Multiscale computational framework for free vibration analysis of boron nitride nanotubes at finite temperature

Abstract

This study proposes a multiscale computational approach that integrates finite temperature effects into the constitutive modeling for evaluating the thermal, mechanical, and free-vibration responses of single-walled boron nitride nanotubes (SWBNNTs). The framework is built upon a temperature-dependent quadratic Cauchy-Born rule, with atomic interactions described using the Tersoff-Brenner potential and various empirical parameter sets. The Helmholtz free energy of the representative unit cell is formulated as the sum of its interatomic potential energy and the thermal energy arising from atomic vibrations at finite temperatures. Stress, moment tensors, and the tangent stiffness matrix are derived by differentiating the Helmholtz free energy density with respect to strain and curvature. A finite element model in cylindrical coordinates is developed using a four-nodded membrane-consistent (4NMC) element, employing a smoothed interpolation technique in the circumferential direction to mitigate membrane locking. The influence of temperature on the natural frequencies of SWBNNTs is thoroughly analyzed, considering changes in nanotube length, radius, and various boundary conditions.