Measurement of Scattering Coefficients

In the previous chapters, we saw how waves in composites behaved under various circumstances, depending on material anisotropy and wave propagation direction. The most important function that describes guided wave propagation, and the plate elastic behavior on which propagation depends, is the reflection coefficient (RC) or transmission coefficient (TC). More generally, we can call either one simply, the scattering coefficient (SC). It is clear that the elastic properties of the composite are closely tied to the SC, and in turn the scattering coefficient determines the dispersion spectrum of the composite plate. Measuring the SC provides a route to the inference of the elastic properties. To measure the SC, we need only observe the reflected or transmitted ultrasonic field of the incident acoustic energy. In doing so, however, the scattered ultrasonic field is influenced by several factors, both intrinsic and extrinsic. Clearly, the scattered ultrasonic field of an incident acoustic beam falling on the plate from a surrounding or contacting fluid will be strongly influenced by the RC or TC of the plate material. The scattering coefficients are in turn dependent on the plate elastic properties and structural composition: fiber and matrix properties, fiber volume fraction, layup geometry, and perhaps other factors. These elements are not, however, the only ones to determine the amplitude and spatial distribution of energy in the scattered ultrasonic field. Extrinsic factors such as the finite transmitting and receiving transducers, their focal lengths, and their placement with respect to the sample under study can make contributions to the signal as important as the SC itself. Therefore, a systematic study of the role of the transducer is essential for a complete understanding and correct interpretation of acoustic signals in the scattered field. The interpretation of these signals leads ultimately to the inference of composite elastic properties. As we pointed out in Chapter 5, the near coincidence under some conditions of guided plate wave modes with the zeroes of the reflection coefficient (or peaks in the transmission coefficient) has been exploited many times to reveal the plate’s guided wave mode spectrum.

Air-Coupled Ultrasonics

Ultrasonic material characterization or inspection for defects is conventionally performed using either liquid coupling (water, usually) or some type of gel or oil in contact-mode coupling. Mechanical waves can be transmitted only through some sound-supporting medium from their source (a transducer) to the object under study, and back again. Using distilled, degassed water to couple ultrasound to an object under test works quite well and has many technical advantages, including relatively low signal loss over laboratory or shop dimensions at typical frequencies, almost zero toxicity, and low cost. For many applications, the use of water is acceptable and preferred. There are, however, certain testing applications for which water can be a disadvantage. These situations include materials that are sensitive to contact with water, such as uncured graphite-epoxy composites or certain electronics. Large objects, whose total immersion is impractical, or objects for which rapid scanning is required might also be unsuitable for water coupling. Recent technological developments are beginning to permit the judicious replacement of water by a far more ubiquitous sound coupling medium—air. Ultrasonic testing in air has been investigated for more than 30 years, but recently there has been an upsurge in interest and application because of the availability of much more efficient sound-generating devices designed specifically for operation in air. In water- or direct-coupled ultrasonics, one typically employs piezoelectric transducers to generate sound waves because they are well suited to the generation of sound in water or in solids because of their high acoustic impedance. In air, however, we need just the opposite. Air is very compliant, so waves from a high-impedance source couple poorly into air. Much effort has been invested in finding suitable impedance matching materials that will render the familiar piezoelectric probe efficient in air-coupled (A-C) ultrasound. The problem, however, is nearly insurmountable because of the large acoustic impedance difference between air and quartz, for example. Quartz has an acoustic impedance of about 15 MRayl, while air’s impedance is about 425 Rayl, a ratio of about 35,000. The challenge is to find a material with an acoustic impedance that nearly equals the geometric average of these two widely disparate values.

Waves in Periodically Layered Composites

Composite materials, unless they are quite thin, often include periodic layering, where laminated plates composed of alternating uniaxial plies in two or more directions result in more evenly distributed in-plane stiffness. The oriented plies can generally be reduced to a unit cell geometry which repeats throughout the laminate and is composed of sublayers each having highly directional in-plane stiffness, but identical out-of-plane properties. As the transverse isotropy of a uniaxial fibrous ply derives from the geometry of the two-phase material, composite laminates of these plies will have microscopic elastic stiffness tensors which change only in the plane of the laminate, as we saw in Chapter 1. The elastic properties normal to the laminate surface remain unchanged from ply to ply. In this chapter we take up the subject of waves in periodically layered plates. Unusual guided wave dispersion effects have been observed experimentally in periodically layered plates. Shull et al. found, for guided waves polarized in the vertical plane in plates of alternating aluminum and aramid–epoxy composites, that dispersion never scales with the frequency–thickness product, as it would in homogeneous isotropic, or layered transversely isotropic, plates. Instead, grouping of the mode curves has been observed. In an attempt to understand this behavior in terms of periodic layering, Auld et al. have analyzed the simpler case of SH wave propagation in periodically layered plates and have demonstrated that these observed phenomena can be attributed to the pass band and stop band structure caused by the periodic layering. In this section, we will show that Floquet modes play a critical role in the behavior of guided waves in plates that are periodically layered. To analyze the problem, we apply an extension of the stiffness matrix method of the previous chapter. Floquet modes, which are the characteristic modes for the infinite periodically layered medium, can be thought of as the analogy—in a periodically layered medium—to the quasilongitudinal and quasishear modes for the infinite homogeneous medium. On the topic of infinite periodic media, many calculations, both approximate and exact, have been performed to model elastic wave propagation in this important class of structures.

Elastic Waves in Multilayer Composites

Expanding on the theme of bulk waves from the previous chapters, we will examine the problem of plane wave sound propagation in layered media. We assume we have an finite stack of planar layers with perfect, rigidly bonded planar interfaces, but infinite in their lateral extent. The problem has significant industrial interest. Most practical composite laminates are composed of layers of uniaxial fibers and plastic, i.e., plies, whose fiber orientation directions vary from ply to ply through the thickness of the laminate. The mechanical purpose of this directional variation is to render the product stiff and strong in all in-plane directions, much as plywood is layered in cross-grain fashion. Almost no practical composite would be fabricated as a uniaxial product, because of the low bending strength normal to the fiber direction. Instead, various types of layering have been devised to give either tailored stiffness for a specific purpose or approximate in-plane isotropy, also known colloquially as a “quasi-isotropic” laminate. In fact, the approximate isotropy is achieved only in the plane of the plies, because the out-of-plane direction still has significant and unavoidable stiffness differences, since it contains no fibers. The scale of the layering is also important. When the laminations are fine, i.e., when each directional lamina is no thicker than an individual ply as we go through the thickness, only acoustic waves of relatively short wavelength will be able to discern the effect of the layering. At longer wavelengths, the laminate may behave more like an effective medium, still anisotropic, but with averaged elastic properties. On the other hand, if each lamina contains multiple numbers of individual 1/8-mm plies, then the frequency at which an acoustic wavelength approaches the layer thickness will be proportionately lower. This is an important distinction, because it suggests the point at which the layering must be treated as a discrete substructure in order to develop an accurate description of waves in a layered medium. The situation is illustrated schematically in Fig. 6.1. The figure illustrates laminations for a quasi-isotropic composite.

Fundamentals of Composite Elastic Properties

In this chapter, we review the mechanical behavior of composites considered from a macroscopic perspective, i.e., the microscopic heterogeneity of the material is ignored in this treatment. Our objective is to provide an overview of the basic composite constitutive behavior and to set the notation for the subsequent chapters. To establish this framework, we draw on concepts from continuum mechanics and elasticity, both of which are also covered by specialized books on these topics. The results in this chapter are important for us because they provide the theoretical framework for all the elastic wave phenomena we describe in detail in the subsequent chapters. Stress in a solid body is measured in force per unit area; there are normal stresses, acting along a normal to the infinitesimal element of the area, and shear (tangential) stresses, acting in the plane of the element. Let us assume that an infinitesimal traction force dT acts on an infinitesimal surface element dA = ndA, where n denotes the normal unit vector of the surface element. In index notation, the stress tensor is then defined through . . . dTi = σijdAj. (1.1) . . . The sign convention for the stress tensor in a Cartesian coordinate system is shown in Fig. 1.1. The choice of coordinate system is arbitrary, but for the sake of simplicity and concreteness, let us develop the relationships in a Cartesian system. They can all be generalized at a later time. Only the stress components acting on the surface elements with positive normal vectors are shown for clarity. On the surface elements with negative normal vectors, the stress directions are opposite. Conventionally, the first index indicates the normal of the surface the stress component is acting upon and the second index indicates the direction of the resulting traction force (however, we will show shortly that equilibrium conditions require that the stress tensor be symmetric, therefore the order of the indices is only of academic importance). For example, σ11 is the normal stress acting on the x2, x3 plane in the x1 direction, σ12 is the shear stress acting on the same plane in the x2 direction, and so forth.

Bulk Ultrasonic Techniques for Evaluation of Elastic Properties

Currently, the design of most composite components is based on stiffness, and therefore methods for static measurement of stiffness are in wide use. The disadvantages of these methods lie in their destructive nature (the samples must be cut from parts of different orientations), in the difficulty of measuring shear properties, and in the need for extra care when measuring Young’s modulus in off-axis directions. Ultrasonic methods are more accurate and have higher spatial resolution than static measurements. As we showed in Chapter 2, by measuring ultrasonic velocities in several predefined directions, all elastic constants can be determined. The generic method described there is also destructive, however, requiring cutting numerous samples with appropriate fiber orientation. Specialized nondestructive methods for determining the elastic moduli of composite materials are more powerful and they can be applied to composite coupons before, during, and after strength or fatigue testing. It is important to have a fast and inexpensive technique to estimate input parameters for composite design. It is even more important to have a technique to evaluate composites during service to verify that the manufactured elastic stiffnesses match those assumed in the design. Several methods that utilize bulk ultrasonic waves for measurement of composite elastic constants are considered in this chapter. By bulk wave methods, we mean quasilongitudinal and quasitransverse ultrasonic wave velocity measurement methods that are applicable when the sample thickness h is larger than both the ultrasonic pulse space length τV and the wavelength λ (τ is the ultrasonic pulse length in time, and V is the wave speed). Other methods, which are applicable in the range h < τV and which account for wave interference with the boundaries of the specimen, will be considered in the following chapters. The most promising way to evaluate composite elastic properties nondestructively is to measure ultrasonic velocities in different directions in the composite material and reconstruct the elastic constants from these values using some kind of an inversion technique. One possible method has been suggested by Markham in the 1970s, who used ultrasonic waves obliquely incident from water onto a composite plate to measure ultrasonic velocities in various directions and evaluated the results to determine elastic constants.

Guided Waves in Plates and Rods

In this chapter we consider elastic wave modes which propagate in composites with finite boundaries. There are those waves that exist between the two plane parallel boundaries of a homogeneous anisotropic solid. We consider that well-known problem, as well as waves in an elastic anisotropic rod, specifically an individual graphite fiber. Composite laminates seen in applications are essentially all multilayered structures, and in many cases can be considered periodically layered. So, we also take up the subject of guided waves in layered plates in later chapters. In a plate geometry, as illustrated in Fig. 5.1, we choose the propagation direction to be parallel to the x1 axis and the x3 axis to be normal to the plate surfaces. This geometry is particularly significant for composite materials since, by design, laminates are often locally planar in nature. While the solutions we find are appropriate for flat plates, with some modifications they describe wave motion in gently curved structures as well. Clear and mathematically straightforward descriptions of the characteristics of plate waves exist for isotropic media. The results obtained for isotropic media are not, however, directly applicable to most composites. We begin by considering the behavior of waves in a uniaxial composite laminate. In later chapters we generalize the calculation to layered orthotropic media, concentrating on the results and physical interpretation rather than the algebraic details. To begin a description of waves in plates, let us consider the possible polarizations of particle motion. Let the plate surfaces lie in the (x1, x2) plane of mirror symmetry with the origin dividing the plate thickness in half, as shown in Fig. 5.1. Then, we will at first assume the wave to be uniform in the x2 direction and propagating in the x1 direction, and (x1, x3) is the plane of symmetry. Particle motion can occur along any axis. Note that in this restricted symmetry, shear partial waves polarized along the x2 axis will have no component of particle motion normal to the plate surfaces. Partial waves are a concept introduced by Rayleigh to acknowledge that a superposition of both shear and longitudinal particle motion is generally needed to produce plate waves polarized in the vertical plane.

Reflection and Refraction of Waves at a Planar Composite Interface

Nondestructive ultrasonic testing of composite materials is affected by several special features of wave propagation that arise from the strong anisotropy and inhomogeneity of these materials. The resulting complexity requires re-examination of old testing methodologies and development of new ones. One of the most fundamental phenomena in ultrasonic nondestructive evaluation is the reflection–refraction of ultrasonic waves at a plane interface. Even the simplest test procedure requires understanding of mode conversion and knowledge of elastic wave reflection and transmission coefficients and refraction angles. Reflection–refraction phenomena, while straightforward and well documented for isotropic materials, are much more complicated for anisotropic materials. When analyzing the oblique incidence inspection method for composite materials, one first has to address the problem of wave propagation through the interface between the coupling medium and the composite material. For example, there is an inherent fluid/composite interface in the immersion technique and a perspex/composite interface in the contact method. In the latter case, assuming that a thin fluid layer is applied to facilitate coupling through the interface, slip rather than welded boundary conditions prevail. Another example of great practical importance is the case of multidirectional fiber plies in a composite laminate, when the reflection and transmission of ultrasonic waves from one ply to another with a different orientation must be considered. Before discussing the general problem of wave refraction in anisotropic composite materials, let us review the simple isotropic case. Consider a plane interface between two isotropic elastic media in “welded” (perfectly bonded) contact, implying continuity of tractions and displacements across the interface, although the boundary conditions are not important at this point. Figure 4.1 shows a schematic diagram of a plane wave with wavenumber ki incident on the interface at angle θi. The parallel lines with spacing equal to the incident wavelength λi correspond to equal-phase planes orthogonal to the incident plane. By definition, the wavenumber ki = 2π/λi is the magnitude of the wave vector ki. The incident wave is converted at the interface into reflected and transmitted waves. The refraction angle of the transmitted wave is θr and its wavenumber is kr.

Elastic Waves in Anisotropic Media

In this chapter, we provide a brief introduction to ultrasonic wave propagation in unbounded anisotropic solids with emphases on examples suitable for ultrasonics of composites. Many excellent books are relevant to the subject addressed in this chapter. Some of them broadly discuss elastic waves in anisotropic solids, including waves in layered anisotropic media. In-depth theoretical description of elastic waves in anisotropic media is given in classical texts, which have influenced and provided guidance to our treatment of some aspects of the theory. Beautiful visualization of ultrasonic waves in crystals (often obtained by laser excitation) is given in reference. The equations of motion for the vibration of an elastic medium are extensions of Newton’s second law for particles. Treating the elastic continuum as a collection of particles, each of which is assumed to obey Newton’s laws, leads to a particularly straightforward argument. We begin by considering a short segment of a bar with length Δx and cross-sectional area S0 as is illustrated in Fig. 2.1. The material is assumed to be linear and elastic, and its deformations can be described by constitutive equations derived in the previous chapter. For simplicity, we assume only uniaxial stress in the x-direction of the continuum.