Construction of piezoelectric and flexoelectric models of composites by asymptotic homogenization and application to laminates

Abstract

The effective piezoelectric and flexoelectric properties of heterogeneous solid bodies with constituents obeying a piezoelectric behavior are evaluated in full generality, based on the asymptotic expansion method. The successive situations of materials obeying a piezoelectric and flexoelectric behavior at the macroscale is envisaged in the present work. Closed-form expressions for the effective flexoelectric properties are obtained for stratified materials. A general theory for laminated piezoelectric plates is formulated on the basis of the formulated asymptotic models, and the response of the homogeneous substitution plate is evaluated for a loading consisting of a pure bending moment, triggering electric fields and strain and electric fields gradients within the plate thickness. The local mechanical and electric fields at the microscopic level within the initial heterogeneous stratified domain are evaluated by solving unit cell boundary value problems for the localization operators. An effective flexoelectric plate model for a stratified composite is constructed, showing the generation of the gradient of an electric field under application of a pure bending moment.

Asymptotic homogenization of flexoelectric composite plates with periodically varying thickness

“…The pioneering contributions with respect to gradient effects are attributed to the nonlocal theory developed by Mindlin [20] who formulated a comprehensive linear model for the deformation of an elastic body based on the functional dependence of the potential energy density on both strain as well as its first and second gradients. Since then, and more systematically in recent years, a significant volume of associated works has been disseminated in the literature pertaining to various modeling aspects such as topology optimization of multimaterial flexoelectric composites [21][22][23], numerical and/or analytical determination of the effective properties of flexoelectric structures [24][25][26][27][28][29], bending and vibration analysis of flexoelectric beams [30][31][32][33], dynamic analysis and control of flexoelectric plates and shells [34][35][36][37][38], and many others. Ever since the rapid growth of the additive manufacturing industry has significantly facilitated fabrication of composites and metamaterials with arbitrary architectures, it stands to reason that the development of models that predict their behavior and properties a priori is a step in the right direction.…”