Size-Dependent Bending and Vibration Analysis of Fractional-Order Piezoelectric Timoshenko Nanobeam

Abstract

In this paper, the size-dependent static bending and free vibration responses of piezoelectric nanobeams is modeled on the basis of fractional-order kinematic relations. The general nonlocal strains in Timoshenko piezoelectric nanobeam are defined by frame-invariant and dimensionally consistent Riesz-Caputo fractional-order derivatives, in which a length scale influence parameter is introduced to obtain a sparse stiffness matrix. The strain energy, work done by external loads and kinetic energy based on fractional-order kinematic relations are derived in explicit form. A finite element model for the fractional-order system is developed in order to achieve the numerical solution. The influence of fractional-order index and length scale influence parameter on the static bending and vibration behaviors of Timoshenko piezoelectric nanobeams is studied numerically. A consistent softening effect is obtained for fractional-order Timoshenko piezoelectric nanobeam.