Thermal vibrations of double-walled carbon nanotubes embedded in elastic foundation using two-phase local/nonlocal integral models

This paper investigates the thermal vibration characteristics of double-walled carbon nanotubes (DWCNTs) embedded in an elastic medium. Unlike most existing studies, we employ both strain-driven (eD) and stress-driven (sD) two-phase local/nonlocal integral models (TPNIMs) to account for nonlocal effects in the beam deformation, elastic foundation response, and thermally induced stresses simultaneously. The governing equations and associated boundary conditions are rigorously derived through Hamilton’s principle, establishing a complete thermomechanical formulation. The constitutive framework transforms integral relations between generalized strain and nonlocal stress fields into equivalent differential forms through systematic incorporation of constitutive boundary constraints. Notably, we provide closed-form solutions for nonlocal thermal stress components, capturing temperature-dependent effects. Numerical solutions for the vibration frequencies are obtained using the generalized differential quadrature method (GDQM). Our results demonstrate the significant influence of the nonlocal parameters, elastic foundation stiffness and thermal stress magnitude on the vibrational response. These effects are quantified for different boundary conditions, providing new insights into the thermo-mechanical behavior of DWCNTs.