Harmonic vibrations of nanosized magnetoelectric bodies with coupled surface and interphase effects: Mathematical models and finite element approaches

The harmonic problems for piezomagnetoelectric nanosized bodies with taking into account the coupled damping and surface effects are considered on the base of the generalized Gurtin-Murdoch model. In the development of previous investigations, the coupled mechanical, electric and magnetic surface effects with surface inertial terms are introduced into the model. For a homogeneous model, the composite material is considered as homogeneous with the suitable effective material properties. The weak or generalized formulation of the steady-state oscillation problem is given together with the suitable formulation of the modal problem. For numerical solution of these problems, the finite element approximations, leading to a symmetric structure of finite element matrices, are present. The procedures of homogenization of piezomagnetoelectric nanostructured composite materials with piezoelectric and piezomagnetic phases are described on the base of the methods of effective moduli and finite elements

Mathematical models and finite element approaches for nanosized piezoelectric bodies with uncoulped and coupled surface effects

In this chapter the dynamic problems for piezoelectric nanosized bodies with account for coupled damping and surface effects are considered. For these problems we propose new mathematical model which generalizes the models of the elastic medium with damping in sense of the Rayleigh approach and with surface effects for the cases of piezoelectric materials. Our model of attenuation and surface effects has coupling properties between mechanical and electric fields, both for the damping terms and constitutive equations for piezoelectric materials on the surface. For solving the problems stated the finite element approximations are discussed. A set of effective finite element schemes is examined for finding numerical solutions of week statements for nonstationary problems, steady-state oscillation problems, modal problems and static problems within the framework of modelling of piezoelectric nanosized materials with damping and surface effects. For transient and harmonic problems, we demonstrate that the proposed models allow the use of the mode superposition method. In addition, we note that for transient and static problems we can use efficient finite element algorithms for solving the systems of linear algebraic equations with symmetric quasi-definite matrices both in the case of uncoupled surface effects and in the case of coupled surface effects