Refined multiscale simulations for constitutive behaviour and free vibration analysis of carbon nanostructures at finite temperature

Abstract

An atomistic–continuum coupled approach in conjunction with finite element method is employed to investigate the free vibration behaviour of the graphene and carbon nanotubes at finite temperature. The coupling of atomic–level deformations (bond length, bond angles, dihedral angles) to the continuum level is established through the kinematics of the quadratic-type Cauchy–Born rule. The governing equations at the continuum scale are solved through the finite element method. In addition, a new set of empirical parameters for the Tersoff–Brenner potential is also proposed and investigated to accurately predict the constitutive behaviour of the carbon nanomaterials. The findings for the constitutive law obtained using a new set of empirical parameters are also compared with the two formerly proposed sets of empirical parameters and DFT-based calculations. The accuracy and efficacy of the new set of empirical parameters are tested for thermal properties, elastic properties, constitutive behaviour at finite strain, and free vibration response of carbon nanostructures at finite temperatures. Compared to the quantum mechanics-based theoretical calculations, the newly proposed set of empirical parameters yields more promising results than those predicted using the other two sets of empirical parameters and are found to be close to those reported in DFT-based continuum models. © 2026 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.