This paper presents a mathematical model and a computational approach for the thermal post-buckling and free vibration in the vicinity of the buckled equilibrium position of microbeams based on the modified couple stress Euler-Bernoulli beam theory and geometrically accurate relation. The governing equations for the whole analysis are established with a intrinsic material length scale parameter to capture the size effect. These equations, in conjunction with the corresponding boundary conditions, are decomposed into two two-point boundary value problems, which are solved using the shooting method. For static deformation, the geometric nonlinearity is involved and the size dependent postbuckling configuration is obtained as a function of temperature rise. For dynamic one, the small amplitude free vibration about the prebuckled position is developed following an assumed harmonic time mode, and the length scale and temperature rise dependent fundamental natural frequency is presented. Numerical computations are executed to illustrate the size dependency in the thermal postbuckling behaviors and fundamental frequency of the vibration around the buckled configuration of microbeams.

Thermal postbuckling analysis is presented for 3D braided composite cylindrical shell of finite length subjected to a uniform temperature rise. Based on a micro-macro-mechanical model, a 3D braided composite may be as a cell system and the geometry of each cell is deeply dependent on its position in the cross-section of the cylindrical shell. The material properties of epoxy are expressed as a linear function of temperature. The governing equations are based on Reddy's higher order shear deformation shell theory with a von Kármán-Donnell-type of kinematic nonlinearity and including thermal effects. A singular perturbation technique is employed to determine the buckling temperatures and postbuckling behaviors of 3D braided composite cylindrical shells. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, braided composite cylindrical shells with different values of geometric parameter and of fiber volume fraction. The results show that the shell has lower buckling temperatures and postbuckling equilibrium paths when the temperature-dependent properties are taken into account. The results reveal that the fiber volume fraction, braiding angle and the shell geometric parameter have a significant effect on the thermal buckling and postbuckling behavior of braided composite cylindrical shells.