  We investigate non-linear elastic deformations in the phase field crystal
model and derived amplitude equations formulations. Two sources of
non-linearity are found, one of them based on geometric non-linearity expressed
through a finite strain tensor. It reflects the Eulerian structure of the
continuum models and correctly describes the strain dependence of the
stiffness. In general, the relevant strain tensor is related to the left
Cauchy-Green deformation tensor. In isotropic one- and two-dimensional
situations the elastic energy can be expressed equivalently through the right
deformation tensor. The predicted isotropic low temperature non-linear elastic
effects are directly related to the Birch-Murnaghan equation of state with bulk
modulus derivative $K'=4$ for bcc. A two-dimensional generalization suggests
$K'_{2D}=5$. These predictions are in agreement with ab initio results for
large strain bulk deformations of various bcc elements and graphene. Physical
non-linearity arises if the strain dependence of the density wave amplitudes is
taken into account and leads to elastic weakening. For anisotropic deformations
the magnitudes of the amplitudes depend on their relative orientation to the
applied strain.
