A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix

Abstract In this study, vibration formulation is presented for nano-scaled beam embedded in an elastic matrix under the effect of thermal environments. The effect of length scale is investigated using Eringen’s nonlocal elasticity theory. The governing equations are obtained by using Hamilton’s principle and variational approach. Finite element formulation has been achieved based on the nonlocal Euler–Bernoulli beam theory for nano-scaled beam. Galerkin method of weighted residuals is considered for development the global stiffness and mass matrices via Hermitian cubic shape functions. The residue is minimized over the elements, after that the shape function is applied to the obtained equation. The influences of the Pasternak foundation parameter, small scale parameter, mechanical properties of material and thermal effect on vibrational frequency are investigated. As a special case, some results have also been given for silicon carbide nanowires.