Freely vibrating nanoplates on nanofoundations

Free vibrations of Kirchhoff axisymmetric nanoplates resting on elastic nano-foundations are investigated accounting for long range interactions through nonlocal methodologies. Size effects are captured by means of the stress-driven integral elasticity. The interaction between plate and supporting medium is modeled by a nonlocal theory according to which the foundation reaction is spatial convolution driven by the displacement field. This formulation represents an advanced methodology with respect to the Winkler (local) approach, leading to a well-posed structure-foundation problem. The governing equation of motion for freely vibrating nanoplates resting on nanofoundations are derived and expressed in nondimensional form, leading to a high-order nonautonomous differential problem. Natural frequencies and mode shapes are computed numerically exploiting the Compound Matrix Method. Case studies involving nanoplates with simply supported and clamped edges are presented to highlight the influence of the characteristic length parameters on the dynamic response. The obtained outcomes show the significant role played by nonlocal effects in accurately predicting the vibrational behavior of nanoplates interacting with elastic substrates.