Dynamic modeling and determination of effective properties of smart composite plates with rapidly varying thickness

Abstract

A new comprehensive micromechanical model for the analysis of the dynamic problem of a smart composite piezo-magneto-thermo-elastic 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. A complete dynamic approach is adhered to beginning with the equations of dynamic equilibrium, the time-varying form of Maxwell’s equations and dynamic thermal balance. The asymptotic homogenization model is derived, the governing equations are determined and subsequently general expressions called unit cell problems that can be used to determine the effective elastic, piezoelectric, piezomagnetic, electrical conductivity etc. properties are presented. Of particular interest in this work is the development of general expressions pertaining to the so-called product properties which are manifested in the macroscopic composite plate via the interaction of the different phases but may be absent from some individual constituents of the composite. Examples of product properties are the magnetoelectric, pyroelectric and pyromagnetic coefficients. The derived expressions pertaining to the unit-cell problems and the resultant effective coefficients are very general and are valid for any geometry of the unit cell. In addition to the effective properties, the developed model also computes the local mechanical displacement and stress, electric displacement, magnetic field, heat flux and free current density. The work is illustrated by means of a 3-layer anisotropic plate consisting of an elastic middle layer sandwiched between thin piezoelectric and piezomagnetic carrier layers.