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In this study, first, the thin cylindrical shell theory was derived from the modified couple stress theory and, afterwards, the vibration of protein microtubules (MTs) was investigated using the developed model. In order to model protein MTs more precisely, the cylindrical micro-shell model was used. Also, to take account of small size effects, equations of motion were obtained on the basis of the modified couple stress theory. For this purpose, first, using Hamilton's principle, vibration equations of cylindrical shell with boundary conditions were derived from the modified couple stress theory. Finally, the effects of size parameters, MT dimensions, and the medium surrounding on the axial and circumferential vibration frequency of the MT were examined. It should be noted that the results obtained from the cylindrical micro-shell model, unlike those from the beam model, have lower dependency on MT length, but they have extreme dependency on MT thickness and radius. In the end, it is worth noting that the model developed in this study can predict experimental results with greater precision compared to classic models. In other words, this model narrows the gap existing between experimental results and previous models and theories.

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The present study focusses on buckling and post-buckling of graphene-reinforced laminated composite plates subjected to uniaxial and biaxial loadings. Poly-methyl-methacrylate (PMMA) is used for matrix. Depending on the type of graphene distribution in each layer, three patterns are considered for the plate cross-section. Graphene sheets are considered in both perfect and defective forms. Kinematics of the plate is modeled using the first shear deformation theory and for large deformation, von Karman nonlinearity is considered. Mechanical properties of each layer are evaluated using the molecular dynamics simulation. Besides, Halpin–Tsai and rule of mixtures are calibrated for graphene PMMA composite. Stability equations are solved based on the incremental-iterative type of Ritz method. In order to validate the solution procedure, comparison studies are conducted on isotropic plates. Numerical results are presented for four different types of boundary conditions. It is shown that, for all types of boundary condition, X-pattern provides higher buckling load. Furthermore, it is found that plates reinforced by defective graphene sheets with 5% vacancy provide lower buckling and post-buckling resistance with respect to those reinforced by pristine graphene.