Nonlocal elasticity and boundary condition paradoxes: a review

Nonclassical continuum mechanics theories have seen a rise in implementation over the past several years due to the increased research into micro-/nanoelectromechanical systems (MEMS/NEMS), micro-/nanoresonators, carbon nanotubes (CNTs), etc. Typically, these systems exist in the range of several nanometers to the micro-scale. There are several available theories that can capture phenomena inherent to nanoscale structures. Of the available theories, researchers utilize Eringen’s nonlocal theory most frequently because of its ease of implementation and seemingly accurate results for specific loading conditions and boundary conditions. Eringen’s integral nonlocal theory, which leads to a set of integro-partial differential equations, is difficult to solve; therefore, the integral form was reduced to a set of singular partial differential equations using a Green-type attenuation function. However, a so-called paradox has arisen between the integral and differential formulations of Eringen’s nonlocal elasticity. For certain boundary and loading conditions, instead of the expected softening effect inherent in nonlocal particle interactions, some researchers have found a stiffening effect. Still, others have found no variation from those results found using classical theories. As such, the discrepancies between the integral and differential forms have been the subject of debate for nearly two decades, with several proposed resolutions published in recent years. This paper serves to review and consolidate existing theories in nonlocal elasticity along with selected theories in nonclassical continuum mechanics, the utilization of Eringen’s nonlocal elasticity in beams, shells, and plates, the existing discrepancies and proposed solutions, and recommendations for future work.