Size-dependent thermal buckling and vibrations of double-walled carbon nanotubes through nonlocal Timoshenko beam model embedded in nonlocal elastic foundation

Abstract

This study offers an in-depth analysis of the thermal buckling and vibrational characteristics of double-walled carbon nanotubes (DWCNTs) embedded within a Winkler-type elastic medium. To address size-dependent effects, strain-driven (eD) and stress-driven (sD) two-phase nonlocal-local integral models (TPNIMs) are employed, considering Timoshenko beam deformation, foundation-structure interactions, and thermally induced stresses. The governing equations and corresponding boundary conditions are systematically derived using Hamilton's principle. The integral constitutive relations linking generalized strain fields to nonlocal stress tensors are reformulated into equivalent differential expressions, incorporating constitutive boundary conditions. A significant methodological contribution of this work lies in the derivation of closed-form analytical solutions for nonlocal thermal stress distributions. The generalized differential quadrature method (GDQM) is utilized to numerically determine critical buckling loads and natural vibration frequencies. Parametric studies are conducted to evaluate the interdependent influences of nonlocal scaling parameters, foundation stiffness, and temperature variation on the mechanical behavior of DWCNTs, revealing pronounced size effects that are dependent on the boundary conditions. These findings provide crucial insights into the stability-performance trade-offs inherent in thermo-mechanically coupled DWCNT systems, thereby establishing foundational design principles for the development of nanotube-based NEMS and MEMS devices operating under high-temperature conditions.