Modeling the flexoelectric effect via the reduced micromorphic model

Abstract

This paper aims to demonstrate how the flexoelectric effect can be incorporated into the reduced micromorphic model, where materials can be designed and fabricated with complex structures that possess electric properties. These kinds of materials are known as composite metamaterials with properties that do not exist in nature such as the flexoelectric effect phenomena. These phenomena take place when both negative and positive charges move to occupy new positions inside the materials where the new positions of these charges depend on the applied external mechanical or electrical fields. The new arrangements of charges produce a spontaneous polarization inside the material that fades when removing the applied fields. According to the classical theory of elasticity, the spontaneous polarization is mathematically related to a macro-scale measure. In the present work, instead of the macro-scale, we use a micro-scale measure to formulate the polarization phenomena. This formulation is achieved by employing the variation technique for the internal energy functional and the virtual work done by the external fields. The field equations and the concomitant boundary conditions are also obtained. Moreover, the constitutive relations are presented in terms of internal state variables. Finally, an application is made in one-dimension and the results are presented in figures with physical explanation.