  Non-hydrostatic stress has a peculiar effect on the phase equilibrium between
solids and liquids. This was already pointed out by Gibbs. Gibbs derived his
formulation of the condition for liquid-solid coexistence applying a surface
accretion process without imposing chemical equilibrium between liquid and
solid. Adding particles to the bulk of a solid was not possible in his view at
the time. Chemical potentials for solids were later introduced by material
scientists. This required extending chemical and mechanical equilibrium with a
third condition involving a relation between grand potential densities
controlling the migration of the interface. These issues are investigated using
a non-linear elastic continuum model (technically an open compressible
neo-Hookean material) developed in a previous publication (M. Sprik, J. Chem.
Phys. 155, (2021) 244701). In common with a liquid, the grand potential density
of the model is equal to minus the mean pressure even if the stress is
non-hydrostatic. Applying isothermal compression normal to a liquid-solid
interface initially in hydrostatic equilibrium drives the system away from
coexistence. We derive the Gibbs-Thomson correction to the pressure of the
liquid required to restore phase equilibrium. We find that the coupling between
chemical potential of the solid and shear stress is a purely non-linear effect.
