  We use first principles calculations to study the influence of
thermomechanical loads on the energetics of precipitation in magnesium-aluminum
alloys. Using Density Functional Theory simulations, we present expressions of
the energy of magnesium-aluminum binary solid solutions as a function of
concentration, strain and temperature. Additionally, from these calculations,
we observe an increase in equilibrium volume (and hence the equilibrium lattice
constants) with the increase in concentration of magnesium. We also observe an
increase in the cohesive energy of solutions with increase in concentration,
and also present their dependence on strain. Calculations also show that the
formation enthalpy of $\beta$ phase solutions to be strongly influenced by
hydrostatic stress, however the formation enthalpy of $\alpha$ phase solutions
remain unaffected by hydrostatic stress. We present an expression of the free
energy of any magnesium aluminum solid solution, that takes into account the
contributions of strain and temperature. We note that these expressions can
serve as input to process models of magnesium-aluminum alloys. We use these
expressions to report the influence of strains and temperature on the
solubility limits and equilibrium chemical potential in Mg-Al alloys. Finally,
we report the influence of thermomechanical loads on the growth of
precipitates, where we observe compressive strains along the $c$ axis promotes
growth, whereas strains along the $a$ and $b$ directions do not influence the
growth of precipitates.
