Introduction
E Mch 521, ACS 521Stress 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
2 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
3 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
4 Table of Contents cont.
2. Dispersion Principles
2.1 Introduction
2.2 Waves in a Taut String
2.2.1 Governing Wave Equation
2.2.2 Solution by Separation of Variables
2.2.3 D’Alembert’s Solution
2.2.4 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
5 Table of Contents cont.
3. Unbounded Isotropic and Anisotropic Media
3.1 Introduction
3.2 Isotropic Media
3.2.1 Equations of Motion
3.2.2 Dilatational and Distortional Waves
3.3 The Christoffel Equation for Anisotropic Media
3.3.1 Sample Problem
3.4 On Velocity, Wave, and Slowness Surfaces
3.5 Exercises
3.6 References
6 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
7 Table of Contents cont.
5. Oblique Incidence
5.1 Introduction
5.2 Reflection and Refraction Factors
5.2.1 Solid-Solid Boundary Conditions
5.2.2 Solid-Liquid Boundary Conditions
5.2.3 Liquid-Solid Boundary Conditions
5.3 Moving Forward
5.4 Exercises
5.5 References
8 Table of Contents cont.
6. Waves in Plates
6.1 Introduction
6.2 The Free Plate Problem
6.2.1 Solution by the Method of Potentials
6.2.2 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
6.7.1 Lamb Waves with Dominant Longitudinal Displacements
6.7.2 Zeros and Poles for a Fluid-Coupled Elastic Layer
6.7.3 Mode Cutoff Frequency
6.8 Exercises
6.9 References
9 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
10 Table of Contents cont.
8. Finite Element Method for Guided Wave Mechanics
8.1 Introduction
8.2 Overview of the Finite Element Method
8.2.1 Using the Finite Element Method to Solve a Problem
8.2.2 Quadratic Elements
8.2.3 Dynamic Problem
8.2.4 Error Control
8.3 FEM Applications for Guided Wave Analysis
8.3.1 2-D Surface Wave Generation in a Plate
8.3.2 Guided Wave Defect Detection in a Two-Inch Steel Tube
8.4 Summary
8.5 Exercises
8.6 References
11 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
9.5.1 Poynting Vector
9.5.2 Energy Velocity
9.5.3 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
12 Table of Contents cont.
10. Guided Waves in Hollow Cylinders
10.1 Introduction
10.2 Guided Waves Propagating in an Axial Direction
10.2.1 Analytic Calculation Approach
10.2.2 Excitation Conditions and Angular Profiles
10.2.3 Source Influence
10.3 Exercises
10.4 References
13 Table of Contents cont.
11. Circumferential Guided Waves
11.1 Development of the Governing Wave Equations for Circumferential Waves
11.1.1 Circumferential Shear Horizontal Waves in a Single-Layer Annulus
11.1.2 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
11.3.1 Numerical Results for CSH-Waves
11.3.2 Numerical Results for CLT-Waves
11.3.3 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
14 Table of Contents cont.12. Guided Waves in Layered Structures
12.1 Introduction
12.2 Interface Waves
12.2.1 Waves at a Solid-Solid Interface: Stoneley Wave
12.2.2 Waves at a Solid-Liquid Interface: Scholte Wave
12.3 Waves in a Layer on a Half Space
12.3.1 Rayleigh-Lamb Type Waves
12.3.2 Love Waves
12.4 Waves in Multiple Layers
12.4.1 The Global Matrix Method
12.4.2 The Transfer Matrix Method
12.4.3 Examples
12.5 Fluid Couples Elastic Layers
12.5.1 Ultrasonic Wave Reflection and Transmission
12.5.2 Leaky Guided Wave Modes
12.5.3 Nonspecular Reflection and Transmission
12.6 Exercises
12.7 References
15 Table of Contents cont.
13 . Source Influence on Guided Wave Excitation
13.1 Introduction
13.2 Integral Transform Method
13.2.1 A Shear Loading Example
13.3 Normal Mode Expansion Method
13.3.1 Normal Mode Expansion in Harmonic Loading
13.3.2 Transient Loading Source Influence
13.4 Exercises
13.5 References
16 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
17 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
18 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
16.10 References
19 Table of Contents cont.17 . Guided Waves in Viscoelastic Media
17.1 Introduction
17.2 Viscoelastic Models
17.2.1 Material Viscoelastic Models
17.2.2 Kelvin-Voight Model
17.2.3 Maxwell Model
17.2.4 Further Aspects of the Hysteretic and Kelvin-Voight Models
17.3 Measuring Viscoelastic Parameters
17.4 Viscoelastic Isotropic Plate
17.5 Viscoelastic Orthotropic Plate
17.5.1 Problem Formulation and Solution
17.5.2 Numerical Results
17.5.3 Summary
17.6 Lamb Waves in a Viscoelastic Layer
17.7 Viscoelastic composite Plate
17.8 Pipes with Viscoelastic Coatings
17.9 Exercises
17.10 References
20 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
21 Table of Contents cont.
19. Guided Wave Array Transducers
19.1 Introduction
19.2 Analytical Development
19.2.1 Linear Comb Array Solution
19.2.2 Annular Array Solution
19.3 Phased Transducer Arrays for Mode Selection
19.3.1 Phased Array Analytical Development
19.3.2 Phased Array Analysis
19.4 Concluding Remarks
19.5 Exercises
19.6 References
22 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
20.6.1 Synchronism
20.6.2 Power Flux
20.6.3 Group Velocity Matching
20.6.4 Sample Laboratory Experiments
20.7 Applications of Higher Harmonic Generation by Guided Waves
20.8 Exercises
20.9 References
23 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
24 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
25 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
26 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
27 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
28 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: 898–908.
29 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.
30 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.
31 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
32 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.
33
Figure 1-1: Comparison of bulk wave and guided wave inspection methods.
34
BULK GUIDED
Phase Velocities Constant Function of frequency
Group Velocities Same as phase velocities Generally not equal to phase velocity
Pulse Shape Non-dispersive Generally dispersive
Table 1.1 – Ultrasonic Bulk vs. Guided Wave Propagation Considerations
35 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…
36 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.
37
ISOTROPIC ANISOTROPIC
Wave Velocities Not function of launch direction
Function of launch direction
Skew Angles No Yes
Table 1.2 – Ultrasonic Wave Considerations for Isotropic vs. Anisotropic Media
38
Bulk Wave Guided Wave
Tedious and time consuming Fast
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 inspection Minimal training
Fixed distance from reflector required Any reasonable distance from reflector acceptable
Reflector must be accessible and seen Reflector can be hidden
Table 1.3 - A Comparison of the Currently Used Ultrasonic Bulk Wave Technique and the Proposed Ultrasonic Guided Wave Procedure for Plate and
Pipe Inspection
39 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
40 Table 1.5. The Difference between SHM and Non-Destructive Testing (NDT)
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
41Increased 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
42
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
43
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
44
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
45Modeling 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
46
Dealing with complex anisotropy and wave velocity and skew angle as a function of directionViscoelastic 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
47
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
48Tees, elbows, bends, and number of elbows and inspection beyond elbows
Quantification in defect location, characterization, sizing, especially depth determinationInspection 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