Non-Local Natural Vibration Behavior of 3D Nanostructures

  • Rodriguez-Herrera, Arturo (Universidad de los Andes)
  • Ramirez, Fernado (Universidad de los Andes)

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Classical local elasticity theory (CLET) is suitable for modeling large scale structural mechanics problems. However, at the nanoscale this theory deviate from experimental results, since it does not take into account long range interaction contributions. An alternative approach is the non-local elasticity theory (NLET) proposed by Eringen and Edelen [1]. In this theory, the stress-strain relationship of a particle within the solid not only depends on itself, but also on all the particles in the material aggregate. Based on Eringen’s theory, Polizzotto [2] proposed a two phase non-local elasticity ap- proach. In this formulation, the volume is composed of CLET and NLET phases. The latter uses an attenuation function to include the long range cohesive forces. This function depends on the internal length of the material (l) which is the main non-local parameter. This approach is convenient as it facilitates the finite element formulation of the problem (NL-FEM). In the present work, the non-local natural vibrations behavior for different structures is studied. For this purpose, the first natural vibration frequency using NLET (ηnl) and CLET (ηl) are calculated using a custom implementation of NL-FEM. The simulations include different geometries (cubes, spheres, plates, and beams), domain sizes (R), mate- rials, non-local parameters (l), and boundary conditions. The obtained normalized frequency response considering the non-local theory (ηnl) does not depend on the material. Furthermore, it does not depend on l or R independently, but instead on their ratio l/R. Therefore, the behavior of ηnl only depends on domain geometry. Additionally, threshold l/R values are determined. A minimum value l/Rmin, from which the CLET holds for vibration analyses, and a maximum l/Rmax when the size of the domain is no longer feasible compared to the internal length of the material. For the range [l/Rmin, l/Rmax] the response for ηnl presents a linear behavior, and the corresponding regressions are presented. References [1] A. Eringen and D. Edelen, “On nonlocal elasticity,” International Journal of Engi- neering Science, vol. 10, no. 3, pp. 233–248, Mar. 1972, issn: 00207225. doi: 10. 1016/0020-7225(72)90039-0. [2] C. Polizzotto, “Nonlocal elasticity and related variational principles,” International Journal of Solids and Structures, vol. 38, no. 42-43, pp. 7359–7380, 2001, issn: 00207683. doi: 10.1016/S0020-7683(01)00039-7.