A Unified Higher-Order Beam Theory for Free Vibration and Buckling of FGCNT-Reinforced Microbeams Embedded in Elastic Medium Based on Unifying Stress–Strain Gradient Framework

The main object of this research is to formulate the linear free vibration as well as static stability of embedded functionally graded carbon nanotube-reinforced composite microbeams in thermal environment. The nonlocal stress–strain gradient theory in conjunction with the unified higher-order beam theory by considering the temperature dependence of material properties and the initial thermal stresses is used to derive nonclassical governing equations. The eigenvalue problems governing the linear vibration and static stability of microbeams are obtained by using the weak form of partial differential equations and employing Chebyshev–Ritz method. The fast rate of convergence of the method is demonstrated numerically, and its accuracy is verified by comparing the results in the limit cases with existing solutions in the literature. The effects of transverse shear stress distribution along the thickness together with the spring constants of Winkler–Pasternak elastic medium, different distribution patterns of CNTs across the thickness, the temperature dependence of material properties, the temperature rise, boundary conditions, nonlocal stress and strain gradient parameters on the frequency parameters and load-bearing capacity are investigated. Findings show that the effects of Pasternak constant of elastic medium on the natural frequency as well as critical buckling load depend on the boundary conditions. It is also shown that the nonlocal stress and strain gradient parameters have opposite effects on the stiffness. The more effective distribution pattern of CNTs across the thickness which enhances the static stability and vibratory behavior of microbeam is determined as well.