Relationships of a micro-structured beam system and a two-phase nonlocal beam

Abstract

Recent advances in the study of thin beams within the framework of nonlocal elasticity demonstrate a close relationship between micro-structured beam systems and Eringen’s differential-type nonlocal beams. In this paper, we establish relationships between a micro-structured beam system and a two-phase nonlocal beam through the equivalence of natural frequencies under two typical boundary conditions: clamped-free and doubly clamped. The analytical characteristic equations of two-phase nonlocal beams are derived and compared to approximately analytical equations to prevent numerical overflow in cases with small volume fractions of material and nonlocal parameters. This work proposes calibrated procedures that may facilitate the equivalence of material mechanics commonly used in engineering applications.