Nanoscale Laws of Motion and Governing Equations for Atomic and Continuum Scales

Abstract The atomic structure of nanomaterials is reviewed, along with the length scales. Nanoscale laws of motion and governing equations with atomic potentials are described for atomic scale. Size effects in nanomaterials and multiscale variations in material properties and structure are discussed, along with the limitations of continuum mechanics. A nanoscale analog of Newton’s friction law is introduced for the interfacial sliding of atomic lattices of two nanomaterials. The dependence of material properties on the length scales is discussed, along with the size effects and nanoscale homogenization. A criterion for nanoscale homogenization is presented for crystalline nanomaterials and carbon nanotubes. The definition of stress for nanostructures and the formulae for its averaging at nanoscale are examined, with the limitations on the size of representative volume elements. Continuum beam and shell models for carbon nanotubes are presented, along with the ranges of their applicability for nanoscale buckling.