Mathematical Modeling and Free Vibration Analysis of Nonhomogeneous Nanobeams with Axially Varying Nonlocal Parameter

This work presents a comprehensive investigation into the free vibration behavior of nonhomogeneous nanobeams, particularly axially functionally graded (AFG) nanobeams, incorporating spatially varying nonlocal effects. The nanobeam is modeled using the Euler–Bernoulli beam theory, while small-scale phenomena are captured through Eringen’s nonlocal elasticity theory. Unlike conventional approaches that assume a constant nonlocal parameter, the present work considers a varying nonlocal parameter along the axial direction, enabling a more realistic representation of nonlocal effects. The material properties of the nanobeam are assumed to vary continuously along the beam’s axis according to a power-law distribution, reflecting the functional gradation of the structure. The governing equations are formulated and solved using the Rayleigh–Ritz method to obtain the frequency parameters. A convergence study is conducted to demonstrate the numerical stability and accuracy of the proposed solution methodology. The obtained results are further validated through comparisons with existing solutions available in the literature for specific limiting cases. Parametric analyses are performed to examine the effects of the power-law exponent and the variable nonlocal parameter on the first four frequency parameters. The results indicate that increasing the power-law exponent or the nonlocal parameters results in a noticeable reduction in the frequency parameters, highlighting the significant roles of material gradation and small-scale effects in the dynamic response. Overall, the study demonstrates that incorporating a spatially varying nonlocal parameter provides enhanced predictive capability for the vibration analysis of AFG nanobeams, offering valuable insights for the design and optimization of advanced nanostructures.