International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 1.3, pp. 9598
Section 1.3.7. Nonlinear dynamic elasticity^{a}Institut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France, and ^{b}Laboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France 
In recent years, the measurements of ultrasonic wave velocities as functions of stresses applied to the sample and the measurements of the amplitude of harmonics generated by the passage of an ultrasonic wave throughout the sample are in current use. These experiments and others, such as the interaction of two ultrasonic waves, are interpreted from the same theoretical basis, namely nonlinear dynamical elasticity.
A first step in the development of nonlinear dynamical elasticity is the derivation of the general equations of motion for elastic waves propagating in a solid under nonlinear elastic conditions. Then, these equations are restricted to elastic waves propagating either in an isotropic or in a cubic medium. The next step is the examination of two important cases:
Finally, the concept of natural velocity is introduced and the experiments that can be used to determine the third and higherorder elastic constants are described.
For generality, these equations will be derived in the X configuration (initial state). It is convenient to obtain the equations of motion with the aid of Lagrange's equations. In the absence of body forces, these equations are or where L is the Lagrangian per unit initial volume and are the elements of the Jacobian matrix.
For adiabatic motionwhere U is the internal energy per unit mass.
Combining (1.3.7.2) and (1.3.7.3), it follows that which can be written since
Using now the equation of continuity or conservation of mass: and the identity of Euler, Piola and Jacobi: we get an expression of Newton's law of motion: with becomes since , the thermodynamic tensor conjugate to the variable , is generally denoted as the `second Piola–Kirchoff stress tensor'.
Using Φ, the strain energy per unit volume, Newton's law (1.3.7.4) takes the form and
As an example, let us consider the case of a plane finite amplitude wave propagating along the axis. The displacement components in this case become Thus, the Jacobian matrix reduces to
The Lagrangian strain matrix is [equation (1.3.6.8)] The only nonvanishing strain components are, therefore, and the strain invariants reduce to
In this case, the strainenergy density becomes Differentiating (1.3.7.6) with respect to the strains, we get All the other .
From (1.3.7.5), we derive the stress components: Note that this tensor is not symmetric.
For the particular problem discussed here, the three components of the equation of motion are
If we retain only terms up to the quadratic order in the displacement gradients, we obtain the following equations of motion:
In this case, the strainenergy density becomes Differentiating (1.3.7.8) with respect to the strain, one obtains All other . From (1.3.7.5), we derive the stress components: In this particular case, the three components of the equation of motion are
If we retain only terms up to the quadratic order in the displacement gradients, we obtain the following equations of motion: which are identical to (1.3.7.7) if we put
The coordinates in the medium free of stress are denoted either a or . The notation is used when we have to discriminate the natural configuration, , from the initial configuration X. Here, the process that we describe refers to the propagation of an elastic wave in a medium free of stress (natural state) and the coordinates will be denoted .
Let us first examine the case of a pure longitudinal mode, i.e.
The equations of motion, (1.3.7.7) and (1.3.7.9), reduce to for an isotropic medium or for a cubic crystal (most symmetrical groups) when a pure longitudinal mode is propagated along [100].
For both cases, we have a onedimensional problem; (1.3.7.7) and (1.3.7.9) can therefore be written
The same equation is also valid when a pure longitudinal mode is propagated along [110] and [111], with the following correspondence: Let us assume that ; a perturbation solution to (1.3.7.10) is where with
If we substitute the trial solutions into (1.3.7.10), we find after one iteration the following approximate solution: which involves secondharmonic generation.
If additional iterations are performed, higher harmonic terms will be obtained. A well known property of the firstorder nonlinear equation (1.3.7.10) is that its solutions exhibit discontinuous behaviour at some point in space and time. It can be seen that such a discontinuity would appear at a distance from the origin given by (Breazeale, 1984) where is the initial value for the particle velocity.
We now consider the propagation of smallamplitude elastic waves in a homogeneously strained medium. As defined previously, or a are the coordinates in the natural or unstressed state. X are the coordinates in the initial or homogeneously strained state. are the components of displacement from the initial state due to the wave.
Starting from (1.3.7.4), we get Its partial derivative is If we expand the state function about the initial configuration, it follows that
The linearized stress derivatives become If we let , the equation of motion in the initial state is The coefficients do not present the symmetry of the coefficients except in the natural state where and are equal.
The simplest solutions of the equation of motion are plane waves. We now assume plane sinusoidal waves of the form where k is the wavevector.
Substitution of (1.3.7.14) into (1.3.7.13) results in or with .
The quantities and A are, respectively, the eigenvalues and eigenvectors of the matrix . Since is a real symmetric matrix, the eigenvalues are real and the eigenvectors are orthogonal.
The main experimental procedures for determining the third and higherorder elastic constants are based on the measurement of stress derivatives of ultrasonic velocities and on harmonic generation experiments. Hydrostatic pressure, which can be accurately measured, has been widely used; however, the measurement of ultrasonic velocities in a solid under hydrostatic pressure cannot lead to the whole set of thirdorder elastic constants, so uniaxial stress measurements or harmonic generation experiments are then necessary.
In order to interpret wavepropagation measurements in stressed crystals, Thurston (1964) and Brugger (1964) introduced the concept of natural velocity with the following comments:
`According to equation of motion, the wave front is a material plane which has unit normal k in the natural state; a wave front moves from the plane to the plane in the time . Thus W, the natural velocity, is the wave speed referred to natural dimensions for propagation normal to a plane of natural normal k.
In a typical ultrasonic experiment, plane waves are reflected between opposite parallel faces of a specimen, the wave fronts being parallel to these faces. One ordinarily measures a repetition frequency F, which is the inverse of the time required for a round trip between the opposite faces.'
In most experiments, the thirdorder elastic constants and higherorder elastic constants are deduced from the stress derivatives of . For instance, Table 1.3.7.1 gives the expressions for and for a cubic crystal. These quantities refer to the natural state free of stress. In this table, p denotes the hydrostatic pressure and the 's are the following linear combinations of thirdorder elastic constants:

References
Breazeale, M. A. (1984). Determination of thirdorder elastic constants from ultrasonic harmonic generation. Physical Acoustics, Vol. 17, edited by R. N. Thurston, pp. 2–75. New York: Academic Press.Brugger, K. (1964). Thermodynamic definition of higherorder elastic coefficients. Phys. Rev. 133, 1611–1612.
Thurston, R. N. (1964). Wave propagation in fluids and normal solids. Physical Acoustics, Vol. 1A, edited by W. P. Mason, pp. 1–109. New York: Academic Press.