Effective elastic constants of solids under stress: Theory and calculations for helium from 11.0 to 23.6 GPa

We extend earlier formulas for calculating elastic constants of crystalline and amorphous solids to the case when the reference state for measuring strain is a stressed state of the system. In the case when the system is stressed, the thermodynamic definition of elastic constants and the coefficients appearing in the equations of motion for small-amplitude elastic waves are not the same, and one defines effective elastic constants. The general thermodynamic definitions of effective elastic constants are presented and related to statistical-mechanics fluctuation formulas which have proven to be of use in calculating elastic constants of unstressed solids in molecular dynamics. In order to illustrate the use of the resulting formulas, we calculate the effective elastic constants for a 576-particle hcp crystal of helium from 11.0 to 23.6 GPa using a potential that is parametrized from shock-wave data. We show that the equation of state on the 300-K isotherm calculated using molecular dynamics gives results equivalent to those obtained using quasiharmonic lattice dynamics. The calculated effective elastic constants of the helium crystal show an increase of 100% over the above pressure range. The molecular-dynamics system is observed to melt as the pressure is lowered towards 10.5 GPa, whereas the observed melting pressure is 11.5 GPa. An interesting aspect of the melting is that it occurs very rapidly and reproducibly as the pressure is lowered to 10.5 GPa, whereas usually in small molecular-dynamics systems with periodic boundary conditions melting is sluggish.