On the resonance of functionally graded nanoplates using bi-Helmholtz nonlocal strain gradient theory

Up to now, there is no a unified model for the force resonance problem of a three-directional functionally graded material (3D-FGM). This contribution tries to present an investigation on the resonance behavior of Kirchhoff nanoplates. To capture the small-scale effects on the resonance deflection of nanoplates, the bi-Helmholtz nonlocal strain gradient theory incorporating three small-scale parameters is adopted. Governing equations of the 3D-FGM nanoplate are derived using the variational approach. Afterwards, an analytical method based on Navier series is utilized to obtain the closed-form solution of the resonance deflection of nanoplates whose properties vary along the thickness direction. The effects of some parameters such as constant material parameters, aspect ratio, and small-scale parameters are investigated in detail. Both low- and high-order nonlocal parameters exhibit a "stiffness-softening" effect; the resonance deflection will move to the lower load frequencies as either low- or high-order nonlocal parameter increases. The strain gradient parameter has, however, a "stiffness-hardening" effect, and the resonance deflection will move to the higher load frequencies if considering an increase in the strain gradient parameter. To the best of the authors' knowledge, it is for the first time that the governing equations of 3D-FGM nanoplates using bi-Helmholtz nonlocal strain gradient theory are derived, and the resonance phenomena is investigated for the bi-Helmholtz nonlocal strain gradient nanoplates. (C) 2019 Elsevier Ltd. All rights reserved.