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The first edition of Acoustic Microscopy ended with a chapter emphasizing the need to understand the contrast in terms of the variation of signal with defocus, V(z). This is a fundamental concept in the contrast from surfaces of stiff materials, and is dominated by excitation of Rayleigh waves in the surface of the sample. The second edition contains a major new chapter on acoustically excited probe microscopy, which changes everything. In ultrasonic force microscopy there is no V(z), no defocus, and no Rayleigh waves. Instead the contrast is dominated by the non‐linear mechanical contact between the tip of an atomic force microscope and the surface of the sample, with its underlying elastic nanostructure.

When the boundary condition is homogeneous, one may think that there is no data at the boundary and that there is no need of boundary estimates in the maximal estimates. This is reminiscent of the case of weakly dissipative symmetric IBVP, and is compatible with the failure of the K.-L. condition at some elliptic boundary frequencies. This chapter constructs a weakly dissipative symmetrizer under appropriate assumptions. This context is the realm of surface waves of finite energy. A paradigm is the Rayleigh waves in linear elasticity.

A good way to start to appreciate what acoustic microscopy can do for you is to look at examples which show unique contrast from the elastic structure of the sample. In glass matrix composites such as borosilicate with silicon carbide fibres, there is contrast between the fibre and the matrix, and also from cracks in the matrix and at the interface between fibre and matrix. A phase of crystobalite is visible where the matrix has devitrified. Samples of granordiorite rock containing plagioclase, biotite, and quartz can be seen, together with boundaries of misorientation which occurred to accommodate strain during the geological processes of deformation of the rock. The composite and rock samples show fringes arising from the Rayleigh waves which are a recurrent motif in acoustic microscopy of stiff materials. Hydroxyapatite, the principal crystalline mineral constituent of bone, can be seen in living cells, with the distinctive contrast arising from the elastic properties and structure of the sample.

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.

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.

The Stroh formalism for two-dimensional elastostatics can be extended to elastodynamics when the problem is a steady state motion. Most of the identities in Chapters 6 and 7 remain applicable. The Barnett-Lothe tensors S, H, L now depend on the speed υ of the steady state motion. However S(υ), H(υ), L(υ) are no longer tensors because they do not obey the laws of tensor transformation when υ≠0. Depending on the problems the speed υ may not be prescribed arbitrarily. This is particularly the case for surface waves in a half-space where υ is the surface wave speed. The problem of the existence and uniqueness of a surface wave speed in anisotropic materials is the crux of surface wave theory. It is a subject that has been extensively studied since the pioneer work of Stroh (1962). Excellent expositions on surface waves for anisotropic elastic materials have been given by Farnell (1970), Chadwick and Smith (1977), Barnett and Lothe (1985), and more recently, by Chadwick (1989d).

Hybrid modes exist as a consequence of acoustic and optical waves having to satisfy the boundary conditions at an interface or at a surface. The author begins the description of hybrid modes in nanostructures with an account of modes in a non-polar, free-standing slab. This chapter includes long-wavelength assumption decouples acoustic and optical modes; isotropy decouples LO and TO modes; s and p modes; acoustic hybrid modes: Love waves, Lamb waves, guided modes, Rayleigh waves; the boundary condition u = 0 for optical modes; the sTO mode; double hybrid: LO and pTO modes; and energy normalization.