Introduction

As discussed in Chapter 9, the modes of wave propagation in an elastic layer are well known from Lamb's (1917) classical work. The Rayleigh–Lamb frequency equations, as well as the corresponding equations for horizontally polarized wave modes, have been analyzed in considerable detail; see Achenbach (1973) and Mindlin (1960). It appears, however, that a simple direct way of expressing wave fields due to the time-harmonic loading of a layer in terms of mode expansions, and a suitable method to obtain the coefficients in the expansions by reciprocity considerations, has so far not been recognized. Of course, wave modes have entered the solutions to problems of the forced wave motion of an elastic layer, at least in the case of surface forces applied normally to the faces of the layer, but via the more cumbersome method of integral transform techniques and the subsequent evaluation of Fourier integrals by contour integration and residue calculus. For examples, we refer to the work of Lyon (1955) for the plane-strain case, and that of Vasudevan and Mal (1985) for axial symmetry.

In this chapter the displacements excited by a time-harmonic point load of arbitrary direction, either applied internally or to one of the surfaces of the layer, are obtained directly as summations over symmetric and/or antisymmetric modes of wave propagation along the layer. This is possible by virtue of an application of the reciprocity relation between time-harmonic elastodynamic states.

Introduction

The scattering of elastic waves by defects, such as cracks, voids and inclusions, located in bodies with boundaries is a challenging topic for analytical and numerical studies in elastodynamics. It is, however, also a topic of great practical interest in the field of quantitative non-destructive evaluation (QNDE), because scattering results can be used to detect and size defects. In the context of QNDE, elastodynamics is referred to as ultrasonics, since it is generally necessary to work with wave signals whose principal frequency components are well above the frequency range audible to the human ear.

For realistic defects it is not possible to obtain solutions of scattering problems by rigorous analytical methods. The best numerical technique is generally the one that employs a Green's function to derive a boundary integral equation, as discussed in Chapter 11, which can then be solved by the boundary element method. This process yields the field variables on the surface of the scattering obstacle (the defect). An integral representation can subsequently be used to calculate the scattered field elsewhere. Of course, as an alternative, the fields on the defect can be approximated. Various approximations are available. We mention the quasistatic approximation for the displacement on the surface of a cavity, the Kirchhoff approximation for a crack and the Born approximation for scattering by an inclusion.

In Section 12.2 the interaction of an incident wave motion with a defect in a waveguide is considered. The incident wave is represented by a summation of modes.

Introduction

In Chapter 1, a formal definition of a reciprocity theorem for elastodynamic states was stated as: “A reciprocity theorem relates, in a specific manner, two admissible elastodynamic states that can occur in the same time-invariant linearly elastic body. Each of the two states can be associated with its own set of time-invariant material parameters and its own set of loading conditions. The domain to which the reciprocity theorem applies may be bounded or unbounded.”

Reciprocity theorems for elastodynamics in one-dimensional geometries were stated in Chapter 5. In the present chapter analogous theorems for three-dimensional elastodynamics are presented, as well as some applications. The most useful reciprocity theorems are for elastodynamic states in the frequency and Laplace transform domains. We also discuss reciprocity in a two-material body and reciprocity theorems for linearly viscoelastic solids.

For the time-harmonic case a number of applications of reciprocity in elastodynamics are considered. Some of the examples are concerned with the reciprocity of fields generated by point forces in bounded and unbounded elastic bodies. Other cases are concerned with the solution of the wave equation with polar symmetry and with reciprocity for plane waves reflected from a free surface.

Another purpose of the chapter is to provide insight on the applicability of reciprocity considerations, together with the use of a virtual wave, as a tool to obtain solutions for elastodynamic problems. Some examples are concerned with two-dimensional cases for anti-plane strain. These examples are very simple.

Introduction

In this chapter we seek solutions to the elastodynamic equations that represent a combination of a carrier wave propagating on a preferred plane and motions that are carried along by that wave. The carrier wave supports standing-wave motions in the direction normal to the plane of the carrier wave. Such combined wave motions include Rayleigh surface waves propagating along the free surface of an elastic half-space and Lamb waves in an elastic layer.

The usual way to construct solutions to the elastodynamic equations of motion for homogeneous, isotropic, linearly elastic solids is to express the components of the displacement vector as the sum of the gradient of a scalar potential and the curl of a vector potential, where the potentials must be solutions of classical wave equations whose propagation velocities are the velocities of longitudinal and transverse waves, respectively. The decomposition of the displacement vector was discussed in Section 3.8.

The approach using displacement potentials has generally been used also for surface waves propagating along a free surface or an interface and Lamb waves propagating along a layer. The particular nature of these guided wave motions suggests, however, an alternative formulation in terms of a membrane-like wave over the guiding plane that acts as a carrier wave of superimposed motions away from the plane. For time-harmonic waves the corresponding formulation, presented in this chapter, shows that the carrier wave satisfies a reduced wave equation, also known as the Helmholtz or membrane equation, in coordinates in the plane of the carrier wave.

Introduction

In this chapter the equations governing linear, isotropic and homogeneous elasticity are used to describe the propagation of mechanical disturbances in elastic solids. Some well-known wave-propagation results are summarized as a preliminary to their use in subsequent chapters.

There are essential differences between waves in elastic solids and acoustic waves in fluids and gases. Some of these differences are exhibited by plane waves. For example, two kinds of plane wave (longitudinal and transverse waves) can propagate in a homogeneous, isotropic, linearly elastic solid. These waves may propagate independently, i.e., uncoupled, in an unbounded solid. Generally, longitudinal and transverse waves are however, coupled by conditions on boundaries. Most boundary conditions or internal source mechanisms generate both kinds of wave simultaneously.

Plane waves in an unbounded domain are discussed in Section 3.2. The flux of energy in plane time-harmonic waves is considered in Section 3.3. The presence of a surface gives rise to reflected waves. The details of the reflection of plane waves incident at an arbitrary angle on a free plane surface are discussed in Section 3.4. As is well known, the incidence of a plane wave, say a longitudinal wave, gives rise to the reflection of both a longitudinal and a transverse wave. This wave-splitting effect happens, of course, also for an incident transverse wave. The reflection coefficients and other relevant results are listed in Section 3.4. Energy partition due to wave splitting is discussed in Section 3.5.

The most important elastic wave field is the basic singular solution.

Introduction

In many applications, waves in an acoustic medium such as water are coupled to wave motion in submerged elastic bodies. There are examples in the area of structural acoustics, which is concerned with the generation of sound in a surrounding acoustic medium by time-variable forces in submerged structures as well as with the detection of submerged bodies by the scattering of incident sound waves. Structural acoustics has been discussed in considerable detail in books by Junger and Feit (1972), Fahy (1985), and Cremer, Heckl and Ungar (1973). Another class of coupled acousto-elastic systems is defined by seismic problems in an oceanic environment, where acoustic waves are used to probe the geological strata under the ocean floor.

In this chapter and the next we distinguish the coupling of wave phenomena as either configurational or physical. For the configurational case, coupling comes about because bodies of different constitutive behaviors are in contact, as is the case for acousto-elastic systems, as described above. However, for physical coupling the wave interaction takes place in the same body when different physical phenomena are coupled by the constitutive equations. That is the case for electromagneto-elastic coupling, specifically piezoelectricity, which will be discussed in the next chapter.

Reciprocity for acousto-elastic systems was already anticipated by Rayleigh (1873), when in his statement of the reciprocity theorem for acoustics, see Chapter 4, he allowed for a space filled with air that is partly bounded by finitely extended fixed bodies.

Introduction

It is shown in this chapter that the reciprocity theorem can be used to calculate in a convenient manner, that is, without the use of integral transform techniques, the surface-wave motion generated by a time-harmonic line load or a time-harmonic point load applied in an arbitrary direction in the interior of a half-space. The virtual wave motion that is used in the reciprocity relation is also a surface wave. Hence the calculation does not include the body waves generated by the loads. For a point load applied normally to the surface of a half-space, it is shown in Section 8.6 that the surface-wave motion is the same as obtained in the conventional manner by the integral transform approach.

It is well known that the dynamic response to a time-harmonic point load normal to the surface of the half-space was solved by Lamb (1904), who also gave explicit expressions for the generated surface-wave motion. The surfacewave motion can be obtained as the contribution from the pole in inverse integral transform representations of the displacement components. The analogous transient time-domain problem for a point load normal to the surface of the half-space was solved by Pekeris (1955). The displacements generated by a transient tangential point load applied to the half-space surface were worked out by Chao (1960).

Introduction

Problems of the motion and deformation of solids are rendered amenable to mathematical analysis by introducing the concept of a continuous medium. In this idealization it is assumed that properties averaged over a very small element, for example the mean mass density and the mean displacement and stress, are continuous functions of position and time. Although it might seem that the microscopic structure of real materials is not consistent with the concept of a continuum, the idealization produces very useful results, simply because the lengths characterizing the microscopic structure of most materials are generally much smaller than any lengths arising in the deformation of the medium. Even if in certain special cases the microstructure gives rise to significant phenomena, these can be taken into account within the framework of the continuum theory by appropriate generalizations.

Continuum mechanics is a classical subject that has been discussed in great generality in numerous treatises. The theory of continuous media is built upon the basic concepts of stress, motion and deformation, upon the laws of conservation of mass, linear momentum, moment of momentum (angular momentum) and energy and on the constitutive relations. The constitutive relations characterize the mechanical and thermal response of a material while the basic conservation laws abstract the common features of mechanical phenomena irrespective of the constitutive relations.

The governing equations used in this book are for homogeneous, isotropic, linearly elastic solids.

Reciprocity relations are among the most interesting and intriguing relations in classical physics. At first acquaintance these relations promise to be a goldmine of useful information. It takes some ingenuity, however, to unearth the nuggets that are not immediately obvious from the formulation. In the theory of elasticity of solid materials the relevant reciprocity theorem emanated from the work of Maxwell, Helmholtz, Lamb, Betti and Rayleigh, towards the end of the nineteenth century, and several applications have appeared in the technical literature since that time. This writer has always believed, however, that more information than is generally assumed can be wrested from reciprocity considerations. I have wondered in particular whether reciprocity considerations could be used to actually determine by analytical means the elastodynamic fields for the high-rate loading of structural configurations. I have explored this question for a number of problems and obtained the actual fields generated by loading from a reciprocity relation in conjunction with an auxiliary solution, a free wave called the “virtual” wave. These recent results comprise an important part of the book.

To my knowledge, the topic of reciprocity in elastodynamics has not been discussed in a comprehensive manner in the technical literature. It is hoped that this book will fill that void. Various forms of the reciprocity theorem are presented, with an emphasis on those for time-harmonic fields, together with numerous applications, general and specific, old and new.

Introduction

The first reciprocity relation specifically for acoustics was stated by von Helmholtz (1859). This relation caught the attention of and was elaborated by Rayleigh (1873) and Lamb (1888). Rayleigh (1873) briefly discussed the reciprocal theorem for acoustics in his paper “Some general theorems relating to vibrations.” In The Theory of Sound (1878, Dover reprint 1945, Vol. II, pp. 145–8), Rayleigh paraphrased this theorem as follows: “If in a space filled with air which is partly bounded by finitely extended fixed bodies and is partly unbounded, sound waves may be excited at any point A, the resulting velocity potential at a second point B is the same both in magnitude and phase, as it would have been at A, had B been the source of sound.” In this statement it is implicitly assumed that sources of the same strength would be applied at both places. In Rayleigh's book (1878) the statement is accompanied by a simple proof. A similar statement of the Helmholtz reciprocity theorem for acoustics can be found in the paper by Lamb (1888). Both Rayleigh and Lamb generalized the theorem to more complicated configurations, and in time the reciprocity theorem became known as Rayleigh's reciprocity theorem.

Most books on acoustics devote attention to the reciprocity theorem; see for example Pierce (1981), Morse and Ingard (1968), Jones (1986), Dowling and Ffowcs Williams (1983) and Crighton et al. (1992). A book by Fokkema and van den Berg (1993) is exclusively concerned with acoustic reciprocity.

The issue of dislocation formation in a strained epitaxial heterostructure was the focus of attention in the preceding chapter. Residual stress was assumed to originate from the combination of a mismatch in lattice parameters between the materials involved and the constraint of epitaxy. The discussion in Chapter 6 led to results in the form of minimal conditions which must be met by a material system, represented by a geometrical configuration and material parameters, for dislocation formation to be possible. Once the values of system parameters are beyond the point of fulfilling such minimal conditions, dislocations begin to form, propagate and interact. The ensemble behavior is usually termed strain relaxation.

There are several practical aspects of strain relaxation that originate from the small size scales involved, the relatively low dislocation densities that are observed, and the fact that kinetic processes occur on a timescale comparable to growth or processing timescales. Can significant strain relaxation be suppressed? Can threading dislocation densities be controlled? Under what conditions can ensemble dislocation behavior be captured by a continuum plasticity representation? How do length scales associated with geometrical configuration and microstructure, such as film thickness and grain size, respectively, influence the process of strain relaxation?

Progress toward resolving such questions is summarized in this chapter. The discussion begins with the issue of fundamental dislocation interaction phenomena and nonequilibrium behavior of interacting dislocations. Attention is then shifted from consideration of films with low dislocation density to the modeling of inelastic deformation of thin films with relatively high densities of dislocations.

Interface delamination and film fracture induced by residual stress in the thin film were the focus of discussion in the preceding chapter where models were developed within the framework of linear elastic fracture mechanics. Such analyses did not account for the actual separation of the film from its substrate. However, there are delamination processes for which transverse deflection of the film away from the substrate becomes an important consideration in a variety of practical applications. Examples include:

1. – transverse buckling instability of a film in compression, as in the case of a ceramic thermal barrier coating or a diamond-like carbon wear resistant coating on a metallic substrate, as illustrated in Figure 5.1,

2. – the bulging of a segment of a thin film away from the substrate under the influence of an applied pressure, which is a deformation mode of interest for the assessment of residual stress, interface fracture energy and mechanical properties in film–substrate systems and MEMS structures, and

3. – the forcible peeling of a film from its substrate as a means of evaluating the adhesion energy of the interface between the film and the substrate.

This chapter deals with the transverse or out-of-plane deflection of a thin film, and it includes quantitative descriptions of the phenomena associated with the buckling, bulging or peeling of a film from its substrate. A common thread throughout the discussion is that system behavior extends beyond the range of geometrically linear deformation.