Nonlinear elastic effects in phase field crystal and amplitude equations: Comparison to ab initio simulations of bcc metals and graphene
C. Hueter, M. Friak, M. Weikamp, J. Neugebauer, N. Goldenfeld, B. Svendsen, R. Spatschek,
Volume: 93. Pages: 21
DOI: 10.1103/PhysRevB.93.214105
Published: 2016
Abstract
We investigate nonlinear elastic deformations in the phase field
crystal model and derived amplitude equation formulations. Two sources
of nonlinearity are found, one of them is based on geometric nonlinearity
expressed through a finite strain tensor. This strain tensor is based
on the inverse right Cauchy-Green deformation tensor and correctly
describes the strain dependence of the stiffness for anisotropic
and isotropic behavior. In isotropic one- and two-dimensional situations,
the elastic energy can be expressed equivalently through the left
deformation tensor. The predicted isotropic low-temperature nonlinear
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 nonlinearity arises if the strain
dependence of the density wave amplitudes is taken into account and
leads to elastic weakening. For anisotropic deformation, the magnitudes
of the amplitudes depend on their relative orientation to the applied
strain.