Free vibration analysis of Euler–Bernoulli curved beams using two-phase nonlocal integral models

Abstract

Eringen’s nonlocal elastic model has been widely applied to address the size-dependent response of micro-/nanostructures, which is observed in experimental tests and molecular dynamics simulation. However, several recent studies have pointed out that some inconsistent results appear while applying it in the analysis of bounded structures, which indicates that it is necessary to adopt other suitable models. In this work, both the well-posed strain-driven and stress-driven two-phase local/nonlocal integral models are used to study the size effect in the free vibration of Euler–Bernoulli curved beams. The governing equations of motion and the associated boundary conditions are derived on the basis of Hamilton’s principle. The two-phase nonlocal integral relation is transformed into an equivalent differential law with two constitutive boundary conditions. Using the generalized differential quadrature method, the governing equation in terms of displacements is solved numerically. The vibration frequencies of the beam under different boundary conditions are obtained and validated by comparing with those existing results. For all boundary conditions, the nonlocal related parameters of the two types of two-phase nonlocal strategies show consistent softening and stiffening effects on vibration response, respectively. Moreover, the effect of the curvature radius of the beam is also investigated.