Place: GAMM 2010 - 81st Annual Meeting of the International Association of Applied Mathematics and Mechanics, Karlsruhe, Germany
Bernhard Eidel, Technical University Darmstadt, Germany
Free surfaces largely determine the mechanical properties and overall materials behaviour of crystalline nanostructures. Atoms residing at free surfaces feature an under-coordination and hence differ in bonding properties from their
bulk counterparts. As a consequence, typical effects are observed at free surfaces like relaxation and reconstruction which play an eminent role in a variety of natural and industrial processes like e.g. in nanoscale contact.
The key challenge in multiscale modelling of crystalline surfaces at the nanoscale is to balance the opposite requirements of accuracy and efficiency. More specifically, three main modelling requirements can be formulated; (i) a nonlocal theory must be employed in order to account for the asymmetry in atomic bonding at or close to free surfaces; (ii) a coarse-grained description using e.g. finite elements is necessary to ensure efficiency; (iii) numerical quadrature in a discretization method must exhibit collocation points for explicit energy-/force-calculation which reside directly at surfaces as the region of interest.
Standard models and methods in discrete mechanics as well as in continuum mechanics exhibit defficiencies with respect to these requirements. Fully atomistic resolution in molecular dynamics/statics can achieve high accuracy by virtue of the nonlocal interatomic potentials employed, the computational burden however may become prohibitive. Continuum models in contrast, which accurately represent bulk or macroscopic material behaviour, fail to account for surface
e ects without the introduction of an internal length scale dependence.
The present contribution aims to achieve predictive simulations of free surfaces of metallic solids. The proposed modelling approach is a fully nonlocal quasicontinuum method which belongs to the class of concurrent multiscale methods. The method's key feature of atomic coarse-graining with adaptive resolution meets the above requirements (i)-(iii). Representative numerical examples illustrate the method's capacity to account for anisotropic surface relaxations in 3d with high accuracy and considerable efficiency.