Frequency domain analysis of nonlocal rods embedded in an elastic medium

• Computational formulation of an elastic medium on nonlocal axial vibration of rods is considered.
• Static and dynamic finite element formulations are proposed.
• The nonlocal parameter impacts both the element mass and stiffness matrices.
• A closed-from exact expression is derived for the upper cut-off natural frequency.
• Nonlocal parameter and the elastic stiffness respectively decrease and increase the natural frequencies.

A novel dynamic finite element method is carried out for a small-scale nonlocal rod which is embedded in an elastic medium and undergoing axial vibration. Eringen's nonlocal elasticity theory is employed. Natural frequencies are derived for general boundary conditions. An asymptotic analysis is carried out. The stiffness and mass matrices of the embedded nonlocal rod are obtained using the proposed finite element method. Nonlocal rods embedded in an elastic medium have an upper cut-off natural frequency which is independent of the boundary conditions and the length of the rod. Dynamic response for the damped case has been obtained using the conventional finite element and dynamic finite element approaches. The present study would be helpful for developing nonlocal finite element models and study of embedded carbon nanotubes for future nanocomposite materials.

Variation of natural frequencies and frequency response function of single walled carbon nanotube embedded in an elastic medium.Figure optionsDownload as PowerPoint slide