Free vibration and wave propagation of thick plates using the generalized nonlocal strain gradient theory

Document Type : Research Article


Faculty of Mechanical Engineering, Urmia University of Technology



In this paper, a size-dependent first-order shear deformation plate model is formulated in the framework of the higher-order generalized nonlocal strain-gradient (GNSG) theory. This model
employs two nonlocal parameters and a strain-gradient coefficient to capture the both higher-order nonlocal stress-gradient and strain-gradient effects in nanostructures. The presence of these different scale parameters renders a unified model, which is able to predict both increase and reduction of stiffness in nanoplates. The governing equations are developed for free
vibration of first-order shear deformation plates using Ritz method. The dispersion relations for the GNSG plate model is also derived. Several numerical examples are studied to show the efficiency, competence and accuracy of the proposed model. To ensure the applicability of the presented GNSG plate model, the results are compared with the experimental data available in the scientific literature. It is found that the effects of scale parameters on the wave frequencies are significant at high wavenumbers and ratio of any pair of these parameters is the main criterion for the correct study of size effects. The results show that the reduced nonlocal strain-gradient (RNSG) model and the GNSG model diverge in higher vibration modes


  • A general nonlocal plate model is developed via nonlocal strain gradient theory.
  • Natural frequencies of the nanoplate are calculated using a Ritz procedure.
  • Three material length scales are used in the model to capture the size effects.
  • Rise and fall in the nanoplate stiffness may happen duo to the length scales.
  • The applicability of the present model is tested by experimental data.


Main Subjects

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