A nonlocal continuum model based on atomistic model at zero temperature

In traditional continuum theories, constitutive relations such as generalized Hooke's law and the Fourier law for heat conduction, are representations of macroscopic phenomenons. These theories have severe limitations as the length scale of the structure becomes close to the atomistic dimensions. The failure of the traditional continuum theories at atomistic scale can be attribute to ignoring of the length scale induced by intrinsic non-locality of atomistic interactions. The primary objective of this work is to construct a continuum model strictly based on the atomistic model by passing any empirical constitutive law and any local homogeneous assumption such as Cauchy-Born rule. The intrinsic non-locality of the atomistic interactions are explicitly built into the model to make the model applicable to cases of strongly nonuniform deformation at the atomistic scales. When the deformation is homogeneous, the model reduces to that derived from Cauchy-Born rule. The present continuum model is expected to work well at atomistic scales. It is also consistent with the traditional continuum as the length scale approach to macro scale.