An atomistic-to-continuum framework for nonlinear crystal mechanics based on asymptotic homogenization

Abstract

Presented is a multiscale methodology enabling description of the fundamental mechanical behavior of crystalline materials at the length scale of a macroscopic continuum (i.e., millimeter resolution) given a characterization of discrete atomic interactions at the nanoscale (i.e., angstrom resolution). Asymptotic homogenization methods permit the calculation of effective mechanical properties (e.g. strain energy, stress, and stiffness) of a representative crystalline volume element containing statistically periodic defect structures. From a numerical standpoint, the theoretical–computational method postulated and implemented here in the context of lattice statics enables prediction of minimum energy configurations of imperfect atomic-scale crystals deformed to finite strain levels. Numerical simulations demonstrate the utility of our framework for the particular case of body-centered-cubic tungsten. Elastic stiffness and energetic properties of periodic unit cells containing vacancies, screw dislocations, and low-angle twist boundaries are computed. Nonlinear aspects of elastic behavior in the context of plastic flow are then modeled from the perspective of atomistic-to-continuum homogenization, following the introduction of a minimal set of kinetic assumptions required to account for the propagation of dislocations across the unit cell at finite deformations.