Derivation of Higher Order Gradient Continuum Models from Atomistic Models for Crystalline Solids

We propose a new upscaling scheme for the passage from atomistic to continuum mechanical models for crystalline solids. It is based on a Taylor expansion of the deformation function and allows us to capture the microscopic properties and the discreteness effects of the underlying atomistic system up to an arbitrary order. The resulting continuum mechanical model involves higher order terms and gives a description of the specimen within the quasi-continuum regime. Furthermore, the convexity of the atomistic potential is retained, which leads to well-posed evolution equations. We numerically compare our technique to other common upscaling schemes for the example of an atomic chain. Then we apply our approach to a physically more realistic many-body potential of crystalline silicon. Here the above-mentioned advantages of our technique hold for the newly obtained macroscopic model as well.