  In this paper, a unified nonlocal rational continuum enrichment technique is
presented for improving the dispersive characteristics of some well known
classical continuum equations on the basis of atomistic dispersion relations.
This type of enrichment can be useful in a wide range of mechanical problems
such as localization of strain and damage in many quasibrittle structures, size
effects in microscale elastoplasticity, and multiscale modeling of materials. A
novel technique of transforming a discrete differential expression into an
exact equivalent rational continuum derivative form is developed considering
the Taylor's series transformation of the continuous field variables and
traveling wave type of solutions for both the discrete and continuum field
variables. An exact equivalent continuum rod representation of the 1D harmonic
lattice with the non-nearest neighbor interactions is developed considering the
lattice details. Using similar enrichment technique in the variational
framework, other useful higher-order equations, namely nonlocal rational
Mindlin-Herrmann rod and nonlocal rational Timoshenko beam equations, are
developed to explore their nonlocal properties in general. Some analytical and
numerical studies on the high frequency dynamic behavior of these novel
nonlocal rational continuum models are presented with their comparison with the
atomistic solutions for the respective physical systems. These enriched
rational continuum equations have crucial use in studying high-frequency
dynamics of many nano-electro-mechanical sensors and devices, dynamics of
phononic metamaterials, and wave propagation in composite structures.
