A layer-wise theory for the dynamic analysis of functionally graded composite nanobeams under thermal environment using nonlocal boundary conditions and Chebyshev collocation technique

Abstract

In this study, a layer-wise theory satisfying displacement continuity at each layer’s interface and the Chebyshev collocation technique are adopted to analyze the vibration behavior of functionally graded composite nanobeams under non-uniform temperature variation in the thickness direction. The temperature-dependent mechanical properties vary according to the rule of mixture. Boundary conditions are modified by including the effect of size parameter and named as nonlocal boundary conditions. These nonlocal boundary conditions along with Eringen’s nonlocal elasticity theory are developed to capture the effect of size-dependency for such a nanobeam based on the first-order shear deformation theory (FSDT) and physical neutral plane. The energy principle and the variational approach are used to obtain force-moment equilibrium equations, which lead to governing differential equations and local boundary conditions. Two types of sandwich constructions are considered, namely, Type 1 (functionally graded-homogeneous-functionally graded) and Type 2 (homogeneous-functionally graded-homogeneous), by maintaining the material continuity at the interfaces. Natural frequencies are analyzed in the fundamental mode for various values of volume fraction index, thickness of the layers, size scale parameter, and thermal environment. The convergence rate of the Chebyshev collocation technique is also shown over the differential quadrature method.