A size-dependent vibration analysis of FG porous microbeam using a quasi-3D beam theory based on a modified couple stress theory with a variable MLSP

The aim of this study is to develop an accurate size-dependent model for the free vibration analysis of functionally graded (FG) porous microbeams. To achieve this, a quasi-3D shear beam formulation incorporating Chebyshev polynomial shape functions is proposed and coupled with the modified strain gradient theory. The third-order Chebyshev function (p=3) is adopted as it naturally satisfies zero transverse shear-stress conditions on the both beam surfaces. Material properties, including variable material length-scale parameters (MLSPs), are defined using the mixing rule and expressed as functions of the thickness, porosity coefficient and gradient index, enabling precise representation of material heterogeneity and porosity. Size-dependent governing equations are derived from Hamilton’s principle, and closed-form solutions for simply supported beams are obtained. Validation against existing results confirms the reliability of the proposed approach. A detailed parametric study demonstrates that the gradient index, porosity coefficient and especially the variation of the MLSP significantly influence the vibration response, highlighting the critical importance of accounting for MLSP variability in FG porous microbeam analysis.