Nonlocal-integro-differential modeling of vibration of elastically supported nanorods

• A novel nonlocal-integro model is developed to study vibrations of nanorods.
• The nonlocality is introduced to both bulk and surface layer via kernel functions.
• The integro-partial differential equations of motion are obtained and solved.
• The efficiency of the proposed meshless model is also validated.
• The roles of both nonlocality and surface effects on frequencies are explored.

In the previously established nonlocal continuum-based models, small characteristic length was commonly incorporated into the mass matrix and the driving force vector which is a bit in contradiction with our sense regarding these factors. Herein, a nonlocal-integro-differential version of the constitutive relations is employed for the bulk and the surface layer of the nanorod. By adopting Hamilton's principle, integro-partial differential equations of motion of elastically supported nanorods are established accounting for both nonlocality and surface energy effects. Then, these are solved by an efficient meshless methodology. For fixed–fixed and fixed–free nanorods, modal analysis of the problem is also performed and the explicit expressions of the mass and stiffness matrices are derived. For these special cases, the obtained results by the meshless technique are successfully verified with those of the modal solution. In the newly developed numerical model, the small-scale parameter is only incorporated into the stiffness matrix which gives us a more realistic sense about the nonlocality effect. Subsequently, the roles of the surface energy, small-scale parameter, elastic supports, and kernel function on natural frequencies of the nanostructure are discussed and explained. This work can be considered as a pivotal step towards a more reasonable nonlocal modeling of vibration of nanoscale structures.

Frequency analysis of elastically supported nanorods is performed via a novel nonlocal integro-differential surface energy-based model. To solve the integro-partial differential equations, reproducing kernel particle method is exploited. For fixed–free and fixed–fixed boundary conditions, assumed mode method is also implemented and the accuracy of the suggested meshless model is examined.Figure optionsDownload as PowerPoint slide