Asymptotic homogenization of flexoelectric composite plates with periodically varying thickness

Abstract

A micromechanical model for the analysis of structurally periodic flexoelectric plates with periodically varying thickness is developed on the basis of asymptotic homogenization. The period of thickness variation is small and comparable to the plate thickness; accordingly, the plate is usually referred to as having a “rapidly varying thickness.” The stipulation for rapidly varying thickness is important because it is envisioned that the developed model can be applied to a broad range of thin plates, both stratified as well as plates endowed with an arbitrary distribution of reinforcements attached to the surface or embedded within the plate. The microscopic problem is implemented in two steps pertaining, respectively, to first- and second-gradient asymptotic homogenization. Each level of homogenization culminates in its own set of unit cell problems from which the effective coefficients of the homogenized structure can eventually be obtained. These effective coefficients couple the force and moment resultants as well as the averaged electric displacement with the first and second gradients of the macroscopic displacement and electric potential. Once the effective coefficients are obtained, the macroscopic problem is invoked, which provides a set of four differential equations from which the macroscopic variables of mechanical displacement and electric potential can be obtained. The model is illustrated by means of laminated flexoelectric composites as well as simple rib-reinforced plates. It is shown that in the limiting case of a thin, purely elastic plate, the derived model converges to the familiar classical plate model.