Higher Order Gradient Continuum Description of Atomistic Models for Crystalline Solids

We propose an upscaling scheme for the passage from atomistic to continuum mechanical models of crystalline solids. It is based on a Taylor expansion of the deformation function up to a given order and describes the material properties to a higher extent than commonly used continuum mechanical models. In particular, the discreteness effects of the underlying atomistic model are captured. The qualitative properties of the technique are numerically analyzed for the model problem of a one-dimensional atomic chain. The approach is then applied to the real three-dimensional physical example of a silicon crystal. The resulting approximation properties are studied in a stationary setting. Finally, a numerical simulation of the time evolution of the elastic response of crystal silicon is presented.