On combination of all small-scale theories for nonlinear free vibrations and thermal buckling of nanobeams under thermal loading

Abstract

Nonlinear free vibration characteristics of nanobeams exposed to thermal loading and surface effects are studied in this paper. Within the framework of the Euler–Bernoulli beam theory in conjunction with the nonlocal strain gradient theory, a size-dependent model for nanobeams subjected to thermal loadings has been developed with considering the von Kármán nonlinearity. In the current investigation, size dependencies are captured in the nonlinear free vibrations of nanobeams by using the strain gradient tensor and the curvature tensor. Moreover, the surface elasticity theory is employed to study the effects of surface elasticity on the dynamic behaviors of nanobeams. Governing equations of nonlocal strain gradient nanobeams are obtained using Hamilton’s principle. The influences of nonlocal parameter, length scale parameter resulting from the strain gradient theory, surface elasticity, and temperature on the nonlinear natural frequency of nanobeams are discussed in detail. Moreover, the hardening-softening behaviors of nanoscale beams are investigated in the presence of the curvature tensor. To validate the results of the proposed model for nanobeams, the results obtained in terms of natural frequencies are compared to those from available well-known references.