A generalized space-time mathematical homogenization theory for bridging atomistic and continuum scales

A generalized space–time mathematical homogenization theory, which constructs an equivalent continuum description directly from molecular dynamics (MD) equations, is developed. The noteworthy theoretical findings of this work are: (i) the coarse-scale continuum equations (PDEs) obtained from the homogenization of MD equations are identical to those obtained from the classical homogenization of the fine-scale continuum, (ii) the lower order effective continuum properties (such as the fourth-order elasticity tensor) have similar characteristics to those resulting from the homogenization of fine-scale continuum, (iii) the higher order effective properties, such as polarization and dispersion tensors, substantially differ from those of continuum due to the discreteness effect. (Both the heterogeneous continuum and discrete media possess the size effect, but only atomistic medium has the discreteness effect.) Some demonstrative numerical examples are presented to verify the formulation