Ab initio calculations of third-order elastic coefficients

Chenxing Luo, Jeroen Tromp, Renata M. Wentzcovitch

Third-order elasticity (TOE) theory is predictive of strain-induced changes
in second-order elastic coefficients (SOECs) and can model elastic wave
propagation in stressed media. Although third-order elastic tensors have been
determined based on first principles in previous studies, their current
definition is based on an expansion of thermodynamic energy in terms of the
Lagrangian strain near the natural, or zero pressure, reference state. This
definition is inconvenient for predictions of SOECs under significant initial
stresses. Therefore, when TOE theory is necessary to study the strain
dependence of elasticity, the seismological community has resorted to an
empirical version of the theory.
This study reviews the thermodynamic definition of the third-order elastic
tensor and proposes using an "effective" third-order elastic tensor. We extend
the ab initio approach to calculate third-order elastic tensors under finite
pressure and apply it to two cubic systems, namely, NaCl and MgO. As
applications and validations, we evaluate (a) strain-induced changes in SOECs
and (b) pressure derivatives of SOECs based on ab initio calculations. Good
agreement between third-order elasticity-based predictions and numerically
calculated values confirms the validity of our theory.