On the definitions of effective stress and deformation gradient for use in MD: Hill's macro-homogeneity and the virial theorem

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 paper 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 micro-mechanics and that based on the virial theorem.