Finite temperature Cauchy-Born rule and stability of crystalline solids with point defects
T. Luo, Y. Xiang and J. Z. Yang
Multiscale Modeling & Simulation, 19, 1710-1735 (2021).

ABSTRACT
We study the convergence of the elastic deformation from an atomistic model to a
continuum model based on the Cauchy--Born rule for crystalline solids, where
point defects are allowed to exist. We prove, under certain sharp stability
conditions at zero temperature of the perfect lattice, that the solids are
stable when the temperature and defect concentration are both low. Based on the
stability conditions at zero/finite temperatures and with/without defects, we
show that the defected version of the Cauchy--Born rule gives a correct
nonlinear elasticity model in the sense that elastically deformed states of the
atomistic model are closely approximated by solutions of the continuum model
with free energy functionals obtained from the Cauchy--Born rule. Both static
and dynamic problems are considered. The results are focused on the simple
crystals and can be easily extended to complex ones.