Third-Order Elastic Constants and the Velocity of Small Amplitude Elastic Waves in Homogeneously Stressed Media

Third-order elastic constants can be determined from the velocity of small amplitude sound waves in statically stressed media. For this purpose exact expressions are derived for the sound velocity and for a natural velocity and their stress derivatives, evaluated at zero stress, in terms of second- and third-order elastic constants. The formulas apply to arbitrary crystal symmetry and to arbitrary stress systems depending on a single scalar variable. Special formulas for hydrostatic pressure and uniaxial stress are listed for the cubic point groups $O$, ${O}_{h}$, ${T}_{d}$, and for isotropic materials. Attention is given to the proper variation of propagation direction with static stress in order to maintain propagation normal to a given crystal face as in ultrasonic experiments, and to the proper separation of isothermal and isentropic coefficients in the results. The simplest and most convenient from of the results employs the natural velocity (natural unstressed length at the same temperature divided by the transit time), which is computed directly from experimental data without correcting the path length for the effect of stress.