Introduction E Mch 521, ACS 521 Stress Waves in Solid Media 3 credit Graduate Course Penn State University Instructors: Dr. Joseph L. Rose Dr. Cliff Lissenden Textbook: Ultrasonic Guided Waves in Solid Media Joseph L. Rose – Cambridge University Press – 2014

Preface  Text: Ultrasonic Waves in Solid Media, 1999  Nondestructive Evaluation  Structural Health Monitoring  Growth of Guided Waves – 1985 to 2014  Publications to 2000 and beyond  University involvement (2 to 40)  Commercialization – piping example  ASNT – working on inspection certification, a new method  ASME, DOT – code requirements/developments 2

Table of Contents Nomenclature Preface Acknowledgments 1. Introduction 1.1 Background 1.2 A Comparison of Bulk versus Guided Waves 1.3 What Is an Ultrasonic Guided Wave? 1.4 The Difference Between Structural Health Monitoring (SHM) and Nondestructive Testing (NDT) 1.5 Text Preview 1.6 Concluding Remarks 1.7 References 3

Table of Contents cont. 2. Dispersion Principles 2.1 Introduction 2.2 Waves in a Taut String Governing Wave Equation Solution by Separation of Variables D’Alembert’s Solution Initial Value Considerations 2.3 String on an Elastic Base 2.4 A Dispersive Wave Propagation Sample Problem 2.5 String on a Viscous Foundation 2.6 String on a Viscoelastic Foundation 2.7 Graphical Representations of a Dispersive System 2.8 Group Velocity Concepts 2.9 Exercises 2.10 References 4

Table of Contents cont. 3. Unbounded Isotropic and Anisotropic Media 3.1 Introduction 3.2 Isotropic Media Equations of Motion Dilatational and Distortional Waves 3.3 The Christoffel Equation for Anisotropic Media Sample Problem 3.4 On Velocity, Wave, and Slowness Surfaces 3.5 Exercises 3.6 References 5

Table of Contents cont. 4. Reflection and Refraction 4.1 Introduction 4.2 Normal Beam Incidence Reflection Factor 4.3 Snell’s Law for Angle Beam Analysis 4.4 Critical Angles and Mode Conversion 4.5 Slowness Profiles for Refraction and Critical Angle Analysis 4.6 Exercises 4.7 References 6

Table of Contents cont. 5. Oblique Incidence 5.1 Introduction 5.2 Reflection and Refraction Factors Solid-Solid Boundary Conditions Solid-Liquid Boundary Conditions Liquid-Solid Boundary Conditions 5.3 Moving Forward 5.4 Exercises 5.5 References 7

Table of Contents cont. 6. Waves in Plates 6.1 Introduction 6.2 The Free Plate Problem Solution by the Method of Potentials The Partial Wave Technique 6.3 Numerical Solution of the Rayleigh-Lamb Frequency Equations 6.4 Group Velocity 6.5 Wave Structure Analysis 6.6 Compressional and Flexural Waves 6.7 Miscellaneous Topics Lamb Waves with Dominant Longitudinal Displacements Zeros and Poles for a Fluid-Coupled Elastic Layer Mode Cutoff Frequency 6.8 Exercises 6.9 References 8

Table of Contents cont. 7. Surface and Subsurface Waves 7.1 Background 7.2 Surface Waves 7.3 Generation and Reception of Surface Waves 7.4 Subsurface Longitudinal Waves 7.5 Exercises 7.6 References 9

Table of Contents cont. 8. Finite Element Method for Guided Wave Mechanics 8.1 Introduction 8.2 Overview of the Finite Element Method Using the Finite Element Method to Solve a Problem Quadratic Elements Dynamic Problem Error Control 8.3 FEM Applications for Guided Wave Analysis D Surface Wave Generation in a Plate Guided Wave Defect Detection in a Two-Inch Steel Tube 8.4 Summary 8.5 Exercises 8.6 References 10

Table of Contents cont. 9. The Semi-Analytical Finite Element Method 9.1 Introduction 9.2 SAFE Formulation for Plate Structures 9.3 Orthogonality-Based Mode Sorting 9.4 Group Velocity Dispersion Curves 9.5 Guided Wave Energy Poynting Vector Energy Velocity Skew Effects in Anisotropic Plates 9.6 Solution Convergence of the SAFE Method 9.7 Free Guided Waves in an Eight-Layer Quasi-Isotropic Plate 9.8 SAFE Formulation for Cylindrical Structures 9.9 Summary 9.10 Exercises 9.11 References 11

Table of Contents cont. 10. Guided Waves in Hollow Cylinders 10.1 Introduction 10.2 Guided Waves Propagating in an Axial Direction Analytic Calculation Approach Excitation Conditions and Angular Profiles Source Influence 10.3 Exercises 10.4 References 12

Table of Contents cont. 11. Circumferential Guided Waves 11.1 Development of the Governing Wave Equations for Circumferential Waves Circumferential Shear Horizontal Waves in a Single-Layer Annulus Circumferential Lamb [Type] Waves in a Single-Layer Annulus 11.2 Extension to Multiple-Layer Annuli 11.3 Numerical Solution of the Governing Wave Equations for Circumferential Guided Waves Numerical Results for CSH-Waves Numerical Results for CLT-Waves Computational Limitations of the Analytical Formulation 11.4 The Effects of Protective Coating on Circumferential Wave Propagation in Pipe 11.5 Exercises 11.6 References 13

Table of Contents cont. 12. Guided Waves in Layered Structures 12.1 Introduction 12.2 Interface Waves Waves at a Solid-Solid Interface: Stoneley Wave Waves at a Solid-Liquid Interface: Scholte Wave 12.3 Waves in a Layer on a Half Space Rayleigh-Lamb Type Waves Love Waves 12.4 Waves in Multiple Layers The Global Matrix Method The Transfer Matrix Method Examples 12.5 Fluid Couples Elastic Layers Ultrasonic Wave Reflection and Transmission Leaky Guided Wave Modes Nonspecular Reflection and Transmission 12.6 Exercises 12.7 References 14

Table of Contents cont. 13. Source Influence on Guided Wave Excitation 13.1 Introduction 13.2 Integral Transform Method A Shear Loading Example 13.3 Normal Mode Expansion Method Normal Mode Expansion in Harmonic Loading Transient Loading Source Influence 13.4 Exercises 13.5 References 15

Table of Contents cont. 14. Horizontal Shear 14.1 Introduction 14.2 Dispersion Curves 14.3 Phase Velocities and Cutoff Frequencies 14.4 Group Velocity 14.5 Summary 14.6 Exercises 14.7 References 16

Table of Contents cont. 15. Guided Waves in Anisotropic Media 15.1 Introduction 15.2 Phase Velocity Dispersion 15.3 Guided Wave Directional Dependency 15.4 Guided Wave Skew Angle 15.5 Guided Waves in Composites with Multiple Layers 15.6 Exercises 15.7 References 17

Table of Contents cont. 16. Guided Wave Phased Arrays in Piping 16.1 Introduction 16.2 Guided Wave Phased Array Focus Theory 16.3 Numerical Calculations 16.4 Finite Element Simulation of Guided Wave Focusing 16.5 Active Focusing Experiment 16.6 Guided Wave Synthetic Focus 16.7 Synthetic Focusing Experiment 16.8 Summary 16.9 Exercises References 18

Table of Contents cont. 17. Guided Waves in Viscoelastic Media 17.1 Introduction 17.2 Viscoelastic Models Material Viscoelastic Models Kelvin-Voight Model Maxwell Model Further Aspects of the Hysteretic and Kelvin-Voight Models 17.3 Measuring Viscoelastic Parameters 17.4 Viscoelastic Isotropic Plate 17.5 Viscoelastic Orthotropic Plate Problem Formulation and Solution Numerical Results Summary 17.6 Lamb Waves in a Viscoelastic Layer 17.7 Viscoelastic composite Plate 17.8 Pipes with Viscoelastic Coatings 17.9 Exercises References 19

Table of Contents cont. 18. Ultrasonic Vibrations 18.1 Introduction 18.2 Practical Insights into the Ultrasonic Vibrations Problem 18.3 Concluding Remarks 18.4 Exercises 18.5 References 20

Table of Contents cont. 19. Guided Wave Array Transducers 19.1 Introduction 19.2 Analytical Development Linear Comb Array Solution Annular Array Solution 19.3 Phased Transducer Arrays for Mode Selection Phased Array Analytical Development Phased Array Analysis 19.4 Concluding Remarks 19.5 Exercises 19.6 References 21

Table of Contents cont. 20. Introduction to Guided Wave Nonlinear Methods 20.1 Introduction 20.2 Bulk Waves in Weakly Nonlinear Elastic Media 20.3 Measurement of the Second Harmonic 20.4 Second Harmonic Generation Related to Microstructure 20.5 Weakly Nonlinear Wave Equation 20.6 Higher Harmonic Generation in Plates Synchronism Power Flux Group Velocity Matching Sample Laboratory Experiments 20.7 Applications of Higher Harmonic Generation by Guided Waves 20.8 Exercises 20.9 References 22

Table of Contents cont. 21. Guided Wave Imaging Methods 21.1 Introduction 21.2 Guided Wave through Transmission Dual Probe Imaging 21.3 A Defect Locus Map 21.4 Guided Wave Tomographic Imaging 21.5 Guided Wave Phased Array in Plates 21.6 Long-Range Ultrasonic Guided Wave Pipe Inspection Images 21.7 Exercises 21.8 References 23

Table of Contents cont. A.1 Physical Principles A.2 Wave Interference A.3 Computational Model for a Single Point Source A.4 Directivity Function for a Cylindrical Element A.5 Ultrasonic Field Presentations A.6 Near-Field Calculations A.7 Angle-of-Divergence Calculations A.8 Ultrasonic Beam Control A.9 A Note of Ultrasonic Field Solution Techniques A.10 Time and Frequency Domain Analysis A.11 Pulsed Ultrasonic Field Effects A.12 Introduction to Display Technology A.13 Amplitude Reduction of an Ultrasonic Waveform A.14 Resolution and Penetration Principles A.14.1 Axial Resolution A.14.2 Lateral Resolution A.15 Phase Arrays and Beam Focusing A.16 Exercises A.17 References Appendix A – Ultrasonic Nondestructive Testing Principles, Analysis, and Display Technology 24

Table of Contents cont. Appendix B – Basic Formulas and Concepts in the Theory of Elasticity B.1 Introduction B.2 Nomenclature B.3 Stress, Strain, and Constitutive Equations B.4 Elastic Constant Relationships B.5 Vector and Tensor Transformation B.6 Principal Stresses and Strains B.7 The Strain Displacement Equations B.8 Derivation of the Governing Wave Equation B.9 Anisotropic Elastic Constants B.10 References 25

Table of Contents cont. Appendix C – Physically Based Signal Processing Concepts for Guided Waves C.1 General Concepts C.2 The Fast Fourier Transform (FFT) C.2.1 Example FFT Use: Analytic Envelope C.2.2 Example FFT Use: Feature Source for Pattern Recognition C.2.3 Discrete Fourier Transform Properties C.3 The Short Time Fourier Transform (STFFT) C.3.1 Example: STFFT to Dispersion Curves C.4 The 2-D Fourier Transform (2DFFT) C.5 The Wavelet Transform (WT) C.6 Exercises C.7 References 26

Table of Contents cont. Appendix D – Guided Wave Mode and Frequency Selection Tips D.1 Introduction D.2 Mode and Frequency Selection Considerations D.2.1 A Surface-Breaking Defect D.2.2 Mild Corrosion and Wall Thinning D.2.3 Transverse Crack Detection in the Head of a Rail D.2.4 Repair Patch Bonded to an Aluminum Layer D.2.5 Water-Loaded Structures D.2.6 Frequency and Other Tuning Possibilities D.2.7 Ice Detection with Ultrasonic Guided Waves D.2.8 Deicing D.2.9 Real Time Phased Array Focusing in Pipe D.2.10 Aircraft, Lap-Splice, Tear Strap, and Skin to Core Delamination Inspection Potential D.2.11 Coating Delamination and Axial Crack Detection D.2.12 Multilayer structures D.3 Exercises D.4 References 27

Background  Preface  To start now with Chapter 1. Let’s see a few references first, of many listed in the book after each chapter.  References Achenbach, J. D. (1976). Generalized continuum theories for directionally reinforced solids, Arch. Mech. 28(3): 257–78. Achenbach, J. D. (1984). Wave Propagation in Elastic Solids. New York: North-Holland. Achenbach, J. D. (1992). Mathematical modeling for quantitative ultrasonics, Nondestr. Test. Eval. 8/9: 363–77. Achenbach, J. D., and Epstein, H. I. (1967). Dynamic interaction of a layer of half space, J. Eng. Mech. Division 5: 27–42. Achenbach, J. D., Gautesen, A. K., and McMaken, H. (1982). Ray Methods for Waves in Elastic Solids. Boston, MA: Pitman. Achenbach, J. D., and Keshava, S. P. (1967). Free waves in a plate supported by a semi- infinite continuum, J. Appl. Mech. 34: 397–404. Auld, B. A. (1990). Acoustic Fields and Waves in Solids. 2nd ed., vols. 1 and 2. Malabar, FL: Krieger. Auld, B. A., and Kino, G. S. (1971). Normal mode theory for acoustic waves and their application to the interdigital transducer, IEEE Trans. ED-18: 898–

Background cont. Auld, B. A., and Tau, M. (1978). Symmetrical Lamb wave scattering at a symmetrical pair of thin slots, in 1977 IEEE Ultrasonic Sympos. Proc. vol. 61. Beranek, L. L. (1990). Acoustics. New York: Acoustical Society of America, American Institute of Physics. Davies, B. (1985). Integral Transforms and Their Applications. 2nd ed. New York: Springer-Verlag. Eringen, A. C., and Suhubi, E. S. (1975). Linear Theory (Elastodynamics, vol. 2). New York: Academic Press. Ewing, W. M., Jardetsky, W. S., and Press, F. (1957). Elastic Waves in Layered Media. New York: McGraw-Hill. Federov, F. I. (1968). Theory of Elastic Waves in Crystals. New York: Plenum. Graff, K. F. (1991). Wave Motion in Elastic Solids. New York: Dover. Kino, C. S. (1987). Acoustic Waves: Devices, Imaging and Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall. Kinsler, L. E., Frey, A. R., Coppens, A. B., and Sanders, J. V. (1982). Fundamentals of Acoustics. New York: Wiley. Kolsky, H. (1963). Stress Waves in Solids. New York: Dover. Love, A. E. H. (1926). Some Problems of Geodynamics. Cambridge University Press. Love, A. E. H. (1944a). Mathematical Theory of Elasticity. 4th ed. New York: Dover. 29

Background cont. Love, A. E. H. (1944b). A Treatise on the Mathematical Theory of Elasticity. New York: Dover. Miklowitz, J. (1978). The Theory of Elastic Waves and Waveguides. New York: North-Holland. Mindlin, R. D. (1955). An Introduction to the Mathematical Theory of Vibrations of Elastic Plates. Fort Monmouth, NJ: U.S. Army Signal Corps Engineers Laboratories. Musgrave, M. J. P. (1970). Crystal Acoustics. San Francisco, CA: Holden-Day. Pollard, H. F. (1977). Sound Waves in Solids. London: Pion Ltd. Rayleigh, J. W. S. (1945). The Theory of Sound. New York: Dover. Redwood, M. (1960). Mechanical Waveguides. New York: Pergamon. Rose, J. L. (1999). Ultrasonic Waves in Solid Media. Cambridge University Press. Rose, J. L. (2002). A baseline and vision of ultrasonic guided wave inspection potential, Journal of Pressure Vessel Technology 124: 273–82. Stokes, G. G. (1876). Smith’s prize examination, Cambridge. [Reprinted 1905 in Mathematics and Physics Papers vol. 5, p. 362, Cambridge University Press.] Viktorov, I. A. (1967). Rayleigh and Lamb Waves – Physical Theory Applications. New York: Plenum. 30

Major Contributors  Michael Avioli  Cody Borigo  Jason Bostron  Huidong Gao  Cliff Lissenden  Yang Liu  Vamshi Chillara  Jing Mu  Jason Van Velsor  Fei Yan  Li Zhang Dedication: Aleksander Pilarski 31

Wave propagation studies are not limited to NDT and SHM, of course. Many major areas of study in elastic wave analysis are under way, including: (1) transient response problems, including dynamic impact loading; (2) stress waves as a tool for studying mechanical properties, such as the modulus of elasticity and other anisotropic constants and constitutive equations (the formulas relating stress with strain and/or strain rate can be computed from the values obtained in various, specially designed, wave propagation experiments); (3) industrial and medical ultrasonics and acoustic-emission nondestructive testing analysis; (4) other creative applications, for example, in gas entrapment determination in a pipeline, ice detection, deicing of various structures, and viscosity measurements of certain liquids; and (5) ultrasonic vibration studies that combine traditional low-frequency vibration analysis tools in structural analysis with high-frequency ultrasonic analysis. 32

Figure 1-1: Comparison of bulk wave and guided wave inspection methods. 33

BULKGUIDED Phase VelocitiesConstantFunction of frequency Group VelocitiesSame as phase velocities Generally not equal to phase velocity Pulse ShapeNon-dispersiveGenerally dispersive Table 1.1 – Ultrasonic Bulk vs. Guided Wave Propagation Considerations 34

The principal advantages of using ultrasonic guided waves analysis techniques can be summarized as follows. Inspection over long distances, as in the length of a pipe, from a single probe position is possible. There’s no need to scan the entire object under consideration; all of the data can be acquired from the single probe position. Often, ultrasonic guided wave analysis techniques provide greater sensitivity, and thus a better picture of the health of the material, than data obtained in standard localized normal beam ultrasonic inspection or other NDT techniques, even when using lower frequency ultrasonic guided wave inspection techniques. Continued on next slide… 35

Continued from previous slide… The ultrasonic guided wave analysis techniques allow the inspection of hidden structures, structures under water, coated structures, structures running under soil, and structures encapsulated in insulation and concrete. The single probe position inspection using wave structure change and wave propagation controlled mode sensitivity over long distances makes these techniques ideal. Guided wave propagation and inspection are cost-effective because the inspection is simple and rapid. In the example described earlier, there would be no need to remove insulation or coating over the length of a pipe or device except at the location of the transducer tool. 36

ISOTROPICANISOTROPIC Wave Velocities Not function of launch direction Function of launch direction Skew AnglesNoYes Table 1.2 – Ultrasonic Wave Considerations for Isotropic vs. Anisotropic Media 37

Bulk WaveGuided Wave Tedious and time consumingFast Point by point scan (accurate rectangular grid scan) Global in nature (approximate line scan) Unreliable (can miss points)Reliable (volumetric coverage) High level training required for inspectionMinimal training Fixed distance from reflector required Any reasonable distance from reflector acceptable Reflector must be accessible and seenReflector can be hidden Table A Comparison of the Currently Used Ultrasonic Bulk Wave Technique and the Proposed Ultrasonic Guided Wave Procedure for Plate and Pipe Inspection 38

Table 1.4. Natural Waveguides  Plates (aircraft skin)  Rods (cylindrical, square, rail, etc.)  Hollow cylinder (pipes, tubing)  Multi-layer structures  An interface  Layer or multiple layers on a half-space 39

Table 1.5. The Difference between SHM and Non-Destructive Testing (NDT) 40 NDT Off-line evaluation Time base maintenance Find existing damage More cost and labor Baseline not available SHM On-line evaluation Condition based maintenance Determine fitness-for-service and remaining useful time Less cost and labor Baseline required Environmental data compensation methods are required

Increased computational efficiency developments and Understanding Basic Principles Phased Array and Focusing developments in plates and pipes Demonstration of optimal mode and frequency selections for penetration power, fluid loading influences, and other defect detection sensitivity requirements Table 1.6. Successes – Guided Waves in General 41

Understanding guided wave behavior in anisotropic media ( Slowness profiles and Skew angle influence) Development of ultrasonic guided wave tomographic imaging methods Comb sensor designs for optimal mode and frequency selection (linear comb and annular arrays) Table 1.7 Successes – Composite Materials 42

Demonstration of feasibility studies in composites and lap splice, tear strap, skin to core delamination, corrosion detection and other applications. Table 1.8 Successes – Aircraft Applications 43

Understanding and utilization of both axisymmetric and non-axisymmetric modes Achieving excellent penetration power with special sensors, focusing, and mode and frequency choices Handling fluid loading with Torsional Modes Defect sizing accomplishments to less than 5% cross sectional area Reduced false alarm calls in inspection due to focusing for confirmation Circumferential location and length of defect estimations with focusing Testing of Pipe under insulation, coatings, and/or soil Table 1.9 Successes – Pipe Inspection 44

Modeling accuracy is critically dependent on accurate input parameters often difficult to obtain – (especially for anisotropic and viscoelastic properties, interface conditions, and defect characteristics.) Signal interpretations often difficult (due to multimode propagation and mode conversion, along with special test structure geometric features) Sensor robustness to environmental situations like temperature, humidity to high stress, mechanical vibrations, shock and radiation Adhesive bonding challenges for mounting sensors and sustainability in an SHM environment Merger of guided wave developments with energy harvesting and wireless technology Penetration power requirements Table 1.10 Practical Challenges – Guided Waves in general 45

Dealing with complex anisotropy and wave velocity and skew angle as a function of direction Viscoelastic influences Penetration power due to anisotropy, viscoelasticity, and inhomogeneity Differentiating critical composite damage such as delamination defects from structural variability during fabrication (including minor fiber misalignments, ply-drops, inaccurate fiber volume fraction, and so on) Guided wave inspection of composites with unknown material properties. Table 1.11 Practical Challenges – Composite Materials 46

Robustness of guided wave sensors under in-flight conditions Influences of aircraft paint and embedded metallic mesh in composite airframes for lightning protection Table 1.12 Practical Challenges – Aircraft Applications 47

Tees, elbows, bends, and number of elbows and inspection beyond elbows Quantification in defect location, characterization, sizing, especially depth determination Inspection reliability and false alarms (due to multimode propagation, mode conversions, and so many pipe features like welds, branches, etc.) Reducers, expanders, unknown layout drawings, cased pipes and sleeves Table 1.13 Challenges – Pipe Inspection 48