Size-dependent free vibration of porous piezoelectric microplate resting on an elastic substrate using MSGT, HSDT, and IGA

This work presents the first comprehensive study combining higher-order shear deformation theory (HSDT), modified strain gradient theory (MSGT), and the isogeometric approach (IGA) to investigate the size-dependent free vibration behavior of functionally graded porous piezoelectric (FGPP) microplates. The microplate is composed of a piezoelectric material, with porosity distributed across the thickness following three distinct patterns: symmetric I, symmetric II, and uniform. Hamilton's principle, combined with HSDT and MSGT, is employed to derive the governing equations of the piezoelectric microplate. The natural frequencies of the FGPP microplates are obtained by solving the explicit governing equations using the isogeometric analysis approach. Furthermore, the effects of length scale parameters, porosity distribution patterns, porosity coefficient, external electric voltage, and boundary conditions on the vibrational behavior of the FGPP microplate are thoroughly investigated. The proposed method offers accurate size-dependent predictions and provides new reference solutions for the optimal design of advanced piezoelectric microstructures.