Well-posed two-phase local/nonlocal integral model for vibration and frequency analysis of edge-cracked functionally graded graphene platelet-reinforced composite microbeams with piezoelectric actuator layers

The well-documented inconsistencies inherent in nonlocal differential models have motivated the development of the two-phase local/nonlocal integral formulation as a theoretically robust alternative. The present work marks the first application of this paradox-free framework to the size-dependent free vibration analysis of functionally graded graphene platelet-reinforced composite microbeams with surface-bonded piezoelectric actuator layers. A distinguishing feature of this work is the simultaneous treatment of size effects in both mechanical bending and electromechanical axial deformation, achieved via an equivalent differential representation of the well-posed two-phase local/nonlocal integral model. The crack effects are incorporated by modeling the cracked section as a massless rotational spring whose stiffness is related to the stress intensity factor at the crack tip. Numerical solutions are obtained via the generalized differential quadrature method along with an interpolation quadrature scheme, which enables accurate frequency determination for beams with different boundary supports. After thorough validation, the study systematically explores the influence of nonlocal parameters, material distribution patterns, weight fraction, geometry, and external voltage on the vibrational response of cracked beams, providing useful guidance for health monitoring and design optimization of functionally graded graphene platelet-reinforced composite material-based micro/nano-electromechanical systems (MEMS/NEMS) devices.