Dynamic Homogenization Of Periodic Magneto-Electro-Elastic Composites

In recent years, dynamic homogenization has become one of the most popular research fields. The target of the research on dynamic homogenization is to predict the effective physical properties and wave propagating characteristics of composite materials under propagation of waves, such as effective moduli and effective mass density of elastic composites under the propagation of elastic waves and effective permittivity and effective dielectric constant of composites under the propagation of electromagnetic waves. Applying the dynamic homogenization to periodic composites we could derive the effective moduli of metamaterials and investigate the mechanism of double-negative feature, furthermore we could design new types of metamaterials. As one kind of widely used smart composites, the piezoelectric/piezomagnetic composite can realize transformation between the elastic wave and the electromagnetic wave due to the piezoelectric/piezomagnetic effect. Under the propagation of waves, the effective properties of piezoelectric/piezomagnetic composites cannot be predicted by traditional static homogenization theory. By applying dynamic homogenization theory, this thesis investigates the effective properties of periodic magneto-electro-elastic composites which are subjected to dynamic loads. In addition, as a new way to manipulating waves and cloaking, transformation physics has received much attention recently. In this thesis, transformation physics is extended to magneto-electro-elastic system and the transformed constitutive equations are obtained for magneto-electro-elastic media which are in the same form of the effective ones of periodic magneto-electro-elastic composites. The specific work of this thesis are:First, the elastodynamic homogenization theories of Willis are extended to the magneto-electro-elastic coupling case. Both elastic waves and electromagnetic waves are considered in an infinite periodic magneto-electro-elastic medium which is subjected to a plane-wave body force and a plane-wave current density as well as an eigenstrain field and a magnetization. The definition of effective fields for Bloch waves in previous literature is followed. The full set of the virtual work principle is presented in order to derive the localization equation by the Green’s function. By the way the meanings of the components of the Green function are interpreted. Then the effective constitutive equations of periodic magneto-electro-elastic media are derived out, which are in the generalized Willis form. It is seen that in magneto-electro-elastic coupling case, not only the effective stress is effective velocity dependent, but the effective electric displacement and the effective magnetic field are also effective velocity related. What’s more, the effective momentum density is related to the effective strain, the electric field and the effective magnetic flux density. The effective constitutive equation is wave number and frequency dependent. In addition, the symmetries of the effective moduli are studied. It is proved that our method can be degenerated to the purely elastodynamic case (which involves acoustic metamaterials) and the purely electromagnetic case (which involves optical metamaterials) successfully. The method is applied to predict the effective electromagnetic/elastodynamic properties of a 1D layered periodic composite which is subjected to a normal-incident electromagnetic wave/elastic SH wave. Both the dispersion relation for free waves and the Green’s function for forced waves are derived and the effective constitutive coefficients are obtained. A numerical example is presented for the 1D problem.Second, Transformation physics has been proven to be a new way to manipulation of physical fields and waves and cloaking. It is investigated that metamaterials can be used to realize this kind of cloaking and manipulation. In this thesis, transformation physics is extended to magneto-electro-elastic system. An arbitrary transformation form of displacement and the mapping transformation form of electric field and magnetic field that proposed by Pendry[2] are adopt to derive the transformed constitutive equations and equilibrium equations for magneto-electro-elastic media. It appears that the transformed constitutive equations have the same form of the effective ones of periodic magneto-electro-elastic composites. In the transformed space, the stress tensor, electric displacement and the magnetic flux are velocity dependent and the momentum density is also related to the strain, the electric field and the magnetic field. According to Milton’s discussion, we may deduce that the periodic magneto-electro-elastic composites with tiny enough microstructure can be used to manipulate the trajectories of waves in magneto-electro-elastic media.