Conventions for the placement of axes in crystal elasticity

A good reference for the symmetry relation in the fourth-rank elasticity tensor ($C_{ijkl}$) components is Brugger (1965), but it relies on a specific placement of Cartesian axes, which is unclear in the paper. A standard for the placement, though seems implicit now, is known by the ultrasound community, exists elsewhere.

R. N. Thurston (Truesdell, 1984):

By material coefficients, we here mean derivatives of a thermodynamic potential, evaluated at some generalized undistorted, zero-field state. In the simplest case, the only nonzero first derivative of any thermodynamic potential is its derivative with respect to the thermal variable, either temperature or entropy. A hydrostatic pressure at the undistorted state is permissible [cf. Sect. 11.4, especially Eq. (11.35)].

The elastic, dielectric, piezoelectric, and other coefficients of crystals are ordinarily specified with reference to a set of rectangular Cartesian axes that are chosen by convention. The conventional method of relating a set of rectangular axes to the crystallographic axes has been standardized through the work of Bond (1949) and others, and is described in publications of Mason (19501947, 1966).

The original text in the book mistakenly refers to Mason (1950) instead of Mason (1947). The Bond (1949) one is a IRE (now IEEE) standard, the placement seems to be clear only in the 1949 version but not in more recent versions. Mason (1947) is a paper and should be quite easy to find; Mason (1966) is a book that seems to be difficult to find¹.

A more recent reference is The AFLOW standard encyclopedia of crystallographic prototypes and its three relevant papers: Mehl et al., (2017), Hicks et al., (2019, 2021).

Reference

• Brugger, K. (1965). Pure Modes for Elastic Waves in Crystals. Journal of Applied Physics, 36(3), 759–768. https://doi.org/10.1063/1.1714215
• Mason, W. P. (1947). First and Second Order Equations for Piezoelectric Crystals Expressed in Tensor Form. Bell System Technical Journal, 26(1), 80–138. https://doi.org/10.1002/j.1538-7305.1947.tb01312.x
• Mason, W. P. (1950). Optical properties and the electro-optic and photoelastic effects in crystals expressed in tensor form. The Bell System Technical Journal, 29(2), 161–188. https://doi.org/10.1002/j.1538-7305.1950.tb00464.x
• Mason, W. P. (1966). Crystal Physics of Interaction Processes. Academic Press.
• Standards on Piezoelectric Crystals, 1949. (1949). Proceedings of the IRE, 37(12), 1378–1395. https://doi.org/10.1109/JRPROC.1949.229975
• Truesdell, C. (Ed.). (1984). Mechanics of Solids Volume IV: Waves in Elastic and Viscoelastic Solids (Theory and Experiment) (Vol. 4). Springer Berlin Heidelberg. https://link.springer.com/book/9783540131632