Dynamic modeling of non-cylindrical curved viscoelastic single-walled carbon nanotubes based on the second gradient theory

This paper is devoted to the theoretical development and dynamic responses of curved viscoelastic single-walled carbon nanotubes (SWCNTs). The material’s viscoelastic damping effect is taken into account by using the Kelvin-Voigt viscoelastic material model. A modified shear deformation beam theory which was developed and verified before, is employed to formulate the governing partial differential equations. When a model is considered in a small scale, quantum impacts are importantly required to be taken into investigation. To consider the quantum impacts in the carbon nanotubes for such small scale models, the well-known nonlocal strain gradient approach is embedded into the governing equations. The extracted equations are solved by utilizing the Galerkin analytical approaches that the obtained partial differential equations are reduced to ordinary differential equations and numerical results are obtained for various boundary conditions. In order to evaluate the proposed theory’s validity, the outcomes in terms of natural frequencies of a nanotube in vibration are compared with those from several available well-known references. Following the validation, several parameters are investigated to show the influences of geometrical properties including different polygons of the nanotube on the dynamic responses. The results can have advantages for mechanical design of carbon nanotubes in nanotechnology.