Airy stress based nonlinear forced vibrations and internal resonances of nonlocal strain gradient nanoplates

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This paper presents a novel investigation of nonlinear forced vibrations and internal resonance in nonlocal strain gradient nanoplates, taking into account in-plane motions. The study comprehensively models the nanoplate structure, employing Kirchhoff's plate theory, nonlocal elasticity theory, and strain gradient elasticity theory. The incorporation of the Airy stress function allows for the consideration of in-plane displacements. The large amplitude deformations in the nanoplate are described using the von Kármán geometric nonlinearity model, and the equations of motion are derived using the force-moment dynamic balance method. The coupled equations of motion are solved using the Galerkin scheme and a dynamic equilibrium technique. Detailed discussions are provided on the influence of nonlocal and strain gradient parameters on the nonlinear vibration response of the Airy stress nanostructure. Moreover, it is demonstrated that specific combinations of nonlocal and strain gradient parameters lead to various types of internal resonance (including two-to-one and three-to-one internal resonances), significantly affecting the nonlinear frequency responses of the nanoplate, resulting in rich nonlinear behaviours. This study contributes to the understanding of the complex dynamics of nanoplates subjected to external time-dependant loads and offers valuable insights for the design of diverse nanoplate systems, including graphene sheets, which are relevant to nanoelectromechanical systems (NEMS) applications.