Second Strain Gradient Finite Element Analysis of Vibratory Nanostructures Based on the Three-Dimensional Elasticity Theory

By introducing the C2 continuous hexahedral element, the free vibration finite element analysis of the nanobeam and nanoplate structures is reported based on the second strain gradient (SSG) theory and three-dimensional elasticity model. The SSG elasticity presents the powerful higher-order continuum theory which can be efficiently used to capture the size-effect on the nano-objects. The finite element discretization procedure is performed within Hamilton's principle where the quadratic matrix version of the strain and kinetic energies are derived on the basis of the three-dimensional SSG elasticity model. In the proposed C2 continuous hexahedral element, the values of the displacement field and the associated higher-order derivatives are considered as the nodal values to satisfy the continuity conditions. As the case studies, the free vibration of the nanobeams and rectangular nanoplates is investigated. Different results are outlined to show the efficiency and convergence of the present model. The influences of the involved parameters on the natural frequencies of nanobeams and plates are also investigated. It is realized that with the increase in the thickness-to-lattice parameter ratio, the difference of the results related to the SSG theory and classical theory decreases.