Uncertainty quantification in free vibration of porous functionally graded micro-beams using double parametric form based Haar wavelet Discretization Method

This study presents a comprehensive investigation into uncertainty quantification in the free vibration analysis of functionally graded (FG) micro-beams using a double parametric form-based Haar Wavelet Discretization Method (HWDM). The spatial variation of Young’s modulus and mass density across the beam’s thickness is characterized by a power-law distribution, with the FG micro-beam composed of two constituent materials of metallic phase and ceramic phase (here aluminum (Al) and alumina (Al₂O₃) are taken), incorporating uniformly distributed porosity to reflect material inhomogeneities. Material uncertainties are modeled using Symmetric Gaussian Fuzzy Numbers (SGFNs) for both the metallic and ceramic constituents. To accurately capture size-dependent mechanical behavior at the microscale, the Modified Couple Stress Theory (MCST) is employed. The numerical robustness and accuracy of HWDM are verified through pointwise convergence studies. To further assess the influence of material uncertainty, a Monte Carlo Simulation Technique (MCS) is utilized, generating a large ensemble of random samples within the defined fuzzy bounds to estimate the natural frequencies. The natural frequencies obtained from HWDM, represented as Lower and Upper Bounds (LB and UB), show excellent agreement with those derived from the MCS, thereby validating the proposed fuzzy-based approach. Additional validation is performed by comparing HWDM results with those from Navier’s method under the Hinged-Hinged (H-H) boundary condition, further demonstrating the accuracy of the present formulation. A detailed parametric study is conducted to explore the effects of the power-law exponent, porosity volume fraction index, and thickness-to-material length scale ratio on the natural frequencies under fuzzy uncertainty. The investigation is carried out across multiple classical boundary conditions—Hinged-Hinged (H-H), Clamped-Hinged (C–H), and Clamped-Clamped (C–C). Physical interpretations of the observed trends are provided, highlighting the complex interplay between material gradation, porosity, size effects, and uncertainty in the dynamic response of FG micro-structures.