Torsional vibration of a flexoelectric nanotube with micro-inertia effect

Abstract

The modified strain gradient theory (MSGT) with flexoelectric and micro-inertia effects is developed to study the free vibration of tube made of elastic centrosymmetric materials. Within this theory, the internal energy density is the function of the strain tensor, deviatoric part of stretch gradient and rotation gradient tensors, dilatation gradient, polarization vector, and electric quadrupoles. The linear coupled governing equations and boundary conditions are derived from the Hamilton variation principle. The torsional vibration problem is analytically solved for non-piezoelectric nanotube with fixed edges. The effect of the nanotube geometry, flexoelectric coefficient, micro-inertia, and micro-stiffness parameters on the natural frequency is studied.

Keywords:

• Modified strain gradient theory
• flexoelectricity
• micro-inertia effect
• micro-stiffness effect
• torsional free vibration
• natural frequency
• size effect

Author contributions

Hrytsyna O.:Investigation, Software, Writing – original draft preparation;

Sladek J.:Conceptualization, Formal analysis, Writing – reviewing and editing;

Sladek V.:Formal analysis, Visualization, Writing – reviewing and editing;

Deng Q.:Investigation, Formal analysis, Writing – reviewing and editing.

Hrytsyna M.:Investigation, Software, Writing- original draft preparation.

Disclosure statement

The authors report there are no competing interests to declare.

Additional information

Funding

This research was supported by the Slovak Research and Development Agency and the Ministry of Education and Science of Ukraine (grant number SK-UA-21-0010) and by the Ministry of Education, Science, Research and Sport of the Slovak Republic (grant number VEGA-2/0061/20).