Elastic Buckling and Free Vibration of Functionally Graded Piezoelectric Nanobeams Using Nonlocal Integral Models

Abstract

Previously it was shown that nonlocal piezoelectric differential models will lead to inconsistent bending response for Euler–Bernoulli beams under different loadings and boundary conditions. In this paper, the general strain- and stress-driven two-phase local/nonlocal integral piezoelectric models are applied to analyze the elastic buckling and free vibration of functionally graded piezoelectric (FGP) Euler–Bernoulli beams under different boundary conditions. The differential governing equations and standard boundary conditions are derived by the Hamilton’s principle. The relations between the general strain and general nonlocal stresses are expressed as integral equations, which are further transformed equivalently to differential forms with constitutive boundary conditions. Several nominal variables are introduced to simplify the differential governing and constitutive equations, as well as the standard and constitutive boundary conditions. The general differential quadrature method is applied to obtain the numerical results of buckling loads and vibration frequencies of the FGP Euler–Bernoulli beam. Numerical results show that general strain- and stress-driven two-phase local/nonlocal integral piezoelectric models will lead to consistently softening and toughening size-dependent responses, respectively.