Analysis of smart piezo-magneto-thermo-elastic composite and reinforced plates: Part I - Model development

Abstract

A comprehensive micromechanical model for the analysis of a smart composite piezo-magneto-thermoelastic thin plate with rapidly-varying thickness is developed in the present paper. A rigorous three-dimensional formulation is used as the basis of multiscale asymptotic homogenization. The asymptotic homogenization model is developed using static equilibrium equations and the quasi-static approximation of Maxwell’s equations. The work culminates in the derivation of a set of differential equations and associated boundary conditions. These systems of equations are called unit cell problems and their solution yields such coefficients as the effective elastic, piezoelectric, piezomagnetic, dielectric permittivity and others. Among these coefficients, the so-called product coefficients are also determined which are present in the behavior of the macroscopic composite as a result of the interactions and strain transfer between the various phases but can be absent from the constitutive behavior of some individual phases of the composite material. The model is comprehensive enough to allow calculation of such local fields as mechanical stress, electric displacement and magnetic induction. In part II of this work, the theory is illustrated by means of examples pertaining to thin laminated magnetoelectric plates of uniform thickness and wafer-type smart composite plates with piezoelectric and piezomagnetic constituents. The practical importance of the model lies in the fact that it can be successfully employed to tailor the effective properties of a smart composite plate to the requirements of a particular engineering application by changing certain geometric or material parameters. The results of the model constitute an important refinement over previously established work. Finally, it is shown that in the limiting case of a thin elastic plate of uniform thickness the derived model converges to the familiar classical plate model.