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

This paper is devoted to the theoretical study of the dynamic response of non-cylindrical curved viscoelastic single-walled carbon nanotubes (SWCNTs). The curved nanotubes are largely used in many engineering applications, but it is challenging in understanding mechanically the dynamic response of these curved SWCNTs when considering the influences of the material viscosity. The viscoelastic damping effect on the dynamic response is considered here by using the Kelvin-Voigt viscoelastic model. A modified shear deformation beam theory is here employed to formulate the governing partial differential equations. When the SWCNTs are considered in a small scale model, quantum impacts are important for a correct evaluation of the mechanical response of the nanosystem. This is here investigated by embedding the well-known nonlocal strain gradient approach into the governing equations. The extracted equations are solved by utilizing the Galerkin analytical approach in which the governing partial differential equations are reduced to ordinary differential equations and numerical findings are achieved for various boundary conditions. In order to evaluate the efficiency of the proposed theory, the outcomes in terms of natural frequencies of the vibrating nanotubes are verified with respect to the available literature. It follows a vast systematic study, where several parameters are varied to investigate the influences of geometrical properties involving different polygons of the SWCNTs on the dynamic response.