Cylindrical thin-shell model based on modified strain gradient theory

In this paper, the cylindrical thin-shell model is developed based on modified strain gradient theory. For this purpose, the study develops the thin shell theory, having considered size effects through modified strain gradient theory. Besides, partial equations of shell motion with classical and non-classical corresponding boundary conditions are derived from Hamilton principle. Finally, by way of example, the free vibration of the single-walled carbon nanotube (SWCNT) is investigated. The study models the SWCNT as a simply-supported shell. Besides, the Navier procedure is used to solve the vibration problem. The results of the new model are compared with those of the couple stress model and the classical theory, leading to the conclusion that the mentioned models are special cases of the modified strain gradient theory. The findings also indicate that the rigidity of the nanoshell in the modified strain gradient theory is greater than that in couple stress model and the classical theory, which leads to the increase in natural frequencies. Furthermore, the effect of the material length scale parameter on the vibration of the nanoshell for different lengths is taken into account.