38 results
The Last Interglacial Ocean
- Rose Marie L. Cline, James D. Hays, Warren L. Prell, William F. Ruddiman, Ted C. Moore, Nilva G. Kipp, Barbara E. Molfino, George H. Denton, Terence J. Hughes, William L. Balsam, Charlotte A. Brunner, Jean-Claude Duplessy, Ann G. Esmay, James L. Fastook, John Imbrie, Lloyd D. Keigwin, Thomas B. Kellogg, Andrew McIntyre, Robley K. Matthews, Alan C. Mix, Joseph J. Morley, Nicholas J. Shackleton, S. Stephen Streeter, Peter R. Thompson
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- Quaternary Research / Volume 21 / Issue 2 / February 1984
- Published online by Cambridge University Press:
- 20 January 2017, pp. 123-224
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The final effort of the CLIMAP project was a study of the last interglaciation, a time of minimum ice volume some 122,000 yr ago coincident with the Substage 5e oxygen isotopic minimum. Based on detailed oxygen isotope analyses and biotic census counts in 52 cores across the world ocean, last interglacial sea-surface temperatures (SST) were compared with those today. There are small SST departures in the mid-latitude North Atlantic (warmer) and the Gulf of Mexico (cooler). The eastern boundary currents of the South Atlantic and Pacific oceans are marked by large SST anomalies in individual cores, but their interpretations are precluded by no-analog problems and by discordancies among estimates from different biotic groups. In general, the last interglacial ocean was not significantly different from the modern ocean. The relative sequencing of ice decay versus oceanic warming on the Stage 6/5 oxygen isotopic transition and of ice growth versus oceanic cooling on the Stage 5e/5d transition was also studied. In most of the Southern Hemisphere, the oceanic response marked by the biotic census counts preceded (led) the global ice-volume response marked by the oxygen-isotope signal by several thousand years. The reverse pattern is evident in the North Atlantic Ocean and the Gulf of Mexico, where the oceanic response lagged that of global ice volume by several thousand years. As a result, the very warm temperatures associated with the last interglaciation were regionally diachronous by several thousand years. These regional lead-lag relationships agree with those observed on other transitions and in long-term phase relationships; they cannot be explained simply as artifacts of bioturbational translations of the original signals.
Contributors
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- By Rony A. Adam, Gloria Bachmann, Nichole M. Barker, Randall B. Barnes, John Bennett, Inbar Ben-Shachar, Jonathan S. Berek, Sarah L. Berga, Monica W. Best, Eric J. Bieber, Frank M. Biro, Shan Biscette, Anita K. Blanchard, Candace Brown, Ronald T. Burkman, Joseph Buscema, John E. Buster, Michael Byas-Smith, Sandra Ann Carson, Judy C. Chang, Annie N. Y. Cheung, Mindy S. Christianson, Karishma Circelli, Daniel L. Clarke-Pearson, Larry J. Copeland, Bryan D. Cowan, Navneet Dhillon, Michael P. Diamond, Conception Diaz-Arrastia, Nicole M. Donnellan, Michael L. Eisenberg, Eric Eisenhauer, Sebastian Faro, J. Stuart Ferriss, Lisa C. Flowers, Susan J. Freeman, Leda Gattoc, Claudine Marie Gayle, Timothy M. Geiger, Jennifer S. Gell, Alan N. Gordon, Victoria L. Green, Jon K. Hathaway, Enrique Hernandez, S. Paige Hertweck, Randall S. Hines, Ira R. Horowitz, Fred M. Howard, William W. Hurd, Fidan Israfilbayli, Denise J. Jamieson, Carolyn R. Jaslow, Erika B. Johnston-MacAnanny, Rohna M. Kearney, Namita Khanna, Caroline C. King, Jeremy A. King, Ira J. Kodner, Tamara Kolev, Athena P. Kourtis, S. Robert Kovac, Ertug Kovanci, William H. Kutteh, Eduardo Lara-Torre, Pallavi Latthe, Herschel W. Lawson, Ronald L. Levine, Frank W. Ling, Larry I. Lipshultz, Steven D. McCarus, Robert McLellan, Shruti Malik, Suketu M. Mansuria, Mohamed K. Mehasseb, Pamela J. Murray, Saloney Nazeer, Farr R. Nezhat, Hextan Y. S. Ngan, Gina M. Northington, Peggy A. Norton, Ruth M. O'Regan, Kristiina Parviainen, Resad P. Pasic, Tanja Pejovic, K. Ulrich Petry, Nancy A. Phillips, Ashish Pradhan, Elizabeth E. Puscheck, Suneetha Rachaneni, Devon M. Ramaeker, David B. Redwine, Robert L. Reid, Carla P. Roberts, Walter Romano, Peter G. Rose, Robert L. Rosenfield, Shon P. Rowan, Mack T. Ruffin, Janice M. Rymer, Evis Sala, Ritu Salani, Joseph S. Sanfilippo, Mahmood I. Shafi, Roger P. Smith, Meredith L. Snook, Thomas E. Snyder, Mary D. Stephenson, Thomas G. Stovall, Richard L. Sweet, Philip M. Toozs-Hobson, Togas Tulandi, Elizabeth R. Unger, Denise S. Uyar, Marion S. Verp, Rahi Victory, Tamara J. Vokes, Michelle J. Washington, Katharine O'Connell White, Paul E. Wise, Frank M. Wittmaack, Miya P. Yamamoto, Christine Yu, Howard A. Zacur
- Edited by Eric J. Bieber, Joseph S. Sanfilippo, University of Pittsburgh, Ira R. Horowitz, Emory University, Atlanta, Mahmood I. Shafi
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- Clinical Gynecology
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- 05 April 2015
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- 23 April 2015, pp viii-xiv
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Plates
- Joseph L. Rose, Pennsylvania State University
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- Ultrasonic Guided Waves in Solid Media
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6 - Waves in Plates
- Joseph L. Rose, Pennsylvania State University
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- Ultrasonic Guided Waves in Solid Media
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- 11 August 2014, pp 76-106
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Summary
Introduction
This chapter presents the governing equations of elastodynamics for waves in plates, along with a series of sample problems and practical discussions. The method of displacement potentials is used to obtain a solution for the case of propagation in a free plate (see e.g., Achenbach 1984 for more detail). Also, we give a brief outline of the method of partial waves (see Auld 1990).
The classical problem of Lamb wave propagation is associated with wave motion in a traction-free homogeneous and isotropic plate. The procedures we use to develop the governing equations and dispersion curve results of phase velocity versus frequency are similar to those used in a countless number of guided wave problems that incorporate bars, tubes, multiple layers, and anisotropic media. In this chapter we shall therefore detail the basic concepts of guided wave analysis. Interpretation procedures and mathematical analysis of phase and group velocity dispersion curves and wave structure can then be extended to a variety of different guided wave problems. An alternative technique of developing dispersion curves is presented Chapter 9.
We will now briefly re-visit the fundamental differences between guided waves and bulk waves. Bulk waves travel in the bulk of the material – hence, away from the boundaries. However, often there is interaction with boundaries by way of reflection and refraction, and mode conversion occurs between longitudinal and shear waves. Although bulk and guided waves are fundamentally different, they are actually governed by the same set of partial differential wave equations. Mathematically, the principal difference is that, for bulk waves, there are no boundary conditions that need to be satisfied by the proposed solution. In contrast, the solution to a guided wave problem must satisfy the governing equations as well as some physical boundary conditions.
Contents
- Joseph L. Rose, Pennsylvania State University
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- Ultrasonic Guided Waves in Solid Media
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11 - Circumferential Guided Waves
- Joseph L. Rose, Pennsylvania State University
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- Ultrasonic Guided Waves in Solid Media
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- 11 August 2014, pp 174-208
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Summary
Introduction
In this chapter the governing equations for circumferential SH waves and circumferential Lamb type waves are developed. For brevity, circumferential SH and Lamb type waves will be abbreviated as CSH-waves and CLT-waves, respectively, from this point onward. Following the development of the single-layer cases, considerations will be made for n-layer annuli. More details on this practical problem can be found in Appendix D, Section 2.11.
Circumferential guided waves are guided waves that propagate in the circumferential direction of a hollow cylinder. They have many practical applications, including the detection of corrosion in piping from in-pipe or in-line inspection (ILI) vehicles. While researchers often use plate-wave solutions to study circumferentially propagating guided waves, this chapter shows that the two cases are quite different, both physically and mathematically, and that significant differences often exist between the plate-wave and circumferential-wave solutions to the governing wave equations. The case in which the two solutions are similar is also discussed in this chapter.
The amount of work published in the area of circumferential guided waves is relatively terse when compared to the body of work relating to wave propagation in the axial direction of hollow cylinders. The treatment of wave propagation in cylindrical layers is first seen in Viktorov (1967). In his text, Viktorov identifies the major physical differences between wave propagation in cylindrical structures and planar structures and specifically addresses the topics of Rayleigh waves on concave and convex surfaces and Lamb type waves in cylindrical layers. He defines the concept of angular wavenumber, which is a physical phenomenon unique to cylindrically curved waveguides. In his treatment of Lamb type waves in a cylindrical layer, Viktorov forms the characteristic equation for an elastic single layer. Because of the limited computational abilities of the time, Viktorov makes several simplifications to arrive at a first approximation.
17 - Guided Waves in Viscoelastic Media
- Joseph L. Rose, Pennsylvania State University
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- Ultrasonic Guided Waves in Solid Media
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- 11 August 2014, pp 323-344
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Summary
Introduction
This chapter outlines basic concepts and analysis of viscoelasticity and its impact on wave propagation. Even though the attenuation due to viscoelastic effects has plagued investigation in ultrasonic NDE for years, limited progress has been made in the study of viscoelasticity, especially in guided wave analysis. However, the reality calls for an understanding of attenuation principles as a function of material properties, wave modes, and frequency. In this chapter, we present possible approaches and discuss a few sample problems.
In many guided wave problems, material viscoelasticity may be ignored in the calculations and analysis of mode selection and wave propagation. However, in this chapter, we examine situations where a consideration of viscoelasticity is important. Significant advances have been made recently in the ability to efficiently model and solve for guided wave modes in viscoelastic waveguides that has aided our understanding of wave propagation.
General elastic theory assumes that, during deformation, a material stores energy with no dissipation. This is accurate for most metals, ceramics, and some other materials. However, many modern artificial materials, including polymers and composites, dissipate a great deal of energy during deformation. The behavior of these materials combines the energy-storing features of elastic media and the dissipating features of viscous liquids; such materials are called viscoelastic. Stresses for viscoelastic materials are functions of strains and derivatives of strains over time. If the stresses and strains and their derivation over time are related linearly, then the material has properties of linear viscoelasticity. It is important to note that viscoelastic material properties are typically very sensitive to temperature changes.
9 - The Semi-Analytical Finite Element Method
- Joseph L. Rose, Pennsylvania State University
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- Ultrasonic Guided Waves in Solid Media
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- 11 August 2014, pp 135-154
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Summary
Introduction
The semi-analytical finite element method (SAFEM) has recently become widely adopted for solving wave propagation problems in waveguides. SAFEM was developed as an alternative approach to more traditional methods such as the global matrix method, primarily because of its benefits of solving arbitrary cross-section waveguide problems (see Hayashi, Song, and Rose 2003). In SAFEM, the waveguide is discretized in the cross section, while an analytical solution is adopted in the wave propagation direction. Based on a variational scheme, a system of linear equations can be constructed with the frequency and wavenumber as unknowns. The unknowns can be solved using standard eigenvalue routines. SAFEM can solve problems of wave propagation in waveguides with complex cross sections, for example, multilayered laminates (Shorter 2004) and rails (Gavrić 1995; Hayashi, Song, and Rose 2003), where it is often difficult to obtain analytical solutions. For waveguides that are infinitely long in one dimension, SAFEM is superior to pure FEM in that exact analytical representations are used for one or two dimensions of the waveguide. Therefore, computational cost is reduced. SAFEM is also advantageous compared to analytical matrix methods because it is less prone to missing roots in developing the dispersion curves. Early employment of SAFEM in solving guided wave propagation problems can be found in Nelson and colleagues (1971) and Dong and colleagues (1972). In recent years, SAFEM was applied to the analysis of wave modes across a pipe elbow (Hayashi et al. 2005) and in materials with viscoelastic properties (Shorter 2004; Bartoli et al. 2006). SAFEM was also utilized to model the composite wing skin-to-spar bonded joints in aerospace structures by Matt and colleagues (2005) and to investigate guided wave propagations in rail structures (Damljanović and Weaver 2004; Lee et al. 2006). Applications of the SAFE technique for guided waves in composite plates can also be found in Liu and Achenbach (1994, 1995), Gao (2007), and Yan (2008).
3 - Unbounded Isotropic and Anisotropic Media
- Joseph L. Rose, Pennsylvania State University
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- Ultrasonic Guided Waves in Solid Media
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- 11 August 2014, pp 36-52
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Summary
Introduction
Bulk wave propagation refers to wave propagation in infinite media; guided waves are those that require a boundary for their existence, such as surface waves, Lamb waves, and interface waves. This chapter will focus on bulk wave propagation in infinite (or semi-infinite) media. Keep in mind that a thin structure can, for all practical purposes, still be considered a half-space or semi-infinite media if the wavelength of excitation is small with respect to the thickness of the test object.
We shall explore some interesting phenomena of phase velocity variation with angle of propagation into solid media. This leads to a dispersive influence as a result of differences in phase velocity and energy velocity. For isotropic materials, phase velocity is independent of entry angle. For lossless media, the energy velocity is equal to the group velocity. However, because of the wave velocity variations with angle, interference phenomena will lead to a skew angle. Trying to send waves or ultrasonic energy in a specific direction may be more difficult than you think!
4 - Reflection and Refraction
- Joseph L. Rose, Pennsylvania State University
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- Ultrasonic Guided Waves in Solid Media
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- 11 August 2014, pp 53-66
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Introduction
Wave reflection and refraction considerations are fundamental to the study of stress wave propagation in solids. This chapter presents basic concepts with an emphasis on physical phenomena. In this chapter we examine normal beam incidence reflection factors as well as computation of refraction angles. Reflection factor concepts are outlined first, followed by angle beam analysis and mode conversion as an ultrasonic wave encounters an interface between two materials. For more details, see Auld (1990), Brekhovskikh (1960), Graff (1991), or Kolsky (1963).
Normal Beam Incidence Reflection Factor
A plane wave encountering an interface between two materials is divided into two components: some energy at the interface is transmitted and some is reflected. The formula allowing us to compute reflection factor at an interface for normal incidence is presented in Figure 4.1.
This equation can be derived by matching normal stress at the interface as well as by matching displacement or particle velocity. Consider an incident harmonic plane wave σI traveling in an x direction to an interface between two media, as shown in Figure 4.1. Stress is reflected σR and transmitted σT. Since the elastic field is independent of the y direction, all derivatives with respect to y will vanish from the equations of motion.
Preface
- Joseph L. Rose, Pennsylvania State University
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Summary
This book builds on my 1999 book, Ultrasonic Waves in Solid Media. Like its predecessor, this book is intended to bring people up to speed with the latest developments in the field, especially new work in ultrasonic guided waves. It is designed for students and for researchers and managers familiar with the field in order to serve as a baseline for further work already under way. I hope to journey with you to provide more breakthroughs in the understanding and application of ultrasonic guided waves. The goal is to improve the health of individuals, industries, and national infrastructures through improved methods of Non-destructive Evaluation (NDE). The purpose of this book is to expand on many of the topics that were introduced in my first book. Several chapters are almost the same, but there are many new fundamental topic chapters with a total emphasis in this book being directed toward the basic principles of ultrasonic guided waves. The field of ultrasonic guided waves itself is treated as a new and separate field compared to ultrasonics and other inspection disciplines as indicated in some of the efforts put forward in inspection certification by the American Society for Non-destructive Testing (ASNT) and also in code requirements in such groups as the American Society for Mechanical Engineers (ASME) and the Department of Transportation (DOT).
The book begins with an overview and background materials in Chapters 1 through 7 and then continues on to more advanced topics in Chapters 8 through 21.
I have had the good fortune to witness the growth of ultrasonic guided waves in Non-destructive Testing (NDT) and Structural Health Monitoring (SHM) since 1985. I have been deeply interested in safety and improved diagnostics utilizing wave propagation concepts. Wave phenomena can be used to evaluate material properties nondestructively as well as to locate and measure defects in critical structures. This work has led to devices that have become valuable quality control tools and/or in-service inspection procedures for structures such as critical aircraft, pipeline, bridge, and nuclear power components whose integrity is vital to public safety.
1 - Introduction
- Joseph L. Rose, Pennsylvania State University
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Summary
Background
The field of ultrasonic guided waves has created much interest this past decade. The number of publications, research activities, and actual product quality control and in-service field inspection applications has increased significantly. Investigators worldwide are considering the possibilities of using ultrasonic guided waves in nondestructive testing (NDT) and structural health monitoring (SHM), and in many other engineering fields. Tremendous opportunities exist because of the hundreds of guided wave modes and frequencies that are available for certain waveguides. Researchers have made tremendous advancements in utilizing mode and frequency selection to solve many problems, for example, in applications for testing pipe, rail, plate, ship hull, aircraft, gas entrapment detection in pipelines, and even ice detection and deicing of rotorcraft and fixed-wing aircraft structures. These have become possible by examining special wave structures that are available via certain modes and frequencies that are capable of effectively carrying out these special work efforts.
Ultrasonic guided waves in solid media have become a critically important subject in NDT and SHM. New faster, more sensitive, and more economical ways of looking at materials and structures have become possible when compared to the previously used normal beam ultrasonic or other inspection techniques. For example, the process of inspecting an insulated pipe required removing all the insulation and using a single probe to check with a normal beam along the length of the pipe with thousands of waveforms. Now, one can use a guided wave probe at a single location, leave the insulation intact, and perhaps inspect the entire pipe by examining just a few waveforms. The knowledge presented in this book will lead to creative ideas that can be used in new inspection developments and procedures.
10 - Guided Waves in Hollow Cylinders
- Joseph L. Rose, Pennsylvania State University
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- 11 August 2014, pp 155-173
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Summary
Introduction
Ultrasonic guided waves are most commonly used in plate, rod, and hollow cylinder (pipeline and tubing) inspections. This subject is receiving much attention recently because of the possibility of inspecting long volumetric lengths of a structure from a single probe position. Components can be inspected if hidden, coated, or under insulation, oil, soil, or concrete. Excellent defect detection sensitivities and long inspection distances have been demonstrated. Guided waves in cylindrical structures may travel in the circumferential or axial direction. Based on boundary conditions, material properties, and geometric properties of the hollow cylinder, the wave behavior can be described by solving the governing wave equations with appropriate boundary conditions. In this chapter, simulations of guided waves propagating in axial directions in cylindrical structures are calculated and evaluated.
Guided Waves Propagating in an Axial Direction
In this section, a calculation approach is developed for guided wave propagation in the axial direction of a hollow cylinder.
Analytic Calculation Approach
When considering the particle motion direction possibilities in a hollow cylinder, the guided waves propagating in the axial direction may involve longitudinal waves and torsional waves. The longitudinal waves have dominant particle motions in either the r and/or z directions and the torsional waves have dominant particle motions in the direction. According to the energy distribution in the circumferential direction, the guided waves contain axisymmetric modes and non-axisymmetric modes (also known as flexural modes). For convenience, a longitudinal mode group will be expressed as L(m, n) and a torsional mode group as T(m, n). Here the integer m denotes the circumferential order of a mode and the integer n represents the group order of a mode. An axisymmetric mode has the circumferential number m = 0.
20 - Introduction to Guided Wave Nonlinear Methods
- Joseph L. Rose, Pennsylvania State University
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- 11 August 2014, pp 378-401
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Introduction
Up to this point we have described linear ultrasonics, that is, where the received signal is at the same frequency as the excitation. Now we consider nonlinear ultrasonics, where the received signal is not at the frequency of the excitation. The material is treated as weakly nonlinear elastic because the amplitude of the signal received at higher harmonics is very small relative to the excitation, which permits the use of a perturbation solution. The generation of higher harmonics in bulk solids has been studied for more than four decades, but the initial studies of higher harmonics in plates are much more recent. These studies are relevant because the amplitudes of higher harmonics have been shown to be sensitive to features of the microstructure of the material, whereas the primary harmonics are generally much less sensitive, or insensitive, to microstructural features such as dislocation density, precipitates, and cavities. This chapter introduces nonlinear methods for guided waves.
To maintain the best possible structural integrity of a component, it is highly desirable to detect damage at the smallest possible scale. Doing so with periodic nondestructive inspection or continuous structural health monitoring (SHM) enables tracking damage evolution over the service life of the structure, which can be used in conjunction with prognostics for condition-based maintenance and improved logistics. Nonlinear systems are known to be very good at indicating damage progression (e.g., Dace, Thompson, and Brashe 1991; Farrar et al. 2007; Worden et al. 2007). Generally speaking, linear ultrasonics with bulk waves can detect anomalies on the order of a wavelength. Ultrasonic guided waves can do significantly better in terms of wavelength, say λ/40 (e.g., Alleyne and Cawley 1992), but longer wavelengths are typically used to enable large penetration lengths. Nonlinear ultrasonics, where the received signal containing the information of interest is at a different frequency than the emitted signal, can provide sensitivity to microstructural changes.
5 - Oblique Incidence
- Joseph L. Rose, Pennsylvania State University
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Summary
Background
One of the most important topics associated with the subject of stress wave propagation in solid materials is the wave reflection and refraction at an interface between two different media. (For more details, see Auld 1990; Graff 1991; Pilarski, Rose, and Balasubramaniam 1990; or Rose 1999.) The subject is important to the study of ultrasonic guided waves since oblique incidence via appropriate angles of incidence and frequency selection can be used to generate guided waves in a variety of different waveguides. Introductory topics and concepts are therefore presented in this chapter. If incident angles are selected properly, long enough wavelengths are used, and the material being inspected has a phase velocity larger than the dilatational velocity in the wedge material, then guided waves can be generated in the test material.
A general introduction to oblique incidence in ultrasonic wave analysis will be presented. The reflection (refraction) factor, or coefficient, is defined as the ratio of the amplitude of the reflected (refracted) wave to the amplitude of the incident wave. The factor depends on the angle of incidence, wave velocities, and possibly frequency, depending on the interface condition. In this chapter, we introduce a boundary condition approach for calculating these factors. We use this approach for the interface between two semi-infinite medium spaces: solid–solid, solid–liquid, and liquid–solid. If the reader would like to calculate reflection and refraction factors for a thin interface solid (and liquid) layer between two different media, it is recommended to follow guidelines established by Jiao and Rose (1991) and from a “spring” model (Pilarski and Rose 1998a,b; Pilarski et al. 1990). These cases are also discussed by Rose (1999).
14 - Horizontal Shear
- Joseph L. Rose, Pennsylvania State University
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Summary
Introduction
Many aspects of horizontal shear wave propagation are intriguing and quite valuable for applications involving wave propagation, including ultrasonic NDT. Traditionally, the longitudinal and vertical shear modes of wave propagation have been the most commonly used – probably because they are simple to understand and to generate. Yet horizontal shear waves can also be generated quite easily through a variety of different transducers. This chapter covers the fundamental concepts of such propagation.
Dispersion Curves
In addition to the Lamb wave modes that exist in flat layers, there also exists a set of time-harmonic wave motions known as shear horizontal (SH) modes. The term “horizontal shear” means that the particle vibrations (displacements and velocities) caused by any of the SH modes are in a plane that is parallel to the surfaces of the layer. This is depicted in Figure 14.1, where the wave propagates in the x1 direction and the particle displacements are in the x3 direction.
Physically, any mode in the SH family can be considered as the superposition of up- and down-reflecting bulk shear waves, polarized along x3, with wavevectors lying in the (x1, x2)-plane and inclined at such an angle that the system of waves satisfies traction-free boundary conditions on the surfaces of the layer.
The dispersion equation governing the SH modes can be derived in several ways, including the use of Helmholtz potentials, partial wave analysis, or transverse resonance (Auld 1990). Because of the simple physical nature of the SH modes, the most straightforward way to solve the problem is to deal directly with the displacement equations of motion. This is the approach taken here; for more discussion of this technique, see Achenbach (1984).
Acknowledgments
- Joseph L. Rose, Pennsylvania State University
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2 - Dispersion Principles
- Joseph L. Rose, Pennsylvania State University
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Summary
Introduction
Before studying stress wave propagation in such waveguide structures as solid rods, bars, plates, hollow cylinders, or multiple layers, it is useful and interesting to review some applicable concepts taken from studies of dispersive wave propagation where wave velocity is a function of frequency. Wave propagation characteristics in waveguides are functions of frequency.
Let’s first, however, consider wave propagation in a taut string where some basic dispersive concepts can be studied. Models of a taut string, a string on an elastic base, a string on a viscoelastic foundation will be discussed.
Even though wave dispersion can be considered for anisotropic media (where wave velocity is a function of direction), the emphasis in this chapter is on dispersion due to structural geometry. Some basic terms are introduced, including wave velocity, wavenumber, wavelength, material and geometrical dispersion, phase velocity, group velocity, attenuation, cutoff frequency, frequency spectrum, and energy transmission, all of which will be useful in further studies. Graphical interpretations and analysis of phase and group velocity are also covered in this chapter. Additional details can be found in other texts including Graff (1991).
15 - Guided Waves in Anisotropic Media
- Joseph L. Rose, Pennsylvania State University
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- 11 August 2014, pp 276-293
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Introduction
The problem of elastic wave propagation in anisotropic layers has received a fair amount of attention in the literature during the past several decades, and recent interest in this subject has increased even more. This is undoubtedly due, at least in part, to the increased use of composite materials in many new facets of structure design. Composite materials that are mechanically anisotropic offer many benefits over more conventional material – a higher stiffness-to-weight ratio, for example. This advantage of composites is in turn due to the fact that their mechanical properties, such as elastic moduli, can be tailored to be high in the directions that are expected to see high loads while remaining considerably lower in other directions. This directional dependence of the mechanical properties of composites classifies them as anisotropic media.
The benefits of using composites come at the cost of a more complicated mechanical response to applied loads, static or dynamic. The anisotropic nature of the solid introduces many interesting wave phenomena not observed in isotropic bodies: a directional dependence of wave speed, a difference between phase and group velocity of the waves, wave skewing, three wave velocities instead of two, and many somewhat more subtle differences. An understanding of the nature of waves in plates made of anisotropic materials is certainly required if one wants to use these materials effectively in structure design or if one wants to inspect them using ultrasonic methods.
Appendix A - Ultrasonic Nondestructive Testing Principles, Analysis, and Display Technology
- Joseph L. Rose, Pennsylvania State University
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Some Physical Principles
It will be useful to review some widely used basic concepts in ultrasonic nondestructive evaluation (NDE) as a complement to the more detailed aspects of the mechanics and mathematics of wave propagation and ultrasonic NDE. Of first concern will be defining such fundamental ultrasonic field parameters as near field and angle of divergence. These will be followed by elements of instrumentation and display technology, along with aspects of axial and lateral resolution of an ultrasonic transducer. An excellent textbook on basic ultrasonics is Krautkramer 1990.
Wave velocity, one of the key parameters of wave propagation study, is the velocity at which a disturbance propagates in some specified material. Its value depends on material, structure, and form of excitation. Many different formulas for wave velocity are presented. The most widely used wave velocity value used in ultrasonic NDE is the bulk longitudinal wave velocity, generally thought of as directly proportional to the square root of the elastic modulus over density. Another common velocity is the bulk shear wave velocity, which is proportional to the square root of the shear modulus over density. These velocities are called bulk velocities. Bulk waves do not require a boundary for support. Guided waves, on the other hand, require a boundary for propagation. Many tables of wave velocity values for different materials are available in the literature.
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