Northwestern University. Theoretical Evaluation of Embedded Plate-Like and Solid Cylindrical Concrete Structures with Guided Waves

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1 Northwestern University Theoretical Evaluation of Embedded Plate-Like and Solid Cylindrical Concrete Structures with Guided Waves A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FUFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Civil and Environmental Engineering By Helsin Wang EVANSTON, ILLINOIS December, 4

2 Abstract Theoretical Evaluation of Embedded Plate-Like and Solid Cylindrical Concrete Structures with Guided Waves Helsin Wang The as-built condition of diaphragm walls or pile foundations affects the performance of a structure supported by these elements. Quality control tests on such elements generally consist of non-destructive tests based on low-frequency onedimensional wave propagation. This approach is limited to finding relatively large anomalies. A three-dimensional guided wave approach can extend the frequency range to allow detection of smaller size defects in foundations when compared to conventional non-destructive evaluation. The objective of this research is to develop theoretical and numerical solutions of Rayleigh-Lamb waves in embedded plate-like concrete structures (diaphragm walls) and flexural mode waves in embedded solid cylindrical concrete structures (pile foundations). The guide wave theory provides information about group velocity, phase velocity, and attenuation as a function of frequency. For any mode, a normalized displacement profile can be computed. The dispersion relations in a complex-valued matrix form are expressed as transcendental solutions between wave number and frequency for given material properties of concrete and soil. Guided waves are graphically illustrated as dispersion and attenuation curves. Group and phase velocities are derived from the dispersion curves and are functions of frequency. i

3 Results of parametric analyses show that the real part of the dispersion curves is not affected by boundary conditions, concrete-to-soil shear modulus ratio, concrete-tosoil density ratio, and soil type. However, these dispersion curves depend upon the Poisson s ratio of concrete. Attenuation curves are affected by boundary conditions, concrete-to-soil shear modulus ratio, density ratio, and soil type. The theory suggests that the usable frequency ranges for conventional NDT tests, defined as the frequency range where the propagation velocity is constant, are inversely proportional to plate thickness or cylinder diameter. Normalized displacement distributions are introduced to allow comparison of mode shapes from different orders/branches of the dispersion curves. From a practical viewpoint, the modes or orders/branches with the lowest attenuation, simplest shape, and largest displacement magnitude should be most easily induced and detected in guided wave experiments. The positions of the maximum relative amplitude of the normalized displacements are suggested as the best locations to install transducers for testing. The guided wave theory was applied to experimental results of impact echo methods on concrete walls. It is shown that the theory can identify resonant frequencies not identified based on interpretations based on one-dimensional wave theory. Recommendations are presented for selecting modes to be used in guided wave experiments in embedded concrete cylinders. ii

4 Acknowledgements I would like to express my deepest appreciation to my advisor, Professor Richard J. Finno, not only for his guidance and continuous support over the years but also for the encouragement and help in courses and my research at Northwestern University. I also acknowledge Professor Charles H. Dowding and Professor Raymond J. Krizek for their valuable suggestions on the research. I would like to thank Professor Tung Dju Lin, who has spent his precious time in helping me with writing of my research paper. I also would like to thank Professor Chien-Min Kao, who has given me his valuable opinions and advice on writing. I would like to express my deepest gratitude to Tony Tsai and Lan Tsai, who encouraged me as their son with their abundant love during my past few years. I would like to thank Rev. Samuel Yang, who opened my heart at an important time of my life. I also would like to thank the people of the Evangelical Taiwan Church, who encouraged and helped me during my study period, especially the Little Donkey Fellowship. Thanks to my colleagues, who helped make my life at Northwestern University a pleasant experience. I wish to thank the non-destructive evaluation research team, Hsiao-Chou Chao and James Lynch, at Northwestern University for their suggestions in the course of this research. The financial support of grants from the Infrastructure Technology Institute at Northwestern University is also gratefully acknowledged. Finally, I would like to express my deepest gratitude to my parents for their love and support in so many ways. iii

5 Dedication Life abundance from God beyond what people can see You, O LORD, keep my lamp burning; my God turns my darkness into light. With your help I can advance against a troop; with my God I can scale a wall. Psalm 18:8 and 9 iv

6 Contents Abstract... i Acknowledgements... iii Dedication... iv Contents... v Contents of Tables... ix Contents of Figures... xi List of Notations... xix Chapter 1. Introduction... 1 Chapter. Review of Guided Waves Fundamentals of Wave Propagation Approach from Kinematics of Deformation Approach from Theory of Elasticity Introduction to Waveguides General Concept Characteristics of Guided Waves Methods of Analysis Phase Velocity and Group Velocity Guided Waves in Half Spaces and Plate-Like Structures Rayleigh Wave SH-Mode Wave Lamb Wave Love Wave Generalized Lamb Wave Guided Waves in Multiple Layers Stoneley Wave Guided Waves in Solid Cylinders Guided Waves in a Traction-Free Solid Cylinders Longitudinal Mode Waves in an Embedded Solid Cylinders Summary Chapter 3. Preliminaries v

7 3.1 Introduction Analysis Method Material Properties Structures Surrounding Soil Values of Material Properties Numerical Analysis Introduction Initial Root Finding Curve Tracing Comments Concerning Solutions of Homogeneous Matrix System of Equations Summary Chapter 4. Rayleigh-Lamb Waves Propagating in an Embedded Plate-Like Concrete Structure Introduction Derivation of Dispersion Relation Governing Equation in a Plate Displacement and Stress Components in a Plate Governing Equation in Soil Displacement and Stress Components in Soil General Frequency Equation Dispersion Relation Solution Interpretation Characteristics of the Dispersion Relation Dispersion Curves Attenuation Parametric Analysis Introduction Effects of Concrete-to-Soil Shear Modulus Ratio and Density Ratio Effect of Soil Type Effect of Poisson s Ratio of Concrete Effect of Plate Thickness Normalization Normalized Displacement Distributions Principle of Analysis Interpretations of Computed Results Universal Frequency Interpretation of Experiments with Impact Echo Method Summary vi

8 Chapter 5. Flexural Mode Waves Propagating in an Embedded Solid Cylindrical Concrete Structure Introduction Derivation of Dispersion Relation Displacement and Stress Components General Frequency Equation Dispersion Relation Solution Interpretation Characteristics of the Dispersion Relation Dispersion Curves Attenuation Parametric Analysis Effects of Concrete-to-Soil Shear Modulus Ratio and Density Ratio Effect of Soil Type Effect of Poisson s Ratio of Concrete Effect of Cylinder Radius Normalization Normalized Displacement Distributions Selected Modes Interpretations of F(1,1) Modes Interpretations of F(1,3) Modes Interpretations of F(1,q) Modes at Local Maximum Group Velocity Practical Implication Universal Frequency Comparison of Flexural Mode Waves and Longitudinal Mode Waves Summary Chapter 6 Summary and Conclusions Summary Conclusions Rayleigh-Lamb Waves Propagating in an Embedded Plate-Like Concrete Structure Flexural Mode Waves Propagating in an Embedded Solid Cylindrical Concrete Structure... 6 References... 9 Appendices vii

9 A Rayleigh-Lamb Waves Propagating in an Embedded Plate-Like Concrete Structure A-1 Program Codes of Dispersion Curves A- Dispersion Solutions for Varying Poisson s Ratios... 4 A-3 Program Codes of Normalization B Flexural Mode Waves Propagating in an Embedded Solid Cylindrical Concrete Structure B-1 Program Code of Dispersion Curves B- Dispersion Solutions for Varying Poisson s Ratios B-3 Program Codes of Normalization... 6 C Dispersion Solutions and Universal Frequencies of Longitudinal Mode Waves Propagating in an Embedded Solid Cylindrical Concrete Structure D Verification of Dispersion Relations for Guided Waves Propagating in Embedded Plate-Like and Solid Cylindrical Concrete Structures D-1 Rayleigh-Lamb Waves Propagating in an Embedded Plate-Like Concrete Structure D- Flexural Mode Waves Propagating in an Embedded Solid Cylindrical Concrete Structure E Characteristics Comparison of Guided Waves Propagating in Embedded Plate-Like and Solid Cylindrical Concrete Structures... 8 E-1 Introduction... 8 E- Anti-Symmetric Mode and Flexural Mode... 8 E-3 Symmetric Mode and Longitudinal Mode E-4 Summary F Practical Examples for Attenuation Evaluation... 9 viii

10 Contents of Tables Table 3-1 Assumed concrete and soil parameters in parametric analysis Table 4-1 Conversion of db values to ordinary figures Table 4- Usable frequency range of the first coupled order for a given thickness of a plate embedded in soft/loose soil (µ/µ =35) Table 4-3 Usable frequency range of the second coupled order for a given thickness of a plate embedded in soft/loose soil (µ/µ =35)... 1 Table 4-4 Frequencies and wave numbers of selected modes for (a) the antisymmetric mode and (b) the symmetric mode of Rayleigh-Lamb waves in a plate embedded in soft/loose soil (µ/µ =35) Table 4-5 Lamé mode universal frequencies and wave numbers of Rayleigh-Lamb waves in an embedded plate Table 4-6 Non-dimensional cut-off frequencies of Rayleigh-Lamb waves for various Poisson s ratios of concrete Table 4-7 Resonant frequencies with IE method and cut-off frequencies with GWA in an intact traction-free prototype diaphragm wall Table 4-8 Cut-off frequencies of Rayleigh-Lamb waves for various Poisson s ratios of concrete for a wall thickness of.15 m Table 4-9 Resonant frequencies with IE method and cut-off frequencies with GWA in a concrete pavement Table 4-1 Cut-off frequencies of Rayleigh-Lamb waves for various Poisson s ratios of concrete for a pavement thickness of.1 m Table 4-11 Resonant frequencies with IE method and cut-off frequencies with GWA in a prototype diaphragm wall embedded in dry sand Table 4-1 Resonant frequencies with IE method and cut-off frequencies with GWA in a prototype diaphragm wall embedded in wet sand Table 4-13 Cut-off frequencies of Rayleigh-Lamb waves for various Poisson s ratios of concrete for a wall thickness of.15 m Table 5-1 Usable frequency range of branch F(1,1) for a given radius of a solid cylinder embedded in soft/loose soil (µ/µ =35) Table 5- Frequencies and wave numbers of selected modes for the flexural mode waves in an embedded solid cylinder Table 5-3 Universal frequencies and wave numbers of flexural mode waves in an embedded solid cylinder... 9 Table 5-4 Theoretically-induced branches and frequency ranges of pile radii of.15 m and m for NDT testing in concrete piles based on the lowest attenuation ix

11 Table C-1 Universal frequencies and wave numbers of longitudinal mode waves in a solid cylinder embedded in soft/loose soil Table F-1 Attenuation and computed initial amplitude (voltage) of mode F(1,1)c for a 1-meter diameter concrete cylinder embedded in soft/loose soil Table F- Attenuation and computed initial amplitude (voltage) of mode F(1,1)c for constant length-to-diameter ratio (L/D = 15) concrete cylinders embedded in soft/loose soil... 9 Table F-3 Attenuation and computed initial amplitude (voltage) of mode F(1,1)c in length-to-diameter ratio (L/D = 15) concrete cylinders embedded in soft/loose soil for a given frequency, f = 1.37 khz x

12 Contents of Figures Figure 1-1 A schematic for the side of a shaft accessible for flexural wave testing... 4 Figure -1 Waves in medium without lateral boundary: (a). Waves on a string and (b). Waves in a medium with the boundary far from observer points... 9 Figure - A schematic representation for (a) an ideally one-dimensional wave and (b) guided waves in a plate Figure -3 Guide waves in guiding systems: (a). two-dimensional and (b). threedimensional Figure -4 Schematic representations for plate-like structures and solid cylindrical structures... Figure -5 Schematic representations of real wave numbers for a longitudinal wave (left side) and a shear wave (right side) in a foundation structure.. Figure -6 Partial wave patterns for the vertical shear waves and longitudinal waves in a traction-free plate... 5 Figure -7 Frequency spectrum... 7 Figure -8 Rayleigh wave: (a). Schematic representation and (b). Wave on Sagittal plane... 9 Figure -9 Variation of non-dimensional phase velocity and group velocity with non-dimensional frequency for first six orders of the SH-mode waves in a traction-free plate... 3 Figure -1 Lamb wave: (a). Anti-symmetric mode and (b). Symmetric mode Figure -11 Dispersion curves for Lamb waves in a plate... 3 Figure -1 Variation of group velocity and phase velocity with modified frequency for Lamb waves Figure -13 Displacement distribution of a Love wave Figure -14 Love waves: (a). Dispersion curves and (b). Variation of nondimensional group velocity and non-dimensional phase velocity with non-dimensional frequency for first five orders Figure -15 Dispersion curves for (a) L(,q) and (b) F(1,q) in a solid cylinder Figure -16 First six branches of longitudinal mode waves in a cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω* space Figure -17 Variation of group velocity and phase velocity with non-dimensional frequency for first six branches of the longitudinal mode waves in an embedded solid cylinder... 4 Figure -18 Variation of (a) attenuation and (b) group velocity with frequency for L(,1) branch in an embedded pile as a function of pile radius for soft/loose soil Figure -19 Variation of comparison of usable frequency: field data and limits xi

13 Figure 3-1 Schematic representation for curve tracing Figure 4-1 Schematic representation for guided waves propagating in an embedded plate-like structure... 6 Figure 4- Figure 4- First five orders of Rayleigh-Lamb waves in a traction-free plate and an embedded plate in soft/loose soil in Re(ξ) Ω space Figure 4-3 Variations of non-dimensional phase velocity and group velocity with non-dimensional frequency for first five orders of Rayleigh-Lamb waves in a plate embedded in soft/loose soil... 8 Figure 4-4 First five orders of Rayleigh-Lamb waves in a plate embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space Figure 4-5 Order arl of Rayleigh-Lamb waves in an embedded plate in Re(ξ) Im(ξ) Ω space for varying concrete-to-soil shear modulus ratios Figure 4-6 Order srl of Rayleigh-Lamb waves in an embedded plate in Re(ξ) Im(ξ) Ω space for varying concrete-to-soil shear modulus ratios Figure 4-7 Order arl of Rayleigh-Lamb waves in an embedded plate in Re(ξ) Im(ξ) Ω space for varying concrete-to-soil density ratios Figure 4-8 Order srl of Rayleigh-Lamb waves in an embedded plate in Re(ξ) Im(ξ) Ω space for varying concrete-to-soil density ratios... 9 Figure 4-9 First five orders of Rayleigh-Lamb waves in an embedded plate in Re(ξ) Ω space for soft/loose soil (µ/µ =35) and hard/dense soil (µ/µ =45) Figure 4-1 First five anti-symmetric modes of Rayleigh-Lamb waves in an embedded plate in Im(ξ) Ω space for soft/loose soil (µ/µ =35) and hard/dense soil (µ/µ =45)... 9 Figure 4-11 First five symmetric modes of Rayleigh-Lamb waves in an embedded plate in Im(ξ) Ω space for soft/loose soil (µ/µ =35) and hard/dense soil (µ/µ =45) Figure 4-1 Order arl of Rayleigh-Lamb waves in a plate embedded in soft/loose soil (µ/µ =35) in Re(ξ) Im(ξ) Ω space for varying Poisson s ratios in concrete Figure 4-13 Effect of Poisson s ratio of concrete on non-dimensional phase velocity and group velocity of order arl of Rayleigh-Lamb waves in a plate embedded in soft/loose soil (µ/µ =35) Figure 4-14 Effects of thickness on group velocity and attenuation curves of order arl of Rayleigh-Lamb waves in a plate embedded in soft/loose soil (µ/µ =35) Figure 4-15 Effects of thickness on group velocity and attenuation curves of order srl of Rayleigh-Lamb waves in a plate embedded in soft/loose soil (µ/µ =35) Figure 4-16 Effects of thickness on group velocity and attenuation curves of order arl1 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil (µ/µ =35)... 1 xii

14 Figure 4-17 Effects of thickness on group velocity and attenuation curves of order srl1 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil (µ/µ =35) Figure 4-18 A schematic of energy flux through a block in an embedded plate Figure 4-19 A schematic of energy flux through a dividable block in an embedded plate Figure 4- Location of selected wave modes for Rayleigh-Lamb waves in a plate embedded in soft/loose soil (µ/µ =35) in Re(ξ) Ω space Figure 4-1 Location of selected wave modes for Rayleigh-Lamb waves in a plate embedded in soft/loose soil (µ/µ =35) in Im(ξ) Ω space Figure 4- Location of selected guided wave modes on non-dimensional phase velocity- and group velocity-frequency curves for first five orders of Rayleigh-Lamb waves in a plate embedded in soft/loose soil (µ/µ =35)... 1 Figure 4-3 Normalized displacement distributions for modes on the srl order... 1 Figure 4-4 Normalized displacement distributions for modes of the symmetric mode of Rayleigh-Lamb waves in an embedded plate with maximum group velocities Figure 4-5 Normalized displacement distributions for modes of the antisymmetric mode of Rayleigh-Lamb waves in an embedded plate with maximum group velocities Figure 4-6 Impact echo frequency spectra of sensors 1 and on an intact traction-free prototype concrete wall Figure 4-7 Impact echo frequency spectrum in a concrete pavement Figure 4-8 Impact echo frequency spectra in a prototype concrete wall embedded in dry sandy soil Figure 4-9 Impact echo frequency spectra in a prototype concrete wall embedded in wet sandy soil Figure 5-1 Schematic representation for the flexural mode waves propagating in an embedded solid cylindrical structure Figure 5- First nine branches of flexural mode waves in a traction-free solid cylinder and embedded solid cylinder in soft/loose soil in Re(ξ*) Ω* space Figure 5-3 Variations of non-dimensional phase velocity and group velocity with non-dimensional frequency for first nine branches of flexural mode waves in a solid cylinder embedded in soft/loose soil Figure 5-4 First nine branches of flexural mode waves in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω* space Figure 5-5 Branch F(1,1) of flexural mode waves in an embedded solid cylinder in Re(ξ*) Im(ξ*) Ω* space for varying concrete-to-soil shear modulus ratios xiii

15 Figure 5-6 Branch F(1,1) of flexural mode waves in an embedded solid cylinder in Re(ξ*) Im(ξ*) Ω* space for varying concrete-to-soil density ratios Figure 5-7 First seven branches of flexural mode waves in an embedded solid cylinder in Re(ξ*) Ω* space for soft/loose soil (µ/µ =35) and hard/dense soil (µ/µ =45) Figure 5-8 First seven branches of flexural mode waves in an embedded solid cylinder in Im(ξ*) Ω* space for soft/loose soil (µ/µ =35) Figure 5-9 First seven branches of flexural mode waves in an embedded solid cylinder in Im(ξ*) Ω* space for hard/dense soil (µ/µ =45) Figure 5-1 Branch F(1,1) of flexural mode waves in a solid cylinder embedded in soft/loose soil (µ/µ =35) in Re(ξ*) Im(ξ*) Ω* space for varying Poisson ratios in concrete Figure 5-11 Effect of Poisson s ratio of concrete on non-dimensional phase velocity and group velocity of branch F(1,1) of flexural mode waves in a solid cylinder embedded in soft/loose soil (µ/µ =35) Figure 5-1 Effects of radius on group velocity and attenuation curves of branch F(1,1) of flexural mode waves in a solid cylinder embedded in soft/loose soil (µ/µ =35) Figure 5-13 A schematic of energy flux through a circular block in an embedded solid cylinder Figure 5-14 Location of selected guided wave modes for flexural mode waves in a solid cylinder embedded in soft/loose soil (µ/µ =35) in Re(ξ*) Im(ξ*) Ω* space Figure 5-15 Location of selected guided wave modes on non-dimensional phase velocity- and group velocity-frequency curves for first five branches of flexural mode waves in a solid cylinder embedded in soft/loose soil (µ/µ =35) Figure 5-16 Three normalized displacement profiles at mode F(1,1)b Figure 5-17 Normalized displacement distributions for mode F(1,1)b Figure 5-18 Normalized displacement distributions for mode F(1,1)c Figure 5-19 Normalized displacement distributions for mode F(1,1)d Figure 5- Normalized displacement distributions for mode F(1,1)e Figure 5-1 Displacement responses for a pile subject to a lateral impact Figure 5- Normalized displacement distributions for mode F(1,3)b... 1 Figure 5-3 Normalized displacement distributions for mode F(1,3)e... Figure 5-4 Normalized displacement distributions for mode F(1,3)f... 3 Figure 5-5 Normalized displacement distributions for mode F(1,)b... 5 Figure 5-6 Normalized displacement distributions for mode F(1,4)b... 6 Figure 5-7 Normalized displacement distributions for mode F(1,5)b... 7 Figure 5-8 First few branches of longitudinal and flexural mode waves in a concrete cylinder embedded in soft/loose soil in Re(ξ*) Ω* space xiv

16 Figure 5-9 Variation of non-dimensional phase velocity with non-dimensional frequency for first few branches of longitudinal and flexural mode waves in an embedded concrete cylinder... 1 Figure 5-3 Variation of non-dimensional group velocity with non-dimensional frequency for first few branches of longitudinal and flexural mode waves in an embedded concrete cylinder Figure 5-31 Attenuation of first few branches of longitudinal and flexural mode waves in a concrete cylinder embedded in soft/loose soil Figure A-1 Order arl of Rayleigh-Lamb waves in a plate embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space for varying Poisson s ratios of concrete... 4 Figure A- Variation of non-dimensional phase velocity and group velocity of order arl of Rayleigh-Lamb waves in a plate embedded in soft/loose soil for varying Poisson s ratios of concrete... 4 Figure A-3 Order srl of Rayleigh-Lamb waves in a plate embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space for varying Poisson s ratios of concrete Figure A-4 Variation of non-dimensional phase velocity and group velocity of order srl of Rayleigh-Lamb waves in a plate embedded in soft/loose soil for varying Poisson s ratios of concrete Figure A-5 Order arl1 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space for varying Poisson s ratios of concrete... 4 Figure A-6 Variation of non-dimensional phase velocity and group velocity of order arl1 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil for varying Poisson s ratios of concrete... 4 Figure A-7 Order srl1 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space for varying Poisson s ratios of concrete Figure A-8 Variation of non-dimensional phase velocity and group velocity of order srl1 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil for varying Poisson s ratios of concrete Figure A-9 Order arl of Rayleigh-Lamb waves in a plate embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space for varying Poisson s ratios of concrete Figure A-1 Variation of non-dimensional phase velocity and group velocity of order arl of Rayleigh-Lamb waves in a plate embedded in soft/loose soil for varying Poisson s ratios of concrete Figure A-11 Order srl of Rayleigh-Lamb waves in a plate embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space for varying Poisson s ratios of concrete xv

17 Figure A-1 Variation of non-dimensional phase velocity and group velocity of order srl of Rayleigh-Lamb waves in a plate embedded in soft/loose soil for varying Poisson s ratios of concrete Figure A-13 Order arl3 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space for varying Poisson s ratios of concrete Figure A-14 Variation of non-dimensional phase velocity and group velocity of order arl3 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil for varying Poisson s ratios of concrete Figure A-15 Order srl3 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space for varying Poisson s ratios of concrete Figure A-16 Variation of non-dimensional phase velocity and group velocity of order srl3 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil for varying Poisson s ratios of concrete Figure A-17 Order arl4 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space for varying Poisson s ratios of concrete Figure A-18 Variation of non-dimensional phase velocity and group velocity of order arl4 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil for varying Poisson s ratios of concrete Figure A-19 Order srl4 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space for varying Poisson s ratios of concrete Figure A- Variation of non-dimensional phase velocity and group velocity of order srl4 of Rayleigh-Lamb waves in a plate embedded in soft/loose soil for varying Poisson s ratios of concrete Figure B-1 Branch F(1,) in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω*space for varying Poisson s ratios of concrete Figure B- Variation of non-dimensional phase velocity and group velocity of branch F(1,) in a solid cylinder embedded in soft/loose soil for varying Poisson s ratios of concrete Figure B-3 Branch F(1,3) in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω*space for varying Poisson s ratios of concrete Figure B-4 Variation of non-dimensional phase velocity and group velocity of branch F(1,3) in a solid cylinder embedded in soft/loose soil for varying Poisson s ratios of concrete Figure B-5 Branch F(1,4) in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω*space for varying Poisson s ratios of concrete Figure B-6 Variation of non-dimensional phase velocity and group velocity of branch F(1,4) in a solid cylinder embedded in soft/loose soil for varying Poisson s ratios of concrete xvi

18 Figure B-7 Branch F(1,5) in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω*space for varying Poisson s ratios of concrete Figure B-8 Variation of non-dimensional phase velocity and group velocity of branch F(1,5) in a solid cylinder embedded in soft/loose soil for varying Poisson s ratios of concrete Figure B-9 Branch F(1,6) in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω*space for varying Poisson s ratios of concrete Figure B-1 Variation of non-dimensional phase velocity and group velocity of branch F(1,6) in a solid cylinder embedded in soft/loose soil for varying Poisson s ratios of concrete Figure B-11 Branch F(1,7) in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω*space for varying Poisson s ratios of concrete... 6 Figure B-1 Variation of non-dimensional phase velocity and group velocity of branch F(1,7) in a solid cylinder embedded in soft/loose soil for varying Poisson s ratios of concrete... 6 Figure C-1 Branch L(,1) in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω* space for varying Poisson s ratios of concrete Figure C- Variation of non-dimensional phase velocity and group velocity of branch L(,1) in a solid cylinder embedded in soft/loose soil for varying Poisson s ratios of concrete Figure C-3 Branch L(,) in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω* space for varying Poisson s ratios of concrete Figure C-4 Variation of non-dimensional phase velocity and group velocity of branch L(,) in a solid cylinder embedded in soft/loose soil for varying Poisson s ratios of concrete Figure C-5 Branch L(,3) in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω* space for varying Poisson s ratios of concrete Figure C-6 Variation of non-dimensional phase velocity and group velocity of branch L(,3) in a solid cylinder embedded in soft/loose soil for varying Poisson s ratios of concrete Figure C-7 Branch L(,4) in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω* space for varying Poisson s ratios of concrete Figure C-8 Variation of non-dimensional phase velocity and group velocity of branch L(,4) in a solid cylinder embedded in soft/loose soil for varying Poisson s ratios of concrete Figure C-9 Branch L(,5) in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω* space for varying Poisson s ratios of concrete... 7 Figure C-1 Variation of non-dimensional phase velocity and group velocity of branch L(,5) in a solid cylinder embedded in soft/loose soil for varying Poisson s ratios of concrete... 7 Figure C-11 Branch L(,6) in a solid cylinder embedded in soft/loose soil in Re(ξ*) Im(ξ*) Ω* space for varying Poisson s ratios of concrete xvii

19 Figure C-1 Variation of non-dimensional phase velocity and group velocity of branch L(,6) in a solid cylinder embedded in soft/loose soil for varying Poisson s ratios of concrete Figure E-1 Dispersion curves of first five orders of anti-symmetric mode waves in a plate and first nine branches of flexural mode waves in a solid cylinder embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space Figure E- Variation of non-dimensional phase velocity with non-dimensional frequency for first five orders of anti-symmetric mode waves in a plate and first nine branches of flexural mode waves in a solid cylinder embedded in soft/loose soil Figure E-3 Variation of non-dimensional group velocity with non-dimensional frequency for first five orders of anti-symmetric mode waves in a plate and first nine branches of flexural mode waves in a solid cylinder embedded in soft/loose soil Figure E-4 Dispersion curves of first five orders of symmetric mode waves in a plate and first five branches of longitudinal mode waves in a solid cylinder embedded in soft/loose soil in Re(ξ) Im(ξ) Ω space Figure E-5 Variation of non-dimensional phase velocity with non-dimensional frequency for first five orders of symmetric mode waves in a plate and first five branches of longitudinal mode waves in a solid cylinder embedded in soft/loose soil Figure E-6 Variation of non-dimensional group velocity with non-dimensional frequency for first five orders of symmetric mode waves in a plate and first five branches of longitudinal mode waves in a solid cylinder embedded in soft/loose soil xviii

20 List of Notations ROMAN Letter Notations A amplitude of a wave or surface A, A 1, A, A 3, A 4, A 5, A 6, A 7, A 8 undetermined coefficients in a plate/cylinder A 1, A, A 4, A 5, A 6, A 7, A 8 undetermined coefficients in a half space/soil B 1, B, B 3, B 4, B 5 undetermined coefficients C wave velocity C bar bar wave velocity C e energy velocity C g group velocity C g non-dimensional group velocity C l longitudinal wave velocity of concrete C l longitudinal wave velocity of a half space/soil C Lamé Lamé velocity C p phase velocity C p non-dimensional phase velocity C plate plate wave velocity C R Rayleigh wave velocity C R Rayleigh wave velocity in a half space/soil C s shear wave velocity C s shear wave velocity in a half space/soil C Stoneley Stoneley wave velocity D diameter E Young s modulus of concrete E d energy density F flexural mode waves in a cylinder G shear modulus of concrete G shear modulus of soil H equivoluminal vector potential in a plate/cylinder H equivoluminal vector potential in a soil H r radial component of equivoluminal vector potential in a cylinder H x x component of equivoluminal vector potential in a plate H y y component of equivoluminal vector potential in a plate H z z/axial component of equivoluminal vector potential in a plate/cylinder H z z/axial component of equivoluminal vector potential in a soil xix

21 H θ angular component of equivoluminal vector potential in a cylinder I voluminal strain <I> average energy flux or intensity per average period <I> normalized average energy flux or intensity per average period <I r > average energy flux or intensity per average period of the sides of a cylinder <I x > average energy flux or intensity per average period of the unit-wide cross-section of a plate <I y > average energy flux or intensity per average period of the interaction sides of a plate <I z > average energy flux or intensity per average period of the edges of a plate or the cross-section of a cylinder L longitudinal mode waves in a cylinder or length of a cylinder R normalized radius of a cylinder RL Rayleigh-Lamb waves (Lamb waves) in a plate S surface area of a plate/cylinder S r surface area of the curved side of a cylinder S x surface area of the end of a plate S y surface area of the interaction sides of surrounding soil and a plate S z surface area of the end of a cylinder or surface area of the edge of a plate T normalized half-thickness of a plate or period T(t) variable function of time X mn m n matrix X(x) variable function of position a radius of a cylinder arl anti-symmetric mode of Rayleigh-Lamb waves in a plate a n1 element of matrix a n1 a n1 n 1 matrix b half thickness of a plate e l unit vector in tensor form e r unit vector in the radial direction in tensor form e r unit vector in the radial direction e x unit vector in the x direction e y unit vector in the y direction e z unit vector in the axial direction in tensor form e z unit vector in the axial or z direction e θ unit vector in the angular direction in tensor form e θ unit vector in the angular direction f frequency f cut-off cut-off frequency f peak peak frequency g normalization constant i 1 k ratio of longitudinal wave velocity to shear wave velocity xx

22 l l w p q r srl t t k u u u u u k ū k u r ū r u r u r u x ū x u x u x u y ū y u y u y u z ū z u z u z u θ ū θ u θ u θ u i,j v v v v w w w position of reflection source wall length circumferential order in a cylinder branch number in a cylinder radius symmetric mode of Rayleigh-Lamb waves in a plate time traction in tensor form displacement vector displacement in the x direction in a plate normalized displacement in the x direction in a plate displacement in the x direction in a half space/soil particle displacement in tensor form particle velocity in tensor form displacement in the radial direction in a cylinder particle velocity in the radial direction in a cylinder normalized displacement in the radial direction in a cylinder displacement in the radial direction in a soil displacement in the x direction in a plate particle velocity in the x direction in a plate normalized displacement in the x direction in a plate displacement in the x direction in a soil displacement in the y direction in a plate particle velocity in the y direction in a plate normalized displacement in the y direction in a plate displacement in the y direction in a soil displacement in the z direction in a plate or the axial direction in a cylinder particle velocity in the z direction in a plate or the axial direction in a cylinder normalized displacement in the z direction in a plate or the axial direction in a cylinder displacement in the axial direction in a soil displacement in the angular direction in a cylinder particle velocity in the angular direction in a cylinder normalized displacement in the angular direction in a cylinder displacement in the angular direction in a soil strain tensor assigned unit vector displacement in the y direction in a plate normalized displacement in the y direction in a plate displacement in the y direction in a half space/soil displacement in the z direction in a plate normalized displacement in the z direction in a plate displacement in the z direction in a half space/soil xxi

23 w x mn y rotation vector element of matrix X mn half thickness GREEK Letter Notations Ω or Ω * non-dimensional frequency Ω cut-off non-dimensional cut-off frequency Φ dilational scalar potential in a plate/cylinder Φ dilational scalar potential in a half space/soil α wave number for longitudinal waves in a plate/cylinder α wave number for longitudinal waves in a half space/soil β wave number for shear waves in a plate/cylinder β wave number for shear waves in a half space/soil ε x stretch strain in the x direction ε y stretch strain in the y direction ε z stretch strain in the z direction λ Lamé constant of concrete λ Lamé constant of soil λ wavelength µ Lamé constant of concrete µ Lamé constant of soil ν Poinsson s ratio of concrete ν Poinsson s ratio of soil ξ wave number in the x/axial direction ξ or ξ * non-dimensional wave number in the x/axial direction ξ i imaginary part of wave number ξ i non-dimensional imaginary part of wave number ξ r real part of wave number ξ r non-dimensional real part of wave number ϑ attenuation coefficient ρ density of concrete ρ density of soil σ stress σ ij or σ kl stress tensor in a plate/cylinder σ ij normalized stress tensor in a plate/cylinder σ ij stress tensor in a half space/soil ω angular frequency xxii

24 Chapter 1 Introduction The as-built condition of diaphragm walls or pile foundations can affect the performance of a structure supported by these elements. For more than thirty years, integrity of concrete piles and drilled shafts has been evaluated in the field by nondestructive surface reflection techniques, such as impact echo and impulse response methods. Impact echo methods recently have been applied to prototype concrete walls to identify the wall dimensions and pre-set defects in traction-free and embedded conditions (Lin 1995; Yang 3). These surface reflection methods are more economical and more conveniently performed than direct transmission methods, such as the sonic logging or parallel seismic methods (Baker et al. 1993; Geo-Institute Deep Foundations Committee ). Since many variables affect the testing results, the main function of surface reflection methods is to help engineers confirm whether the as-built condition of the foundation element meets its design intent (Davis 1998). A number of study cases evaluated the capability of the impulse response and sonic echo techniques to provide reliable assessments of foundation quality (e.g., Baker et al. 1993; Davis 1995; Finno et al. 1997; Finno and Gassman 1998; Geo- 1

25 Institute Deep Foundations Committee ). In concrete pile foundations, the length-to-diameter ratio is limited empirically to less than 3 for medium stiff clay and less than 5 for soft soils (Davis 1995) primarily because of the energy leakage of wave propagation into surrounding soils. However, it is difficult for one to distinguish among different types of defects based on these test results. For example, the gradual changes in concrete quality and the cross section of a pile or defects near the bottom of drilled shafts cannot be reliably identified. Moreover, the highest induced frequencies in such tests are several kilohertz, and depend upon the contact time of a hammer blow on a pile. This input frequency limits the wavelength introduced into piles. A defect size less than 3% of the cross-sectional area of a pile cannot be detected with these surface reflection methods (Geo-Institute Deep Foundations Committee ). One-dimensional wave propagation is assumed implicitly when interpreting results of conventional surface reflection tests. In a given pile, the propagation velocity of the input stress waves is regarded as a constant, and is not a function of frequency. The distance between reflection sources, l, either the bottom of a foundation structure or a defect, and sensors is computed by l = C/ f, (1-1) where C is the bar wave velocity and f is the peak resonance frequency. The induced wavelength generally is larger than the dimension of the foundation structure. As such, a plane wave theoretically is induced into a pile, resulting in a uniform

26 3 deformation throughout the cross section. These stress waves thus pass through small size defects. No signals, or at best, very low reflections, come from such defects in conventional non-destructive evaluation (NDE) methods. More generally, waves traveling in diaphragm walls and piles include different clusters of waves, named guided waves. These three-dimensional wave clusters arise from the incidences and reflections of a variety of compression, shear, and surface waves along the boundaries of foundation structures. The true wave velocity is a function of frequency, and the displacement magnitudes vary along a pile or wall cross-section. Popovics (1994) showed that the predominant waves in concrete slabs excited by impact echo methods correspond to low frequencies of the lowest branch of guided waves. Chao (), Finno et al. (1), and Hu (1999) reached the same conclusions in concrete piles and drilled shafts. Finno et al. (1) showed that the usable frequency range (upper frequency limit) for surface reflection techniques theoretically varies from 5 Hz to 1 Hz for 4-m- to 1-meter-diameter concrete piles, respectively. Guided waves at higher frequencies induce smaller wavelengths than conventional surface reflection techniques, and thus, in principle, one should able to detect smaller sized defects in guided wave experiments than in conventional methods. Using longitudinal guided waves, Chao () identified a 7% cross-section notch in a prototype concrete pile. Presumably, the applicability of guided waves can be extended to the non-destructive evaluations of diaphragm walls, wharf shafts, and bridge piers by inducing stress waves with high frequencies.

27 4 Bridge or wharf slab Flexural mode waves Shaker Portable data acquisition Sensors Shafts/ piers Water Soil Figure 1-1 A schematic for the side of a shaft accessible for flexural wave testing The objective of this research is to develop analytic formulations and numerical solutions for guided waves in an embedded plate-like structure (i.e., a structural slurry wall) and the flexural mode waves in an embedded solid cylindrical structure (i.e., a concrete pile or drilled shaft). The flexural mode wave theory provides solutions for the cases where only the sides of wharf shafts (Finno et al. 1; Finno and Lynch ) and bridge piers (Douglas and Holt 1994; Rix et al. 1995) are accessible for non-destructive testing (NDT), as shown in Figure 1-1. The results of this research provide information about the group and phase wave velocities, attenuation, and the displacement profiles for selected modes of wave propagation so that guided wave NDE can be employed efficiently. This thesis is divided into six chapters. Chapter presents the fundamentals of

28 5 wave propagation with emphasis on waveguides and summarizes different types of guided waves. Fundamentals of wave propagation illustrate the essential difference between the kinematics of deformation used in one-dimensional waves and the theory of elasticity applied to three-dimensional waves (guided waves). Two approaches can be used to develop dispersion relations for guided waves. This thesis uses the displacement potential theory because this method provides not only the required dispersion relations, but also displacement distributions associated with each mode. Dispersion characteristics are discussed and phase and group velocities are defined. A variety of guided waves are described, including Rayleigh and Stoneley waves in half spaces, SH-mode wave, Lamb wave, Love wave, and generalized Lamb wave in platelike structures, and longitudinal, flexural, and torsional waves in solid cylinders. Chapter 3 summarizes numerical analysis methods used in developing solutions to the dispersion equations, assumptions for material properties, and values of material properties for concrete and soil used herein. Techniques used for initial root finding and curve tracing of derived dispersion relations are discussed. Comments concerning solutions for a homogeneous matrix system also are made. Chapter 4 develops the solutions for guided waves propagating in an embedded plate-like concrete structure. The frequency equation is solved with the displacement potential theory. The dispersion relation derived from the frequency equation is expressed as an implicit relation between frequency and wave number. The

29 6 characteristics of the first few dispersion solutions are illustrated by results of numerical analysis with Maple programs for typical material properties of concrete and soil. Results of parametric analyses illustrate the effects of concrete-to-soil shear modulus ratio, concrete-to-soil density ratio, soil type, Poisson s ratio of concrete, and plate thickness on dispersion and attenuation behaviors. A normalization concept is developed and applied to determine displacement profiles for any mode. Analysis of displacement distributions provides recommendations for performing guided wave experiments in concrete plates. The universal frequencies are discussed and identified. The experimental results of impact echo methods published in literature are interpreted with the guided wave theory. Chapter 5 describes the solution for the flexural mode waves propagating in an embedded solid cylindrical concrete structure. The dispersion relation derived from the frequency equation is expressed as an implicit relation between frequency and wave number. The characteristics of the first few dispersion solutions again are shown by results of numerical analysis using Maple programs for typical material properties of concrete and soil. Results of parametric analyses illustrate the effects of concrete-to-soil shear modulus ratio, concrete-to-soil density ratio, soil type, Poisson s ratio of concrete, and cylinder radius on dispersion and attenuation behaviors. Normalization formulations are derived and applied to determine displacement distributions for any mode. Analysis of displacement distributions provides guidelines for performing flexural mode wave experiments in concrete piles or drilled shafts. A

30 7 comparison between the flexural mode and longitudinal mode waves is made to help identify conditions wherein either test is more suited to conduct guided wave experiments in piles or drilled shafts. Chapter 6 summarizes this thesis and presents conclusions related to the Rayleigh-Lamb waves propagating in an embedded plate-like concrete structure and the flexural mode waves propagating in an embedded solid cylindrical concrete structure.

31 Chapter Review of Guided Waves.1 Fundamentals of Wave Propagation The study of the characteristics of wave propagation in an ideal linear elastic medium can start from either the kinematics of deformation or the theory of elasticity. The method of kinematics of deformation is used in one-dimensional wave propagation and the method of theory of elasticity is applied to three-dimensional wave (guided wave) propagation..1.1 Approach from Kinematics of Deformation The method of kinematics of deformation is easy to apply to one-dimensional wave propagation. The boundary condition (either traction-free surface or bonding with other medium) alters the properties of waves in a medium that is not infinite. Therefore, the method of kinematics of deformation generally is considered as an approximation of the true deformation for wave propagation in structures, even though extensions to the method, such as the Timoshenko beam theory, can provide additional 8

32 information (Abramson 1956; Graff 1975). 9 Direction of Propagation r>>x >X 1 boundary (a). explosion Direction of propagation Figure -1 Waves in medium without lateral boundary: (a). Waves on a string and (b). Waves in a medium with the boundary far from observer points x (b). A B X 1 X r The method of kinematics of deformation describes a one-dimensional wave, or a plane wave, traveling on a string in a specific direction (Figure -1 (a)). In this situation, the derivation of the appropriate wave equation does not consider the effect of lateral deformation (Poisson s effect). In this case, the initial condition will dominate the wave behavior. Alternatively, waves in a medium with boundaries far from two close-by observation points will also meet the required one-dimensional conditions (see Figure -1 (b)). For example, a spherical wave generated by an explosion passes through two points A and B. The observed wave behavior between these two points could be regarded as a one-dimensional wave because its response does not depend upon the boundary effect. Furthermore, the waves on a string or plane waves in an infinite space could be reflected and transmitted if a material interface exists. Such a boundary condition just on the direction of propagation could change the wave amplitude but not its velocity.

33 1 The partial differential equation for a one-dimensional wave in medium is wellknown and can be found in many mathematical or wave mechanics books (Achenbach 1973; Kolsky 1963; Love 197). The displacement equation for a wave which propagates along an x-direction is a function of position, x, and time, t, is u t = C u, x (-1) where u is the displacement and C is the wave velocity. The waves described by this equation can be either longitudinal or shear waves. The longitudinal wave velocity, C l, and shear wave velocity, C s, are expressed in terms of material properties as and C l E = (-) ρ G C s =, (-3) ρ where E is the Young s modulus, G is the shear modulus, and ρ is the density of the material that comprises the medium. Two methods can be used to solve the wave equation: Separation of Variables and D Alembert s Method. Separation of variables decomposes the displacement into the multiplication of two independent variable functions, position, X(x), and time, T(t):

34 11 uxt (, ) = XxTt ( ) ( ). (-4) D Alembert s Method decomposes the displacement into the summation of two waves, positive-direction-movement wave, f(x-ct), and negative-direction-movement wave, g(x+ct): uxt (, ) = f( x Ct) + gx ( + Ct). (-5) The solution can be obtained by substituting equation (-4) or (-5) into equation (- 1), and is given as uxt (, ) = Ae + Ae, (-6) i( ξx+ ωt) i( ξx ωt) 1 where A 1 and A are the coefficients determined by the initial conditions, ξ is the wave number defined as π/λ, ω is the angular frequency equal to πf, λ is the wavelength, and f is the frequency..1. Approach from Theory of Elasticity The theory of elasticity provides the exact equations for three-dimensional wave propagation and extends the applicability of the analysis to the interaction of elastic waves with surfaces and boundaries. The boundary condition depends on the stress or/and displacement constraints, the Poisson s effect, and the bonding at the media interfaces. These factors will not only influence the wave amplitude, energy dissipation and wave velocity, but also will substantially change the wave equation

35 formulation. 1 The method based on theory of elasticity describes a wave traveling in any direction within or on a bounded medium, such as a plate or cylinder. The derivation process can be found in either theory of elasticity or introductory books on wave mechanics (Sokolnikoff 1956; Kolsky 1963). The displacement equations of motion in isotropic elastic media in the absence of body forces are expressed by the Navier- Stokes equation as: u ( + ) u+ u =, (-7) t λ µ µ ρ where u is the particle displacement vector in the x, y, and z directions and µ and λ are Lamé constants. Assuming small strains, the voluminal strain of a material is defined by I = u = u = u + u + u = ε + ε + ε, (-8) i, i x, x y, y z, z x y z where u i,j is the strain tensor and ε x, ε y, and ε z are the normal strains in the x, y, and z directions, respectively, so that equation (-7) may be re-written as: λ + µ + µ = ρ. ( ) I u u (-9) t An alternation form for equation (-7) can be obtainable by using the vector identity: u = u u. (-1)

36 13 Substituting equation (-1) into equation (-7) gives: u ( λ + µ ) u µ u = ρ. (-11) t The definition for the rotation vector, w, is 1 w = u, (-1) and equation (-11) can be transformed to u ( λ + µ ) I µ w = ρ. (-13) t The advantage of equation (-13) is that it explicitly writes the displacement equation for wave propagation in terms of dilation and rotation parts. There are two kinds of waves that travel at different speeds in a medium a longitudinal wave and a shear wave. The longitudinal waves, without rotation, propagate at speed C l, and the shear waves, without volume change, propagate at speed C s. The longitudinal wave velocity from theory of elasticity is faster than that from kinematics of deformation due to lateral volume deformation. The shear wave velocities from both methods are the same value as a result of constant volume conditions. The ratio of the longitudinal to shear wave velocity may be expressed as:

37 14 Cl ν k = =. (-14) C 1 ν s Since Poisson s ratios of materials are within and.5, longitudinal wave velocity is always more than shear wave velocity. To evaluate a dilational wave (longitudinal wave) propagating through a medium, the divergence operation is performed on both sides of equation (-7) with ; ( u ) = ( u) = I. (-15) Hence, the dilational wave equation can be simplified as: λ + µ I = ρ ( ) I t or ( λ + µ ) I ρ t I =. (-16) To evaluate an equivoliminal wave (shear wave) propagating through a medium, the curl operation is performed on the both sides of equation (-7). Since the curl of the gradient of a scalar is zero, the equivoliminal wave equation can be simplified as: w w or t µ = ρ µ ρ w =. w (-17) t The longitudinal wave velocity, C l, and shear wave velocity, C s, of a medium can express in terms of its material properties:

38 15 ( λ + µ ) (1 ν ) E C l = =, (-18) ρ (1 + ν )(1 ν ) ρ µ G E C s = = =, (-19) ρ ρ (1 + ν ) ρ where ν is Poisson s ratio. There exists some differences in the descriptions of wave propagation between the methods based upon kinematics of deformation and theory of elasticity. Equations (-1), (-16) and (-17) have similar PDE formations. However, for a longitudinal wave, the displacement in equation (-1) is replaced by voluminal strain, I, in equation (-16) for a shear wave, its displacement is replaced by rotation vector, w, in equation (-17).. Introduction to Waveguides..1 General Concept When a wave travels in a medium without bound or on an infinite string, it is called a free wave (Krautkrämer and Krautkrämer 199). Its physical behavior is described as that of waves in a medium without lateral boundaries. When a homogeneous, isotropic, elastic medium is not infinite, for example, a plate or

39 16 cylinder, the boundary imposes mechanical variables, such as stress and/or strain constraints, causing the waves to be reflected at the boundary, often yielding a change of wave type and their directions (Royer and Dieulesaint ). As an example, the waves in a traction-free plate (parallel free surfaces) can propagate by alternatively reflecting at the two surfaces. The oriented waves are called guided waves. Wave velocity = constant Wave velocity = f(frequency, geometric and boundary conditions) (a). (b). Figure - Schematic representation for (a) an ideally one-dimensional wave and (b) guided waves in a plate The different characteristics of an ideally one-dimensional wave and a guided wave in a plate are graphically demonstrated in Figure -. The assumption of a onedimensional wave propagation is usually made for conventional surface reflection techniques for use as quality control for drilled shafts, such as impact echo or impulse response tests. The wave velocity of an ideally one-dimensional wave is constant. Its displacement distribution in the direction of propagation is uniform, implying that the boundary condition does not affect the wave shape. However, the wave velocity of guided waves is a function of frequency, geometric conditions and boundary conditions. Their displacement distribution across the structure is a combination of

40 17 different types of wave shapes due to the reflections of dilational and transverse waves at the boundaries (Graff 1975; Redwood 196). Indeed, guided waves are threedimensional waves. Furthermore, for a specific medium shape with given boundary conditions, the wave velocity of guided waves is a function of frequency, and hence is dispersive. (a). (b). Figure -3 Guide waves in guiding systems: (a). two-dimensional and (b). threedimensional (after Auld 199) Another simple explanation for guided waves can be found in electromagnetism. As illustrated in Figure -3, guided waves can be imagined as energy flow mainly along the direction of guiding configuration or waveguide in a two- or threedimensional guiding system without geometric attenuation. The properties of waves in the guiding system can be altered, but their direction will be confined along the guiding system (Auld 199)... Characteristics of Guided Waves Existence of Guided Waves: In a broad view, guided waves exist in any finite

41 18 medium, including both elastic and inelastic materials. The difference for the guided waves in elastic or inelastic materials is that the waves in an elastic medium follow the theory of elasticity (Graff 1975) and those in an inelastic medium follow some kind of elastic-viscous constitutive law. In general, all media are dispersive, other than light in vacuum (Brillouin 196). This means that the wave velocity of a guided wave is a function of frequency, and is called dispersive. Solutions: For every guiding structure, an infinite number of different guided wave solutions are possible (Auld 199). These multiple guided wave solutions can be dispersive in a homogeneous, isotropic, elastic medium. Each of the solutions has a unique behavior in terms of dispersion relation, variations of velocity, and displacement and stress distributions. Solving Process: The theoretical derivation of guided waves depends on the dynamic equations of motion and certain boundary conditions. For instance, the analytic formulation for wave propagation in an infinite plate with finite thickness is a function of the wave type (i.e. longitudinal waves) and boundary conditions. An appropriate wave equation is simplified from equation (-7) and the boundary conditions are substituted into the wave equation to solve the undetermined coefficients. Finally, their dispersion relation and exact displacement and stress distributions can be derived in explicit forms.

42 19 Relation between Guided Waves and One-Dimensional Waves: Due to the limitation of most excited waves within a low and narrow frequency range associated with conventional NDT techniques used in deep foundations, the velocities in any finite elastic medium do not change much with frequency; therefore, wave velocity in these narrow ranges usually are regarded as a constant that does not change with frequency (Finno et al 1). The assumption of constant wave velocity is accepted (Kolsky 1963) and the concept of one-dimensional waves (based on kinematics of deformation), and not guided waves, is applied in general. Therefore, the case analyzed with the one-dimensional wave theory is one special case of guided wave theory, usually valid at low frequencies..3 Methods of Analysis The methods to solve wave propagation in a plate-like structure and solid cylinder are found in Auld (199), Popovics (1994), and Tolstoy and Usdin (1953). The details of the derivation process and corresponding formulations for different types of guided waves are published in Armenakas (1967), Chree (1889), Gazis (1959), Graff (1975), Hanifah (1999), Lamb (1917), Meeker and Meitzler (1964), Mindlin (196), Royer and Dieulesaint (), and Zemanek (196). Plate-like and cylindrical structures can be represented in rectangular coordinate, x y z, and orthogonal cylindrical coordinate, r θ z, systems, respectively (see Figure

43 C l, C s, λ, µ, ρ, α, β z x b y (a) C l, C s, λ, µ, ρ, α, β z x z a b (b) y C l, C s, λ, µ, ρ, α, β a C l, C s, λ, µ, ρ, α, β C l, C s, λ, µ, ρ, α, β soil θ r θ r C l, C s, λ, µ, ρ, α, β (c) z (d) Figure -4 Schematic representations for plate-like structures and solid cylindrical structures

44 1-4). The physical properties of the plate, half space, and cylindrical structure also are shown in the figure. In a structure composed of elastic material, C l is the longitudinal wave velocity, C s is the shear wave velocity, λ and µ are Lamé constants, ρ is the density, α is the wave number for longitudinal waves, and β is the wave number for shear waves. In an elastic half space, C l is the longitudinal wave velocity, C s is the shear wave velocity, λ and µ are Lamé constants, ρ is the density, α is the wave number for longitudinal waves, and β is the wave number for shear waves. Required geometric parameters include, b, half the thickness of a plate, and a, radius of a cylinder. Part A Displacement Potential Theory The Navier-Stokes equation for the displacement equations of motion in isotropic elastic media in the absence of body force is re-called: ( λ + µ ) u + µ u = ρ& u&. (-7) The Helmholtz s theorem is introduced by u= Φ + H, (-) where Φ is the dilational scalar potential and H is the curl of a equivoluminal vector potential (Phillips 1933). Substituting equation (-) into equation (-7) and regrouping gives:

45 Φ ( λ + µ ) Φ= ρ ; t (-1) µ ρ H H =. t (-) Rayleigh (1889) also introduced the concept of wave numbers to solve the Navier-Stokes equation. Lamb (194) formally applied this concept to solve the tremors over the surface of a half space. Then this concept was accepted by many scholars (Love 1911; Lamb 1917; Stoneley 194; Sezawa and Nishimura 198) and was used to study issues related to guided waves in media. At that time, these scholars regarded guided waves as free waves or free modes, where no energy leakage from a structure into the surrounding air. When a structure is attached to a half space or embedded in a space, attenuation due to leaky modes is seen in the form of outgoing waves from the plate into the half space (Lowe 1995). Direction of propagation α ω C l β ω C s ξ Figure -5 Schematic representations of real wave numbers for a longitudinal wave (left side) and a shear wave (right side) in a foundation structure ξ Real-valued wave numbers, α, β, and ξ are schematically shown in Figure -5 and related by:

46 3 ω + and α ξ = C l ω β + ξ = (-3), C s where α is the wave number for longitudinal waves, β is the wave number for shear waves, and ξ is the wave number in the direction of propagation. The functional forms of Φ and H in the rectangular coordinate, x y z, system are taken as an example and can be expressed as: i( ξx ωt ) Φ = ( A cosαy + A sinαy) e, (-4) H H H x y z 1 i( ξx ωt ) = ( A cos βy + A sin βy) e, (-5) 3 4 i( ξx ωt ) = ( A cos βy + A sin βy) e, (-6) 5 6 i( ξx ωt ) = ( A cos βy + A sin βy) e, (-7) 7 8 where A 1, A, A 3, A 4, A 5, A 6, A 7, and A 8 are the indeterminate coefficients. These functions are appropriate for a plate extending in the x direction and independent on the z direction, i.e., a plane-strain condition. Based on a plane-strain condition, / z =, three displacement and stress components related to the boundary conditions are given by: Φ H z u = +, x y (-8) Φ H z v =, y x (-9) H H x y w = +. y x (-3)

47 4 v u σ yy = ( λ + µ ) + λ, y x (-31) v u σ yx = µ ( + ), x y (-3) w σ yz = µ. y (-33) Substituting the potentials (-4) to (-7) into displacement and stress component equations (-8) to (-33) permit the displacement and stress components to be expressed in terms of wave numbers and indeterminate coefficients. The boundary conditions for a plate-like structure at y = ±b depend upon what kind of wave travels in the plate and what kind of mechanical restriction exists at the medium interface. These boundary conditions can include stress components or displacement components, or both. After the boundary conditions and divh = are fulfilled, the dispersion relations can be determined. Part B Superposition of Partial Wave Techniques An alternative method, the Superposition of Partial Wave Technique, to obtain the dispersion relations of wave propagation in a plate-like structure was initially developed in the early 196 s. The guided waves are constructed as hybrid waves consisting of longitudinal waves and shear waves (see Figure -6). The relations between incident waves and reflected waves are based on the reflection and refraction equations at the boundary.

48 5 b y x b + b Longitudinal (L) Partial Wave Vertical Shear (SV) Partial Wave Figure -6 Partial wave patterns for the vertical shear waves and longitudinal waves in a traction-free plate (after Auld 199) The example in Figure -6 consists of the well-known Lamb waves in a plate, pertaining to the two coupling longitudinal (L) and vertical shear (SV) waves. The guided waves travel along the x direction and constitute a resonance in the transverse direction (y direction) of the plate. The incident longitudinal waves and vertical shear waves interact with the boundaries of the plate and yield reflected longitudinal waves and vertical shear waves. The superposition of these incident and reflected

49 longitudinal and shear waves yields guided waves in a plate as shown at top of Figure The advantages to this approach include simple solution techniques, insight into the physical nature of the waves, and applicability to waves in an anisotropic free plate or an isotropic plate over an anisotropic half space (Solie and Auld 1973; Auld 199). On the other hand, the main disadvantage of the superposition of partial wave technique is that it provides no information about the stress and displacement distributions in a plate. This information is quite important to an experimentalist when deciding where to place transducers to measure the effects of guided waves..4 Phase Velocity and Group Velocity The dispersion relation of guided waves can be presented as curves in a plot of non-dimensional frequency versus non-dimensional wave number. From these curves, the phase velocity and group velocity can be derived. The fundamental definitions of phase velocity, C p, non-dimensional phase velocity, C p, group velocity, C g, and nondimensional group velocity, C g, are given as: C p = ω C p ω ; C p, Re( ξ ) = C = C Re( ξ ) (-34) s s

50 7 C g = dω Cg dω ; Cg, Re( dξ ) = C = C Re( dξ ) (-35) s s where ξ is the wave number, ω is the angular frequency. A plot of angular frequency versus wave number (see Figure -7) shows the relation between group velocity and phase velocity. A dispersion curve is plotted as a solid curve. The dotted and dashed lines represent the secant and tangent at points on the dispersion curve, respectively. The dotted line indicates the group velocity with gradient dω / d Re( ξ ) and the dashed line indicates the phase velocity with secant ω / Re( ξ ). ω B A Phase velocity Group velocity Re(ξ) Figure -7 Frequency spectrum Group velocity plays an important role in experiments because it is the velocity of energy transportation (Rose 1999) and important physical quality from a measurement perspective (Chao ). A simple formulation was presented by Havelock (1914) and Brillouin (196), wherein the energy velocity, C e, is defined as the ratio of energy flux per average period I to energy density, E d (Brillouin 1953):

51 8 Ce = I / Ed, (-36) Achenbach (1973) gave a complete and rigorous proof for the final solution C e ω dω = Cg =. ξ dξ (-37).5 Guided Waves in Half Spaces and Plate-Like Structures.5.1 Rayleigh Wave The simplest guided wave, a Rayleigh wave can occur on one free surface (Royer and Dieulesaint ). The particle path in a Rayleigh wave has the shape of an elliptical rotation (see Figure -8). If a Rayleigh wave propagation moves from the left to the right, the direction of its particle rotation is counterclockwise. The two displacement components (x and y directions) have a phase difference of π/, a combination of an x-direction wave and a y-direction wave, and remain on the Sagittal plane (Royer and Dieulesaint ). The Sagittal plane is defined as a plane perpendicular to the surface of a half space and parallel to the direction of wave propagation. The amplitudes decrease with depth at different rates. Their wave type changes into a transverse wave at about the depth of.λ and vanishes at the depth of

52 λ (see Figure -8 (a)). 9 Sagittal plane Free surface x z Direction of propagation air x Half space y (a). (b). Figure -8 Rayleigh wave: (a). Schematic representation (after Royer and Dieulesaint ) and (b). Wave on Sagittal plane (after Krautkrämer and Krautkrämer 199) The velocity of a Rayleigh wave in an elastic half space, always somewhat slower than a shear wave, is free of velocity dispersion and approximately is given as (Krautkrämer and Krautkrämer 199): C R ν G ν = 1+ ν ρ 1+ ν C s. (-38) The Rayleigh wave velocity is 87 to 95 percent of the shear wave velocity.

53 .5. SH-Mode Waves in a Free Plate 3 A horizontal shear (SH) wave occurs only when particle displacement is parallel to the z direction and out of the Sagittal plane (Figure -4 (a)). The derived dispersion relation for SH-mode waves is (Meeker and Meitzler 1964; Mindlin 196): Ω = n + ξ, as Ω= bω bξ ; ξ ; πc = π n=natural numbers and zero (-39) s Non-dimensional phase and group velocity curves are plotted in Figure -9. The first order is non-dispersive, whereas all the other higher orders are dispersive since their velocity is a function of frequency non-dimensional phase velocity, C p n= n=1 n= n=3 n=4 n=5 non-dimensional group velocity, Cg non-dimensional frequency, Ω non-dimensional frequency, Ω Figure -9 Variations of non-dimensional phase velocity and group velocity with nondimensional frequency for first six orders of the SH-mode waves in a traction-free plate

54 .5.3 Lamb Wave 31 A wave propagating in a traction-free plate (see Figure -4 (a)), called Rayleigh- Lamb waves (or Lamb waves), was independently found by Rayleigh in 1889 and Lamb in When the thickness of the plate becomes comparable to wavelength, its longitudinal and vertical shear components are coupled (Royer and Dieulesaint ). Particle path Neutral plane (a) Direction of propagation Particle path Neutral plane (b) Direction of propagation Figure -1 Lamb wave: (a). Anti-symmetric mode and (b). Symmetric mode (adapted after Krautkrämer and Krautkrämer 199) There are two kinds of Lamb waves: anti-symmetrical and symmetrical modes (see Figure -1). The deformation of the anti-symmetric mode is anti-symmetrical to the neutral plane of the plate (see Figure -1 (a)). The particle path of the antisymmetric mode has the shape of either elliptical rotation or purely vertical movements. Particles on the neutral plane are subjected to pure transverse

55 oscillations. The other particles have elliptical oscillations. Periodic flexing of the boundaries occurs in a plate (Auld 199). 3 The deformation of the symmetric mode is symmetrical to the neutral plane of the plate (see Figure -1 (b)). The particle path of the symmetric mode has the shape of either elliptical rotation or purely horizontal movements. Particles on the neutral plane are subjected to pure longitudinal oscillations. The other particles have elliptical oscillations. The boundaries of the plate periodically dilate and contract (Auld 199). L Lamé S R Figure -11 Dispersion curves for Lamb waves in a plate (fine solid lines represent uncoupled longitudinal waves and vertical shear waves; thick solid lines represent the symmetric mode; thick dashed lines represent the anti-symmetric modes; L- longitudinal wave line; S- shear wave line; R- Rayleigh wave line; Lamé- Lamé modes line; Poisson s ratio, ν =.31; k = 1.91) (after Mindlin 196) The dispersion solution of Lamb waves is expressed as (Meeker and Meitzler

56 1964; Mindlin 196): 33 tan βb 4αβξ ± = { } 1, tanαb ( ξ β ) (-4) where +1 is for the symmetric mode and 1 is for the anti-symmetric mode. The dispersion curves on Figure -11 are functions of frequency, wave number and Poisson s ratio. All modes are dispersive (see Figure -1) since their phase and group velocities are a function of frequency. The velocities of the two lowest orders approach the Rayleigh wave velocity at high frequencies. The velocities of the higher orders approach the shear wave velocity at high frequencies. C plate C s C R (a) C plate (b) Figure -1 Variation of (a) group velocity and (b) phase velocity with modified frequency for Lamb waves in a plate (S indicates symmetric; A indicates antisymmetric) (after Rose 1999) C R C l C s

57 Love Wave A horizontal shear (SH) wave propagating in an isotropic plate (film or layer) attached at a half space (substrate) is called a Love wave, found by Love in 1911 and verified by many researchers in geophysical applications (Ewing et al 1957; Sezawa and Kanai 1935). A Love wave must satisfy two conditions: (1) the shear wave velocity in the plate, C s, should be less than the shear wave velocity in the half space, C s, and () the wavelength should be short compared with the thickness of the plate. Its energy will propagate in both the layer and the substrate (see Figures -13). x Free surface Layer z Substrate y Figure -13 Displacement distribution of a Love wave (after Royer and Dieulesaint ) All dispersion curves start at the shear wave line in the half space and asymptotically approach the shear wave line in the plate (see Figures -14 (a)). Both

58 35 group and phase velocities start at the shear wave velocity in the half space at low frequencies (see Figure -14 (b)). At high frequencies, both velocities approach the shear wave velocity in the plate, and displacement will concentrate in the layer at high frequencies (Royer and Dieulesaint ). non-dimesional frequency, Ω plate shear wave half space shear wave n= n=1 n= n=3 n= non-dimensional wave number, ξ non-dimensional group velocity, Cg; non-dimensional phase velocity, Cp n= group velocity n= phase velocity n=1 group velocity n=1 phase velocity n= group velocity n= phase velocity C s non-dimensional frequency, Ω (a). (b). Figure -14 Love waves: (a). Dispersion curves and (b). Variation of non-dimensional group velocity and non-dimensional phase velocity with non-dimensional frequency for first five orders C s.5.5 Generalized Lamb Wave A generalized Lamb wave propagating in an isotropic plate attached at a half space, similar to a Lamb wave, is composed with coupling longitudinal (L) waves and vertical shear (SV) waves (Figure -4 (b)). Many scholars (Achenbach and Epstein 1967; Achenbach and Keshava 1967; Ewing et al. 1957; Kanai 1951; Sezawa and

59 36 Kanai 1935; Tolstoy and Usdin 1953) presented the dispersion relations for different material properties and boundary conditions. There are several types of solutions for the generalized Lamb waves in a plate. The classification depends upon the relative relations among the shear wave velocities and Rayleigh wave velocities of both the plate and half space. Achenbach and Epstein (1967) solved the dispersion relation in a transcendental form in a coating film and the solution implies that generalized Lamb waves are dispersive..5.6 Guided Waves in Multiple Layers Guided waves propagating in multiple layers have been studied since the late 19 s. Two common types have application to seismology: a weak stratum embedded two hard half spaces (Sezawa and Nishimura 198; Sezawa and Kanai 1939) and a layered half space submerged in seawater (Jardetzky and Press 1953). The two lowest orders of phase velocity were developed and applied to interpret earthquake records. The analysis of multiple-layered media was first theoretically introduced by Thomson in 195 and improved by Haskell in A transfer matrix describes the velocity and stress components at the bottom of a layer with respect to those at the top of the layer. Then the velocity and stress components at the bottom of a multiplelayered system can be related to those at the top of a multiple-layered system (Lowe

60 1995) Stoneley Wave A Stoneley wave (Stoneley 194) is a Rayleigh-like wave propagating along the interface of two half spaces or an interface wave whose wavelength is much smaller than the dimensions of two finite media. The displacement components decrease on the either side of the interface. In fact, Stoneley waves could be regarded as a generalized non-dispersive Rayleigh wave (Stoneley 194). The velocity of a Stoneley wave is determined by the Lamé constants of the two media. Stoneley waves only occur at an interface when the ratio of shear wave velocity in the two media is.97 to.99 (Auld 199; Rose 1999)..6 Guided Waves in Solid Cylinders.6.1 Guided Waves in Traction-Free Solid Cylinders The frequency equations of the longitudinal and torsional waves propagating in a solid cylinder (see Figure -4 (c)) were independently published by Pochhammer in 1876 and Chree in 1889 based on the method of theory of elasticity. Hudson (1943) solved the first branch of the first circumferential order of the flexural mode waves.

61 38 Then Abramson (1956, 1957) extended the solutions to the third branch. In the early 196 s, the complete solutions were graphically developed (Pao and Mindlin 196; Pao 196; Zemanek 196; Meeker and Meitzler 1964). Zemanek published both theoretical formulations and results of experimental investigations of the torsional, longitudinal, and flexural mode waves in a solid cylinder. The dispersion relations for the torsional, longitudinal, and flexural mode waves are given, respectively, as: β aj βa) = J ( β ), (-41) ( 1 a α ( β a + ξ ) J ( αa) J ( βa) ( β ξ ) J ( αa) J ( βa) 4ξ αβj ( αa) J ( βa) =, (-4) and λ α ξ α α [ ( + ) a + ( a p )] Jp( a) µ ( β a p ) J p( βa) + βaj p( βa) p[ βaj p( βa) J p( βa)] + αaj ( αa) p pαaj αa J αa p J βa + βaj βa p β a J αa + βaj αa = [ p( ) p( )] [ p( ) p( )] ( ) p( ) p( ), ξ β αaj p( αa) ( ) βaj ( ) ( ) p βa pj p βa ξ (-43) where p is the circumferential order of a solid cylinder. The torsional mode dispersion relation, T(,q), where represents torsion and q is the branch number, is a function of frequency and wave number, and only has an

62 angular displacement component, u θ. Its dispersion curves, group velocity and phase velocity are similar to the SH-mode waves in a plate (Meeker and Meitzler 1964). 39 L S R Ω* (a) Imaginary ξ* L Real ξ* S R Ω* (b) Imaginary ξ* Real ξ* Figure -15 Dispersion curves for (a) L(,q) and (b) F(1,q) in a solid cylinder (solid lines represent the real or imaginary; dashed line represent complex; Poisson s ratio, ν=.3317) (after Zemanek 197)