Fractional nonlocal strain-gradient isogeometric analysis of porous functionally graded piezoelectric microplates resting on elastic substrates

Abstract

Existing strain-gradient models for piezoelectric microplates often capture local microstructural size effects but do not simultaneously represent long-range spatial interactions, porosity-induced material degradation, electromechanical coupling, and elastic-substrate effects within a unified computational framework. This paper develops a fractional nonlocal strain-gradient isogeometric formulation for the free vibration analysis of porous functionally graded piezoelectric microplates resting on elastic substrates. The plate kinematics are described using higher-order shear deformation theory, while the fractional Laplacian operator is introduced to model spatial nonlocality and the strain-gradient term accounts for intrinsic length-scale stiffening. The governing equations are derived from Hamilton's variational principle by incorporating thickness-dependent elastic, piezoelectric, dielectric, and density properties. The Winkler-Pasternak foundation model is used to represent substrate interaction, and isogeometric analysis is adopted to satisfy the higher-order continuity requirements of the enriched formulation. The proposed model is validated against published benchmark results in the classical and strain-gradient limits, with percentage-error comparisons included. Parametric results show that porosity reduces the natural frequencies, whereas strain-gradient effects and elastic foundation stiffness increase them. The fractional order, nonlocal interaction length, applied electric voltage, and porosity pattern significantly influence the effective stiffness and vibration response. The proposed formulation provides a physically consistent framework for designing smart porous piezoelectric microplates used in microelectromechanical systems, sensors, actuators, and micro-energy harvesting systems.