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.

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.

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.

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).

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!

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.

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.

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.