introduction e mch 521, acs 521 stress waves in solid media 3 credit graduate course penn state...

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


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


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


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


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


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


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


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


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


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


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


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


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


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


18. Ultrasonic Vibrations

18.1 Introduction

18.2 Practical Insights into the Ultrasonic Vibrations Problem


18.4 Exercises

18.5 References


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


19.5 Exercises

19.6 References


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


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


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


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


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

introduction e mch 521, acs 521 stress waves in solid media 3 credit graduate course penn state...

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