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.

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 todetect 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.

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).

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).

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).

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.

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.

Introduction

Guided wave dispersion curve calculations are based on the assumption of an infinite continuous plane wave excitation producing a set of particular phase velocity values at specific frequencies. However, in real applications, the excitation sources are of a finite size over a finite frequency spectrum. This chapter establishes the guidelines for evaluating the effects of excitation sources on wave excitation. In particular, we address the problem of guided wave excitation and propagation in a traction-free plate.

Many excitation mechanisms can be used to generate ultrasonic waves in a solid medium. The commonly used sources for guided wave excitation are piezoelectric transducers, electromagnetic acoustic transducers (EMAT), magnetostrictive devices, physical impact, and laser ultrasonics. The piezoelectric transducer can be used in a normal incident or oblique incidence situation with an angle wedge. Instead of propagating the waves from the transducer to the structure, the EMAT and laser, for example, generates ultrasound within the structure either as a surface loading or a body loading. Most recently, for purposes of structural health monitoring (SHM), piezoelectric wafer transducers are attached directly to the structure to generate specific kinds of ultrasonic guided waves in the structure.

General Concepts

Interesting processing concepts are discussed quite often in guided wave studies. In guided wave analysis, it is often useful to think on a frequency basis rather than to employ more common time domain thinking. As a result, Fourier transform analysis is commonplace. An analytic envelope for a Hilbert transform is also useful. The short time Fourier transform (STFFT) can also provide significant insight into the studies of guided wave response functions encountered in different situations. Physical insight into a resulting image of a spectrogram and its relationship to a group velocity dispersion curve along with the wavelet transform and its ability to see when certain frequency packets arrives as a function of time, as well as its relationship to a group velocity dispersion curve. On the other hand, the 2-D Fourier transform (2DFFT) relates to portions of a phase velocity dispersion curve. Quite often, the partial images generated by these transform techniques can provide us with an indication of damage in a structure by shifting and other indicators.

This appendix gives an overview of some of these transform techniques that can also be used as a basis for extracting features that provide insight into important characteristics of guided waves. In particular, they provide data for constructing portions of the relevant dispersion curves, identifying the modes that are actually propagating, and providing a physical explanation of certain aspects of ultrasonic guided wave propagation.

Introduction

Multiple element array transducers are extremely useful and popular in today’s inspection environment. The first applications were to carry out electronic B and C scans opposed to earlier developed mechanical scans. This was followed by phasing of the elements in a bulk wave problem where beam steering and focusing were possible during the electronic scanning process. Today, array transducers are being used in guided wave inspection. Linear comb and annular array sensors are two possibilities. Beyond electronic scanning and focusing it is also now possible to select time delay profiles for guided wave mode and frequency selection to optimize sensitivity to certain defects and penetration power in special situations. Thus, time delays for mode selection and electronic scanning can be superimposed for rapid and efficient NDT and SHM. As a consequence, such items as the excitation spectrum and the mode excitability function will be studied along with phasing principles for linear combs and annular arrays.

To employ guided waves for nondestructive evaluation (NDE) or structural health monitoring (SHM) purposes, these waves must first be generated in the structure of interest. Accordingly, to fully reap the benefits guided waves can offer, such wave generation should be performed in a well-designed, highly controlled manner, which is only possible through deliberate transducer design. Proper guided wave mode control can provide distinct advantages in terms of sensitivity to particular defects, sensitivity to environmental variables, penetration power, and other factors in guided wave inspection. Additionally, the suppression of spurious modes and/or the excitation of a particularly nondispersive mode can greatly enhance the potential signal analysis of gathered data by simplifying the waveforms.

Background

The existence of surface waves was predicted theoretically over a century ago. Elastic waves propagating along the surface of a half-space were first predicted by Lord Rayleigh in 1885 in his paper “On waves propagating along the plane surface of an elastic bar,” submitted to the Proceedings of the London Mathematical Society. It is telling that this paper was submitted to a mathematical society and not a physical society, as such surface waves were primarily a mathematical concept, although Lord Rayleigh did suspect that they would be relevant to seismology. By the middle of the twentieth century, however, surface waves began to enter into mainstream technological applications. These waves, often referred to as Rayleigh waves or surface acoustic waves (SAW), are now being employed in a number of areas of science and technology, including ultrasonic NDE and SHM, seismology, and electronic circuitry. There is much literature on this subject, including for example Chadwick and Smith (1977), Farnell (1970), Pollard (1977), and Viktorov (1967). Experimental evidence was first obtained in observing wave propagation over the surface of the earth (as a result of earthquakes) and subsequent mode conversion at the earth’s surface. Observations were made regarding the unusual behavior of energy decay with increased depth and the ability of waves to travel along curved surfaces.

This chapter examines surface waves on an isotropic, homogeneous, linear elastic semi-space. We take a rather classical approach to-the problem, one that is based on potential functions and boundary conditions for a free surface. Assumptions of isotropy, homogeneity, and linear elastic response will also be made. For more detail, see Auld (1990), Basatskaya and Ermolov (1980), Couchman and Bell (1978), Heelan (1953), Kolsky (1963), Nikiforov and Kharitonov (1981), Pilarski and Rose (1989), Uberall (1973), and Viktorov (1967).

Introduction

Ultrasonic vibrations have often been used in the past on various structures without any real understanding of the impact of a loading function on structural resonances or resulting vibrational patterns at different frequencies. The method was used to obtain an ultrasonic signal signature of a part being inspected, most often in quality control after manufacture. A purpose of this chapter is therefore to establish an understanding of the ultrasonic vibration as a superposition of guided wave modes traveling in a structure and ways to optimize sensitivity to certain defects by way of loading function choice. In more traditional low-frequency modal analysis, the loading function choice is not so critical; results depend primarily on excitation frequency. With ultrasonic vibration, we will see that the choice of a loading function plays a major role in the design of an inspection system for developing various quality control and in-service inspection solutions.

This chapter now examines the subject of ultrasonic vibrations, the topic of which is a logical extension to ultrasonic guided waves. A few applications are considered.

Let’s consider the long time solution to a wave propagation problem. In the bulk wave case, because waves are traveling in infinite space, there is no vibration aspect of the problem to be considered as there are no wave reflection and transmission factors. On the other hand, for guided wave propagation, the long time solution in many cases leads to a vibrations problem. This may not occur if no boundary exists in a particular direction as the wave is transmitted to infinity. In examining a closed structure, such as a finite plate or tube, the reflection and transmission factors for each entry of a wave onto a boundary leads to a variety of constructive and destructive interference phenomena. The long time solution therefore leads to a vibrations or modal analysis problem. There will be specific resonant frequency values for a structure as well as specific vibrational patterns at on and off resonance. Researchers are carrying out work examining the transition from the initial transient response to a long time vibrations solution. As a consequence, ultrasonic nondestructive evaluation (NDE) and structural health monitoring (SHM) is being developed by considering many aspects of ultrasonic bulk wave analysis, ultrasonic guided wave analysis, and modal vibration analysis.

Introduction

Many engineering structures consist of multiple layers. Examples include plates with coatings, painted structures, diffusion bonded or adhesively bonded structures, ice or contaminant accreted aircraft structures, and laminated composites. To achieve long-distance inspection and monitoring of these structures using ultrasonic guided waves, scholars must study the guided wave propagation characteristics in such structures. This chapter examines the wave propagation problem in layered plate structures consisting of isotropic materials. More advanced studies in complicated structures involving material anisotropy and viscoelasticity are addressed later.

Wave propagation in layered plate structures can be abstracted into several models for different layer thicknesses. When a layer thickness is much larger than the selected wavelength, a half-space model can be used to approximate the thick layer. Interface guided wave modes may exist at the interface between two thick materials. In this case, these two layers are modeled as two half-spaces in the classic problems associated with the Stoneley and Scholte wave solutions. When one of the layers is much thicker than the other layers, guided wave modes exist within the thin layers and the upper region of the thick layer. The thick layer can be modeled as a half-space. One classic problem that falls into this category is the Love wave problem, which studies shear horizontal (SH) guided waves in a layer on a half-space. When all of the layers are of compatible thickness, and the wavelengths of the guided waves are also of a compatible scale, a multilayer model can be established by using finite thicknesses for all of the layers.

Guided wave imaging methods are extremely popular and useful for evaluating defect locations and sizes in guided wave inspection. Interpretation is easier and preferred when compared to studying complex RF waveforms. Images are generally reconstructed from the RF waveforms in a variety of ways. A few imaging methods are discussed here, including the following:

1. Guided Wave through Transmission Dual Probe Imaging

2. Defect Locus Map

3. Guided Wave Tomographic Imaging

4. Guided Wave Phased Array in Plates

5. Long-Range Ultrasonic Guided Wave Pipe Inspection Images

Introduction

Imaging methods in nondestructive evaluation (NHM) are preferred whenever possible, compared to results presented in RF waveform format. Imaging results are easier to read, to interpret, and to convince others of a result. RF waveforms can be complex and difficult to analyze, often calling for sophisticated signal processing and pattern recognition analysis. As a result, a discussion of several guided wave imaging methods are presented in this chapter.

Guided Wave through Transmission Dual Probe Imaging

A number of guided wave through transmission dual probe configurations have been used for defect imaging purposes. The two probes are fairly close together and scan from one to eight inches apart. The idea stems from an acousto-ultrasonic approach by Vary (1987). At that time, two normal beam probes were placed close to each other and an ultrasonic signature of the test piece was taken that often depicted a good or bad character of the component being tested. The method was also developed to assist in acoustic emission development programs where a sending transducer was used to imitate an acousto emission signal for subsequent capture and analysis by an acoustic emission sensor and analysis system.

Introduction

The finite element method (FEM), also known as finite element analysis (FEA), is a widely used numerical solution approach for solving problems of wave propagation and vibration. For some complex differential or integral equations without analytical solutions, FEM can provide accurate and computationally efficient solutions that may be improved by refining the elements used in the study. FEM may be combined with analytical analysis or other numerical methods to achieve optimum solutions.

Cook and colleagues (1992) concluded that a successful FEM simulation usually involves the items listed in Table 8.1. Although commercial FEM software has been widely used, the only step commercial software can accomplish is the finite element analysis step. To achieve an accurate FEM simulation, the analyst needs to appropriately understand the physical problem, conduct a basic preliminary study, and select a correct mathematical model. The analyst should also check the calculation results based on either theoretical research or experimental data.

Overview of the Finite Element Method

Using the Finite Element Method to Solve a Problem

Simple geometry, loading, and boundary conditions are required to achieve an exact analytical solution for a field problem. For a complicated problem that cannot be solved analytically while considering if the entire structure can be separated into small, discrete elements with simple loading and boundary conditions, one may simplify this problem as a field problem in each simple element.

Introduction

Computation plays a critical role in the development of any ultrasonic guided wave inspection system. Beyond hardware and software development, modeling analysis is essential for evaluating various designs and ultimately relates to making a good choice from the hundreds of test points available on the dispersion curves. The process is illustrated in Figure D.1.

First, phase and group velocity dispersion curves must be calculated for a particular waveguide structure being considered. Wave structure computation is also critical. This early part of the process is the analytical section. This part leads to a mode and frequency choice based on experience, further modeling of wave interactions with certain defects in specific structures, or an intuitive selection and evaluation.

The actuator design could be, for example, a normal or angle beam device, or a comb-type array. A piezoelectric, electromagnetic acoustic, magnetostrictive, laser, or controlled mechanical impact could be considered for closeness to other mode points in an attempt at mode isolation, or phase velocity spectrum influences from a source influence, or of course a center frequency and frequency spectrum. See Figures D.2 and D.3.