  One of the advantages of the phase-field method (PFM) is its ability to
incorporate elastic interactions that dominate solid-state processes including
phase transformations and plastic deformation. As mechanical equilibrium is
attained much faster than chemical equilibrium, the former should be used as a
constraint explicitly in deriving the governing equations of time-evolution of
PFM order parameters. Current models for elastically anisotropic and
inhomogeneous media in the literature do not impose such a constraint in their
governing equations. In particular, they ignore the dependence of the total
strain on the order parameters while evaluating the variational derivative of
the elastic energy (VDEE). There is no mathematical proof to support this
treatment and the fundamental thermodynamic consistency of such a models could
be challenged. In this work, we present a rigorous and physically transparent
formulation of PFM governing equations for elastically anisotropic and
inhomogeneous media in which the mechanical equilibrium is explicitly used as a
constraint. From our formulation, we show that VDEE computed in the
Leo-Lowengrub-Jou (1998) (LLJ) model is thermodynamically consistent. We also
show that the Wang-Jin-Khachaturyan (2003) (WJK) model has significant error in
VDEE calculation. We proposed a first-order correction term to the WJK model
which nullifies the error in VDEE and makes the model thermodynamically
consistent. Also, we computed the VDEE by a numerical method with-out using any
assumptions and compared the numerical VDEE with the LLJ, WJK and modified WJK
models.
