  A general procedure to investigate the elastic response and calculate the
elastic constants of stressed and unstressed materials through continuum field
modeling, particularly the phase field crystal (PFC) models, is presented. It
is found that for a complete description of system response to elastic
deformation, the variations of all the quantities of lattice wave vectors,
their density amplitudes (including the corresponding anisotropic variation and
degeneracy breaking), the average atomic density, and system volume should be
incorporated. The quantitative and qualitative results of elastic constant
calculations highly depend on the physical interpretation of the density field
used in the model, and also importantly, on the intrinsic pressure that usually
pre-exists in the model system. A formulation based on thermodynamics is
constructed to account for the effects caused by constant pre-existing stress
during the homogeneous elastic deformation, through the introducing of a
generalized Gibbs free energy and an effective finite strain tensor used for
determining the elastic constants. The elastic properties of both solid and
liquid states can be well produced by this unified approach, as demonstrated by
an analysis for the liquid state and numerical evaluations for the bcc solid
phase. The numerical calculations of bcc elastic constants and Poisson's ratio
through this method generate results that are consistent with experimental
conditions, and better match the data of bcc Fe given by molecular dynamics
simulations as compared to previous work. The general theory developed here is
applicable to the study of different types of stressed or unstressed material
systems under elastic deformation.
