Timoshenko Elastic and Electroelastic Beam Models Incorporating the Local Mass Displacement Effect

Abstract

The objective of the paper is to model the size-dependent mechanical behaviors of elastic and electroelastic micro-/nanosize beams. To this end, the relations of local gradient theory of polarized continua are utilized. This higher-grade theory is based on taking account of non-diffusive and non-convective mass flux related to the changes in the material microstructure. The above mass flux is associated with the process of local mass displacement. In this paper, the general non-classical governing equations and associated boundary conditions for linear local gradient theory of piezoelectric media are obtained by making use of a variational approach. The local gradient elastic and electroelastic bending theories for plane-strain beams are developed as well. The variational equation for the thin beam bending problem was formulated by considering the contribution of the local mass displacement to the free energy density. The kinematic hypotheses of Timoshenko beam theory were used to obtain the coupled ordinary differential equations governing the states of elastic and dielectric beams. The problem of a thin elastic cantilever beam under a point load at the free end was solved to illustrate the theory efficiency. The effect of the piezoelectricity on the beam behavior is studied as well. It is shown that the local mass displacement and material piezoelectric properties being taken into account stiffens the nanocantilever beam. It is found that the size effect induced by the local mass displacement is significant when the beam thickness is comparable in size to the material length scale parameter.