A Comprehensive Study of Bending and Stability Responses of 2D-FG Nanobeams Using a Microstructure-Surface Energy-Based Model under Various Boundary Conditions

The size-dependent bending and static stability characteristics of nanobeams made of bi-directional functionally graded materials (2D-FGMs) under different boundary conditions are comprehensively investigated. Based on the modified couple stress theory and surface elasticity theory, the size-dependent model is formulated for 2D-FG Euler-Bernoulli beam. The material properties of the beam smoothly change along both the axial and thickness directions according to power-law distribution. The continuous spatial variations of the single material length scale parameter and the three surface constants are incorporated to describe the effects of microstructure and surface energy, respectively. This model accounts for the axial and transverse displacements, the exact position of the physical neutral plane, and Poisson’s effect. To obtain the static response of the present model, Ritz method is employed by approximating the axial and transverse displacements in terms of polynomial forms. Different boundary conditions, i.e., Simply-simply (S-S), Clamped-clamped (C-C), Clamped-simply (C-S), and Clamped-free (C-F), are considered and satisfied by adding auxiliary functions to the displacement functions. Numerical results with various cases of boundary conditions are performed with an insight to explore the effects of gradient indices in thickness and length directions, surface energy, material length scale parameter, slenderness ratio, and thickness on the static deflection and buckling responses of 2D-FG nanobeams. Results disclose that, the material properties, the surface energy, and microstructure effects have a significant effect on the bending, and buckling responses of 2D-FG nanobeams. Hence, this study can be helpful in the design and optimization of 2D-FG nanobeams in bending and buckling responses.