Nonlinear vibration analysis of FG nano-beams resting on elastic foundation in thermal environment using stress-driven nonlocal integral model

Abstract

This paper comprehensively studies the nonlinear vibration of functionally graded nano-beams resting on elastic foundation and subjected to uniform temperature rise. The small-size effect, playing an essential role in the dynamical behavior of nano-beams, is considered here applying the innovative stress driven nonlocal integral model due to Romano and Barretta. The governing partial differential equations are derived from the Bernoulli–Euler beam theory utilizing the von Karman strain–displacement relations. Using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural frequency for four different boundary conditions is then established employing the Homotopy Analysis Method. The nonlinear natural frequencies, evaluated according to the stress-driven nonlocal integral model, are compared with those obtained by Eringen differential model. Finally, the effects of different parameters such as length, elastic foundation parameter, thermal loading and nonlocal characteristic parameter are investigated. The emergent results establish that when the nonlocal characteristic parameter increases, the nonlinear natural frequencies obtained by the stress-driven nonlocal integral model reveal a stiffness-hardening effect. On the other hand, Eringen's differential law reveals a stiffness-softening effect excepting the case of cantilever nano-beam. Also, increase in temperature and the elastic foundation parameter leads to increase in the nonlinear frequency ratios in Eringen differential model but decrease in the frequency ratios in the stress-driven nonlocal integral model.