Thermal buckling and free vibration of Timoshenko nanobeams using bi-Helmholtz kernel based integral nonlocal model

This study develops a novel finite element formulation for analyzing thermal buckling and free vibration behaviors of Timoshenko nanobeams, employing a unified strain- and stress-driven two-phase local/nonlocal integral model with bi-Helmholtz kernel. The governing equations are derived using Hamilton’s principle, with nonlocal effects incorporated through equivalent differential formulations and constitutive boundary conditions. Key methodological innovations include explicit formulations of thermal-induced forces (axial, bending, and shear components), transformation of constitutive boundary conditions into equivalent external forces via the minimum potential energy principle, and implementation of a Lagrange multiplier-enhanced finite element approach to handle higher-order boundary variables. The model’s validity is established through comparison with benchmark solutions, followed by systematic parametric studies examining nonlocal effects, thermal loading, and geometric influences. Numerical results demonstrate the model’s effectiveness in capturing size-dependent behaviors across various boundary conditions, providing a robust computational tool for nanobeam analysis.