Size-dependent vibration in two-directional functionally graded porous nanobeams under hygro-thermo-mechanical loading

In this study, vibrational behavior of a two-directional functionally graded (2D-FG) porous nanobeam is investigated under hygro-thermo-mechanical loading, based on the Euler-Bernoulli beam theory for various boundary conditions. The properties of the 2D-FG nanobeam are assumed to be in the form of power-law theory. Furthermore, using Eringen's nonlocal elasticity theory, Hamilton's principle is utilized to derive the governing equation. Subsequently, the governing equation is converted to an ordinary differential equation using the Galerkin method. The obtained equation is then solved through the Ritz averaging technique for simply-simply supported (S-S), clamped-simply supported (C-S), and clamped-clamped (C-C) boundary conditions to achieve the frequency response of the 2D-FG nanobeam. To validate the obtained equation and the solution procedure, the obtained numerical results are compared with those of a well-known reference in the literature, under similar boundary conditions, material properties, and the values of porosity index and nonlocal parameter, through which the good agreement between the results is demonstrated. Finally, influences of various parameters including moisture (hygro-)effect, porosity, thermal change, nonlocal parameters, and power-law indices in the x - and z -directions are examined for various boundary conditions.