Abstract
This study presents an analytical framework for uncertainty quantification in the free vibration analysis of porous functionally graded (FG) micro-beams using a double parametric form of Navier’s method. The micro-beam consists of aluminum and alumina phases, with material gradation described by a power-law distribution and uniformly distributed porosity representing inherent material heterogeneity. To capture uncertainties in material properties, Symmetric Gaussian Fuzzy Numbers (SGFNs) are applied to both constituents. Scale-dependent effects at the microscale are modeled using the Modified Couple Stress Theory (MCST). Closed-form expressions for the natural frequencies are derived for the Hinged–Hinged (H–H) boundary condition, yielding fuzzy bounds that reflect the uncertain nature of the material system. A detailed parametric study is performed to investigate the influence of the material gradation index, porosity volume fraction, and the thickness-to-material length scale ratio—on the vibrational characteristics of the micro-beam under uncertainty. The results elucidate the combined impact of size effects, porosity, and material gradation in shaping the dynamic response of FG micro-structures.