Harmonizing continuum and discrete theories for monatomic graphene: Uncertainties and challenges

Any nanomaterials with periodic, discrete structure exhibit scale effects, thus a common belief is that direct application of classical continuum theories is skeptical. Many studies reveal that there is significant difference between the classical continuum model and discrete model and thus a variety of modified continuum models have been proposed. Is it really impossible to harmonize the classical continuum and discrete theories? Here we show that there exist two distinct aspects for the concept of material thickness: intrinsic thickness and structural thickness, which correspond to the occupied space by physical particles and non-particle physical effect such as long-range force. For a suspended stacked-layer graphene, the most representative nanomaterial, the structural thickness produced by long-range force becomes a quantity that has a similar order with intrinsic thickness in terms of physical effects. While for monolayer graphene, the structural thickness does not exist because any long-range force vanishes. This discontinuity from mono- to multi-layer graphene leads to a highly controversial issue of applicability for the classical continuum theories. We thus reexamine the feasibility with respect to monolayer graphene and carbon nanotube, and devote to harmonize a missing bridge between the classical continuum mechanics and discrete mechanics models.