Vibration analysis of AFG nanobeams with elastic boundary constraints carrying a concentrated mass at an arbitrary position based on the stress-driven nonlocal integral model with discontinuity

We present a study on the size-dependent free vibration and frequency shifts of axially functionally graded (AFG) nanobeams functioning as resonant mass sensors. The nanobeam resonators are subject to general elastic boundary constraints, and the influence of axial-lateral vibration coupling induced by an attached concentrated mass is taken into account. Unlike traditional analyses that rely on the mathematically ill-posed strain-driven nonlocal theory, the present work adopts the well-posed stress-driven nonlocal integral elasticity model for effectively capturing the size-dependent mechanical behavior (stiffness-hardening effect), while eliminating the inherent conflict between the constitutive relation and structural equilibrium. Owing to the domain discontinuity introduced by the attached mass, both the standard and constitutive matching conditions at the mass attachment point are established. The generalized differential quadrature method (GDQM) is subsequently employed to discretize the differential problem governing the dimensionless equations of motion and the associated boundary/continuity constraints. After validating the proposed modeling framework against available literature for simplified cases, extensive parametric investigations are conducted. The combined effects of the nonlocal parameter, AFG index, magnitude and position of the concentrated mass, and the stiffness of the elastic boundary constraints on the resonant frequencies and sensor sensitivity (frequency shift ratios) are thoroughly explored. The results offer rigorous theoretical guidance for evaluating and optimizing advanced AFG micro/nanomechanical mass-sensing devices.