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.