Unified nonlocal rational continuum models developed from discrete atomistic equations

摘要

In this paper, a unified nonlocal rational continuum enrichment technique ispresented for improving the dispersive characteristics of some well knownclassical continuum equations on the basis of atomistic dispersion relations.This type of enrichment can be useful in a wide range of mechanical problemssuch as localization of strain and damage in many quasibrittle structures, sizeeffects in microscale elastoplasticity, and multiscale modeling of materials. Anovel technique of transforming a discrete differential expression into anexact equivalent rational continuum derivative form is developed consideringthe Taylor's series transformation of the continuous field variables andtraveling wave type of solutions for both the discrete and continuum fieldvariables. An exact equivalent continuum rod representation of the 1D harmoniclattice with the non-nearest neighbor interactions is developed considering thelattice details. Using similar enrichment technique in the variationalframework, other useful higher-order equations, namely nonlocal rationalMindlin-Herrmann rod and nonlocal rational Timoshenko beam equations, aredeveloped to explore their nonlocal properties in general. Some analytical andnumerical studies on the high frequency dynamic behavior of these novelnonlocal rational continuum models are presented with their comparison with theatomistic solutions for the respective physical systems. These enrichedrational continuum equations have crucial use in studying high-frequencydynamics of many nano-electro-mechanical sensors and devices, dynamics ofphononic metamaterials, and wave propagation in composite structures.