  The quasiharmonic approximation (QHA), in its simplest form also called the
statically constrained (SC) QHA, has been shown to be a straightforward method
to compute thermoelastic properties of crystals. Recently we showed that for
non-cubic solids SC-QHA calculations develop deviatoric thermal stresses at
high temperatures. Relaxation of these stresses leads to a series of
corrections to the free energy that may be taken to any desired order, up to
self-consistency. Here we show how to correct the elastic constants obtained
using the SC-QHA. We exemplify the procedure by correcting to first order the
elastic constants of MgSiO$_3$-perovskite and MgSiO$_3$-post-perovskite, the
major phases of the Earth's lower mantle. We show that this first order
correction is quite satisfactory for obtaining the aggregated elastic averages
of these minerals and their velocities in the lower mantle. This type of
correction is also shown to be applicable to experimental measurements of
elastic constants in situations where deviatoric stresses can develop, such as
in diamond anvil cells.
