  Athermal resistance to the motion of a phase interface due to a precipitate
is investigated. The coupled phase field and elasticity equations are solved
for the phase transformation (PT). The volumetric misfit strain due to the
precipitate is included using the error and rectangular functions. Due to the
presence of precipitates, the critical thermal driving forces remarkably differ
between the direct and reverse PTs, resulting in a hysteresis behavior. For the
precipitate radius small compared to the interface width, the misfit strain
does not practically show any effect on the critical thermal driving force.
Also, the critical thermal driving force value nonlinearly increases vs. the
precipitate concentration for both the direct and reverse PTs. Change in the
precipitate surface energy significantly changes the PT morphology and the
critical thermal driving forces. The critical thermal driving force shows
dependence on the misfit strain for large precipitate sizes compared to the
interface width. For both the constant surface energy (CSE) and variable
surface energy (VSE) boundary conditions (BCs) at the precipitate surface, the
critical thermal driving force linearly increases vs. the misfit strain
coefficient for the direct PT while it is almost independent of it for the
reverse PT. For larger precipitates, the critical thermal driving force
nonlinearly increases vs. the precipitate concentration for the direct PT. For
the reverse PT, its value for the CSE BCs linearly increases vs. the
precipitate concentration while it is independent of the precipitate
concentration for the VSE BCs. Also, for any concentration, the VSE BCs result
in higher thermal critical driving forces, a smaller hysteresis range, and a
larger transformation rate. The critical microstructure and thermal driving
forces are validated using the thermodynamic phase equilibrium condition for
stationary interfaces.
