1
Propagation in an Unbounded Solid

The objective of this chapter - divided into five sections - is to describe the propagation of mechanical vibrations in an unbounded solid. As elastic wavelengths are very long, compared to interatomic distances, the medium is considered to be continuous. The phenomena studied are macroscopic: we do not consider the individual motion of the molecules that constitute the medium, but that of a solid particle. This term designates an element of infinitesimal volume at the scale of the physical dimensions of the medium, which nonetheless contains a large number of molecules. The propagation equations for an elastic wave are, thus, deduced from the conservation equations of matter and momentum, complemented by the constitutive equation of the propagation medium.

In a deformable solid, two tensors play a fundamental role. The mechanical stress, whose tensor character comes from the fact that a force has three components and that the surface element on which it is exerted is defined by the components of its normal. The strain, which expresses the relative difference between the displacements of two neighboring material points, that is, the extremities of an element of infinitesimal length (section 1.1). A solid is elastic if it returns to its initial state when the external forces that deformed it are removed. This return to the state of rest is the work of internal stresses, which vanish with the strains. When these strains are small, the linear relationship between stresses and strains, which generalizes Hooke's law, defines a fourth rank tensor: the stiffness tensor.

The equation of propagation of elastic waves can be obtained by writing the generalized Hooke's law in the fundamental relation of dynamics. In an unbounded isotropic solid, a longitudinal displacement and a transverse displacement propagate independently, at two different velocities (section 1.2). In an anisotropic medium, for a given direction, three waves can propagate and the direction of propagation of energy is, in general, not parallel to the wave vector: it is given by the direction of the Poynting vector (section 1.3). The solutions are represented by a slowness surface, analogous to the index surface in optics. This surface, composed of two sheets in optics, is formed here of three sheets. This representation reveals the importance of the symmetry axes, along which the propagation modes are pure, with the wave vector and energy vector generally being carried by this axis. In a piezoelectric crystal, at least one of the elastic waves is accompanied by an electric field (section 1.4). The importance of this electromechanical coupling, depending on the direction, can also be deduced from the slowness surface. A dimensionless parameter that expresses the ability of piezoelectric materials to generate or detect elastic waves is defined.

In practice, elastic waves propagate in any material medium: gaseous, liquid, homogeneous or inhomogeneous solid, isotropic or anisotropic solid. However, their amplitude decreases during the propagation because the bonds between atoms or molecules are not purely elastic (section 1.5). The attenuation of the waves is smaller when the medium is more ordered. Thus, losses are larger in a liquid than in a solid, while losses are larger in an amorphous or polycrystalline solid, compared to a single crystal. On the other hand, these losses increase rapidly with frequency, so that liquids are rarely used beyond 50 MHz and only single crystals are used at frequencies in the GHz domain.

1.1. Reviewing the mechanics of continuous media

In this section, we will establish the equations for the stress and the mechanical displacement fields that are independent of the medium (linear or nonlinear, isotropic or anisotropic, elastic or viscoelastic), supporting the elastic waves. The passage of an elastic wave in a solid is accompanied by a transport of energy without any permanent displacement of the matter. As in electromagnetism, the acoustic power crossing a surface element is equal to the flux of a vector called the Poynting vector.

1.1.1. Conservation equations

The fundamental equations of the mechanics of continuous media may be derived from the four laws of conservation of classical physics: the conservation of mass, the conservation of linear momentum, the conservation of angular momentum and the conservation of energy:

• - the first law states that in the absence of a flow source, the mass of a given set of particles is invariant;
• - Newton's second law indicates that in an inertial frame of reference, the change per unit time of linear momentum mv is equal to the external force applied to the particle with mass m:

• - according to the theorem of angular momentum, the change per unit time of the angular momentum with respect to a fixed point, is equal to the moment of the external forces:

• - the law of conservation of energy expresses, for an isolated mechanical system, the integral conversion of kinetic energy into potential energy and vice versa, so that their sum remains constant. However, a complete energy balance must include the power supplied by external sources and the energy dissipated by internal friction processes, through which the mechanical energy linked to the overall movement1 progressively disappears, giving rise to the thermal excitation of molecules. This excitation is manifested through a (local) increase in the temperature of the medium. To extend the law of conservation of energy to non-conservative systems, it is necessary to introduce the notion of internal energy: the dissipation of part of the mechanical energy increases the internal energy of the system by a quantity dq, equal to the amount of heat released. It must be noted that other transformations are also possible, for example, structural changes.

When writing conservation equations, we have the choice between two equivalent descriptions of motion (Salençon 1988). In the Lagrangian description, each particle is identified by its initial position in a configuration taken as the reference. The motion is then described by the position of each particle over time (trajectory). The Eulerian description (chosen here) defines the motion by the knowledge, at each time t, of the velocity field of the particles v (x, t).

Similarly, all physical quantities are represented by functions of position x and time t in the reference frame, which is assumed to be Galilean. To establish the integral forms of the conservation laws, let us consider any fixed volume V , inside the medium, bounded by a closed surface S of unit normal l oriented toward the exterior. The variation per unit time of the considered quantity consists of two terms: the derivative of the quantity contained in the volume V and the flux across the surface S arising from the transport of matter. The first term is expressed by a volume integral, the second one by a surface integral involving the scalar product v .l, which is written as:

taking into account the summation rule over dummy indices (Einstein convention). To convert a surface integral into a volume integral and vice versa, we will use Green's theorem:

1.1.1.1. Integral equations

The mass density ? of the medium varies according to the law of conservation of matter. The material flow at the point x, with coordinates xi (i = 1, 2, 3) and at time t, is equal to the product of the density ?(x, t) and the velocity vector of the particles v (x, t). The mass that crosses per unit time the surface element dS of unit normal is equal to The integral of this quantity2 on the surface S represents the decrease per unit time of the mass contained in the fixed volume V :

According to Newton's second law, the time derivative of the linear momentum is equal to the resultant force acting on the fixed volume V. These forces are exerted directly in the volume, like gravity, with a density F (x, t) per unit mass, or through the intermediary of the surface S delimiting the volume V with a surface density T. This is the case with actions exerted by the matter situated outside V, which are progressively transmitted by the bond strengths between molecules. Since their radius of action is very small on a macroscopic scale, their resultant is expressed by the integral on the surface S of the elementary forces T () dS. The vector T (), called mechanical traction or stress vector, depends on x and t, but also on the orientation of the surface element dS, defined by the unit vector normal to S (Figure 1.1).

In the case of a perfect fluid, this force is normal to the surface element and directed toward the interior of the volume V. It is expressed as a function of the hydrostatic pressure p:

In a solid or in a viscous fluid, tangential forces appear. Let us write the condition for equilibrium of the volume V, taking into...