New Mathematical and Finite Element Models with Dissipations for Multiferroic Media with Voids

摘要

In this chapter, for modeling magnetoelectric or piezomagnetoelectric media and transducers, the special mathematical model, which generalize the model of the magnetoelectric material with dissipation, and the Cowin-Nunziato approach for the elastic and piezoelectric media with voids are proposed. Using these models for magnetoelectric bodies with voids or pores, the effective moduli of porous magnetoelectric composite material could be defined more precisely. In this theory the mechanical displacements, electric and magneticic potentials and porosity change function are the main unknown functions. On the base of this model, we have obtained the setting of the generalized continual formulations for magnetoelectric solids with voids or porous and numerical approximation in the general and reduced statements. We have studied the mathematical properties of the eigenfre-quencies and eigenvectors for magnetoelectric solids with voids for different mechanical, electric, magnetic and "porous" boundary conditions, including the contact type boundary conditions. We have established some properties of changes of the eigenfrequencies with changes of the boundary conditions and material moduli. For numerical analysis of magnetoelectric solids with voids, we have obtained the finite element approximations with symmetric quasi-definite matrices.