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Non-linear elastic effects in phase field crystal and amplitude equations: Comparison to ab initio simulations of bcc metals and graphene

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.