Introduction

An important application of the reciprocity relation is its use to generate integral representations. With the aid of the basic singular elastodynamic solution for an unbounded solid, an integral representation can be derived that provides the displacement field at a point of observation in terms of the displacements and tractions on the boundary of a body. In the limit as the point of observation approaches the boundary, a boundary integral equation is obtained. This equation can be solved numerically for the unknown displacements or tractions. The calculated boundary values are subsequently substituted in the original integral representation to yield the desired field variables at an arbitrary point of observation.

The boundary element method is often used for the numerical solution of boundary integral equations. The advantage of the boundary element method for solving boundary integral equations is that the dimensionality of the problem is reduced by one. Rather than calculations in a two- or three-dimensional discretized space, we have calculations for discretized curves or surfaces. For detailed discussions of boundary element methods in elastodynamics we refer to the review papers by Beskos (1987) and Kobayashi (1987). These papers contain numerous additional references. We also mention the book edited by Banerjee and Kobayashi (1992), and a recent book by Bonnet (1995) that has several sections on dynamic problems.

We start this chapter with an exposition of the basic ideas for the simpler, two-dimensional, case of anti-plane strain in Section 11.2.

Modelling of technical production facilities, plants and processes is an integral part of engineering and process technology development, planning and construction. The successful implementation of modelling tools is strongly related to one's understanding of the physical processes involved. Most important in the context of chemical and process technologies are momentum, heat and mass transfer during production. Projection, or scaling, of the unit operations of a complex production plant or process, from laboratory-scale or pilot-plant-scale to production-scale, based on operational models (in connection with well-known scaling-up problems) as well as abstract planning models, is a traditional but important development tool in process technology and chemical engineering. In a proper modelling approach, important features and the complex coupled behaviour of engineering processes and plants may be simulated from process and safety aspects viewpoints, as well as from economic and ecologic aspects. Model applications, in addition, allow subdivision of complex processes into single steps and enable definition of their interfaces, as well as sequential investigation of the interaction between these processes in a complex plant. From here, realization conditions and optimization potentials of a complex process or facility may be evaluated and tested. These days, in addition to classical modelling methods, increased input from mathematical models and numerical simulations based on computer tools and programs is to be found in engineering practice. The increasing importance of these techniques is reflected by their incorporation into educational programmes at universities within mechanical and chemical engineering courses.

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).

Having divided the atomization and spray process modelling procedure into three main areas:

1. atomization (disintegration),

2. spray, and

3. compaction,

in this chapter we look specifically at the disintegration process as it applies to the case of molten metal atomization for spray forming. We begin by breaking down disintegration into a number of steps:

• the melt flow field inside the tundish and the tundish melt nozzle,

• the melt flow field in the emerging and excited fluid jet,

• the gas flow field in the vicinity of the twin-atomizer,

• interaction of gas and melt flow fields, and

• resulting primary and secondary disintegration processes of the liquid melt.

Several principal atomization mechanisms and devices exist for disintegration of molten metals. An overview of molten metal atomization techniques and devices is given, for example, in Lawley (1992), Bauckhage (1992), Yule and Dunkley (1994) and Nasr et al. (2002). In the area of metal powder production by atomization of molten metals, or in the area of spray forming of metals, especially, twin-fluid atomization by means of inert gases is used. The main reasons for using this specific atomization technique are:

• the possibility of high throughputs and disintegration of high mass flow rates;

• a greater amount of heat transfer between gas and particles allows rapid, partial cooling of particles;

• direct delivery of kinetic energy to accelerate the particles towards the substrate/deposit for compaction;

• minimization of oxidation risks to the atomized materials within the spray process by use of inert gases.

A common characteristic of the various types of twin-fluid atomizers used for molten metal atomization is the gravitational, vertical exit of the melt jet from the tundish via the (often cylindrical) melt nozzle.

Coupling of several mechanisms into an integrated model for the spray forming process is the final aim of simulation. Such an integral spray forming model has been investigated, for example, by Bergmann (2000), Minisandram et al. (2000) (which has already been introduced in Section 6.3) and Pedersen (2003).

Connection between those submodels aforementioned has been performed, for transient temperature material behaviour, from melt superheating in the tundish to room temperature in the preform via cooling and solidification, in a three-stage approach in Bergmann (2000). Tundish melt flow and a thermal model are accounted for in the first stage. The local separation method is employed in the second stage to determine temperature-averaged properties of the spray and solid fractions in the particle mass at the centre-line of the spray. The temporal behaviour of a melt element is derived in terms of the averaged mean residence time of the particle mass. By combination of these data with calculated temporal cooling and solidification distributions of a fixed volume element inside the deposit, one yields the mean thermal history of the material at a specified location in the deposit.

The transient thermal and solidification distributions are shown in Figures 7.1 and 7.2. Results are illustrated separately for the three different modelling areas (with different time scales) for: (a) melt flow in the tundish, (b) particle cooling in the spray, (c) growth and cooling of the deposit. The process conditions used are for a steel spray forming of a Gaussian-shaped deposit (see Table 6.5).