A Lagrangian-based continuum homogenization approach applicable to molecular dynamics simulations

The continuum notions of effective mechanical quantities as well as the conditions that give meaningful deformation processes for homogenization problems with large deformations are reviewed. A continuum homogenization model is presented and recast as a Lagrangian-based approach for heterogeneous media that allows for an extension to discrete systems simulated via molecular dynamics (MD). A novel constitutive relation for the effective stress is derived so that the proposed Lagrangian-based approach can be used for the determination of the "stress-deformation" behavior of particle systems. The paper is concluded with a careful comparison between the proposed method and the Parrinello-Rahman approach to the determination of the "stress-deformation" behavior for MD systems. When compared with the Parrinello-Rahman method, the proposed approach clearly delineates under what conditions the Parrinello-Rahman scheme is valid.