Dynamics of two-dimensional functionally graded tapered Timoshenko nanobeam in thermal environment using nonlocal strain gradient theory

Abstract

In this contribution, we study the free vibration of non-uniform nano-size beams in thermal environment. In order to capture the size-dependent effects, we adopt nonlocal strain gradient theory within a Timoshenko beam model. The nanobeam is made of functionally graded materials, which properties are both temperature and porosity dependent and vary continuously along the length and thickness directions. The governing equations as well as boundary conditions are obtained according to a variational approach, which is solved numerically using the generalized differential quadrature method. We compute the natural frequencies and analyze the sensitivity of the vibration response for different non-uniformity, power indices along x and z-directions, porosity coefficients, small-scaling parameters, thermal effect, and geometry conditions. We also study the influence of various boundary conditions including simply-supported, clamped or a combination of them. We find for all boundaries that an increasing nonlocal (strain gradient) parameter leads to decreasing (increasing) natural frequencies.