The continuum notions of effective deformation gradient and effective stress for homogenization problems with large deformations are reviewed. The local problem to be homogenized can include inertia effects to allow for a link between continuum homogenization and the estimation of average properties for particle ensembles via molecular dynamics. The focus of this presentation is on the role played by boundary conditions in: defining a meaningful space average of deformation, defining a meaningful space average of stress, and establishing a connection between the idea of effective stress from micromechanics and that based on the virial theorem.
We show that the notion of effective stress borrowed from micromechanics can be used in molecular dynamics (MD). Furthermore, we show that the notion of stress frequently used in MD called virial stress may or may not be considered a mechanical notion of stress depending on how it is defined. In particular we show that using an accepted definition of virial stress, its time average coincides with the time average of the effective stress proposed herein.
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