Events
Place: Materials Science and Engineering 2008, Nürnberg
Bernhard Eidel
This talk presents a novel version of the quasicontinuum (QC) method coupling fully atomistic resolution with continuum length scales of crystalline solids at zero temperature. The QC method and variants belong to the family of concurrent multiscale methods which perform an upscaling via coarse-graining with adaptive resolution. The present approach draws on the seminal work of Knap and Ortiz [1] but di®ers from this precursor in some notable aspects. Opposed to Knap and Ortiz, the energy instead of forces is subject to a cluster-based sampling scheme [2]. We show that only the present ansatz endows the QC theory with a variational structure leading to conservative forces as indicated by symmetric sti®ness matrices. As an implication of this property we show that energy-sampling strictly preserves the symmetry in atomic interactions in the entire crystal, whereas force-sampling generally does not. Starting from a well dened total potential guarantees the existence of an equilibrium state provided that the total potential exhibits a minimum. Moreover, standard minimization methods like CG methods can directly be applied. We analyze the numerical errors in the present approximation scheme, which have their origin in the main approxi- mation steps, in discretization and in numerical quadrature.We show that the method enables a truly seamless scale transition between fully atomistic resolution and coarse-grained regions and formulate conditions to achieve this goal. Nanoindentation is chosen as a paradigmatic problem for concurrent multiscale methods in order to demonstrate the performance of the present method and to assess it in comparison with lattice statics. We show that the novel QC approach accurately captures significant details of the material's behavior (e®ect of free surfaces, force-depth curve, load level and locus of dislocation nucleation) but for a small fraction of the computational costs of its fully atomistic counterpart. We conclude by an outlook to future research directions, where the QC method is expected to achieve reliable predictions of material behavior at the nano- and micro-scale. References 1 Knap, J. and Ortiz, M. (2001): Analysis of the quasicontinuum method. J. Mech. Phys. Solids 49, 1899{1923. 2 Eidel, B. and Stukowski, A. (2007): A variational formulation of the qua- sicontinuum method based on energy sampling in clusters. submitted for publication.
