Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory

In this study, free vibration behavior of piezoelectric Timoshenko nanobeams in the vicinity of postbuckling domain is investigated based on the nonlocal elasticity theory. It is assumed that the piezoelectric nanobeam is subjected to an axial compression force, an applied voltage and a uniform temperature change. Using Hamilton principle, the governing differential equations of motion incorporating von Kármán geometric nonlinearity and the corresponding boundary conditions are derived and then discretized on the basis of generalized differential quadrature (GDQ) scheme. After solving the parameterized equations using Newton–Raphson technique, a dynamic analysis based on a numerical solution strategy is performed to predict the natural frequencies of piezoelectric nanobeams associated with both prebuckling and postbuckling domains. Numerical results are presented to study the effects of nonlocal parameter, temperature rise and external electric voltage on the size-dependent vibration behavior of piezoelectric nanobeams with clamped–clamped (C–C), clamped-simply supported (C-SS) and simply supported-simply supported (SS-SS) end conditions. It is demonstrated that these parameters may shift the postbuckling domain to higher or lower applied axial loads.