Dynamic analysis of FG porous nano-beams using HOSDT and various porosity and volume fraction models

This study presents a comprehensive framework for analyzing the free vibration behavior of functionally graded (FG) porous nanobeams. A high-order shear deformation theory is employed to formulate the governing equations of motion, incorporating Eringen’s nonlocal differential constitutive relations within the context of a refined three-variable beam theory. The formulation captures small-scale effects through the length scale parameter and accounts for porosity distributions through various models, including uniform, non-uniform, logarithmic non-uniform, and mass-density-based approaches. Additionally, different volume fraction profiles, such as the power-law, Viola–Tornabene four-parameter, and trigonometric models, are considered to accurately represent the material gradation within the nanobeam. A parametric investigation is conducted to elucidate the influence of critical factors, including the nonlocal parameter, the material index, the length-to-thickness ratio, the porosity coefficient, and porosity distribution patterns, on the dynamic response of the nanobeam. The study provides valuable insights into the interplay between small-scale effects, material heterogeneity, and porosity, offering a comprehensive understanding of their collective impact on the vibration characteristics of FG porous nanobeams.