogy, k = is the radian period.) The ω focused signal C(¯r0) becomes

πB k + c c 2π Z Z N cRs sin θi X C(¯r ) = dk dφ f −kˆ , kˆ 0 v i j i i πB 0 j=1 k − c c · exp [2ik (sin θi cos φi∆x + sin θi sin φi∆y + h∆z)] (5.66)

173 The circular ﬂight path of ISAR measurements suggests

∆x = ∆r cos φ (5.67)

∆y = ∆r sin φ (5.68)

where φ is the angular variable locally deﬁned for (∆x, ∆y) and ∆r = p∆x2 + ∆y2.

Substitute these relations into C(¯r0).

πB k + c c 2π Z Z N cRs sin θi X C(¯r ) = dk dφ f −kˆ , kˆ 0 v i j i i πB 0 j=1 k − c c · exp [2ik (sin θi cos(φi − φ)∆r + h∆z)] (5.69)

Integration over φi provides

πB k + c c 2π Z Z N cRs sin θi X C(¯r ) = dk dφ f −kˆ , kˆ 0 v i j i i πB 0 j=1 k − c c · exp [2ik (sin θi cos(φi − φ)∆r + h∆z)] (5.70) πB k + c c Z N cRs sin θi X = dk f −kˆ , kˆ exp [2ikh∆z] v j i i πB j=1 k − c c 2π Z · dφi exp [2ik∆r sin θi cos(φi − φ)] (5.71) 0 πB k + c c Z N cRs sin θi X = dk f −kˆ , kˆ exp [2ikh∆z] · (2π)J (2k∆r sin θ ) (5.72) v j i i 0 i πB j=1 k − c c

174 where J0 is the Bessel function of zero order. The integral representation of the Bessel function is 1 Z 2π h nπ i Jn(x) = dφ exp −ix cos φ − inφ + i (5.73) 2π 0 2

J0(x) can be deﬁned by an inﬁnite power series.

x2 x4 x6 J (x) = 1 − + − + ··· (5.74) 0 22 22 · 42 22 · 42 · 62

The combined range and cross-range resolution, ∆r, is constrained by

λ c 1 1 2k∆r sin θi ≤ π ⇒ ∆r ≤ = · · (5.75) 4 sin θi 2∆f 2nf sin θi

2π c since k = and λ = . ∆f is the frequency resolution, and n is the length of the λ f f discrete frequency vector (f = nf ∆f). Considering the previous example in which 1 π λ = m (f = 12 GHz) and assuming z = h << y (θ ∼= ), the spatial resolution of 40 s s i 2 ISAR measurements is approximately 0.00625 m (0.25 in), a ﬁgure much smaller than the range and cross-range resolutions of traditional radar and SAR measurements. This proves the eﬃcacy of ISAR over traditional radar measurements.

5.3 Backprojection Algorithms

5.3.1 Range Compression

In this section, derivation of backprojection algorithms is provided. Recall the time- independent SAR point response S(x, y sin θi) as

πy sin θi πx S(x, y sin θi) = sinc sinc (5.76) ρr ρxr

It is equivalent to express S(x, y sin θi) in S(¯rs,j) in whichr ¯s,j is the distance of the point scatterer in the slant plane.

πR S(¯r ) = sinc s,j (5.77) s,j ρ

175 where Rs,j = |r¯s,j|. The time-dependent S(¯rs,j) can be written by

ωc + πB 1 Z S(¯rs,j, t) = 2 dω · exp [iωt] (5.78) Rs,j ωc − πB

2R The range compression on S(¯r , t) is conducted by shifting the time t to tˆ= t− s,j , s,j c leading to

ω + πB c Z 2Rs,j ˆ 1 2Rs,j S(¯rs,j, t − ) = S(¯rs,j, t) = 2 dω · exp iω t − (5.79) c Rs,j c ωc − πB

which gives ˆ B ˆ ˆ S(¯rs,j, t) = 2 exp iωct · sinc Bt (5.80) Rs,j 2R where the term t − s,j shifts the origin of traveling time to the time receiving the c reﬂected signals. By shifting the origin of time to the center of the scatterer, the range distance Rs,j has been compressed in the way that the center of the reconstructed image coincides with the center of the scatterer.

In this time-shifting process, it is assumed that the traveling speed of radar signals is the speed of light in free space (c = 3 × 108 m/s), which is only true when the propagation and scattering of radar signals entirely occur in free space. However, in the circumstance of scattering from multi-layered dielectrics, some received signals are reflected from in-depth scatterers/interfaces as the result of multiple reflections in c √ multi-layered dielectrics, whose traveling speed is 0 for non-magnetic dielectrics, r 0 where r is the dielectric constant or relative permittivity. Consider two cases of the reflection of radar signals in Figure 5-5. Two radar antennas transmit signals toward a PEC (perfect electric conductor) plate of thickness d. In the propagation path of radar signals, there is only Medium 1 for Antenna A, and there are Media 1 and 2 for

0 0 Antenna B. Assume non-magnetic dielectrics (relative permeativity µr1 = µr2 = 1) 0 and Medium 1 to be free space (r1 = 1). Considering only the ﬁrst reﬂection from

176 the PEC plate, the diﬀerence in the traveling time of Antennas A and B is

d ∆t = 2p0 − 1 · (5.81) r2 c

Should the surface of the PEC plate facing the radar be the center of a scatterer,

0 the time difference ∆t becomes significant when (1) r2 is large, or (2) thickness d is large. Since the footprint is assumed to be a Gaussian amplitude distribution, the correlation decays as the amplitude of a Gaussian distribution decays from its peak. Consequently, this time difference will project the center of the scatterer to a shifted position on the time axis as illustrated in Figure 5-6, leading to the shifting of scattering signals in reconstructed images. This creates an error for locating the center of defect using reconstructed images. To rectify or minimize this ∆t error, accurate estimation on the dielectric properties of target material is necessary, leading to the need for a comprehensive dielectric modeling of multi-layered cementitious composites.

Express the radar location vectorr ¯s by its position on the synthetic aperture, ξ, as shown in Figure 5-7. It is clear that

ξ = |r¯s|φi = Rsφi (5.82)

The total range-compressed focused signal collected at ξ on the aperture, D(ξ, tˆ), is the integration of the range-compressed focused signal received from all scatterers in the domain Ωs. In cylindrical coordinate systems,

R 2π Z s Z D(ξ, tˆ) = dr¯j dφj · G(¯rj, φj)S(¯rs,j, tˆ) (5.83) 0 0 where G(¯rj, φj) is a scattering amplitude density function at (¯rj, φj). Or it can be

177 written to be

xmax ymax Z Z q 2 2 D(ξ, tˆ) = dx dy · G(x, y)S( ((xs − xj) + (ys − yj) ), tˆ) (5.84)

xmin ymin

in Cartesian coordinate systems. D(ξ, tˆ) is the range-compressed focused signal for 2R the radar located at ξ on the aperture, whose time being shifted by s,j . c

5.3.2 Backprojection Processing

The backprojection processing can be conducted by deﬁning the backprojected signal

BBP (ξ, t) as ∂D(ξ, tˆ) B (ξ, t) = C · (5.85) BP BP ∂t

where CBP is the backprojection coeﬃcient hereby deﬁned in order to yield an ideal bandpass transfer function and

1 4π 2 C = (5.86) BP i c

when an ideal point scatterer with unit cross section atr ¯j is considered and the frequency integration is performed by ω. When the frequency integration is carried out by f, 1 4π 2 C = (5.87) BP 2πi c

∂D(ξ, tˆ) is deﬁned continuously, while D(ξ, t) is usually discrete. To prevent a po- ∂t ∂D(ξ, tˆ) tential discontinuity in from occurring, a matched ﬁlter M(tˆ) is applied on ∂t the derivative.

tˆ tˆ ∂D(ξ, tˆ) ∂ Z Z ∂M(tˆ− t0) = dt0 · D(ξ, tˆ)M(tˆ− t0) = dt0 · D(ξ, tˆ) (5.88) ∂t ∂t ∂t 0 0

178 This convolution operation is also advantageous for computational eﬃciency since ∂M(tˆ− t0) can be calculated in advance. In the frequency domain, convolution is ∂t performed by multiplying a phase factor. Finally, the backprojection image is given by R θ Zs int I(¯r, φ) = y dξ · BBP (ξ, tˆ) (5.89) 0

which is the integration of BBP (ξ, tˆ) over the entire aperture length Rsθint. I(¯r, φ) is a two-dimensional, spatial image (range vs. cross-range) of the structure.

5.3.3 Support Band Analysis – Method of Stationary Phase

The purpose of support band analysis is to deﬁne a band support region Ω in which the spatial spectrum of backprojection image I(x, y) is guaranteed non-trivial, using the method of stationary phase [22]. This analysis is performed in the following paragraphs. 2R Consider a time-dependent SAR point response S(¯r , t − s,j ) as s,j c

ω + πB c Z 1 2Rs,j S(¯rs,j, t) = 2 dω · exp iω t − (5.90) Rs,j c ωc − πB

The scattering amplitude density function G(x, y) is δ(x − xj)δ(y − yj). The ranged- compressed signal D(ξ, t) is

ω + πB c Z 1 2Rs,j D(ξ, t) = 2 dω · exp iω t − (5.91) 2πRs,j c ωc − πB f Zmax 1 2Rs,j = 2 df · exp i2πf t − (5.92) Rs,j c fmin

179 Therefore, f Zmax ∂D(ξ, t) i2π 2Rs,j = 2 df · f · exp i2πf t − (5.93) ∂t Rs,j c fmin

The backprojected signal BBP (ξ, t) is

f 2 Zmax 4π 1 2Rs,j BBP (ξ, t) = · 2 df · f · exp i2πf t − , (5.94) c Rs,j c fmin

and the backprojection image I(x, y) is

R θ f 2 Zs int Zmax 4π yf 2Rs,j I(x, y) = dξ df · 2 · exp i2πf t − (5.95) c Rs,j c 0 fmin

where Rs,j is a function of ξ since Rs (orr ¯s) depends on ξ. The spatial spectrum of I(x, y) is obtained by

xmax ymax 1 Z Z I˜(k , k ) = dx dy · I(x, y) · exp [i (2πft − k x − k y)] (5.96) x y (2π)2 x y xmin ymin

This leads to

xmax ymax Rsθint fmax 22 Z Z Z Z yf I˜(k , k ) = dx dy dξ df · · x y c R2 x y s,j min min 0 fmin 2R exp i2πf t − s,j + i (2πft − k x − k y) (5.97) c x y

Consider the far-ﬁeld ISAR measurements being conducted at a constant elevation ˜ ˜ (zs = constant). The variables in I(kx, ky) are illustrated in Figure 5-7. I(kx, ky) is

the reconstructed image assembled byr ¯ = {r¯(x, y)|(x, y) ∈ ΩBP } in account of the

scattering from the scatterer j atr ¯j observed by the monostatic radar atr ¯s whose

position is deﬁned by ξ.ΩBP is the domain of the image to be reconstructed, typically

180 ˜ to be rectangular. The phase factor Φ(x, y, ξ, f) in I(kx, ky) is

2R Φ(x, y, ξ, f) = 4πft − 2πf s,j − (k x + k y) (5.98) c x y

where

2q t = (x − x )2 + (y − y )2 (5.99) c s s q 2 2 Rs,j = (xs − xj) + (ys − yj) (5.100) ξ xs = Rs · cos φi = · cos φi (5.101) φi ξ ys = Rs · sin φi = · sin φi (5.102) φi

Assuming that the monostatic radar operates at constant range Rs, the radar position

(xs, ys) is only a function of φi. Replacing 2πf by ω, the phase factor Φ(x, y, φi, ω) can be written to be

4ω q Φ(x, y, φ , ω) = (x − R cos φ )2 + (y − R sin φ )2 i c s i s i 2ω q − (R cos φ − x )2 + (R sin φ − y )2 − k x − k y (5.103) c s i j s i j x y

where the wave numbers kx and ky remain unchanged in the four-dimensional integral over (x, y, φi, ω). The principle of stationary phase suggests that the condi- ˜ tion of Φ(x, y, φi, ω) determines the condition of I(kx, ky). The ﬁrst derivatives of

181 Φ(x, y, φi, ω) are

∂Φ 4ω x − Rs cos φi = · − kx (5.104) ∂x c q 2 2 (x − Rs cos φi) + (y − Rs sin φi)

∂Φ 4ω y − Rs sin φi = · − ky (5.105) ∂y c q 2 2 (x − Rs cos φi) + (y − Rs sin φi)

∂Φ 4ωRs (sin φi − cos φi) = q ∂φi 2 2 c (x − Rs cos φi) + (y − Rs sin φi) 4ωR (cos φ − sin φ ) − s i i (5.106) q 2 2 c (Rs cos φi − xj) + (Rs sin φi − yj) ∂Φ 4q = (x − R cos φ )2 + (y − R sin φ )2 ∂ω c s i s i 2q − (R cos φ − x )2 + (R sin φ − y )2 (5.107) c s i j s i j

∂Φ ∂Φ ∂Φ ∂Φ The stationary point at which ∇Φ = 0 occurs when , , , = 0. Let ∂x ∂y ∂φi ∂ω ˆ the stationary point occur at (ˆx, y,ˆ φi, ωˆ) which are determined by

ckx 4ω xˆ = Rs cos φi + s · (y − Rs sin φi) (5.108) ck 2 1 − x 4ω

cky 4ω yˆ = Rs sin φi + s · (x − Rs cos φi) (5.109) ck 2 1 − y 4ω (x2 + y2) − (x2 + y2) ˆ j j y − yj ˆ cos φi = − sin φi (5.110) 2Rs(x − xj) x − xj 4(x2 + y2) − (x2 + y2) ˆ j j 4y − yj ˆ cos φi = − sin φi (5.111) 2Rs(4x − xj) 4x − xj

182 xs ys ∂Φ Since cos φi = and sin φi = , the condition of = 0 is further expressed by Rs Rs ∂φi

2 2 2 2 (x + y ) − (xj + yj ) y − yj xs = − ys 2(x − xj) x − xj ˆ −1 ys ⇒ φi = tan (5.112) xs

∂Φ and the condition of = 0 provides ∂ω

2 2 2 2 4(x + y ) − (xj + yj ) 4y − yj xs = − ys 2(4x − xj) 4x − xj ˆ −1 ys ⇒ φi = tan (5.113) xs

˜ The stationary points must fall in the integration limits of I(kx, ky) in order to obtain ˜ non-zero values of I(kx, ky). In other words, (ˆx, yˆ) > 0. The conditions on (ˆx, yˆ)

provide the constraints on (kx, ky). Conditionx ˆ > 0 provides

s 2 C1 4ω kx > 2 (5.114) 1 + C1 c

Rs cos φi where C1 = . Conditiony ˆ > 0 provides Rs sin φi − y

s 2 C2 4ω ky > 2 (5.115) 1 + C2 c

Rs sin φi where C2 = . Therefore, the stationary point is inside the limits of Rs cos φi − x integration if and only if (kx, ky) ∈ Ω where

s 2 s 2 c C1 4ω C2 4ω Ω = {(kx, ky)|fmin < k < fmax, kx > 2 , ky > 2 } (5.116) 4π 1 + C1 c 1 + C2 c

p 2 2 and k = kx + ky. Ω is the band support region for I(kx, ky).

183 z

Flight path rxyzs = ()sss,, Ending position Starting position

θ ˆ i ki

Footprint domain G

x (a) Inverse synthetic aperture radar (ISAR)

Flight path Footprint domain G

g y

Starting position Ending position φi

Synthetic aperture θint x (b) ISAR – Plane view

Figure 5-4: Conﬁguration of ISAR measurements

184 Medium 1 (ε r′1 , µr′1 )

Rs, j

PEC

Rsj, − d d

Medium 1 (ε r′1 , µr′1 ) Medium 2 (ε′ , µ′ ) r 2 r 2 Figure 5-5: Two cases of the reﬂection of radar signals

d ∆t =−⋅21ε′ r 2 c

Actual origin Projected origin

Figure 5-6: Time shifting error

185 y

rrrs, jsj= −

r r ξ = r ⋅φ j s φ s i i x

rxy= ( , )

˜ Figure 5-7: Variables in I(k x, ky) and their relationship

186 5.3.4 Advantages of Backprojection Algorithms

Main advantages of the backprojection algorithm include: (1) lower and localized artifact levels than frequency-domain algorithms, (2) easy adjustment to an approximate inverse formula for perturbed problems, (3) readily for parallel computing with limited interprocessor communications, and (4) simple motion compensation by time- shift operation [177, 194]. As a result of the advantages mentioned above, sub-images with different bandwidths and angular ranges can be available before the physical inspection (far-field ISAR measurements) is completed. This enables the technique for different purposes of inspection. For instance, sub-images of narrow bandwidths and angular ranges (although poor resolution) are advantageous for preliminary inspections due to their rapid inspection and processing. Final images which utilize full bandwidth and full angular range data are useful for detailed inspections once a suspicious local area can be determined. The superposing process of sub-images provides not only the gradually-improved image resolutions (range and cross-range), but also the evolution of scattering signals in the image. For example, angular sensitivity of defect scattering signals is revealed by processing ISAR measurements at each sub-aperture. It is evident that understanding the pattern (evolution, convergence) of defect scattering signals in backprojection images is beneficial to studying actual defects or damages. Knowledge about the needed frequency bandwidth and angular range for revealing certain types of defects in backprojection images can also be established based on such understanding.

5.4 Implementation and Coding Procedure

In this dissertation, the backprojection processing is implemented mainly by the Matlab r (MATrix LABoratory) language (by The MathWorks), a high-performance technical computing environment, and some by the FORTRAN (FORmula TRANs- lator) 90 (or F90) language. While Matlab r is a scripting language similar to the C++ language, its extensibility (Toolboxes), eﬃciency (matrix computations, graph-

187 ics generation), and ability to directly communicate with other computing languages (e.g., FORTRAN and C++) potentially ease the effort for future modifications and expansions. The original codes were developed by the MIT LL and modified by the author. In what follows, implementation of the backprojection processing is described in the coding procedure using a sample measurement signal from the specimen AD1

◦ (8GHz∼12GHz, HH polarization, φi = −15 , oblique incidence) in which basic SAR processing steps are also included.

1. Input the one-dimensional (1D), steady-state, complex far-ﬁeld ISAR measure-

ment/signal (amplitude and phase) collected at the azimuth angle φi and over

the frequency range of [fmin, fmax]. The length of the signal is n. Amplitude

(A = A(fi)) and phase (ϑ = ϑ(fi)) of the sample far-ﬁeld ISAR data used are shown in Figure 5-8.

3 -20 2 -22 1 -24 0 -26

Phase (rad) -1 Amplitude (dBsm) -28 -2

-30 -3 8 9 10 11 12 8 9 10 11 12 Frequency (GHz) Frequency (GHz) Figure 5-8: Far-ﬁeld ISAR measurement of the specimen AD1 – HH polarization, ◦ φi = −15

2. Combine the amplitude and phase of the signal to obtain the complex form of the signal by

A ℘(fi) = 10 20 · exp [iϑ] (5.117)

188 The combined complex signal is shown in Figure 5-9

0.08 0.1 0.06 0.08 0.04 0.06 0.02 0.04

0.02 0 -0.02 Real part 0

-0.02 part Imaginary -0.04

-0.04 -0.06

-0.06 -0.08 8 9 10 11 12 8 9 10 11 12 Frequency (GHz) Frequency (GHz) Figure 5-9: Complex form of the far-ﬁeld ISAR measurement

3. Perform SAR processing by choosing a desired sidelobe pattern. Reduction of sidelobe levels (SLLs) is achieved by applying a weighting function to the raw data as shown in Figure 5-8). A weighting function used in this dissertation is

lj w = 1 + 2Σm f · cos 2π · (5.118) j l=1 l 2

where m is the order the weighting function and the shape function

−1 " 2 2 2 1 2 # " 2 # 1 l a + π m − 2 l fl = 1 − · · 1 − (5.119) 2 m 2 2 1 2 (l − 1)! m−1 a + π l − 2 (l+1)! √ where a = log(SLL + SLL2 − 1) and SLL is the sidelobe levels. Figure 5-10 wj shows the normalized weighting function P with diﬀerent SLLs ranging [wj] SLL/20 from 30dB to 90dB. The SLLs in dB is determined by SLLdB = 10 . The weighted complex signal is shown in Figure 5-11.

4. Prepare an empty range-cross-range image as a base image for backprojection. Default size of the base image is 1 m-by-1 m. Since the size of used GFRP-

189 0.014 30dB 0.012 40dB 50dB 60dB 0.01 70dB 80dB 0.008 90dB

0.006 Weighting

0.004

0.002

0 50 100 150 200 Data point

Figure 5-10: Desired sidelobe pattern with diﬀerent sidelobe levels (SLLs) ranging from 30dB to 90dB

wrapped concrete specimens is less than 0.5 m, 1 m-by-1 m is capable of covering the specimen and the scattering signals. Another size, 2 m-by-2 m, is also used in some cases. Increase the image resolutions by oversampling the grids to prepare for the coordinate transformation after performing one-dimensional (1D), complex discrete Fourier transform (DFT) of the weighted data. The

3 used oversampling ratio (rOS) is to be greater than 2 = 8 for less interpolation errors during the coordinate transformation. 24 = 16 and 25 = 32 have been used for obtaining satisfactory results. Perform the center-shifting operation again on the obtained 1D Fourier conjugate of the elongated data. The amount n of shifting index is 1 + DFT . The length of the oversampled/elongated signal 2 for 1D complex DFT is determined by

log2(n·rOS ) nDFT = 2 (5.120)

5. Shift the center of the weighted data of length n to the origin by truncating

190 -8 -8 x 10 x 10 4

3 1

2 0

1 -1

0 -2 Real part -1

Imaginary part Imaginary -3 -2 -4 -3

8 9 10 11 12 8 9 10 11 12 Frequency (GHz) Frequency (GHz) Figure 5-11: Weighted complex form of the far-ﬁeld ISAR measurement

the data points of indices [nc : n] and moving it to [1 : n − nc + 1] of an n elongated/oversampled data with length n0 = r n, where n = + 1. The OS c 2 other half of the weighted data, [1 : nc −1], is placed at the end of the elongated data. Perform zero-padding for the increased points of the elongated data. Conduct 1D complex discrete Fourier transform on the elongated data collected

at (φi, f). The 1D Fourier conjugate (also complex) of the elongated data is obtained. The shifted 1D DFT of the complex signal is shown in Figure 5-12.

6. Prepare the radiation pattern projection of the radar from its local coordinate system to the image plane. The radiation pattern at the local coordinate system ¯ T of the radar is deﬁned by the Poynting vector Pl = (Pxr,Pr,Ph) as shown in Figure 5-13. The nominal Poynting vector of the radar at the local coordinate ¯ T system is Pl = (0, 1, 0) or Pr = 1 (far-ﬁeld). The coordinate transformation matrix from the local coordinate system of the radar to the image plane is

    cos φi − sin φi 0 1 0 0     P¯ =   ·  0 0  · P¯ (5.121) g  sin φi cos φi 0   0 cos θi − sin θi  l     0 0 0 0 1 0 sin θi cos θi

191 -7 -6 x 10 x 10 0 0 -0.5 -2 -1 -4 -1.5 -6 -2 Real part -8 Imaginary part Imaginary -10 -2.5

-12 -3

1000 2000 3000 4000 1000 2000 3000 4000 Data point Data point Figure 5-12: Shifted 1D DFT of the weighted complex signal

0 π where θi = − θi is the grazing angle. φi is the azimuth angle. In the case 2 π considered in this research, θ = . i 2

Ph z

Pr θi Image plane Pxr

φi

Figure 5-13: Poynting vector of the radar

7. Multiply the shifted 1D DFT signal with the phasor ΦR which is

f Φ = exp −i4π R (5.122) R c s,j

and with the transformed Poynting vector based on the azimuth angle. Form the BP image by projecting the multiplied signal onto the base image using the

192 range distance of each pixel (Rs,j) (with respect to the center of the image) and

the azimuth angle (φi). The projected image is shown in Figure 5-14.

Figure 5-14: Projected image of the transformed far-ﬁeld ISAR measurement

5.5 Eﬀects of Aperture Size and Bandwidth

5.5.1 Aperture Size

As shown in Figure 5-14, the 1D signal at a given azimuth angle generates an image with constant scattering signals along the cross-range direction since physically no angular variation can be discovered if the measurement is only made at one azimuth angle. Should we expand the azimuth vector of input measurements, for example, from (−15◦) to (−14.8◦, −15◦, −15.2◦) (the far-ﬁeld ISAR measurements are shown in Figure 5-15) and follow the same procedure described in the previous section, the projected image becomes Figure 5-16. In Figure 5-16, it is found that variation of the scattering signals in the cross-range direction occurs as expected. Projected images using azimuth vectors (−14.6◦, −14.8◦, −15◦, −15.2◦, −15.4◦) and (−14.4◦, −14.6◦, −14.8◦, −15◦, −15.2◦, −15.4◦, −15.6◦) are shown in Figures 5-17 and 5- 18, respectively. Ultimately, in this example, the half-aperture and full-aperture measurements produce images shown in Figures 5-19 and 5-20, respectively.

193 12 12 3 -1 11.5 11.5 -2 2 11 11 -3 1 10.5 -4 10.5

10 -5 10 0

9.5 -6 9.5 -1 Frequency (GHz) Frequency -7 (GHz) Frequency 9 9 -8 -2 8.5 8.5 -9 8 8 -3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 ∆Φ i ∆Φ i

Figure 5-15: Full-bandwidth far-ﬁeld ISAR measurements with an azimuth vector of (−14.8◦, −15◦, −15.2◦)

5.5.2 Bandwidth

The effect of different bandwidths on reconstructed BP images is studied by using the same measurement as shown in Figure 5-8. The full-bandwidth of the collected far-field ISAR measurement is 4 GHz (8 GHz ∼ 12 GHz). The bandwidth of collected measurements represents the probing ability of the radar in the range direction. As a result of it, wider bandwidths offer better detectability in the range direction. Consequently, range resolution of the radar and of BP images can be improved or reduced by the use of an increasing or decreasing bandwidth.

Figures 5-21, 5-22, and 5-23 show the full-aperture BP images using diﬀerent bandwidths (in a descending order). It is clear that reducing the bandwidth of input measurements results in poorer range resolution in the BP images. Equivalently, better detectability in the range direction can be provided, given wider bandwidths of the measurement.

194

Figure 5-16: Full-bandwidth far-ﬁeld ISAR measurements with an azimuth vector of (−14.8◦, −15◦, −15.2◦)

5.6 Summary

In this chapter, theoretical background of SAR and ISAR processing for image reconstruction is provided. Fast backprojection algorithm is used for numerically implementing the concept of tomographic reconstruction in this work. The implementation and coding procedure is also described with far-field ISAR measurement examples. Effects of aperture size and bandwidth in reconstructed images are addressed. Re- search findings are summarized as follows.

• As demonstrated in this chapter, ISAR processing offers a flexibility for improving the resolution of radar measurements by incorporating additional measurements with respect to frequency (range and cross-range resolutions) and incident angle (cross-range resolution). The basis of ISAR processing is the movement of radar and the integration of measurements at different azimuth angles. The bandwidth of measurements is proportional to the resolution in the range direction, and the aperture size decides the focusing capability in the cross-range direction.

• The importance of dielectric properties of materials is identiﬁed in this chapter.

195

Figure 5-17: Full-bandwidth far-ﬁeld ISAR measurements with an azimuth vector of (−14.6◦, −14.8◦, −15◦, −15.2◦, −15.4◦)

While the dielectric properties of the structure are not required as input for rendering images containing near-surface defects, they become important when the center of backprojection is deep inside the dielectric medium; the backprojected center of scattering signals in reconstructed images will be shifted from its actual location if the diﬀerence in traveling time is not accounted (defect locating error). This leads to the research need for better understanding the dielectric properties of GFRP-concrete systems including concrete, a porous cementitious composite, as described in Chapter 6 Dielectric Modeling of GFRP-concrete Systems.

196

Figure 5-18: Full-bandwidth far-ﬁeld ISAR measurements with an azimuth vector of (−14.4◦, −14.6◦, −14.8◦, −15◦, −15.2◦, −15.4◦, −15.6◦)

Figure 5-19: Full-bandwidth half-aperture backprojection image – ∆φi ×[−150 : 1 : 1] ◦ (∆φi = 0.2 )

197

Figure 5-20: Full-bandwidth full-aperture backprojection image – ∆φi × [−150 : 1 : ◦ 150] (∆φi = 0.2 )

Figure 5-21: Sub-bandwidth full-aperture backprojection image – [8, 11] (GHz)

198

Figure 5-22: Sub-bandwidth full-aperture backprojection image – [8, 10] (GHz)

-0.4 0.05 ) -0.2 2 0.04 0.03 0 0.02 0.01 Power (W/m Power

Cross-range (m) 0.2 0.4 0.2 0.4 0.4 0 0.2 0 -0.2 -0.2 -0.4 -0.4 -0.4 -0.2 0 0.2 0.4 Cross-range (m) Range (m) Range (m) Figure 5-23: Sub-bandwidth full-aperture backprojection image – [8, 9] (GHz)

199 200 Chapter 6

Dielectric Modeling of GFRP-concrete Systems

”Different men can be affected by one and the same object in different ways, and one and the same man can be affected in different ways by one and the same object at different times.” —– Ethics, Prop. LII., Baruch de Spinoza (1632∼1677)

Any use of electromagnetic waves for investigating materials must deal with the dielectric properties of materials, this includes the case of radar NDT for construction materials. As indicated by Maxwell’s equations materials are characterized by electric permittivity , electric conductivity σ, magnetic permeability µ, and magnetic conductivity σ∗. Materials are described by these properties and classified accordingly. The frequency-dependence of these properties is termed dielectric dispersion. The essential goal of dielectric modeling aims at the development of mathematical models/equations for the determination of dielectric properties at different measurement frequencies. In Chapter 3 Numerical Simulation concrete is assumed as a homogeneous, isotropic, and lossless dielectric medium. Knowing the heterogeneity, multi-phase nature of concrete, it is realized such assumption is highly simplified. To improve the accuracy of material modeling in numerical simulation, appropriate dielectric

201 models must be developed. Additionally, in Chapter 5 Image Reconstruction, it is also identiﬁed that the knowledge of dielectric properties can further improve the precision of backprojection processing when deep defects are to be detected and located. Therefore, the purpose of this chapter is to model the dielectric dispersion of considered materials (water, epoxy, E-glass fabric, GFRP, and oven-dried hydrated Portland cement paste) in the frequency range of 8 GHz to 18 GHz using reported experimental measurements in literature [75, 107, 153, 96, 113, 119, 107, 154, 271]. In view of the novel use of dielectric properties in civil engineering, deﬁnition and physics of dielectric properties and dielectric dispersion are introduced as the background information. Applications of dielectric properties in civil engineering are provided, indicating the increasing importance of dielectric properties in the profes- sion. A geometrical analysis technique used for dielectric modeling is introduced. Dielectric models of considered materials (water, epoxy, E-glass fabric, GFRP, and oven-dried hydrated cement paste) are provided. Challenges in the dielectric modeling of concrete as a multi-phased cementitious composite are also addressed.

6.1 Background

Dielectric properties of a material can be used to determine other material properties such as moisture content, bulk density, bio-content, chemical concentration, and stress-strain relationship. Such knowledge can be utilized for research and application in food science, medicine, biology, agriculture, chemistry, electrical devices, defense industry (security), and various engineering disciplines including civil engineering. This section provides the background information regarding the impacts of dielectric properties of materials in general and in the NDT problem of the GFRP-concrete systems.

6.1.1 Deﬁnition and Physics of Dielectric Properties

Dielectric properties of materials can be interpreted both microscopically and macroscopically. Microscopically, dielectric properties represent the polarization ability of

202 molecules in the material corresponding to an externally applied electric ﬁeld E¯. Macroscopically, dielectric properties are the relationship between the applied electric ﬁeld strength E¯ (volts/m2) and the electric displacement D¯ (coulombs/m2), both externally measured.

Dielectric properties are the collective term of electric permittivity (farads/m or F/m), electric conductivity σ (siemens/m or S/m), magnetic permeability µ (henrys/m or H/m), and magnetic conductivity σ∗ (ohms/m or O/m). Materials are described and classified by these properties into various types such as metals and dielectrics. In this research work, only isotropic materials are considered, meaning that all these properties are described by first-order tensor. In other words, modeling using second-order (or rank-two) tensors for anisotropic materials is not considered in this research. These quantities can be real or complex, depending on the nature of the material. When alternating current (a.c.) fields are applied on the material under investigation, generally these quantities are complex. Definition of dielectric properties of materials is introduced in the following paragraphs.

• Complex electrical permittivity (F/m) describes the ability a material interacts with an applied electric field. It is defined as the ratio between the electric displacement field D¯ (coulombs/m2 or C/m2) and the electric field E¯ (volts/m or V/m). Generally, D¯ = E¯ (6.1)

where = (ω) for dielectric materials. It represents the ability a material to permit an electric ﬁeld passing through the material. For non-metallic materials (or dielectrics) their frequency-dependent response subject to applied a.c. ﬁelds is the result of the molecular polarizability [28]. Their delayed and attenuated response is also observed and described as the dielectric dispersion phenomenon attributing to several polarization phenomena in the microscopic level [113]. To account for these energy absorption and losses an imaginary part is needed in the dielectric description of the material property. Therefore, the complex

203 electrical permittivity (or complex permittivity) is deﬁned as

= 0 + i(−00) = 0 − i00 (6.2) where 0 is the real part of representing the ability a material can store the energy carried by the electromagnetic ﬁeld transmitting through it, and 00 is the imaginary part representing energy absorption and loss. The negative sign deﬁnition of 00 is applied since energy dissipation/loss occurs to the materials considered in this research. Positive sign of 00, on the other hand, suggests energy creation.

The measured values of 0 and 00 mainly depend on measurement frequency and temperature, while in some cases, as well as on pressure [126]. For example, the Debye equations provide a frequency-dependent representation of 0 and 00, satisfying the Kramers-Kronig relations [140], as [65]

− 0(ω) = + s ∞ (6.3) ∞ 1 + (ωτ)2 ωτ( − ) 00(ω) = s ∞ (6.4) 1 + (ωτ)2 where ω = 2πf is the angular frequency (rad/s or rad-Hz), f is the temporal frequency (1/s or Hz), ∞ is the permittivity measured by the alternating current

(a.c.) field at frequency ω = ∞ (infinite frequency), s is the permittivity measured by the direct current (d.c.) field at frequency ω = 0 (static frequency), and τ is the characteristic relaxation time (s/rad). The reported relaxation time τ is usually represented by τ = (s). Materials whose response can be described t 2π by the Debye equations are called the Debye material. Details about the Debye equations will be introduced in the modeling section of this chapter.

A dimensionless representation is also used for deﬁning the complex permittivity. The complex relative permittivity is deﬁned by

0 00 − 0 00 r = = = r − ir (6.5) 0 0

204 −12 where 0 is the electrical permittivity of free space and 0 = 8.85 × 10 F/m. 0 00 r is the dimensionless dielectric constant and r is the dimensionless loss factor. It is only the dimensionless nature leading to the name ”dielectric

0 constant” since r is not a constant when considered over a range of frequencies. Diﬀerent symbols were adopted for the dielectric constant such as κ [113] and Σ [75]. In this research, the representation is adopted.

0 00 The ratio between r and r is the loss tangent (or dissipation factor).

00 00 r tanδ = 0 = 0 (6.6) r

This dimensionless representation , tanδ is simpler than and advantageous 0 over the original (0, 00) representation because it clearly shows how the material is diﬀerent from free space (or vacuum).

• Complex magnetic permeability µ (H/m) describes the ability a material interacts with an applied magnetic field. It is the ratio between the magnetic flux density B¯ (webers/m2 or W/m2 or Tesla) and the magnetic field H¯ (amperes/m or A/m). B¯ = µH¯ (6.7)

µ is a scalar for isotropic materials. The complex magnetic permeability (or complex permeability) is used when magnetic losses are present in the material.

µ = µ0 − iµ00 (6.8)

where µ0 and µ00 are the real and imaginary parts of the complex permeability (henrys/m, or H/m), respectively. The negative imaginary part of µ suggests the energy dissipation. A dimensionless, relative complex permeability can be further deﬁned as 0 00 µ − iµ 0 00 µr = = µr − iµr (6.9) µ0

−7 where µ0 is the permeability of free space and µ0 = 4π × 10 H/m. Deﬁnition

205 of various materials based on is listed in Table 6.1. Threshold values were suggested by Epstein (1962) [79]. It was reported in the literature that Debye-

Table 6.1: Types of magnetic materials

Description Criterion

Ferromagnetic µr > 10 Magnetic Paramagnetic 1 < µr < 10 Diamagnetic µr < 1 0 00 Non-magnetic µr = µr = 1, µr = 0

type models can be used for modeling the complex permeability of magnetic materials [79].

• Apparent electrical permittivity a (F/m) is deﬁned by accounting for the direct current (d.c.) conductivity loss in the representation of complex permittivity in this research. Since the imaginary part of the complex permittivity represents the dielectric loss of the material, which is only part of the total energy dissipation from EM waves transmitting through the material. The total energy dissipation in the material can be expressed by the dissipated or absorbed power using the complex Poynting vector theorem [52] and the Maxwell’s equations. The complex Poynting vector P¯ is

1 I P¯ = E¯ × H¯ ∗ dS¯ (6.10) 2 S

where S is a closed surface and dS¯ is a vector element of area directed outward from the volume V , H¯ ∗ is the conjugate of the magnetic ﬁeld H¯ (A/m). The divergence theorem provides

I Z E¯ × H¯ ∗ dS¯ = ∇ · E¯ × H¯ ∗ dV¯ (6.11) S V

From vector calculus we have

∇ · E¯ × H¯ ∗ = ∇ × E¯ · H¯ ∗ − ∇ × H¯ ∗ · E¯ (6.12)

206 With Maxwell’s equations ∇ × E¯ = −iωB¯ and ∇× = −iωD¯ + J¯, Ohm’s law J¯ = σ · E¯, and B¯ = µ · H¯ , the complex Poynting vector becomes

1 I 1 Z E¯ × H¯ ∗ dS¯ = −iωµ · H¯ · H¯ ∗ + iω · E∗ − σ · E¯ E¯ dV¯ (6.13) 2 S 2 V

Since = 0 − i00 and µ = µ0 − iµ00, reorganizing the real and imaginary parts of the complex Poynting vector gives

1 I 1 Z E¯ × H¯ ∗ dS¯ = −iωµ0H¯ · H¯ ∗ − ωµ00H¯ · H¯ ∗ dV¯ + 2 S 2 V 1 Z ω0E¯ · E¯∗ − iω00E¯ · E∗ − σE¯ · E¯∗ dV¯ (6.14) 2 V

Rearranging the above equation provides

1 I Re E¯ × H¯ ∗ −dS¯ = 2 S ω Z σ µ00H¯ · H¯ ∗ + 00E¯ · E¯∗ + E¯ · E¯∗ d¯(V ) (6.15) 2 V ω 1 I Im E¯ × H¯ ∗ −dS¯ = 2 S ω Z µ0H¯ · H¯ ∗ − 0E¯ · E¯∗ dV¯ (6.16) 2 V where −dS¯ is the vector pointing toward the closed surface S and tangent to the surface boundary. The real part of the complex Poynting vector is the dissipated energy Pdis absorbed by the material, which consists of three parts; magnetic loss, electric loss, and conductivity loss. Since µ00 = 0 for non-magnetic materials, Pdis becomes

Z Z ω 00 σ ¯ ¯∗ ¯ ω 00 ¯ ¯∗ ¯ Pdis = + E · E dV = e E · E dV (6.17) 2 V ω 2 V and the eﬀective dielectric loss factor is deﬁned as

σ 00 = 00 + (6.18) e ω

207 Equivalently, an eﬀective conductivity can be deﬁned as

00 00 σe = ωe = ω + σ (6.19)

where σ = σs is the electrical conductivity (S/m) measured by the d.c. current at static frequency (ω = 0). This conductivity is the real part of the complex conductivity. With the eﬀective dielectric loss factor at hand, an apparent complex permittivity is deﬁned.

σ = 0 − i00 = 0 − i 00 + s (6.20) a e ω

Deﬁnitions of materials based on this apparent complex permittivity are listed in Table 6.2.

Table 6.2: Types of dielectric materials

Description Criterion 0 00 Perfect dielectric > 0, e = 0 0 00 Imperfect dielectric > 0, e > 0

• Complex electrical conductivity σ (S/m) describes the ability a material to conduct an applied electric current, represented by the ratio between the current ﬂux density J¯ (amperes/m2 or A/m2), and the electric ﬁeld E¯ (V/m2).

J¯ = σ · E¯ (6.21)

where σ is the scalar electrical conductivity (S/m) for isotropic materials. Fol- lowing the previously shown energy treatment for the deﬁnition of the complex permittivity, the complex electrical conductivity (or complex apparent conductivity) is deﬁned as

0 00 0 00 σa = iωa = iω ( − ie ) = σe + iσ (6.22)

208 where

0 00 00 σe = ωe = ω + σs (6.23) σ00 = ω0 (6.24)

The deﬁned eﬀective conductivity is actually the real part of the complex apparent conductivity.

0 00 σe = ωe = σe (6.25)

The d.c. conductivity, an frequency-independent term, is part of the apparent complex conductivity in the definition. The d.c. conduction effect is significant only in low-frequency or high-temperature situations, while it is insignificant in

0 microwave frequency range since in σe

00 ω > σs (6.26)

By constructing this relationship the deﬁnition regarding conductivity in complex apparent permittivity is connected to the complex apparent conductivity. This also suggests that the behavior of this complex apparent conductivity is, by deﬁnition, similar to the one the complex apparent permittivity exhibits. For the example of the Debye model, the real and imaginary parts of the complex apparent conductivity are

0 00 σa = σe + iσ (6.27) 2 0 σ∞ − σs (ωτ) (σ∞ − σs) σ = σs + = σs + (6.28) e 1 + (ωτ)−2 1 + (ωτ)2 −1 00 (ωτ) (σ∞ − σ s) ωτ (σ∞ − σs) σ = ω∞ + = ω∞ + (6.29) 1 + (ωτ)−2 1 + (ωτ)2 where σ∞ is the permittivity measured by the alternating current (a.c.) field at frequency ω = ∞ (infinite frequency), and σs is the permittivity measured by the direct current (d.c.) field at frequency ω = 0 (static frequency). Comparing

209 0 00 σe with ωe provides

s − ∞ = τ (σ∞ − σs) (6.30)

The relaxation time can be determined by

− τ = s ∞ (6.31) σ∞ − σs

The Debye-type behavior of complex electrical conductivity was observed by Grant (1958) in his measurements. It is evident that the loss tangent can also be expressed by 00 σ0 σ0 ω00 tan δ = e = e = e = e (6.32) 0 σ00 ω0 σ00 Those with a high conductivity are considered to be conductors such as metals, and those without conductivity are insulators such as glass. Nonconductive materials are also termed loss-less. Materials with slight conductivity are called low-loss. Table 6.3 summaries various materials deﬁned by conductivity.

Table 6.3: Types of materials deﬁned by conductivity

Description Criterion Note

Lossy 0 < tan δ When the electrical conductivity is present in the material (1) tan δ 1 General deﬁnition Low-loss (2) tan δ < 0.2 For concrete of dielectric constant in the range of 2.5 and 8 [10] (3) 10−4 ≤ tan δ ≤ [188] 10−3 (4) tan δ < 5 × 10−4 For electronic packing materials [123] Lossless tan δ = 0 When the electrical conductivity is not present in the material

210 In this dissertation, the complex electrical permittivity is used as the characteristic parameter in the dielectric modeling of materials for the following reasons.

• All the materials considered in this research are not magnetically dispersive, suggesting constant magnetic properties of the materials.

• Complex electrical conductivity is not directly measured from the received EM waves, but can be indirectly related to the complex electrical permittivity.

Therefore, emphasis of the dielectric modeling is placed on the development of dielectric models for determining complex permittivity. Only isotropic, non-magnetic dielectrics are considered, suggesting that the complex permittivity is a tensor of rank zero.

6.1.2 Dielectric Spectroscopy and Dielectric Dispersion

From the perspective of electromagnetism materials can be generally classified into metal and dielectric. Metals are usually perfect conductors, in which spontaneous response occurs when subject to the application of an external electric field, meaning that the creation and diminishing of internal field in metals occur without delay in time. It is because that the molecular dipoles in metals can freely rotate to align with the direction of the external field instantaneously without consuming extra energy. Consequently, the dielectric properties of metals are frequency-independent. Dielectric materials (or dielectrics), on the other hand, display delayed response when subject to an external electric field. It is because the rotation of molecular dipoles in dielectrics suffers from the inherent ”friction” around the dipoles. This ”hindered rotation” [65] further leads to a hysteretic response which serves as the energy absorption and storage mechanism of dielectrics, making dielectrics an ”insu- lating material” or an ”insulator” as called by Faraday (1965) [81]. The frequency-dependence behavior of complex permittivity of dielectric materials is called the dielectric dispersion. This behavior can be explained by different types of dispersion theories such Debye dispersion [65], Drude dispersion [147], Lorentz

211 dispersion [15], and other multiple poles dispersions. Among these dispersion theories, Debye and Drude dispersions characterize single-pole response, while Lorentz dispersion describes double-pole response. In the frequency range of investigation (8 GHz∼18 GHz) the considered materials in this research exhibit single-pole response in their Cole-Cole diagram (to be explained later), hence the use of multiple poles dispersion theories is excluded. The single pole dispersion theory or the single relaxation time theory was developed by several pioneer researchers in the early development of this field [65, 50, 51, 90, 147]. The physical meaning of the relaxation time is explained as follows: When a dielectric is subjected to the application of an external field, an internal field is generated inside the dielectric. When the external field is removed, the field inside the dielectric will decrease gradually. The time the internal field drops to 1/e of its original magnitude is called the relaxation time and usually denoted by τ. For gaseous substances there is no relaxation behavior since all the charges are free to rotate upon the application of external field. However, for liquid and solid dielectrics, the free rotation of bond charges is dampened by the ”friction” around the molecules, as the hindered rotation hypothesis proposed by Debye (1929) [65]. The study of the dispersion behavior of dielectrics is termed dielectric spectroscopy. The response of dielectrics is usually represented by an effective complex permittivity, characterized by the frequency-dependent fluctuation of the complex permittivity, and originated at several energy storage and dissipation mechanisms which the fluctuation is attributed to.

6.1.3 Energy Storage and Dissipation Mechanisms

A typical dielectric response of the real part of the complex permittivity is provided in Figure 6-1. Two fundamental mechanisms dominate the response of a material when it is placed in an electric ﬁeld; the energy storage and dissipation mechanisms. The energy storage mechanism is a capacitance eﬀect, caused by the polarizibility of the material or the molecular/molar polarizibility [28], which can be explained by the following polarization phenomena.

212

ε '

Frequency (Hz) AM/FM radio Microwave Infrared UV

Visual light

Figure 6-1: Dielectric dispersion of several types of polarization (Modiﬁed after Knight and Nur (1987) [135])

1. Electronic and atomic polarizations - These polarizations are attributed to the displacement of binding electrons/atoms from their equilibrium position on a molecular level. These polarization processes are rapid and considered as frequency-dependent with characteristic frequency ranging between 1012 ∼1016 Hz (infrared, visual light, and ultraviolet (UV)).

2. Dipolar/orientational polarization - The dipolar process is the rotation and alinement of permanent dipoles along electric ﬁeld lines. Its characteristic frequency ranges between 109 ∼1011 Hz (microwave).

3. Interfacial/ionic polarization - The interfacial polarization is also called the Maxwell-Wagner polarization. This process happens at the internal interfaces between two materials with diﬀerent conductivities with characteristic frequency less than 106 Hz (amplitude modulation (AM)).

213 In Figure 6-1, it is clear that, in the frequency range of interest (8 GHz∼18 GHz), only the dipolar polarization is dominant. The energy dissipation/loss mechanism in materials is attributed to two main components; conductivity dissipations and dielectric loss.

1. Conductivity dissipations - Conductivity dissipation process occurs when free-charge conduction is present in a dielectric. The dissipated energy is transformed into the heat of conduction. There are two types of conductivity dissipation: [270]

(a) d.c. conductivity or conductivity in the interfacial polarization – At low frequencies including direct current (d.c.) condition, the conduction in materials may be attributed to bond conduction [245] and d.c. hopping conduction [77]. To account for the electrical conduction, the imaginary part (loss factor) of complex permittivity needs to add one more term in the Debye’s dispersion equation as

σ (ω) = 0(ω) − i 00(ω) + s (6.33) ωs

where σs is the d.c. (static) electrical conductivity when σ = 0 (S/m) [219]. Modiﬁed dispersion equations by [50] and [63] can also be treated the same way to account for d.c. conduction. d.c. conductivity is critical at low frequency and high temperature situations.

(b) Conductivity in electrode polarization – The polarization effect takes place at electrodes when a dielectric displays significant bulk conductivity. Un- like the Maxwell-Wagner effect, the measured electrical conductivity increases at low frequencies. It is because of the presence of a high-impedance layer on the electrode surface. The reasons for such layer may be attributed to imperfect contact between the electrode and the dielectric. It is postulated that at low frequencies the conduction has sufficient time to transfer the applied field through the dielectric and the electrode layer, resulting in an increase in the measured capacitance. Electrode polarization occurs

214 in the frequency range up to 10 MHz and the conductivity induced by electrode polarization can be modeled as

2 σω(ωτe) σ(ω) = 2 2 (6.34) 1 + ω (τc + τe)

where σω is the a.c. electrical conductivity when ω = ∞ (S/m), τe is the

relaxation time of the electrode layer (s), τc is the relaxation time of the material (s) [270].

2. Dielectric loss - Field energy is dissipated by the friction between molecules/atoms when the bound-charge molecular/atomic displacement takes place. The dissipated energy is transformed into the heat of friction due to the drift of electrons or free ions in the applied ﬁeld. The dielectric loss is proportional to the total measured a.c. conductivity minus the d.c. conductivity.

The loss factor representing energy loss mechanism has two loss components. It is possible to distinguish these two loss components experimentally by using the Kramers-Kronig relations which depicts the interdependence of 0(ω) and 00(ω) [147]. Further information about dielectric polarizations and theory can be found at [65, 90, 113, 236, 245, 151]. In the case where perfect dielectric is considered, electrical conductivity is not present or no free-ion conduction occurs. However, most materials used in civil engineering are not perfect dielectric. When an imperfect dielectric is subject to an external ﬁeld, the resulting total current density ﬂowing through the dielectric is consisted of three parts; the instantaneous current density due to capacitance (dielectric storage), the polarization current density due to the polarization of the material, and the conduction current density from the electrical conductivity of the material. Elec- trically conductive materials are also called the lossy material and the magnitude of electrical conductivity is called the lossness.

215 6.1.4 Dielectric Properties in Microwave and Radar NDT

Applications of dielectric properties in civil engineering have been increased during the past decade. The contrast in the dielectric properties of diﬀerent materials is useful for distinguishing the presence of an inclusion in a matrix dielectric, as well as for monitoring the change of material property during a period of time. In civil engineering, reported applications of dielectric properties are summarized in Table 6.4.

Table 6.4: Civil engineering applications of dielectric properties

Application Reference Quality control of concrete/cement curing [117, 263, 30, 139] Mechanical strength determination [114] Corrosion detection of steel reinforcement [224] Pavement thickness detection [6] Structural health monitoring [162] NDT and damage detection [66, 35, 168, 167, 88, 217, 185, 4]

6.2 Approaches for the Determination of Dielec- tric Properties

Dielectric properties can be determined by various existing approaches, depending on the chosen experimental conﬁguration. These approaches include

2 1. Approach I – Use total current density J (A/m ), geometrical capacitance C0

(F), electric potential diﬀerence V0 (V), and phase diﬀerence φ (rad) [74].

J (ω) = (sin φ − i cos φ) (6.35) C0V0ω

This is the parallel plate capacitor technique which is based on a capacitor model for determining the permittivity of the material. It requires the specimen to

216 possess certain shape with ﬂat surfaces on the two sides contacting the two parallel plates. The constraints on the shape of specimen and measurement condition limit the use of the technique, making it a laboratory method rather than an in-situ method.

2. Approach II – Use average electric displacement ﬁeld vector E (V/m) and electric ﬁeld [12]. Z 1 2 (ω) = 2 ˆ|∇V | dΩ (6.36) ΩE Ω

3. Approach III – Use capacitance C (F) and equivalent resistance R (O/m).

0(ω) = Q = ωCR (6.37) 00(ω)

where Q is the cavity quality factor or the Q-parameter [236, 244]. This is the resonate cavity technique which is based on a resonator/oscillator model. It provides accurate results than other broadband techniques. Only results at one frequency are available at a time, suggesting the need for signiﬁcant measurement eﬀorts when a wide range of frequency response is needed. Additionally, it cannot measure sample sizes greater then the cavity volume/capacity.

4. Approach IV – Use waveguide wavelengths λ (m) and resonant frequencies ω (rad/s) [184].

λ 2 λ 2 ω 2 ω 2 0(ω) = + = g + c (6.38) λg λc ω ω

This is a waveguide technique based on the transmission line model. Waveguides can only operate in designed frequency bands associated with certain wave propagation modes. Several diﬀerent samples are needed when the measurement is conducted over a large frequency range. Inaccuracy in the measurements may occur due to the air gap between the waveguide and the specimen [178]. Waveguide techniques can be tedious in terms of sample preparation when the designated frequency band is not available in advance.

217 5. Approach V – Use average absorption microwave power W (W/m3) and average electric ﬁeld |E| (V/m) [70].

2W 00(ω) = (6.39) ω |E|2

6. Approach VI – Use the amplitude and phase measurements of transmitted EM waves through plate specimens (usually the reﬂection is needed as well). The

amplitude transmission coeﬃcient T0 is deﬁned by

p 0 d p 0 d T0 = exp −π r · tan δ · · exp i2π r · (6.40) λ0 λ0

where d is the thickness of the plate specimen (m), λ0 is the wavelength of the incident wave in free space (m). For low-loss dielectrics (tan δ ≤ 10−4 ∼ 10−3) [188], the specimen thickness d should be less than the eﬀective wavelength λ √0 within the dielectric; d < λe = 0 , for the ease of phase estimation. In the r case where d > λe is inevitable or preferred, such as the use of high frequencies, the real phase φ of EM waves transmitted through the specimen is

2π p 0 φ = s · 2π + φ1 = r · d (6.41) λ0

where s is an integer, and φ1 ∈ h0, 2πi. For the estimation of unknown dielectric 0 constant r, two plate specimens with diﬀerent thicknesses are prepared. The phase measurements provides two equations.

p 0 λ0 r = (s1 · 2π + φ1) 2πd1

p 0 λ0 r = (s1 · 2π + φ2) (6.42) 2πd2

0 Solving for the integers s1 and s2 leads to the evaluation of r. With the ampli-

tude measurements T0(d1) and T0(d2), the loss tangent tan δ can be determined. Therefore, the complex permittivity is found.

218 6.3 Integrated Methodology for the Determina- tion of Dielectric Properties

In this research work, an integrated methodology for the determination of complex permittivity of dielectric materials based on transmission-only, coherent, wide- bandwidth free-space measurements [36]. This proposed methodology, as a time- domain method, integrates the time difference of arrival (TDOA) measurement of plate specimens and a root-searching optimization scheme using parametric system identification (PSI) and the error sum of squares (SSE) criterion for the unique determination of complex permittivity. Consider the normal incidence of a TE wave upon a plate specimen in the configuration as shown in Figure 6-2. In Figure 6-2, a dielectric plate specimen is placed in the middle of two horn antennas in free space. Construction materials including Portland cement concrete (PCC) slabs and a glass fiber reinforced polymer (GFRP) sheet are subjected to the free space measurement. Validation of this methodology is performed using Teflon, Lexan, and Bakelite whose dielectric properties are well- documented in literature. Procedure of the integrated methodology consists of the following steps: (1) collec- tion of free-space measurements of complex transmission coefficients, (2) estimation of the real part of complex permittivity using TDOA, and (3) determination of the imaginary part of the complex permittivity using PSI and SSE. The overview of the proposed methodology is provided in Figure 6-3. In the transmission model shown in Figure 6-2 and considering the normal incidence of a transverse electric (TE) wave, the complex transmission coefficient T is given by [138]

4 exp [i(k − k )d ] T ∗ = 1z 0z 1 (6.43) (1 + p01)(1 + p10)(1 + R01)R10 exp i2k1zd1

√ √ ∗ ∗ k1z 1 µ0k1z 1 − p01 where k0z = ω µ00, k1z = ω µ11, p01 = = = ∗ , R01 = , k0z p10 µ1k0z 1 + p01 ∗ 0 00 0 00 d1 = thickness of the specimen, and 1 = 1 − i1 = 0 (r − ir ). For nonmagnetic ∗ specimens, µ = µ0. Components of the methodology are further described in the

219 Network Analyzer

Region 0 Region 1 Region 2 * * ε0 , µ0 ε1 , µ1 ε0 , µ0 Transmitting horn antenna Receiving horn antenna

1 2

d0 d1 d0

--z-direction

Figure 6-2: A two-dimensional free-space transmission model

following sections.

6.3.1 Time Diﬀerence of Arrival (TDOA)

The TDOA technique is used to estimate the dielectric constant of a low-loss (less conductive) material using experimental measurements of transmission coefficients. This technique is conceptually based on the same two-dimensional model for EM wave propagation shown in Figure 6-2. Under the assumptions of normal incidence, first-peak response, and minor transmission losses, the time difference of arrival of an EM plane wave due to the presence of the specimen is [241]

d ∆t = p0 − 1 (6.44) c r

where ∆t = additional propagation time (time diﬀerence of arrival) between the transmitting and receiving horn antennas when the specimen is present compared to

220 Free-space measurements of complex transmission Unique combination of the real and coefficients imaginary parts of complex permittivity

Real part of Imaginary part of Estimation of real part of complex complex complex permittivity using TDOA ' " permittivity,ε r permittivity, ε r

Parametric System Identification

and Error Sum of Squares (SSE) 1 2.2

0.5 ' 0.8 ε 2.1 r

0.4 0.6 " 0.3 2 ε r log(SSE) 0.4 log (ESS) 0.2 Loss Factor 1.9 0.2 0.1 ' ε r 1.8 0 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 ' Loss Factor Dieletric Constantε r

Figure 6-3: Overview of the proposed methodology

the measurement when the specimen is not present, d = thickness of the plate/slab specimen, and c = speed of light in free-space. The estimation of ∆t is achieved by processing the measured transmission coefficient from frequency-domain (recorded by the network analyzer) to time-domain using inverse Fourier transformation. Eq.6.44 can be used as a tool for assessing the dielectric constant of the specimen by estimating its time difference of arrival using a set of experimentally measured transmission coefficients. With Eq. (6.44), the real part of complex permittivity or dielectric constant is found. c∆t2 0 = 1 + (6.45) r d The accuracy of the ∆t estimation depends on the bandwidth of the signal being processed and the accurate measurement of the specimen thickness. Note that Eqs.(6.44) and (6.45) must be modified when the loss factor is significant. This estimate will be the basis for the identification of the imaginary part (or loss tangent) of complex

221 permittivity in the root-searching optimization scheme.

6.3.2 Root-searching Optimization Scheme

Complex permittivity cannot be explicitly represented in terms of the transmission

∗ ∗ coeﬃcient T or the scattering parameter (S-parameter),S21, which is the forward transmission gain measured by the transmitting and the receiving radar antennas. Furthermore, the exact solution for the complex permittivity is not straightforward due to the multiple roots associated with Eq.(6.43) for lossy materials. In order to resolve the problem, a root-searching procedure involving the use of PSI and the SSE criterion is proposed. PSI refers to the use of a mathematical model to characterize the behavior of a system [266], and in this application it is used to provide the theoretical estimation of transmission coeﬃcients. Thereafter, SSE criterion is introduced for evaluating the optimal combination of complex permittivity. Experimental measurements were conducted in coherent condition where mea-

∗ ∗ sured complex S-parameter, S21, equals the complex transmission coeﬃcient, T . ∗ Coherent condition provides a non-distorted phase in the measurement of S21 (between antennas 2 (receiver) and 1 (transmitter)) such that amplitude attenuation within the specimen does not contribute to the measured transmission coeﬃcient. Experimental facilities for the free-space measurement are shown in Figures 6-4 and

∗ 6-5. In coherent condition, the magnitude of T expressed in decibel (dB), TdB, is ∗ related to S21 by ∗ ¯∗ TdB = 10 · log S21 · S21 (6.46)

¯∗ ∗ where TdB is a real number, and S21 is the complex conjugate of S21. TdB is the m measured transmission coefficient or TdB = T Eq. (6.43) provides the form of theoretical/predicted transmission coefficient collected in decibel, or T ∗ = T p. With the correct value of complex permittivity, the predicted T p will be identical to the measured T m. The root-searching optimization scheme is introduced to determine the closest value of T p to T m by minimizing the SSE between T p and T m. PSI (parametric system identification) is carried out by generating a set of T p from the

222

(a) Transmitting horn antenna (b) Network analyzer

Figure 6-4: Transmitting horn and the network analyzer for the free-space measurement [Courtesy of the MIT Lincoln Laboratory (MIT LL)]

0 00 0 00 possible combinations of r and r . From the possible combinations of r and r , those resulting in minimum diﬀerence between T m (measurement) and T p (theory) will be the estimates closest to the actual values.

SSE (error sum of squares) criterion is applied in this root-searching procedure.

0 00 Each combination of the estimated r and r provides a corresponding SSE value, which is calculated by

n 0 00 X m p 0 00 2 SSE (r, r ) = |T (ω) − T (ω, r, r )| (6.47) j=1

where n is the number of measurement frequencies. It is obvious that SSE evaluates the absolute error in an energetic manner; the closest prediction possess the minimum energy with the measurement. With the SSE criterion, an error surface is generated

0 00 for various combinations of r and r . Figure 6-6 shows the error surfaces generated by the free-space measurements of four dielectric materials including Teﬂon, Lexan, Bakelite, and Portland cement concrete. Discussion of the result is provided in the validation section. Using the estimated dielectric constant (from TDOA) a corresponding error curve containing various combinations of loss factor can be lo-

223

(a) Transmitting horn (b) Plate specimen (c) Receiving horn

Figure 6-5: Transmitting and receiving horns for the free-space measurement [Cour- tesy of the MIT LL] cated from the error surface, as shown in Figure 6-7. The most possible loss factor is determined by selecting the one with minimum error on the curve. Hence, the root-searching procedure is accomplished.

224 1 1 6 5 0.8 0.8 5

0.6 4 0.6 3 3 0.4 0.4 2 Loss Factor Loss 2 Factor Loss

0.2 0.2 1 1

0 0 0 2 4 6 8 10 0 2 4 6 8 10 Dieletric Constant Dieletric Constant Dielectric Constant Dielectric Constant (a) Teflon (b) Lexan

1 1

0.6 0.4 0.8 0.8 0.5

0.6 0.3 0.6 0.4

0.3 0.4 0.2 0.4

Loss Factor Loss Factor 0.2

0.2 0.1 0.2 0.1

0 0 0 2 4 6 8 10 0 2 4 6 8 10 DielectricDieletric Constant Constant DielectricDieletric Constant Constant

Figure 6-6: Error surfaces of the dielectric measurements collected from Teﬂon, Lexan, Bakelite, and Portland cement concrete

225

7 7

6.5 6.6

6 6.2

5.5

5.8 log (SSE) log (SSE) log 5

5.4 4.5

4 5 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 Loss Factor Loss Factor

(a) Teflon (b) Lexan

2.2 2.6

2.15

2.1 2.4

2.05 log(SSE) log (SEE) log log (SSE) log 2 2.2

1.95

1.9 2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Loss Factor Loss Factor

Figure 6-7: Error surfaces of the dielectric measurements collected from Teﬂon, Lexan, Bakelite, and Portland cement concrete

226 6.3.3 Validation of the Methodology

In order to evaluate the predictions from the proposed methodology, experimental measurements of transmission coeﬃcients were conducted for several materials whose dielectric properties are known from the literature. These materials are Teﬂon, Lexan, and Bakelite. In addition, Portland cement concrete and GFRP specimens were also measured as examples of construction materials.

Sample Description and Experimental Conﬁguration

For the measurements, slab-type material specimens of 12in-by-12in (305mm-by- 305mm) cross-section with varying thicknesses were used. The selected width and height of the specimens meet the requirement of radar measurements in the far-field condition. Table 6.5 shows the various specimen thicknesses for the materials used. The concrete sample was manufactured with a cement/sand/aggregate mix ratio of 1:2.25:3.2 by weight. The water-to-cement ratio (w/c) was 0.60. Portland cement of Type I was used. The uniaxial compression strength of the concrete was 3480 psi (24 MPa) at 28 days. The GFRP sample was manufactured by extruding a 12in-by-12in square from a unidirectional fiber woven sheet, which was then saturated with an epoxy thermoset polymer. The GFRP sample was cured for 7 days before testing. The experimental configuration is shown in Figures 6-2 and 6-4. The network ana-

Table 6.5: Thicknesses of the specimens

Material Thickness (mm) Teﬂon 6 Lexan 6 Bakelite 6 Portland cement concrete 50 GFRP 1.5 lyzer was a Hewlett Packard Model 8510C that was operated in step frequency mode. At each frequency the network analyzer performs a measurement of the complex

227 transmission coefficient. Transmission measurements were collected from X-band through Ku-band (8 to 18 GHz) with a frequency step of 12.5 MHz. A typical set of measurements consisted of amplitudes (dB), phase angles (deg), and their corresponding frequency. Raw data was then calibrated using free-range measurements, which were conducted in the absence of the specimen, to obtain the TDOA information. Far-field test conditions were ensured during measurements. The far-field condition is required to ensure that the wave front is approximately plane, which is directly related to the theoretical methodology for dielectric property characterization. Far-field conditions also minimize complex wave behavior in near-field between the horn antennas and the specimen. Considering the highest frequency of 18 GHz for the proposed experiments, it was calculated that the specimen should be placed at least 1.8 in. (45.72 mm) away from the horn antennas, and the minimum area of the slab specimens should be greater than 25 in2 (161.29 cm2) for adequate illumination to satisfy the far-field condition. The applied experimental set-up meets these requirements. Di- rect coupling between the horn antennas is also eliminated in this experimental set-up due to the use of the network analyzer. The unwanted coupling between antennas is measured in free-range and stored in the phase angle information collected by the network analyzer. This information is then used in the calibration of all consequent measurements.

Estimation of the dielectric constant using TDOA

The TDOA technique was used to estimate the dielectric constant of materials under investigation from both X- and Ku- bands. After performing inverse Fourier transformation of the experimental transmission coeﬃcient data, the time diﬀerences of arrival (∆t) for the materials considered in this study were calculated. These results are tabulated in Table 6.6. Using the TDOA information, application of Eq.(6.45) yielded estimates for dielectric constants, which are also tabulated in Table 6.6. For the case of GFRP the TDOA technique could not provide a reliable result due to the very small thickness of the specimen.

228 0 Table 6.6: TDOA measurements and dielectric constants r of the specimens

0 0 Material ∆t (ns) Estimated r Reported r Teﬂon 0.007 1.79 2.0 Lexan 0.012 2.48 2.77 Bakelite 0.030 5.95 5.0 Portland cement concrete 0.235 5.95 4.44∼7.22

Root-searching results of loss factor

Following the previously described procedure, the loss factor can be found by extracting an error curve using TDOA from the error surface generated by PSI and SSE criterion. The loss factor corresponding to the minimum SSE on the error curve is identiﬁed as the most appropriate value. Estimated loss factors thus found are given in Table 6.7 for the tested materials.

00 Table 6.7: Loss factors r of the specimens

00 Material Estimated r Teﬂon 0.04 Lexan 0.04 Bakelite 0.28 Portland cement concrete 0.62

Discussion

From the results, it is found that a unique combination of real and imaginary parts of the complex permittivity can be found through the proposed methodology for low-loss materials, such as Teﬂon, Bakelite, Lexan, and dried/hardened concrete in this study. The non-uniqueness problem of real and imaginary parts of complex permittivity has been resolved, and appropriate values were predicted using the integrated methodology. The method has potential for in-situ measurement of dielectric properties for

229 construction materials.

The estimated results of complex permittivity using the proposed methodology are summarized in Tables 6.6 and 6.7. The estimated dielectric constant values of Teﬂon, Lexan, Bakelite, and concrete are close to the reported values or within the ranges reported in the literature. It is emphasized that the values found using the proposed methodology for dielectric constants and loss factors are not expected to exactly correlate with the reported values in the literature because of the diﬀerences in frequency ranges and measurement techniques. Note that in Table 6.7 the loss factor comparison could not be made because of the lack of information on loss factors in the literature. In addition, loss factor values are generally more sensitive to measurement frequency and experimental parameters.

Two main issues in this methodology are addressed as follows.

• Effect of selected frequency bands in TDOA TDOA estimates for the dielectric property characterization of the materials presented in this study were obtained using the entire set of measurements for the X- and Ku-band. For practical applications of this methodology, the question arises as to what would be the optimum frequency range to provide a sound estimate for the dielectric constant. Calculations were conducted in 1 GHz, 2 GHz, 3.3 GHz, and 5 GHz intervals covering the range from 8 GHz to 18 GHz to study the effect of frequency bandwidth. Table 6.8 presents a summary of different estimates of dielectric constant using TDOA that were calculated in 1 GHz bandwidth. The convergence of these estimate curves is evaluated by the variance shown in Figure 6-8. It is observed that estimates calculated from narrow frequency bandwidths may provide poor estimates for dielectric constant. Accurate estimates using the TDOA technique require the use of wide frequency bandwidths for uniquely identifying the dielectric constant. Reliable estimation was achieved when frequency bandwidth used exceeded 3.3 GHz in our measurements. The accuracy of such estimation is critical to the subsequent determination of the loss factor.

230 0 Table 6.8: Estimated r by TDOA using diﬀerent frequency bandwidths (GHz)

Material 8-9 9- 10- 11- 12- 13- 14- 15- 16- 17- 10 11 12 13 14 15 16 17 18 Teﬂon 1.42 5.80 1.89 1.89 2.37 1.18 1.32 2.28 2.28 2.34 Lexan – 6.05 6.96 2.25 2.53 2.06 4.53 3.61 2.41 3.23 Bakelite – 0.96 14.75 5.38 8.86 8.17 5.45 4.69 9.76 – Concrete 4.80 6.03 6.64 5.30 5.01 4.66 8.04 6.76 5.99 4.47

• Limitations of the use of TDOA

1. Eﬀective thickness of specimen - From the results for GFRP, it is observed that the thickness of the sample aﬀects the performance of TDOA technique. Thus, a minimum thickness limitation might apply to the accurate characterization of dielectric constant of construction materials using the proposed methodology. Our preliminary investigations indicate that measurements of specimens of 6 mm or larger would provide acceptable results.

2. Accuracy of the use of simpliﬁed wave velocity - From EM wave theory it is known that the wave (phase) velocity of EM waves is a function of dielectric properties of materials. For example, for lossy materials, the phase velocity within the medium is [138]

" r !#−1/2 ω 1 1 σ 2 vp = = √ 1 + + 1 (6.48) kR µ 2 ω

where kR is the real part of the complex wavenumber. The ﬁrst-order

expansion of vp provides its approximation as

1 1 σ 2−1 v ∼= √ 1 + (6.49) p µ 8 ω

231 25 Teflon Lexan Bakelite 20 Concrete

15 ance i ar

V 10

0 1 2 3 4 5 6 7 8 9 10 Frequency Bandwidth Frequency bandwidth (GHz) 0 Figure 6-8: Convergence of estimates of r at diﬀerent frequency bandwidths using TDOA

For lossless materials σ = 0, and the wave velocity becomes

1 c √ vp = √ = 0 0 (6.50) µ µrr

Eq.(6.50) is used in the TDOA procedure described in this paper because of its simplicity, and it is considered as a convenient approximation of σ Eq.(6.49) when 1. To study the accuracy of this approximation, a ω quantitative study was performed. Wave velocities using Eqs.(6.48), (6.49), σ and (6.50) are calculated with respect to different values of ranging ω from 0 to 20, shown in Figure 6-9. It is observed that the differences between Eqs.(6.48) and (6.50) or between Eqs.(6.49) and (6.50) may be σ substantial when is greater than 1. In the range of investigation where σ ω < 0.1, the difference between either Eqs.(6.48) and (6.50) or Eqs.(6.49) ω and (6.50) is insignificant as demonstrated in Figure 6-9(b). Hence, the use

232 of Eq.(6.50) in the proposed methodology for low-loss materials is justiﬁed.

1.001 Theoretical v in lossy media 1 p Approximated v in lossy media p v in lossless media 1 p 0.8 0.999

0.6 0.998

0.4 0.997

Theoretical v in lossy media 0.2 p 0.996 Approximated v in lossy media

Normalized Phase Velocity Normalized Phase Velocity p v in lossless media p 0 0.995 0 5 10 15 20 0 0.02 0.04 0.06 0.08 0.1 σ / (ωε) σ / (ωε)

(a) σ ωε = 0 ~ 20 (b) σ ωε = 0 ~ 0.1

00 Figure 6-9: Normalized phase velocity vs. loss factor r

6.4 Modeling Approach for the Dielectric Proper- ties of Materials

In the microwave frequency range the predominant polarization is the dipolar polarization which can be derived from the internal ﬁeld and the external ﬁeld of the dielectric. A geometrical analysis technique proposed by Cole and Cole (1941) [50] called the Cole-Cole diagram is also introduced for illustrating the dispersion response in a complex plane. Equivalent circuits of these models are also provided for better understanding of their physics.

6.4.1 Internal Field Approach versus External Field Approach

Internal Field Approach – Debye’s Treatment

A systematic approach for the modeling of dipolar polarization for homogeneous and isotropic liquid and solid dielectrics in microwave/radar frequencies was proposed by

233 Debye [65]. The molar polarization P in liquids is derived and deﬁned as

− 1 M 4πN µ2 1 P (ω) = = α + (6.51) + 2 ρ 3 0 3kT 1 + iωτ

where is the electrical permittivity (F/m), M is the molecular weight (kg), ρ is the density, N is Avogadro’s constant (N = 6.06 × 1023 = number of particles in

one mole), α0 is a factor accounting for the distortion eﬀect of the molecule, µ is the intrinsic electric moment (e.s.u.=300V/cm), T is the absolute temperature (K), and

−16 M k is Boltzman’s constant (k = 1.37 × 10 ergs). ρ is also called the molar volume (cc or cm3). The electrical permittivity is found to be [65]

ρ 1 + 2 P (ω) M = ρ (6.52) 1 − P (ω) M

Two relationships were also reported by Debye (1929).

− 1 M 4πNα ∞ = 0 (6.53) ∞ + 2 ρ 3 2 s − 1 M 4πN µ = α0 + (6.54) s + 2 ρ 3 3kT

With these relationships, becomes

s + iωτ ∞ + 2 + 2 = s ∞ (6.55) 1 + iωτ s s + 2 ∞ + 2

Assuming + 2 χ = s ωτ (6.56) ∞ + 2 then the representation of becomes

1 + i ∞ χ 0 − i00 = s = (6.57) s 1 + iχ s

234 It yields to − − = s ∞ (6.58) ∞ 1 + iχ where

− 0 = + s ∞ = 0(ω) (6.59) ∞ 1 + χ2 χ ( − ) 00 = s ∞ = 00(ω) (6.60) 1 + χ2

and 00 ∞ + 2 ωτ = 0 (6.61) s + 2 − ∞ is a characteristic parameter in Debye’s model. Eqs. (6.59) and (6.60) are called Debye’s dispersion equations.

Debye’s treatment on deriving the complex permittivity was based on the internal field with Mosotti’s hypothesis. Mosotti considered the molecule as a polarizable system, in which the polar moment in an electric field is set up by the force causing a displacement of the charge in the material. Debye found that, in his dispersion equations for ideal liquids, Mosotti’s hypothesis about the internal field was not to- tally correct, especially when dense dielectrics are considered. However, the difference between his theory and experimental results was only a quantitative but qualitative deviation. Since Debye’s model provides a good estimation for dilute solvents in general, it is been widely applied as the paradigm model for gaseous and liquid dielectrics ever since.

External Field Approach – Fr¨ohlich’s Treatment

Unlike Debye’s treatment whose relaxation time is made based on the microscopic or molecular relaxation process, Fr¨ohlich (1949) [90] chose another approach in which the relaxation time is based on a macroscopic process. The time-dependent relaxation process of dielectric dispersion can described as the convolution between the electric

235 ﬁeld and a decay function. For t > u + du,

D(t − u) = α(t − u)E(u)du (6.62)

where D(t − u) is the electric displacement (C/m2) or dielectric induction, α(t − u) is the decay function describing the gradual decrease of D(t − u) and α(t − u) → 0 when t → ∞, and E(u) is the electric ﬁeld (V/m). The decay function α(t − u)

also contains an instantaneous response component which is assumed to be ∞E(u). Thus, for u < t < u + du,

D(t − u) = ∞E(u) + α(u)du (6.63)

Should another ﬁeld E(u0) be applied at a later time interval between u0 and u0 + du0, the corresponding response is linearly superimposed onto the former one (the principle of superposition). Consider a general case in which D(t) and E(t) may not vanish for t < 0. The response D(t) at the time t is given by

Z t D(t) = ∞E(t) + E(u)α(t − u)du (6.64) −∞

Assume that the decay function α(t) is an exponential function.

t − α(t) ∝ e τ (6.65)

where τ is the relaxation time observed in the macroscopic process and

dα(t) 1 = − α(t) (6.66) dt τ

Diﬀerentiating the response function provides

dD(t) dE(t) 1 Z t = ∞ + α(0)E(t) − E(u)α(t − u)du (6.67) dt dt τ 0

236 After rearranging, it yields

d τ [D(t) − E(t)] + [D(t) − E(t)] = τα(0)E(t) (6.68) dt ∞ ∞

Consider the attained equilibrium under the application of a constant ﬁeld. It provides

d (D − E) = 0 (6.69) dt ∞

D = sE (6.70)

Therefore, we have

τα(0) = s − ∞ (6.71)

Since Eq.(6.65), α(t) is found to be

t − − α(t) = s ∞ · e τ (6.72) τ

With these relationships,

d τ (D − E) + D − E = ( − ) E (6.73) dt ∞ ∞ s ∞

Consider an alternating ﬁeld of the form E ∝ e−iωt. With

dE = −iωE (6.74) dt and Eq.(6.1), it yields dD = −iω(ω)E (6.75) dt Substituting Eq.(6.65) into Eq.(6.64), along with E ∝ e−iωt, we have

t − u Z t s − ∞ −iωu − (ω)E = ∞E + e τ du (6.76) τ −∞

237 Let t − u = x. The limit of the integral is changed from [−∞, t] to [∞, 0]. Also, du = −dx. Then,

x Z 0 s − ∞ iω(x − t) − (ω)E = ∞E − e τ dx τ ∞ x Z ∞ s − ∞ −iωt iωx − ⇒ (ω)E = ∞E + e · e τ dx τ 0 x Z ∞ s − ∞ iωx − ⇒ (ω)E = ∞E + E e τ dx τ 0 x Z ∞ s − ∞ iωx − ⇒ (ω) = ∞ + e τ dx (6.77) τ 0

The response function becomes

− (ω) − = s ∞ (6.78) ∞ 1 + iωτ and the real and imaginary parts are

0 s − ∞ 0 = ∞ + = (ω) (6.79) 1 + (ωτ)2 ωτ ( − ) 00 = s ∞ = 00(ω) (6.80) 1 + (ωτ)2

00 where ωτ = 0 . The above equations are Fröhlich’s dispersion equations derived − ∞ from the equilibrium of an external field E. Frohlich’s equations are also known as the Debye-Drude equations [107]. Fröhlich further generalized the variation of (ω) by accounting the contribution to the static dielectric constant of the groups of dipoles having individual relaxation times in a range dτ near τ. He also assumed that the contributions of various groups of dipoles can be linearly superposed and modeled by a function y(τ)dτ. The total contribution to the static dielectric constant is then determined by ∞ Z s − ∞ = y(τ)dτ (6.81) 0

238 where y(τ) is termed the distribution function. Thus, Eqs.(6.79) and (6.80) can be generalized as

∞ Z 0 y(τ) (ω) = ∞ + dτ (6.82) 1 + (ωτ)2 0 ∞ Z ωτy(τ) 00(ω) = dτ (6.83) 1 + (ωτ)2 0

Cole and Cole (1941) [50] proposed a modiﬁed model for liquid and solid dielectrics: s − ∞ = ∞ + (6.84) 1 + (iωτ)1 − α1

where α1 is an empirical parameter and 0 ≤ α1 ≤ 1. Davidson and Cole (1950) proposed another model also based on their dielectric measurements on liquids and dielectrics: s − ∞ = ∞ + (6.85) (1 + iωτ)β where β is an empirical parameter and 0 ≤ β ≤ 1. A parametric study is conducted

0 00 to understand the inﬂuence of α1 on and . By assuming s = 2 and ∞ = 1 the 0 00 variations of and with respect to parameters α1 (0 ≤ α1 ≤ 1) and β (0 ≤ β ≤ 1) in the Cole-Cole model are shown in Figures 6-10 and 6-11 .

The Cole-Cole and Davidson-Cole models were generalized by Havriliak-Negami’s model (1966)[108]: − = + s ∞ (6.86) ∞ h iγ 1 + (iωτ)α2

where α2 and γ are empirical parameters and 0 ≤ α2 ≤ 1, 0 ≤ γ ≤ 1. Havril- iak and Negami’s model has been also used for materials exhibiting non-Debye-type relaxations such as ionic conductors and magnetic ﬂuids [282].

Other models such as the Fuoss-Kirkwood model (1941) [91](for giant molecule systems such as long-chain polymers), the Williams-Watts model (1970) [269] (for general polymer systems), the Jonscher model (1975)[125] (for temperature-dependent

239

Figure 6-10: Behavior of the Cole-Cole model (s = 2, ∞ = 1) relaxation response of polymers), and the Dissado-Hill model (1979) [89] (for general polymers) are not investigated in this research because (1) they are developed pri- marily for polymers, and (2) they require the measurement of empirical parameters used in the model.

Comparison of Internal Field and External Field Approaches

Dispersion equations derived by the internal field approach (Debye’s treatment) and external field approach (Fröhlich’s treatment) are compared in Table 6.9. Since ∞ < ∞ + 2 s, < 1. This suggests that the relaxation time derived from internal fields is s + 2 greater than the one from external fields. Many derivative models have been proposed and reported based on experimental measurements on different dielectrics in the following years. Most of the models basically follow Debye’s approach due to the sound physical foundation and the concise mathematical expression offered by the approach.

240

Figure 6-11: Behavior of the Davidson-Cole model (s = 2, ∞ = 1)

However, in Debye’s treatment, a prior knowledge on the distribution of internal fields is needed for determining the molar polarization P (ω). This is difficult to obtain in the situation where close-range interactions between two different molecules become significant. The distribution of internal fields, as well as the relationship between internal and external fields, tends to be complicated. In addition, the molecular-level variables considered in Debye’s treatment of the complex permittivity are not easy to obtain experimentally, although the picture depicted in Debye’s treatment provides a clear understanding on how dielectric dispersion occurs associated with the inherent properties of the materials (molecules).

The external field approach, on the other hand, avoids the difficulty on assuming the distribution of internal fields. It is adopted in this research since the experimental radar measurements were conducted in macroscopic level, rather than in molecular level. While there are many wide-bandwidth dielectric models currently available to use, this research is interested in the dielectric behavior of the materials in GFRP-

241 Table 6.9: Comparison of internal ﬁeld approach and external ﬁeld approach

Variable Internal field External field (Debye’s model) (Fröhlich’s model)

0 s − ∞ s − ∞ ∞ + ∞ + + 2 2 1 + (ωτ)2 1 + s ωτ ∞ + 2

00 s − ∞ s + 2 ωτ (s − ∞) 2 ωτ 2 + 2 ∞ + 2 1 + (ωτ) 1 + s ωτ ∞ + 2

00 00 ∞ + 2 ωτ 0 0 s + 2 − ∞ − ∞ concrete systems in microwave range (X-band (8 ∼ 12 GHz) and Ku-band (8 ∼ 18 GHz). The entire dielectric spectroscopy of the considered materials is not in the scope of this modeling eﬀort as to capture all dielectric dispersion mechanisms within the frequency range of interest. Knowing that in the frequency range of 8 ∼ 18 GHz only dipolar polarization is most signiﬁcant, the Havriliak-Negami model is chosen as the prototype model to capture the monopole (single peak) dispersion of the materials.

6.4.2 Geometrical Analysis

An illustration of (0, 00) in complex domain was proposed by Cole and Cole (1941) for demonstrating the dispersion behavior of dielectrics, which is thereafter termed the Cole-Cole diagram. The Cole-Cole diagram is formed by assembling 0 and 00 on a complex plane using frequency as an implicit variable. For dielectrics satisfying Fr¨ohlich’s equations their Cole-Cole diagram is a semicircular plot; for those obeying Cole-Cole’s and Davidson-Cole’s models, their Cole-Cole diagrams are the deformed version of a semicircle, as shown in Figures 6-12 and 6-13 in which s = 2 and

∞ = 1. It is obvious that Cole-Cole’s model and Davidson-Cole’s model degrade to

Fr¨ohlich’s model when 1 − α1 = 1 and β = 1. In addition, the variation of α1 can

242 be depicted by the equation of circle with diﬀerent radiuses and centers, while the curves with a changing β are no longer circular. Consider the Fr¨ohlich model. Since

1- =1 α1 1- =0.2 0.5 α1 1- =0.5 α1 0.4 1- =0.7 α1 1- =0 α1 0.3 ) ω "( ε 0.2

0.1

-0.1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 ε'(ω)

Figure 6-12: The Cole-Cole diagram of Cole-Cole’s model

0 0 0 max = (0) = s and min = (∞) = ∞, the corresponding imaginary parts are both d00 diminished. It can be seen that 00 occurs at ω = 0, and 00 occurs at = 0. min max d0 d00 = 0 can be determined by d0

d00 d00 d0 −1 1 − (ωτ)2 = · = (6.87) d0 dτ dτ 2ωτ

00 It is evident that max occurs at ωτ = 1 (ωτ must ≤ 0). Knowing that ωτ = 00 00 0 0 1 s + ∞ 00 1 s − ∞ 0 , we have = − ∞ or = and = . The − ∞ τ 2 τ 2 governing equation for the semicircles in the Cole-Cole diagram for Fr¨ohlich’s model is + 2 − 2 0 (ω) − s ∞ + 00 (ω)2 = s ∞ (6.88) 2 2 − + The semicircles are of radius s ∞ and centered at s ∞ , 0 An equivalent 2 2 circuit to Fr¨ohlich’s model consisting of two capacitors and one resistor is provided

243 β=0 =0.2 0.5 β β=0.5 β=0.7 0.4 β=1

0.3 ) ω "( ε 0.2

0.1

-0.1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 ε'(ω)

Figure 6-13: The Cole-Cole diagram of Davidson-Cole’s model

00 1 in Figure 6-14. The relaxation time is determined as τ = 0 when ωτ = 1. − ∞ ω Two auxiliary vectors are deﬁned for the semicircles in the Cole-Cole diagram; ~u and

ε −ε Radius = s ∞ 2 ε = (εε′, ′′ ) π ε −ε ε " ()ω G 2 s ∞ u ε G ∞ τ v ε s −ε ∞ ε ' (ω) ε ε + ε ε ∞ s ∞ s 2

Figure 6-14: The Cole-Cole diagram and the equivalent circuit of Fr¨ohlich’s model

~v. From the geometry it is known that |~u + ~v| = s − ∞. Since ~u is the diﬀerence 0 00 between points ( , ) and (∞, 0), u = − ∞ and v = s − . With ~u and ~v, the

244 governing equation for semicircles can be represented by

v − u + 22 u + v 2 0 − + 002 = (6.89) 2 2 and Fr¨ohlich’s model is represented by

− u + v v s ∞ = 1 + iωτ = = 1 + (6.90) − ∞ u u

Therefore, v − = iωτ = s (6.91) u − ∞

1−α iπ(1−α1) As for Cole-Cole’s model, with i 1 = e 2 , it can be rationalized to yield

0 − 1 + (ωτ)1−α1 sin α1π ∞ = 2 (6.92) 2(1−α1) 1−α1 α π s − ∞ 1 1 + (ωτ) + 2 (ωτ) sin 2 00 (ωτ)1−α1 cos α1π = 2 (6.93) 2(1−α1) 1−α1 α π s − ∞ 1 1 + (ωτ) + 2 (ωτ) sin 2

To eliminate ωτ, the above equations are rearranged.

 1 h 2(1−α1) i  0 (ωτ) − 1 − ∞ 1 2 = 1 − h i  s − ∞ 2 1 2(1−α1) 1−α1 α1π 2 (ωτ) + 1 + (ωτ) sin 2

 1 h 1−α1 −(1−α1)i  1 2 (ωτ) − (ωτ) = 1 − h i  2 1 1−α1 −(1−α1) α1π 2 (ωτ) + (ωτ) + sin 2 " # 1 sinh x = 1 − (6.94) 2 α1π cosh x + sin 2 00 1 cos α1π = 2 (6.95) − 2 α1π s ∞ cosh x + sin 2

245 where x = (1 − α1) ln (ωτ). Therefore,

  0 s − ∞ sinh x = ∞ + 1 − α π  (6.96) 2 cosh x + cos 1 2  α1π  cos 00 s − ∞ 2 =  α π  (6.97) 2 cosh x + sin 1 2

It can be shown that

+ − 20 α π sinh x = s ∞ · cos 1 (6.98) 200 2 − α π α π cosh x = s ∞ · cos 1 − sin 1 (6.99) 200 2 2

With cosh2 x − sinh2 x = 1, the governing equation of the semicircles is

+ 2 − α π ( − )2 α π s ∞ − 0 + s ∞ · tan 1 + 00 = s ∞ sec2 1 (6.100) 2 2 2 4 2

+ − α π − α π Its center is located at s ∞ , − s ∞ tan 1 . with radius s ∞ sec 1 . 2 2 2 2 2 The ratio between two auxiliary vectors is

s v ( − 0)2 + 002 = s (6.101) 0 2 002 u ( − ∞) + and the relaxation time is determined by

v 1−α1 ωτ = (6.102) u

The Cole-Cole diagram and the equivalent circuit of the Cole-Cole model are shown in Figure 6-15. For Davidson-Cole’s model,

s − ∞ = ∞ + (6.103) (1 + iωτ)β

246 ε −ε Radius = s ∞ 2 ε = ()εε′′′,

" π ε s −ε ∞ ε ()ω G ()1−α1 u 2 ε G ∞ τωτ()i −α1 v ⋅ ' ε (ω) εεs − ∞ ε ∞ ε ε + ε s α π s ∞ 1 2 2

Figure 6-15: The Cole-Cole diagram and the equivalent circuit of Cole-Cole’s model

Following similar rationalization process done to Cole-Cole’s model, it yields

0 β = ∞ + (s − ∞) · cos y · cos (βy) (6.104)

00 β = (s − ∞) · cos y · sin (βy) (6.105)

where y = tan−1 (ωτ). Figure 6-16 shows the behavior of Davidson-Cole’s model and its equivalent circuit. The slope on the curve in the Cole-Cole diagram is

d00 = − cot (1 + β) y (6.106) d0

In the following sections, geometrical analysis is applied for developing dielectric models of the considered materials.

6.5 Dielectric Properties of Water

Dielectric properties of water have been a major research topic in physics and chemistry ever since the work by H. Cavendish and A. Lavoisier in 1780s. It is because that, through the dielectric properties of water, the structure and behavior of water molecules can be better understood. While salient research results can be found in the literatures, due to diﬀerent interests from various disciplines [65, 76, 113], the purpose of this investigation is to develop a dielectric model for water speciﬁcally in

247 ()iωτ 1−β

0.4

0.35

0.3 ε s −ε ∞ 0.25 ε ) ∞ " ω 0.2 "( ε ()ω ε 0.15 R 0.1 ()1− β π

0.05 2 0 1 1.2 1.4 1.6 1.8 2 ' ε'(ω) ε ∞ ε ()ω ε s Figure 6-16: The Cole-Cole diagram and the equivalent circuit of Davidson-Cole’s model the frequency range of 8 GHz∼18 GHz. In this dissertation, dielectric modeling of water is conducted for the water molecules inside a cementitious composite like concrete. Therefore, it is important to know the water molecules reside in concrete and what they are. The following paragraphs brieﬂy deﬁnes the types of water molecules in concrete according to the microstructure of concrete.

Hydrated concrete is a porous, colloidal material (hydration products) whose pore structure permits air (gaseous) and water (liquid) molecules to reside inside concrete, making concrete a multi-phase cementitious composite. Sources of liquid water inside concrete include (i) extra water remaining from hydration, being trapped in the pore structure of concrete, and (ii) absorbed water from the environment (e.g., rain water). Typically, the porous, colloidal structure of concrete is characterized by the undiffer- entiated amorphous C-S-H (Calcium-Silicate-Hydrate), a hydration product having a major contribution to the binding of other components of concrete, including fine (sand) and coarse (gravel) aggregates. C-S-H gel is recognized as a solid made up of small particles of colloidal dimensions (at least one dimension). These particles are mainly the hydrates of calcium silicate, which are poorly crystalized. The formation of pore structure of C-S-H is generally regarded as a function of various factors (e.g., duration of hydration, cement composition, cement fineness, water/cement (w/c) ratio, temperature, and admixtures of concrete), and certainly it varies with time during

248 the setting and hardening of concrete. Since the C-S-H gel (cement paste) is considered the major element binding other components in concrete, it is also considered covering the surface of other components in concrete. As a result, variation of water contents in other components (ﬁne and coarse aggregates) is assumed to be constant once the cement paste is hardened/hydrated. Therefore, in this research, it is assumed that the variation of water content in the C-S-H gel or cement gel results in the change of bulk dielectric properties of concrete. Following this assumption, the next step is to understand how hardened/hydrated cement paste (hcp) can be modeled and what types of water can present in the pore structure of hcp. Three classical models on hardened/hydrated cement paste are reviewed and summarized as follows.

• Powers’ model [209, 206, 208] –

1. Cement gel is a rigid substance made up of colloid-size particles, possessing a characteristic porosity of 28% which is limited to normally cured cement pastes.

2. Two types of pores are modeled; (i) gel pores (average width = 15A˚) and (ii) capillary pores (bigger pores).

3. Three types of water are modeled; (i) chemical/combined/non-evaporable water, (ii) gel water, and (iii) capillary water.

4. At later stages of hydration, the speciﬁc surface area is independent of the degree of hydration.

5. Based on experimental water-vapour isotherms. On heating at 23◦C, gel and capillary water can be fully removed. On heating at 105◦C, part of the non-evaporable water can be removed. On heating at 105◦C, the non- evaporable water is removed.

• Ishai’s model [120] –

249 1. In addition to the chemical water, four types of evaporable water are modeled; (i)intracrystalline(zeolitic) water (thickness less than 4A˚), (ii) water adsorbed on the surface of the crystallites (thickness between 4A˚ and 8A˚), (iii) water conﬁned between adjacent crystallite surfaces (thickness less than 8A˚), and (iv) capillary water (at a distance greater 10A˚ ∼20A˚).

2. Capillary water is outside the range of surface forces .

• Feldman and Sereda’s model [84]–

1. In addition to the chemical water, two types of evaporable water are modeled; (i) interlayer hydrate water, and (ii) physically-adsorbed water in gel and capillary pores.

2. Water-vapour isotherms for hardened cement pastes are irreversible. The BET method is not applicable.

3. Variation of measured Asp by diﬀerent methods is recognized. The use of

nitrogen for Asp determination results in much lower values of Asp than using water since the size of nitrogen molecules (4.05A˚) is greater than the one of water molecules (3.25A˚).

4. Evaporable water can reside in the interlayer pores of the C-S-H.

While many other hcp models have been proposed since these classical models became available, essential differences among the models are: (1) classification of the water held in the C-S-H structure, (2) assumed importance of different types of water in the properties of the C-S-H structure and cement pastes, and (3) distribution of the pore sizes and configuration of the pores. In this research, two types of water inside concrete are considered:

1. Free water – Water molecules that are subjected to gravitational ﬁeld only and can respond to the application of an external EM ﬁeld without any delay in their relaxation time.

2. Bound water – Water molecules that are subjected to both gravitational ﬁeld and the attraction ﬁeld (chemical potential for adsorption) from the solid phase

250 of cementitious composites, displaying certain amount of delay in their relaxation time when subjected to the application of an EM ﬁeld.

The amount of bound water is determined by the product of the specific surface area (Asp) of hcp and the thickness of adsorbed, bound water layer/film. Adsorption isotherms are typically used for determining the specific surface area of a material. The BET (Brunauer-Emmett-Teller) method is usually applied for hardened cementitious composites [221]. In the dielectric modeling of water, it is also assumed that (1) Only liquid water is considered. Water vapour and ice are not considered; (2) Water can only reside in concrete, not in GFRP-epoxy composite; (3) Only the water presenting in the pore structure of C-S-H is considered for modeling. The water in fine and coarse aggregates is not considered; (4) Chemical or non-evaporable water has no impacts on the measured dielectric properties of concrete with different water contents; (5) Free water and bound water are both evaporable water. These assumptions are considered reasonable for the dielectric modeling of materials in the frequency range of 8 GHz to 18 GHz in room temperature. In what follows, dielectric modeling of free water and bound water is provided.

6.5.1 Free Water

In this research, free water is deﬁned as the water molecules sorely subjected to the gravitational ﬁeld externally and the O-H bond energy internally. The O-H bond energy is half the energy of formation of water molecule since water has two O-H bonds, whose value is 109.7 kcal/mol at 0◦K (-273◦C). This bond energy is related to the relaxation time τ (rad-sec) of water. Our interest of free water lays in its frequency-dependent dielectric dispersion. For the dielectric modeling of free water using Havriliak-Negami’s model, parameters to be determined are s, ∞, τ, and α1. Reported dielectric properties (complex 0 ” permittivity; r and r) in the frequency range of 0.58 GHz and 134.9 GHz are quoted from the literature and used for obtaining the parameters. These measurements are

251 listed in Table 6.10.

Table 6.10: Measured complex permittivity of water at 20◦C

0 Frequency (GHz) r(0) r”(0) 0.581 80.3 2.75 1.741 79.3 7.9 32 77.2 13.1 3.651 76.3 15.6 9.32 61.5 31.4 9.6253 63.5 29.1 19.253 41.5 36.5 23.812 30.8 35.2 33.93 23.1 31.2 344 19.5 30 70.23 9.5 19.97 96.772 8.5 12.0 134.93 5.64 10.42

Using Havriliak-Negami’s model and the mean-square error (MSE) criterion for curve ﬁtting, the ﬁtted parameters are obtained at the minimum MSE.

(ω) 71.84 = 9.34 + (6.107) 0.93 1 0 1 + (i9.23ω) where the relaxation time τ=9.23 rad-ps (1 ps = 1 picosecond = 10−12 seconds) when ω (rad/sec) is used, or τ=9.23 ps when f (1/sec) is used (ωτ is dimensionless). Obtained τ is close to the value (9.26 rad-ps or 9.26 ps) reported by Grant et al.(1957) [75] in 20◦C. This model becomes a Cole-Cole-type model since γ = 1. Result of curve- ﬁtting is in good agreement with measurements as shown in Figure 6-17. Reported model parameters in Debye’s model and Cole-Cole’s model are also summarized in

1[75] 2[107] 3[153] 4[96]

252 40

30 r

" 20 ε Grant et al. (1957) Hasted & Shah (1964) 10 Grant & Shack (1967) von Loon & Finsy (1975) Havriliak-Negami' model 0 0 10 20 30 40 50 60 70 80 90 ε' r Figure 6-17: Performance of the Cole-Cole model for free water

Table 6.11 for comparison.

Table 6.11: Fitted parameters in Debye’s and Cole-Cole’s models of free water

◦ Source T( C) s /0 ∞/0 τ(ps/rad) α1 f (GHz) [235] 19 81 1.8 – 0 – [20] 19 81 1.8 9.4 0.09 – [75] 0∼60 – 4.5 3.87∼ 0.02±0.007 0.58∼ 5.5 17.66 0 30.8 [220] 20 – 5.5∼6.2 – 0 96.77 [107] 20 78.5 5.2 – 0 3∼23.8 [143] – 81 1.8 11.5 0 – [82] – 81.836 23.461 9.4 0 0.5∼3

6.5.2 Bound Water

The interaction between the adsorbed molecules and the surface may be classiﬁed into the following types: (1) Chemical adsorption – Or chemisorption. e.g., stearic acid from benzene solution on metal powders; (2) Hydrogen bonding – e.g., long-chain alcohols from hydrocarbon solution on dry oxide surfaces; (3) Hydrophobic bonding – e.g., acids on polystryrene; and (4) van der Waal’s forces – Long-range attraction.

253 In addition to these forces, capillary condensation also plays an important role in porous structures. When water molecules (absorbate) enter the pore structure of hydration products (absorbent), they tend to be adsorbed onto the surface of the pores once they contact the surface. This type of physisorption (reversible process) can develop an adsorbed layer on the surface of the pores. Thickness of the adsorbed layer can grow up to a threshold value at which the surface attraction is no longer capable of holding any free water molecules onto the layer. In this dissertation, the ”bound water” is referred to the water molecules within this layer. Figure 6-18 illustrates the considered sorption model.

solid phase of optional concrete boundary t

bound water c.d. region free water chemical water 3.25 Å tmax

water molecule

Figure 6-18: Considered sorption model in the pore structure of hcp

Dielectric properties of the bound water subjected to surface attraction are diﬀer- ent from the ones of the free water since the attraction constrains the movement of water molecules. Under the application of external ﬁelds (radar signals), the rotation of bound water molecules is hindered, resulting in the change of individual relaxation times of bound water molecules. It is believed that such surface attraction, as a

254 binding energy between the molecule and the surface, is important for the dielectric dispersion of molecules possessing permanent electric dipole moments such as water and can be modeled by a potential function. Since the variation of individual relaxation times is described by the distribution function y(τ) and recall Eqs.(6.82) and (6.83), the dielectric properties of the bound water are described by

∞ Z 0 1 bw(ω) = ∞ + dτ 2 · y(τ) (6.108) 1 + (ωτbw) 0 ∞ Z 00 ωτbw bw(ω) = dτ 2 · y(τ) (6.109) 1 + (ωτbw) 0

where τbw is the relaxation time for each molecule layer of bound water, which is related to the potential barrier determining the strength of attraction.

Relaxation Time and the Reaction Rate of Relaxation

It is obvious that the determination of complex permittivity depends on the knowledge of the distribution function of relaxation time, y(τ). Fr¨ohlich (1949) [90](page 93) had shown that, in dilute solutions, the y(τ) of dielectric materials displaying Debye’s dispersion can be expressed by

kT 1 y(τ) = (s − ∞) (6.110) v0 τ

−23 −1 where k is the Boltzmann’s constant (or kB, 1.3806504×10 J ·K =8.617343 −1 ◦ eV·K ), T is the Kelvin temperature ( K), v0 is the diﬀerential energy barrier of

the energy barrier H0 measured from a normal (permanent) position of the water

molecule. (H0 + v0) is the range of inﬂuence to the complex permittivity of the molecule under investigation. The relationship between the relaxation time and the sorption process can be explained by the rate theory in chemistry. Kauzmann (1942) [130] suggested that the reaction rate of relaxation (or simply the relaxation rate) k (rad/s) of a chemical

255 process can be determined by

1 kT dF k = = exp − (6.111) τ h RT where h is Plank’s constant (6.626069×10−34J·s = 4.135667×10−15eV·s), dF = dH − SdT is the molar (Helmholtz) free energy, dH is the heat of activation for dipole relaxation, S is the entropy of activation for dipole relaxation, and R is the gas

m3·Pa J constant (8.3144 K·mole = K·mole ). In porous media like concrete, the Gibbs free energy is considered instead.

X dG = V dp + ΦdA + µjdnj − SdT (6.112) j where V is the volume, p is the pressure, Φ is the surface tension (capillary condensation), A is the interfacial area, µj is the chemical potential of component j (kJ/mole or kcal/mole), nj is the number of moles of component j (One gram-mole of water has a mass of approximately 18.016 grams, and one ml of water has 3.011022 molecules.)

The chemical potential µj is given by

∂G µj = (6.113) ∂nj T ;p;A

Measurements Conducted by Badmann et al.(1981)

Badmann et al. (1981) [212] conducted a series of experiments on the determination of chemical potential, µ, for porous hardened cement paste (hcp) in which capillary condensation (cc) is signiﬁcant. The BET C constant was obtained from their measurements on hcp (C3S=59.33%, C2S=12.86%, C3A=16.33%, C4AF=8.92%) from 1%

to 97% relative humidity (r.h.). The chemical potential µcc(t) was given by

t t µcc(t) = −µ0 exp − − µ1 exp − (6.114) h0 h1 where t is the thickness of the adsorbed water layer or the bound water layer (A˚), √ 3 µ0 = 5.9 C kJ/mole = 18.5 kJ/mole when C = 31 as the experimentally-determined

256 BET C constant, h0 = 1.91A˚, µ1 = 0.142 kJ/mole for h1 = 26A˚. An expression of

µvdW accounting for the van der Waals forces from silica surfaces was also proposed by Badmann et al. (1981) [212].

AVm µvdW(t) = − 3 (6.115) 6π (h + ∆w) where A is the Hamaker constant (A < 1.4 × 10−13 erg = 1.4 × 10−23 kJ for the water

3 ◦ ◦ on silica surfaces), Vm = 18 cm /mole is the molar volume of water at 25 C (298 K), h is the distance from the center of the ﬁrst-layer water molecule to the center of the water molecule at other layers (A˚), ∆w is the radius of water molecules (∆w is taken as 1.625A˚). Figure 6-19 shows the determination of h in the sorption model. The

bound water h

chemical water

Figure 6-19: Jellium distance in the sorption model

chemical potential µhcp (shown in Figure 6-20) between the bound water and the pore structure of hcp is modeled by

µhcp(t) = µcc(t) + µvdW(t) (6.116)

In Figure 6-20, the decay of µhcp(t) is essentially an exponential function as suggested by other researchers [105]. Additionally, the semi-empirical expression of statistical thickness of the adsorbed water layer (ﬁlm) on hcp was also reported by Badmann et

257 1 10 µcc

µvdW

0 10

-1 10 Potential energy (kJ/mole) energy Potential

-2 10

0 5 10 15 20 Thickness (angstrom) Figure 6-20: Used chemical potential accounting for the formation of the bound water on the surface of hcp

al. (1981) [212]. " √ # 5.9 3 C P t = −h0 ln − h0 ln − ln (6.117) RT P0

where P is the pressure at a given r.h., and P0 is the pressure at r.h.=0%. This expression was obtained from the measurements (shown in Table 6.12). The perfor-

mance of Eq.(6.117) is illustrated in Figure 6-21. In Figure 6-21, samples of C2S

and C3S were prepared as non-porous specimen and hcp as porous specimen. The maximum average thickness of the adsorbed water layer on hcp was reported to be

around 18A˚. The maximum thickness of adsorbed water layer on SiO2 measured by Harkins and Jura (1944) [105] was 12.8A˚ when P = 80% at which Eq.(6.117) pre- P0 dicts 12.22A˚. T.C. Powers (1965) [207] suggested the maximum thickness on cement paste at r.h.=100% to be 13A˚. Considering the similarity the chemical composition between SiO2 and hcp, it is believed that Eq.(6.117) can be used to generally describe the adsorption of water molecules on hardened hcp.

258 Table 6.12: Averaged statistical thickness, t (A˚), of adsorbed water layer on hcp t(A˚) 0.92 1.33 1.78 2.28 2.64 2.94 3.24 3.55 3.83 4.06 4.32

P/P0(%) 1 2.5 5 10 15 20 25 30 35 40 45 t(A˚) 4.53 4.79 5.17 5.37 5.68 6.08 6.61 7.27 8.20 9.87 —

P/P0(%) 50 55 60 65 70 75 80 85 90 95 —

20 Data on C S and C S by Badmann et al.(1981) 18 2 3 Model for C S and C S (BET C = 31) 2 3 16 Model for hcp (BET C = 31) 14

Statistical thickness, t (A) t thickness, Statistical 4

0 0 0.2 0.4 0.6 0.8 1 P/P 0 Figure 6-21: Measurements and model prediction of non-porous and porous specimens of hydration products

Proposed Dielectric Model for Bound Water

The purpose of referring the measurements conducted by Badmann et al.(1981) [212] in this modeling work is to develop an expression for the complex permittivity of bound water in hcp based on reported experimental measurements made on hcp.

Considering an adiabatic process (dT = 0) for unit mole of water molecules (nj = 1), in which no surface change and pressure change occur (dA = 0 and dp = 0), we have ∂G = dG. Recall the reaction rate equation, Eq.(6.111). The formation energy of

259 free water molecules can be calculated from the measured relaxation time, τw.

τ (T )kT µ (T ) = RT · ln w (6.118) w h where h is Planck’s constant, k is Boltzmann’s constant, T is the Kelvin temperature, and R is the gas constant. Table 6.13 lists the reported measurements of the relaxation time of free water molecules in the temperature range from 0◦C to 60◦C (or 273◦K to

◦ 333 K) [106]. With this relation, the calculated bonding potential, µw(T ), is provided

Table 6.13: Relaxation time, τw (ps), of free water molecules T(◦C) 0 10 20 30 40 50 60

τw(ps) 17.7 12.6 9.2 7.1 5.7 4.8 3.9

in Figure 6-22. This bonding potential macroscopically accounts for the multiple

10.5

10 , (kJ/mole) µ

9.5 Bonding potential, potential, Bonding

9 270 280 290 300 310 320 330 340 Absolute temperature (K) Figure 6-22: Calculated bonding potential of free water molecules

interactions between one free water molecule and its neighbor molecules when they all subject to the application of an external ﬁeld. Within the adsorbed water layer on

260 hcp, relation of water molecules is believed to be hindered by the attraction potential

from hcp (µhcp). Equivalently, the relaxation time is prolonged due to the presence of

µhcp. Outside the adsorbed water layer, the relaxation time of bound water molecules should reduce to the one of free water molecules. The general form of the bonding potential is given by

µ(T, t) = µw(T ) + µhcp(T, t) (6.119)

in which the thickness of adsorbed water layers, t, mainly depends on temperature, T , and pressure ratio, P . Therefore, the relaxation time of bound water can be P0 calculated by h µ(T, t) τ (T, t) = exp (6.120) bw kT RT This result is illustrated in Figure 6-23 for T = 293◦K or 20◦C. Note that, in this model, the relaxation time of water molecules outside the statistical thickness of the bound water layer, t, converges to the one of free water molecules. Therefore, the boundary condition of τbw is satisﬁed in this model. Substitute Eq. (6.110) into

4 x 10 2 150

1.8

1.6

1.4 100 1.2

0.8 50 0.6 Relaxation time (ps) time Relaxation (ps) time Relaxation 0.4

0.2

0 0 0 5 10 15 20 5 10 15 Thickness (A) Thickness (A)

Figure 6-23: Calculated relaxation time distribution over the bound water later

261 Eqs.(6.108) and (6.109). We have

( − ) kT ∞ 0 (ω) = + s ∞ · ln τ (6.121) bw ∞ 2 1 + (ωτbw) vo 0 ( − )ωτ kT ∞ 00 (ω) = s ∞ bw · ln τ (6.122) bw 2 1 + (ωτbw) vo 0

In the above equations, v0 represents the diﬀerential energy barrier of the energy barrier H0 within the adsorbed water layer. This diﬀerential energy barrier is considered to be attributed to the variation in the µhcp(t), which is experimentally determined.

Equivalently, v0 is numerically calculated from Figure 6-20. It is noteworthy to point out that the value of τ should not exceed the relaxation time at the center of adsorbed water molecules in the ﬁrst layer. This constitutes the upper bound of τbw. Also, τbw should not less than the relaxation time of free water molecules, τw. Thus, the above equations now become

0 (s − ∞) kT τbw(T, t = ∆w) bw(ω) = ∞ + 2 · ln (6.123) 1 + (ωτbw) vo τw(T ) 00 (s − ∞)ωτbw kT τbw(T, t = ∆w) bw(ω) = 2 · ln (6.124) 1 + (ωτbw) vo τw(T )

Eqs.(6.123) and (6.124) are the proposed dielectric model for determining the complex permittivity of bound water adsorbed on hardened hcp.

An information needed for determining the contribution of bound water in the dielectric properties of hcp is the volumetric fraction of bound water in hcp, which can be experimentally determined by the BET method. This can be done either by (1) measuring the weight change of specimens or by (2) multiplying the surface area of hcp by the thickness of bound water (adsorbed water molecules). The volume computed from both approaches should converge to each other if the measurements are conducted at perfect precision. However, since the formation of hcp (or concrete) structure depends on many factors (e.g., w/c ratio, curing temperature, degree of hydration) and the variability in measured surface areas using diﬀerent adsorbates (Nitrogen at 77◦K and liquid water at 293◦K, typically), obtaining a general equation

262 capable of estimating the amount of adsorbed water molecules on the surface of hcp is still challenging. While the derivation of such equation is beyond the scope of this research, a simpliﬁed model is proposed, with several assumptions, for approximately estimating the amount (volume) of bound water. The measurements using Nitrogen adsorption by Feldman (1972) [83] on the spe-

ciﬁc surface area (Ssp) of hcp specimens with diﬀerent w/c ratios are adopted. It is assumed that (1)ordinary Portland cement (Type I) used and no admixtures, (2) stabilized hydration, and (3) regular curing process and temperature. The reported

2 Ssp values of hcp are listed in Table 6.14. The curve-ﬁtted model for Ssp (m /g) is

Table 6.14: Speciﬁc surface area of hcp [83] W/c ratio 0.4 0.5 0.6 0.8 1.0 Ssp 30 55 51 57 57

Ψ − 0.4 S = 9.2986 + 18.1437 · tan−1 (6.125) sp 0.05

where Ψ is the w/c ratio. The main reason for using Nitrogen (N2) as the adsorbate is due to its sensitivity to the change of w/c ratio in the Ssp of hcp determined by

the BET method. Knowing that the Ssp found by N2 molecules is typically much less than the value found by water molecules (H2O), the Ssp(N2) is indeed the speciﬁc ”surface” area since it does not include the area between hcp gels/layers/pores. Not

only because of the well-known incapability of N2 of penetrating into hcp gel pores, but ◦ also because of the low thermal activation state when using N2 at 77 K. Performance of the model is shown in Figure 6-24. This model can be further improved with additional measurements.

6.6 Dielectric Properties of GFRP

Glass ﬁber reinforced polymer (GFRP) consists of glass fabric and epoxy resin; glass fabric is for providing tensile strength of the assembly and epoxy resin for molding purpose. Mechanical properties, such as Young’s modulus, Poisson ratio, tensile strength,

263 60

50 /g) 2

10 Specificsurface area (m N2 adsorption by Feldman (1972) Fitted model 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 w/c ratio Figure 6-24: Performance of the curve-ﬁtted model for the speciﬁc surface area of hcp by nitrogen adsorption

shear strength, and ﬂexural strength of GFRP, are typically concerned among its other properties in civil engineering applications. Durability is also crucial for the long-term performance of GFRP, such as thermal coeﬃcient, thermal conductivity, and glass transition temperature. Usually, these properties are laboratory-tested using ASTM (American Society for Testing and Materials) methods and provided by manufacturers/suppliers.

However, dielectric properties of GFRP have yet become required information in the property description of GFRP for its use in strengthening of concrete structures. The manufacturer/supplier of GFRP and epoxy (Fyfe Co. LLC) in this research did not perform standard measurements of GFRP over a range of frequencies. To overcome this diﬃculty, reported dielectric measurements of the GFRP and epoxy similar to the ones used in this research are adopted for modeling, which are taken from the experimental work done by von Hippel (1954) [113] in the Laboratory for Insulation Research at MIT.

264 6.6.1 Epoxy Resin

Chemical molecules are classiﬁed by the functional groups they contain. An epoxide is a molecule containing the epoxide group (one oxygen and two carbon molecules) as part of its structure. An epoxy resin system is formed when other types of molecules are added to the epoxide to formulate a thermosetting system. The formation of epoxy resin systems undergoes a curing reaction to achieve a rigid state, in which curing agents are also included for required curing rate in practice. Sometimes diluents/ﬁllers are used for improving the viscosity of uncured epoxy resin systems. Dielectric properties of an epoxy resin system depend on their chemical composition or functional groups as the resultant macromolecular system. Complex permittivity of the araldite adhesive Type I Natural manufactured by Chiba-Geigy Ltd.(Japan) whose composition is similar to the used Tyfo r S Epoxy was measured in the frequency range 100 Hz∼10 GHz in room temperature (25◦C) [113]. Four dielectric models have been tested for their applicability on the epoxy; Debye model, Cole-Cole model, Davidson-Cole model, and Havriliak-Negami model. It is found that

1. The Debye model cannot describe the dispersion behavior of this epoxy resin system simply because the Cole-Cole diagram of the epoxy is not a semicircle.

2. The Davidson-Cole model can describe the dispersion behavior of this epoxy resin system but with relatively high error comparing to the Cole-Cole model.

3. The Havriliak-Negami model can describe the dispersion behavior of this epoxy resin system but the coeﬃcient γ is almost unity, suggesting the use of the Cole-Cole model instead.

The obtained Cole-Cole model for epoxy is

(ω) 1.06 = 2.96 + 0.93 (6.126) 0 1 + (i99.47ω) where the unit of constant 99.47 is rad-ns (1 ns = 1 nanosecond = 10−9 seconds). Its performance is shown in Figure 6-25.

265 0.14

0.12

0.1

0.08 " r ε 0.06

0.04

0.02 von Hippel (1954) Model 0 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 ε ' r Figure 6-25: Performance of the Cole-Cole model for epoxy

6.6.2 E-glass Fabric

A unidirectional glass fabric system Tyfo r SEH-51A (Fyfe Co. LLC) was used in the GFRP-retroﬁtted concrete specimens. Dielectric measurements of this GFRP system (using E-glass; for electrical applications) are not available from the material supplier. Similar dielectric measurements on E-glass fabric (Owens-Corning Corp.) were reported in the literature [113], and they are used for modeling the dielectric properties of E-glass fabric in the frequency range of 8 GHz to 18 GHz.

Recall the dispersion equations of the Cole-Cole model.

  0 s − ∞ sinh(1 − α1)x = ∞ + 1 − α π  (6.127) 2 cosh(1 − α )x + cos 1 1 2  α1π  cosh 00 s − ∞ 2 =  α π  (6.128) 2 cosh(1 − α )x + sin 1 1 2

266 where x = ln (ωτ). The relationship between 0 and 00 is derived to be

− α π s ∞ · cos 1 · 0 2 2 00 = (6.129) − α π α π + s ∞ · sinh (1 − α ) x + cos 1 − sin 1 · s ∞ − 0 2 1 2 2 2

To perform the curve-ﬁtting as a parametric system identiﬁcation, a model similar to Eq.(6.129) is obtained.

0 00 0.9485 = " # + 1.0391 (6.130) 0 − 6.32 0.4406 0.4599 − 455 + 0.5578 − 1.04140 6.3

With these parameters the ﬁtted model has a standard error of 0.004242 and a cor-

0 relation coeﬃcient of 0.9855 with the measured data. The expressions of r and ”r as a function of frequency are also determined.

0 r = 4.2027 − 0.022 [log(f + 107.74) − 1676.45] (6.131)

”r = 0.004732 [log(f − 25, 903, 094) − 13.72] (6.132) where the frequency f is in hertz. The performance of the model is shown in Figures 6-26 and 6-27. Figure 6-27 illustrates the performance of Eqs.(6.131) and (6.132). It is seen that, although the performance of the model for the glass fabric in lower frequency ranges is relatively poor, the model provides reasonable estimation in the high frequency range (GHz) which is the frequency range of interest. Therefore, the use of this model is considered feasible in the frequency range of 8 GHz to 18 GHz.

6.6.3 GFRP Layer/Sheet

The unidirectional, one-layered GFRP sheet consisting of glass fabric and epoxy is illustrated in Figure 6-28. Such material is generally considered as a dielectric mixture since both glass fabric and epoxy are dielectric. Knowing that the mixing ratio of epoxy to glass fabric is 0.8:1 by mass and densities of epoxy and glass fabric are 0.0011

267 0.1 von Hippel (1954) Model 0.08

0.06 r " ε 0.04

0.02

0 6 6.05 6.1 6.15 6.2 6.25 6.3 6.35 6.4 6.45 6.5 ε' r Figure 6-26: Performance of the Cole-Cole model for E-glass fabric

kg/cm3 and 0.00255 kg/cm3 [152], the volumetric ratio of epoxy to glass fabric in the GFRP layer is

me mg 0.8 1 Ve : Vg = : = : = 1.82 : 1 (6.133) ρe ρg 0.0011 0.0025

where V denotes volume, W denotes weight, and ρ for density. The volumetric fractions of epoxy and glass fabric are

1.82 1 v : v = : = 0.645 : 0.355 (6.134) e g 2.82 2.82

In order to determine the dielectric properties (namely, complex permittivity) of such dielectric mixture, six two-phase (two-component) dielectric mixing models/laws are considered [256], in which is the eﬀective or the overall complex permittivity of

the mixture, h is the complex permittivity of host dielectric, and i is the complex permittivity of inclusion dielectric. In the conﬁguration as shown in Figure 6-28, epoxy is the host dielectric and GFRP the inclusion dielectric.

268 6.5 0.08 von Hippel (1954) von Hippel (1954) Model 0.07 Model 6.4

0.06

6.3 0.05 r r ' 6.2 " 0.04 ε ε

0.03 6.1

0.02 6 0.01

5.9 0 0 5 10 15 0 5 10 15 10 10 10 10 10 10 10 10 log(f) (Hz) log(f) (Hz) Figure 6-27: Performance of the Cole-Cole model for E-glass fabric – Real part and imaginary part

Epoxy resin

E-glass fabric

Figure 6-28: Unidirectional GFRP layer

1. Maxwell (1891) [174] – Spherical inclusion (Maxwell-Garnett formula)

3vih = h + (6.135) i + 2h − vi i − h

2. Rayleigh (1892) [242] – Spherical inclusion (Average t-matrix approximation)

− 1 + 2v i h i + 2 h = i h (6.136) i − h 1 − vi i + 2h

3. Wiener (1912) [268] – Arbitrary shape inclusion; u = 2h for spherical, u = 2i

269 − 3 for disk-like inclusion, u = i h for needle-like inclusion (used). 2

− + v i h u h i + 2 = i h (6.137) i − h 1 − vi i + u

4. Lichtenecker (1926) [150] – Arbitrary shape inclusion

= evi ln i + vh ln h (6.138)

5. B¨ottcher (1945) [21] – Spherical inclusion √ C2 + 8 − C = i h (6.139) 4

where C = (1 − 3vi)i + (3vi − 2)h.

6. Looyenga (1965) [155] – Spherical inclusion

√ √ 3 3 3 = (vi i + vh h) (6.140)

In order to evaluate their performance, reported GFRP-epoxy dielectric measurements are needed. Unfortunately, extremely limited amount of dielectric measurements made on E-glass GFRP-epoxy was reported. Only one publication is found in the literature to be comparable. Seo et al. (2004) [119] measured the complex permittivity of a E-glass GFRP and similar epoxy sheet (UGN150, SK Chemical,

0 South Korea) in the frequency range of 8.2 GHz to 12.4 GHz. The measured r was 5.004 at 10 GHz and 4.986 at 10.4 GHz. The case of ﬁber orientation at θ = 90◦

0 in their work corresponds to the conﬁguration in this research. Predicted r values of the GFRP-epoxy system (vi = 0.33) are provided in Figure 6-29. Absolute errors 0 are calculated for the measured r at 10 GHz and 10.4 GHz, as shown in Table 6.16. 0 While all six models provide acceptable estimation on r in a reasonable range of 0 error, their performance on ”r is in general much poorer than on r. Reported ”r

270 Table 6.15: Comparison of two GFRP-epoxy systems

Parameter Seo et al. (2004) This research

Density ρepoxy 10.01 lb/gal 9.17 lb/gal Density ρGF RP 21.20 lb/gal 21.28 lb/gal Thickness of GFRP-epoxy layer 0.33 cm (0.25∼0.4) cm Vepoxy : VGF RP 0.67 : 0.33 0.645 : 0.355 vi 0.3308 0.355

4.94 0.05

0.045 4.92 0.04 4.9 Maxwell (1981) 0.035 Rayleigh (1982) 4.88 0.03 Wiener (1912) ' " r Lichtenecher (1926) r ε ε 4.86 0.025 Bottcher (1945) Looyenga (1965) 0.02 4.84 0.015 4.82 0.01

4.8 0.005 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 8 9 10 11 12 Frequency (GHz) Frequency (GHz)

Figure 6-29: Performance of six mixing models for GFRP-epoxy

values were 0.1112 at 10GHz and 0.1111 at 10.4 GHz. On the other hand, predicted

”r values fall in the range of 0.007∼0.042, all in underestimation. This is attributed to the diﬀerence in the used material (mostly epoxy). However, it is found that, among these models, only Wiener’s mixing model dis-

plays same descending behavior of ”r as reported by Seo et al. (2004) [119] when fre-

quency increases. Wiener’s model also provides the least error on ”r, compared with other mixing models. In addition, the needle-like inclusion assumption in Wiener’s model is physically analogous to the GFRP inclusion in a epoxy resin system. There- fore, Wiener’s mixing model is applied for modeling the GFRP-epoxy system in this research. i − h i − 3h h + vi i + 2h 2 = − (6.141) 1 − v i h i − 3 + i h i 2

271 Table 6.16: Performance of six mixing models for a GFRP-epoxy system

0 0 Model Predicted r Error (%) Predicted r Error (%) (10GHz) (10.4GHz) Maxwell (1811) 4.8602 2.80 4.4950 9.85 Rayleigh (1892) 4.8602 2.80 4.4950 9.85 Wiener (1912) 4.8574 2.86 4.3592 12.57 Lichtenecher (1926) 4.8221 3.57 4.4520 10.71 B¨ottcher (1945) 4.9154 1.70 4.5444 8.86 Looyenga (1965) 4.9064 1.88 4.5406 8.93

6.7 Dielectric Properties of Concrete

Dielectric modeling of multi-phased, heterogeneous composites like concrete is a challenging task. Concrete, as a cementitious composite, is a mixture of

• Portland cement hydration products — Including calcium silicate hy-

drate (C-S-H or C3S2H8 or 3CaO · 2SiO2 · 8H2O), calcium hydroxide (CH or ¯ CaO · H2O), ettringite (C6AS3H32 or 3CaOAl2O3 · 3CaOSO3 · 32H2O), and mono- ¯ sulfoaluminate (C3A · CS · H12 or 3CaOAl2O3 · CaOSO3 · 12H2O). ASTM spec- iﬁcations such as C150 deﬁne the properties of Portland cement.

• Water – ASTM C94 requires tests of mixing water when its quality is ques- tionable.

• Aggregates – Including ﬁne aggregate (sand) and coarse aggregate gravels). ASTM C33 and C294 describe the properties of acceptable aggregates.

Chemical admixtures are also used in practice as a processing addition to aid in manufacturing and handling of concrete or as a functional addition to modify the properties of concrete. In this research work, chemical admixtures are not considered in the dielectric modeling, nor used in the manufacturing of concrete specimens. In addition to ASTM speciﬁcations, other building codes such as ACI 318M also provide requirements on the quality of the components used in concrete. Three approaches may be applied to model the dielectric properties of concrete.

272 1. Phenomenological approach – Describe the complex permittivity of concrete as a homogeneous composite by considering one or two major parameters (such as frequency) only. This can lead to only the global/macroscopic description on the dielectric dielectric dispersion of concrete.

2. Physical approach – Describe the complex permittivities of the components of concrete individually using the causality relation (or the Kramers-Kronig relation), and assemble them by the dielectric mixture theory. This approach can explain the dielectric dispersion of concrete from a microscopic perspective, although the composition of concrete must be fully understood.

3. Statistical-physical approach – Describe the complex permittivities of the components of concrete individually using the causality relation, and assemble them by statistical distribution functions. This approach can explain he dielectric dispersion of concrete from a microscopic perspective, while signiﬁ- cant amount of experimental data must be provided in order to obtain reliable statistical functions.

Phenomenological approach has been widely used for engineering applications since it avoids the direct difficulties on dealing with the microscopic structure of concrete. In many cases, the complex permittivity of concrete is treated as a function of frequency and the water/cement ratio. Physical approach requires a full knowledge on the composition of concrete at the state of investigation, which is not easy to acquire considering the spatial variability of concrete’s composition. Microanal- ysis methods such as SEM (scanning electron microscopy), NMR (nuclear magnetic resonance), and X-ray diffraction techniques are typically used for determining the micro-structure of concrete. At the present time, comprehensive experimental data on the micro-structure of concrete specimens made of various water/cement ratios and other manufacturing factors are not reported in literature. Statistical-physical approach even demands more measurements than the physical approach to result in reliable distribution functions with sufficient confidence level. It is obvious that experimental measurements on the complex permittivity of concrete are key to the

273 success of dielectric modeling on concrete. In the context of this doctoral research, the purpose of this modeling work is to estimate/predict the complex permittivity of concrete in order to model the concrete in numerical simulation and to improve the resolutions in the reconstructed images for condition assessment. In this development, two major challenges arises.

1. Incomplete information regarding the spatial composition of concrete – The microscopic structure of concrete is mainly determined by the water/cement ratio, degree of hydration, and the conditions (temperature and moisture) under which the concrete is cured. It is diﬃcult to completely describe the components (hydration products, pores, moisture, ﬁne and coarse aggregates) of concrete and their spatial distributions in concrete even if all the information during the manufacturing of concrete becomes available. In addition, shape of the pores in concrete is also important since it determines the distributions of air and water molecules in concrete.

2. Lack of the full dielectric dispersion spectra for the components in concrete – Even if the composition of concrete can be fully aware, dielectric dispersion spectra of the components in concrete, such as hydration products (e.g.,C-S-H) and aggregates, are still needed in order to calculate the bulk complex permittivity of concrete either physically or statistically. This information is also not complete in literature.

These constraints limit the development of a full-spectrum physical model of concrete. As the result of it, dielectric models of concrete and cementitious composites (cement paste and cement mortar) can only be developed based on specimens with certain parameters (e.g., frequency, water/cement ratio), on which some dielectric measurements have been reported. In view of the limited amount of dielectric measurements of concrete in the frequency range of or nearby 8 GHz∼18 GHz, dielectric modeling is conducted on an- hydrous (oven-dried) hydrated cement paste (hcp) in this research.

274 6.7.1 Determination of Volumetric Fractions

In the process of mixing two or more ingredients/components into one mixture, it is the volumetric fraction of each component pivotal in determining the properties of the mixture. One major difference distinguishing concrete as a hydrate is the formation of a porous structure with high surface area. This porous structure is the remaining space left by the evaporated mixing water during the hydration process. Such evaporation can be achieved either by internal thermal gradient originated from the hydration heat (room-temperature drying) or by artificial heating (oven drying, defined by ASTM C29). To what extent the weight loss of hydrated cementitious composites achieve re- gards the classification of mixing water in the hydration process. Typically, the mixing water can be categorized into the following two types. This classification is based on the research works by [209], Copeland, Kantro, and Verbeck (1960) [148], Ishai (1965) [120], Feldman and Sereda (1970) [85], Feldman (1972) [83], and [238]. Definitions in recent texts [179, 223] are essentially based on the findings found in the early work.

1. Non-evaporable water – The water that participates the hydration reactions and absorbed by the hydration products is considered non-evaporable.

2. Evaporable water – The water that participates in the casting of cementitious composites for workability purpose but not being absorbed by the hydration products after the hydration reactions stabilized is considered evaporable. This includes:

• Free water – The water residing in the pore structure not subjected to the surface traction of the hydration products, whose volume is denoted

by Vfw. This type of water can be removed when the composite is heated up to 105◦C. Free water is also termed capillary water.

• Bound water – The water residing in the pore structure and subjected to the surface traction of the hydration products, whose volume is denoted

by Vbw. This type of water can be removed when the composite is heated

275 up to 105◦C. Bound water is also termed gel water.

• Chemical water – The water underneath the surface of hydration products, attached by a strong chemical potential, whose volume is denoted by

Vcw. This type of water can be removed when the composite is heated up to 1010◦C. Chemical water is also termed combined water.

Since the chemical water is embedded inside the hydration products, it is considered that the pore structure in cementitious composites is formed by the liberation of the free and bound water. The oven-drying scheme used in the specimen preparation of most experiments in literature can achieve 105◦C, suggesting the removal of both the free and bound water in their measurements. The removal of the chemical water requires a heating temperature up to 1000◦C which is not a situation commonly encountered in practice. This scheme is illustrated in Figure 6-18. Considering two extreme cases (fully-saturated and oven-dried specimens), the volumetric fractions of cement paste, cement mortar, and concrete are deﬁned in

Table 6.17 in which vs denotes the volumetric fraction of the solid phase, vg is the

volumetric fraction of the gaseous phase, and vl is the volumetric fraction of the liquid m m phase, all in the multi-phased composite. In Table 6.17, s = hcp for cement paste ρs ρhcp ms mhcp mfa (purely hydration cement products or hcp), = vhcp · + vfa · for cement ρs ρhcp ρfa ms mhcp mfa mca mortar, and = vhcp · + vfa · + vca · for concrete, where vhcp, vfa, and ρs ρhcp ρfa ρca vca denote the volumetric fractions of cement paste (hcp), ﬁne aggregate (fa), and coarse aggregate (ca), respectively.

As indicated in Table 6.17, the mass of free and bound water, mfw + mbw, is the diﬀerential mass between the fully-saturated mass and the oven-dried mass of the specimen, while the total amount of water determined by the water/cement ratio is mfw + mbw + mcw. The mass measurement is the ratio between the weight, W , and m gravitational constant, g (∼9.81 s2 ). The density of the solid phase of hydrated cement paste (hcp) depends on the structure of hcp and ,thus, is a function of the w/c ratio and other factors. Feldman

(1972) [83] measured the variation of density of cement paste, ρhcp, with respect to

276 Table 6.17: Volumetric fractions of cementitious composites

Vol. frac. Fully-Saturated Oven-dried

ms ms ρ ρ Solid, v s s s m m + m + m ms mcw s + fw bw cw + ρs ρw ρs ρw

mfw + mbw

ρw Gaseous, vg 0 m m s + cw ρs ρw

mfw + mbw

ρw Liquid, vl m m + m + m 0 s + fw bw cw ρs ρw

the w/c ratio , as shown in Table 6.18. Other researchers suggested an average value of 2.5 g/cm3 [238](page 70). The density of water at room temperature is 1 g/cm3. For normal weight concrete the average density of ﬁne and coarse aggregate is about 1.5 g/cm3. However, maximum density can be achieved when the volumetric ratio between ﬁne and coarse aggregates is taken to be vfa = 2 [223], which is around 1.75 vca 3 g/cm3.

Table 6.18: Density of cement paste specimens [83]

w/c ratio 0.4 0.5 0.6 0.8 1.0 Density (g/cm3) 2.19 — 2.28 2.30 2.29

6.7.2 Dielectric Modeling of Oven-Dried Hydrated Cement Paste

Cement paste is made by mixing cement and water at a given water/cement ratio. It consists only the hydration products. Reported dielectric measurements used in

277 the dielectric modeling on cement paste specimens in the frequency range between 8 GHz and 18 GHz are by Hasted and Shah (1964) [107], De Loor (1961) [154], and Wittmann and Schlude (1975) [271]. Other measurements made on cement pastes outside the frequency range of interest are not referred here.

Hydrated cement paste (hcp) is the fundamental structure of other cementitious composites such as cement mortar and concrete. When hcp is oven-dried, the free and bound water presenting in the capillary and gel pores of hcp is considered all liberated, resulting in a two-phase (solid and gaseous) composite with a maximum void ratio or

(vg)max and a minimum moisture content or (vl)min = 0. Such state also determines

the volumetric fraction of the solid phase of hcp, vs. Later, the moisture ingress from external environment will decrease the void ratio and increase the moisture content, but the volume of solid phase is considered unchanged during the drying-and-wetting

process. In other words, only vg and vl vary in hcp. vs is considered a constant once the age of the material reaches 28 days. In brief, the water/cement ratio is the

fundamental parameter determining vs and the structure of hcp. Our interest here is to investigate how the dielectric properties of hcp change with respect to not only the measurement frequency but also the water/cement ratio of hcp. To simplify the influence due to many factors (e.g.,moisture content, different polarizations), oven-dried hcp is first considered. For oven-dried hcp, generally the higher the w/c ratio, the lower the dielectric constant. This is because that the

0 0 increase of the w/c ratio produces more air voids (r = 1) in hcp (r = 5 ∼ 15), 0 resulting in lower bulk r for the porous medium. This observation is conﬁrmed by dielectric measurements on oven-dried (hydrous) hcp of diﬀerent w/c ratios reported by several researchers in the literature, as shown in Figure 6-30. As can be seen in

0 Figure 6-30, the product of the w/c ratio Ψ and the dielectric constant r at high frequency range (GHz range) is approximately constant when there is no moisture present in the specimen. It is postulated that the dielectric constant at extremely high frequency range can be represented by

γ ∞ = (6.142) (Ψ + C1) C2

278 where parameters γ, C1, and C2 are experimentally determined. Following this as-

sumption, the term (s − inf ) is represented to be

γ s − ∞ = (6.143) (Ψ + C3) C4

A modiﬁed Debye’s model is thus proposed for characterizing the dielectric properties

5 de Loor (1961), 3GHz Hasted & Shah (1964), 3GHz 4 Hasted & Shah (1964), 9GHz Hasted & Shah (1964), 24GHz 3 Wittman & Schlude (1975), 3GHz ) r ' ε x

Ψ ( 2

0 0.3 0.32 0.34 0.36 0.38 0.4 w/c ratio, Ψ

Figure 6-30: Relationship between the w/c ratio and product of the w/c ratio and dielectric constant of oven-dried cement paste specimens

of oven-dried hcp. 2γ

γ (Ψ + C3) C4 (ω, Ψ) = + αΨ (6.144) (Ψ + C1) C2 1 + (iωτ) In this model, the relation time τ is also a function of the w/c ratio Ψ, accounting for the change of material structure. The curve-ﬁtting results using experimental data on oven-dried hcp in the frequency range of 3 GHz∼24 GHz and the w/c ratio range 0.25∼0.4 are shown in Figure 6-31, with determined values of the parameters listed in Table 6.19.

279 Table 6.19: Parameters in the oven-dried hcp model

Coef. α τ γ C1 C2 C3 C4 Value 0.76 859 ns 1.625 0.55 0.4 0.3 0.5

6.7.3 Challenges in the dielectric modeling of concrete

As generally considered a poorly crystalized mineral, concrete has an amorphous structure that poses a challenge in describing its spatial composition with accuracy. Not only such amorphous structure is a traditional problem when the mechanical properties and fracture of concrete are concerned, but also it creates a problem when the dielectric properties of concrete are to be modeled. As previously mentioned in this chapter, the full spectrum dielectric models for the components in concrete including hydrated cement paste (hcp), ﬁne aggregate (sand), and coarse aggregate (gravels) also need to be available for the use and development of a dielectric mixing law on concrete. Factors that are inﬂuential to the measured dielectric properties of concrete are summarized as follows.

• Manufacturing factors – Factors determining the manufacturing of concrete need to be incorporated into the modeling process since they aﬀect the spatial composition of concrete, including the w/c ratio, mixing proportion, cement properties (cement composition, ﬁneness of the cement), the use of admixtures, and curing conditions (curing temperature, curing rate, curing method). These factors are the reason why predicting the formation of concrete is still a challenging task at present time. Some researchers applied numerical tools to model the chemical reactions involved in the hydration process with preliminary results [23], while a general model with sound predictability and ease for application is still lacking.

• Measurement factors – Even if the spatial composition of concrete is completely known, the measured dielectric properties of concrete can still vary at

280 different measurement frequencies and temperatures using different measurement techniques. The frequency-dependence and temperature-dependence are the characteristics of dielectric dispersion as the work by P. Debye (1929) [65] suggested around 80 years ago, accounting for the combined effect of various polarizations at different scales (see previous sections on dielectric dispersion in this chapter). Variation in the measured dielectric properties due to the selection of measurement technique originates from the difference in the interpretation of dielectric properties. In some techniques additional polarization may be involved (e.g., interface polarization in contact techniques). This is why the dielectric measurements at low frequencies can not be directly used for the modeling in high frequencies and same with temperature.

In view of the above arguments, a comprehensive experimental program is apparently needed to obtain a complete description of the dielectric dispersion of concrete as a basis for developing a full-spectrum physical model accounting for the heterogeneity in concrete. To the best knowledge of the author, such complete description is not available at present time. However, with increasing appreciation on the usefulness and importance of dielectric properties as a mutual interest in the communities of civil engineering, material sciences and engineering, chemistry, and physics, it is expected that more experimental measurements will be made and reported in the near future.

6.8 Summary

Research ﬁndings are summarized as follows.

• From the theoretical perspective, in the frequency range of 8 GHz to 18 GHz, the real part of complex permittivity (dielectric constant) is the combined result of the dipolar, atomic, and electronic polarizations, while the imaginary part (loss factor) is only aﬀected by the dipolar polarization. This suggests the feasibility of using single-pole dispersion models such as Debye’s model for the modeling work in this speciﬁc frequency range.

281 • An integrated methodology for determining the unique combination of complex permittivity of materials is developed. This methodology is based on transmission-only, coherent, wide-bandwidth free-space measurements as a time- domain method, integrating the time diﬀerence of arrival (TDOA) measurement of plate specimens and a root-searching optimization scheme using parametric system identiﬁcation (PSI) and the error sum of squares (SSE) criterion for the unique determination of complex permittivity [36].

• Fundamental diﬀerence in the dielectric modeling based on internal and external ﬁelds is described. The geometrical analysis technique used for constructing dielectric models for the materials considered in this research is explained. Sev- eral representations (analytical equation, the Cole-Cole diagram, and equivalent circuit) of dielectric dispersion models are provided as the basis for constructing dielectric models from measurements. Dielectric modeling is conducted for the use in the frequency range of 8 GHz to 18 GHz and room temperature (20◦C∼25◦C).

• A Cole-Cole type model for the dielectric properties of free water is proposed. As for bound water, the relaxation time function of bound water molecules is derived based on the statistical thickness and bonding potential of adsorbed water layer in hcp by Badmann et al. (1981) [212] and the reaction rate theory. Complex permittivity of bound water molecules at diﬀerent levels within the adsorbed water layer is calculated by this model.

• Dielectric models of epoxy resin and E-glass fabric are developed using the dielectric measurements reported in the literature [113]. Model parameters were determined by geometric analysis using the Cole-Cole diagram. GFRP layer, as a dielectric mixture of epoxy resin and E-glass fabric, is modeled by Wiener’s model, out of six two-phase dielectric mixing laws considered in this chapter.

• Dielectric properties of oven-dried hydrated cement paste (hcp) are determined by a modiﬁed Debye’s model incorporating the water/cement ratio as a major

282 material parameter. The challenges of dielectric modeling on concrete originate from the combined eﬀect of manufacturing and measurement factors in concrete. A physically-sound dielectric model for concrete must be validated by suﬃcient amount of experimental data to account for all possible combinations of those factors.

In this chapter, dielectric modeling of materials including water, GFRP, and oven- dried hydrated cement paste is performed. The knowledge of dielectric properties of materials is advantageous for better modeling materials in numerical simulation (e.g., Chapter 3). In the investigation of dielectric modeling of multi-phase cementitious composites such as concrete, complexities and challenges are addressed, as well as demonstrated by the variation of radar measurements on a plain concrete cylinder shown in Chapter 4 Laboratory Radar Measurements. It is also indicated in Chapter 5 Image Reconstruction that dielectric properties are important for improving the resolution of reconstructed images when the defects deep from the surface are to be located.

283 9 1 3GHz / H&S (1964) 9GHz 8 0.8 24GHz r r ' 7 " 0.6 hcp model: w/c=0.28 ε ε

6 0.4

5 0.2 0 5 10 15 20 25 30 0 5 10 15 20 25 30

8 1 3GHz / de Loor (1961) 3.75GHz 7 7.45GHz

r 9.37GHz r ' " 0.5 ε ε hcp model: w/c=0.31 6

5 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30

8 1 3 GHz / H&S (1964) 9 GHz 7 hcp model: w/c=0.325 r r ' " 0.5 ε ε 6

5 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30

8 1 3 GHz / H&S (1964) 9 GHz 7 24 GHz r r ' 6 " 0.5 hcp model: w/c=0.34 ε ε

4 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30

7 0.8 3 GHz / H&S (1964) 10 GHz / W&S (1975) 0.6 6 hcp model: w/c=0.4 r r ' " 0.4 ε ε 5 0.2

4 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Frequency (GHz) Frequency (GHz)

Figure 6-31: Curve-ﬁtting results of the oven-dried hcp in the frequency range of 3 GHz to 24 GHz

284 Chapter 7

Condition Assessment of GFRP-concrete Systems – FAR NDT

A good engineer should posses a scientist’s brain, and a good scientist should own an engineer’s hands.

In this chapter, a non-contact, far-field radar NDT method capable of detecting defects and damages that may attribute to either GFRP delamination, or GFRP debonding from concrete substrate, or sizeable air pockets trapped between GFRP sheets and concrete substrate during manufacturing in near-surface region of concrete structures wrapped and/or bonded with GFRP sheets/plates (Chapter 4 Labora- tory Radar Measurements), or the combination of above, using monostatic inverse synthetic aperture radar (ISAR) measurements and tomographic image reconstruction algorithms (Chapter 5 Image Reconstruction) is described. High-frequency continuous waves (time harmonic) in the frequency range of 8 GHz to 18 GHz are used as incident signals. One or more electronic signals, in terms of amplitude and phase, reflected from the GFRP-retrofitted concrete structure are collected by radar antenna in the far-field region, which represent the electromagnetic response of the

285 structure. The proposed NDT capability consists of three components; a physical inspection component allowing distant assessment of civil infrastructures, a numerical processing component transforming reflection response to spatial profile of the structure, and a pattern recognition component quantitatively evaluating the presence of damage. In the following sections, technical details of three components in the FAR NDT are addressed. Validation of the methodology is conducted using the far-field ISAR measurements of laboratory GFRP-confined concrete specimens.

7.1 Components of FAR NDT

The developed FAR NDT technique consists of a far-ﬁeld monostatic ISAR inspection with two measurement schemes, and a data processing algorithm. Instrumentation of the technique mainly consists of an airborne horn antenna, a signal generator, a signal modulator, and an analyzer. In principle, modulated radar signals or EM waves are designed and generated by the signal generator, modulated by the modulator, and transmitted by the radar antenna (a horn antenna was used in the experimental validation). Assessment procedure and the information obtained in each component are described in Figure 7-1. This technique is essentially characterized by the following features:

1. Normal and oblique incidence inspection schemes - The signal transmitter and receiver can operate in two inspection schemes for measuring the reflection response of the target structure; normal and oblique incidence schemes, as shown in Figure 7-2. Distant (near-field and far-field) measurements conducted in these schemes construct the ISAR frequency-angle data, also as shown in Figure 7-2.

2. Data processing algorithm - The measured ISAR frequency-angle data are processed by tomographic reconstruction methods. The spatial proﬁle of the

286 Component Procedure Information

Physical Far-field ISAR Frequency-angle Inspection Measurements Measurements

Numerical Image Range – Cross-range Processing Reconstruction Images (spatial profile)

Pattern Damage Feature-Extracted Recognition Assessment Images with Quantitative Indices

Figure 7-1: Overview of the FAR NDT

structure is characterized by the range-cross-range imagery, and is used for condition assessment.

3. Progressive image focusing - Image resolution is progressively improved by assembling all subaperture images in this technique. The spatial profile of the structure can be first reconstructed by using the data in one subaperture for preliminary evaluation. The resolution of imagery can be enhanced finishing all planned frequency bandwidth and the range of azimuth angle. This computational focusing scheme enables the technique to be used for different purposes, such as preliminary inspection and detailed inspection.

The frequency-angle data is processed by an image reconstruction algorithm which is based on tomographic reconstruction methods to generate range-cross-range imagery for visualization. Spatial proﬁle of the inspected structure is characterized by the range-cross-range imagery in which geometric features of the structure and unseen near-surface defects are revealed by scattering signals. Image resolution is in general proportional to (a) the frequency bandwidth and (b) the angular range of incident angle. Presence of anomalies (defects, damages, and rebars) are detected and represented by scattering signals in the reconstructed images (Figure 7-3). Locations of anomalies

287 GFRP- retrofitted concrete cylinder

fname: CYLAD1FV, HH Pol., max = -3.59 dBsm fname: CYLAD1BH, HH Pol., max = -6.41 dBsm 12 0 12 0

11.5 11.5 -2 11 11 -5

10.5 10.5 -4

10 10 -10

-6 9.5 9.5 Frequency (GHz) Frequency Frequency (GHz) Frequency x-y plane 9 y-z plane 9 -15 -8 8.5 8.5

8 -10 8 -20 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 φ (deg) φ (deg) (a) Normal incidence (b) Oblique incidence

Figure 7-2: Normal and oblique incidence inspection schemes of the FAR NDT technique

can also be determined in the images. In Figure 7-3 (a), the presence of an artiﬁcial defect is illustrated by a scattering signal centered at the location where the defect is placed. In Figure 7-3 (b), rebars are discovered by the appearance of two scattering signals.

288

(a) Defect detection – Oblique incidence inspection

(b) Rebar detection – Normal incidence inspection

Figure 7-3: Defect and rebar detection

7.2 Physical Inspection – Far-ﬁeld ISAR Measure- ments

For the far-field monostatic ISAR inspection as described in Chapter 4 Laboratory Radar Measurements, an airborne horn antenna is configured in normal incidence (specular dominant, Figure 7-4) or oblique incidence (specular recessive, Figure 7-5) measurement scheme. Coherent reflection responses are collected at different frequencies and azimuth angles in ISAR mode to form the frequency-angle data. The frequency range of 8 GHz to 18 GHz is chosen for its relevancy and abilities in assessing the unseen damages and defects hidden behind the GFRP sheets in the interface region between GFRP sheets and concrete substrate. The standoff or inspection distance, depending on the aperture size of the antenna, can be practically as far as 10 meters and more in the chosen measurement frequency range, enabling a real dis-

289 tant inspection capability. Figure 7-6 shows the far-ﬁeld distance calculated in the frequency range of 8 GHz to 18 GHz using two sizes of antenna aperture. The

Antenna GFRP-concrete ξ structure

−θint/2

θ = 0 θint/2 GFRP-concrete r structure

(θ = −θint/2 ∼ θint/2) (a) Top view (b) Front view

Figure 7-4: Specular dominant circumstance

Antenna ξ GFRP-concrete structure

−θint/2

φ = 0 θint/2

r GFRP-concrete structure (φ = −θint/2 ∼ θint/2) (a) Front view (b) Top view

Figure 7-5: Specular recessive circumstance inspection procedure of the technique is described in Figure 7-7.

290 20

15 (m) ff

D = 0.4 m 2 10 D = 0.2 m 1

5 Far-field distance, d distance, Far-field

0 8 10 12 14 16 18 Frequency (GHz)

Figure 7-6: Computed far-ﬁeld distances at various frequencies and two antenna apertures

291 Installation of the inspection apparatus

Defect detection Rebar detection or rebar detection? Normal incidence Defect detection ISAR measurements

Oblique incidence ISAR measurements Image reconstruction

Image reconstruction

YES Improve the YES image resolution?

Condition assessment

Figure 7-7: Inspection procedure of the FAR NDT technique

292 The inspection is conducted by deploying a radar antenna, sending radar signals by the antenna, and collecting the reflected radar signals (responses, measurements) by the same antenna. The radar antenna can be placed beyond the far-field distance from the target structure. Steady-state responses are measured at particular frequency and azimuth angle. Incident frequency is shifted from the starting frequency (8GHz) to the ending frequency (18GHz) at a frequency increment of 0.2GHz. Azimuth angle is shifted from the starting angle (φsta or θsta) to the ending angle (φend or θend) after the required frequency bandwidth is explored at each azimuth angle. Since the radar can operate in two inspection schemes, the possible trajectory of the radar can be described as a sub-domain of a sphere as shown in Figure 7-2. Two sets of far-field ISAR measurements collected from a GFRP-retrofitted concrete cylinder using linearly HH-polarized signals are also provided in Figure 7-2 as examples of the frequency-angle data. The developed technique allows convenient inspections from distance for highway and cross-river bridge columns. Figure 7-8 illustrates several scenarios for bridge inspection.

Radar unit Radar unit

(a) Beam inspection (b) Column inspection

Figure 7-8: Bridge inspection – Beam and column

293 7.3 Numerical Processing – Image Reconstruction

In-depth profiles or range-cross-range images of the target structure are generated by the image reconstruction method as described in Chapter 5 Image Reconstruc- tion, using the far-field ISAR measurements. The used tomographic reconstruction method is implemented by a time-domain backprojection algorithm for the efficiency and flexibility offered by the algorithm. Each pixel in the backprojection images is reconstructed by coherently integrating the far-field ISAR measurements over the ranges of incident frequency and azimuth angle. Processing efficiency and inspection flexibility are provided by using the progressive image focusing scheme. This scheme enables the improvement of image resolution by increasing the amount of measurements, rather than sacrificing inspection convenience by shortening the inspection range. With this flexibility in signal processing, different purposes of inspection can be chosen with respect to desired resolution. In this section, the physical meaning of scattering signals in the reconstructed backprojection images is first explained. The progressive image focusing scheme of the FAR NDT technique is described with two types of integration for improving damage detectability. Damage detectability using backprojection images is defined as the ability defect signals can be separated/distinguished from background signals in the images.

7.3.1 Physical Meaning of the Scattering Signals in the Im- ages

In the monostatic mode of radar operation the scattering signals in backprojection images are proportional to the magnitude of far-ﬁeld ISAR measurements. The backprojection processing of ISAR measurements distinguishes the contributions of various scatterers in the physical coordinate system and reconstructs these contributions as the scattering signals in the backprojection image. Therefore, the shape and magnitude of a scattering signal in backprojection images indicate the inﬂuence of a scatterer (defect) in the structure. In other words, the presence of defects or damages

294 triggers the scattering effect and is revealed by the backprojection processing. The backprojection images contains two types of scattering signals; defect scattering signal (or defect signal) and background scattering signal (or background signal). Defect scattering signals are due to the presence of defects, while background scattering signals are attributed to the direction reflection from the surface of the structure, and therefore, are related to the geometry of the structure. Background scattering signals can be identified providing the information about the geometry of the structure. This is usually the case in field inspection. After excluding background scattering signals, the remaining scattering signals are believed related to the presence (location, size, and orientation) of defects.

7.3.2 Progressive Image Focusing

As a result of the sub-aperture and sub-band processing nature of the backprojection algorithm, sub-aperture and/or sub-band images can be rendered before the far-field ISAR measurements are completed. Summation of sub-aperture and/or sub-band images can be conducted in part or in whole to progressively focus the reconstructed image. Coherent summation/integration is conducted by time-shifting and space- aligning the signal collected at each aperture position for every pixel in the image. Complete reconstruction is accomplished after the integration is done throughout the entire data plane. During this summation process, advantages are offered for the convenience of field inspection because:

1. Processing eﬃciency is provided by dividing the entire data plane (Figure 7- 2) into several sub-domains and by performing one-dimensional (1D) inverse Fourier transformation (IFT) for each sub-domain in the data plane.

2. Sub-aperture and sub-band images can be used for the purpose of preliminary inspection which can be rapidly completed. This is especially advantageous for the inspection of civil infrastructures. Inspection ﬂexibility is oﬀered by the option of rendering images at sub-aperture or sub-band levels.

295 3. The convergence of image pattern can be used to understand the features of the structure (background signals). With this knowledge, better identiﬁcation of defect signals can be achieved.

Two types of integration, frequency and angular, are described in the following sections, using the oblique incidence (specular recessive) ISAR frequency-angle data of the specimen AD1 as an example.

Frequency Integration

Integration of the far-ﬁeld ISAR frequency-angle data over certain frequency bandwidth can be conducted at constant center frequency (Figure 7-9 (a)) or at shifting center frequency (Figure 7-9 (b)). Improvement of image resolutions using both con-

fmax fmax

fc fc

fmin fmin

Bmin Bmax Bmin Bmax (a) Constant center frequency (b) Shifting center frequency

Figure 7-9: Two types of frequency bandwidth integration stant center frequency and shifting center frequency schemes are provided in Figure

7-10 for the images produced using HH polarized radar signals, in which Bmin =

0.2 GHz and Bmax = 4 GHz. Image resolutions, range and cross-range, are better improved when shifting center frequency scheme is used. Formulae are provided in Table 7.1 for estimating image resolutions (range and cross-range) through the backprojection processing in this development. In Table 7.1, c is the speed of light in free

8 space (c = 3 × 10 m/s), B is the bandwidth (GHz), and λc is the wavelength of

296 Range Range Cross-range Cross-range 1 1

0.8 0.8

0.6 0.6 Resolution (m) 0.4 Resolution (m) 0.4

0.2 0.2

0 0 0 1 2 3 4 0 1 2 3 4 Bandwidth (GHz) Bandwidth (GHz) (a) Constant center frequency (b) Shifting center frequency

Figure 7-10: Improvement of image resolutions Progressive image focusing – Fre- quency integration, HH polarization

c center frequency (λc = ) (m). Range resolution is accurately estimated in general, fc while the empirical constants in Table 7.1 are associated with various degrees of error as shown in Figure 7-11. The approximation error of cross-range resolution using the empirical formulae is less than 3 cm when the bandwidth is greater than 1 GHz.

Table 7.1: Image resolution formulae

Resolution Constant Shifting center center frequency frequency c c Range 2B 2B λ λ Cross-range 5.72 c 5 c B B

Figure 7-12 shows the performance of bandwidth integration using the frequency- data of the specimen AD1. In this case, shifting center frequency is adopted. The center frequency ranges from 8.2 GHz (bandwidth = 0.4 GHz) to 9.8 GHz (bandwidth = 3.6 GHz). It is found that the contrast of defect signals with background signals

297 0.25 Shifting f c Constant f c 0.2

0.15

Error (m) 0.1

0.05

0 0 0.5 1 1.5 2 2.5 3 3.5 4 Bandwidth (GHz)

Figure 7-11: Prediction error of cross-range resolution formulae improves signiﬁcantly when the used bandwidth increases. The presence of defect is gradually revealed in the backprojection images with B > 1.6 GHz (Figure 7-12 (d), B = 1.6 GHz). The example in Figure 7-12 demonstrates the eﬀectiveness of frequency integration on improving image resolutions for damage detection.

Angular Integration

Unlike the bandwidth integration, integration of the far-field ISAR frequency-angle data over certain angular range may not guarantee the improvement of damage detectability due to the angular-dependent detectability of defects. For instance, in the specular dominant circumstance, background signals cover defect signals in the reflection response. On the other hand, the weakening/recessing of background signals reveals defect signals in the specular recessive circumstance. Averaging images in these circumstances will not improve the damage detectability in the specular dominant circumstance to a significant extent, but will reduce the one in the specular recessive circumstance dramatically. Therefore, angular integration should be conducted by incorporating images only at incident angles in the specular recessive

298 circumstance. These angles are considered eﬀective on detecting near-surface defects in the GFRP-wrapped concrete columns.

Another issue is the variation of cross-range resolution, ρxr, with respect to bandwidth and angular support. Both constant and shifting center frequency schemes are investigated. Figures 7-13(a) and (b) show the variation of ρxr(B, θ) in the angular range of θ ∈ [−30, 30] (◦ or deg.) and in the bandwidth range of B ∈ [0.2, 4] (GHz). In these figures, ρxr is improved with the increase of bandwidth B as expected. Figures 7-13(c) and (d) show the increase of required angular support for an increasing B. In order to achieve the maximum ρxr at a given combination of fc and B, a minimum amount of angular measurements is needed, which is the size of angular support. The fundamental principle behind the effectiveness of detecting near-surface defects in the GFRP-concrete systems is the scattering effect at oblique incidence. Such effectiveness is evaluated by the magnitude of scattered signals, which is also a function of incident angle (or azimuth angle). Figures 7-14 and 7-15 show the comparison of reconstructed images of the intact(without defect) and damaged (with defect) surfaces of the specimen AD1 using full bandwidth signals (8GHz to 12GHz).

7.4 Image Resolutions and Damage Detectability

In the FAR NDT technique, it is the reconstructed backprojection images that are used as the basis for damage detection. Therefore, the image resolutions (range and cross-range) are crucial for the damage detectability of the technique and for revealing the presence of near-surface defects by reconstructing scattering signals. However, it is noteworthy to point out the following:

1. Near-surface defects can be detected only when the scattering signals can be physically reconstructed in a backprojection image. This is associated with the penetration depth of far-ﬁeld radar signals, which is related to the incident frequency and the dielectric properties of the medium; higher frequencies and lossy media result in poor penetration and vice versa.

299 2. While the image resolutions can be estimated by the formulas provided in Chapter 5 Image Reconstruction, it is believed that the angular-dependent nature of the scattering signals due to near-surface defects plays a critical role in the damage detectability of the technique. Image resolutions may be related to the damage detectability only when (1) eﬀective incident angles are used, and (2) suﬃcient frequency bandwidth is provided.

3. When eﬀective incident angles are used with suﬃcient frequency bandwidth in the reconstructed images, image resolutions represent the capability of the images for damage detection. This capability is (1) to distinguish defect scattering signals (due to near-surface defects) from background scattering signals (due to the edges or geometrical features of the structure) and (2) to distinguish two adjacent defect scattering signals.

With the above discussions, it is clear that the damage detectability of the FAR NDT technique should be interpreted in conjunction with the following:

• The ability of far-ﬁeld radar signals to penetrate/reach near-surface defects (penetration depth)

• The condition of eﬀective incident angles

• The condition of suﬃcient frequency bandwidth

For the practical use and ﬁeld applications of the FAR NDT technique, it is realized that quantitative indices are usually more useful than qualitative guidance. Therefore, quantitative interpretations on the backprojection images are believed to be of great potential for practical use and, therefore, to be addressed in the following sections.

7.5 Pattern Recognition – Damage Detection

Damage detection for the structural assessment of GFRP-wrapped concrete systems is conducted by interpreting scattering signals in the reconstructed images. To demonstrate this, reconstructed images of the intact side (without defect) and the damaged

300 side (with defect) of the specimen AD1 (Figure 7-16 (a) and (b)) are rendered. Im- ages reconstructed from the ISAR measurements (Figure 7-16 (c)) of the specimen AD2 are generated for demonstrating the effectiveness of damage detection. In this research, the maximum amplitude (local index) and the pattern of backprojection images (global index) are used for quantifying the presence of defects. While the maximum amplitude of a backprojection image locally indicates the significance (size, angular sensitivity) of scattering signals (defect or background), the morphological pattern of a backprojection image captures the global feature of the image. Knowing that the backprojection image of a damaged structure contains defect signals in addition to background signals, the image pattern should be distinguishable from the one of an intact structure. This difference is globally characterized by mathematical morphology using a quantitative index (Euler’s number) as a damage indicator. In what follows, these two approaches are to be explained.

7.5.1 Local Index - Maximum Amplitude

Since the presence of defects leads to the appearance of scattering signals in the backprojection images, no defect signals should be expected in the image of an intact structure after excluding background signals. However, in some circumstances such as the specular dominant case as shown in Figure 7-4, background signals become dominant and cover the defect signal. This is the case when low signal-to-noise ratio (SNR) is encountered, and the maximum amplitude is associated with background signals. On the other hand, in the specular recessive case as shown in Figure 7-5, high SNR can be expected due to the alleviation of background signals (noise), and the maximum amplitude is likely to be associated with defect signals. The maximum amplitude in the backprojection images is deﬁned by

i i Imax = max I (x, y) |(x, y) ∈ Ωs (7.1) or d d Imax = max I (x, y) |(x, y) ∈ Ωs (7.2)

301 i d where Imax and Imax are the maximum amplitude in the backprojection images of i d intact structure, I (x, y), and of damaged structure, I (x, y), respectively. Ωs is the physical domain of the specimen in the backprojection images deﬁned by (x, y). The diﬀerence between Ii(x, y) and Id(x, y) is the defect signal.

The images of the specimen AD1 are used as an example. In the use of maximum amplitude as a damage indicator, identifying background signals is important (in order to exclude them), and it is also important to know how background signals affects the total response. Maximum amplitudes are extracted from the backprojection images in two scenarios; (i) including background signals (blind test), and (ii) excluding background signals (a priori test). Evaluating a backprojection image without excluding background signals is a blind test, from which the overall maximum amplitude is obtained. Excluding background signals requires a priori knowledge about the structure. Figures 7-17 (a) and (b) show the maximum amplitudes of backprojection images produced from the intact and damaged sides of the specimen AD1 in both scenarios using HH polarized signals. In this example, it is observed that the defect signal is detectable in two angular regions, θ ∈ [−30◦, −7◦] and θ ∈ [7◦, 30◦], in blind test scenario in Figure 7-17(a). Two similar regions are also found in Figure 7-17(b) where background signals are excluded, but with better detectability due to the significant separation between the curves. In the angular region θ ∈ (−7◦, 7◦), specular effects are dominant, and the difference is not distinguishable.

Diﬀerential amplitude between maximum amplitude curves is calculated for both blind test and a priori test scenarios, as shown in Figure 7-18. The diﬀerential amplitude, ∆A, is determined by

A − A ∆A = d i (7.3) Ai where Ad represents the damage curve and Ai the intact curve in Figure 7-17. In Figure 7-18, ∆A is shown in percentage, whose pattern demonstrates the sensitivity and effectiveness of incident angle on damage assessment. The greater the difference is, the more effective the chosen incident angle is. Figure 7-18 also indicates that,

302 in the example presented in this section, the optimal incident angle is approximately 15◦ deviating from perpendicular (normal) incidence. Findings obtained from the example presented in this section include:

1. Eﬀectiveness of incident angle on damage assessment is aﬀected by the measurement scheme. In the specular recessive scheme as shown in Figure 7-4, background signals dominate the total response, making defect signals unde- tectable. This is the case of θ ∈ (−7◦, 7◦) in Figures 7-17(a) and (b).

2. With a priori knowledge about background signals, detectability can be greatly improved.

3. The shape of the maximum amplitude curve of the intact side indicates the pattern of background signals.

4. Detectability of defects is sensitive to the selection of incident angles. A defect or damage is more detectable in some ranges of incident angle than in other ranges.

7.5.2 Global Index - Mathematical Morphology

Pattern recognition of the backprojection images is conducted by extracting the features (edges) from the images and by evaluating the features quantitatively using mathematical morphology. Global features (e.g., shape of scattering signals) are characterized and quantitatively evaluated using techniques in mathematical morphology [Shirai 1987; Marchand-Maillet and Sharaiha 2000; Nixon and Aguado 2002]. In this research, the backprojection images are evaluated by mathematical morphology in the following two steps:

1. Feature extraction - The backprojection images are rendered with continuous response levels, in which both background and defect signals are involved. In order to extract the characteristic shape of a backprojection image, the image

is ﬁrst transformed into a binary image based on a threshold value nthv. Two morphological operations, erosion and dilation, are subsequently applied to the

303 binary image to obtain a feature-extracted version of the original backprojection image. These morphological operations are deﬁned by

K (I) = {r¯|Kr ⊂ I(x, y)} (7.4)

δV (I) = {r¯|Vr ∩ I(x, y) 6= ∅} (7.5)

where I(x, y) is the backprojection image, K is the erosion operation function-

ing with the erosion structure K, Kr is the eroded set operating at positionr ¯,

δV is the dilation operator functioning with the dilation structure V , Vr is the dilated set operating atr ¯, and ∅ is the empty set. An eight-node element is adopted for both erosion and dilation structures, as shown in Figure 7-19. The feature extraction operation on I(x, y) is performed on the binary version of

I(x, y) in this research, denoted by IBW (x, y|nthv). The operation is deﬁned by

ˆ I (x, y|nthv) = δV [K [IBW (x, y|nthv)]] (7.6)

ˆ where I (x, y|nthv) is the feature-extracted binary image characterized by a

threshold value nthv related to the level of the extracted edge in the image.

2. Feature quantiﬁcation - A quantitative index used in this research to globally ˆ characterize I (x, y|nthv) is Euler’s number, nE. The variation of nE with respect ˆ to the incident angle is investigated. For each I (x, y|nthv) obtained at a given

incident angle θ, nE is deﬁned by

nE (θ|nthv) = nobj (θ|nthv) − nhol (θ|nthv) (7.7)

ˆ where nobj (θ|nthv) is the number of objects in I (x, y|nthv), and nhol (θ|nthv) is ˆ the number of holes within the objects in I (x, y|nthv). With a ﬁxed value of

nthv, nE (θ) can be obtained.

The presence of damages introduces additional defect signals into backprojection images globally, and changes the maximum amplitude locally. Logically, the presence

304 of scatterers (defect or background) leads to an increasing nobj. The value of nEθ is

subsequently altered. Given same nthv and same inspection domain Ωs, the ﬂuctuation of defect scattering signals will create more holes than objects, thus, resulting in

small nE(θ). The purpose of using mathematical morphology is to quantify such change. Additionally, in view of the angular sensitivity of defect signals, it is believed that damage assessment based on single measurement (or image) is unlikely reliable. Multiple images (more information) are needed to conﬁrm the speculation on one suspicious image. For this reason, an averaging (low-pass) ﬁlter is applied to the

nE(θ) curve, which is deﬁned by

θint/2 X nE (θ) nf (θ) = (7.8) E L θ=−θint/2

f where nE (θ) is the filtered nE (θ) curve, and L is the length of the filter. The purpose of this filter is to remove local fluctuations from the original nE (θ) curve in order to (i) avoid false alarm at local level and (ii) obtain a globally consistent result. Additionally, the length of the filter suggests the required amount of angular measurements. The length of the filter also relies on the resolution of the image. For high resolution images, small L is expected. The backprojection images used for morphological processing are shown in Figures 7-20 and 7-21. In these figures, the physical location of the specimen is indicated by a solid-line rectangle which is the

i inspection domain Ωs. I (x, y)

In applying mathematical morphology for damage detection, the value of nthv must be determined. The value of nthv is decided when the variation of Euler’s number nE is at its critical stage. When small values of nthv are chosen, all or most of the signals are preserved, leading to a feature-extracted image dominated by low amplitude signals. ˆ When large values of nthv are chosen, I (x, y|nthv) will be dominated by high amplitude

signals. Consequently, the computed Euler’s number nE in these two extreme cases does not represent/reﬂect the main feature of the image. To avoid these misleading

circumstances, the pattern of nE is ﬁrst investigated as shown in Figures 7-22 and 7- ˆ 23 in which nthv = 1 corresponds to the maximum amplitude of I (x, y|nthv). Figure

305 7-22 shows the variation of nE with the indication of critical nthv = 0.81 for both HH and VV polarized signals in the intact-side images. Same pattern is also found in the

damaged-side images as shown in Figure 7-23 with a critical value of nthv = 0.73. Figure 7-22(a) is produced from Figure 7-20(a) and Figure 7-22(b) from Figure 7-21(a). Figure 7-23(a) is produced from Figure 7-20(b) and Figure 7-23(b) from Fig- ure 7-21(b). Figure 7-24 shows the reconstructed backprojection images I (x, y) and ˆ their feature-extracted version I (x, y|nthv) of the specimen AD1 using HH polarized ˆ signals at full bandwidth. With the selection of critical nthv,I (x, y|nthv) captures the main feature of the original I (x, y). The Euler’s number of the intact-side image ˆ ˆ I (x, y|nthv = 0.81) is nthv = −1 . For the damaged-side image I (x, y|nthv = 0.73),

nthv = −2.The feature-extracted images using VV polarized signals also provide similar result and are not repeatedly shown here. Following the same procedure described ˆ above, I (x, y|nthv) can be produced for other incident angles, resulting in the intact-

side and damaged-side curves nE(θ) of the specimen AD1 as shown in Figure 7-25.

The nE(θ) curves in Figure 7-25 demonstrate the sensitivity and eﬀectiveness of incident angle with respect to the damage indication using Euler’s number. Since the scattering due to defects is angle-dependent, evaluating the structure using images at several incident angles is needed. This leads to the application of an averaging ﬁlter f to obtain nE(θ) curves. Findings obtained in the use of mathematical morphology for damage detection are summarized in the following.

f 1. Figure 7-26 shows the nE(θ) curves using a ﬁlter length of L = 3. In Figure 7-26, the ﬁltering produces a clear separation between the intact-side and damaged-

◦ ◦ side nE(θ) curves, except in the θ ∈ (−10 , 10 ) region. The nE(θ) values of the intact-side images are in general greater than the ones of the damaged-side images since the presence of defect signals creates more holes in the images,

resulting in smaller values of nE(θ).

2. The ﬁlter length L is related to the required amount of angular measurements for achieving a globally consistent assessment. Knowing that specular eﬀects are dominant in the angular range of θ ∈ (−7◦, 7◦) as found in the use of

306 maximum amplitude for damage detection, separation between the nE(θ) curves of intact and damaged images is required outside the range of θ ∈ (−7◦, 7◦). L is subsequently determined in order to achieve this goal. In the example presented in this section, at least three angular measurements at an interval of 1◦ (3◦ of angular range) are needed for each comparison between the images of intact and damaged structures.

3. In the case of the absence of background signals in backprojection images, vari-

ation of nobj and nhol can be used instead.

7.6 Summary

In this chapter, a non-contact, far-field radar NDT method capable of detecting (1) GFRP (glass fiber reinforced polymer) delamination, (2) GFRP debonding from concrete substrate, and (3) sizeable air pockets trapped between GFRP sheets and concrete substrate during manufacturing in near-surface region of concrete and masonry structures wrapped and/or bonded with GFRP sheets/plates, using monostatic inverse synthetic aperture radar (ISAR) measurements and tomographic image reconstruction algorithms is described. The proposed NDT capability consists of three components; a physical inspection component allowing distant assessment of civil infrastructures, a numerical processing component transforming reflection response to spatial profile of the structure, and a pattern recognition component quantitatively evaluating the presence of damage. In physical inspection, far-field ISAR measurements of the structure under investigation are collected at different frequencies and inspection (azimuth) angles. Steady- state reflection responses (amplitude and phase) of the structure are collected at each frequency and angle, forming the frequency-angle measurements. Frequency-angle measurements are processed by an image reconstruction algorithm based on tomographic reconstruction methods to render the transformed in-depth profiles (range-

307 cross-range images) of the structure. Finally, a pattern recognition technique based on mathematical morphology analyzes the range-cross-range images with a quantitative index for condition assessment. Information regarding the presence and location of anomaly is obtained from the result of condition assessment and can be used for routine maintenance, as well as for special purpose safety assessment, of GFRP-retrofitted concrete structures including bridges, buildings, and walls, as well as masonry walls strengthened by GFRP sheets. This information is also beneficial to the management of civil infrastructures in the way resources for repair and rehabilitation can be effectively allocated.

308 1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

-0.2 -0.2 -0.2 Cross-range (m) Cross-range (m) Cross-range -0.4 (m) Cross-range -0.4 -0.4

-0.6 -0.6 -0.6

-0.8 -0.8 -0.8

-1 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Range (m) Range (m) Range (m)

(a) fc = 8.2GHz, B = 0.4 GHz (b) fc = 8.4GHz, B = 0.8GHz (c) fc = 8.6GHz, B = 1.2GHz

1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

-0.2 -0.2 -0.2 Cross-range (m) Cross-range (m) Cross-range -0.4 (m) Cross-range -0.4 -0.4

-0.6 -0.6 -0.6

-0.8 -0.8 -0.8

-1 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Range (m) Range (m) Range (m)

(d) fc = 8.8GHz, B = 1.6GHz (e) fc = 9.0GHz, B = 2.0GHz (f) fc = 9.2GHz, B = 2.4GHz

1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0

-0.2 -0.2 -0.2 Cross-range (m) Cross-range (m) -0.4 -0.4 (m) Cross-range -0.4

-0.6 -0.6 -0.6

-0.8 -0.8 -0.8

-1 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Range (m) Range (m) Range (m) (g) f = 9.4GHz, B = 2.8GHz (h) f = 9.6GHz, B = 3.2GHz (i) f = 9.8GHz, B = 3.6GHz c c c

Figure 7-12: Progressive image focusing – Frequency integration using shifting center frequency, HH polarization, θ = 15◦

309

(a) Constant fc – ρxr ()B,θ (b) Shifting fc – ρxr ()B,θ

0.35 0.35 B = 1 GHz B = 1 GHz 0.3 B = 2 GHz 0.3 B = 2 GHz B = 4 GHz B = 4 GHz

0.25 0.25

0.2 0.2

0.15 0.15

0.1 0.1 Cross-range resolution (m) resolution Cross-range Cross-range resolution (m) resolution Cross-range 0.05 0.05

0 0 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 Incident angle, θ (deg.) Incident angle, θ (deg.)

(c) Constant fc – ρxr ()B ∈[1, 2, 4] ,θ (d) Shifting fc – ρxr ()B ∈[1, 2, 4] ,θ Figure 7-13: Improvement of image resolutions progressive image focusing – Angular integration, HH polarization

310 0.5 0.5 0.5

0.4 0.4 0.4

0.3 0.3 0.3

0.2 0.2 0.2

0.1 0.1 0.1

0 0 0

-0.1 -0.1 -0.1 Cross-range (m) Cross-range (m) Cross-range -0.2 (m)Cross-range -0.2 -0.2

-0.3 -0.3 -0.3

-0.4 -0.4 -0.4

-0.5 -0.5 -0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 Range (m) Range (m) Range (m) (a-1) 30º (a-2) 20º (a-3) 10º (a) Intact surface

0.5 0.5 0.5

0.4 0.4 0.4

0.3 0.3 0.3

0.2 0.2 0.2

0.1 0.1 0.1

0 0 0

-0.1 -0.1 -0.1 Cross-range (m) Cross-range (m) Cross-range -0.2 -0.2 (m) Cross-range -0.2

-0.3 -0.3 -0.3

-0.4 -0.4 -0.4

-0.5 -0.5 -0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 Range (m) Range (m) Range (m) (b-1) 30º (b-2) 20º (b-3) 10º (b) Damaged surface

Figure 7-14: Comparison of images of the intact and damaged surfaces of the specimen AD1 at diﬀerent incident angles (30◦ ∼ 10◦)

311 0.5 0.5 0.5

0.4 0.4 0.4

0.3 0.3 0.3

0.2 0.2 0.2

0.1 0.1 0.1

0 0 0

-0.1 -0.1 -0.1

Cross-range (m) Cross-range -0.2 (m) Cross-range -0.2 (m) Cross-range -0.2

-0.3 -0.3 -0.3

-0.4 -0.4 -0.4

-0.5 -0.5 -0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 Range (m) Range (m) Range (m) (a-4) -10º (a-5) -20º (a-6) -30º (a) Intact surface

0.5 0.5 0.5

0.4 0.4 0.4

0.3 0.3 0.3

0.2 0.2 0.2

0.1 0.1 0.1

0 0 0

-0.1 -0.1 -0.1

Cross-range (m) Cross-range -0.2 (m) Cross-range -0.2 (m) Cross-range -0.2

-0.3 -0.3 -0.3

-0.4 -0.4 -0.4

-0.5 -0.5 -0.5 -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 Range (m) Range (m) Range (m) (b-4) -10º (b-5) -20º (b-6) -30º (b) Damaged surface

Figure 7-15: Comparison of images of the intact and damaged surfaces of the specimen AD1 at diﬀerent incident angles (−10◦ ∼ −30◦)

312 30º θ = –30º 30º

0º 0º 0º

30º θ = –30º θ = –30º

(a) Specimen AD1 (b) Specimen AD1 (c) Specimen AD2 – Intact side – Damaged side – Damaged side

Figure 7-16: Description of the used far-ﬁeld ISAR measurements and specimens for damage detection

-15 -15

-20 -20

-25 -25

-30 -30

-35 -35

-40 -40

-45 -45

Max. amplitude (dBsm) -50 Max. amplitude (dBsm) -50

-55 Damaged -55 Damaged Intact Intact -60 -60 -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 Incident angle, θ (deg.) Incident angle, θ (deg.) (a) Blind test (b) A priori test

Figure 7-17: Maximum amplitudes of the backprojection images of the specimen AD1 – Full bandwidth (8GHz∼12GHz), HH polarization

313 12 Blind test A priori test 10

4 Differential amplitude (%) amplitude Differential 2

0 -30 -20 -10 0 10 20 30 Incident angle, θ (deg.)

Figure 7-18: Diﬀerential maximum amplitudes of the backprojection images of the specimen AD1 – Full bandwidth (8GHz∼12GHz), HH polarization

rx( jj, y)

Figure 7-19: An eight-node element for morphological operations

314 (dBsm) (dBsm) -25 -25 0.3 0.3 -30 -30

0.2 -35 0.2 -35

-40 -40 0.1 0.1 -45 -45

0 -50 0 -50

-55 -55 Cross-range (m) Cross-range (m) -0.1 -0.1 -60 -60

-65 -0.2 -65 -0.2

-70 -70 -0.3 -0.3 -75 -75 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Range (m) Range (m) (a) Intact side (b) Damaged side

Figure 7-20: Backprojection images of the specimen AD1 – HH polarization, θ = −15◦

(dBsm) (dBsm) -25 -25 0.3 0.3 -30 -30

0.2 -35 0.2 -35

-40 -40 0.1 0.1 -45 -45

0 -50 0 -50

-55 -55 Cross-range (m) Cross-range -0.1 Cross-range (m) -0.1 -60 -60

-0.2 -65 -0.2 -65

-70 -70 -0.3 -0.3 -75 -75 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Range (m) Range (m)

(a) Intact side (b) Damaged side

Figure 7-21: Backprojection images of the specimen AD1 – VV polarization, θ = −15◦

315 20 15 E E n n 15 10

5 Euler's number, Euler's Euler's number, Euler's 5

0 0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Threshold value, n Threshold value, n thv thv (a) HH polarization (b) VV polarization

Figure 7-22: Variation of nE with respect to nthv of the intact-side of the specimen AD1

30 25

25 E E 20 n n 20 15 15 10 10 Euler's number, number, Euler's Euler's number,Euler's 5 5

0 0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Threshold value, n Threshold value, n thv thv (a) HH polarization (b) VV polarization

Figure 7-23: Variation of nE with respect to nthv of the damaged-side of the specimen AD1

316

(a) Intact-side images – nthv = 0.81 (b) Damaged-side images – nthv = 0.73

Figure 7-24: Backprojection images and their feature-extracted version of the specimen AD1 (HH polarization, full bandwidth, θ = −15◦)

2 AD1 intact 1.5 AD1 damaged 1

0.5

r 0

-0.5

-1 Euler' numbe

-1.5

-2

-2.5

-3 -30 -20 -10 0 10 20 30 Inspection angle (deg.)

Figure 7-25: Original nE(θ) curves of the intact and damaged surfaces of the specimen AD1 (HH polarization, full bandwidth)

317 1.5 AD1 intact AD1 1 damaged

0.5

-0.5

Averaged Euler' number -1

-1.5

-2 -30 -20 -10 0 10 20 30 Inspection angle, θ (deg.)

Figure 7-26: Filtered nE(θ) curves of the intact and damaged surfaces of the specimen AD1 (HH polarization, full bandwidth, L = 3)

318 Chapter 8

Conclusions

In this thesis, a novel NDT technique termed FAR NDT using far-field monostatic radar measurements for the distant inspection of GFRP-retrofitted concrete cylinders is developed. In particular, the investigation of dielectric modeling of construction materials, application of the FDTD method to the far-field response of dielectric cylinders with local cavity, and processing of far-field radar measurements into spatial images using the fast backprojection algorithm are presented. In what follows, obtained research findings are first concluded. The contributions of this research are summarized, and possible future work is suggested. The research findings from this work include:

• Current NDT techniques and their applicability on GFRP-concrete systems are reviewed in this research. A distant NDT technique that is capable of providing in-depth information of GFRP-concrete systems in ﬁeld conditions is lacking. There is a need to develop such technique for the condition assessment (safety inspection) and proper maintenance of GFRP-concrete systems and potentially for the damage detection of concrete and reinforced concrete structures.

• Numerical simulation –

1. FDTD simulation can demonstrate the multiple reﬂections within and the scattering response of dielectric media, providing an accurate description to material properties and geometry.

319 2. The simulated scattering response of a concrete cylinder (dielectric) with four rebars indicates strong reﬂections not only in the line-of-sight direction of the radar but also in other directions. This is the reason why far-ﬁeld monostatic ISAR normal incidence measurements can be used for detecting rebars.

3. FDTD simulation can be used as a tool for understanding the pattern of scattering signals due to the presence of near-surface defects in dielectric media. The knowledge learned from FDTD simulation is beneﬁcial for planning an eﬀective experimental program and for identifying key system design parameters including measurement scheme, measurement frequency, and the range of incident angles.

• Measurement scheme –

1. From laboratory radar measurements, far-ﬁeld monostatic ISAR oblique incidence scheme can be conclusively used for detecting near-surface defects such as GFRP debonding in GFRP-retroﬁtted concrete cylinders, while the normal incidence scheme cannot provide a conclusive indication.

2. The magnitude of far-ﬁeld monostatic ISAR normal incidence measurements is sensitive to the geometry, surface roughness, and dielectric properties of the target structure. Variation of these properties in normal incidence measurements can obstruct the detection of near-surface defects.

• Measurement frequency –

1. Continuous wave radar signals in the frequency range of 8 GHz to 18 GHz are found to be eﬀective for detecting near-surface defects in GFRP- retroﬁtted concrete cylinders.

2. The use of higher frequency signals covers more angular regions than the use of lower frequency signals, in which defect signals appear.

• Range of incident angles –

320 1. In the reconstructed images of the specimen AD1, two angular regions, [−10◦ ∼ −25◦] and [10◦ ∼ 20◦], are found to be eﬀective for detecting the presence of a cubic-like defect. Detectability can be further improved if the background signal can be reduced/removed. Incident angles outside the range [−30◦ ∼ 30◦] are not eﬀective for damage detection.

2. The result by mathematical morphology reveals the observation in laboratory radar measurements on the multiple angular regions for damage detection. The angular region [−7◦ ∼ 8◦] is found not eﬀective for detecting near-surface defects.

• Image reconstruction –

1. The backprojection processing of monostatic ISAR oblique incidence measurements is capable of visualizing and locating the presence of near- surface defects; with a bandwidth [8GHz∼12GHz], a 1.5”-by-1.5”-by-1” (cubic-like) near-surface defect can be detected, and a 3”-by-3”-by-0.2” (delamination-like) near-surface defect can be detected with detailed features (edges).

2. The use of a shifting center frequency provides better convergence of cross- range resolution than the use of a constant center frequency.

3. Maximum amplitude of the reconstructed images can be used as an index for damage detection.

4. In the image reconstruction algorithm, dielectric properties of the target structure (GFRP-concrete cylinder) are not necessary; radar signals are assumed propagating in free space. This is the reason why the backprojection processing can provide satisfactory results for detecting near-surface defects; changes in the phase velocity of radar signals are insigniﬁcant when only a small portion of propagation distance in dielectric media (non-free space) is involved.

• Dielectric properties and modeling –

321 1. Dielectric properties are the key information for the correct modeling of materials in numerical simulation, as well as for improving the accuracy of backprojection processing for detecting deep defects.

2. In the proposed methodology for determining dielectric properties of materials, the estimation of TDOA is crucial; thickness of the specimen must be greater than 0.236 in. (6 mm) for a reliable estimate of TDOA.

3. In the frequency range of 8 GHz to 18 GHz, signal-pole models can be used for water, epoxy resin, and E-glass fabric considered as homogeneous media. Dipolar polarization is the dominant dispersion mechanism in this frequency range.

4. Wiener’s dielectric mixing law is applicable for GFRP (epoxy resin and E-glass fabric).

5. Dielectric properties of cementitious (porous) composites such as concrete depend on not only the volumetric fractions of each component but also the distribution of each component. At the present time, lack of a complete dielectric database for cementitious composites is a major obstacle for developing a physically-sound dielectric model for multi-phase cementitious composites.

Contributions of this research are summarized in the following:

1. A distant NDT capability (FAR NDT) utilizing far-ﬁeld monostatic ISAR measurements for the in-depth inspection of GFRP-concrete structures is developed for the safety assessment of the structures [280]. This capability is applicable for ﬁeld applications such as highway bridge piers, cross-river bridge piers and cross-valley bridge piers strengthened with GFRP composites.

2. The feasibility of using the backprojection algorithm for reconstructing the spatial proﬁle of multi-layered dielectric systems is demonstrated. The representation of near-surface defects in the backprojection images is provided.

322 3. The use of FDTD simulation as a predictive tool to estimate/explain the reﬂec- tion response of dielectric media with local features (near-surface defects and rebars) is demonstrated.

4. An integrated methodology for the determination of unique combination of complex permittivity using transmission-only, coherent, wide-bandwidth free- space measurements is developed [36].

5. Dielectric models of materials including water, epoxy resin, E-glass fabric, GFRP, and oven-dried hydrated cement paste are developed for their use in the frequency range of 8 GHz to 18 GHz.

Upon the research ﬁndings based on preliminary laboratory testing and imaging results and addressed in this research, suggested future work is summarized in the following.

1. Field performance of the FAR NDT technique – Upon the excellent performance of the FAR NDT technique in a laboratory environment, it would be of interest to know how the technique performs on real structures in field conditions. Not only the background electromagnetic noises (stationary and/or non-stationary) could present, but also the operational constraints in the field could lead to the need for further optimization of the technique. These issues cannot be fully understood without conducting field measurements using the technique. Consequently, the development of a portable phototype system of the FAR NDT technique is needed, as well as the development of denoising strategy using finite support (e.g., wavelet transforms) and infinite support (e.g., Fourier transforms) basis functions.

2. Dielectric modeling of multi-phase cementitious composites – The dielectric modeling work presented in this dissertation is based on limited dielectric measurements reported in the literature [113]. Further experiments on the cyclic absorption-and-desorption (of water molecules) behavior of cement and

323 concrete specimens are advantageous for better understanding the microstructure of cementitious composites and for the improvement of the proposed dielectric models in this research.

3. Image reconstruction – Further investigation of the backprojection images on the features of defect signals with respect to the properties (e.g., size, shape, and orientation) of the defect can lead to a better interpretation of the images. Additionally, modiﬁcation to the image reconstruction algorithm by considering the inﬂuence of dielectric properties is advantageous for improving the accuracy of locating defects.

324 Appendix A

Phase Velocity of Love Waves in A Layer Underlain by A Half Space Medium

This appendix derives the analytical expression of the phase velocity of Love waves in a layer underlain by a solid half space, which is mentioned in Chapter 2 Literature Review. A.E.H. Love (1911) [156] discovered the existence of one type of surface waves (diﬀerent from the Rayleigh wave) that can be hospitalized/trapped within layered media when he investigated the propagation data of seismic waves. Ewing, Jardetzky, and Press (1957) [272] followed the same approach provided by Love and derived the period of Love waves in a thin layer underlain by a solid half space, as shown in Figure A-1. In Figure A-1, a planar layer (Medium 2) with thickness of H is placed above a half space medium (Medium 1). Only the solid medium is considered here. The incident wave propagates from a source located in a distance far from the boundary between Media 1 and 2, resulting in the case of plane wave incidence at the boundary. After impinging the boundary, the transmitted wave bounces back-and-forth within Medium 2, being trapped within the layer and forming Love waves.

325 z= - H

E2, ν2, ρ2 Medium 2 z= 0 x E1, ν1, ρ1 ki Medium 1

Incident wave z

Figure A-1: A layer underlain a solid half space

326 Consider the period equation by Ewing, Jardetzky, and Press (1957) (Sec. 4-5, p.210) as follows. s c 2 1 − G2 vs2 tan kγˆ1H = · s (A.1) G1 c 2 − 1 vs1 ω where k = is the wavenumber of Love waves, ω = 2πf is the radian frequency, vL vL is the phase velocity of Love waves, f is the temporal frequency in Hertz,γ ˆ1 = s 2 r c Gi − 1 , c is the speed of light, vsi = is the shear wave velocity, and vs1 ρi r Ei Gi = is the modulus of rigidity, νi is Poisson’s ratio, ρ is the density, and 2(1 + νi) H is the thickness of the layer. Knowing that

x3 2x5 17x7 tan x = x + + + ··· , (A.2) 3 15 315

the ﬁrst-two-term approximation is

x3 tan x ∼= x + (A.3) 3

s s s 2 2 2 c G2 c c Letting C1 =γ ˆ1H = H · − 1 and C2 = · 1 − / − 1, vs1 G1 vs2 vs1 the approximated tangent function becomes

ω ω 1 ω 3 tan C1 = C1 + · C1 = C2 (A.4) vL vL 3 vL

After re-arrangement, an inhomogeneous cubic equation is yielded.

C2 3 2 3 3 · vL − 3ω · vL − ω = 0 C1 3 3 C1 2 ω C1 =⇒ vL − ω · vL − · = 0 (A.5) C2 3 C2

Using the method by Scipione del Ferro (1465-1526) and assuming vs1 < c < vs2, the

327 discriminant equation for determining the characteristic of roots is

" # 4 C 2 ω3 C 2 C 3 ∆ = − 1 − 27 · · 1 − · 1 > 0 3 C2 3 C2 27 C2 (A.6) suggesting there is one real root and a pair of conjugate roots. The inhomogeneous cubic equation is in the form of

3 2 a · vL + b · vL + c · vL + d = 0 (A.7) with coeﬃcients

a = 1 C b = −ω 1 C2 c = 0 (A.8) ω3 C d = − · 1 3 C2

The parameters deﬁned in del Ferro’s method are

a3 C 1 p = b − = −ω 1 − (A.9) 3 C2 3 2 · a3 − 9ab 1 C q = c + = 2 + 9ω 1 (A.10) 27 27 C2 " #1/3 q q2 p3 1/2 u = + + (A.11) 2 4 27

Therefore, the phase velocity of Love waves in this model is

p a p 1 v = − u − = − u − (A.12) L 3u 3 3u 3

328 where the coeﬃcient C1 as C2

c 2 c 2 − 1 − 1 C G v E (1 + ν )1/2 v 1 = H · 1 · s1 = H · 1 2 · s1 C G " 2#1/2 E (1 + ν ) " 2#1/2 2 2 c 2 1 c 1 − 1 − vs2 vs2 E (1 + ν )1/2 2ρ (1 + ν )c2 2ρ (1 + ν )c2 −1/2 = H · 1 2 · 1 1 − 1 · 1 − 2 2 (A.13) E2(1 + ν1) E1 E2

Since

C1 C1 = (H,E1,E2, ν1, ν2, ρ1, ρ2, c) C2 C2

p = p (ω, H, E1,E2, ν1, ν2, ρ1, ρ2, c)

q = q (ω, H, E1,E2, ν1, ν2, ρ1, ρ2, c)

u = u(p, q) = u (ω, H, E1,E2, ν1, ν2, ρ1, ρ2, c)

are determined by material properties (E1,E2, ν1, ν2, ρ1, ρ2), geometry (layer thickness

H), and incident wave property (frequency ω), vL can be fully determined once these properties are provided.

329 330 Appendix B

Analytical Approach to Several Plane Wave Incidence Problems

This appendix addresses the analytical approach to several electromagnetic scattering problems in which plane wave incidence is considered. In the FAR NDT technique, reflection response of the target structure is used for image reconstruction. The evaluation of reflected radar signals from a GFRP-wrapped concrete column is in nature an electromagnetic scattering problem in which the use of far-field measurements indicates the incidence of plane waves. The purpose of this appendix is to evaluate the reflected/scattered fields in two plane wave incidence problems. The default geometry of the target structure/scatterer in the problems is circular, and the target scatterer is a non-conductive dielectric. Def- inition of reflection coefficient and reflectivity is first provided. One two-dimensional scattering problem and one three-dimensional scattering problem are studied.

B.1 Reflection Coefficient and Reflectivity

Reflection responses or signals can be evaluated by the reflection coefficient (field intensity) or reflectivity (power intensity). Consider a two-layer model with one infinite boundary subjected to the impinging of a TE polarized plane wave traveling toward the boundary, as shown in Figure B-1 . The dielectric properties of regions 0

331 and 1 are characterized by (0, µ0) and (1, µ1), respectively. The plane wave impinges the boundary at an incident angle θi, creating a reﬂected wave and a transmitted wave. The incident ﬁelds are considered as [138]

Hi Hr Region 0 Incidence Reflection ε0, µ0 Ei Er Hi Hr kr E k i i θi θr kr E ki r

x Region 1 Transmission ε1, µ1 Ht

θ E t kt t

Ht kt

Et : Incidence plan e

Figure B-1: A two-dimensional two-layer model with inﬁnite boundary (TE waves)

¯ ikixx − ikizz Ei =yE ˆ 0e (B.1) ¯ 1 ¯ Hi = ∇ × Ei iωµ0 k k iz ikixx − ikizz ix ikixx − ikizz =x ˆ E0e +z ˆ E0e (B.2) ωµ0 ωµ0 indicating an electric field propagating in (ˆx, −zˆ) direction with a +ˆy component, and a magnetic field propagating in (ˆx, −zˆ) direction with +ˆx and +ˆz components. The reflected fields are

¯ ikrxx + ikrzz Er =yR ˆ 01E0e (B.3) k k ¯ rz ikrxx + ikrzz rx ikrxx + ikrzz Hr = −xˆ R01E0e +z ˆ R01E0e (B.4) ωµ0 ωµ0

332 The transmitted ﬁelds are

¯ iktxx − iktzz Et =yT ˆ 01E0e (B.5) k k ¯ tz iktxx + iktzz tx iktxx + iktzz Ht =x ˆ T01E0e − zˆ T01E0e (B.6) ωµ0 ωµ0

¯ ¯ The continuity of tangential ﬁelds (Ey and Hx) at the boundary (z = 0) must be satisﬁed. Therefore, it provides two conditions for two unknowns.

¯ Ey : Eiy + Ery = Ety =⇒ 1 + R01 = T01 (B.7) ¯ kiz krzR ktzT Hx : Hix + Hrx = Htx =⇒ − = (B.8) µ0 µ0 µ1

With the two conditions, the Fresnel’s reﬂection and transmission coeﬃcients of TE waves can be determined as follows.

k µ 1 − tz 0 k µ R¯TE = iz 1 (B.9) 01 k µ 1 + tz 0 kiz µ1 2 T¯TE = (B.10) 01 k µ 1 + tz 0 kiz µ1

¯ ¯ For TM waves the continuity conditions are held on Hy and Ex at z = 0. Following same approach, the Fresnel’s reﬂection and transmission coeﬃcients of TM waves are obtained.

k + k µ tz ix 0 − 1 k + k µ R¯TM = iz ix 1 (B.11) 01 k + k µ tz ix 0 + 1 kiz + kix µ1 2 T¯TM = (B.12) 01 k + k µ tz ix 0 + 1 kiz + kix µ1

Both TE and TM wave modes are illustrated in Figure B-2, along with the incident ¯ wave vector ki. The power reflection coefficient or the reflectivity is defined to be

333

z (TM)

Ei ki

Ei θ 0 (TE)

φ0

Figure B-2: TE and TM waves and incident wave vector

TE TE 2 r01 = R01 (B.13)

The power transmission coeﬃcient or the transmissibility is then deﬁned to be

TE TE µ0ktx TE 2 t01 = 1 − r01 = T01 (B.14) µ1kix

Notice that ktx and kix are the x component of wave vectors in Regions 1 and 0, respectively. It is the reﬂectivity and the transmissibility of a target that are measured by radar in laboratory or in ﬁeld.

B.2 A Two-dimensional Three-layer Model

The influence of the presence of a thin layer to the reflectivity of layered systems can be first evaluated by a two-dimensional three-layer model as shown in Figure B-3. The reflection coefficient of the three-layer model is determined following the same approach mentioned for the two-layer model, using the transmitted wave in Region 2 as the incident wave for Region 3. The general reflection coefficient for the three-layer

334 z

H i 1st reflection Region 0 ε0, µ0 E 2nd reflection i Interface 3rd reflection coefficients ki θi θr

R01, T01 x R10, T10 Region 1 ε1, µ1 θ t R12, T12

R21, T21 Region 2 ε2, µ2

Figure B-3: A two-dimensional three-layer model (TE waves)

model is [47] T R T e2ik1zd1 R¯ = R + 01 12 10 (B.15) 01 01 2ik d 1 − R10R12e 1z 1 where TE waves:

TE µnkmz − µmknz Rmn = (B.16) µnkmz + µmknz TM waves:

TM nkmz − mknz Rmn = (B.17) nkmz + mknz Wave vectors are determined by

kmz = kz cos θm (B.18)

knz = kz cos θn (B.19)

335 where θm and θn are determined by Snell’s law:

sin θ rµ m = m m (B.20) sin θn µnn

Reﬂection coeﬃcients of multi-layer models can be derived by the propagation matrix approach [138], and by the transmission line theory in which the impedance of materials is used [188].

B.3 A Three-dimensional Inﬁnite Dielectric Cylinder Model

In this section, the scattering problem of an infinite dielectric cylinder of radius a impinged by a plane wave is considered. The problem is a three-dimensional one since the wave vector of incident waves is not required to be perpendicular to the axis of the cylinder (Figure B-4). This problem characterizes the scattering of far-field radar signals (plane waves) from GFRP-wrapped concrete columns. Derivation of the scattered fields in this problem is performed using the Hertzian potentials for solving the Helmholtz wave equations in cylindrical coordinates. Unknown coefficients are determined by matching the boundary condition at the surface of the dielectric cylinder as the result of continuation of tangential fields at the interface. For such problems, TE and TM wave modes are usually coupled and needed in the representation of scattered fields, except in some special cases (e.g., a PEC cylinder impinged by plane waves). First, the TE and TM incident waves are considered in the cylindrical coordinates (ρ, φ, z). TE waves:

¯ iki · r¯ Hzi = H0 sin θ0e (B.21) ¯ iki · r¯ Hρi = −H0 cos θ0 cos(φ − φ0)e (B.22) ¯ iki · r¯ Hφi = H0 cos θ0 sin(φ − φ0)e (B.23)

336 z’

z ki θ0

x φ0

y’

x’

Figure B-4: A three-dimensional inﬁnite dielectric cylinder impinged by plane waves

TM waves:

¯ iki · r¯ Ezi = H0 sin θ0e (B.24) ¯ iki · r¯ Eρi = −E0 cos θ0 cos(φ − φ0)e (B.25) ¯ iki · r¯ Eφi = E0 cos θ0 sin(φ − φ0)e (B.26)

¯ where ki = k(sin θ0 cos φ0xˆ + sin θ0 sin φ0yˆ + cos θ0zˆ) is the wave vector of incident ˆ waves,r ¯ = ρ cos φxˆ + ρ sin φyˆ + zzˆ is the observation position vector, and ki · rˆ = kz cos θ0 + kρ sin θ0 cos(φ − φ0). Substituting these ﬁelds into Maxwell’s curl equations yields the relationship between

337 electromagnetic ﬁelds and Hertzian potentials.

∂2 H = + k2 ΠTE (B.27) zi ∂z2 i zi ¯ 2 2 iki · r¯ = Aiki sin θ0e (B.28)

∂2 E = + k2 ΠTM (B.29) zi ∂z2 i zi ¯ 2 2 iki · r¯ = Biki sin θ0e (B.30) where

H0 Ai = 2 (B.31) k sin θ0 E0 Bi = 2 (B.32) k sin θ0

The Hertzian potentials are then expanded in Fourier series with respect to φ for the mathematical convenience of satisfying the boundary condition at the surface of the cylinder.

∞ TE X ikz cos θ0 in(φ − φ0 + π/2) Πzi = Aie Jn(kρ sin θ0)e (B.33) n=−∞ ∞ TM X ikz cos θ0 in(φ − φ0 + π/2) Πzi = Bie Jn(kρ sin θ0)e (B.34) n=−∞

where Jn(kρ sin θ0) is the Bessel function with argument (kρ sin θ0). The scattered

ﬁelds are written in similar form with unknown coeﬃcients Asn and Bsn.

∞ TE X ikiz cos θ0 (2) in(φ − φ0 + π/2) Πzs = Asne Hn (kiρ sin θ0)e (B.35) n=−∞ ∞ TM X ikiz cos θ0 (2) in(φ − φ0 + π/2) Πzs = Bsne Hn (kiρ sin θ0)e (B.36) n=−∞

338 (2) where where Hn (kiρ sin θ0) is the Hankel function of the second kind with argument

(kiρ sin θ0). The transmitted ﬁelds inside the cylinder are

∞ TE X iktz cos θ0 in(φ − φ0 + π/2) Πzt = Atne Jn(ktρ sin θ0)e (B.37) n=−∞ ∞ TM X iktz cos θ0 in(φ − φ0 + π/2) Πzt = Btne Jn(ktρ sin θ0)e (B.38) n=−∞

Unknown coeﬃcients Asn, Atn, Bsn, and Btn are determined by the boundary condition which is the continuity of tangential ﬁelds (z and φ components) at ρ = a. The continuity conditions are

Ez:

2 2 (2) 2 2 (k sin θ0) BiJn(ka sin θ0) + BsnHn (kia sin θ0) = kt sin θtBtnJn(kta sin θt) (B.39)

Hz:

2 2 (2) 2 2 (k sin θ0) AiJn(ka sin θ0) + AsnHn (kia sin θ0) = kt sin θtAtnJn(kta sin θt) (B.40)

Eφ:

k n cos θ i 0 B J (ka sin θ ) + B H(2)(k a sin θ ) a i n 0 sn n i 0 h 0 (2)0 i +iωµki sin θ0 AiJn(kia sin θ0) + AsnHn (kia sin θ0) k n cos θ = i 0 [B J (k a sin θ )] + iωµ sin θ [A J 0 (k a sin θ )] (B.41) a tn n t t t tn n t t

Hφ:

k n cos θ i 0 A J (k a sin θ ) + A H(2)(k a sin θ ) a i n i 0 sn n i 0 h 0 (2)0 i +iωki sin θ0 AiJn(kia sin θ0) + AsnHn (kia sin θ0) k n cos θ = − i 0 [B J (k a sin θ )] + iωµ sin θ [A J 0 (k a sin θ )] (B.42) a tn n t t t tn n t t

339 where θt is determined by Snell’s law, which is

−1 ki θt = cos cos θ0 (B.43) kt

With these four conditions, the four unknowns can be evaluated completely. Consid- ering only non-magnetic materials provides

√ √ √ √ √ kt = ω tµt = ω tµ0 = ω 0µ0 r = k0 r (B.44)

pi pi Furthermore, choosing θ = leads to θ = and therefore, 0 2 t 2

0 E0 0 E0r Jn(kta) Jn(ka) − Jn(k0a) ki kt Jn(kta) Asn = 2 0 (B.45) ki r Jn(kta) ( (2)0 Hn2)(kia) − kiHn (kia) kt Jn(kta)

2 kt Jn(kta) Atn = 2 (2) (B.46) E0Jn(kia) + ki Hn (kia)Asn

0 H0 0 µrH0 Jn(kta) Jn(kia) − Jn(kia) ki kt Jn(kta) Bsn = 2 0 (B.47) ki Jn(kta) (2) (2)0 Hn − kiHn (kia) kt Jn(kta)

2 kt Jn(kta) Btn = (B.48) 2 (2) H0Jn(kia) + ki Hn (kia)Bsn

340 The scattered ﬁelds outside the cylinder are evaluated.

2 TM Ezs = k0Πzs (B.49)

2 TE Hzs = k0Πzs (B.50) ∞ X ∂ E = A H(2)(kρ)e−jn(φ−φ0+π/2) ρs sn ∂ρ n n=−∞ ∞ (2) ωµ X Hn (kρ) − B e−jn(φ−φ0+π/2) (B.51) ρ sn n n=−∞ ∞ 1 X E = A H(2)(kρ)(−jn)e−jn(φ−φ0+π/2) φs ρ sn n n=−∞ ∞ X ∂ +jωµ B H(2)(kρ)e−jn(φ−φ0+π/2) (B.52) sn ∂ρ n n=−∞ ∞ jω X H = A H(2)(kρ)(−jn)e−jn(φ−φ0+π/2) ρs ρ sn n n=−∞ ∞ X ∂ + B H(2)(kρ)e−jn(φ−φ0+π/2) (B.53) sn ∂ρ n n=−∞ ∞ X ∂ H = −jω A H(2)(kρ)e−jn(φ−φ0+π/2) φs sn ∂ρ n n=−∞ ∞ 1 X + B H(2)(kρ)(−jn)e−jn(φ−φ0+π/2) (B.54) ρ sn n n=−∞

B.4 Summary

This appendix describes the analytical investigation of two scattering problems in which a circular scatterer is considered. It is known that the analytical form of scattered fields from a three-dimensional scatterer of finite size and with localized anomaly may not be available. Mathematical difficulty arises in finding a relevant special function to analytically describe the scattered fields. Approximation methods such as perturbation analysis [22] and numerical methods such as FDTD [251] are usually applied instead.

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