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.