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.